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ch 1
Indicate whether the statement is true or false. 1. Defining a problem well will often make it much easier to solve. a. True b. False 2. Because they simplify reality, models are generally not helpful in examining things that would be impossible to
do in reality. a. True b. False 3. In spreadsheet modeling of a problem, there is no direct correspondence between mathematical equation and the spreadsheet. a. True b. False 4. Humans usually do not make errors in estimation due to anchoring and framing effects. a. True b. False 5. A mathematical model uses mathematical relationships to describe or represent an object or decision problem. a. True b. False 6. The proliferation of powerful PCs and the development of easy-to-use electronic spreadsheets have made the tools of business analytics far more practical and available to a much larger audience. a. True b. False 7. OR/MS specialists do not deliver business value. a. True b. False 8. Good decisions always result in good outcomes. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. Virtually everyone who uses a spreadsheet today for model building and decision making a. is a practitioner of business analytics. b. possesses an advanced knowledge of mathematics and computer programming languages. c. is a CPA. d. is in a position to influence decision makers. 10. A situation when decision quality is good and the resulting outcome quality is good is referred to as Copyright Cengage Learning. Powered by Cognero.
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ch 1 a. pure luck. b. deserved success. c. dumb luck. d. poetic justice. 11. Operations Research got its start a. during World War II. b. with the first Univac computers in the early 1950's. c. from roots in Operations Management. d. from Frederick Taylor's Scientific Management. 12. If we do not identify the correct problem, the best we can hope for is: a. wasted time and effort. b. useful experience in problem definition efforts. c. a descriptive model. d. the right answer to the wrong question. 13. The best models a. accurately reflect relevant characteristics of the real-world object or decision. b. are mathematical models. c. replicate all aspects of the real-world object or decision. d. replicate the characteristics of a component in isolation from the rest of the system. 14. In which of the following categories of modeling techniques do the independent variables have unknown or uncertain values or coefficients? a. Descriptive models b. Predictive models c. Prescriptive models d. Probabilistic models 15. The Chapter One "The World of Business Analytics" case reading discusses the relationship between OR/MS and IS professionals. Which of the following statements is NOT true? a. OR/MS analysts need IS professionals' data for their models. b. OR/MS analysts need to take many of the IS customers. c. The IS professional cannot use OR/MS tools in their applications. d. The IS tools can start to recommend solutions using OR/MS skills. 16. The goal of the modeling approach to problem solving is to a. help individuals make good decisions. b. ensure optimality of decisions. c. determine a set of optimal decisions. d. determine feasibility of decisions. 17. If results testing produces unsatisfactory results a. the problem-solving process requires new formulation and implementation. Copyright Cengage Learning. Powered by Cognero.
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ch 1 b. minor adjustments to the existing model. c. checking the solution algorithm. d. repeated testing. 18. In a decision-making framework presented in Chapter One, the term "poetic justice" refers to a situation when the following occur: a. Good decision quality and good outcome quality. b. Good decision quality and bad outcome quality. c. Bad decision quality and good outcome quality. d. Bad decision quality and bad outcome quality. 19. Business analytics focuses on a. identifying and leveraging business opportunities. b. formulating analytical models. c. using models to analyze problem. d. testing and implementing results. 20. The Chapter One "The World of Business Analytics" case reading offers the CEO alternatives to start the OR/MS collaboration process. All the following are alternatives offered except: a. Require the OR/MS group to save their yearly salary in every study. b. Use OR/MS personnel as consultants. c. Hire some OR/MS professionals and give them a problem to work. d. Institute more participation from OR analysts. 21. Two of the effects associated with decision problems are: a. anchoring and framing. b. anchoring and loading. c. framing and complacency. d. anchoring and luck 22. Which of the following categories of modeling techniques addresses uncertainty in the values of the independent variables? a. Descriptive models b. Predictive models c. Prescriptive models d. Scale models 23. Consider the spreadsheet model shown in the figure below. This is an example of a A
B
C
1 2 3 4
Purchase price less:
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$32,500
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$ 4,000
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8 9 10 12 13 Monthly payment
$481.08
14 a. descriptive model. b. predictive model. c. prescriptive model. d. preventive model. 24. In a spreadsheet, input cells correspond conceptually to a. dependent variables. b. functions. c. independent variables. d. output cells. 25. A valid model: a. accurately represents a decision problem being studied. b. produces an optimal solution. c. produces a good solution. d. produces a feasible solution. 26. Which of the following fields of study is defined in Chapter One as the one that "uses computers, statistics, and mathematics to solve business problems"? a. Accounting b. Information systems c. Business analytics d. Scientific management 27. A situation when decision quality is bad and the resulting outcome quality is bad is referred to as a. pure luck. b. deserved success. c. bad luck. d. poetic justice. 28. In the following expression, which is (are) the dependent variable(s)? PROFIT = REVENUE − EXPENSES Copyright Cengage Learning. Powered by Cognero.
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ch 1 a. Profit b. Revenue c. Expenses d. (b) and (c) 29. In a decision-making framework presented in Chapter One, the term "dumb luck" refers to a situation when the following occur: a. Good decision quality and good outcome quality. b. Good decision quality and bad outcome quality. c. Bad decision quality and good outcome quality. d. Bad decision quality and bad outcome quality. 30. Anchoring occurs when: a. a trivial factor is used as a starting point for estimations in a decision-making problem. b. a difficult factor is incorporated in a problem. c. an easy solution is obtained to a difficult problem. d. obtaining a solution is trivial. 31. A situation when decision quality is good and the resulting outcome quality is bad is referred to as a. pure luck. b. deserved success. c. bad luck. d. poetic justice. 32. The mathematical modeling approaches presented in the textbook a. are a subset of the total problem-solving process. b. cover the entire spectrum of decision support approaches. c. are exhaustive. d. are complementary. 33. Business opportunities can be viewed and formulated as a. decision problems. b. analytical models. c. empirical models. d. testing tools. 34. To illustrate how a complex system will be built, an engineer will likely use a a. mathematical model. b. mental model. c. physical model. d. visual model. 35. In order to be useful to a decision-maker, decision problems need to be a. valid. Copyright Cengage Learning. Powered by Cognero.
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ch 1 b. analyzed. c. simplified. d. tested. 36. Identifying the real problems faced by the decision maker a. is not important since the decision maker has already defined the problem. b. requires insight, some imagination, time and a good bit of detective work. c. first requires a well-defined problem statement. d. will lead to developing the best model. 37. The specification or description of the relationship between the dependent and independent variables is generally called a. a constraint. b. a declaration. c. a function. d. a mathematical model. 38. In this text we use the term "mathematics" to encompass i. ii.
familiar elements of math such as algebra. logic. a. i only b. ii only c. Both i and ii d. Neither i nor ii
39. The ultimate goal of the problem identification step of the problem-solving process is a. collecting lots of information. b. helping the decision maker realize there is a problem. c. identifying the root problem or problems causing the mess. d. convincing the decision maker the mess is really a problem that can be solved. 40. Which step of the problem-solving process is considered the most important? a. Identify problem. b. Analyze model. c. Test results. d. Implement solution. 41. Variables are termed independent when they satisfy which of the following? a. The function value depends upon their values. b. The decision maker has no control over them. c. The variables have no relationship to one another. d. The variable is described as an output of the spreadsheet model. Copyright Cengage Learning. Powered by Cognero.
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ch 1 42. A factor that plays a role in determining whether a good or bad outcome occurs is called a. luck. b. intuition. c. certainty. d. predictability. 43. In the textbook the words "opportunity" and "problem" are a. disjoint. b. used interchangeably. c. mutually exclusive. d. complementary. 44. Consistently using a structured, model based process to make decisions a. should produce good outcomes more frequently. b. is less effective than making decisions in a haphazard manner. c. is evidence that luck plays an important role in decision making. d. always leads to well-deserved success in managerial decision making. 45. In which step of the problem-solving process is the main focus to generate and evaluate alternatives? a. Identify problem b. Formulate model c. Use model to analyze problem d. Test results 46. Which of the following statements is true of using models in problem solving and decision analysis? a. It is a fairly new idea. b. It is required in order to find good solutions. c. It is something everyone has done before. d. It is tied to the use of computers. 47. Why would someone wish to use a spreadsheet model? a. To implement a computer model. b. Because spreadsheets are convenient. c. To analyze decision alternatives. d. All of these. 48. Which of the following steps in the problem-solving process is most likely to incur resistance from people affected by the proposed solution? a. Formulate model b. Use model to analyze problem c. Test results d. Implement solution 49. Solutions to which of the following categories of modeling techniques indicate a course of action to the decision maker? Copyright Cengage Learning. Powered by Cognero.
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ch 1 a. Descriptive models b. Predictive models c. Prescriptive models d. Preventive models 50. The notion that every problem is also an opportunity is reflected in the term a. probortunity. b. formulation. c. simulation. d. business opportunity. 51. A purely rational decision maker should a. consistently select the same alternative, regardless of how the problem is framed. b. disregard the consequences of his/her choices. c. always select optimal action. d. allow emotions influence the decision. 52. There are a variety of problems a manager might face. While presenting and defending your approach, how would you complete this thought? Several different modeling techniques are available to solve managerial decision problems, a. the wrong choice of modeling technique is a common source of implementation difficulties. b. students should develop a strong preference and expertise in one technique so when faced with problems as managers they can formulate them as a model that can be solved by their favorite technique. c. fundamental characteristics of the problem guide the selection of an appropriate modeling technique. d. most problems faced by managers are fundamentally the same. 53. Which of the following is the type of model used throughout this textbook? a. Mathematical model b. Mental model c. Physical model d. Visual model 54. The main point brought forward in the Chapter One "The World of Business Analytics" case reading is: a. At a cocktail party, it is more efficient to divide the dip into several bowls and place them around the room. b. Competitive rivalry between IS and OR/MS groups can be turned to advantage when tackling business process re-engineering projects. c. Information system analysts trained in management science can help turn ordinary information systems into money-saving decision-support systems. d. OR/MS professionals lack communication skills and tend to focus on "rigor without relevance". 55. Which of the following problem-solving steps is often considered the most difficult? a. Identify the problem. b. Analyze the model. c. Test results. Copyright Cengage Learning. Powered by Cognero.
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ch 1 d. Implement the solution. 56. Which of the following categories of modeling techniques involves determining the value of a dependent variable based on specific values of independent variables? a. Biased models. b. Descriptive models. c. Predictive models. d. Prescriptive models. 57. A road map is an example of a. a mathematical model. b. a mental model. c. a physical model. d. a visual model. 58. In a model Y=f(x1, x2), Y is called: a. a dependent variable. b. an independent variable. c. a confounded variable. d. a convoluted variable. 59. In which of the following categories of modeling techniques are the specifications of the relationships between dependent and independent variables unknown or ill-defined? a. Descriptive models b. Open models c. Predictive models d. Prescriptive models 60. A mathematical model is considered to be "valid" when a. it accurately represents the relevant characteristics of the object or decision. b. it has passed a validation test. c. it replicates all aspects of the object or decision. d. the left-hand and right-hand sides of expressions are equal. 61. The concept of "probortunity" is a. the first step in the problem-solving process. b. a decision support method. c. part of testing results. d. part of solution implementation. 62. In a model Y=f(x1, x2), x1 is called: a. an independent variable. b. a dependent variable. c. a confounded variable. Copyright Cengage Learning. Powered by Cognero.
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ch 1 d. a convoluted variable. 63. Which of the following is true of "What if?" analysis? a. A well-designed spreadsheet facilitates "What if?" analysis. b. It is not very useful when working with non mathematical models. c. "What if?" analysis is an efficient optimization technique. d. "What if?" analysis is useful in creating a well-defined problem statement. 64. All of the following are benefits of modeling except: a. Modeling delivers needed information on a more timely basis. b. Modeling finds the right answers to incorrect or flawed problem statements. c. Modeling is helpful in examining things that would be impossible to do in reality. d. Modeling is less expensive than implementing several alternative solutions. 65. The textbook figure of the problem-solving process is an example of a a. mental model. b. prescriptive model. c. graphical model. d. visual model. 66. The categories of modeling techniques presented in this book include all of the following except: a. descriptive models. b. predictive models. c. prescriptive models. d. preventive models. 67. Better decision making due to using a modeling process is achieved due to a. the interaction with the spreadsheet. b. the visualization of the system being studied. c. the insight gained through the process. d. the timeliness of the results obtained. 68. To be effective, a modeler must a. be an effective presenter of results. b. collect the proper input data for the model. c. understand how modeling fits into the problem-solving process. d. apply the correct modeling technique. 69. Chapter One discussed all of the following except: a. how models of decision problems differ in a number of important characteristics. b. how spreadsheet modeling and analysis fit into the problem-solving process. c. how spreadsheet models of decision problems can be used to analyze the consequences of possible courses of action. d. how to implement a problem formulation as a spreadsheet model. Copyright Cengage Learning. Powered by Cognero.
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ch 1 70. Which of the following categories of modeling techniques includes optimization techniques? a. Capitalistic models b. Descriptive models c. Predictive models d. Prescriptive models 71. In a decision-making problem, anchoring effects occur when a. decision makers are tied too closely to previous decisions. b. organizations refuse to consider new alternatives. c. a seemingly trivial factor serves as a starting point for estimations. d. a person in a position of authority exerts his or her opinion very forcefully. 72. In which step of the problem-solving process is the concept of "probortunity" introduced? a. Identify problem b. Formulate model c. Use model to analyze problem d. Test results 73. Implementing solutions to problems involves people and change. Which of the following is a suggested approach to effectively implement solutions? a. Decision-making authority centralized to those who have specialized training in decision making. b. Involve anyone affected by the decision in all steps of the problem-solving process. c. Making decisions according to majority vote. d. More skillful communication of management decisions. 74. In a decision-making framework presented in Chapter One, the term "deserved success" refers to a situation when the following occur: a. Good decision quality and good outcome quality. b. Good decision quality and bad outcome quality. c. Bad decision quality and good outcome quality. d. Bad decision quality and bad outcome quality. 75. Framing effect refers to: a. how a decision maker views the alternatives in a decision problem. b. how difficult the decision is. c. whether a software program can be used to obtain an optimal solution to a decision problem. d. how structured the decision problem is. 76. The essence of decision analysis is: a. breaking down complex situations into manageable elements. b. choosing the best course of action among alternatives. c. finding the root cause of why something has gone wrong. d. thinking ahead to avoid negative consequences. 77. In a decision-making framework presented in Chapter One, the term "bad luck" refers to a situation when the Copyright Cengage Learning. Powered by Cognero.
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ch 1 following occur: a. Good decision quality and good outcome quality. b. Good decision quality and bad outcome quality. c. Bad decision quality and good outcome quality. d. Bad decision quality and bad outcome quality. 78. Which of the following is most likely to be used when faced with the decision of how to arrange furniture in a room? a. Mathematical model b. Mental model c. Physical model d. Visual model 79. Beneficial uses of the testing process include all of the following except: a. double checking the validity the model. b. finding that some important assumption has been left out of the model. c. giving no new insights into the nature of the problem. d. improving solutions after the implementation step. 80. A situation when decision quality is bad and the resulting outcome quality is good is referred to as a. dumb luck. b. deserved success. c. bad luck. d. poetic justice.
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ch 1 Answer Key 1. True 2. False 3. False 4. False 5. True 6. True 7. False 8. False 9. a 10. b 11. a 12. d 13. a 14. a 15. d 16. a 17. a 18. d 19. a 20. a 21. a 22. a 23. c 24. c 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 1 26. c 27. d 28. a 29. c 30. a 31. c 32. a 33. a 34. d 35. a 36. b 37. c 38. c 39. c 40. a 41. a 42. a 43. b 44. a 45. c 46. c 47. d 48. d 49. c 50. a 51. a Copyright Cengage Learning. Powered by Cognero.
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ch 1 52. c 53. a 54. c 55. d 56. c 57. d 58. a 59. c 60. a 61. a 62. a 63. a 64. b 65. d 66. d 67. c 68. c 69. d 70. d 71. c 72. a 73. b 74. a 75. a 76. b Copyright Cengage Learning. Powered by Cognero.
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ch 1 77. b 78. b 79. d 80. a
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ch 2
Indicate whether the statement is true or false. 1. Mathematical programming is an approach that involves determining how to allocate the resources in such a way as to maximize profits or minimize costs.
a. True b. False 2. The first step in formulating an LP model is determining the decision variables. a. True b. False 3. How much money should an individual withdraw each year from various retirement accounts is an example of a constraint. a. True b. False 4. An extreme point of the feasible region can include negative values of coordinates. a. True b. False 5. The best way of solving LP problems is to apply managerial intuition regarding the levels of decision variables. a. True b. False 6. The decisions in an optimization problem are often represented in a mathematical model by the symbols X1, X2, …, Xn. a. True b. False 7. The objective function coefficients represent per unit objective function contributions from one unit of the associated decision variables. a. True b. False 8. In mathematical programming formulations the objective function may contain cubic terms. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. A company makes two products, X1 and X2. They require at least 20 of each be produced. Which set of lower bound constraints reflect this requirement? a. X1 ≥ 20, X2 ≥ 20 b. X1 + X2 ≥ 20 c. X1 + X2 ≥ 40 Copyright Cengage Learning. Powered by Cognero.
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ch 2 d. X1 ≥ 20, X2 ≥ 20, X1 + X2 ≤ 40 10. A diet is being developed which must contain at least 100 mg of vitamin C. Two fruits are used in this diet. Bananas contain 30 mg of vitamin C and Apples contain 20 mg of vitamin C. The diet must contain at least 100 mg of vitamin C. Which of the following constraints reflects the relationship between Bananas, Apples and vitamin C? a. 20 A + 30 B ≥ 100 b. 20 A + 30 B ≤ 100 c. 20 A + 30 B = 100 d. 20 A = 100 11. The objective function for a LP model is 3 X1 + 2 X2. If X1 = 20 and X2 = 30, what is the value of the objective function? a. 0 b. 50 c. 60 d. 120 12. The second step in formulating a linear programming problem is a. Identify any upper or lower bounds on the decision variables. b. State the constraints as linear combinations of the decision variables. c. Understand the problem. d. Identify the decision variables. e. State the objective function as a linear combination of the decision variables. 13. The number of units to ship from Chicago to Memphis is an example of a(n) a. decision. b. constraint. c. objective. d. parameter. 14. Suppose that a constraint 2x1+3x2 ≥ 900 is binding. Then, a constraint 4x1+6x2 ≥ 600 is a. redundant. b. binding. c. limiting. d. infeasible. 15. Which of the following is the general format of an objective function? a. f(X1, X2, ..., Xn) ≤ b b. f(X1, X2, ..., Xn) ≥ b c. f(X1, X2, ..., Xn) = b d. f(X1, X2, ..., Xn) 16. Level curves are used when solving LP models using the graphical method. To what part of the model do level curves Copyright Cengage Learning. Powered by Cognero.
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ch 2 relate? a. constraints b. boundaries c. right hand sides d. objective function 17. A facility produces two products and wants to maximize profit. The objective function to maximize is z=350x1+300x2. The number 350 means that: a. one unit of product 1 contributes $350 to the objective function b. one unit of product 1 contributes $300 to the objective function c. the problem is unbounded d. the problem has no constraints 18. Most individuals manage their individual retirement accounts (IRAs) so they a. maximize the amount of money they withdraw. b. minimize the amount of taxes they must pay. c. retire with a minimum amount of money. d. leave all their money to the government. 19. A set of values for the decision variables that satisfy all the constraints and yields the best objective function value is a. a feasible solution. b. an optimal solution. c. a corner point solution. d. both (a) and (c). 20. A manager has only 200 tons of plastic for his company. This is an example of a(n) a. decision. b. constraint. c. objective. d. parameter. 21. Suppose that the left side of the constraint cannot take a specific value, b. This can be expressed mathematically as a. f(X1, X2, ..., Xn) ≤ b. b. f(X1, X2, ..., Xn) ≥ b. c. f(X1, X2, ..., Xn) = b. d. f(X1, X2, ..., Xn) ≠ b. 22. Mathematical programming is referred to as a. optimization. b. satisficing. c. approximation. d. simulation. Copyright Cengage Learning. Powered by Cognero.
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ch 2 23. The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch MAX: Subject to:
150 X1 + 250 X2 2 X1 + 5 X2 ≤ 200 − resource 1 3 X1 + 7 X2 ≤ 175 − resource 2 X1, X2 ≥ 0 How many units of resource 1 are consumed by each unit of product 1 produced? a. 1 b. 2 c. 3 d. 5 24. The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch
MAX: Subject to:
150 X1 + 250 X2 2 X1 + 5 X2 ≤ 200 3 X1 + 7 X2 ≤ 175 X1, X2 ≥ 0 How many units of resource one (the first constraint) are used if the company produces 10 units of product 1 and 5 units of product 2? a. 45 b. 15 c. 55 d. 50 25. A common objective when manufacturing printed circuit boards is a. maximizing the number of holes drilled. b. maximizing the number of drill bit changes. c. minimizing the number of holes drilled. d. minimizing the total distance the drill bit must be moved. 26. A production optimization problem has 4 decision variables and resource 1 limits how many of the 4 products can be produced. Which of the following constraints reflects this fact? a. f(X1, X2, X3, X4) ≤ b1 b. f(X1, X2, X3, X4) ≥ b1 Copyright Cengage Learning. Powered by Cognero.
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ch 2 c. f(X1, X2, X3, X4) = b1 d. f(X1, X2, X3, X4) ≠ b1 27. In a mathematical formulation of an optimization problem, the objective function is written as z=2x1+3x2. Then: a. x1 is a decision variable b. x2 is a parameter c. z needs to be maximized d. 2 is a first decision variable level 28. The constraints of an LP model define the a. feasible region b. practical region c. maximal region d. opportunity region 29. If a problem has infinite number solutions, the objective function a. is parallel to one of the binding constraints. b. goes through exactly one corner point of the feasible region. c. cannot identify a feasible region. d. is infeasible. 30. The constraint for resource 1 is 5 X1 + 4 X2 ≤ 200. If X1 = 20 and X2 = 5, how much of resource 1 is unused? a. 0 b. 80 c. 100 d. 200 31. A redundant constraint is one which a. plays no role in determining the feasible region of the problem. b. is parallel to the level curve. c. is added after the problem is already formulated. d. can only increase the objective function value. 32. A common objective in the product mix problem is a. maximizing cost. b. maximizing profit. c. minimizing production time. d. maximizing production volume. 33. The constraint for resource 1 is 5 X1 + 4 X2 ≥ 200. If X2 = 20, what it the minimum value for X1? a. 20 b. 24 c. 40 Copyright Cengage Learning. Powered by Cognero.
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ch 2 d. 50 34. For an infeasible problem, the feasible region: a. is an empty set b. has infinite number of feasible solutions c. has only one optimal solution d. is unbounded 35. The constraint for resource 1 is 5 X1 + 4 X2 ≤ 200. If X1 = 20, what it the maximum value for X2? a. 20 b. 25 c. 40 d. 50 36. Which of the following fields of business analytics finds the optimal method of using resources to achieve the objectives of a business? a. Simulation b. Regression c. Mathematical programming d. Discriminant analysis 37. A production optimization problem has 4 decision variables and a requirement that at least b1 units of material 1 are consumed. Which of the following constraints reflects this fact? a. f(X1, X2, X3, X4) ≤ b1 b. f(X1, X2, X3, X4) ≥ b1 c. f(X1, X2, X3, X4) = b1 d. f(X1, X2, X3, X4) ≠ b1 38. Linear programming problems have a. linear objective functions, non-linear constraints. b. non-linear objective functions, non-linear constraints. c. non-linear objective functions, linear constraints. d. linear objective functions, linear constraints. 39. The constraint for resource 1 is 5 X1 + 4 X2 ≥ 200. If X1 = 40 and X2 = 20, how many additional units, if any, of resource 1 are employed above the minimum of 200? a. 0 b. 20 c. 40 d. 80 40. The following diagram shows the constraints for a LP model. Assume the point (0,0) satisfies constraint (B,J) but does not satisfy constraints (D,H) or (C,I). Which set of points on this diagram defines the feasible solution space? Copyright Cengage Learning. Powered by Cognero.
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ch 2
a. A, B, E, F, H b. A, D, G, J c. F, G, H, J d. I, F, G, J 41. If constraints are added to an LP model the feasible solution space will generally a. decrease. b. increase. c. remain the same. d. become more feasible. 42. A greater than or equal to constraint can be expressed mathematically as a. f(X1, X2, ..., Xn) ≤ b. b. f(X1, X2, ..., Xn) ≥ b. c. f(X1, X2, ..., Xn) = b. d. f(X1, X2, ..., Xn) ≠ b. 43. The symbols X1, Z1, Dog are all examples of a. decision variables. b. constraints. c. objectives. d. parameters. 44. What is the goal in optimization? a. Find the decision variable values that result in the best objective function and satisfy all constraints. b. Find the values of the decision variables that use all available resources. c. Find the values of the decision variables that satisfy all constraints. d. None of these. 45. If there is no way to simultaneously satisfy all the constraints in an LP model the problem is said to be Copyright Cengage Learning. Powered by Cognero.
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ch 2 a. infeasible. b. open ended. c. multi-optimal. d. unbounded. 46. Limited resources are modeled in optimization problems as a. an objective function. b. constraints. c. decision variables. d. alternatives. 47. This graph shows the feasible region (defined by points ACDEF) and objective function level curve (BG) for a maximization problem. Which point corresponds to the optimal solution to the problem?
a. A b. B c. C d. D e. E 48. The third step in formulating a linear programming problem is a. Identify any upper or lower bounds on the decision variables. b. State the constraints as linear combinations of the decision variables. c. Understand the problem. d. Identify the decision variables. e. State the objective function as a linear combination of the decision variables. 49. Why do we study the graphical method of solving LP problems? a. Lines are easy to draw on paper. b. To develop an understanding of the linear programming strategy. c. It is faster than computerized methods. d. It provides better solutions than computerized methods.
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ch 2 50. The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch MAX: Subject to:
150 X1 + 250 X2 2 X1 + 5 X2 ≤ 200 3 X1 + 7 X2 ≤ 175 X1, X2 ≥ 0 How much profit is earned per each unit of product 2 produced? a. 150 b. 175 c. 200 d. 250 51. Which of the following special conditions in an LP model represent potential errors in the mathematical formulation? a. Alternate optimum solutions and infeasibility b. Redundant constraints and unbounded solutions c. Infeasibility and unbounded solutions d. Alternate optimum solutions and redundant constraints 52. Some resources (i.e. meat and dairy products, pharmaceuticals, a can of paint) are perishable. This means that once a package (e.g. a can or a bag) is open the content should be used in its entirety. Which of the following constraints reflects this fact? a. f(X1, X2, X3, X4) ≤ b1 b. f(X1, X2, X3, X4) ≥ b1 c. f(X1, X2, X3, X4) = b1 d. f(X1, X2, X3, X4) ≠ b1 53. The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch MAX: Subject to:
150 X1 + 250 X2 2 X1 + 5 X2 ≤ 200 3 X1 + 7 X2 ≤ 175 X1, X2 ≥ 0 How much profit is earned if the company produces 10 units of product 1 and 5 units of product 2? a. 750 b. 2500 c. 2750 Copyright Cengage Learning. Powered by Cognero.
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ch 2 d. 3250 54. A facility produces two products. The labor constraint (in hours) is formulated as: 350x1+300x2 ≤ 10,000. The number 10,000 represents a. a profit contribution of one unit of product 1. b. one unit of product 1 uses 10,000 hours of labor. c. there are 10,000 hours of labor available for use. d. the problem has no objective function. 55. What are the three common elements of an optimization problem? a. objectives, resources, goals. b. decisions, constraints, an objective. c. decision variables, profit levels, costs. d. decisions, resource requirements, a profit function. 56. When do alternate optimal solutions occur in LP models? a. When a binding constraint is parallel to a level curve. b. When a non-binding constraint is perpendicular to a level curve. c. When a constraint is parallel to another constraint. d. Alternate optimal solutions indicate an infeasible condition. 57. The desire to maximize profits is an example of a(n) a. decision. b. constraint. c. objective. d. parameter. 58. A company uses 4 pounds of resource 1 to make each unit of X1 and 3 pounds of resource 1 to make each unit of X2. There are only 150 pounds of resource 1 available. Which of the following constraints reflects the relationship between X1, X2 and resource 1? a. 4 X1 + 3 X2 ≥ 150 b. 4 X1 + 3 X2 ≤ 150 c. 4 X1 + 3 X2 = 150 d. 4 X1 ≤ 150 59. The following linear programming problem has been written to plan the production of two products. The company wants to maximize its profits. X1 = number of product 1 produced in each batch X2 = number of product 2 produced in each batch
MAX:
150 X1 + 250 X2
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ch 2 2 X1 + 5 X2 ≤ 200 3 X1 + 7 X2 ≤ 175 X1, X2 ≥ 0 How many units of resource two (the second constraint) are unutilized if the company produces 10 units of product 1 and 5 units of product 2? a. 110 b. 150 c. 155 d. 100 Subject to:
60. When the objective function can increase without ever contacting a constraint the LP model is said to be a. infeasible. b. open ended. c. multi-optimal. d. unbounded. 61. Which of the following actions would expand the feasible region of an LP model? a. Loosening the constraints. b. Tightening the constraints. c. Multiplying each constraint by 2. d. Adding an additional constraint. 62. Retail companies try to find a. the least costly method of transferring goods from warehouses to stores. b. the most costly method of transferring goods from warehouses to stores. c. the largest number of goods to transfer from warehouses to stores. d. the least profitable method of transferring goods from warehouses to stores. 63. The first step in formulating a linear programming problem is a. Identify any upper or lower bounds on the decision variables. b. State the constraints as linear combinations of the decision variables. c. Understand the problem. d. Identify the decision variables. e. State the objective function as a linear combination of the decision variables. 64. A mathematical programming application employed by a shipping company is most likely a. a product mix problem. b. a manufacturing problem. c. a routing and logistics problem. d. a financial planning problem. 65. What most motivates a business to be concerned with efficient use of their resources? a. Resources are limited and valuable. b. Efficient resource use increases business costs. Copyright Cengage Learning. Powered by Cognero.
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ch 2 c. Efficient resources use means more free time. d. Inefficient resource use means hiring more workers. 66. A facility produces two products. The labor constraint (in hours) is formulated as: 350x1+300x2 ≤ 10,000. The number 350 means that a. one unit of product 1 contributes $350 to the objective function. b. one unit of product 1 uses 350 hours of labor. c. the problem is unbounded. d. the problem has no objective function. 67. A linear formulation means that: a. the objective function and all constraints must be linear b. only the objective function must be linear c. at least one constraint must be linear d. no more than 50% of the constraints must be linear
68. Solve the following LP problem graphically using level curves. MAX: Subject to:
5 X1 + 6 X2 3 X1 + 8 X2 ≤ 48 12 X1 + 11 X2 ≤ 132 2 X1 + 3 X2 ≤ 24 X1, X2 ≥ 0
69. Solve the following LP problem graphically by enumerating the corner points. MAX: Subject to:
4 X1 + 3 X2 6 X1 + 7 X2 ≤ 84 X1 ≤ 10 X2 ≤ 8 X1, X2 ≥ 0
70. Solve the following LP problem graphically using level curves. MIN: Subject to:
8 X1 + 12 X2 2 X1 + 1 X2 ≥ 16 2 X1 + 3 X2 ≥ 36 7 X1 + 8 X2 ≥ 112 X1, X2 ≥ 0
71. Bob and Dora Sweet wish to start investing $1,000 each month. The Sweets are looking at five investment plans and wish to maximize their expected return each month. Assume interest rates remain fixed and once their investment plan is selected they do not change their mind. The investment plans offered are: Copyright Cengage Learning. Powered by Cognero.
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ch 2 Fidelity 9.1% return per year Optima 16.1% return per year CaseWay 7.3% return per year Safeway 5.6% return per year National 12.3% return per year Since Optima and National are riskier, the Sweets want a limit of 30% per month of their total investments placed in these two investments. Since Safeway and Fidelity are low risk, they want at least 40% of their investment total placed in these investments. Formulate the LP model for this problem. 72. Solve the following LP problem graphically by enumerating the corner points. MIN: Subject to:
8 X1 + 3 X2 X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0
73. Solve the following LP problem graphically using level curves. MIN: Subject to:
5 X1 + 7 X2 4 X1 + 1 X2 ≥ 16 6 X1 + 5 X2 ≥ 60 5 X1 + 8 X2 ≥ 80 X1, X2 ≥ 0
74. Project 2.1 Joey Koons runs a small custom computer parts company. As a sideline he offers customized and pre-built computer system packages. In preparation for the upcoming school year, he has decided to offer two custom computer packages tailored for what he believes are current student needs. System A provides a strong computing capability at a reasonable cost while System B provides a much more powerful computing capability, but at a higher cost. Joey has a fairly robust parts inventory but is concerned about his stock of those components that are common to each proposed system. A portion of his inventory, the item cost, and inventory level is provided in the table below.
Part Processor Memory Hard Drive Monitor Graphics Card CD-
Type / Cost 366 MHZ $175 64 MB $95 4 GB $89 14 " $95 Stock $100 24X
On Hand 40 40 10 3 100 5
Type / Cost 500 MHZ $239 96 MB $189 6 GB $133 15 " $160 3-D $250 40X
On Hand 40 40 25 65
Type / Cost 650 MHZ $500 128 MB $250 13 GB $196 17 " $280
On Hand 40
72X
50
15 35 25
Type / Cost 700 MHZ $742 256 MB $496 20 GB $350 19 " $480
On Hand 40
DVD
45
15 50 10
15 25
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ch 2 ROM Sound Card Speakers Modem Mouse Keyboard Game Devices
$30 Stock $99 Stock $29 Stock $99 Stock $39 Stock $59 Stock $165
100 75
$58 Sound II $150 60 W $69
50 75
$125 Plat II $195 120 W $119
$178 25 25
125 125 100
Ergo $69 Ergo $129
35 35
25
The requirements for each system are provided in the following table:
System A System B Processor 366 MHZ 700 MHZ Memory 64 MB 96 MB Hard Drive 6 GB 20 GB Monitor 15 " 15 " Graphics Card Stock Stock CD-ROM 40X 72X Sound Card Stock Stock Speakers Stock 60W Modem Stock Stock Mouse Stock Stock Keyboard Stock Stock Each system requires assembly, testing and packaging. The requirements per system built and resources available are summarized in the table below.
System A System B Total Hours Available Assembly (hours) 2.25 2.50 200 Testing (hours) 1.25 2.00 150 Packaging (hours) 0.50 0.50 75 Joey is uncertain about product demand. In the past he has put together similar types of computer packages but his sales results vary. As a result is unwilling to commit all his in-house labor force to building the computer packages. He is confident he can sell all he can build and is not overly concerned with lost sales due to stock-outs. Based on his market survey, he has completed his advertising flyer and will offer System A for $ 1250 and will offer system B for $ 2325. Joey now needs to let his workers know how many of each system to build and he wants that mix to maximize his profits. Formulate an LP for Dave's problem. Solve the model using the graphical method. What is Dave's preferred product mix? What profit does Dave expect to make from this product mix? 75. Solve the following LP problem graphically by enumerating the corner points. MIN: Subject to:
8 X1 + 5 X2 6 X1 + 7 X2 ≥ 84 X1 ≥ 4 X2 ≥ 6
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ch 2 X1, X2 ≥ 0 76. Solve the following LP problem graphically using level curves. MAX: Subject to:
5 X1 + 3 X2 2 X1 − 1 X2 ≤ 2 6 X1 + 6 X2 ≥ 12 1 X1 + 3 X2 ≤ 5 X1, X2 ≥ 0
77. Solve the following LP problem graphically by enumerating the corner points. MAX: Subject to:
2 X1 + 7 X2 5 X1 + 9 X2 ≤ 90 9 X1 + 8 X2 ≤ 144 X2 ≤ 8 X1, X2 ≥ 0
78. Solve the following LP problem graphically using level curves. MAX: Subject to:
7 X1 + 4 X2 2 X1 + X2 ≤ 16 X1 + X2 ≤ 10 2 X1 + 5 X2 ≤ 40 X1, X2 ≥ 0
79. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced. a.
Formulate the LP model for this problem.
b.
Solve the problem using the graphical method.
80. The Big Bang explosives company produces customized blasting compounds for use in the mining industry. The two ingredients for these explosives are agent A and agent B. Big Bang just received an order for 1400 pounds of explosive. Agent A costs $5 per pound and agent B costs $6 per pound. The customer's mixture must contain at least 20% agent A and at least 50% agent B. The company wants to provide the least expensive mixture which will satisfy the customers requirements. a.
Formulate the LP model for this problem.
b.
Solve the problem using the graphical method.
81. Jim's winery blends fine wines for local restaurants. One of his customers has requested a special blend of two burgundy wines, call them A and B. The customer wants 500 gallons of wine and it must contain at least 100 gallons of A and be at least 45% B. The customer also specified that the wine have an alcohol content of at least 12%. Wine A contains Copyright Cengage Learning. Powered by Cognero.
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ch 2 14% alcohol while wine B contains 10%. The blend is sold for $10 per gallon. Wine A costs $4 per gallon and B costs $3 per gallon. The company wants to determine the blend that will meet the customer's requirements and maximize profit. a.
Formulate the LP model for this problem.
b.
Solve the problem using the graphical method.
c.
How much profit will Jim make on the order?
82. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces a limited demand. There are a limited number of wiring, assembly and inspection hours available next month. The data for this problem is summarized in the following table. Maximum
Assembly
Inspection
Computer
Profit per
demand for
Wiring Hours
Hours
Hours
Model
Model ($)
product
Required
Required
Required
Plain
30
80
0.4
0.5
0.2
Fancy
40
90
0.5
0.4
0.3
Hours Available
50
50
22
a.
Formulate the LP model for this problem.
b.
Solve the problem using the graphical method.
83. The Happy Pet pet food company produces dog and cat food. Each food is comprised of meat, soybeans and fillers. The company earns a profit on each product but there is a limited demand for them. The pounds of ingredients required and available, profits and demand are summarized in the following table. The company wants to plan their product mix, in terms of the number of bags produced, in order to maximize profit. Profit per
Demand for
Pounds of
Pounds of
Pounds of
Bag ($)
product
Meat per bag
Soybeans per bag
Filler per bag
Dog food
4
40
4
6
4
Cat food
5
30
5
3
10
100
120
160
Product
Material available (pounds) a.
Formulate the LP model for this problem.
b.
Solve the problem using the graphical method.
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ch 2 Answer Key 1. True 2. False 3. False 4. False 5. False 6. True 7. True 8. False 9. a 10. a 11. d 12. d 13. a 14. a 15. d 16. d 17. a 18. b 19. b 20. b 21. d 22. a 23. b 24. a 25. d Copyright Cengage Learning. Powered by Cognero.
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ch 2 26. a 27. a 28. a 29. a 30. b 31. a 32. b 33. b 34. a 35. b 36. c 37. b 38. d 39. d 40. d 41. a 42. b 43. a 44. a 45. a 46. b 47. d 48. e 49. b 50. d 51. c Copyright Cengage Learning. Powered by Cognero.
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ch 2 52. c 53. c 54. c 55. b 56. a 57. c 58. b 59. a 60. d 61. a 62. a 63. c 64. c 65. a 66. b 67. a 68. Obj = 57.43 X1 = 9.43 X2 = 1.71 69. Obj = 50.29 X1 = 10.00 X2 = 3.43 70. Alternate optima solutions exist between the corner points. The value of the objective function for both corner points is 144.00. X1 = 9.6 X2 = 5.6
X1 = 18 X2 = 0
71. Copyright Cengage Learning. Powered by Cognero.
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ch 2 MAX: Subject to:
0.091X1 + 0.161X2 + 0.073X3 + 0.056X4 + 0.123X5 X1 + X2 + X3 + X4 + X5 = 1000 X2 + X5 ≤ 300 X1 + X4 ≥ 400 X1, X2, X3, X4, X5 ≥ 0
72. Obj = 48.00 X1 = 0.00 X2 = 16.00 73. Obj = 72.17 X1 = 3.48 X2 = 7.83 74. The cost to make System A is $1007.00 while the cost to make System B is $1992.00. The inventory levels for hard drives limit System A production to 25 while the 700 MHZ processor inventory limits System B production to 40. The common monitor is the 15 " unit and its inventory limits total production to 60. Coupled with the assembly, testing, and packaging constraints, the LP formulation is: Maximize
$243 X1 + $333 X2 2.25 X1 + 2.50 X2 ≤ 200 1.25 X1 + 2.00 X2 ≤ 150 0.50 X1 + 0.50 X2 ≤ 75 X1 ≤ 25 X2 ≤ 40 X1 + X2 ≤ 60 X1 , X2 ≥ 0
{assembly hours} {testing hours} {packaging hours} {hard drive limits} {processor limits} {monitor limits}
Build 20 System A and 40 System B, Total profit $18,180. 75. Obj = 74.86 X1 = 4.00 X2 = 8.57 76. Obj = 11.29 X1 = 1.57 X2 = 1.14 77. Obj = 63.20 X1 = 3.6 X2 = 8 78. Obj = 58.00 X1 = 6.00 Copyright Cengage Learning. Powered by Cognero.
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ch 2 X2 = 4.00 79. a. Let
MAX: Subject to:
b.
MIN: Subject to:
X1 = Pounds of agent A used X2 = Pounds of agent B used 5 X1 + 6 X2 X1 ≥ 280 (Agent A requirement) X2 ≥ 700 (Agent B requirement) X1 + X2 = 1400 (Total pounds) X1, X2 ≥ 0
Obj = 7700.00 X1 = 700.00 X2 = 700.00
81. a. Let
MIN: Subject to:
b.
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for X2) X1, X2 ≥ 0
Obj = 240.00 X1 = 4.00 X2 = 3.00
80. a. Let
b.
X1 = Number of Beds to produce X2 = Number of Desks to produce
X1 = Gallons of wine A in mix X2 = Gallons of wine B in mix 4 X1 + 3 X2 X1 + X2 ≥ 500 (Total gallons of mix) X1 ≥ 100 (X1 minimum) X2 ≥ 225 (X2 minimum) .14 X1 + .10 X2 ≥ 60 (12% alcohol minimum) X1, X2 ≥ 0
Obj = 1750.00 X1 = 250.00 X2 = 250.00
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ch 2 c.
$3250.00 total profit.
82. a. Let
MAX: Subject to:
b.
30 X1 + 40 X2 .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0
Obj = 3975.00 X1 = 12.50 X2 = 90.00
83. a. Let
MAX: Subject to:
b.
X1 = Number of Plain computers produce X2 = Number of Fancy computers to produce
X1 = bags of Dog food to produce X2 = bags of Cat food to produce 4 X1 + 5 X2 4 X1 + 5 X2 ≤ 100 (meat) 6 X1 + 3 X2 ≤ 120 (soybeans) 4 X1 + 10 X2 ≤ 160 (filler) X1 ≤ 40 (Dog food demand) X2 ≤ 30 (Cat food demand)
Obj = 100.00 X1 = 10.00 X2 = 12.00
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ch 3
Indicate whether the statement is true or false. 1. The LHS value of a constraint represents the usage of an associated resource by the decision variables. a. True b. False 2. Decision variables are sometimes referred to as problem parameters. a. True b. False 3. Solving LP problems in Excel requires only a copy and paste operation. a. True b. False 4. Analytic Solver Platform (ASP) is functionally similar to Excel Solver but it offers many useful extensions. a. True b. False 5. Objective cell, variable cells and constraint cells are terms used in Excel solver to describe the purpose of the cells. a. True b. False 6. Bounds on the decision variables are known as nonnegativity constraints. a. True b. False 7. Excel and other spreadsheets contain an add-on called solver. a. True b. False 8. In modeling a problem it is usually best to start by entering all equations in a spreadsheet. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. Exhibit 3.3 The following questions are based on this problem and accompanying Excel windows. Jack's distillery blends scotches for local bars and saloons. One of his customers has requested a special blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products, call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50 1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B. The customer also specified that the blend have an alcohol content of at least 85%. Product A contains 95% alcohol while product B contains 78%. The blend is sold for $12.50 per bottle. Product A costs $7 per liter and product B costs $3 per liter. The company wants to determine the blend that will meet the customer's Copyright Cengage Learning. Powered by Cognero.
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ch 3 requirements and maximize profit.
Let
X1 = Number of liters of product A in total blend delivered X2 = Number of liters of product B in total blend delivered
MIN: Subject to:
7 X1 + 3 X2 X1 + X2 = 1.5 * 50 (Total liters of mix) X1 ≥ 0.48 * 1.5 * 50 (X1 minimum) X2 ≥ 0.5 * 50 (X2 minimum) .0.95 X1 + 0.78 X2 ≥ 0.85 * 1.5 * 50 (85% alcohol minimum) X1, X2 ≥ 0 A
1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11
B
C
D
E
Jacks' Distillery
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
A
B
10.5
4.5
1 1
1
Total Cost:
Supplied
0.95
1 0.78
A
B
10.5
4.5
1 1
1
Requirement 75 36 25 63.75
Total Cost:
Supplied
0.95
1 0.78
Requirement 75 36 25 63.75
9. Refer to Exhibit 3.3. What formula should be entered in cell D11 in the accompanying Excel spreadsheet to compute the total liters of alcohol supplied? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B11:C11,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10) 10. Refer to Exhibit 3.3. Which cells should be changing cells in this problem? Copyright Cengage Learning. Powered by Cognero.
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ch 3 a. B4:C4 b. E5 c. D8:D10 d. E8:E10 Exhibit 3.4 The following questions are based on this problem and accompanying Excel windows. A financial planner wants to design a portfolio of investments for a client. The client has $300,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 25% of the money in any one investment, at least one third should be invested in long-term bonds which mature in seven or more years, and no more than 25% of the total money should be invested in C or D since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.
Investment A B C D
Return 6.45% 7.10% 8.20% 9.00%
Years to Maturity 9 8 5 8
Rating 1-Excellent 2-Very Good 4-Fair 3-Good
Let
X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D
MAX: Subject to:
.0645 X1 + .071 X2 + .082 X3 + .09 X4 X1 + X2 + X3 + X4 ≤ 300000 X1 ≤ 75000 X2 ≤ 75000 X3 ≤ 75000 X4 ≤ 75000 X1 + X2 + X4 ≥ 100000 X3 + X4 ≤ 75000 X1, X2, X3, X4 ≥ 0 A
1 2 3 4 5 6
Bond A B C D
B Amount Invested $0 $0 $0 $0
C Maximum 25.00% $75,000 $75,000 $75,000 $75,000
D Return 6.45% 7.10% 8.20% 9.00%
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ch 3 7 8
< < < < < < < < <
Total Invested: Total Available:
E 1 Years to 2 Maturity 3 9 4 8 5 5 6 8 7 Total: 8 Required:
$0 $300,000
Total:
$0
> >
F G H 7+ years? Good or worse? (1-yes, 0-no) Rating (1-yes, 0-no) 1 1-Excellent 0 1 2-Very Good 0 0 4-Fair 1 1 3-Good 1 $0 Total: $0 $100,000 Allowed: $75,000
11. Refer to Exhibit 3.4. What formula should be entered in cell B7 in the accompanying Excel spreadsheet to compute total dollars invested? a. =ADD(B3:B6) b. =SUM(B3:B6) c. =TOTAL(B3:B6) d. =TALLY(B3:B6) 12. Data Envelopment Analysis (DEA) is an LP-based methodology in which weighted sums of inputs and outputs are calculated and a. the constraints capture the maximum effectiveness of each unit. b. the objective is to maximize every units output. c. the constraints ensure the sum of the weighted outputs is one. d. the objective for each unit is to maximize the weighted sum of its outputs. 13. The "Objective Value of" option in the Analytic Solver Platform task pane may be used to a. find a solution at a maximum value. b. find a solution at a minimum value. c. find a solution for a specific objective function value. d. returns the best feasible solution. Exhibit 3.1 The following questions are based on this problem and accompanying Excel windows. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let
X1 = Number of Beds to produce X2 = Number of Desks to produce
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ch 3 The LP model for the problem is
MAX: Subject to:
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for desks) X1, X2 ≥ 0 A
1 2 3 4 5 6 7 8 9 10
B
C
D
E
Jones Furniture
Number to make: Unit profit: Constraints: Carpentry Varnishing Desk demand
Beds
Desks
30
40
Total Profit:
Used 6 4
4 8 1
Available 36 40 8
14. Refer to Exhibit 3.1. Which of the following statements represent the carpentry, varnishing and limited demand for desks constraints? a. B4:C4 ≤ B5:C5 b. E5 ≤ 0 c. D8:D10 ≤ E8:E10 d. E8:E10 ≤ D8:D10 15. How many constraints are there in a transportation problem which has 5 supply points and 4 demand points? (ignore the non-negativity constraints) a. 4 b. 5 c. 9 d. 20 16. In the Analytic Solver Platform dialog box simple upper and lower bounds for decision variables are specified by a. referring directly to the decision variable cells in the Constraints-Bound area. b. requiring the addition of the bounds above and below the variable cells. c. resolving the problem with the bounds added. d. incorporating the bounds in the objective function. 17. The built-in Solver in Excel is found under which tab on the ribbon? a. Tools b. Insert Copyright Cengage Learning. Powered by Cognero.
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ch 3 c. Data d. Window Exhibit 3.5 The following questions are based on this problem and accompanying Excel windows. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost is stable but demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand. Quarterly inventory holding cost is 3% of the cost of production. The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month. There are currently 5,000 units in inventory. The company wants to produce at no less than one half of its maximum capacity in any quarter.
Quarter 1
2
3
4
Unit Production Cost
$ 300
$ 300
$ 300
$
Units Demanded Maximum Production
2,000 8,000
9,000 7,000
12,000 8,000
11,000 9,000
Let
Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i
MIN:
300 P1 + 300 P2 + 300 P3 + 300 P4 + 9(B1 + B2)/2 + 9(B2 + B3)/2 + 9(B3 + B4)/2 + 9(B4 + B5)/2 4000 ≤ P1 ≤ 8000 3500 ≤ P2 ≤ 7000 4000 ≤ P3 ≤ 8000 4500 ≤ P4 ≤ 9000 3000 ≤ B1 + P1 − 2000 ≤ 12000 3000 ≤ B2 + P2 − 9000 ≤ 12000 3000 ≤ B3 + P3 − 12000 ≤ 12000 3000 ≤ B4 + P4 − 11000 ≤ 12000 B2 = B1 + P1 − 2000 B3 = B2 + P2 − 9000 B4 = B3 + P3 − 12000 B5 = B4 + P4 − 11000 Pi, Bi ≥ 0
Subject to:
A
B
1 2
C
D
1
2
300
E
F
3
4
Quarter
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ch 3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Beginning Inventory Units Produced Units Demanded Ending Inventory
5,000 8,000 2,000 11,000
11,000 7,000 9,000 9,000
9,000 8,000 12,000 5,000
5,000 9,000 11,000 3,000
Minimum Production Maximum Production
4,000 8,000
3,500 7,000
4,000 8,000
4,500 9,000
Minimum Inventory Maximum Inventory
3,000 12,000
3,000 12,000
3,000 12,000
3,000 12,000
$300 $9.00
$300 $9.00
$300 $9.00
$300 $9.00
$2,400,000 $72,000
$2,100,000 $90,000
$2,400,000 $63,000
$2,700,000 $36,000
Unit Production Cost Unit Carrying Cost
3.0%
Quarterly Production Cost Quarterly Carrying Cost
Total Cost
$9,861,000
18. Refer to Exhibit 3.5. What formula should be entered in cell C6 in the accompanying Excel spreadsheet to compute ending inventory? a. =C3-C4+C5 b. =C3+C4-C5 c. =C3-(C4-C5) d. =C5-C4-C3 19. Which function is equivalent to =SUMPRODUCT(A1:A3,B1:B3)? a. =SUM(PRODUCT((A1:A3,B1:B3)) b. =PRODUCT(SUM((A1:A3,B1:B3)) c. =PRODUCT(A1+B1,A2+B2,A3+B3)) d. =A1*B1+A2*B2+A3*B3 20. Models which are setup in an intuitively appealing, logical layout tend to be the most a. Reliable b. Modifiable c. Auditable d. Organized Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table. Copyright Cengage Learning. Powered by Cognero.
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ch 3 Maximum demand for Wiring Hours product Required 80 .4 90 .5 Hours Available 50
Computer Model Plain Fancy
Profit per Model ($) 30 40
Let
X1 = Number of Plain computers to produce X2 = Number of Fancy computers to produce
MAX: Subject to:
30 X1 + 40 X2 .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0
A 1
B
C
D
Assembly Hours Required .5 .4 50
Inspection Hours Required .2 .3 22
E
Byte Computer Company
2 3
Plain
4
Number to make:
5
Unit profit:
Fancy Total Profit:
30
40
6 7
Constraints:
8
Wiring
0.4
0.5
50
9
Assembly
0.5
0.4
50
10 Inspection
0.2
0.3
22
11 Plain Demand
1
12 Fancy Demand
Used
Available
80 1
90
21. Refer to Exhibit 3.2. Which cells should be changing cells in this problem? a. B4:C4 b. E5 c. D8:D10 d. E8:E10 22. A heuristic solution is a. used by Analytic Solver Platform (ASP) when the Guess button is used. b. guaranteed to produce an optimal solution. Copyright Cengage Learning. Powered by Cognero.
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ch 3 c. used by Analytic Solver Platform (ASP) if Standard GRG Nonlinear method is selected. d. a rule-of-thumb for making decisions. 23. Which tab in the Analytic Solver Platform task pane is used to define an optimization problem? a. Guess b. Model c. Change d. Delete Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table. Maximum demand for Wiring Hours product Required 80 .4 90 .5 Hours Available 50
Computer Model Plain Fancy
Profit per Model ($) 30 40
Let
X1 = Number of Plain computers to produce X2 = Number of Fancy computers to produce
MAX: Subject to:
30 X1 + 40 X2 .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0 A
1
B
C
D
Assembly Hours Required .5 .4 50
Inspection Hours Required .2 .3 22
E
Byte Computer Company
2 3
Plain
4
Number to make:
5
Unit profit:
Fancy Total Profit:
30
40
6 Copyright Cengage Learning. Powered by Cognero.
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ch 3 7
Constraints:
Used
Available
8
Wiring
0.4
0.5
50
9
Assembly
0.5
0.4
50
10 Inspection
0.2
0.3
22
11 Plain Demand
1
80
12 Fancy Demand
1
90
24. Refer to Exhibit 3.2. Which of the following statements will represent the constraint for just assembly hours? a. B4:C4 ≤ B5:C5 b. D9 ≤ E9 c. D8:D10 ≤ E8:E10 d. E8:E10 ≤ D8:D10 Exhibit 3.1 The following questions are based on this problem and accompanying Excel windows. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let
X1 = Number of Beds to produce X2 = Number of Desks to produce
The LP model for the problem is
MAX: Subject to:
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for desks) X1, X2 ≥ 0 A
1 2 3 4 5 6 7 8 9
Number to make: Unit profit: Constraints: Carpentry Varnishing
B
C Jones Furniture
Beds
Desks
30
40
D
E
Total Profit:
Used 6 4
4 8
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ch 3 10 Desk demand
1
8
25. Refer to Exhibit 3.1. What formula should be entered in cell D8 in the accompanying Excel spreadsheet to compute the amount of carpentry used? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B8:C8,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10) 26. Spreadsheet modeling is an acquired skill because a. there is generally only one correct way to build a model. b. the spreadsheet is free-form providing many modeling options. c. using Analytic Solver Platform requires lots of experience. d. spreadsheets are not very easy to use. 27. What action is required to make Analytic Solver Platform (ASP) solve a specified problem? a. Type go in cell A1. b. Click the "Optimize" button on the ASP Ribbon, or the green arrow "Solve" in the Task Pane. c. Click the Close button in the ASP Parameters dialog box. d. Click the Guess button in the ASP Parameters dialog box. 28. Which of the following describes Data Envelopment Analysis (DEA). a. DEA finds the most effective company among some set of companies. b. DEA determines if a company is converting inputs to outputs as effectively as possible. c. DEA determines how effective a company converts inputs to outputs compared to other companies. d. DEA compares how effective a company converts inputs to outputs compared to a benchmark composite of all companies. Exhibit 3.1 The following questions are based on this problem and accompanying Excel windows. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
Let
X1 = Number of Beds to produce X2 = Number of Desks to produce
The LP model for the problem is
MAX: Subject to:
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry)
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ch 3 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for desks) X1, X2 ≥ 0 A 1 2 3 4 5 6 7 8 9 10
B
C
D
E
Jones Furniture Beds Number to make: Unit profit: Constraints: Carpentry Varnishing Desk demand
Desks Total Profit:
30
40
6 4
4 8 1
Used
Available 36 40 8
29. Refer to Exhibit 3.1. Which cells should be changing cells in this problem? a. B4:C4 b. E5 c. D8:D10 d. E8:E10 30. Refer to Exhibit 3.1. What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total profit? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B8:C8,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10) 31. What is the significance of an absolute cell reference in Excel? a. The cell reference will not change if the formula containing the reference is copied to another location b. The cell will always contain the absolute value of any number entered into it c. The cell reference changes if the formula containing the reference is copied to another location d. It is the only formula used to refer to a cell on another spreadsheet Exhibit 3.1 The following questions are based on this problem and accompanying Excel windows. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited, so at most 8 will be produced.
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ch 3 Let
X1 = Number of Beds to produce X2 = Number of Desks to produce
The LP model for the problem is
MAX: Subject to:
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for desks) X1, X2 ≥ 0 A
1 2 3 4 5 6 7 8 9 10
B
C Jones Furniture
Beds Number to make: Unit profit: Constraints: Carpentry Varnishing Desk demand
D
E
Desks Total Profit:
30
40
6 4
4 8 1
Used
Available 36 40 8
32. Refer to Exhibit 3.1. Which cells should be the constraint cells in this problem? a. B4:C4 b. E5 c. D8:D10 d. E8:E10 33. Which type of spreadsheet cell represents the decision variables in an LP model? a. Target or set cell b. Variable cell c. Constraint cell d. Constant cell Exhibit 3.3 The following questions are based on this problem and accompanying Excel windows. Jack's distillery blends scotches for local bars and saloons. One of his customers has requested a special blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products, call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50 1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B. The customer also specified that the blend have an alcohol content of at least 85%. Product A contains 95% alcohol while product B contains 78%. The blend is sold for $12.50 per bottle. Product A costs Copyright Cengage Learning. Powered by Cognero.
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ch 3 $7 per liter and product B costs $3 per liter. The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let
X1 = Number of liters of product A in total blend delivered X2 = Number of liters of product B in total blend delivered
MIN: Subject to:
7 X1 + 3 X2 X1 + X2 = 1.5 * 50 (Total liters of mix) X1 ≥ 0.48 * 1.5 * 50 (X1 minimum) X2 ≥ 0.5 * 50 (X2 minimum) .0.95 X1 + 0.78 X2 ≥ 0.85 * 1.5 * 50 (85% alcohol minimum) X1, X2 ≥ 0 A
1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11
B C Jacks' Distillery A
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
D
E
B Total Cost:
10.5
4.5
1 1
1
Supplied
0.95
1 0.78
A
B
Requirement 75 36 25 63.75
Total Cost: 10.5
4.5
1 1
1
Supplied
0.95
1 0.78
Requirement 75 36 25 63.75
34. Refer to Exhibit 3.3. What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total cost? a. =B4*C4+B5*C5 b. =SUMPRODUCT(B4:C4,B5:C5) c. =SUM(B5:C5) d. =SUM(E8:E10)
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ch 3 Exhibit 3.5 The following questions are based on this problem and accompanying Excel windows. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost is stable but demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand. Quarterly inventory holding cost is 3% of the cost of production. The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month. There are currently 5,000 units in inventory. The company wants to produce at no less than one half of its maximum capacity in any quarter.
Quarter Unit Production Cost Units Demanded Maximum Production
1
2
3
4
$ 300 2,000 8,000
$ 300 9,000 7,000
$ 300 12,000 8,000
$ 300 11,000 9,000
Let
Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i
MIN:
300 P1 + 300 P2 + 300 P3 + 300 P4 + 9(B1 + B2)/2 + 9(B2 + B3)/2 + 9(B3 + B4)/2 + 9(B4 + B5)/2 4000 ≤ P1 ≤ 8000 3500 ≤ P2 ≤ 7000 4000 ≤ P3 ≤ 8000 4500 ≤ P4 ≤ 9000 3000 ≤ B1 + P1 − 2000 ≤ 12000 3000 ≤ B2 + P2 − 9000 ≤ 12000 3000 ≤ B3 + P3 − 12000 ≤ 12000 3000 ≤ B4 + P4 − 11000 ≤ 12000 B2 = B1 + P1 − 2000 B3 = B2 + P2 − 9000 B4 = B3 + P3 − 12000 B5 = B4 + P4 − 11000 Pi, Bi ≥ 0
Subject to:
A 1 2 3 4 5
B
C
D
1 5,000 8,000 2,000
2 11,000 7,000 9,000
E
F
3 9,000 8,000 12,000
4 5,000 9,000 11,000
Quarter Beginning Inventory Units Produced Units Demanded
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ch 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ending Inventory
11,000
9,000
5,000
3,000
Minimum Production Maximum Production
4,000 8,000
3,500 7,000
4,000 8,000
4,500 9,000
Minimum Inventory Maximum Inventory
3,000 12,000
3,000 12,000
3,000 12,000
3,000 12,000
$300 $9.00
$300 $9.00
$300 $9.00
$300 $9.00
$2,400,000 $72,000
$2,100,000 $90,000
$2,400,000 $63,000
$2,700,000 $36,000
Unit Production Cost Unit Carrying Cost
3.0%
Quarterly Production Cost Quarterly Carrying Cost
Total Cost
$9,861,000
35. Refer to Exhibit 3.5. What formula should be entered in cell C18 in the accompanying Excel spreadsheet to compute the quarterly carrying costs? a. =C15*C3+C6 b. =C15*(C3+C6) c. =C15*C3/2 d. =C15*(C3+C6)/2 36. The "Analyze Without Solving" tool in Analytic Solver Platform is useful for a. verifying the equations in a spreadsheet model. b. toggling between absolute and relative cell referencing. c. executing the Excel spreadsheet layout Wizard. d. naming cells and cell ranges for easier modifiability. Exhibit 3.3 The following questions are based on this problem and accompanying Excel windows. Jack's distillery blends scotches for local bars and saloons. One of his customers has requested a special blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products, call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50 1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B. The customer also specified that the blend have an alcohol content of at least 85%. Product A contains 95% alcohol while product B contains 78%. The blend is sold for $12.50 per bottle. Product A costs $7 per liter and product B costs $3 per liter. The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let
X1 = Number of liters of product A in total blend delivered X2 = Number of liters of product B in total blend delivered
MIN: Subject to:
7 X1 + 3 X2 X1 + X2 = 1.5 * 50 (Total liters of mix)
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ch 3 X1 ≥ 0.48 * 1.5 * 50 (X1 minimum) X2 ≥ 0.5 * 50 (X2 minimum) .0.95 X1 + 0.78 X2 ≥ 0.85 * 1.5 * 50 (85% alcohol minimum) X1, X2 ≥ 0 A 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11
B C Jacks' Distillery A
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
D
E
B Total Cost:
10.5
4.5
1 1
1
Supplied
0.95
1 0.78
A
B
Requirement 75 36 25 63.75
Total Cost: 10.5
4.5
1 1
1
Supplied
0.95
1 0.78
Requirement 75 36 25 63.75
37. Refer to Exhibit 3.3. Which cells should be the constraint cells in this problem? a. B4:C4 b. E5 c. D8:D11 d. E8:E10 Exhibit 3.4 The following questions are based on this problem and accompanying Excel windows. A financial planner wants to design a portfolio of investments for a client. The client has $300,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 25% of the money in any one investment, at least one third should be invested in long-term bonds which mature in seven or more years, and no more than 25% of the total money should be invested in C or D since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.
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ch 3 Investment A B C D
Return 6.45% 7.10% 8.20% 9.00%
Years to Maturity 9 8 5 8
Rating 1-Excellent 2-Very Good 4-Fair 3-Good
Let
X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D
MAX: Subject to:
.0645 X1 + .071 X2 + .082 X3 + .09 X4 X1 + X2 + X3 + X4 ≤ 300000 X1 ≤ 75000 X2 ≤ 75000 X3 ≤ 75000 X4 ≤ 75000 X1 + X2 + X4 ≥ 100000 X3 + X4 ≤ 75000 X1, X2, X3, X4 ≥ 0 A
1 2 3 4 5 6 7 8
< < < < < < < < <
Bond A B C D Total Invested: Total Available:
1 2 3 4 5 6 7 8
E Years to Maturity 9 8 5 8 Total: Required:
B Amount Invested $0 $0 $0 $0 $0 $300,000
C Maximum 25.00% $75,000 $75,000 $75,000 $75,000 Total:
D Return 6.45% 7.10% 8.20% 9.00% $0
> > > > > > > > >
F G H 7+ years? Good or worse? (1-yes, 0-no) Rating (1-yes, 0-no) 1 1-Excellent 0 1 2-Very Good 0 0 4-Fair 1 1 3-Good 1 $0 Total: $0 $100,000 Allowed: $75,000
38. Refer to Exhibit 3.4. What formula should be entered in cell D7 in the accompanying Excel spreadsheet to compute the total return? Copyright Cengage Learning. Powered by Cognero.
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ch 3 a. =B7*SUM(D3:D6) b. =SUMPRODUCT(B3:B6,D3:D6) c. =SUM(B3:B6) d. =SUMPRODUCT(B3:E3,B6:E6) 39. Numeric constants should be a. embedded in formulas. b. placed in individual cells c. placed in separate workbooks. d. entered manually every time a model is solved. Exhibit 3.5 The following questions are based on this problem and accompanying Excel windows. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost is stable but demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand. Quarterly inventory holding cost is 3% of the cost of production. The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month. There are currently 5,000 units in inventory. The company wants to produce at no less than one half of its maximum capacity in any quarter.
Quarter 1
2
3
4
Unit Production Cost Units Demanded
$ 300 2,000
$ 300 9,000
$ 300 12,000
$ 300 11,000
Maximum Production
8,000
7,000
8,000
9,000
Let
Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i
MIN:
300 P1 + 300 P2 + 300 P3 + 300 P4 + 9(B1 + B2)/2 + 9(B2 + B3)/2 + 9(B3 + B4)/2 + 9(B4 + B5)/2 4000 ≤ P1 ≤ 8000 3500 ≤ P2 ≤ 7000 4000 ≤ P3 ≤ 8000 4500 ≤ P4 ≤ 9000 3000 ≤ B1 + P1 − 2000 ≤ 12000 3000 ≤ B2 + P2 − 9000 ≤ 12000 3000 ≤ B3 + P3 − 12000 ≤ 12000 3000 ≤ B4 + P4 − 11000 ≤ 12000 B2 = B1 + P1 − 2000
Subject to:
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ch 3 B3 = B2 + P2 − 9000 B4 = B3 + P3 − 12000 B5 = B4 + P4 − 11000 Pi, Bi ≥ 0 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
B
C
D
E
F
Beginning Inventory Units Produced Units Demanded Ending Inventory
1 5,000 8,000 2,000 11,000
2 11,000 7,000 9,000 9,000
3 9,000 8,000 12,000 5,000
4 5,000 9,000 11,000 3,000
Minimum Production Maximum Production
4,000 8,000
3,500 7,000
4,000 8,000
4,500 9,000
Minimum Inventory Maximum Inventory
3,000 12,000
3,000 12,000
3,000 12,000
3,000 12,000
Unit Production Cost Unit Carrying Cost
$300 $9.00
$300 $9.00
$300 $9.00
$300 $9.00
$2,400,000 $72,000
$2,100,000 $90,000
$2,400,000 $63,000
$2,700,000 $36,000
Quarter
3.0%
Quarterly Production Cost Quarterly Carrying Cost
Total Cost
$9,861,000
40. Refer to Exhibit 3.5. Which cells are changing cells in the accompanying Excel spreadsheet? a. C4:F4 b. C9:F9 c. F20 d. C12:F12 41. Microsoft Excel contains a built-in optimizer called a. what-if engines. b. calculators. c. solvers. d. risk analyzers. Exhibit 3.4 The following questions are based on this problem and accompanying Excel windows. A financial planner wants to design a portfolio of investments for a client. The client has $300,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 25% of the money in any one investment, at least one third should be invested in long-term bonds which mature in seven or more years, and no more than 25% of the total money should be invested in C or D since they Copyright Cengage Learning. Powered by Cognero.
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ch 3 are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.
Investment A B C D
Return 6.45% 7.10% 8.20% 9.00%
Years to Maturity 9 8 5 8
Rating 1-Excellent 2-Very Good 4-Fair 3-Good
Let
X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D
MAX: Subject to:
.0645 X1 + .071 X2 + .082 X3 + .09 X4 X1 + X2 + X3 + X4 ≤ 300000 X1 ≤ 75000 X2 ≤ 75000 X3 ≤ 75000 X4 ≤ 75000 X1 + X2 + X4 ≥ 100000 X3 + X4 ≤ 75000 X1, X2, X3, X4 ≥ 0 A
1 2 3 4 5 6 7 8
< < < < < < < < <
Bond A B C D Total Invested: Total Available:
E 1 Years to 2 Maturity 3 9 4 8 5 5 6 8 7 Total: 8 Required:
B Amount Invested $0 $0 $0 $0 $0 $300,000
C Maximum 25.00% $75,000 $75,000 $75,000 $75,000 Total:
D Return 6.45% 7.10% 8.20% 9.00% $0
> > > > > > > > >
F G H 7+ years? Good or worse? (1-yes, 0-no) Rating (1-yes, 0-no) 1 1-Excellent 0 1 2-Very Good 0 0 4-Fair 1 1 3-Good 1 $0 Total: $0 $100,000 Allowed: $75,000
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ch 3
42. Refer to Exhibit 3.4. Which cells are changing cells in the accompanying Excel spreadsheet? a. B3:B6 b. B7:I7 c. C7 d. E7 Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table. Maximum demand for Wiring Hours product Required 80 .4 90 .5 Hours Available 50
Computer Model Plain Fancy
Profit per Model ($) 30 40
Let
X1 = Number of Plain computers to produce X2 = Number of Fancy computers to produce
MAX: Subject to:
30 X1 + 40 X2 .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0
A 1
B
C
D
Assembly Hours Required .5 .4 50
Inspection Hours Required .2 .3 22
E
Byte Computer Company
2 3
Plain
4
Number to make:
5
Unit profit:
Fancy Total Profit:
30
40
6 7
Constraints:
Used
Available
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ch 3 8
Wiring
0.4
0.5
50
9
Assembly
0.5
0.4
50
10 Inspection
0.2
0.3
22
11 Plain Demand
1
12 Fancy Demand
80 1
90
43. Refer to Exhibit 3.2. What formula should be entered in cell D8 in the accompanying Excel spreadsheet to compute the amount of wiring used? a. =B4*B5+C4*C5 b. =SUMPRODUCT(B8:C8,$B$4:$C$4) c. =SUM(B5:C5) d. =SUM(E8:E10) 44. Using Data Envelopment Analysis (DEA) for an inefficient unit, a more efficient composite unit can be found by a. Solving its DEA problem and retrieving the weights from the answer report. b. Solving its DEA problem and examining those units whose final value is non-zero. c. Solving its DEA problem and using the resulting shadow prices as composite weights. d. Solving its DEA problem and using the positive resulting shadow prices as composite weights. 45. What function is used to add the contents of cells A1, A2 and A3? a. =A1+A2+A3 b. =ADD(A1:A3) c. =TOTAL(A1:A3) d. =PRODUCT(A1:A3) Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table. Maximum demand for Wiring Hours product Required 80 .4 90 .5 Hours Available 50
Computer Model Plain Fancy
Profit per Model ($) 30 40
Let
X1 = Number of Plain computers to produce X2 = Number of Fancy computers to produce
Assembly Hours Required .5 .4 50
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Inspection Hours Required .2 .3 22
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ch 3 MAX: Subject to:
30 X1 + 40 X2 .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0 A
B
1
C
D
E
Byte Computer Company
2 3
Plain
4
Number to make:
5
Unit profit:
Fancy Total Profit:
30
40
6 7
Constraints:
8
Wiring
0.4
0.5
50
9
Assembly
0.5
0.4
50
10 Inspection
0.2
0.3
22
11 Plain Demand
1
12 Fancy Demand
Used
Available
80 1
90
46. Refer to Exhibit 3.2. Which cells should be the constraint cells in this problem? a. B4:C4 b. E5 c. D8:D12 d. E8:E12 Exhibit 3.3 The following questions are based on this problem and accompanying Excel windows. Jack's distillery blends scotches for local bars and saloons. One of his customers has requested a special blend of scotch targeted as a bar scotch. The customer wants the blend to involve two scotch products, call them A and B. Product A is a higher quality scotch while product B is a cheaper brand. The customer wants to make the claim the blend is closer to high quality than the alternative. The customer wants 50 1500 ml bottles of the blend. Each bottle must contain at least 48% of Product A and at least 500 ml of B. The customer also specified that the blend have an alcohol content of at least 85%. Product A contains 95% alcohol while product B contains 78%. The blend is sold for $12.50 per bottle. Product A costs $7 per liter and product B costs $3 per liter. The company wants to determine the blend that will meet the customer's requirements and maximize profit.
Let
X1 = Number of liters of product A in total blend delivered
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ch 3 X2 = Number of liters of product B in total blend delivered MIN: Subject to:
7 X1 + 3 X2 X1 + X2 = 1.5 * 50 (Total liters of mix) X1 ≥ 0.48 * 1.5 * 50 (X1 minimum) X2 ≥ 0.5 * 50 (X2 minimum) .0.95 X1 + 0.78 X2 ≥ 0.85 * 1.5 * 50 (85% alcohol minimum) X1, X2 ≥ 0 A
1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10 11
B C Jacks' Distillery A
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
Liters to use Unit cost: Constraints: Total Liters A required B required 85% alcohol
D
E
B Total Cost:
10.5
4.5 Supplied
1 1
1
0.95
1 0.78
A
B
Requirement 75 36 25 63.75
Total Cost: 10.5
4.5 Supplied
1 1 0.95
1 1 0.78
Requirement 75 36 25 63.75
47. Refer to Exhibit 3.3. Which of the following statements could represent a constraint in this problem? a. B4:C4 ≤ B5:C5 b. E5 ≤ 0 c. D8 = E8 d. E8:E11 ≤ D8:D11 Exhibit 3.2 The following questions are based on this problem and accompanying Excel windows. The Byte computer company produces two models of computers, Plain and Fancy. It wants to plan how many computers to produce next month to maximize profits. Producing these computers requires wiring, assembly and inspection time. Each computer produces a certain level of profits but faces only a limited demand. There are also a limited number of Copyright Cengage Learning. Powered by Cognero.
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ch 3 wiring, assembly and inspection hours available in each month. The data for this problem is summarized in the following table. Maximum demand for Wiring Hours product Required 80 .4 90 .5 Hours Available 50
Computer Model Plain Fancy
Profit per Model ($) 30 40
Let
X1 = Number of Plain computers to produce X2 = Number of Fancy computers to produce
MAX: Subject to:
30 X1 + 40 X2 .4 X1 + .5 X2 ≤ 50 (wiring hours) .5 X1 + .4 X2 ≤ 50 (assembly hours) .2 X1 + .2 X2 ≤ 22 (inspection hours) X1 ≤ 80 (Plain computers demand) X2 ≤ 90 (Fancy computers demand) X1, X2 ≥ 0
A 1
B
C
D
Assembly Hours Required .5 .4 50
Inspection Hours Required .2 .3 22
E
Byte Computer Company
2 3
Plain
4
Number to make:
5
Unit profit:
Fancy Total Profit:
30
40
6 7
Constraints:
8
Wiring
0.4
0.5
50
9
Assembly
0.5
0.4
50
10 Inspection
0.2
0.3
22
11 Plain Demand
1
12 Fancy Demand
Used
Available
80 1
90
48. Refer to Exhibit 3.2. What formula should be entered in cell E5 in the accompanying Excel spreadsheet to compute total profit? a. =B4*C4+B5*C5 b. =SUMPRODUCT(B4:C4,B5:C5) c. =SUM(B5:C5) d. =SUM(E8:E10) Copyright Cengage Learning. Powered by Cognero.
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ch 3 Exhibit 3.5 The following questions are based on this problem and accompanying Excel windows. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost is stable but demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 12,000 units and management wants to keep at least 3,000 units on hand. Quarterly inventory holding cost is 3% of the cost of production. The company estimates the number of units carried in inventory each month by averaging the beginning and ending inventory for each month. There are currently 5,000 units in inventory. The company wants to produce at no less than one half of its maximum capacity in any quarter.
Quarter Unit Production Cost Units Demanded Maximum Production
1
2
3
4
$ 300 2,000 8,000
$ 300 9,000 7,000
$ 300 12,000 8,000
$ 300 11,000 9,000
Let
Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i
MIN:
300 P1 + 300 P2 + 300 P3 + 300 P4 + 9(B1 + B2)/2 + 9(B2 + B3)/2 + 9(B3 + B4)/2 + 9(B4 + B5)/2 4000 ≤ P1 ≤ 8000 3500 ≤ P2 ≤ 7000 4000 ≤ P3 ≤ 8000 4500 ≤ P4 ≤ 9000 3000 ≤ B1 + P1 − 2000 ≤ 12000 3000 ≤ B2 + P2 − 9000 ≤ 12000 3000 ≤ B3 + P3 − 12000 ≤ 12000 3000 ≤ B4 + P4 − 11000 ≤ 12000 B2 = B1 + P1 − 2000 B3 = B2 + P2 − 9000 B4 = B3 + P3 − 12000 B5 = B4 + P4 − 11000 Pi, Bi ≥ 0
Subject to:
A 1 2 3 4 5
B
C
D
1 5,000 8,000 2,000
2 11,000 7,000 9,000
E
F
3 9,000 8,000 12,000
4 5,000 9,000 11,000
Quarter Beginning Inventory Units Produced Units Demanded
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ch 3 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ending Inventory
11,000
9,000
5,000
3,000
Minimum Production Maximum Production
4,000 8,000
3,500 7,000
4,000 8,000
4,500 9,000
Minimum Inventory Maximum Inventory
3,000 12,000
3,000 12,000
3,000 12,000
3,000 12,000
$300 $9.00
$300 $9.00
$300 $9.00
$300 $9.00
$2,400,000 $72,000
$2,100,000 $90,000
$2,400,000 $63,000
$2,700,000 $36,000
Unit Production Cost Unit Carrying Cost
3.0%
Quarterly Production Cost Quarterly Carrying Cost
Total Cost
$9,861,000
49. Refer to Exhibit 3.5. What formula could be entered in cell F20 in the accompanying Excel spreadsheet to compute the Total Cost for all four quarters? a. SUMPRODUCT($C$4:$F$4,C17:F17) b. SUM(C17:F17) + SUM(C18:F18) c. PRODUCT(SUM(C14:F15,C17:F18) d. SUMPRODUCT(C4:F4,C14:F14) + SUMPRODUCT(C6:F6,C15:F15) 50. What does the Excel "=SUMPRODUCT(A1:A5,C6;C10)" function do? a. Sums each range and multiplies the sums. b. Sum each pair of cells and multiples each sum. c. Multiplies the contents of cells containing the =SUM() command. d. Multiplies each pair of cells in two arrays matched by position and sums the products. 51. How many decision variables are there in a transportation problem which has 5 supply points and 4 demand points? a. 4 b. 5 c. 9 d. 20 52. Which type of spreadsheet cell represents the left hand sides (LHS) formulas in an LP model? a. Target or set cell b. Changing variable cell c. Constraint cell d. Constant cell 53. Which type of spreadsheet cell represents the objective function in an LP model? a. Objective cell b. Changing variable cell c. Constraint cell d. Constant cell Copyright Cengage Learning. Powered by Cognero.
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ch 3 54. The constraints X1 ≥ 0 and X2 ≥ 0 are referred to as a. positivity constraints. b. optimality conditions. c. left hand sides. d. nonnegativity conditions. 55. Scaling problems a. can cause Analytic Solver Platform to consider a linear problem as nonlinear. b. can cause problems in accuracy of solutions returned. c. are caused by small numbers and large numbers used in the same problem. d. all of these. 56. A solvable problem must have: a. a feasible region that is not an empty set. b. the best solution. c. no more than two constraints. d. no more than two decision variables.
57. A company needs to purchase several new machines to meet its future production needs. It can purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. What values would you enter in the Analytic Solver Platform (ASP) task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem? Objective Cell: Variables Cells: Constraints Cells:
Let
Xi = number of machines of type i purchased
MAX: Subject to:
200X1 + 250X2 + 300X3 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0 A
B
1
C Capital Expansion
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D
E
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ch 3 2 3 4 5 6 7 8 9 10
Machine 1
Machine Types Machine 2
Number to buy Machine output
200
250
300
Requirements: Square feet Cost
2,000 80,000
3,000 50,000
5,000 40,000
Machine 3 Total Output:
Used
Available 20,000 500,000
58. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week Trucking Limits Railway Limits 1 45 60 2 50 55 3 55 45 Costs ($ per 1000 tons) $200 $140 The following is the LP model for this logistics problem.
Let
Air Cargo Limits 15 10 5 $400
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3 WLij = weekly limit of mode i in week j (as provided in above table) 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
MIN: Subject to: Xij ≤ WLij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 X11 + X21 + X31 ≥ 120 X11 + X12 + X13 ≥ 0.45*250 X21 + X22 + X23 ≥ 0.40*250 X31 + X32 + X33 ≤ 0.15*250 Xij ≥ 0 for all i and j
Weekly limits by mode Total at end of three weeks Total at end of two weeks Total at end of first week Truck mix requirement Rail mix requirement Air mix limit
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ch 3 A 1 Costs 2 3 Week 1 4 Week 2 5 Week 3 6 Shipped by 7 Percentage 8 Total Limit 9 10 11 12 13 Week 1 14 Week 2 15 Week 3
B $200.00 by Truck 45 50 13 108 45% 108
C $140.00 by Rail 60 55 12 127 40% 100
D $500.00 by Air 15 0 0 15 15% 37.5
E
F
Totals 120 225 250
Required 120 200 250
Total Cost Weekly Limits Truck 45 50 55
Rail 60 55 45
$46,880.00
Air 15 10 5
What formula goes in cells F10, E3, E4, E5, and B6 of this Excel spreadsheet? 59. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What numbers should be entered into cells B5:C5 and B8:C10 to implement this model?
MAX: Subject to:
4 X1 + 3 X2 6 X1 + 7 X2 ≤ 84 X1 ≤ 10 X2 ≤ 8 X1, X2 ≥ 0
A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE
Used
Available 84 10 8
60. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What numbers should be entered into cells B5:C5 and B8:C10 to implement this model?
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ch 3 MIN: Subject to:
8 X1 + 3 X2 X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0 A
1 2 3
B
C
X1
X2
D
4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
E
OBJ. FN. VALUE
Used
Available 8 80 60
61. A financial planner wants to design a portfolio of investments for a client. The client has $400,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 30% of the money in any one investment, at least one half should be invested in long-term bonds which mature in six or more years, and no more than 40% of the total money should be invested in B or C since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.
Investment
Return
Years to Maturity
Rating
A B C D
6.45% 8.5% 9.00% 7.75%
6 5 8 4
1-Excellent 3-Good 4-Fair 2-Very Good
Let
X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D
MAX: Subject to:
.0645 X1 + .085 X2 + .090 X3 + .0775 X4 X1 + X2 + X3 + X4 ≤ 400000 X1 ≤ 120000 X2 ≤ 120000 X3 ≤ 120000 X4 ≤ 120000 X1 + X3 ≥ 200000
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ch 3 X2 + X3 ≤ 160000 X1, X2, X3, X4 ≥ 0
A 1 2 3 4 5 6 7 8
Bond A B C D Total Invested: Total Available:
< < < < < < < < <
B Amount Invested $0 $0 $0 $0 $0 $400,000
E Years to Maturity 6 5 8 4 Total: Required:
1 2 3 4 5 6 7 8
C Maximum 30.0% $120,000 $120,000 $120,000 $120,000 Total:
F 6+ years? (1-yes, 0-no) 1 0 1 0 $0 $200,000
D Return 6.45% 8.5% 9.00% 7.75% $0
G Rating 1-Excellent 3-Good 4-Fair 2-Very Good Total: Allowed:
> > > > > > > > >
H Good or worse? (1-yes, 0-no) 0 0 1 1 $0 $160,000
What formulas are required for the following cells in the Excel spreadsheet implementation of the formulation? B7 D7 F7 H7 62. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What formulas should be entered into cells E5 and D8:D10 to implement this model?
MIN: Subject to:
8 X1 + 3 X2 X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0 A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit:
OBJ. FN. VALUE 8
3
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ch 3 6 7 Constraints: 8 1 9 2 10 3
Used 8 3
1 5 5
Available 8 80 60
63. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: ∙ ∙
homework can account for up to 25% of the grade, but must be at least 5% of the grade; the project can account for up to 25% of the grade, but must be at least 5% of the grade; the mid-term and final must each account for between 10% and 40% of the grade but cannot ∙ account for more than 70% of the grade when the percentages are combined; and the project and final exam grades may not collectively constitute more than 50% of the ∙ grade. Formulate an LP model for Robert to maximize his numerical grade. 64. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What numbers should be entered into cells B5:C5 and B8:C10 to implement this model?
MAX: Subject to:
2 X1 + 7 X2 5 X1 + 9 X2 ≤ 90 9 X1 + 8 X2 ≤ 144 X2 ≤ 8 X1, X2 ≥ 0 A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE
Used
Available 90 144 8
65. A company is planning production for the next 4 quarters. They want to minimize the cost of production. The production cost, demand and production capacity vary from quarter to quarter. The maximum amount of inventory which can be held is 100 units and management wants to keep at least 50 units on hand. Quarterly inventory holding cost is 4% of the cost of production. There are currently 50 units in inventory. The company wants to produce at no less than one half Copyright Cengage Learning. Powered by Cognero.
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ch 3 of its maximum capacity in any quarter.
1 $55 100 150 $2.2
Unit Production Cost Units Demanded Maximum Production Holding cost
2 $50 150 150 $2
Quarter 3 $50 180 160 $2
Let
Pi = number of units produced in quarter i, i = 1, ..., 4 Bi = beginning inventory for quarter i
MIN:
55 P1 + 50 P 2 + 50 P3 + 45 P4 + 2.2 (B1 + B2)/2 + 2 (B2 + B3)/2 + 2 (B3 + B4)/2 + 1.8 (B4 + B5)/2 75 ≤ P1 ≤ 150 75 ≤ P2 ≤ 150 80 ≤ P3 ≤ 160 65 ≤ P4 ≤ 130 50 ≤ B1 + P1 − 100 ≤ 100 50 ≤ B2 + P2 − 150 ≤ 100 50 ≤ B3 + P3 − 180 ≤ 100 50 ≤ B4 + P4 − 120 ≤ 100 B2 = B1 + P1 − 100 B3 = B2 + P2 − 150 B4 = B3 + P3 − 180 B5 = B4 + P4 − 120 Pi, Bi ≥ 0
Subject to:
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Beginning Inventory Units Produced Units Demanded Ending Inventory
1 50 120 100 70
D Quarter 2 70 150 150 70
Minimum Production Maximum Production
75 150
75 150
80 160
65 130
Minimum Inventory Maximum Inventory
50 100
50 100
50 100
50 100
$55 $2.20
$50 $2.00
$50 $2.00
$45 $1.80
Unit Production Cost Unit Carrying Cost
B
4.0%
C
4 $45 120 130 $1.8
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E
F
3 70 160 180 50
4 50 120 120 50
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ch 3 17 Monthly Production Cost 18 Monthly Carrying Cost 19 20
$6,600 $132
$7,500 $140
$8,000 $120
$5,400 $90
Total Cost
$27,982
What formulas are required for cells D3, D6, D8, D15, D17 and D18 in the Excel spreadsheet implementation of the formulation? 66. A financial planner wants to design a portfolio of investments for a client. The client has $400,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 30% of the money in any one investment, at least one half should be invested in long-term bonds which mature in six or more years, and no more than 40% of the total money should be invested in B or C since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio.
Investment
Return
Years to Maturity
Rating
A B C D
6.45% 8.50% 9.00% 7.75%
6 5 8 4
1-Excellent 3-Good 4-Fair 2-Very Good
Formulate the LP for this problem. 67. The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
New Hope
Hospital County General
City East
83.0 123.8 225.0
105.0 162.3 200.0
104.1 154.0 231.0
Input Measures Bed days unused (1000s) Supply expense ($1000s) Full-time staff Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 Customer satisfaction 98.0 88.0 83.0 Based on the following formulation, is City East efficient? If not, what input and output values should they aspire to in order to become efficient?
Let
wi = weight assigned to output j, j = 1, ..., 4 vi = weight assigned to input i, i = 1,...,3
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ch 3 MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0 68. The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
New Hope
Hospital County General
City East
83.0 123.8 225.0
105.0 162.3 200.0
104.1 154.0 231.0
Input Measures Bed days unused (1000s) Supply expense ($1000s) Full-time staff Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 Customer satisfaction 98.0 88.0 83.0 Enter the numbers in the appropriate cells of ranges B4:H6 in the Excel spreadsheet to solve this problem based on the following formulation.
Let
wi = weight assigned to output j, j = 1, ..., 4 vi = weight assigned to input i, i = 1,...,3
MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0 A 1 2 3 4
Hospital New Hope
B Patient Days (1000s)
C Nurses Qual.
D Asst on Staff
E Cust Sat.
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ch 3 5 6 7 8 9 10 11 12 < < < < < < < < < < < < <
Cnty. General City East
1 2 3 4 5 6 7 8 9 10 11 12
Weights
0
0
UNIT Output Input
3 0.81 1.0
0
F Bed-Days Unused (1000s)
G Supply Expense ($1000s)
H Full Time Staff
0
0
0
> > > > > > > >
0
I
J
K
Wgt. Output 97% 83% 81%
Wgt. Input 97% 87% 100%
Diff 0.0000 −0.0381 −0.1877
69. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available. The following table summarizes the data for the problem.
Profit per Yield per Crop Acre ($) Acre (lb) Corn 2,100 21,000 Pumpkin 900 10,000 Beans 1,050 3,500 Formulate the LP for this problem.
Maximum Demand (lb) 200,000 180,000 80,000
Irrigation (acre ft) 2 3 1
Fertilizer (pounds/acre) 500 400 300
70. Pete's Plastics manufactures plastic at plants in Miami, St. Louis and Cleveland. Pete needs to ship plastic to customers in Pittsburgh, Atlanta and Chicago. He wants to minimize the cost of shipping the plastic from his plants to his customers. The data for the problem is summarized in the following table.
Distance From Plants to Customers Plant
Pittsburgh
Atlanta
Chicago
Supply
Miami
1,200
700
1,300
30
St. Louis
700
550
300
40
Cleveland
125
675
350
50
Demand
40
60
20
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ch 3 Formulate the LP for this problem. 71. The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
New Hope
Hospital County General
City East
83.0 123.8 225.0
105.0 162.3 200.0
104.1 154.0 231.0
Input Measures Bed days unused (1000s) Supply expense ($1000s) Full-time staff Output Measures Patient-days (1000s) 105.0 71.0 82.7 Nurses qualified 253.0 92.0 175.0 Assistants on staff 125.0 45.0 65.0 satisfaction 98.0 88.0 83.0 What are the key formulas in cells I4, J4, K4, B11 and B12 for this Excel spreadsheet implementation of the following formulation?
Let
wi = weight assigned to output j, j = 1, ..., 4 vi = weight assigned to input i, i = 1,...,3
MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0 A 1 2 3 4 5 6 7 8 9 10 11 12
Hospital New Hope Cnty. General City East
B Patient Days (1000s) 105.10 71.00 82.70
Weights
0.002009
UNIT Output Input
3 0.812259 1
C Nurses Qual. 253.00 92.00 175.00
D Asst on Staff 125.0 45.0 65.0
E Cust Sat. 98 88 83
0
0
0.00778
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ch 3 < < < < < < < < < < < < <
1 2 3 4 5 6 7 8 9 10 11 12
F Bed-Days Unused (1000s) 83.00 105.00 104.10
G Supply Expense ($1000s) 123.80 162.30 154.00
H Full Time Staff 225.00 200.00 231.00
0
0
0.004329
I
J
K
Wgt. Output 97% 83% 81%
Wgt. Input 97% 87% 100%
Diff 0.0000 −0.0381 −0.1877
72. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week Trucking Limits 1 45 2 50 3 55 Costs ($ per 1000 tons) $200 Formulate an LP model for this logistics problem.
Railway Limits 60 55 45 $140
Air Cargo Limits 15 10 5 $400
73. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: ∙ ∙
homework can account for up to 25% of the grade, but must be at least 5% of the grade; the project can account for up to 25% of the grade, but must be at least 5% of the grade; the mid-term and final must each account for between 10% and 40% of the grade but cannot ∙ account for more than 70% of the grade when the percentages are combined; and the project and final exam grades may not collectively constitute more than 50% of the ∙ grade. The following LP model allows Robert to maximize his numerical grade.
Let
W1 = weight assigned to homework W2 = weight assigned to the project
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ch 3 W3 = weight assigned to the mid-term W4 = weight assigned to the final MAX: Subject to:
75W1 + 94W2 + 85W3 + 92W4 W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40 A
1 2 3 4 5 6 7 8 9 10 11 12 13
Mid Term Final Project Homework
Both Exams Final & Project
B
100%
C Percentage to grade 0.40 0.25 0.25 0.10 1.00 1.00
Grade Total 0.65 0.5
88.00 Limit 0.70 0.50
Grade 85 92 94 75
D
E Limits
Lower 0.10 0.10 0.05 0.05
Upper 0.40 0.40 0.25 0.25
What values would you enter in the Analytic Solver Platform (ASP) task pane for the cells in this Excel spreadsheet implementation of this problem? Objective Cell: Variables Cells: Constraints Cells: 74. A company needs to purchase several new machines to meet its future production needs. It can purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. What are the key formulas in cells G4 and E7 for this Excel spreadsheet implementation of the following formulation?
Let
Xi = number of machines of type i purchased
MAX:
200X1 + 250X2 + 300X3
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ch 3 Subject to:
2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0 A
B Machine 1
C Machine 2
D Machine 3
1 2
Number to buy
3 4
Production Possible
200
250
300
5 6 7 8
Resources Floor Space Req'd Assemble
Hours Required 2 80
Used 3 50
Available 4 40
E
F
G
Total:
20 500
75. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What formulas should be entered into cells E5 and D8:D10 to implement this model?
MAX: Subject to:
4 X1 + 3 X2 6 X1 + 7 X2 ≤ 84 X1 ≤ 10 X2 ≤ 8 X1, X2 ≥ 0 A
B
C
X1
X2
4
3
6 1 0
7 0 1
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE
Used
Available 84 10 8
76. A financial planner wants to design a portfolio of investments for a client. The client has $400,000 to invest and the planner has identified four investment options for the money. The following requirements have been placed on the planner. No more than 30% of the money in any one investment, at least one half should be invested in long-term bonds which mature in six or more years, and no more than 40% of the total money should be invested in B or C since they are riskier investments. The planner has developed the following LP model based on the data in this table and the requirements of the client. The objective is to maximize the total return of the portfolio. Copyright Cengage Learning. Powered by Cognero.
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ch 3 Investment
Return
Years to Maturity
Rating
A B
6.45% 8.50%
6 5
1-Excellent 3-Good
C D
9.00% 7.75%
8 4
4-Fair 2-Very Good
Let
X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D
MAX: Subject to:
.0645 X1 + .085 X2 + .090 X3 + .0775 X4 X1 + X2 + X3 + X4 ≤ 400000 X1 ≤ 120000 X2 ≤ 120000 X3 ≤ 120000 X4 ≤ 120000 X1 + X3 ≥ 200000 X2 + X3 ≤ 160000 X1, X2, X3, X4 ≥ 0
A 1 2 3 4 5 6 7 8
< < < < < < < < <
Bond A B C D Total Invested: Total Available:
1 2 3 4 5 6 7 8
E Years to Maturity 6 5 8 4 Total: Required:
B Amount Invested $0 $0 $0 $0 $0 $400,000
C Maximum 30.0% $120,000 $120,000 $120,000 $120,000 Total:
F 6+ years? (1-yes, 0-no) 1 0 1 0 $0 $200,000
D Return 6.45% 8.5% 9.00% 7.75% $0
G Rating 1-Excellent 3-Good 4-Fair 2-Very Good Total: Allowed:
> > > > > > > > >
H Good or worse? (1-yes, 0-no) 0 0 1 1 $0 $160,000
What values would you enter in the Analytic Solver Platform (ASP) task pane for the following cells for this Excel spreadsheet implementation of this problem? Copyright Cengage Learning. Powered by Cognero.
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ch 3 Objective Cell: Variables Cells: Constraints Cells: 77. A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
Time period Required # of Nurses Wage ($/hr) 12 am − 4 am 20 15 4 am − 8 am 30 16 8 am − 12 pm 40 13 12 pm − 4 pm 50 13 4 pm − 8 pm 40 14 8 pm − 12 am 30 15 What values would you enter in the Analytic Solver Platform (ASP) task pane for the following cells for this Excel spreadsheet implementation of the formulation for this problem? Objective Cell: Variables Cells: Constraints Cells:
Let
Xi = number of nurses working in time period i; i = 1,6
MIN: Subject to:
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0
1 2 3 4 5 6 7 8
A
B
C
Shift 1 2 3
Mid 4am 1 0 0
4am 8am 1 1 0
D Nurse
E Hiring
Hours for each shift 8am Noon Noon 4pm 0 0 1 0 1 1
F
G
H
I
4pm 8pm 0 0 0
8pm Mid 0 0 0
Nurses Scheduled
Wages per Nurse $15 $16 $13
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ch 3 9 10 11 12 13
4 5 6 Available: Required:
0 0 1
0 0 0
0 0 0
1 0 0
1 1 0
0 1 1
20
30
40
50
40
30
$13 $14 $15 Total Wages:
78. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What cell references would you enter in the Analytic Solver Platform (ASP) task pane for the following? Objective Cell: Variables Cells: Constraints Cells:
MAX: Subject to:
12 X1 + 9 X2 9 X1 + 10.5 X2 ≤ 126 X1 ≥ 5 X2 ≥ 6 X1, X2 ≥ 0 A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE 12
9 Used
9 1 0
10.5 0 1
Available 126 5 6
79. A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
Time period Required # of Nurses Wage ($/hr) 12 am − 4 am 20 15 4 am − 8 am 30 16 8 am − 12 pm 40 13 12 pm − 4 pm 50 13 4 pm − 8 pm 40 14 8 pm − 12 am 30 15 What are the key formulas in cells I12 and B12 for this Excel spreadsheet implementation of the following formulation? Copyright Cengage Learning. Powered by Cognero.
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ch 3 Let
Xi = number of nurses working in time period i; i = 1,6
MIN: Subject to:
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0 A
1 2 3 4 5 6 7 8 9 10 11 12 13
Shift 1 2 3 4 5 6 Available: Required:
B
C
Mid 4am
4am 8am
D Nurse
E Hiring
Hours for each shift 8am Noon Noon 4pm
F
G
H
I
4pm 8pm
8pm Mid
Nurses Scheduled
Wages per Nurse $15 $16 $13 $13 $14 $15
Total Wages:
80. A company needs to purchase several new machines to meet its future production needs. It can purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. Formulate the LP for this problem. 81. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What formulas should be entered into cells E5 and D8:D10 to implement this model?
MAX: Subject to:
2 X1 + 7 X2 5 X1 + 9 X2 ≤ 90 9 X1 + 8 X2 ≤ 144 X2 ≤ 8 X1, X2 ≥ 0
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ch 3 A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE 2
7
5 9 0
9 8 1
Used
Available 90 144 8
82. A company needs to purchase several new machines to meet its future production needs. It can purchase three different types of machines A, B, and C. Each machine A costs $80,000 and requires 2,000 square feet of floor space. Each machine B costs $50,000 and requires 3,000 square feet of floor space. Each machine C costs $40,000 and requires 5,000 square feet of floor space. The machines can produce 200, 250 and 350 units per day respectively. The plant can only afford $500,000 for all the machines and has at most 20,000 square feet of room for the machines. The company wants to buy as many machines as possible to maximize daily production. Enter the numbers in the appropriate cells of range B5:F10 in the Excel spreadsheet to solve this problem based on the following formulation.
Let
Xi = number of machines of type i purchased
MAX: Subject to:
200X1 + 250X2 + 300X3 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0 A
1 2 3 4 5 6 7 8 9 10
B
Machine 1
C Capital Expansion Machine Types Machine 2
D
E
F
Machine 3
Number to buy Machine output
Total Output:
Requirements: Square feet Cost
Used
Available
83. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. There are only 50 acre ft of irrigation available and only 8,000 pounds/acre of fertilizer available. The following table summarizes the data for the problem. Copyright Cengage Learning. Powered by Cognero.
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ch 3 Profit per Yield per Maximum Irrigation Fertilizer Crop Acre ($) Acre (lb) Demand (lb) (acre ft) (pounds/acre) Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 Enter the numbers in the appropriate cells of ranges B12:D12 and E8:F12 in the Excel spreadsheet to solve this problem based on the following formulation.
Let
X1 = aces of corn X2 = acres of pumpkin X3 = acres of beans
MAX: Subject to:
2100X1 + 900X2 + 1050X3 21X1 ≤ 200 10X2 ≤ 180 3.5X3 ≤ 80 X1 + X2 + X3 ≤ 20 2X1 + 3X2 + 1X3 ≤ 50 5X1 + 4X2 + 3X3 ≤ 80 X1, X2, X3 ≥ 0 A
1 2 3 4 5 6 7 8 9 10 11 12
B Farm
C Planning
D Problem
Corn
Pumpkin
Beans
E
Acres to plant Profit per acre
F
Total Profit:
Constraints: Corn demand Pumpkin demand Bean demand Water Fertilizer
Used
Available
84. A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
Time period 12 am − 4 am 4 am − 8 am 8 am − 12 pm 12 pm − 4 pm 4 pm − 8 pm
Required # of Nurses 20 30 40 50 40
Wage ($/hr) 15 16 13 13 14
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ch 3 8 pm − 12 am 30 15 Enter the numbers in the appropriate cells of ranges B6:G11 and B13:G13 in the Excel spreadsheet to solve this problem based on the following formulation.
Let
Xi = number of nurses working in time period i; i = 1,6
MIN: Subject to:
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0 A
1 2 3 4 5 Shift 6 1 7 2 8 3 9 4 10 5 11 6 12 Available: 13 Required:
B
C
Mid 4am
4am 8am
D Nurse
E Hiring
Hours for each shift 8am Noon Noon 4pm
F
G
H
I
4pm 8pm
8pm Mid
Nurses Scheduled
Wages per Nurse $15 $16 $13 $13 $14 $15
Total Wages:
85. State Farm Supply has just received an order for 10,000 pounds of chicken feed. The farmer has specified certain that the feed meet minimum requirements for Protein, Carbohydrate, Fat and Vitamins. State Farm can blend four different feeds to produce the required mix. The farmer would like to pay the lowest possible price for the feed. The data for the problem is summarized in the following table. State Farm Supply Percent of Nutrient in: Nutrient
Minimum
Feed 1
Feed 2
Feed 3
Feed 4
Req'd Amt
Protein Carbohydrate
15 20
20 10
30 10
15 15
18 12
Fat
20
30
15
20
20
Vitamin
1
1.50
0.75
0.50
1
$500
$600
$550
$450
Cost/1,000 lbs
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ch 3 Formulate the LP for this problem. 86. Project 3.1 -- The Diet Problem: Ordering Meals from McDonald's Based on: Robert A. Bosch, "Big Mac Attack: The Diet Problem revisited, Eating at McDonald's," OR/MS Today, August 1993, pp 30-31. Tina Simpson is a new fourth-grade teacher at Forest Ridge Elementary. The first teacher workshop for the upcoming school year is next Monday and by majority vote, McDonald's was selected as the food of choice. As the new person, Tina is tasked with developing the meal for the workshop. McDonald's has graciously offered to deliver whatever food Tina decides to order, along with a variety of condiments applicable to whatever is ordered. Rather than offer a menu choice, Tina has decided to simply order the same meal for each person in the workshop. To get started, Tina took a trip to McDonald's and obtained their published information on the nutritional content of their food. That data is summarized in the table below. Price Menu Item
Calories
Protein
($)
Fat
Sodium
>
(mg)
>
(grams)
Hamburger
0.59
255
12
9
490
>
McLean Dlx
1.79
320
22
10
670
>
Big Mac
1.65
500
25
26
890
>
Small Fries
0.68
220
3
12
110
>
McNuggets
1.56
270
20
15
580
>
Honey
0.00
45
0
0
0
>
Chef Salad
2.69
170
17
9
400
>
Garden Salad
1.96
50
4
2
70
>
Egg McMuffin
1.36
280
18
11
710
>
Wheaties
1.09
90
2
1
220
>
Yogurt Cone
0.63
105
4
1
80
>
Milk
0.56
110
9
2
130
>
Orange Juice
0.88
80
1
0
0
>
Grapefruit juice
0.68
80
1
0
0
>
Apple Juice
0.68
90
0
0
5
>
Prices recorded August, 1991 in Oberlin Ohio <
Vit A
Vit C
Vit B1
> Vit B2
Niacin
Calcium
Iron
<
Menu Item
% U.S. Recommended Daily Allowance (RDA)
<
Hamburger
4
4
20
10
20
10
15
<
McLean Dlx
10
10
25
20
35
15
20
<
Big Mac
6
2
30
25
35
25
20
<
Small Fries
*
15
10
*
10
*
2
<
McNuggets
*
*
8
8
40
*
6
<
Honey
*
*
*
*
*
*
*
<
Chef Salad
100
35
20
15
20
15
8
<
Garden
90
35
6
6
2
4
8
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ch 3 Salad < < < < < < <
Egg McMuffin Wheaties Yogurt Cone Milk Orange Juice Grapefruit juice Apple Juice
10
*
30
20
20
25
15
20
20
20
20
20
2
20
2
*
2
10
2
10
*
10
4
8
30
*
30
*
*
120
10
*
*
*
*
*
100
4
2
2
*
*
*
2
2
*
*
*
4
<
Prices recorded August, 1991 in Oberlin Ohio
<
* Contains less than 2% of the U.S. RDA of these nutrients
Tina wants the meal to be nutritionally complete. The National Research Council publishes their Recommended Daily Allowances. In this publication, they contend that a diet (in this case the meal) should provide at least 100 percent of the U.S. RDA of numerous nutrients. The specific amount of the RDA depends on such factors as age, weight and gender. In addition, the council recommends daily sodium and cholesterol intakes be kept to at most 2.4 grams of sodium and 300 milligrams of cholesterol. Further, at most 30 percent of the calories consumed should come from fat, and at most 10 percent from saturated fat. Each gram of fat contains 9 calories. Based on the above information, Tina wants to design a least-cost meal that provides at least 100% of the U.S. RDA of vitamins A, C, B1, B2, niacin, calcium, and iron; supplies at least 55 grams of protein; contains at most 3 grams of sodium; and contains at most 30 percent of its calories from fat. Only those foods list in the table above are available for the meal. Formulate the LP model for Tina's problem. Develop a spreadsheet model of the problem and use Excel Solver to determine the least-cost meal that meets all the stated requirements. What is the recommended meal? Is this meal reasonable? If not, modify the model to obtain what you believe to be a reasonable meal that meets the stated requirements. 87. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What cell references would you enter in the Analytic Solver Platform (ASP) task pane for the following? Objective Cell: Variables Cells: Constraints Cells:
MIN: Subject to:
8 X1 + 3 X2 X2 ≥ 8 8 X1 + 5 X2 ≥ 80 3 X1 + 5 X2 ≥ 60 X1, X2 ≥ 0
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ch 3 A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE
Used
Available 8 80 60
1 5 5
8 3
88. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week Trucking Limits Railway Limits 1 45 60 2 50 55 3 55 45 Costs ($ per 1000 tons) $200 $140 The following is the LP model for this logistics problem.
Let
Air Cargo Limits 15 10 5 $400
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3 WLij = weekly limit of mode i in week j (as provided in above table) 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
MIN: Subject to: Xij ≤ WLij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 X11 + X21 + X31 ≥ 120 X11 + X12 + X13 ≥ 0.45*250 X21 + X22 + X23 ≥ 0.40*250
Weekly limits by mode Total at end of three weeks Total at end of two weeks Total at end of first week Truck mix requirement Rail mix requirement
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ch 3 X31 + X32 + X33 ≤ 0.15*250 Xij ≥ 0 for all i and j A 1 Costs 2 3 Week 1 4 Week 2 5 Week 3 6 Shipped by 7 Percentage 8 Total Limit 9 10 11 12 13 Week 1 14 Week 2 15 Week 3
Air mix limit
B $200.00 by Truck 45 50 13 108 45% 108
C $140.00 by Rail 60 55 12 127 40% 100
Truck 45 50 55
Weekly Limits Rail 60 55 45
D $500.00 by Air 15 0 0 15 15% 37.5
E
F
Totals 120 225 250
Required 120 200 250
Total Cost
$46,880.00
Air 15 10 5
What values would you enter in the Analytic Solver Platform (ASP) task pane for the cells in this Excel spreadsheet implementation of this problem? Objective Cell: Variables Cells: Constraints Cells: 89. A paper mill has received an order for rolls of paper. The customer wants 400 12" wide rolls, 300 18" rolls and 200 24" rolls. The company has 40" wide rolls of paper which it can slit to the appropriate width. The company wants to minimize the number of rolls it must use to fill the order. Formulate the LP for this problem. 90. You have been given the following linear programming model and Excel spreadsheet to solve this problem. What cell references would you enter in the Analytic Solver Platform (ASP) task pane for the following? Objective Cell: Variables Cells: Constraints Cells:
MAX: Subject to:
8 X1 + 5 X2 3 X1 + 5 X2 = 54 11 X1 + 10 X2 ≤ 144 X1 ≥12 X1, X2 ≥ 0
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ch 3 A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE 8
5
3 11 1
5 10 0
Used
Available 54 144 12
91. A grain store has six types of grain, each varying in cost, quality, and nutritional content. Periodically, excess inventory of these grains are consolidated into two local products, Feed-M-All and Supreme-Feed. Feed-M-All sells for $6.50 for a 10-pound bag while Supreme-Feed sells for $8.50 for a 10-pound bag. These feeds are advertised as having the following nutritional content:
Grain Minimum Protein Minimum Fat Feed-M-All 16% 18% Supreme-Feed 18% 18% The component grains have the following content characteristics:
Grain
Cost/10 lbs
Quality
Protein
Fat
Maximum Carbohydrates 10% 9%
Carbohydrates
Pounds Avail.
A $4.75 4 15% 10% 10% 90 B $4.00 2 20% 20% 8% 120 C $3.75 1 10% 25% 5% 150 D $4.25 3 15% 20% 10% 125 E $4.50 3 20% 20% 10% 85 F $5.00 4 25% 15% 12% 165 Targets for Feed-M-All are a cost of $ 4.35 per 10-pound bag, a quality rating of 2.25, along with the minimum percentages of protein and fat, and the maximum percentage of carbohydrates. Similar targets are set for Supreme-Feed with cost set at $ 4.60 and quality at 2.45. There must be at least a 70%-30% mix among these two local feeds. Formulate an LP model for this product mix problem. 92. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. The following table summarizes the data for the problem.
Profit per Yield per Maximum Irrigation Fertilizer Crop Acre ($) Acre (lb) Demand (lb) (acre ft) (pounds/acre) Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 What are the key formulas in cells F5 and E8 for this Excel spreadsheet implementation of the following formulation? Copyright Cengage Learning. Powered by Cognero.
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ch 3
Let
X1 = aces of corn X2 = acres of pumpkin X3 = acres of beans
MAX: Subject to:
2100X1 + 900X2 + 1050X3 21X1 ≤ 200 10X2 ≤ 180 3.5X3 ≤ 80 2X1 + 3X2 + 1X3 ≤ 50 5X1 + 4X2 + 3X3 ≤ 80 X1, X2, X3 ≥ 0 A
1 2 3 4 5 6 7 8 9 10 11 12
Acres to plant Profit per acre Constraints: Corn demand Pumpkin demand Bean demand Water Fertilizer
B Farm
C Planning
D Problem
Corn
Pumpkin
Beans
E
F
Total Profit: 2100
900
1050 Used
21000 10000 2 500
3 400
3500 1 300
Available 200000 180000 80000 50 8000
93. A hospital needs to determine how many nurses to hire to cover a 24 hour period. The nurses must work 8 consecutive hours but can start work at the start of 6 different shifts. They are paid different wages depending on when they start their shifts. The number of nurses required per 4-hour time period and their wages are shown in the following table.
Time period Required # of Nurses 12 am − 4 am 20 4 am − 8 am 30 8 am − 12 pm 40 12 pm − 4 pm 50 4 pm − 8 pm 40 8 pm − 12 am 30 Formulate the LP for this problem.
Wage ($/hr) 15 16 13 13 14 15
94. The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months. Copyright Cengage Learning. Powered by Cognero.
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ch 3
New Hope
Hospital County General
City East
Bed days unused (1000s) Supply expense ($1000s) Full-time staff
83.0 123.8 225.0
105.0 162.3 200.0
104.1 154.0 231.0
Patient-days (1000s) Nurses qualified Assistants on staff Customer satisfaction
105.0 253.0 125.0 98.0
71.0 92.0 45.0 88.0
82.7 175.0 65.0 83.0
Input Measures
Output Measures
a.
Formulate a DEA LP model to evaluate the efficiency of City East.
b.
Implement a spreadsheet model for this problem and compute the DEA efficiency for each facility. Which facilities are efficient?
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ch 3 Answer Key 1. True 2. False 3. False 4. True 5. True 6. False 7. True 8. False 9. b 10. a 11. b 12. d 13. c 14. c 15. c 16. a 17. c 18. b 19. d 20. c 21. a 22. d 23. b 24. b 25. b Copyright Cengage Learning. Powered by Cognero.
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ch 3 26. b 27. b 28. c 29. a 30. a 31. a 32. c 33. b 34. b 35. d 36. a 37. c 38. b 39. b 40. a 41. c 42. a 43. b 44. c 45. a 46. c 47. c 48. b 49. b 50. d 51. d Copyright Cengage Learning. Powered by Cognero.
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ch 3 52. c 53. a 54. d 55. d 56. a 57. Objective Cell: $F$6 Variables Cells: $B$5:$D$5 Constraints Cells: $B$5:$D$5 ≥ 0 $E$9:$E$10 ≤ $F$9:$F$10 58. F10 E3 E4 E5 B6
=SUMPRODUCT($B$1:$D$1,$B$6:$D$6) =SUM($B$3:$D$3) =SUM($B$4:$D$4) =SUM($B$5:$D$5) =SUM($B$3:$B$5)
59. A
B
C
X1
X2
4
3
6 1 0
7 0 1
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE
Used
Available 84 10 8
D
E
60. A 1 2 3 4 Number to make:
OBJ. FN. VALUE
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ch 3 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
61. B7 D7 F7 H7
8
3 Used
0 8 3
1 5 5
Available 8 80 60
=SUM(B3:B6) =SUMPRODUCT($B$3:$B$6,D3:D6) =SUMPRODUCT($B$3:$B$6,F3:F6) =SUMPRODUCT($B$3:$B$6,H3:H6)
62. Cell E5 D8 63. Let
MAX: Subject to:
Formula =SUMPRODUCT(B4:C4,B5:C5) =SUMPRODUCT($B$4:$C$4,B8:C8)
Copied to D9:D10
W1 = weight assigned to homework W2 = weight assigned to the project W3 = weight assigned to the mid-term W4 = weight assigned to the final 75W1 + 94W2 + 85W3 + 92W4 W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40
64. A
B
C
X1
X2
D
E
1 2 3 4 Number to make: 5 Unit profit: 6 7 Constraints: 8 1 9 2 10 3
OBJ. FN. VALUE 2
7
5 9 0
9 8 1
Used
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Available 90 144 8 Page 60
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ch 3
65. D3 D6 D8 D15 D17 D18
=C6 =D3+D4-D5 =D9/2 =$B$15*D1 =D14*D4) =D15*(D3+D6)/2)
66. Let
MAX: Subject to:
X1 = Dollars invested in A X2 = Dollars invested in B X3 = Dollars invested in C X4 = Dollars invested in D .0645 X1 + .085 X2 + .090 X3 + .0775 X4 X1 + X2 + X3 + X4 ≤ 400000 X1 ≤ 120000 X2 ≤ 120000 X3 ≤ 120000 X4 ≤ 120000 X1 + X3 ≥ 200000 X2 + X3 ≤ 160000 X1, X2, X3, X4 ≥ 0
67. No, City East is not efficient. The following shows that 78.69% of New Hope input and outputs produces a composite unit with outputs greater than or equal to those of City East requiring less input than City East.
--- Outputs ---
Unit New Hope County General City East
Patient Days (1000s) 105.10 71.00 82.70
Comp Vals
82.7
< < < < Unit < New Hope < County General
Nurses Qual. 253.00 92.00 175.00
Asst on Staff 125.0 45.0 65.0
Cust Sat. 98 88 83
199.1
98.4
77.1
--- Inputs --Bed-Days Unused (1000s) 83.00 105.00
Supply Expense ($1000s) 123.80 162.30
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Full Time Staff 225.00 200.00
> > > > > > > > >
Composite Weight 0.7869 0 Page 61
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ch 3 < City East < < Comp Vals
104.10
154.00
231.00
65.3
97.4
177.0
0
Note, however, the drop in customer satisfaction. City East will not want to aspire to that particular level. These composite values will make City East efficient.
A 1 2 3 Hospital 4 New Hope 5 Cnty. General 6 City East 7 8 Weights 9 10 UNIT 11 Output 12 Input
< < < < < < < < < < < < <
1 2 3 4 5 6 7 8 9 10 11 12
B Patient Days (1000s) 105.10 71.00 82.70 0.004279
C Nurses Qual. 253.00 92.00 199.0
D Asst on Staff 125.0 45.0 98.4
E Cust Sat. 98 88 83
0
0
0.007784
3 1 1
F Bed-Days Unused (1000s) 83.00 105.00 65.3
G Supply Expense ($1000s) 123.80 162.30 97.4
H Full Time Staff 225.00 200.00 177
0
0
0.005649
Hospital New Hope Cnty. General City East
B Patient Days (1000s) 105.10 71.00 82.70
Weights
0.002009
> > > > > > > > > > > > >
I
J
K
Wgt. Output 121% 99% 100%
Wgt. Input 127% 113% 100%
Diff −0.0586 −0.1411 0.0000
68. A 1 2 3 4 5 6 7 8 9
C Nurses Qual. 253.00 92.00 175.00
D Asst on Staff 125.0 45.0 65.0
E Cust Sat. 98 88 83
0
0
0.00778
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ch 3 10 11 12
< < < < < < < < < < < < <
1 2 3 4 5 6 7 8 9 10 11 12
69. Let
MAX: Subject to:
70. Let MIN: Subject to:
UNIT Output Input
3 0.812259 1
> > >
F Bed-Days Unused (1000s) 83.00 105.00 104.10
G Supply Expense ($1000s) 123.80 162.30 154.00
H Full Time Staff 225.00 200.00 231.00
0
0
0.004329
I
J
K
Wgt. Output 97% 83% 81%
Wgt. Input 97% 87% 100%
Diff 0.0000 −0.0381 −0.1877
X1 = aces of corn X2 = acres of pumpkin X3 = acres of beans 2100X1 + 900X2 + 1050X3 21X1 ≤ 200 10X2 ≤ 180 3.5X3 ≤ 80 X1 + X2 + X3 ≤ 20 2X1 + 3X2 + 1X3 ≤ 50 5X1 + 4X2 + 3X3 ≤ 80 X1, X2, X3 ≥ 0
Xij = tons shipped from plant i to customer j (i and j = 1, 2, 3) 1200X11 + 700X12 + 1300X13 + 700X21 + 550X22 + 300X23 + 125X31 + 675X32 + 350X33 X11 + X12 + X13 = 30 X21 + X22 + X23 = 40 X31 + X32 + X33 = 50 X11 + X21 + X31 ≥ 40 X12 + X22 + X32 ≥ 60 X13 + X23 + X33 ≥ 20
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ch 3 Xij ≥ 0 71. Cell I4 J4 K4 B11 B12
Formula =SUMPRODUCT($B$8:$E$8,B4:E4) =SUMPRODUCT($F$8:$H$8,F4:H4) =I4-J4 =INDEX(I4:I6,B10,1) =INDEX(J4:J6, B10,1)
72. Let
Copied to I5:I6 J5:J6 K5:K6
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3 WLij = weekly limit of mode i in week j (as provided in above table) 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
MIN: Subject to: Xij ≤ WLij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 ≥ 250 X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 X11 + X21 + X31 ≥ 120 X11 + X12 + X13 ≥ 0.45*250 X21 + X22 + X23 ≥ 0.40*250 X31 + X32 + X33 ≤ 0.15*250 Xij ≥ 0 for all i and j
Weekly limits by mode Total at end of three weeks Total at end of two weeks Total at end of first week Truck mix requirement Rail mix requirement Air mix limit
73. Objective Cell: C10 Variables Cells: C3:C6 Constraints Cells: C3:C6 ≤ D3:D6 C3:C6 ≥ E3:E6 C7 = C8 B12 ≤ C12 B13 ≥ C13 74. Cell G4 E7
Formula =SUMPRODUCT(B2:D2,B4:D4) =SUMPRODUCT($B$2:$D$2,B7:D7)
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ch 3 75. Cell E5 D8
Formula =SUMPRODUCT(B4:C4,B5:C5) =SUMPRODUCT($B$4:$C$4,B8:C8)
Copied to D9:D10
76. Objective Cell: D7 Variables Cells: B3:B6 Constraints Cells: B3:B6 ≤ C3:C6 B3:B6 ≥ 0 B7 ≤ B8 F7 ≥ F8 H7 ≤ H8 77. Objective Cell: $I$12 Variables Cells: $H$6:$H$11 Constraints Cells: $H$6:$H$11 ≥ 0 $B$12:$G$12 ≥ $B$13:$G$13 78. Objective Cell: $E$5 Variables Cells: $B$4:$C$4 Constraints Cells: $B$4:$C$4 ≥ 0 $D$8 ≤ $E$8 $D$9:$D$10 ≥ $E$9:$E$10 79. Cell I12 B12
Formula =SUMPRODUCT(H6:H11,I6:I11) =SUMPRODUCT(B6:B11,$H$6:$H$11)
80. Let
Xi = number of machines of type i purchased
MAX: Subject to:
Copied to C12:G12
200X1 + 250X2 + 300X3 2X1 + 3X2 + 5X3 ≤ 20 80X1 + 50X2 + 40X3 ≤ 500 X1, X2, X3 ≥ 0
81. Copyright Cengage Learning. Powered by Cognero.
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ch 3 Cell E5 D8
Formula =SUMPRODUCT(B4:C4,B5:C5) =SUMPRODUCT($B$4:$C$4,B8:C8)
Copied to D9:D10
82. A 1 2 3 4 5 6 7 8 9 10
B
C Capital Expansion
D
Machine Types Machine 2
Machine 1
E
F
Machine 3
Number to buy Machine output
Total Output: 200
250
300
Requirements: Square feet Cost
2,000 80,000
3,000 50,000
5,000 40,000
Used
Available 20,000 500,000
83. A 1 2 3 4 5 6 7 8 9 10 11 12
Acres to plant Profit per acre Constraints: Corn demand Pumpkin demand Bean demand Water Fertilizer
B Farm
C Planning
D Problem
Corn
Pumpkin
Beans
E
F
Total Profit: 2100
900
1050 Used
Available 200000 180000 80000 50 8000
21000 10000 2 500
3500 1 300
3 400
84.
1 2 3 4 5 6 7 8 9 10 11
A
B
C
Shift 1 2 3 4 5 6
Mid 4am 1 0 0 0 0 1
4am 8am 1 1 0 0 0 0
D Nurse
E Hiring
Hours for each shift 8am Noon Noon 4pm 0 0 1 0 1 1 0 1 0 0 0 0
F
G
H
I
4pm 8pm 0 0 0 1 1 0
8pm Mid 0 0 0 0 1 1
Nurses Scheduled
Wages per Nurse $15 $16 $13 $13 $14 $15
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ch 3 12 Available: 13 Required:
85. Let MIN: Subject to:
Total Wages: 20
30
40
50
40
30
Xi = pounds of feed i used in mixture .5X1 + .6X2 + .55X3 + .45X4 .15X1 + .20X2 + .3X3 + .15X4 ≥ 1800 .20X1 + .10X2 + .1X3 + .15X4 ≥ 1200 .20X1 + .30X2 + .15X3 + .20X4 ≥ 2000 .01X1 + .015X2 + .0075X3 + .005X4 ≥ 100 1X1 + 1X2 + 1X3 + 1X4 = 10000 Xi ≥ 0
86. Answer not provided. 87. Objective Cell: $E$5 Variables Cells: $B$4:$C$4 Constraints Cells: $B$4:$C$4 ≥ 0 $D$8:$D$10 ≥ $E$8:$E$10 88. Objective Cell: F10 Variables Cells: B3:D5 Constraints Cells: B3:D3 ≤ B13:D13 E3:E5 ≥ F3:F5 B6:C6 ≥ B8:C8 D6 ≤ D8 89. Define the following cutting patterns. Number of widths in roll Cutting pattern
12"
18"
24"
1
3
0
0
2
1
1
0
3
1
0
1
4
0
2
0
400
300
200
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ch 3 Let
Xi = number of rolls cut in pattern i
MIN: Subject to:
1X1 + 1X2 + 1X3 + 1X4 3X1 + 1X2 + 1X3 ≥ 400 1X2 + 2X4 ≥ 300 1X3 ≥ 200 Xi ≥ 0
90. Objective Cell: $E$5 Variables Cells: $B$4:$C$4 Constraints Cells: $B$4:$C$4 ≥ 0 $D$8 = $E$8 $D$9 ≤ $E$9 $D$10 ≥ $E$10 (or $B$4 ≥ $E$10) 91. Let
Xij = amount of grain i in feed j where i = A, B, C, D, E, F and j = 1(Feed-M-All), 2(Supreme-Feed) Yj = total amount of feed j produced
MAX: $6.50Y1 + $8.50Y2 Subject to: Y1 = X11 + X21 + X31 + X41 + X51 + X61 Define Yj values Y2 = X12 + X22 + X32 + X42 + X52 + X62 Grain availability X11 + X12 ≤ 90 X21 + X22 ≤ 120 X31 + X32 ≤ 150 X41 + X42 ≤ 125 X51 + X52 ≤ 85 X61 + X62 ≤ 165 Mix requirements 220.5 ≤ Y1 ≤ 514.5 220.5 ≤ Y2 ≤ 514.5 Quality targets 4X11 + 2X21 + X31 + 3X41 + 3X51 + 4X61 ≥ 2.25Y1 4X12 + 2X22 + X32 + 3X42 + 3X52 + 4X62 ≥ 2.45Y2 4.75X11 + 4X21 + 3.75X31 + 4.25X41 + 4.5X51 + 5X61 ≤ Cost targets 4.35Y1 4.75X12 + 4X22 + 3.75X32 + 4.25X42 + 4.5X52 + 5X62 ≤ 4.60Y2 Protein targets 10X11 + 20X21 + 10X31 + 15X41 + 20X51 + 25X61 ≥ Copyright Cengage Learning. Powered by Cognero.
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ch 3 16Y1 10X12 + 20X22 + 10X32 + 15X42 + 20X52 + 25X62 ≥ 18Y2 10X11 + 20X21 + 25X31 + 20X41 + 20X51 + 15X61 ≥ Fat targets 18Y1 10X12 + 20X22 + 25X32 + 20X42 + 20X52 + 15X62 ≥ 18Y2 10X11 + 8X21 + 5X31 + 10X41 + 10X51 + 12X61 ≤ 10Y1 Carbohydrate targets 10X12 + 8X22 + 5X32 + 10X42 + 10X52 + 12X62 ≤ 9Y2 Xij ≥ 0 for all i and j, Yj ≥ 0 for all j. 92. Cell F5 E8
Formula =SUMPRODUCT(B4:D4,B5:D5) =SUMPRODUCT($B$4:$D$4,B8:D8)
93. Let
Xi = number of nurses working in time period i; i = 1,6
MIN: Subject to:
Copied to E:E12
1X1 + 1X2 + 1X3 + 1X4 + 1X5 + 1X6 1X1 + 1X2 ≥ 30 1X2 + 1X3 ≥ 40 1X3 + 1X4 ≥ 50 1X4 + 1X5 ≥ 40 1X5 + 1X6 ≥ 30 1X1 + 1X6 ≥ 20 Xi ≥ 0
94. a. Let
wi = weight assigned to output j, j = 1, ...,4 vi = weight assigned to input i, i = 1,...,3
MAX: 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 Subject to: 105.1 w1 + 253.0 w2 + 125.0 w3 + 98.0 w4 − 83.0 v1 − 123.8 v2 − 225.0 v3 ≤ 0 71.0 w1 + 92.0 w2 + 45.0 w3 + 88.0 w4 − 105.0 v1 − 162.3 v2 − 200 v3 ≤ 0 82.7 w1 + 175.0 w2 + 65.0 w3 + 83.0 w4 − 104.1 v1 − 154.0 v2 − 231.0 v3 ≤ 0 104.1 v1 + 154.0 v2 + 231.0 v3 = 1 w1, w2, w3, w4, v1, v2, v3 ≥ 0
b.
A 1
B Patient
C
D Asst
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ch 3 2 3 4 5 6 7 8 9 10 11 12
< < < < < < < < < < < < <
> > > > > > > > > > >
Hospital New Hope Cnty. General City East
Days (1000s) 105.10 71.00 82.70
Nurses Qual. 253.00 92.00 175.00
on Staff 125.0 45.0 65.0
Cust Sat. 98 88 83
Weights
0.002009
0
0
0.00778
UNIT Output Input
3 0.812259 1
G Supply Expense ($1000s) 123.80 162.30 154.00
H Full Time Staff 225.00 200.00 231.00
I
J
K
Wgt. Output 97% 83% 81%
Wgt. Input 97% 87% 100%
Diff 0.0000 −0.0381 −0.1877
0
0.004329
F 1 Bed-Days 2 Unused 3 (1000s) 4 83.00 5 105.00 6 104.10 7 8 0 9 10 11 12
Unit New Hope County General City East
Results: DEA Efficiency 1.0000 0.9297 0.8123 New Hope is an efficient facility.
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ch 4
Indicate whether the statement is true or false. 1. If the allowable increase or allowable decrease for the objective function coefficient for one or more variables is equal zero, the solution is degenerate. a. True b. False 2. One of the benefits of using the simplex method to solve LP problems is its speed. a. True b. False 3. If a shadow price is positive for a maximization problem, a unit increase in the RHS value of the associated constraint results in a decrease in the optimal objective function value.
a. True b. False 4. A constraint is binding if it is not satisfied as a strict equality in the optimal solution. a. True b. False 5. Shadow prices represent the marginal values of the resources in an LP problem, a. True b. False 6. Sensitivity analysis is useful in considering real-world uncertainties without resolving the entire formulation. a. True b. False 7. Examining the effect of changes in the RHS values of constraints is part of the answer report. a. True b. False 8. The first section of the Answer Report summarizes the original and final (optimal) value of the objective cell. The second section summarizes the original and final (optimal) values of the decision variable cells. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. The reduced cost for a changing cell (decision variable) is a. the amount by which the objective function value changes if the variable is increased by one unit. b. how many more units to product to maximize profits. c. the per unit profits minus the per unit costs for that variable. d. equal to zero for variables at their optimal values. 10. When the allowable increase or allowable decrease for the objective function coefficient of one or more variables is Copyright Cengage Learning. Powered by Cognero.
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ch 4 zero it indicates (in the absence of degeneracy) that a. the problem is infeasible. b. alternate optimal solutions exist. c. there is only one optimal solution. d. no optimal solution can be found. 11. Analytic Solver Platform provides all of the following reports except a. Answer b. Sensitivity c. Cost performance d. Limits 12. The difference between the right-hand side (RHS) values of the constraints and the final (optimal) value assumed by the left-hand side (LHS) formula for each constraint is called the slack and is found in the a. Status report b. Slack report c. Answer report d. Cell Value report 13. The sensitivity analysis provides information about which of the following? a. the impact of a change to an objective function coefficient. b. the impact of a change in a resource level. c. the impact of adding simple upper or lower bounds on a decision variable. d. all of these. 14. A binding greater than or equal to (≥) constraint in a minimization problem means that a. the variable is up against an upper limit. b. the minimum requirement for the constraint has just been met. c. another constraint is limiting the solution. d. the shadow price for the constraint will be positive. 15. A change in the right hand side of a constraint changes a. the slope of the objective function b. objective function coefficients c. other right hand sides d. the feasible region 16. If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the objective function value? a. increase of 150 b. increases more than 0 but less than 150 c. no change d. increases but by an unknown amount 17. If the allowable increase for a constraint is 100 and we add 110 units of the resource what happens to the objective Copyright Cengage Learning. Powered by Cognero.
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ch 4 function value? a. increase of 100 b. increase of 110 c. decrease of 100 d. increases but by unknown amount 18. When a solution is degenerate the reduced costs for the changing cells a. is always equal to zero. b. may not be unique. c. may be set to any value the manager needs. d. is equal to infinity. 19. Binding constraints have a. zero slack. b. negative slack. c. positive slack. d. surplus resources. 20. A solution to the system of equations using a set of basic variables is called a. a feasible solution. b. basic feasible solution. c. a nonbasic solution. d. a nonbasic feasible solution 21. What is the value of the slack variable in the following constraint when X1 and X2 are nonbasic and only nonnegativity is used as simple bounds? X1 + X2 + S1 = 100 a. 0 b. 50 c. 100 d. can't be determined from the given information 22. Consider the formulation below. How many decision variables will be there after the problem has been converted to a standard form? MAX: Subject to:
8 X1 + 4 X2 5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≤ 18 X1, X2 ≥ 0
a. 4 b. 2 c. 3 Copyright Cengage Learning. Powered by Cognero.
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ch 4 d. 5 23. What needs to be done to the two constraints in order to convert the problem to a standard form? MAX: Subject to:
8 X1 + 4 X2 5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≤ 18 X1, X2 ≥ 0 a. a slack variable needs to be added to each constraint to convert them to equalities. b. nothing. c. they need to be combined to a single constraint. d. they need to be subtracted side-by-side
24. If the shadow price for a resource is 0 and 150 units of the resource are added what happens to the optimal solution? a. increases by an unknown amount b. increases more than 0 but less than 150 c. no change d. decreases by an unknown amount 25. The simplex method of linear programming (LP): a. moves to better and better corner point solution of the feasible region until no further objective function improvement can be achieved b. explicitly enumerates all corner points of the feasible region and selects the best objective function value c. evaluates all constraints for feasibility d. does not covert all constraints to equalities 26. The allowable decrease for a constraint is a. how many more units of resource to purchase to maximize profits. b. the amount by which the resource can decrease given shadow price. c. how much resource to use to get the optimal solution. d. the amount by which constraint coefficient can increase without changing the final optimal value. 27. What is the value of the objective function if X1 is set to 0 in the following Limits Report?
Cell
Target Name
Value
$E$5
Unit profit: Total Profit:
3200
Cell
Adjustable Name
Value
Lower Limit
Target Result
Upper Limit
Target Result
Number to make: X1 Number to make: X2
80 20
0 0
800 2400
79.99999999 20
3200 3200
$B$4 $C$4 a. 80 b. 800
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ch 4 c. 2400 d. 3200 28. The absolute value of the shadow price indicates the amount by which the objective function will be a. improved if the corresponding constraint is loosened. b. improved if the corresponding constraint is tightened. c. made worse if the corresponding constraint is loosened. d. improved if the corresponding constraint is unchanged. 29. The optimization technique that locates solutions in the interior of the feasible region is known as a. sub-optimal optimization b. sensitivity analysis c. robust optimization d. USET optimization 30. The allowable increase for a changing cell (decision variable) is a. how many more units to produce to maximize profits. b. the amount by which the objective function coefficient can increase without changing the optimal solution. c. how much to charge to get the optimal solution. d. the amount by which constraint coefficient can increase without changing the optimal solution. 31. When a solution is degenerate the shadow prices and their ranges a. may be interpreted in the usual way but they may not be unique. b. must be disregarded. c. are always valid and unique. d. are always understated 32. Given an objective function value of 150 and a shadow price for resource 1 of 5, if 10 more units of resource 1 are added (assuming the allowable increase is greater than 10), what is the impact on the objective function value? a. increase of 50 b. increase of unknown amount c. decrease of 50 d. increase of 10 33. Which of the following statements is false concerning either of the Allowable Increase and Allowable Decrease columns in the Sensitivity Report? a. The values equate the decision variable profit to the cost of resources expended. b. The values give the range over which a shadow price is accurate. c. The values give the range over which an objective function coefficient can change without changing the optimal solution. d. The values provide a means to recognize when alternate optimal solution exist. 34. The Simplex method works by first a. identifying any basic feasible solution. Copyright Cengage Learning. Powered by Cognero.
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ch 4 b. choosing the largest value for X1. c. setting X1 at one-half of the its maximum value. d. going directly to the optimal solution. 35. A formulation has 20 variables and 8 constraints (not counting non-negativity). How many variables are nonbasic? a. 8 b. 12 c. 20 d. 28 36. For a minimization problem, if a decision variable's final value is 0, and its reduced cost is negative, which of the following is true? a. Alternate optimal solutions exist. b. There is evidence of degeneracy. c. No feasible solution was found. d. The variable has a non-negativity constraint. 37. The shadow price of a nonbinding constraint is a. positive b. zero c. negative d. indeterminate 38. A manager should consider how sensitive the model is to changes in all of the following except a. differential coefficients. b. objective function coefficients. c. constraint coefficients. d. right-hand side values for constraints. 39. The allowable decrease for a changing cell (decision variable) is a. the amount by which the constraint coefficient can decrease without changing final optimal solution. b. an indication of how many more units to produce to maximize profits. c. the amount by which objective function coefficient can decrease without changing the final optimal solution. d. an indication of how much to charge to get the optimal solution. 40. The solution to an LP problem is degenerate if a. the right hand sides of any of the constraints have an allowable increase or allowable decrease of zero. b. the shadow prices of any of the constraints have an allowable increase or allowable decrease of infinity. c. the objective coefficients of any of the variables have an allowable increase or allowable decrease of zero. d. the shadow prices of any of the constraints have an allowable increase or allowable decrease of zero. 41. A spider plot a. is a graphical representation of multiple optimization runs b. requires multiple runs of the problem Copyright Cengage Learning. Powered by Cognero.
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ch 4 c. is computationally expensive to generate for problems with many constraints and variables d. all of the above 42. Consider the formulation below. The standard form of the second constraint is: MAX: Subject to:
8 X1 + 4 X2 5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≤ 18 X1, X2 ≥ 0
a. 5 X1 + 5 X2 = 20 b. 6 X1 + 2 X2 = 18 c. 6 X1 + 2 X2 + S2 = 18 d. 6 X1 + 2 X2 - S2 = 18 43. A binding less than or equal to (≤) constraint in a maximization problem means a. that all of the resource represented by the constraint is consumed in the solution. b. it is not a constraint that the level curve contacts. c. another constraint is limiting the solution. d. the requirement for the constraint has been exceeded. 44. How many basic variables are there in a linear programming model which has n variables and m constraints? a. n b. m c. n + m d. n − m 45. Finding a robust solution to an LP problem a. is one of the many useful features of the Analytic Solver Platform b. is trivial c. is a guessing procedure d. all of the above 46. The simplex method of linear programming (LP): a. considers only the extreme points of the feasible region to achieve efficiency in solving LP problems b. explicitly enumerates all corner points of the feasible region and selects the best objective function value c. evaluates all constraints for feasibility d. is not used in the Analytic Solver Platform software 47. Analytic Solver Platform provides sensitivity analysis information on all of the following except the a. range of values for objective function coefficients which do not change optimal solution. b. impact on optimal objective function value of changes in constrained resources. c. impact on optimal objective function value of changes in value of decision variables. Copyright Cengage Learning. Powered by Cognero.
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ch 4 d. impact on right hand sides of changes in constraint coefficients. 48. The Simplex method uses which of the following values to determine if the objective function value can be improved? a. shadow price b. target value c. reduced cost d. basic cost 49. A formulation has 10 variables and 4 constraints (not counting non-negativity). How many variables are basic? a. 4 b. 10 c. 14 d. 6 50. When a variable is basic a. it is present in the solution. b. it may not be unique. c. it is not present in the solution. d. it's value is equal to zero. 51. The Cell Value column in the Solver Answer Report shows a. which constraints are binding. b. final (optimal) value assumed by each constraint cell. c. objective function values. d. Right hand sides of constraints. 52. To convert ≤ constraints into = constraints the Simplex method adds what type of variable to the constraint? a. slack b. dummy c. redundant d. spreading 53. A robust solution to an LP problem a. is interior of the feasible region b. has a reasonably good objective function value c. is sub-optimal d. all of the above 54. Consider the formulation below. The standard form of the second constraint is: MAX: Subject to:
8 X1 + 4 X2 5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≥ 18 X1, X2 ≥ 0
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ch 4 a. 5 X1 + 5 X2 = 20 b. 6 X1 + 2 X2 = 18 c. 6 X1 + 2 X2 + S2 = 18 d. 6 X1 + 2 X2 - S2 = 18 55. Meaningful Analytic Solver Platform (ASP) sensitivity report headings can be defined a. by adding cell notes to spreadsheet cells. b. by using the Guess button in the Analytic Solver Platform (ASP) dialog box. c. by carefully labeling rows and columns in the spreadsheet model. d. naming cells in the spreadsheet model. 56. The slope of the level curve for the objective function value can be changed by a. increasing the value of the decision variables. b. doubling all the coefficients in the objective function. c. increasing the right hand sides of constraints. d. changing a coefficient in the objective function. 57. Slack variables a. are always equal to zero. b. are usually negative. c. are always positive. d. can be positive or negative. 58. A change in the right hand side of a binding constraint may change all of the following except a. optimal value of the decision variables b. slack values c. other right hand sides d. objective function value 59. Benefits of sensitivity analysis include all the following except: a. provides a better picture of how solutions change as model factors change. b. fosters managerial acceptance of the optimal solution. c. overcomes management skepticism of optimal solutions. d. answers potential managerial questions regarding the solution to an LP problem. 60. The coefficients in an LP model (cj, aij, bj) represent a. random variables. b. numeric constants. c. random constants. d. numeric variables. 61. The allowable increase for a constraint is a. how many more units of resource to purchase to maximize profits. Copyright Cengage Learning. Powered by Cognero.
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ch 4 b. the amount by which the resource can increase given shadow price. c. how much resource to use to get the optimal solution. d. the amount by which the constraint coefficient can increase without changing the final optimal value. 62. A variable with a final value equal to its simple lower or upper bound and a reduced cost of zero indicates that a. an alternate optimal solution exists. b. an error in formulation has been made. c. the right hand sides should be increased. d. the objective function needs new coefficients. 63. When performing sensitivity analysis, which of the following assumptions must apply? a. All other coefficients remain constant. b. Only right hand side changes really mean anything. c. The X1 variable change is the most important. d. The non-negativity assumption can be relaxed 64. When a manager considers the effect of changes in an LP model's coefficients he/she is performing a. a random analysis. b. a coefficient analysis. c. a sensitivity analysis. d. a qualitative analysis.
65. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water which are in short supply. The following table summarizes the data for the problem.
Crop
Profit per Acre ($)
Yield per Acre (lb)
Maximum Demand (lb)
Irrigation (acre ft)
Fertilizer (pounds/acre)
Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 Suppose the farmer can purchase more fertilizer for $2.50 per pound, should he purchase it and how much can he buy and still be sure of the value of the additional fertilizer? Base your response on the following Analytic Solver Platform sensitivity output.
Changing Cells Cell $B$4 $C$4 $D$4
Name Acres of Corn Acres of Pumpkin Acres of Beans
Final Value 9.52 0 10.79
Reduced Cost 0 −500.01 0
Objective Coefficient 2100 899.99 1050
Allowable Increase 1E+30 500.01 210
Allowable Decrease 350 1E+30 375.00
Final
Shadow
Constraint
Allowable
Allowable
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ch 4 Cell $E$8 $E$9 $E$10 $E$11 $E$12
Name Corn demand Used Pumpkin demand Used Bean demand Used Water Used Fertilizer Used
Value 200000 0 37777.78 29.84 8000
Price 0.017 0 0 0 3.5
R.H. Side 200000 180000 80000 50 8000
Increase 136000 1E+30 1E+30 1E+30 3619.04
Decrease 152000 180000 42222.22 20.15 3238.09
66. Use slack variables to rewrite this problem so that all its constraints are equality constraints. MIN: Subject to:
2.5 X1 + 1.5 X2 4 X1 + 3 X2 ≥ 24 2 X1 + 4 X2 ≥ 24 X1, X2 ≥ 0
67. What is the smallest value of the objective function coefficient X1 can assume without changing the optimal solution?
MAX: Subject to:
7 X1 + 4 X2 2 X1 + X2 ≤ 16 X1 + X2 ≤ 10 2 X1 + 5 X2 ≤ 40 X1, X2 ≥ 0 Changing Cells Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4
Number to make: X1 Number to make: X2
6 4
0 0
7 4
1 3
3 0.5
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8 $D$9 $D$10
Used Used Used
16 10 32
3 1 0
16 10 40
4 1 1E+30
2.67 2 8
Constraints
Exhibit 4.1 The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
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ch 4 Week 1 2 3 Costs ($ per 1000 tons)
Trucking Limits 45 50 55 $200
Railway Limits 60 55 45 $140
Air Cargo Limits 15 10 5 $400
The following is the LP model for this logistics problem.
Let
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3
Let
WLij = weekly limit of mode i in week j (as provided in above table)
MIN:
200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to: Weekly limits by mode Xij ≤ WL ij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 Total at end of three weeks ≥ 250 Total at end of two weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of first week X11 + X21 + X31 ≥ 120 Truck mix requirement X11 + X12 + X13 ≥ 0.45*250 Rail mix requirement X21 + X22 + X23 ≥ 0.40*250 Air mix limit X31 + X32 + X33 ≤ 0.15*250 Xij ≥ 0 for all i and j
Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$D$6 $E$6 $F$6 $D$7 $E$7 $F$7 $D$8 $E$8 $F$8
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air
45 60 15 50 55 0 13 12 0
0 0 0 0 0 360 0 0 360
200 140 500 200 140 500 200 140 500
360 360 1E+30 0 0 1E+30 1E+30 60 1E+30
1E+30 1E+30 360 1E+30 1E+30 360 0 0 360
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$18 $E$18 $F$18 $D$19
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck
45 60 15 50
−360 −360 0 0
45 60 15 50
13 15 1E+30 13
0 0 0 25
Constraints
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ch 4 $E$19 $F$19 $D$20 $E$20 $F$20 $D$9 $E$9 $F$13 $F$9 $G$6 $G$7 $G$8
Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air Shipped by Truck Shipped by Rail Total Shipped Tons Shipped by Air Week 1 Totals Week 2 Totals Week 3 Totals
55 0 13 12 0 108 127 250 15 120 225 250
0 0 0 0 0 60 0 140 0 360 0 0
55 10 55 45 5 108 100 250 37.5 120 200 250
12 1E+30 1E+30 1E+30 1E+30 12 27 33 1E+30 0 25 0
25 10 42 33 5 13 1E+30 0 22.5 15 1E+30 1E+30
68. Refer to Exhibit 4.1. The Week 1 by Truck and Week 1 by Rail constraints each have a shadow price of −360. What do these values imply? Exhibit 4.2 The following questions correspond to the problem below and associated Analytic Solver Platform sensitivity report. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: ∙ ∙
homework can account for up to 25% of the grade, but must be at least 5% of the grade; the project can account for up to 25% of the grade, but must be at least 5% of the grade; the mid-term and final must each account for between 10% and 40% of the grade but cannot account for more than 70% of the grade when the percentages are combined; and the project and final exam grades may not collectively constitute more than 50% of the grade.
∙ ∙
The following LP model allows Robert to maximize his numerical grade. Let
W1 = weight assigned to homework W2 = weight assigned to the project W3 = weight assigned to the mid-term W4 = weight assigned to the final
MAX: Subject to:
75W1 + 94W2 + 85W3 + 92W4 W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40
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ch 4 Adjustable Cells Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
Project to grade Homework to grade
0.40 0.25 0.25 0.10
10.00 0.00 2.00 0.00
85 92 94 75
1E+30 2 1E+30 10
10 17 2 1E+30
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$E$14 $E$15 $F$9
Both Exams Total Final & Project Total 100% to grade
0.65 0.5 1.00
0 17 75.00
0.7 0.5 1
1E+30 0.05 0.15
0.05 0.15 0.05
Cell
Name
$F$5
Mid Term to grade
$F$6
Final to grade
$F$7 $F$8 Constraints
69. Refer to Exhibit 4.2. Constraint cell F9 corresponds to the constraint, W1 + W2 + W3 + W4 = 1, and has a shadow price of 75. Armed with this information, what can Robert request of his instructor regarding this constraint? 70. Identify the different sets of basic variables that might be used to obtain a solution to this problem. MIN: Subject to:
2.5 X1 + 1.5 X2 4 X1 + 3 X2 ≥ 24 2 X1 + 4 X2 ≥ 24 X1, X2 ≥ 0
71. What are the objective function coefficients for X1 and X2 based on the following Analytic Solver Platform sensitivity output?
Cell
Target Name
Value
$E$5
Unit profit: OBJ. FN. VALUE
58
Cell
Adjustable Name
Value
Lower Limit
Target Result
Upper Limit
Target Result
$B$4 $C$4
Number to make: X1 Number to make: X2
6 4
0 0
16 42
6 4
58 58
72. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced. How much can the price of Desks drop before it is no longer profitable to produce them? Base your response on the following Analytic Solver Platform sensitivity output.
Let
X1 = Number of Beds to produce
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ch 4 X2 = Number of Desks to produce The LP model for the problem is
MAX: Subject to:
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for X2) X1, X2 ≥ 0 Changing Cells Cell
$B$4 $C$4
Name Number to make: Beds Number to make: Desks
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
4 3
0 0
30 40
30 20
10 20
Constraints Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8 $D$9 $D$10
Carpentry Used Varnishing Used Desk demand Used
36 40 3
2.5 3.75 0
36 40 8
24 26.67 1E+30
16 16 5
73. Jones Furniture Company produces beds and desks for college students. The production process requires carpentry and varnishing. Each bed requires 6 hours of carpentry and 4 hour of varnishing. Each desk requires 4 hours of carpentry and 8 hours of varnishing. There are 36 hours of carpentry time and 40 hours of varnishing time available. Beds generate $30 of profit and desks generate $40 of profit. Demand for desks is limited so at most 8 will be produced. Suppose the company can purchase more varnishing time for $3.00, should it be purchased and how much can be bought before the value of the additional time is uncertain? Base your response on the following Analytic Solver Platform sensitivity output.
Let
X1 = Number of Beds to produce X2 = Number of Desks to produce The LP model for the problem is
MAX: Subject to:
30 X1 + 40 X2 6 X1 + 4 X2 ≤ 36 (carpentry) 4 X1 + 8 X2 ≤ 40 (varnishing) X2 ≤ 8 (demand for X2) X1, X2 ≥ 0 Changing Cells Cell
$B$4 $C$4
Name Number to make: Beds Number to make: Desks
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
4 3
0 0
30 40
30 20
10 20
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ch 4 Constraints Cell
$D$8 $D$9 $D$10
Name Carpentry Used Varnishing Used Desk demand Used
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
36 40 3
2.5 3.75 0
36 40 8
24 26.67 1E+30
16 16 5
Exhibit 4.1 The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week 1 2 3 Costs ($ per 1000 tons)
Trucking Limits 45 50 55 $200
Railway Limits 60 55 45 $140
Air Cargo Limits 15 10 5 $400
The following is the LP model for this logistics problem.
Let
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3
Let
WLij = weekly limit of mode i in week j (as provided in above table)
MIN:
200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to: Weekly limits by mode Xij ≤ WL ij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 Total at end of three weeks ≥ 250 Total at end of two weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of first week X11 + X21 + X31 ≥ 120 Truck mix requirement X11 + X12 + X13 ≥ 0.45*250 Rail mix requirement X21 + X22 + X23 ≥ 0.40*250 Air mix limit X31 + X32 + X33 ≤ 0.15*250 Copyright Cengage Learning. Powered by Cognero.
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ch 4 Xij ≥ 0 for all i and j
Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$D$6 $E$6 $F$6 $D$7 $E$7 $F$7 $D$8 $E$8 $F$8
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air
45 60 15 50 55 0 13 12 0
0 0 0 0 0 360 0 0 360
200 140 500 200 140 500 200 140 500
360 360 1E+30 0 0 1E+30 1E+30 60 1E+30
1E+30 1E+30 360 1E+30 1E+30 360 0 0 360
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$18 $E$18 $F$18 $D$19 $E$19 $F$19 $D$20 $E$20 $F$20 $D$9 $E$9 $F$13 $F$9 $G$6 $G$7 $G$8
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air Shipped by Truck Shipped by Rail Total Shipped Tons Shipped by Air Week 1 Totals Week 2 Totals Week 3 Totals
45 60 15 50 55 0 13 12 0 108 127 250 15 120 225 250
−360 −360 0 0 0 0 0 0 0 60 0 140 0 360 0 0
45 60 15 50 55 10 55 45 5 108 100 250 37.5 120 200 250
13 15 1E+30 13 12 1E+30 1E+30 1E+30 1E+30 12 27 33 1E+30 0 25 0
0 0 0 25 25 10 42 33 5 13 1E+30 0 22.5 15 1E+30 1E+30
Constraints
74. Refer to Exhibit 4.1. Of the three percentage of effort constraints, Shipped by Truck, Shipped by Rail, and Shipped by Air, which should be examined for potential cost reduction? 75. Constraint 3 is a non-binding constraint in the final solution to a maximization problem. Complete the following entry for the Analytic Solver Platform sensitivity report. Cell labels are included to ease of reference.
Constraints Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8
Constraint 3
6
??
10
??
??
76. Consider the following linear programming model and Analytic Solver Platform sensitivity output. What is the optimal objective function value if the RHS of the first constraint increases to 18?
MAX:
7 X1 + 4 X2
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ch 4 Subject to:
2 X1 + X2 ≤ 16 X1 + X2 ≤ 10 2 X1 + 5 X2 ≤ 40 X1, X2 ≥ 0 Changing Cells Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4
Number to make: X1 Number to make: X2
6 4
0 0
7 4
1 3
3 0.5
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8 $D$9 $D$10
Used Used Used
16 10 32
3 1 0
16 10 40
4 1 1E+30
2.67 2 8
Constraints
77. Use slack variables to rewrite this problem so that all its constraints are equality constraints. MAX: Subject to:
2 X1 + 7 X2 5 X1 + 9 X2 ≤ 90 9 X1 + 8 X2 ≤ 144 X2 ≤ 8 X1, X2 ≥ 0
78. Given the following Analytic Solver Platform sensitivity output what range of values can the objective function coefficient for variable X1 assume without changing the optimal solution?
Changing Cells Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4
Number to make: X1 Number to make: X2
9.49 1.74
0 0
5 6
1.54 1.5
1 1.47
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8 $D$9 $D$10
Used Used Used
42 132 24
0 0.24 1.24
48 132 24
1E+30 12 1.33
6 12 2
Constraints
Exhibit 4.2 The following questions correspond to the problem below and associated Analytic Solver Platform sensitivity report. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, Copyright Cengage Learning. Powered by Cognero.
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ch 4 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: ∙ ∙
homework can account for up to 25% of the grade, but must be at least 5% of the grade; the project can account for up to 25% of the grade, but must be at least 5% of the grade; the mid-term and final must each account for between 10% and 40% of the grade but cannot account for more than 70% of the grade when the percentages are combined; and the project and final exam grades may not collectively constitute more than 50% of the grade.
∙ ∙
The following LP model allows Robert to maximize his numerical grade. Let
W1 = weight assigned to homework W2 = weight assigned to the project W3 = weight assigned to the mid-term W4 = weight assigned to the final
MAX: Subject to:
75W1 + 94W2 + 85W3 + 92W4 W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40 Adjustable Cells Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
Project to grade Homework to grade
0.40 0.25 0.25 0.10
10.00 0.00 2.00 0.00
85 92 94 75
1E+30 2 1E+30 10
10 17 2 1E+30
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$E$14 $E$15 $F$9
Both Exams Total Final & Project Total 100% to grade
0.65 0.5 1.00
0 17 75.00
0.7 0.5 1
1E+30 0.05 0.15
0.05 0.15 0.05
Cell
Name
$F$5
Mid Term to grade
$F$6
Final to grade
$F$7 $F$8 Constraints
79. Refer to Exhibit 4.2. Based on the Analytic Solver Platform sensitivity report information, Robert has been approved by his instructor to increase the total weight allowed for the project and final exam to 0.50 plus the allowable increase. When Robert re-solves his model, what will his new final grade score be? 80. Identify the different sets of basic variables that might be used to obtain a solution to this problem. MAX:
8 X1 + 4 X2
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ch 4 Subject to:
5 X1 + 5 X2 ≤ 20 6 X1 + 2 X2 ≤ 18 X1, X2 ≥ 0
81. Which of the constraints are binding at the optimal solution for the following problem and Analytic Solver Platform sensitivity output?
MAX: Subject to:
7 X1 + 4 X2 2 X1 + X2 ≤ 16 X1 + X2 ≤ 10 2 X1 + 5 X2 ≤ 40 X1, X2 ≥ 0 Changing Cells Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4
Number to make: X1 Number to make: X2
6 4
0 0
7 4
1 3
3 0.5
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8 $D$9 $D$10
Used Used Used
16 10 32
3 1 0
16 10 40
4 1 1E+30
2.67 2 8
Constraints
Exhibit 4.1 The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week 1 2 3 Costs ($ per 1000 tons)
Trucking Limits 45 50 55 $200
Railway Limits 60 55 45 $140
Air Cargo Limits 15 10 5 $400
The following is the LP model for this logistics problem.
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ch 4 Let
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3
Let
WLij = weekly limit of mode i in week j (as provided in above table)
MIN:
200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
Subject to: Weekly limits by mode Xij ≤ WL ij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 Total at end of three weeks ≥ 250 Total at end of two weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of first week X11 + X21 + X31 ≥ 120 Truck mix requirement X11 + X12 + X13 ≥ 0.45*250 Rail mix requirement X21 + X22 + X23 ≥ 0.40*250 Air mix limit X31 + X32 + X33 ≤ 0.15*250 Xij ≥ 0 for all i and j
Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$D$6 $E$6 $F$6 $D$7 $E$7 $F$7 $D$8 $E$8 $F$8
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air
45 60 15 50 55 0 13 12 0
0 0 0 0 0 360 0 0 360
200 140 500 200 140 500 200 140 500
360 360 1E+30 0 0 1E+30 1E+30 60 1E+30
1E+30 1E+30 360 1E+30 1E+30 360 0 0 360
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$18 $E$18 $F$18 $D$19 $E$19 $F$19 $D$20 $E$20 $F$20 $D$9 $E$9 $F$13 $F$9
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air Shipped by Truck Shipped by Rail Total Shipped Tons Shipped by Air
45 60 15 50 55 0 13 12 0 108 127 250 15
−360 −360 0 0 0 0 0 0 0 60 0 140 0
45 60 15 50 55 10 55 45 5 108 100 250 37.5
13 15 1E+30 13 12 1E+30 1E+30 1E+30 1E+30 12 27 33 1E+30
0 0 0 25 25 10 42 33 5 13 1E+30 0 22.5
Constraints
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ch 4 $G$6 $G$7 $G$8
Week 1 Totals Week 2 Totals Week 3 Totals
120 225 250
360 0 0
120 200 250
0 25 0
15 1E+30 1E+30
82. Refer to Exhibit 4.1. Should the company negotiate for additional air delivery capacity? 83. Given the following Analytic Solver Platform sensitivity output how much does the objective function coefficient for X2 have to increase before it enters the optimal solution at a strictly positive value?
Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4 $D$4
X1 X2 X3
9.52 0 10.79
0 −500.01 0
2100 899.99 1050
1E+30 500.01 210
350 1E+30 375.01
Exhibit 4.2 The following questions correspond to the problem below and associated Analytic Solver Platform sensitivity report. Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations: ∙ ∙
homework can account for up to 25% of the grade, but must be at least 5% of the grade; the project can account for up to 25% of the grade, but must be at least 5% of the grade; the mid-term and final must each account for between 10% and 40% of the grade but cannot account for more than 70% of the grade when the percentages are combined; and the project and final exam grades may not collectively constitute more than 50% of the grade.
∙ ∙
The following LP model allows Robert to maximize his numerical grade. Let
W1 = weight assigned to homework W2 = weight assigned to the project W3 = weight assigned to the mid-term W4 = weight assigned to the final
MAX: Subject to:
75W1 + 94W2 + 85W3 + 92W4 W1 + W2 + W3 + W4 = 1 W3 + W4 ≤ 0.70 W3 + W4 ≥ 0.50 0.05 ≤ W1 ≤ 0.25 0.05 ≤ W2 ≤ 0.25 0.10 ≤ W3 ≤ 0.40 0.10 ≤ W4 ≤ 0.40
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ch 4 Adjustable Cells Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
Project to grade Homework to grade
0.40 0.25 0.25 0.10
10.00 0.00 2.00 0.00
85 92 94 75
1E+30 2 1E+30 10
10 17 2 1E+30
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$E$14 $E$15 $F$9
Both Exams Total Final & Project Total 100% to grade
0.65 0.5 1.00
0 17 75.00
0.7 0.5 1
1E+30 0.05 0.15
0.05 0.15 0.05
Cell
Name
$F$5
Mid Term to grade
$F$6
Final to grade
$F$7 $F$8 Constraints
84. Refer to Exhibit 4.2. Based on the Analytic Solver Platform sensitivity report information, is there anything Robert can request of his instructor to improve his final grade? 85. What is the optimal objective function value if X1 is at its lower limit in the following Analytic Solver Platform sensitivity output?
Cell
Target Name
Value
$E$5
Unit profit: OBJ. FN. VALUE
58
Cell
Adjustable Name
Value
Lower Limit
Target Result
Upper Limit
Target Result
$B$4 $C$4
Number to make: X1 Number to make: X2
6 4
0 0
16 42
6 4
58 58
86. Is the optimal solution to this problem unique, or is there an alternate optimal solution? Explain your reasoning.
MAX Subject to:
5 X1 + 2 X2 3 X1 + 5 X2 ≤ 15 10 X1 + 4 X2 ≤ 20 X1, X2 ≥ 0 Changing Cells
Final Cell
Name
Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4
Number to make: X1 Number to make: X2
2 0
0 0
5 2
1E+30 0
0 1E+30
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8
Used
6
0
15
1E+30
9
Constraints
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ch 4 $D$9
Used
20
0.5
20
30
20
87. A farmer is planning his spring planting. He has 20 acres on which he can plant a combination of Corn, Pumpkins and Beans. He wants to maximize his profit but there is a limited demand for each crop. Each crop also requires fertilizer and irrigation water both of which are in short supply. The following table summarizes the data for the problem.
Crop
Profit per Acre ($)
Yield per Acre (lb)
Maximum Demand (lb)
Irrigation (acre ft)
Fertilizer (pounds/acre)
Corn 2,100 21,000 200,000 2 500 Pumpkin 900 10,000 180,000 3 400 Beans 1,050 3,500 80,000 1 300 Based on the following Analytic Solver Platform sensitivity output, how much can the price of Corn drop before it is no longer profitable to plant corn?
Changing Cells Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$B$4 $C$4 $D$4
Acres of Corn Acres of Pumpkin Acres of Beans
9.52 0 10.79
0 −500.01 0
2100 899.99 1050
1E+30 500.01 210
350 1E+30 375.00
Exhibit 4.1 The following questions are based on the problem below and accompanying Analytic Solver Platform sensitivity report. Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
Week 1 2 3 Costs ($ per 1000 tons)
Trucking Limits 45 50 55 $200
Railway Limits 60 55 45 $140
Air Cargo Limits 15 10 5 $400
The following is the LP model for this logistics problem.
Let
Xij = amount shipped by mode i in week j where i = 1(Truck), 2(Rail), 3(Air) and j = 1, 2, 3
Let
WLij = weekly limit of mode i in week j (as provided in above table)
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ch 4 200(X11 + X12 + X13) + 140(X21 + X22 + X23) + 500(X31 + X32 + X33)
MIN:
Subject to: Weekly limits by mode Xij ≤ WL ij for all i and j X11 + X12 + X13 + X21 + X22 + X23 + X31 + X32 + X33 Total at end of three weeks ≥ 250 Total at end of two weeks X11 + X21 + X31 + X12 + X22 + X32 ≥ 200 Total at end of first week X11 + X21 + X31 ≥ 120 Truck mix requirement X11 + X12 + X13 ≥ 0.45*250 Rail mix requirement X21 + X22 + X23 ≥ 0.40*250 Air mix limit X31 + X32 + X33 ≤ 0.15*250 Xij ≥ 0 for all i and j
Cell
Name
Final Value
Reduced Cost
Objective Coefficient
Allowable Increase
Allowable Decrease
$D$6 $E$6 $F$6 $D$7 $E$7 $F$7 $D$8 $E$8 $F$8
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air
45 60 15 50 55 0 13 12 0
0 0 0 0 0 360 0 0 360
200 140 500 200 140 500 200 140 500
360 360 1E+30 0 0 1E+30 1E+30 60 1E+30
1E+30 1E+30 360 1E+30 1E+30 360 0 0 360
Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$18 $E$18 $F$18 $D$19 $E$19 $F$19 $D$20 $E$20 $F$20 $D$9 $E$9 $F$13 $F$9 $G$6 $G$7 $G$8
Week 1 by Truck Week 1 by Rail Week 1 by Air Week 2 by Truck Week 2 by Rail Week 2 by Air Week 3 by Truck Week 3 by Rail Week 3 by Air Shipped by Truck Shipped by Rail Total Shipped Tons Shipped by Air Week 1 Totals Week 2 Totals Week 3 Totals
45 60 15 50 55 0 13 12 0 108 127 250 15 120 225 250
−360 −360 0 0 0 0 0 0 0 60 0 140 0 360 0 0
45 60 15 50 55 10 55 45 5 108 100 250 37.5 120 200 250
13 15 1E+30 13 12 1E+30 1E+30 1E+30 1E+30 12 27 33 1E+30 0 25 0
0 0 0 25 25 10 42 33 5 13 1E+30 0 22.5 15 1E+30 1E+30
Constraints
88. Refer to Exhibit 4.1. Are there alternate optimal solutions to this problem? Copyright Cengage Learning. Powered by Cognero.
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ch 4 Answer Key 1. False 2. True 3. False 4. False 5. True 6. True 7. False 8. True 9. c 10. b 11. c 12. c 13. d 14. b 15. d 16. c 17. d 18. b 19. a 20. b 21. c 22. a 23. a 24. c 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 4 26. b 27. b 28. a 29. c 30. b 31. a 32. a 33. a 34. a 35. b 36. d 37. b 38. a 39. c 40. a 41. d 42. c 43. a 44. b 45. a 46. a 47. d 48. c 49. a 50. a 51. b Copyright Cengage Learning. Powered by Cognero.
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ch 4 52. a 53. d 54. d 55. c 56. d 57. c 58. c 59. b 60. b 61. b 62. a 63. a 64. c 65. Yes, because the cost of $2.50 is less than the shadow price of $3.50. The allowable increase is 3619.04 pounds. 66. MIN Subject to:
2.5 X1 + 1.5 X2 4 X1 + 3 X2 − S1 = 24 2 X1 + 4 X2 − S2 = 24 X1, X2 ≥ 0
67. Coefficient − allowable decrease = 7 − 3 = 4 68. Increase the weekly limits on these two modes to reduce total cost by $360 per unit increase in limit. 69. Nothing. The constraint has the largest shadow price but enforces the total percentages to equal 1, thus nothing can be changed. 70. X1 0 0 12 2.4
X2 0 8 0 4.8
S1 24 0 24 0
S2 24 8 0 0
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ch 4 71. Coefficient for X1 is 7 and coefficient for X2 is 4 72. The allowable decrease is 20. 73. Yes, because the cost of $3.00 is less than the shadow price of $3.75. The allowable increase is 26.67 hours. 74. The percentage by Truck, Shipped by Truck, should be examined. Decreasing the percentage by truck (from 45%) will decrease cost as the shadow price is 60. 75. Constraints Cell
Name
Final Value
Shadow Price
Constraint R.H. Side
Allowable Increase
Allowable Decrease
$D$8
Constraint 3
6
0
10
1E+30
4
76. Shadow price of first constraint is 3 with an allowable increase of 4. A 2-unit increase in RHS value increases objective function by 6. New objective function value is 6 * 7 + 4 * 4 + 2 * 3 = 64. 77. MAX: Subject to:
2 X1 + 7 X2 5 X1 + 9 X2 + S1 = 90 9 X1 + 8 X2 + S2 = 144 X2 + S3 = 8 X1, X2 ≥ 0
78. 4 − 6.54 79. 88.85 since shadow price of 17 and increase of 0.05 equates to 0.85. 80. X1 0 0 3 2.5
X2 0 4 0 1.5
S1 20 0 5 0
S2 18 10 0 0
81. X1 = 6, X2 = 4 2 * 6 + 4 = 16 6 + 4 = 10 2 * 6 + 5 * 4 = 32
binding binding non-binding
82. No. The shadow prices for each week of air delivery are zero. 83. 500.01 Copyright Cengage Learning. Powered by Cognero.
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ch 4 84. Robert can request an increase in the total weight allowed for the project and final exam combined since this has a positive shadow price. 85. 16 86. Alternate optimal solutions exist because variable X2 has a final value of 0 and a reduced cost of 0. 87. The allowable decrease for corn is 350. 88. Cannot tell because we cannot rule out degeneracy according to our guidelines due to the zero values in the Allowable Increase and Allowable Decrease columns of the constraint portion of the Analytic Solver Platform sensitivity report.
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ch 5
Indicate whether the statement is true or false. 1. Any shortest path problem can be modeled as a transshipment problem. a. True b. False 2. In a network flow problem, the value +80 next to a node indicates that the number of units needs to decrease by 80. a. True b. False 3. Most of the other types of network flow problems can be viewed as simple variations of the transshipment problem. a. True b. False 4. For minimum cost network flow problems where Total Supply > Total Demand, the balance of flow rule to use at each node is Inflow – Outflow >= Supply or Demand. a. True b. False 5. Transportation/assignment problems are sparse or not fully interconnected, meaning all the supply nodes have arcs connecting them to all the demand nodes. a. True b. False 6. The equipment replacement problem is a common type of business problem that can be modeled as a shortest path problem. a. True b. False 7. When the lines connecting the nodes in a network are arrows that indicate a direction, the arcs in the network are called directed arcs. a. True b. False 8. A number of practical decision problems in business fall into a category known as network flow problems. These problems can be described or displayed in a graphical form known as a network. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. Maximal flow problems are converted to transshipment problems by a. connecting the supply and demand nodes with a return arc b. adding extra supply nodes c. adding supply limits on the supply nodes Copyright Cengage Learning. Powered by Cognero.
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ch 5 d. requiring integer solutions 10. In the generalized network flow problem solver could not find a feasible solution. This means that: a. the total supply is not capable of meeting the total demand b. the total supply is capable of meeting the total demand c. the total demand is not capable of meeting the total supply d. dummy demand is needed in the formulation 11. What is the correct constraint for node 2 in the following diagram?
a. X12 + X23 = 100 b. X12 − X23 ≤ 100 c. −X12 + X23 ≥ −100 d. X12 − X23 ≥ 100 12. What is missing from transportation problems compared to transshipment problems? a. arcs b. demand nodes c. transshipment nodes d. supply nodes 13. What is the objective function in the following maximal flow problem?
a. MIN X41 b. MAX X12 + X13 c. MAX X14 d. MAX X41 Copyright Cengage Learning. Powered by Cognero.
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ch 5 14. In the shortest path problem, the originating and terminating network nodes are called: a. source and sink b. source and end c. beginning and end d. beginning and sink 15. The number of constraints in network flow problems is determined by the number of a. nodes. b. arcs. c. demands. d. supplies. 16. The transshipment nodes in the graphical representation of the transshipment problem: a. have total demanded quantities expressed as positive numbers b. have all directed arcs originating at them c. have some unidirectional arcs d. have all directed arcs terminating at them 17. In a transshipment problem, which of the following statements is a correct representation of the balance-of-flow rule if Total Supply < Total Demand? a. Inflow − Outflow ≥ Supply or Demand b. Inflow + Outflow ≥ Supply or Demand c. Inflow − Outflow ≤ Supply or Demand d. Inflow + Outflow ≤ Supply or Demand 18. Supply quantities for supply nodes in a transshipment problem are customarily indicated by a. positive numbers. b. negative numbers. c. imaginary numbers. d. either positive or negative numbers. 19. How could a network be modified if demand exceeds supply? a. add extra supply arcs b. remove the extra demand arcs c. add a dummy supply d. add a dummy demand 20. What formula would be entered in cell G18 in this Excel model?
A
B
C
D
E
F
G
H
I
1 2 3 4
J
K
L
Supply/
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ch 5 Ship From To 5 6 55 1 LAV 2 PHO 7 45 1 LAV 4 REN 8 5 2 PHO 3 LAX 9 0 3 LAX 5 SAN 10 25 5 SAN 3 LAX 11 0 5 SAN 4 REN 12 0 5 SAN 6 DEN 13 0 6 DEN 5 SAN 14 0 7 SLC 4 REN 15 115 7 SLC 5 SAN 16 35 7 SLC 6 DEN 17 18 Total a. SUMPRODUCT(K6:K12,L6:L12) b. SUMPRODUCT(B6:B16,G6:G16) c. SUMPRODUCT(G6:G16,K6:K12) d. SUMPRODUCT(B6:G16,L6:L12)
Unit Cost
60 120 160 70 90 70 90 50 190 90 100
Nodes
1 2 3 4 5 6 7
LAV PHO LAX REN SAN DEN SLC
Net Flow
Demand
−100 50 30 45 90 35 −150
−100 50 30 45 90 35 −150
25600
21. When might a network flow model for a transportation/assignment problem be preferable to a matrix form for the problem? a. When an integer solution is required. b. When the problem is large and not fully connected. c. When the problem is large and fully connected. d. When supply exceeds demand. 22. A network flow problem that allows gains or losses along the arcs is called a a. non-constant network flow model. b. non-directional, shortest path model. c. generalized network flow model. d. transshipment model with linear side constraints. 23. Which property of network flow models guarantees integer solutions? a. linear constraints and balance of flow equation format b. linear objective function coefficients c. integer objective function coefficients d. integer constraint RHS values and balance of flow equation format 24. What is the interpretation of units "shipped" along arcs from dummy supply nodes to demand nodes? a. Indicates unmet demand at demand nodes b. Indicates unmet supply at demand nodes c. Indicates unmet demand at supply nodes d. Indicates unmet supply at supply nodes 25. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is $11 and the yield Copyright Cengage Learning. Powered by Cognero.
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ch 5 is 85 percent. The following flowchart depicts this problem. What is the balance of flow constraint for node 7 (Diesel)?
a. X35 + X36 + X37 = 75 b. X37 + X47 ≥ 75 c. .90 X37 + .95 X47 = 75 d. X37 + X47 −X36 − X35 − X45 − X46 ≥ 75 26. A node which can both send to and receive from other nodes is a a. demand node. b. supply node. c. random node. d. transshipment node. 27. The supply nodes in the graphical representation of the transshipment problem: a. have total available quantities expressed as negative numbers b. have all directed arcs terminating at them c. have some unidirectional arcs d. have total demanded quantities expressed as positive numbers 28. The right hand side value for the ending node in a shortest path problem has a value of a. −1 b. 0 c. 1 d. 2 29. How many arcs are required to make a spanning tree in a network with n nodes and m arcs? a. n b. n − 1 c. m d. m − 1 Copyright Cengage Learning. Powered by Cognero.
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ch 5 30. How many constraints are there in a transshipment problem which has n nodes and m arcs? a. n b. m c. (n + m) d. (m − n) 31. The equipment replacement problem is an example of which network problem? a. transportation problem. b. shortest path problem. c. maximal flow problem. d. minimal spanning tree problem. 32. Which method is preferred for solving fully connected transportation problems? a. linear programming b. network flow methods c. trial and error d. simulation 33. What is the constraint for node 2 in the following maximal flow problem?
a. X12 − X23 − X24 = 0 b. X12 + X23 + X24 = 0 c. X12 ≤ 4 d. X12 + X13 − X23 = 0 34. Demand quantities for demand nodes in a transshipment problem are customarily indicated by a. positive numbers. b. negative numbers. c. imaginary numbers. d. either positive or negative numbers. 35. The street intersections in a city road network represent a. nodes. Copyright Cengage Learning. Powered by Cognero.
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ch 5 b. arcs. c. resources. d. expenses. 36. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. Consider arc 2-4. The per unit shipping cost of crude B from source 2 (node 2) to refinery 2 (node 4) is $11 and the yield is 85 percent. The following network representation depicts this problem. What is the balance of flow constraint for node 3 (Refinery 1)?
a. X13 + X23 − .95 X35 − .90 X36 − .90 X37 = 0 b. .80 X13 + .95 X23 − X35 − X36 − X37 = 0 c. .80 X13 + .95 X23 − .90 X36 − .90 X37 ≥ 0 d. X13 + X23 − X35 − X36 − X37 ≥ 0 37. Which balance of flow rule should be applied at each node in a network flow problem when Total Supply > Total Demand? a. Inflow − Outflow ≤ Supply or Demand b. Inflow − Outflow ≥ Supply or Demand c. Inflow − Outflow = Supply or Demand d. Inflow − Supply ≥ Outflow or Demand 38. Almost all network problems can be viewed as special cases of the a. transshipment problem. b. shortest path problem. c. maximal flow problem. d. minimal spanning tree problem. 39. Which method is preferred for solving minimal spanning tree problems? a. linear programming b. transshipment models c. simulation Copyright Cengage Learning. Powered by Cognero.
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ch 5 d. manual algorithms 40. In generalized network flow problems a. solutions may not be integer values. b. flows along arcs may increase or decrease. c. it can be difficult to tell if total supply is adequate to meet total demand. d. all of these. 41. The demand nodes in the graphical representation of the transshipment problem: a. have total demanded quantities expressed as positive numbers b. have all directed arcs originating at them c. have some unidirectional arcs d. have total available quantities expressed as negative numbers 42. What happens to the solution of a network flow model if side constraints are added that do not obey the balance of flow rules? a. The model solution is not guaranteed to be integer. b. The model solution will more accurately reflect reality. c. The model solution will be integer but more accurate. d. The model solution is not guaranteed to be feasible. 43. Which formula should be used to determine the Net Flow values in cell K6 in the following spreadsheet model?
A B C D E F G H I J K 1 2 3 4 Ship From To Unit Cost Nodes Net Flow 5 6 55 1 LAV 2 PHO 60 1 LAV −100 7 45 1 LAV 4 REN 120 2 PHO 50 8 5 2 PHO 3 LAX 160 3 LAX 30 9 0 3 LAX 5 SAN 70 4 REN 45 10 25 5 SAN 3 LAX 90 5 SAN 90 11 0 5 SAN 4 REN 70 6 DEN 35 12 0 5 SAN 6 DEN 90 7 SLC −150 13 0 6 DEN 5 SAN 50 14 0 7 SLC 4 REN 190 15 115 7 SLC 5 SAN 90 16 35 7 SLC 6 DEN 100 17 18 Total 25600 a. SUMIF($C$6:$C$16,I6,$B$6:$B$16)−SUMIF($E$6:$E$16,I6,$B$6:$B$16) b. SUMIF($I$6:$I$12,B6,$B$6:$B$16)−SUMIF($I$6:$I$12,I6,$B$6:$B$16) c. SUMIF($E$6:$E$16,I6,$B$6:$B$16)−SUMIF($C$6:$C$16,I6,$B$6:$B$16) Copyright Cengage Learning. Powered by Cognero.
L
Supply/ Demand
−100 50 30 45 90 35 −150
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ch 5 d. SUMPRODUCT(B6:B16,G6:G16) 44. Decision variables in network flow problems are represented by a. nodes. b. arcs. c. demands. d. supplies. 45. A maximal flow problem differs from other network models in which way? a. arcs are two directional b. multiple supply nodes are used c. arcs have limited capacity d. arcs have unlimited capacity 46. What is the objective function for the following shortest path problem?
a. −X12 − X13 = 0 b. MIN −50 X12 − 200 X13 + 100 X24 + 35 X34 c. MIN 50 X12 + 200 X13 + 100 X24 + 35 X34 d. MAX −50 X12 − 200 X13 + 100 X24 + 35 X34 47. If a side constraint for a network flow model cannot be avoided, and non-integer solutions result, how can the solution be expressed as an integer solution? a. Apply integer programming techniques. b. Round off all the non-integer arc flow decision variables. c. Increase the supply until the solutions are all integer using a dummy supply node. d. Increase the demand until the solutions are all integer using a dummy demand node. 48. What is the constraint for node 2 in the following shortest path problem?
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ch 5
a. −X12 − X13 = 0 b. −X12 − X24 = 1 c. X12 + X13 = 0 d. −X12 + X24 = 0 49. The right hand side value for the starting node in a shortest path problem has a value of a. −1 b. 0 c. 1 d. 2 50. A minimum or maximum flow restriction in the network flow problem can be modeled by a. adding dummy nodes b. adding dummy arcs c. adding additional flow restrictions on affected arcs d. all of the above 51. The minimal spanning tree solution algorithm works by defining a subnetwork and a. adding the least expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork. b. adding the most expensive arc which connects any node in the current subnetwork to any node not in the current subnetwork. c. adding the least expensive arc which connects unconnected nodes in the current subnetwork. d. adding the least expensive arc which connects the most recently added node in the current subnetwork to the closest node not in the current subnetwork. 52. Consider the equipment replacement problem presented in the chapter. Recall that in the network model formulation of this problem a node represents a year when the equipment was purchased. An arc from node i to node j indicates that the equipment purchased in year i can be replaced at the beginning of year j. How could the network model below be modified to depict an equipment purchase in year 4 and operating costs only through the remainder of the planning window?
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ch 5
a. Modify the cost on arc 4-5 to account for only operating costs. b. Add a second arc 4-5 to represent just the operating costs. c. Add a dummy node, 6, so that arc 4-6 represents just the operating costs. d. Add a dummy node, 6, so that arc 4-5 represents operating costs and 5-6 represents new equipment purchase. 53. The demand nodes in the graphical representation of the transshipment problem: a. have all directed arcs originating at them b. have all directed arcs terminating at them c. have some unidirectional arcs d. are greater than the demand nodes 54. The supply nodes in the graphical representation of the transshipment problem: a. have all directed arcs originating at them b. have all directed arcs terminating at them c. have some unidirectional arcs d. are greater than the demand nodes 55. A factory which ships items through the network would be represented by which type of node? a. demand b. supply c. random d. decision 56. In the assignment problem: a. the sums of all rows and columns must be equal to one. b. the number of rows is greater than the number of columns c. the number of rows is smaller than the number of columns d. there is no limit on the sum of all rows 57. Consider modeling a warehouse with three in-flow arcs and three outflow arcs. The warehouse node is a transshipment node but has a capacity of 100. How would one modify the network model to avoid adding a side constraint that limits either the sum of in-flows or the sum of the out-flows to 100? a. Place a limit of 34 on each in-flow arc. b. Add a side constraint limiting the out-flow arcs sum to 100. c. Separate the warehouse node into two nodes, connected by a single arc, with capacity of 100. d. It cannot be accomplished, a side constraint must be added. Copyright Cengage Learning. Powered by Cognero.
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ch 5 58. The arcs in a network indicate all of the following except? a. routes b. paths c. constraints d. connections 59. The idea that the total flow into a node must be consumed at a node and the remainder must flow out of a node is referred to as: a. the conservation of flow principle b. the node-arc incidence matrix c. a directed chain d. integrality constraint 60. For a network with n nodes, a spanning tree is a. a set of (n-1) arcs that connects all nodes and contains no loops b. a set of dummy arcs c. a set of n arcs that connects all nodes d. a random subset of arcs covering all nodes 61. The assignment problem is equivalent to a transportation problem with a. the sums of all rows and columns equal to one. b. binary decision variables. c. non-negativity constraints removed. d. all of the above. 62. The constraint X13 + X23 − X34 ≥ 50 indicates that a. 50 units are required at node 3. b. 50 units will be shipped from node 3. c. 50 units will be shipped in from node 1. d. 50 units must pass through node 3.
63. A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs $25 per hour while contracted labor costs $45 per hour. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. Draw the network flow model that captures this problem. 64. An oil company wants to produce lube oil, gasoline and diesel fuel at two refineries at the minimum cost. There are two sources of crude oil. The following network representation depicts this problem.
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ch 5
Write out the LP formulation for this problem. 65. A company needs to ship 100 units from Roanoke to Washington at the lowest possible cost. The costs associated with shipping between the cities are:
To From Roanoke Lexington Charlottesville
Lexington
Washington
Charlottesville
50 − −
− 50 30
80 40
Draw the network representation of this problem. 66. Draw the network and indicate how many units are flowing along each arc based on the following Analytic Solver Platform solution.
Units of Flow 5 35 0 25 5 15 0
From 1 1 2 3 3 4 5
A A B C C D E
B C D D E F F
Unit Cost 20 15 30 10 25 10 30
Total
1150
To 2 3 4 4 5 6 6
Nodes 1 2 3 4 5 6
A B C D E F
Net Flow −40 5 5 10 5 15
Supply/ Demand −40 5 5 10 5 15
67. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem. Copyright Cengage Learning. Powered by Cognero.
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ch 5 What values would you enter in the Analytic Solver Platform task pane for the following Excel spreadsheet? Objective Cell: Variables Cells: Constraints Cells:
A
B
C
D
E
F
G
H
I
J
K
L
M
Net Flow
Supply/ Demand −120 −60 0 0 75 50 25
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Flow from Node 1 Crude A 1 Crude A 2 Crude B 2 Crude B 3 Refinery 1 3 Refinery 1 3 Refinery 1 4 Refinery 2 4 Refinery 2 4 Refinery 2
Yield 0.90 0.85 0.80 0.85 0.95 0.90 0.90 0.90 0.95 0.95
Flow into Node 3 Refinery 1 4 Refinery 2 3 Refinery 1 4 Refinery 2 5 Lube Oil 6 Gasoline 7 Diesel 5 Lube Oil 6 Gasoline 7 Diesel
Unit Cost 15 13 9 11 4 7 8 3 9 6
1 2 3 4 5 6 7
Nodes Crude A Crude B Refinery 1 Refinery 2 Lube Oil Gasoline Diesel
Total cost
68. A company wants to determine the optimal replacement policy for its delivery truck. New trucks cost $30,000. The company does not keep trucks longer than 2 years and has estimated the annual operating costs and trade-in values for trucks during each of the 2 years as:
Age in years 0-1
1-2
Operating Cost
$15,000
$16,500
Trade-in Value
$20,000
$16,000
Draw the network representation of this problem. 69. Solve the following minimal spanning tree problem starting at node 1.
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70. A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are:
To From
Portland
Spokane
Salt Lake City
Denver
Seattle
100
500
600
−
Portland
−
350
300
−
Spokane
−
−
250
200
Salt Lake City
−
−
−
200
What values would you enter in the Analytic Solver Platform task pane for the following Excel spreadsheet? Objective Cell: Variables Cells: Constraints Cells:
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B
C
Ship
D
E
From
1 1 1 2 2 3 3 4
SEA SEA SEA POR POR SPO SPO SLC
F
To
2 3 4 3 4 4 5 5
G
H
I
Unit Cost
POR SPO SLC SPO SLC SLC DEN DEN
100 500 600 350 300 250 200 200
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J
Nodes
1 2 3 4 5
SEA POR SPO SLC DEN
K
L
Net Flow
Supply/ Demand
−100 0 0 0 100
−100 0 0 0 100
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ch 5 15
Total cost
71. A company wants to manage its distribution network which is depicted below. Identify the supply, demand and transshipment nodes in this problem.
72. Draw the network representation of the following network flow problem. MIN: Subject to:
5 X12 + 3 X13 + 2 X14 + 3 X24 + 2 X34 −X12 − X13 − X14 = −10 X12 − X24 = 2 X13 − X34 = 3 X14 + X24 + X34 = 5 Xij ≥ 0 for all i and j
73. Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table:
Distance Center 1 Center 2 Center 3 Plant A 45 60 53 Plant B 81 27 49 Plant C 55 40 35 Demand 350 325 400 Draw the transportation network for Clifton's distribution problem.
Center 4 75 62 60 375
Supply 500 700 650
74. Solve the following minimal spanning tree problem starting at node 1.
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75. Solve the following minimal spanning tree problem starting at node 1.
76. A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are listed below. Also, the decision variable associated with each pair of cities is shown next to the cost. To From
Portland
Spokane
Salt Lake City
Denver
Seattle
100 (X12)
500 (X13)
600 (X14)
−
Portland
−
350 (X23)
300 (X24)
−
Spokane Salt Lake City
−
−
250 (X34)
200 (X35)
−
−
−
200 (X45)
Write out the LP formulation for this problem. 77. The following network depicts a transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.
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78. The following network depicts a balanced transportation/distribution problem for Clifton Distributing. Formulate the LP for Clifton assuming they wish to minimize the total product-miles incurred.
79. Project 5.1 − Recruit Training You are a military training analyst in charge of initial training for the XXX career field and must decide how to best train the new recruits to satisfy the requirements for skilled recruits. There are six different courses (A, B, C, D, E, F) used for training in the XXX career field and four different sequences of courses that can be taken to achieve the required skill level. These sequences are A-E, B, C-F, and A-D-F. The table below provides information on the six courses.
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ch 5 Course Cost Per Student Min. Num. of Trainees Max. Num. of Trainees A 25 15 40 B 55 10 50 C 30 15 50 D 10 15 50 E 20 10 50 F 15 10 50 There are 100 recruits available for training and a demand for 100 skilled recruits. Assume all recruits pass each course and that you are trying to put students in classes in order to minimize the total cost of training. Assume non-integer solutions are acceptable. Further, assume each course will be held.
a.
Draw a network flow diagram describing the problem.
b.
Formulate the associated network flow linear program.
c.
Implement a spreadsheet model and use Risk Solver Platform (RSP) to obtain a solution to the problem. Use your model to answer the following questions.
What is the expected student load for each course? Should any course be expanded? Should any course or sequence be considered for elimination? Next, assume that not all students pass each course. In fact only 90% of the students pass courses A, E, and F and only 95% of the students pass courses B, C, and D. Each course is considered independent. The requirement for 100 skilled recruits remains. Your job is now to determine the number of recruits to place into the training program to obtain the 100 trained recruits while continuing to minimize the total cost of training.
d.
Re-draw the network flow diagram describing the problem to accommodate the above changes.
e.
Formulate the associated generalized network flow linear program.
f.
Implement a spreadsheet model of this changed model and use Risk Solver Platform (RSP) to obtain a solution to the expanded problem. How many recruits are needed and what is the change in total training cost?
80. The following network depicts a balanced assignment/transportation problem for Joe Fix's repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences.
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81. Draw the network representation of this LP model. What type of problem is it? MAX Subject to:
X41 X41 − X12 − X13 = 0 X12 − X24 = 0 X13 − X34 = 0 X24 + X34 − X41 = 0 0 ≤ X12 ≤ 5, 0 ≤ X13 ≤ 4, 0 ≤ X24 ≤ 3, 0 ≤ X34 ≤ 2, 0 ≤ X41
82. A company wants to determine the optimal replacement policy for its photocopier. The company does not keep photocopiers longer than 4 years. The company has estimated the annual costs for photocopiers during each of the 4 years and developed the following network representation of the problem. Write out the LP formulation for this problem.
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83. Clifton Distributing has three plants and four distribution centers. The plants, their supply, the distribution centers, their demands, and the distance between each location is summarized in the following table:
Distance Center 1 Center 2 Center 3 Center 4 Plant A 45 60 53 75 Plant B 81 27 49 62 Plant C 55 40 35 60 Demand 350 325 400 375 Draw the balanced transportation network for Clifton's distribution problem.
Supply 500 700 650
84. Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is:
Plane 1 Plane 2 Plane 3 Repair Person 1 11 9 21 Repair Person 2 17 7 13 Repair Person 3 9 12 17 Repair Person 4 14 8 28 Repair Person 5 12 5 12 Draw the balanced network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule. 85. A manufacturing company has a pool of 50 labor hours. A customer has requested two products, Product A and Product B, and has requested 15 and 20 of each respectively. It requires 2 hours of labor to produce Product A and 3 hours of labor to produce Product B. The company can obtain up to 50 additional hours of labor if required. In-house labor costs $25 per hour while contracted labor costs $45 per hour. The following network flow model captures this problem.
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Write out the LP formulation for this problem. 86. A railroad needs to move the maximum amount of material through its rail network. The numbers on arcs represent maximum flow. Formulate the LP model to determine this maximum amount based on the following network diagram.
87. An oil company wants to create lube oil, gasoline and diesel fuel at two refineries. There are two sources of crude oil. The following Excel spreadsheet shows this problem. What formula should be entered in cell E6 (and copied to cells E7:E15) in this spreadsheet?
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
B
C
D
Flow from Node 1 Crude A 1 Crude A 2 Crude B 2 Crude B 3 Refinery 1 3 Refinery 1 3 Refinery 1 4 Refinery 2 4 Refinery 2 4 Refinery 2
Yield 0.90 0.85 0.80 0.85 0.95 0.90 0.90 0.90 0.95 0.95
E
F
G
Flow into Node 3 Refinery 1 4 Refinery 2 3 Refinery 1 4 Refinery 2 5 Lube Oil 6 Gasoline 7 Diesel 5 Lube Oil 6 Gasoline 7 Diesel
H
Unit Cost 15 13 9 11 4 7 8 3 9 6
I
J
K
1 2 3 4 5 6 7
Nodes Crude A Crude B Refinery 1 Refinery 2 Lube Oil Gasoline Diesel
L
M
Net Flow
Supply/ Demand −120 −60 0 0 75 50 25
Total cost
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ch 5 88. Joe Fix plans the repair schedules each day for the Freeway Airline. Joe has 3 planes in need of repair and 5 repair personnel at his disposal. Each plane requires a single repairperson, except plane 3, which needs 2 personnel. Anyone not assigned to maintaining an airplane works in the maintenance shop for the day (not modeled). Each repairperson has different likes and dislikes regarding the types of repairs they prefer. For each plane, Joe has pulled the expected maintenance and determined the total preference matrix for his repair personnel. The preference matrix is:
Plane 1 Plane 2 Plane 3 Repair Person 1 11 9 21 Repair Person 2 17 7 13 Repair Person 3 9 12 17 Repair Person 4 14 8 28 Repair Person 5 12 5 12 Draw the network flow for this assignment problem assuming Joe would like to maximize the total preference in his worker-to-aircraft schedule. 89. A company needs to ship 100 units from Seattle to Denver at the lowest possible cost. The costs associated with shipping between the cities are: To From
Portland
Spokane
Salt Lake City
Denver
Seattle Portland Spokane
100 − −
500 350 −
600 300 250
− − 200
Salt Lake City
−
−
−
200
What values should go into cells G6:L13 in the following Excel spreadsheet?
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
Ship
D
E
From
1 1 1 2 2 3 3 4
SEA SEA SEA POR POR SPO SPO SLC
F
To
2 3 4 3 4 4 5 5
G
H
I
Unit Cost
POR SPO SLC SPO SLC SLC DEN DEN
J
Nodes
1 2 3 4 5
K
L
Net Flow
Supply/ Demand
SEA POR SPO SLC DEN
Total cost
90. The following network depicts an assignment/transportation problem for Joe Fix's repair scheduling problem. Formulate the LP for Joe assuming he wishes to maximize the total repairperson to plane assignment preferences. Copyright Cengage Learning. Powered by Cognero.
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91. Draw the network and show the solution for the maximal flow problem represented by the following Excel spreadsheet.
Units of Flow 4 8 4 0 4 4 8 12
From 1 1 2 2 3 3 4 5
12
Maximal flow
To 2 3 4 5 4 5 5 1
A A B B C C D E
B C D E D E E A
Upper Bound 4 8 6 2 4 5 9 999
Nodes 1 2 3 4 5
A B C D E
Net Flow 0 0 0 0 0
Supply/ Demand 0 0 0 0 0
92. A trucking company wants to find the quickest route from Seattle to Denver. What values should be placed in cells L6:L10 of the following Excel spreadsheet?
A 1 2 3 4 5 6
B
C
Select Route
0
D
E
From
1
F
To
SEA
2
POR
G
H
I
Driving Time
3
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J
Nodes
1
SEA
K
L
Net Flow
Supply/ Demand
−1 Page 24
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ch 5 7 8 9 10 11 12 13 14 15 16 17
0 1 0 0 0 0 0 0 1
1 1 1 2 2 2 3 3 4
SEA SEA SEA POR POR POR SPO SPO SLC
Total Driving Time
3 4 5 3 4 5 4 5 5
SPO SLC DEN SPO SLC DEN SLC DEN DEN
4 12 18 9 12 16 10 15 5
2 3 4 5
POR SPO SLC DEN
0 0 0 1
17
93. Project 5.2 − Small Production Planning Project (Fixed Charge Problem via Network Flow with Side Constraints) Jack Small Enterprises runs two factories in Ohio, one in Toledo and one in Centerville. His factories produce a variety of products. Two of his product lines are polished wood clocks which he adorns with a regional theme. Naturally, clocks popular in the southwest are not as popular in the northeast, and vice versa. Each plant makes both of the clocks. These clocks are shipped to St Louis for distribution to the southeast and western states and to Pittsburgh for distribution to the south and northeast. Jack is considering streamlining his plants by removing certain production lines from certain plants. Among his options is potentially eliminating the clock production line at either the Toledo or the Centerville plant. Each plant carries a fixed operating cost for setting up the line and a unit production cost, both in terms of money and factory worker hours. This information is summarized in the table below. Production Cost
Clocks Produced
per Clock
per Hour
Available
Fixed Cost for
Southwest
Northeast
Southwest
Northwest
Hours per
Plant
Line
Clock
Clock
Clock
Clock
Month
Toledo Centerville
$20,000 $24,000
$10 $9
$12 $13
5 5.5
6 6.2
500 675
The Southwest clocks are sold for $23 each and the Northwest clocks are sold for $25 each. Demand rates used for production planning are 1875 Southwest clocks for sale out of the St Louis distribution center and 2000 Northeast clocks for sale out of the Pittsburgh distribution center. Assume all these units are sold. The per clock transportation costs from plant to distribution center is given in the following table.
(cost per clock shipped) Cost to Ship to Distribution Center Plant St Louis Pittsburgh Toledo $2 $4 Centerville $3 $2 Develop a generalized network flow model for this problem and implement this model in solver. Use the model to answer the following questions.
a.
Should any of the production lines be shut down?
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b.
How should worker hours be allocated to produce the clocks to meet the demand forecasts? Are there any excess hours, and if so how many?
c.
What is the expected monthly profit?
d.
If a plant is closed, what are the estimated monthly savings?
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ch 5 Answer Key 1. True 2. False 3. True 4. True 5. False 6. True 7. True 8. True 9. a 10. a 11. d 12. c 13. d 14. a 15. a 16. a 17. c 18. b 19. c 20. b 21. b 22. c 23. d 24. a 25. c Copyright Cengage Learning. Powered by Cognero.
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ch 5 26. d 27. a 28. c 29. b 30. a 31. b 32. a 33. a 34. a 35. a 36. b 37. b 38. a 39. d 40. d 41. a 42. a 43. c 44. b 45. c 46. c 47. a 48. d 49. a 50. d 51. a Copyright Cengage Learning. Powered by Cognero.
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ch 5 52. c 53. b 54. a 55. b 56. a 57. c 58. c 59. a 60. a 61. d 62. a
63. 64. MIN: Subject to:
15X13 + 13X14 + 9X23 + 11X24 + 4X35 + 7X36 + 8X37 + 3X45 + 9X46 + 6X47 −X13 − X14 = −100 −X23 − X24 = −50 0.80X13 + 0.95X23 − X35 − X36 − X37 = 0 0.85X14 + 0.85X24 − X45 − X46 − X47 = 0 0.95X35 + 0.90X45 = 50 0.90X36 + 0.95X46 = 25 0.90X37 + 0.95X47 = 75 Xij ≥ 0
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65.
66. 67. Objective Cell: H17 Variables Cells: A6:A15 Constraints Cells: A6:A15 ≥ 0 L6:L12 ≥ M6:M12
68. 69. Copyright Cengage Learning. Powered by Cognero.
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ch 5 Arc 1−2 2−4 3−5 4−5 5−6 Total
Value 4 3 2 4 6 19
70. Objective Cell: G15 Variables Cells: B6:B13 Constraints Cells: B6:B13 ≥ 0 K6:K10 = L6:L10 71. Supply Demand Transsshipment
1 6 2, 3, 4, 5
72.
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73. 74. Arc 1−2 1−3 2−4 3−5 5−6 Total
Value 5 4 6 4 2 21
75. Arc 1−2 2−5 3−5 3−4 Total
Value 4 2 5 4 15
76. MIN: Subject to:
77. Let
100 X12 + 500 X13 + 600 X14 + 350 X23 + 300 X24 + 250 X34 + 200 X35 + 200 X45 −X12 − X13 − X14 ≥ −100 X12 − X23 − X24 = 0 X13 − X34 − X35 = 0 X14 + X24 + X34 − X45 = 0 X35 + X45 ≥ 100 Xij ≥ 0
Xij = flow from plant i (A, B, or C) to distribution center j (Center 1, 2, 3, or 4).
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ch 5 MIN: Subject to:
78. Let MIN: Subject to:
45 X14 + 60X15 + 53X16 + 75X17 + 81X24 + 27X25 + 49X26 + 62X27 + 55X34 + 40X35 + 35X36 + 60X37 −X14 − X15 − X16 − X17 ≥ −500 −X24 − X25 − X26 − X27 ≥ −700 −X34 − X35 − X36 − X37 ≥ −650 X14 + X24 + X34 ≥ 350 X15 + X25 + X35 ≥ 325 X16 + X26 + X36 ≥ 400 X17 + X27 + X37 ≥ 375 All Xij ≥ 0
Xij = flow from plant i (A, B, or C) to distribution center j (Center 1, 2, 3, 4 and 5 (dummy)). 45X14 + 60X15 + 53X16 + 75X17 + 81X24 + 27X25 + 49X26 + 62X27 + 55X34 + 40X35 + 35X36 + 60X37 −X14 − X15 − X16 − X17 − X18 = −500 −X24 − X25 − X26 − X27 − X28 = −700 −X34 − X35 − X36 − X37 − X38 = −650 X14 + X24 + X34 = 350 X15 + X25 + X35 = 325 X16 + X26 + X36 = 400 X17 + X27 + X37 = 375 X18 + X28 + X38 = 400 All Xij ≥ 0
79. a. Draw a network flow diagram describing the problem.
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b.
Formulate the associated network flow linear program. Minimize Subject to:
c.
25 X1A + 55 X1B + 30 X1C + 10 X1D + 20 X1E + 15 X1F −X1A − X1B − X1C = −100 X1A − X1E − X1D = 0 X1B − XB,TP = 0 X1C − X1F = 0 XAD − XDF = 0 XAE − XE.TP = 0 XCF + XDF − XF,TP = 0 XB,TP + XE,TP + XF,TP = 100 10 ≤ X1A ≤ 50 15 ≤ X1A ≤ 40 10 ≤ X1B ≤ 50 15 ≤ X1C ≤ 50 10 ≤ XAE ≤ 50 15 ≤ XAD ≤ 50 10 ≤ XB,TP ≤ 50 10 ≤ XCF ≤ 50 10 ≤ XDF ≤ 50 10 ≤ XE,TP ≤ 50 10 ≤ XF,TP ≤ 50 Implement your model in Excel and solve the model. Answer each of the following: Load for each course: A − 40 D − 15
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ch 5 B − 25 C − 35
E − 25 F − 50
Should any course be expanded: Courses A and F are running at capacity Should any course or sequence be considered for elimination: Sequence A-D-F. Course D is at a minimum level. This minimum forces underutilization of course B.
d.
(5 points) Re-draw and properly label the network flow diagram of part (a) to accommodate the above changes.
e.
Formulate this modified model. Minimize
25 X1A + 55 X1B + 30 X1C + 10 X1D + 20 X1E + 15 X1F
Subject to:
−X1A − X1B − X1C ≥ 0 0.90 X1A − X1E − X1D = 0 0.95 X1B − XB,TP = 0 0.95 X1C − X1F = 0
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ch 5 0.95 XAD − XDF = 0 0.90 XAE − XE.TP = 0 0.90 XCF + XDF − XF,TP = 0 XB,TP + XE,TP + XF,TP = 100 15 ≤ X1A ≤ 40 10 ≤ X1B ≤ 50 15 ≤ X1C ≤ 50 10 ≤ XAE ≤ 50 15 ≤ XAD ≤ 50 10 ≤ XCF + XDF ≤ 50 f.
Implement this changed model in Excel and solve. How many recruits are needed and what is the change in total training cost? A total of 116 recruits are needed, increasing costs to $5,538.95.
80. Let MIN: Subject to:
Xij = assignment of repairperson i (1, 2, 3, 4, or 5) to plane j (1, 2, or 3). 11X16 + 9X17 + 21X18 + 17X26 + 7X27 + 13X28 + 9X36 + 12X37 + 17X38 + 14X46 + 8X47 + 28X48 + 12X56 + 5X57 + 12X58 −X16 − X17 − X18 − X19 = −1 −X26 − X27 − X28 − X29 = −1 −X36 − X37 − X38 − X39 = −1 −X46 − X47 − X48 − X49 = −1 −X56 − X57 − X58 − X59 = −1 X16 + X26 + X36 + X46 + X56 = 1 X17 + X27 + X37 + X47 + X57 = 1 X18 + X28 + X38 + X48 + X58 = 2 X19 + X29 + X39 + X49 + X59 = 1 All Xij ≥ 0
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ch 5 81. It is a maximal flow problem.
82. MIN: Subject to:
26 X12 + 50 X13 + 25 X23 + 50 X24 + 28 X34 + 40 X35 + 17 X45 −X12 − X13 = −1 X12 − X23 − X24 = 0 X13 + X23 − X34 − X35 = 0 X24 + X34 − X45 = 0 X45 + X35 = 1
83.
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ch 5
84. 85. MINIMIZE Subject to:
86. MAX: Subject to:
25X13 + 25X14 + 45X23 + 45X24 −X13 − X14 ≥ −50 −X23 − X24 ≥ −50 0.50X13 + 0.50X23 ≥ 15 0.33X14 + 0.33X24 ≥ 20 Xij ≥ 0 for i = 1,2; j = 3,4
X61 X61 − X12 − X13 = 0 X12 − X25 = 0 X13 − X35 − X34 = 0 X34 − X46 = 0 X25 + X35 − X56 = 0 X56 + X46 − X61 = 0 0 ≤ X12 ≤ 5 0 ≤ X25 ≤ 6 0 ≤ X34 ≤ 6 0 ≤ X46 ≤ 2
0 ≤ X13 ≤ 4 0 ≤ X35 ≤ 6 0 ≤ X56 ≤ 8 0 ≤ X61
87. D6*A6, copied to E7:E15
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88. 89. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
90. Let MIN: Subject to:
B
C
Ship
D
E
From
1 1 1 2 2 3 3 4
F
To
SEA SEA SEA POR POR SPO SPO SLC
2 3 4 3 4 4 5 5
G
H
I
Unit Cost
POR SPO SLC SPO SLC SLC DEN DEN
100 500 600 350 300 250 200 200
J
Nodes
1 2 3 4 5
SEA POR SPO SLC DEN
K
L
Net Flow
Supply/ Demand
−100 0 0 0 100
−100 0 0 0 100
Total cost
Xij = assignment of repairperson i (1, 2, 3, 4, or 5) to plane j (1, 2, or 3). 11X16 + 9X17 + 21X18 + 17X26 + 7X27 + 13X28 + 9X36 + 12X37 + 17X38 + 14X46 + 8X47 + 28X48 + 12X56 + 5X57 + 12X58 −X16 − X17 − X18 ≥ −1 −X26 − X27 − X28 ≥ −1 −X36 − X37 − X38 ≥ −1 −X46 − X47 − X48 ≥ −1
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ch 5 −X56 − X57 − X58 ≥ −1 X16 + X26 + X36 + X46 + X56 = 1 X17 + X27 + X37 + X47 + X57 = 1 X18 + X28 + X38 + X48 + X58 = 2 All Xij ≥ 0
91. 92. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
B
C
Select Route
0 0 1 0 0 0 0 0 0 1
D
E
From
1 1 1 1 2 2 2 3 3 4 Total Driving Time
F
To
SEA SEA SEA SEA POR POR POR SPO SPO SLC
2 3 4 5 3 4 5 4 5 5
POR SPO SLC DEN SPO SLC DEN SLC DEN DEN
G
H
I
Driving Time
3 4 12 18 9 12 16 10 15 5
J
Nodes
1 2 3 4 5
SEA POR SPO SLC DEN
K
L
Net Flow
Supply/ Demand
−1 0 0 0 1
−1 0 0 0 1
17
93. The following network diagram captures the Small Production problem:
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The decision whether to keep open or close a plant is captured in the early arcs, which are modeled as binary, flow equal zero or flow equal one. Once opened, the flow opening that plant is transformed into the hours available at that plant. These hours are then used to produce either the Southwest Clock (product 1) or the Northeast Clock (product 2). Any unused hours flow into the collection node, Unused Hours. These products has a per unit production cost captured by the next set of flows. Distribution costs and the distribution plan are captured in the arcs between the Prod # Cost nodes and the St Louis and Pittsburgh nodes. Finally, sales of the products are captured in the flows from the warehouses to the Prod # Sales nodes. These final nodes indicate the product demand set for the problem. This problem is essentially a generalized network flow problem with side constraints. Analysis of this problem indicates closing Toledo. Both clock products are produced at Centerville. The 675 hours available at Centerville are allocated as follows: 340.9 hours for the Southwest Clock, 322.6 hours for the Northeast Clock, and the remaining 11.5 hours unused. This yields an expected monthly profit of $16,625 on these clock products. This is a savings of $20,000. With both plants open, the expected loss per month is $3,375 as Toledo will produce the Southwest Clocks leaving an excess of 125 hours and Centerville will produce the Northeast Clocks leaving an excess of 352.4 hours.
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ch 6
Indicate whether the statement is true or false. 1. Hard constraints can be violated, if necessary a. True b. False 2. Linear programming problems cannot have multiple objectives a. True b. False 3. In the goal programming problem, the weights, wi, attached to deviational variables must decay exponentially a. True b. False 4. The second step in Goal Programming is defining the goals a. True b. False 5. Trade-offs in goal programming can be made by modifying the weights to satisfy the priorities of a decision maker a. True b. False 6. The deviational variables represent the amount by which the goal's target is underachieved a. True b. False 7. In MOLP, a decision alternative is dominated if another alternative produces a better value of at least one objective without worsening the value of other objectives a. True b. False 8. Goal Programming and Multiple Objective Optimization are not related a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. Suppose that the first goal in a GP problem is to make 3 X1 + 4 X2 approximately equal to 36. Using the deviational variables d1− and d1+, the following constraint can be used to express this goal. 3 X1 + 4 X2 + d1− − d1+ = 36 If we obtain a solution where X1 = 6 and X2 = 2, what values do the deviational variables assume? a. d1− = 0, d1+ = 10 b. d1− = 10, d1+ = 0 Copyright Cengage Learning. Powered by Cognero.
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ch 6 c. d1− = 5, d1+ = 5 d. d1− = 6, d1+ = 0 Exhibit 7.1 The following questions are based on the problem below. A company wants to advertise on TV and radio. The company wants to produce about 6 TV ads and 12 radio ads. Each TV ad costs $20,000 and is viewed by 10 million people. Radio ads cost $10,000 and are heard by 7 million people. The company wants to reach about 140 million people, and spend about $200,000 for all the ads. The problem has been set up in the following Excel spreadsheet.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A Problem Data Cost Coverage
B TV 20 10
C Radio 10 7
D
E
Goal Constraints Actual Amount +Under − Over = Goal Target Value
TV 0 0 0 0 6
Radio 0 0 0 0 12
Cost
Coverage
0 0 0 200
0 0 0 140
Percentage Deviation: Under Over
1 0
1 0
1 0
1 0
Weights Under Over Objective
0
10. Refer to Exhibit 7.1. What formula goes in cell B9? a. =SUM(B6:B8) b. =B6+B7-B8 c. =B6-B7+B8 d. =B10-B8 11. Refer to Exhibit 7.1. Which of the following is a constraint specified to Analytic Solver Platform for this model? a. $B$9:$E$9=$B$6:$E$6 b. $B$9:$E$9<$B$10:$E$10 c. $B$9:$E$9=$B$10:$E$10 d. $B$9:$E$9>$B$10:$E$10 12. One major advantage of goal programming (GP) is that the technique: Copyright Cengage Learning. Powered by Cognero.
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ch 6 a. allows a decision-maker to jointly examine several objectives b. multiple objectives can be assigned different weights depending on their relative importance c. can focus on a single objective, if necessary d. all of the above 13. Goal programming problems a. typically include a set of multiple goals b. cannot include hard constraints c. consist of soft constraints only d. must have a single objective function 14. A manager wants to ensure that he does not exceed his budget by more than $1000 in a goal programming problem. If the budget constraint is the third constraint in the goal programming problem which of the following formulas will best ensure that the manager's objective is met? a. MIN d3+ b. d3− ≥ 1000 c. d3+ = 1000 d. d3+ ≤ 1000 15. What is the meaning of the ti term in this objective function for a goal programming problem?
a. The time required for each decision variable. b. The percent of goal i met. c. The coefficient for the ith decision variable d. The target value for goal i. 16. A hard constraint a. cannot be violated b. may be violated c. is always binding d. is always part of the feasible solution 17. MINIMAX solutions to multi-objective linear programming (MOLP) problems are a. dually optimal. b. Pareto optimal. c. suboptimal. d. maximally optimal. 18. A constraint which represents a target value for a problem is called a a. fuzzy constraint. b. vague constraint. Copyright Cengage Learning. Powered by Cognero.
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ch 6 c. preference constraint d. soft constraint Exhibit 7.3 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The following minimax formulation of the problem has been solved in Excel.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
Goals Average return Average risk
Actual 0 0
Target 11.8% 0.22
Weights 1 1
Objective:
0
E
Weighted % Deviation 0 0
19. Refer to Exhibit 7.3. Which value should the investor change, and in what direction, if he wants to reduce the risk of the portfolio? a. D11, increase b. D12, increase c. C12, increase d. D12, decrease 20. If no other feasible solution to a multi-objective linear programming (MOLP) problem allows an increase in any objective without decreasing at least one other objective, the solution is said to be a. dually optimal. b. Pareto optimal. c. suboptimal. d. maximally optimal. Exhibit 7.2 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% Copyright Cengage Learning. Powered by Cognero.
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ch 6 and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The following multi-objective linear programming (MOLP) has been solved in Excel.
1 2 3 4 5 6 7 8 9 10 11
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
Objectives: Average return Average risk
0 0
21. Refer to Exhibit 7.2. Which cells are the changing cells in this model? a. $B$6:$C$6, $B$10:$B$11 b. $B$6:$C$6 c. $B$6:$D$6 d. $B$10:$B$11 22. Suppose that environmental and human variables are assigned the weight of zero. Then the "triple bottom line" approach reduces to: a. profit maximization b. environmental issues c. HR objectives d. achieving social equilibrium Exhibit 7.3 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The following minimax formulation of the problem has been solved in Excel.
1 2 3 4 5 6 7 8
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
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ch 6 9 10 11 12 13 14
Goals Average return Average risk
Actual 0 0
Objective:
0
Target 11.8% 0.22
Weights 1 1
Weighted % Deviation 0 0
23. Refer to Exhibit 7.3. What formula goes in cell E11? a. =D11*(C11−B11)/C11 b. =(C11−B11)/C11 c. =D11*C11 d. =D11*(C11−B11) 24. The primary benefit of a MINIMAX objective function is a. it yields any feasible solution by changing the weights. b. it is limited to all corner points. c. it yields a larger variety of solutions than generally available using an LP method. d. it makes many of the deviational variables equal to zero. 25. Deviational variables a. are added to constraints to indicate acceptable departures from the target values of their corresponding goals b. are negative c. are positive for underachievement only d. are positive for overachievement only 26. The "triple bottom line" incorporates multiple objective decision-making by: a. simultaneously considering profit, people and planet b. environmental issues only c. financial objectives only d. wealth redistribution in the society 27. An optimization technique useful for solving problems with more than one objective function is a. dual programming. b. sensitivity analysis. c. multi-objective linear programming. d. goal programming. 28. Goal programming (GP) is: a. iterative b. inaccurate c. static d. all of the above 29. Suppose that the first goal in a GP problem is to make 3 X1 + 4 X2 approximately equal to 36. Using the deviational variables d1− and d1+, what constraint can be used to express this goal? Copyright Cengage Learning. Powered by Cognero.
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ch 6 a. 3 X1 + 4 X2 + d1− − d1+ ≤ 36 b. 3 X1 + 4 X2 − d1− − d1+ = 36 c. 3 X1 + 4 X2 + d1− + d1+ = 36 d. 3 X1 + 4 X2 + d1− − d1+ = 36 30. In the "triple bottom line" the term "people" refers to: a. social responsibility issues b. environmental issues c. financial objectives d. all of the above Exhibit 7.2 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The following multi-objective linear programming (MOLP) has been solved in Excel.
1 2 3 4 5 6 7 8 9 10 11
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
Objectives: Average return Average risk
0 0
31. Refer to Exhibit 7.2. What Analytic Solver Platform constraint involves cells $B$6:$C$6? a. $B$6:$C$6=$B$7:$C$7 b. $B$6:$C$6≥$B$7:$C$7 c. $B$6:$C$6≤$B$7:$C$7 d. $B$6:$C$6=$D$7 32. Goal programming solution feedback indicates that the d4+ level of 50 should not be exceeded in future solution iterations. How should you modify your goal constraint 40 X1 + 20 X2 + d4− + d4+ = 300 to accommodate this requirement? a. Increase the RHS value from 300 to 350. Copyright Cengage Learning. Powered by Cognero.
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ch 6 b. Replace the constraint with 40 X1 + 20 X2 ≤ 350. c. Do not modify the constraint, add a constraint d4+ ≤ 50. d. Do not modify the constraint, add a constraint d4+ = 50. 33. The RHS value of a goal constraint is referred to as the a. target value. b. constraint value. c. objective value. d. desired value. Exhibit 7.1 The following questions are based on the problem below. A company wants to advertise on TV and radio. The company wants to produce about 6 TV ads and 12 radio ads. Each TV ad costs $20,000 and is viewed by 10 million people. Radio ads cost $10,000 and are heard by 7 million people. The company wants to reach about 140 million people, and spend about $200,000 for all the ads. The problem has been set up in the following Excel spreadsheet.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A Problem Data Cost Coverage
B TV 20 10
C Radio 10 7
D
E
Goal Constraints Actual Amount +Under − Over = Goal Target Value
TV 0 0 0 0 6
Radio 0 0 0 0 12
Cost
Coverage
0 0 0 200
0 0 0 140
Percentage Deviation: Under Over
1 0
1 0
1 0
1 0
Weights Under Over Objective
0
34. Refer to Exhibit 7.1. If the company is very concerned about going over the $200,000 budget, which cell value should change and how should it change? a. D13, increase b. D13, decrease c. D14, increase d. D14, decrease Copyright Cengage Learning. Powered by Cognero.
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ch 6 35. The decision maker has expressed concern with Goal 1, budget, achievement. He indicated that future candidate solutions should stay under budget. How can you modify your goal programming model to accommodate this change? a. Make budget a hard constraint in the model. b. Give d1+ an extremely large weight in the objective function. c. Remove d1+ from the goal constraint. d. All of these. Exhibit 7.2 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The following multi-objective linear programming (MOLP) has been solved in Excel.
1 2 3 4 5 6 7 8 9 10 11
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
Objectives: Average return Average risk
0 0
36. Refer to Exhibit 7.2. Which cell(s) is(are) the target cells in this model? a. $B$6:$C$6, $B$10:$B$11 b. $B$6:$C$6 c. $B$6:$D$6 d. $B$10:$B$11 37. The di+ variable indicates the amount by which each goal's target value is a. missed. b. underachieved. c. overachieved. d. overstated. 38. Decision-making problems which can be stated as a collection of desired objectives are known as what type of problem? a. A non-linear programming problem. b. An unconstrained programming problem. Copyright Cengage Learning. Powered by Cognero.
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ch 6 c. A goal programming problem. d. An integer programming problem. 39. Which of the following is false regarding a goal constraint? a. A goal constraint allows us to determine how close a given solution comes to achieving a goal. b. A goal constraint will always contain two deviational variables. c. Deviation variables are non-negative. d. If two deviation variables are used in a constraint at least one will have a value of zero. 40. Which of the following formulas is a deviation-minimizing objective function for a goal programming problem? a. b. c. d. 41. Goal programming differs from linear programming or integer linear programming is that a. goal programming provides for multiple objectives. b. goal programming excludes hard constraints. c. with goal programming we iterate until an acceptable solution is obtained. d. goal programming requires fewer variables. Exhibit 7.2 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The following multi-objective linear programming (MOLP) has been solved in Excel.
1 2 3 4 5 6 7 8 9 10 11
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
Objectives: Average return Average risk
0 0
42. Refer to Exhibit 7.2. What formula goes in cell B11? a. =SUMPRODUCT(B2:C2,$B$6:$C$6)/$D$7 Copyright Cengage Learning. Powered by Cognero.
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ch 6 b. =B2*C2+B3*C3 c. =SUMPRODUCT(B3:C3,$B$6:$C$6)/$D$7 d. =SUMPRODUCT(B3:C3,$B$6:$C$6) Exhibit 7.1 The following questions are based on the problem below. A company wants to advertise on TV and radio. The company wants to produce about 6 TV ads and 12 radio ads. Each TV ad costs $20,000 and is viewed by 10 million people. Radio ads cost $10,000 and are heard by 7 million people. The company wants to reach about 140 million people, and spend about $200,000 for all the ads. The problem has been set up in the following Excel spreadsheet.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A Problem Data Cost Coverage
B TV 20 10
C Radio 10 7
D
E
Goal Constraints Actual Amount +Under − Over = Goal Target Value
TV 0 0 0 0 6
Radio 0 0 0 0 12
Cost
Coverage
0 0 0 200
0 0 0 140
Percentage Deviation: Under Over
1 0
1 0
1 0
1 0
Weights Under Over Objective
0
43. Refer to Exhibit 7.1. Which cell(s) is(are) the objective cell(s) in this model? a. $B$20 b. $D$6 c. $E$6 d. $B$13:$E$14, $B$9:$E$9 44. What weight would be assigned to a neutral deviational variable? a. 0 b. 1 c. 10 d. 100 45. Goal programming (GP) is typically Copyright Cengage Learning. Powered by Cognero.
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ch 6 a. a minimization problem of the sum of weighted percentage deviations b. a maximization problem of positive deviations only c. a minimization problem of negative deviations only d. a maximization problem of continuous goals Exhibit 7.1 The following questions are based on the problem below. A company wants to advertise on TV and radio. The company wants to produce about 6 TV ads and 12 radio ads. Each TV ad costs $20,000 and is viewed by 10 million people. Radio ads cost $10,000 and are heard by 7 million people. The company wants to reach about 140 million people, and spend about $200,000 for all the ads. The problem has been set up in the following Excel spreadsheet.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A Problem Data Cost Coverage
B TV 20 10
C Radio 10 7
D
E
Goal Constraints Actual Amount +Under − Over = Goal Target Value
TV 0 0 0 0 6
Radio 0 0 0 0 12
Cost
Coverage
0 0 0 200
0 0 0 140
Percentage Deviation: Under Over
1 0
1 0
1 0
1 0
Weights Under Over Objective
0
46. Refer to Exhibit 7.1. What formula goes in cell D6? a. =SUMPRODUCT(B2:B3,B6:B7) b. =B2*C2+B6*C6 c. =SUMPRODUCT(B2:C2,B10:C10) d. =SUMPRODUCT(B2:C2,B6:C6) Exhibit 7.2 The following questions are based on the problem below. An investor has $150,000 to invest in investments A and B. Investment A requires a $10,000 minimum investment, pays a return of 12% and has a risk factor of .50. Investment B requires a $15,000 minimum investment, pays a return of 10% and has a risk factor of .20. The investor wants to maximize the return while minimizing the risk of the portfolio. The Copyright Cengage Learning. Powered by Cognero.
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ch 6 following multi-objective linear programming (MOLP) has been solved in Excel.
1 2 3 4 5 6 7 8 9 10 11
A Problem data Expected return Risk rating
B A 12% 0.50
C B 10% 0.20
D
Variables Amount invested Minimum required
A 0 $10,000
B 0 $15,000
Total 0 $150,000
Objectives: Average return Average risk
0 0
47. Refer to Exhibit 7.2. What formula goes in cell B10? a. =SUMPRODUCT(B2:C2,$B$6:$C$6)/$D$7 b. =B2*C2+B3*C3 c. =SUMPRODUCT(B3:C3,$B$6:$C$6)/$D$7 d. =SUMPRODUCT(B2:C2,$B$6:$C$6) 48. Suppose that profit and human variables are assigned the weight of zero. Then the "triple bottom line" approach reduces to: a. profit maximization only b. environmental considerations only c. HR objectives only d. achieving social happiness Exhibit 7.1 The following questions are based on the problem below. A company wants to advertise on TV and radio. The company wants to produce about 6 TV ads and 12 radio ads. Each TV ad costs $20,000 and is viewed by 10 million people. Radio ads cost $10,000 and are heard by 7 million people. The company wants to reach about 140 million people, and spend about $200,000 for all the ads. The problem has been set up in the following Excel spreadsheet.
1 2 3 4 5 6 7 8 9
A Problem Data Cost Coverage
B TV 20 10
C Radio 10 7
D
E
Goal Constraints Actual Amount +Under − Over = Goal
TV 0 0 0 0
Radio 0 0 0 0
Cost
Coverage
0 0 0
0 0 0
Copyright Cengage Learning. Powered by Cognero.
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ch 6 10 11 12 13 14 15 16 17 18 19 20
Target Value
6
12
200
140
Percentage Deviation: Under Over
1 0
1 0
1 0
1 0
Weights Under Over Objective
0
49. Refer to Exhibit 7.1. Which cells are the variable cells in this model? a. $B$6:$C$6, $B$7:$E$8 b. $B$6:$C$6 c. $B$9:$E$9 d. $B$6:$E$8 50. Which of the following is true regarding goal programming? a. The objective function is not useful when comparing goal programming solutions. b. We can place upper bounds on any of the deviation variables. c. A preemptive goal program involves deviations with arbitrarily large weights. d. All of these are true. 51. Which of the following are true regarding weights assigned to deviational variables? a. Larger weights are assigned to undesirable deviations from the respective goals. b. The weights assigned must sum to one. c. The weight assigned to the deviation under a particular goal must be the same as the weight assigned to the deviation above that particular goal. d. All weights must be nonzero. 52. A constraint which cannot be violated is called a a. binding constraint. b. hard constraint. c. definite constraint. d. required constraint. 53. What is the soft constraint form of the following hard constraint? 3X1 + 2 X2 ≤ 10 a. 3X1 + 2 X2 + d1− − d1+ = 10 b. 3X1 + 2 X2 + d1− + d1+ = 10 c. 3X1 + 2 X2 − d1− − d1+ ≤ 10 d. 3X1 + 2 X2 + d1− − d1+ ≥ 10 Copyright Cengage Learning. Powered by Cognero.
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ch 6 54. Suppose that X1 equals 4. What are the values for d1+ and d1− in the following constraint? X1 + d1−− d1+ = 8 a. d1− = 4, d1+ = 0 b. d1− = 0, d1+ = 4 c. d1− = 4, d1+ = 4 d. d1− = 8, d1+ = 0 55. Multi-objective linear programming (MOLP) provide a. a way to analyze LP problems with multiple conflicting objectives b. a way to incorporate soft constraints c. a way to incorporate hard constraints d. a simple way to solve the problem as a relaxed LP 56. A soft constraint a. represents a target a decision-maker would like to achieve b. is always tight c. cannot be violated d. typically represents a single goal 57. A MINIMAX objective function in goal programming (GP): a. is used to minimize the maximum deviation from a goal b. is captured in a decision table c. is estimated by trial-and-error d. often produces an infeasible solution 58. The MINIMAX objective a. yields the smallest possible deviations. b. minimizes the maximum deviation from any goal. c. chooses the deviation which has the largest value. d. maximizes the minimum value of goal attainment. 59. The di+, di− variables are referred to as a. objective variables. b. goal variables. c. target variables. d. deviational variables.
60. A company makes 2 products A and B from 2 resources. The products have the following resource requirements and produce the accompanying profits. The available quantity of resources is also shown in the table. Copyright Cengage Learning. Powered by Cognero.
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ch 6 Product 1 2 Labor (hr/unit) 3 2 Material (ounces/unit) 1 2 Profit($/unit) 7 6 Management has developed the following set of goals
Available resources 150 200
Goal 1: Produce approximately 40 units of product 1. Goal 2: Produce approximately 70 units of product 2. Goal 3: Achieve a profit over $400. Goal 4: Consume less than 150 hours of labor Goal 5: Consume less than 200 ounces of material Based on this GP formulation of the problem and the associated optimal integer solution what values should go in cells B2:F16 of the following Excel spreadsheet?
Let
X1 = number of product 1 X2 = number of product 2
MIN: Subject to:
d1− + d1+ + d2− + d2+ + d3− + d4+ + d5+ X1 + d 1 − − d 1 + = 40 X2 + d 2 − − d 2 + = 70 7 X1 + 6 X2 + d3− − d3+ = 400 3 X1 + 2 X2 + d4− − d4+ = 150 1 X1 + 2 X2 + d5− − d5+ = 200 Xi, di−, di+ ≥ 0 for all i
product 1 product 2 profit labor material
(X1, X2) = (4, 69) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A Problem Data Labor Material Profit Goal Constraints Actual Amount + Under − Over = Goal Target Value
B A
C B
D
E
F
A
B
Labor
Material
Profit
Weights Under Over Objective
Exhibit 7.4 Copyright Cengage Learning. Powered by Cognero.
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ch 6 The following questions are based on the problem below. Robert Gardner runs a small, local-only delivery service. His fleet consists of three smaller panel trucks. He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers. The box weights are: 210, 160, 320, 90, 110, 70, 410, 260, 170, 240, 80 and 180 for boxes 1 through 12, respectively. Since each truck differs each truck has different load capacities as given below:
Truck 1 2 3
Weight Capacity 800 pounds 900 pounds 700 pounds
Box Capacity 5 6 4
Cost per pound $0.34 $0.42 $0.25
Robert would like each truck equally loaded, both in terms of number of boxes and in terms of total weight, while minimizing his shipping costs. Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight. The following integer goal programming formulation applies to his problem. Y1 = weight loaded in truck 1; Y2 = weight loaded in truck 2; Y3 = weight loaded in truck 3; Xi,j = 0 if truck i not loaded with box j; 1 if truck i loaded with box j.
MIN 0.34Y1 + 0.42Y2 + 0.25Y3 + 50*(d1+ + d2+ + d3+) + 0.10*(d4− + d5− + d6−)*100 Subject to: Y1 = 210X11 + 160X12 + 320X13 + 90X14 + 110X15 + 70X16 + 410X17 + 260X18 + 170X19 + 240X1,10 + 80X1,11 + 180X1,12 Y2 = 210X21 + 160X22 + 320X23 + 90X24 + 110X25 + 70X26 + 410X27 + 260X28 + 170X29 + 240X2,10 + 80X2,11 + 180X2,12 Y3 = 210X31 + 160X32 + 320X33 + 90X34 + 110X35 + 70X36 + 410X37 + 260X38 + 170X39 + 240X3,10 + 80X3,11 + 180X3,12 Y1 ≤ 800 Y2 ≤ 900 Y3 ≤ 700 Y1 + d1− − d1+ = 767 Y2 + d2− − d2+ = 767 Y3 + d3− − d3+ = 767 X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 X13 + X23 + X33 = 1 X14 + X24 + X34 = 1 X15 + X25 + X35 = 1 X16 + X26 + X36 = 1 X17 + X27 + X37 = 1 Copyright Cengage Learning. Powered by Cognero.
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ch 6 X18 + X28 + X38 = 1 X19 + X29 + X39 = 1 X1,10 + X2,10 + X3,10 = 1 X1,11 + X2,11 + X3,11 = 1 X1,12 + X2,12 + X3,12 = 1 X11 + X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X1,10 + X1,11 + X1,12 + d4− − d4+ = 4 X21 + X22 + X23 + X24 + X25 + X26 + X27 + X28 + X29 + X2,10 + X2,11 + X2,12 + d5− − d5+ = 4 X31 + X32 + X33 + X34 + X35 + X36 + X37 + X38 + X39 + X3,10 + X3,11 + X3,12 + d6− − d6+ = 4 Xij ≥ 0, dk−, dk+ ≥ 0 for k = 1,2,3,4,5,6 Given the following spreadsheet solution of this integer goal programming formulation, answer the following questions.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A Cost Capacity 1 2 3 4 5 6 7 8 9 10 11 12 Total Items Under Over Actual Target Items Total Wt Under Over Actual Wt Target Wt
B $0.34 8.0 1 0 1 1 0 0 0 0 0 0 1 1 0 4 0 0 4 4 8 0.00 0.33 7.67 7.67
C $0.42 9.0 2 1 0 0 0 1 1 1 0 0 0 0 0 4 0 0 4 4 8 0.00 0.33 7.67 7.67
D $0.25 7.0 3 0 0 0 1 0 0 0 1 1 0 0 1 4 0 0 4 4 7 0.67 0.00 7.67 7.67 Cost
E
F
G
Weight 2.1 1.6 3.2 0.9 1.1 0.7 4.1 2.6 1.7 2.4 0.8 1.8
Assigned 1 1 1 1 1 1 1 1 1 1 1 1
Required 1 1 1 1 1 1 1 1 1 1 1 1
Item Cost Wt. Cost
$50 $0.25
$7.8967
61. Refer to Exhibit 7.4. The spreadsheet model has scaled all the weights from pounds into 100s pounds. How does this scaling effect the solution obtained using the Analytic Solver Platform (ASP)? 62. Refer to Exhibit 7.4. Based on the integer goal programming formulation, the associated solution, and spreadsheet model, what formulas should go in cells B19:E19 and B24:E24 of the spreadsheet? 63. Refer to Exhibit 7.4. The solution indicates Truck 3 is under the target weight by 67 pounds. What if anything can be done to this model to provide a solution in which Truck 3 is closer to the target weight? 64. A company needs to supply customers in 3 cities from its 3 warehouses. The supplies, demands and shipping costs are Copyright Cengage Learning. Powered by Cognero.
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ch 6 shown below.
Destination Warehouse 1 2 1 34 60 2 70 40 3 56 40 Demand 500 300 The company has identified the following goals:
3 36 50 32 200
Supply 400 300 200
The company would like to come as close as possible to satisfying its customers demand. Goal 2: It would also like to ensure that the cost is approximately $290,000. Formulate a goal programming model of this problem. Goal 1:
65. An investor wants to invest $50,000 in two mutual funds, A and B. The rates of return, risks and minimum investment requirements for each fund are:
Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 Note that a low Risk rating means a less risky investment. The investor wants to maximize the expected rate of return while minimizing his risk. Any money beyond the minimum investment requirements can be invested in either fund. The investor has found that the maximum possible expected rate of return is 11.4% and the minimum possible risk is 0.32. Formulate a goal programming model with a MINIMAX objective function. 66. A dietician wants to formulate a low cost, high calorie food product for a customer. The following information is available about the 2 ingredients which can be combined to make the food. The customer wants 1000 pounds of the food product and it should contain 250 pounds of Food 1 and 300 pounds of Food 2. The final cost of the blend should be about $1.15 and contain about 2500 calories per pound. The percent of fat, protein, carbohydrate in each food is summarized below with the target values for the goals. The dietician would prefer the food product be low in fat while also high in protein and carbohydrates.
Cost ($/pound) Fat Protein Carbohydrate Calories/pound Pounds of food 1 Pounds of food 2 Formulate the GP for this problem
Food 1 $1.00 15% 35% 50% 3000
Food 2 $1.25 25% 40% 35% 2000
TARGET $1.15 300 pounds 370 pounds 400 pounds 2500 250 300
67. A company makes 2 products A and B from 2 resources, labor and material. The products have the following resource requirements and produce the accompanying profits. The available quantity of resources is also shown in the table. Copyright Cengage Learning. Powered by Cognero.
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ch 6 Product A B Labor (hr/unit) 3 2 Material (ounces/unit) 1 2 Profit($/unit) 7 6 Management has developed the following set of goals
Available resources 150 200
Goal 1: Produce approximately 40 units of product 1. Goal 2: Produce approximately 70 units of product 2. Goal 3: Achieve a profit over $400. Goal 4: Consume less than 150 hours of labor Goal 5: Consume less than 200 ounces of material Formulate a goal programming model of this problem. Exhibit 7.4 The following questions are based on the problem below. Robert Gardner runs a small, local-only delivery service. His fleet consists of three smaller panel trucks. He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers. The box weights are: 210, 160, 320, 90, 110, 70, 410, 260, 170, 240, 80 and 180 for boxes 1 through 12, respectively. Since each truck differs each truck has different load capacities as given below:
Truck 1 2 3
Weight Capacity 800 pounds 900 pounds 700 pounds
Box Capacity 5 6 4
Cost per pound $0.34 $0.42 $0.25
Robert would like each truck equally loaded, both in terms of number of boxes and in terms of total weight, while minimizing his shipping costs. Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight. The following integer goal programming formulation applies to his problem. Y1 = weight loaded in truck 1; Y2 = weight loaded in truck 2; Y3 = weight loaded in truck 3; Xi,j = 0 if truck i not loaded with box j; 1 if truck i loaded with box j.
MIN 0.34Y1 + 0.42Y2 + 0.25Y3 + 50*(d1+ + d2+ + d3+) + 0.10*(d4− + d5− + d6−)*100 Subject to: Y1 = 210X11 + 160X12 + 320X13 + 90X14 + 110X15 + 70X16 + 410X17 + 260X18 + 170X19 + 240X1,10 + 80X1,11 + 180X1,12 Y2 = 210X21 + 160X22 + 320X23 + 90X24 + 110X25 + 70X26 + 410X27 + 260X28 + 170X29 + 240X2,10 + 80X2,11 + 180X2,12 Y3 = 210X31 + 160X32 + 320X33 + 90X34 + 110X35 + 70X36 + 410X37 + 260X38 + 170X39 + 240X3,10 + 80X3,11 + 180X3,12 Copyright Cengage Learning. Powered by Cognero.
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ch 6 Y1 ≤ 800 Y2 ≤ 900 Y3 ≤ 700 Y1 + d1− − d1+ = 767 Y2 + d2− − d2+ = 767 Y3 + d3− − d3+ = 767 X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 X13 + X23 + X33 = 1 X14 + X24 + X34 = 1 X15 + X25 + X35 = 1 X16 + X26 + X36 = 1 X17 + X27 + X37 = 1 X18 + X28 + X38 = 1 X19 + X29 + X39 = 1 X1,10 + X2,10 + X3,10 = 1 X1,11 + X2,11 + X3,11 = 1 X1,12 + X2,12 + X3,12 = 1 X11 + X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X1,10 + X1,11 + X1,12 + d4− − d4+ = 4 X21 + X22 + X23 + X24 + X25 + X26 + X27 + X28 + X29 + X2,10 + X2,11 + X2,12 + d5− − d5+ = 4 X31 + X32 + X33 + X34 + X35 + X36 + X37 + X38 + X39 + X3,10 + X3,11 + X3,12 + d6− − d6+ = 4 Xij ≥ 0, dk−, dk+ ≥ 0 for k = 1,2,3,4,5,6 Given the following spreadsheet solution of this integer goal programming formulation, answer the following questions.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A Cost Capacity 1 2 3 4 5 6 7 8 9 10 11 12 Total Items Under Over Actual Target Items
B $0.34 8.0 1 0 1 1 0 0 0 0 0 0 1 1 0 4 0 0 4 4
C $0.42 9.0 2 1 0 0 0 1 1 1 0 0 0 0 0 4 0 0 4 4
D $0.25 7.0 3 0 0 0 1 0 0 0 1 1 0 0 1 4 0 0 4 4
E
F
G
Weight 2.1 1.6 3.2 0.9 1.1 0.7 4.1 2.6 1.7 2.4 0.8 1.8
Assigned 1 1 1 1 1 1 1 1 1 1 1 1
Required 1 1 1 1 1 1 1 1 1 1 1 1
Item Cost Wt. Cost
$50 $0.25
Copyright Cengage Learning. Powered by Cognero.
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ch 6 21 22 23 24 25 26
Total Wt Under Over Actual Wt Target Wt
8 0.00 0.33 7.67 7.67
8 0.00 0.33 7.67 7.67
7 0.67 0.00 7.67 7.67 Cost
$7.8967
68. Refer to Exhibit 7.4. Given the solution indicated in the spreadsheet, which trucks, if any, are under an equal weight amount, and which trucks are over an equal weight amount? 69. Consider the following multi-objective linear programming problem (MOLP):
MAX: MAX: Subject to:
3 X1 + 4 X2 2 X1 + X2 6 X1 + 13 X2 ≤ 78 12 X1 + 9 X2 ≤ 108 8 X1 + 10 X2 ≤ 80 X1, X2 ≥ 0 Graph the feasible region for this problem and compute the value of each objective at each extreme point. What are the solutions to each of the component LPs?
Exhibit 7.4 The following questions are based on the problem below. Robert Gardner runs a small, local-only delivery service. His fleet consists of three smaller panel trucks. He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers. The box weights are: 210, 160, 320, 90, 110, 70, 410, 260, 170, 240, 80 and 180 for boxes 1 through 12, respectively. Since each truck differs each truck has different load capacities as given below: Copyright Cengage Learning. Powered by Cognero.
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ch 6 Truck 1 2 3
Weight Capacity 800 pounds 900 pounds 700 pounds
Box Capacity 5 6 4
Cost per pound $0.34 $0.42 $0.25
Robert would like each truck equally loaded, both in terms of number of boxes and in terms of total weight, while minimizing his shipping costs. Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight. The following integer goal programming formulation applies to his problem. Y1 = weight loaded in truck 1; Y2 = weight loaded in truck 2; Y3 = weight loaded in truck 3; Xi,j = 0 if truck i not loaded with box j; 1 if truck i loaded with box j.
MIN 0.34Y1 + 0.42Y2 + 0.25Y3 + 50*(d1+ + d2+ + d3+) + 0.10*(d4− + d5− + d6−)*100 Subject to: Y1 = 210X11 + 160X12 + 320X13 + 90X14 + 110X15 + 70X16 + 410X17 + 260X18 + 170X19 + 240X1,10 + 80X1,11 + 180X1,12 Y2 = 210X21 + 160X22 + 320X23 + 90X24 + 110X25 + 70X26 + 410X27 + 260X28 + 170X29 + 240X2,10 + 80X2,11 + 180X2,12 Y3 = 210X31 + 160X32 + 320X33 + 90X34 + 110X35 + 70X36 + 410X37 + 260X38 + 170X39 + 240X3,10 + 80X3,11 + 180X3,12 Y1 ≤ 800 Y2 ≤ 900 Y3 ≤ 700 Y1 + d1− − d1+ = 767 Y2 + d2− − d2+ = 767 Y3 + d3− − d3+ = 767 X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 X13 + X23 + X33 = 1 X14 + X24 + X34 = 1 X15 + X25 + X35 = 1 X16 + X26 + X36 = 1 X17 + X27 + X37 = 1 X18 + X28 + X38 = 1 X19 + X29 + X39 = 1 X1,10 + X2,10 + X3,10 = 1 X1,11 + X2,11 + X3,11 = 1 X1,12 + X2,12 + X3,12 = 1 X11 + X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X1,10 + X1,11 + X1,12 + d4− − d4+ = 4 Copyright Cengage Learning. Powered by Cognero.
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ch 6 X21 + X22 + X23 + X24 + X25 + X26 + X27 + X28 + X29 + X2,10 + X2,11 + X2,12 + d5− − d5+ = 4 X31 + X32 + X33 + X34 + X35 + X36 + X37 + X38 + X39 + X3,10 + X3,11 + X3,12 + d6− − d6+ = 4 Xij ≥ 0, dk−, dk+ ≥ 0 for k = 1,2,3,4,5,6 Given the following spreadsheet solution of this integer goal programming formulation, answer the following questions.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A Cost Capacity 1 2 3 4 5 6 7 8 9 10 11 12 Total Items Under Over Actual Target Items Total Wt Under Over Actual Wt Target Wt
B $0.34 8.0 1 0 1 1 0 0 0 0 0 0 1 1 0 4 0 0 4 4 8 0.00 0.33 7.67 7.67
C $0.42 9.0 2 1 0 0 0 1 1 1 0 0 0 0 0 4 0 0 4 4 8 0.00 0.33 7.67 7.67
D $0.25 7.0 3 0 0 0 1 0 0 0 1 1 0 0 1 4 0 0 4 4 7 0.67 0.00 7.67 7.67 Cost
E
F
G
Weight 2.1 1.6 3.2 0.9 1.1 0.7 4.1 2.6 1.7 2.4 0.8 1.8
Assigned 1 1 1 1 1 1 1 1 1 1 1 1
Required 1 1 1 1 1 1 1 1 1 1 1 1
Item Cost Wt. Cost
$50 $0.25
$7.8967
70. Refer to Exhibit 7.4. What formulas should go in cell E26 of the spreadsheet? 71. A dietitian wants to formulate a low cost, high calorie food product for a customer. The following information is available about the 2 ingredients which can be combined to make the food. The customer wants 1000 pounds of the food product and it must contain at least 250 pounds of Food 1 and 300 pounds of Food 2.
Cost ($/pound) % Fat % Protein % Carbohydrate Calories/pound Minimum pounds of food Formulate the MOLP for this problem.
Food 1 1.00 15 35 50 3000 250
Food 2 1.25 25 40 35 2000 300
Requirements Minimize Less than 30% At least 37% At least 40% Maximize
72. A company makes 2 products A and B from 2 resources. The products have the following resource requirements and Copyright Cengage Learning. Powered by Cognero.
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ch 6 produce the accompanying profits. The available quantity of resources is also shown in the table.
Product 1 2 Labor (hr/unit) 3 2 Material (ounces/unit) 1 2 Profit($/unit) 7 6 Management has developed the following set of goals
Available resources 150 200
Goal 1: Produce approximately 40 units of product 1. Goal 2: Produce approximately 70 units of product 2. Goal 3: Achieve a profit over $400. Goal 4: Consume less than 150 hours of labor Goal 5: Consume less than 200 ounces of material Based on the following GP formulation of the problem, and the associated optimal solution, what formulas should go in cells D6:F6, B9:F9, and B16 of the following Excel spreadsheet? NOTE: Formulas are not required in all of these cells.
Let
X1 = number of product 1 X2 = number of product 2
MIN: Subject to:
d1− + d1+ + d2− + d2+ + d3− + d4+ + d5+ X1 + d1− − d1+ = 40 X2 + d2− − d2+ = 70 7 X1 + 6 X2 + d3− − d3+ = 400 3 X1 + 2 X2 + d4− − d4+ = 150 1 X1 + 2 X2 + d5− − d5+ = 200 Xi, di−, di+ ≥ 0 for all i
product 1 product 2 profit labor material
(X1, X2) = (4, 69) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A Problem Data Labor Material Profit Goal Constraints Actual Amount + Under − Over = Goal Target Value
B A 3 1 7 A 4 36 0 40 40
C B 2 2 6 B 69 1 0 70 70
D
E
F
Labor 150 0 0 150 150
Material 142 58 0 200 200
Profit 442 0 42 400 400
Weights Under Over
1 1
1 1
0 1
0 1
1 0
Objective
37
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ch 6 73. A company wants to purchase large and small delivery trucks. The company wants to purchase about 10 large and 15 small trucks. Each large truck costs $30,000 and has a 10 ton capacity. Each small truck costs $20,000 and has a 7 ton capacity. The company wants to have about 200 tons of capacity and spend about $600,000. Based on the following goal programming formulation, associated solution, and spreadsheet model, what formulas should go in cells D6:E6, B9:E9, and B16 of the spreadsheet?
Let
X1 = number of large trucks X2 = number of small trucks
MIN: Subject to:
d1− + d1+ + d2− + d2+ + d3− + d3+ + d4− − d4+ X1 + d1− − d1+ = 10 X2 + d2− − d2+ = 15 10 X1 + 7 X2 + d3− − d3+ = 200 30 X1 + 20 X2 + d4− − d4+ = 600 Xi, di−, di+ ≥ 0 for all I
large trucks small trucks capacity cost
(X1, X2) = (10, 15) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A Problem Data Cost Capacity
B Large 10 30
C Small 7 20
D
E
Goal Constraints Actual Amount + Under − Over = Goal Target Value
Large 10 0 0 10 10
Small 15 0 0 15 15
Cost 205 0 5 200 200
Capacity 600 0 0 600 600
Weights Under Over
1 1
1 1
1 1
1 1
Objective
5
74. An investor wants to invest $50,000 in two mutual funds, A and B. The rates of return, risks and minimum investment requirements for each fund are:
Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 Note that a low Risk rating means a less risky investment. The investor can invest to maximize the expected rate of return or minimize risk. Any money beyond the minimum investment requirements can be invested in either fund. Copyright Cengage Learning. Powered by Cognero.
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ch 6 The following is the MOLP formulation for this problem:
Let
X1 = dollars in investment A X2 = dollars in investment B
MAX: MIN: Subject to:
0.12 X1/50000 + 0.09 X2/50000 0.5 X1/50000 + 0.3 X2/50000 X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000 Xi ≥ 0 for all i The solution for the second LP is (X1, X2) = (20,000, 30,000). What formulas should go in cells B2:D11 of the spreadsheet? NOTE: Formulas are not required in all of these cells.
1 2 3 4 5 6 7 8 9 10 11
A Problem data Expected return Risk rating
B A 12% 0.50
C B 9% 0.20
D
Variables Amount invested Minimum required
A $20,000 $20,000
B $30,000 $10,000
Total $50,000 $50,000
Objectives: Average return Average risk
10.2% 0.32
75. Robert Gardner runs a small, local-only delivery service. His fleet consists of three smaller panel trucks. He recently accepted a contract to deliver 12 shipping boxes of goods for delivery to 12 different customers. The box weights are: 210, 160, 320, 90, 110, 70, 410, 260, 170, 240, 80 and 180 for boxes 1 through 12, respectively. Since each truck differs each truck has different load capacities as given below:
Truck Weight Capacity Box Capacity Cost per pound 1 800 pounds 5 $0.34 2 900 pounds 6 $0.42 3 700 pounds 4 $0.25 Robert would like each truck equally loaded, both in terms of number of boxes and in terms of total weight, while minimizing his shipping costs. Assume a cost of $50 per item for trucks carrying extra boxes and $0.10 per pound cost for trucks carrying less weight. Formulate the integer goal programming problem for Robert. (Hint: objective function involves decision and deviation variables.) 76. An investor wants to invest $50,000 in two mutual funds, A and B. The rates of return, risks and minimum investment Copyright Cengage Learning. Powered by Cognero.
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ch 6 requirements for each fund are:
Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 Note that a low Risk rating means a less risky investment. The investor can invest to maximize the expected rate of return or minimize risk. Any money beyond the minimum investment requirements can be invested in either fund. Formulate the MOLP for this investor. 77. An investor wants to invest $50,000 in two mutual funds, A and B. The rates of return, risks and minimum investment requirements for each fund are:
Fund Rate of return Risk Minimum investment A 12% .5 $20,000 B 9% .3 $10,000 Note that a low Risk rating means a less risky investment. The investor can invest to maximize the expected rate of return or minimize risk. Any money beyond the minimum investment requirements can be invested in either fund. The following is the multi-objective linear programming (MOLP) formulation for this problem:
Let
X1 = dollars in investment A X2 = dollars in investment B
MAX: MIN: Subject to:
0.12 X1/50000 + 0.09 X2/50000 0.5 X1/50000 + 0.3 X2/50000 X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000 Xi ≥ 0 for all i The solution for the second LP is (X1, X2) = (20,000, 30,000). Based on this solution, what values should go in cells B2:D11 of the spreadsheet?
1 2 3 4 5 6 7 8 9
A Problem data Expected return Risk rating
B A
C B
D
Variables Amount invested Minimum required
A
B
Total
Objectives:
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ch 6 10 11
Average return Average risk
78. An investor wants to invest $50,000 in two mutual funds, A and B. The rates of return, risks and minimum investment requirements for each fund are:
Fund Rate of return Risk Minimum investment A 12% 0.5 $20,000 B 9% 0.3 $10,000 Note that a low Risk rating means a less risky investment. The investor wants to maximize the expected rate of return while minimizing his risk. Any money beyond the minimum investment requirements can be invested in either fund. The investor has found that the maximum possible expected rate of return is 11.4% and the minimum possible risk is 0.32. The following Excel spreadsheet has been created to solve a goal programming problem with a MINIMAX objective based on the following goal programming formulation with MINIMAX objective and corresponding solution.
MINIMIZE Subject to:
Q X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000
Xi ≥ 0 for all i, Q ≥ 0 with solution (X1, X2) = (15,370, 34,630). What values should go in cells B2:D14 of the spreadsheet?
1 2 3 4 5 6 7 8 9 10 11 12 13
A Problem data Expected return Risk rating
B A
C B
D
Variables Amount invested Minimum required
A
B
Total
Goals Average return Average risk
Actual
Target
Weights 1 1
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E
Weighted % Deviation
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ch 6 14
Objective:
79. A company wants to purchase large and small delivery trucks. The company wants to purchase about 10 large and 15 small trucks. Each large truck costs $30,000 and has a 10 ton capacity. Each small truck costs $20,000 and has a 7 ton capacity. The company wants to have about 200 tons of capacity and spend about $600,000. Formulate a goal programming model of this problem. 80. A company wants to purchase large and small delivery trucks. The company wants to purchase about 10 large and 15 small trucks. Each large truck costs $30,000 and has a 10 ton capacity. Each small truck costs $20,000 and has a 7 ton capacity. The company wants to have about 200 tons of capacity and spend about $600,000. Based on the following formulation and associated integer solution, what values should go in cells B2:E16 of the spreadsheet?
Let
X1 = number of large trucks X2 = number of small trucks
MIN: Subject to:
d1− + d1+ + d2− + d2+ + d3− + d3+ + d4− − d4+ X1 + d1− − d1+ = 10 X2 + d2− − d2+ = 15 10 X1 + 7 X2 + d3− − d3+ = 200 30 X1 + 20 X2 + d4− − d4+ = 600 Xi, di−, di+ ≥ 0 for all I
large trucks small trucks capacity cost
(X1, X2) = (10, 15) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A Problem Data Cost Capacity
B Large
C Small
D
E
Goal Constraints Actual Amount + Under − Over = Goal Target Value
Large
Small
Cost
Capacity
Weights Under Over Objective
81. Given the following goal constraints Copyright Cengage Learning. Powered by Cognero.
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ch 6 5 X1 + 6 X2 + 7 X3 + d1− − d1+ = 87 3 X1 + X2 + 4 X3 + d2− − d2+ = 37 7 X1 + 3 X2 + 2 X3 + d3− − d3+ = 72 and solution (X1, X2, X3) = (7, 2, 5), what values do the deviational variables assume?
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ch 6 Answer Key 1. False 2. False 3. False 4. True 5. True 6. False 7. True 8. False 9. b 10. b 11. c 12. d 13. a 14. d 15. d 16. a 17. b 18. d 19. b 20. b 21. b 22. a 23. a 24. c 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 6 26. a 27. c 28. a 29. d 30. a 31. b 32. c 33. a 34. c 35. d 36. d 37. c 38. c 39. b 40. d 41. c 42. c 43. a 44. a 45. a 46. d 47. a 48. b 49. a 50. d 51. a Copyright Cengage Learning. Powered by Cognero.
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ch 6 52. b 53. a 54. a 55. a 56. a 57. a 58. b 59. d 60.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A Problem Data Labor Material Profit Goal Constraints Actual Amount + Under − Over = Goal Target Value
B A 3 1 7 A 4 36 0 40 40
C B 2 2 6 B 69 1 0 70 70
D
E
F
Labor 150 0 0 150 150
Material 142 58 0 200 200
Profit 442 0 42 400 400
Weights Under Over
1 1
1 1
0 1
0 1
1 0
Objective
37
61. The solution obtained is the same regardless of scaling. In terms of pounds, the total cost would be multiplied by 100, to obtain $78,967. 62. Cell B19 B24
Formula =B16 + B17 − B18 =B21 + B22 − B23
Copied to C19:E19 C24:E24
63. Nothing. Truck 3 is at its capacity of 700 pounds. 64. Let
Xij = number of units shipped from warehouse i to destination j
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ch 6 MIN: Subject to: X11 + X12 + X13 + d 1 − − d 1 + = 400 X21 + X22 + X23 + d 2 − − d 2 + = 300 X31 + X32 + X33 + d 3 − − d 3 + = 200 X11 + X21 + X31 + d 4 − − d 4 + = 500 X12 + X22 + X32 + d 5 − − d 5 + = 300 X13 + X23 + X33 + d 6 − − d 6 + = 200 34 X11 + 60 X12 + 36 X13 + 70 X21 + 40 X22 + 50 X23 + 56 X31 + 40 X32 + 32 X33 + d7− + d7+ = 290000 Xij, di−, di+ ≥ 0 for all i,j 65. Let
MINIMIZE Subject to:
X1 = dollars in investment A X2 = dollars in investment B Q X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000
Xi ≥ 0 for all i, Q ≥ 0 66. Let
MIN: Subject to:
X1 = pounds of food 1 X2 = pounds of food 2 d1− + d1+ + d2+ + d3− + d4− + d5− + d5+ + d6− + d6+ + d7− + d7+ cost 1 X1 + 1.25 X2 + d1− − d1+ = 1.15 − + fat .15 X1 + .25 X2 + d2 − d2 = 300 − + protein .35 X1 + .40 X2 + d3 − d3 = 370 − + carbohydrate .50 X1 + .35 X2 + d4 − d4 = 400 − + calories 3000 X1 + 2000 X2 + d5 − d5 = 2500 − + food 1 X1 + d6 − d6 = 250 − + food 2 X2 + d7 − d7 = 300 food product required X1 + X2 = 1000 Xi, di−, di+ ≥ 0 for all i
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ch 6 67. Let
MIN: Subject to:
X1 = number of product 1 X2 = number of product 2 d1− + d1+ + d2 − + d2+ + d3− + d4+ + d5+ X1 + d1− − d1+ = 40 X2 + d2− − d2+ = 70 7 X1 + 6 X2 + d3− − d3+ = 400 3 X1 + 2 X2 + d4− − d4+ = 150 1 X1 + 2 X2 + d5− − d5+ = 200 Xi, di−, di+ ≥ 0 for all i
product 1 product 2 profit labor material
68. Trucks 1 and 2 are over the target weight by 33 pounds. Truck 3 is under the target weight by 67 pounds.
69. OBJ 1 OBJ 2 X1 X2 0 0 0 0 0 6 24 6 9 0 27 18 5.9 3.27 30.78 15.07 7.5 2 30.5 17 The solution to MAX 3 X1 + 4 X2 is (X1, X2) = (5.9, 3.27) and objective function value of 30.78. The solution to MAX 2 X1 + X2 is (X1, X2) = (9, 0) and objective function value of 18. 70. =SUMPRODUCT(B21:D21, B1:D1) + F17*SUM(B18:D18) + F18*SUM(B22:D22) 71. Let
X1 = pounds of food 1 X2 = pounds of food 2
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ch 6 MIN: MAX: Subject to:
minimize cost or food product maximize the calories fat protein carbohydrate food 1 food 2 total food product required
1 X1 + 1.25 X2 3000 X1 + 2000 X2 .15 X1 + .25 X2 ≤ 300 .35 X1 + .40 X2 ≥ 370 .50 X1 + .35 X2 ≥ 400 X1 ≥ 250 X2 ≥ 300 X1 + X2 = 1000 Xi ≥ 0 for all i
72. Cell D6 E6 F6 B9 B16
Formula =SUMPRODUCT(B2:C2,$B$6:$C$6) =SUMPRODUCT(B3:C3, $B$6:$C$6) =SUMPRODUCT(B4:C4, $B$6:$C$6) =B6+B7-B8 =SUMPRODUCT(B7:F8,B13:F14)
Copied to
73. Cell D6 E6 B9 B16
Formula =SUMPRODUCT(B2:C2,$B$6:$C$6) =SUMPRODUCT(B3:C3, $B$6:$C$6) =B6+B7-B8 =SUMPRODUCT(B13:E14,B7:E8)
Copied to
74. Cell D6 B10 B11
Formula =B6+C6 =SUMPRODUCT(B2:C2,B6:C6)/D7 =SUMPRODUCT(B3:C3,B6:C6)/D7
C9:F9
C9:E9
75. Y1 = weight loaded in truck 1; Y2 = weight loaded in truck 2; Y3 = weight loaded in truck 3; Xi,j = 0 if truck i not loaded with box j; 1 if truck i loaded with box j. MIN 0.34Y1 + 0.42Y2 + 0.25Y3 + 50*(d1+ + d2+ + d3+) + 0.10*(d4− + d5− + d6−)*100 Subject to: Y1 = 210X11 + 160X12 + 320X13 + 90X14 + 110X15 + 70X16 + 410X17 + 260X18 + 170X19 + 240X1,10 + 80X1,11 + 180X1,12 Y2 = 210X21 + 160X22 + 320X23 + 90X24 + 110X25 + 70X26 + 410X27 + 260X28 + 170X29 + 240X2,10 + 80X2,11 + 180X2,12 Y3 = 210X31 + 160X32 + 320X33 + 90X34 + 110X35 + 70X36 + 410X37 + 260X38 + 170X39 + 240X3,10 + 80X3,11 + 180X3,12 Y1 ≤ 800 Y2 ≤ 900 Y3 ≤ 700 Copyright Cengage Learning. Powered by Cognero.
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ch 6 Y1 + d1− − d1+ = 767 Y2 + d2− − d2+ = 767 Y3 + d3− − d3+ = 767 X11 + X21 + X31 = 1 X12 + X22 + X32 = 1 X13 + X23 + X33 = 1 X14 + X24 + X34 = 1 X15 + X25 + X35 = 1 X16 + X26 + X36 = 1 X17 + X27 + X37 = 1 X18 + X28 + X38 = 1 X19 + X29 + X39 = 1 X1,10 + X2,10 + X3,10 = 1 X1,11 + X2,11 + X3,11 = 1 X1,12 + X2,12 + X3,12 = 1 X11 + X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X1,10 + X1,11 + X1,12 + d4− − d4+ = 4 X21 + X22 + X23 + X24 + X25 + X26 + X27 + X28 + X29 + X2,10 + X2,11 + X2,12 + d5− − d5+ = 4 X31 + X32 + X33 + X34 + X35 + X36 + X37 + X38 + X39 + X3,10 + X3,11 + X3,12 + d6− − d6+ = 4 Xij ≥ 0, dk−, dk+ ≥ 0 for k = 1,2,3,4,5,6 76. Let
X1 = dollars in investment A X2 = dollars in investment B
MAX: MIN: Subject to:
0.12 X1/50000 + 0.09 X2/50000 0.5 X1/50000 + 0.3 X2/50000 X1 + X2 = 50000 X1 ≥ 20000 X2 ≥ 10000 Xi ≥ 0 for all i
77.
1 2 3 4 5 6 7
A Problem data Expected return Risk rating
B A 12% 0.50
C B 9% 0.20
D
Variables Amount invested Minimum required
A $20,000 $20,000
B $30,000 $10,000
Total $50,000 $50,000
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ch 6 8 9 10 11
Objectives: Average return Average risk
10.2% 0.32
78.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
79. Let
MIN: Subject to:
A Problem data Expected return Risk rating
B A 0.12 0.5
C B 0.09 0.3
D
Variables Amount invested Minimum required
A 15370 10000
B 34630 20000
Total 50000 50000
Goals Average return Average risk
Actual 0.09922 0.36148
Target 0.114 0.320
Weights 1 1
Objective:
0.12963
E
Weighted % Deviation 0.12963 0.12963
X1 = number of large trucks X2 = number of small trucks d1− + d1+ + d2− + d2+ + d3− + d3+ + d4− − d4+ X1 + d1− − d1+ = 10 X2 + d2− − d2+ = 15 10 X1 + 7 X2 + d3− − d3+ = 200 30 X1 + 20 X2 + d4− − d4+ = 600 Xi, di−, di+ ≥ 0 for all i
large trucks small trucks capacity cost
80.
1 2 3 4 5 6 7 8 9 10
A Problem Data Cost Capacity
B Large 10 30
C Small 7 20
D
E
Goal Constraints Actual Amount + Under − Over = Goal Target Value
Large 10 0 0 10 10
Small 15 0 0 15 15
Cost 205 0 5 200 200
Capacity 600 0 0 600 600
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ch 6 11 12 13 14 15 16
Weights Under Over
1 1
Objective
5
1 1
1 1
1 1
81. d1− = 5, d2+ = 6, d3− = 7, all others equal to zero.
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ch 7
Indicate whether the statement is true or false. 1. A NLP problem can have nonlinear objective and a nonlinear constraints a. True b. False 2. The main difference between shadow prices and Lagrange multipliers involves the range of RHS values over which the shadow price or Lagrange multiplier remains valid a. True b. False 3. The solution procedure Solver uses to solve NLP problems is called the generalized reduced gradient (GRG) algorithm a. True b. False 4. Nonzero reduced gradient values indicate the exact impact on the objective function value of large changes in the value of a given variable a. True b. False 5. The process of formulating an NLP problem is virtually the same as formulating an LP problem a. True b. False 6. A given local optimal solution is also a global optimal solution to a NLP problem a. True b. False 7. This selection of the initial feasible solution, often referred to as starting point, has no impact on the determination of the optimal solution to the NLP problem a. True b. False 8. If Solver produces the completion message: "Solver found a solution. All constraints and optimality conditions are satisfied,” a global optimal solution has been found a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. The total annual cost for the economic order quantity model is a. purchase cost + ordering cost + holding cost. b. fixed ordering cost + holding cost. c. purchase cost + fixed ordering cost. d. unit cost + variable ordering cost + holding cost. Copyright Cengage Learning. Powered by Cognero.
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ch 7 10. In solving the NLP problem, Solver produced a message "Solver cannot improve the current solution. All constraints are satisfied." This means that Solver found: a. a local optimal solution b. a global optimal solution c. the objective function changed very slowly for the last few iterations d. a degenerate model 11. How much must the objective function coefficient of the variable Pumpkin change before any Pumpkins are produced based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4 $D$4
Final Value 9.52 0 10.79
Reduced Gradient 0 −499.99 0
Final Value 200000.00 99.00 37777.78 29.84 8000.00
Lagrange Multiplier 0.016 12.000 0 0 3.490
Name Corn Pumpkin Beans
Constraints Cell Name $E$8 Corn $E$9 Pumpkin $E$10 Beans $E$11 Water $E$12 Fertilizer a. increase by 12 b. decrease by 12 c. increase by 499.99 d. decrease by 499.99
12. The Reduced Gradient is similar to which of these terms from linear programming? a. Shadow Price b. Allowable Decrease c. Allowable Increase d. Reduced Cost 13. Why does the GRG algorithm not provide allowable increase or allowable decrease information with the Reduced Gradient and Lagrange multiplier information? a. Because all constraints have slack. b. Because the constraints may not be linear. c. Because the solution is not necessarily a global optimum. d. Because the solutions are not necessarily corner points. 14. What is the search path for the following feasible solution space? The dashed line represents the objective function and the objective is to maximize the value of the objective function. Copyright Cengage Learning. Powered by Cognero.
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ch 7
a. A, D b. A, F, E, D c. A, B, C, D d. A, F, E 15. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What constraint would you impose on this problem to ensure that at least one order is placed per year?
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
Total Cost:
$502,738.61
a. MAX B9 b. B9 = 1 c. B9 ≥ 1 d. MIN B11 16. In solving the NLP problem, Solver produced a message "Solver has converged to the current solution. All constraints are satisfied." This means that: a. Solver found a local optimal solution b. Solver found a global optimal solution c. the objective function changed very slowly for the last few iterations d. Your model is degenerate 17. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 5% return but with minimum risk.
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ch 7 Let:
Pi = proportion of total funds invested in stock i σi2 = variance of stock i σij = covariance between stocks i and j. Ri = average return on stock i What is the objective function for the NLP formulation of this problem? a. b. c. d. 18. How much are additional units of Labor worth based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4
Name Number to make: X1 Number to make: X2
Final Value 9.42 1.71
Reduced Gradient 0 0
Name Wood Labor Plywood
Final Value 42 132 24
Lagrange Multiplier 0 1.21 2.57
Constraints Cell $D$8 $D$9 $D$10 a. 0 b. 1.21 c. 2.57 d. 9.42 19. The GRG algorithm operates by a. moving in the direction of most rapid improvement in the objective function. b. choosing a search direction at random. c. searching directly for the optimum solution. d. moving in a clockwise direction. 20. In solving the NLP problem, Solver produced a message "Solver found a solution. All constraints and optimality conditions are satisfied." This means that Solver found: a. a local optimal solution b. a global optimal solution c. the objective function changed very slowly for the last few iterations d. a degenerate model 21. Which point or points are local optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function. Copyright Cengage Learning. Powered by Cognero.
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ch 7
a. D b. E c. F d. D, F 22. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of product A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of product B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − 0.10*P1 Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs.
B 2 2 18 30 − 0.07*P2
Available resources 150 200
The objective function for this problem is? a. MAX 25 X1 + 18 X2 b. MAX (20 − 0.10*P1) X1 + (30 − 0.07*P2) X2 c. MAX 20 P1 − 0.1 P12 + 30 P2 − 0.07 P22 − 1040 d. MAX 22.5 P1 − 0.1 P12 + 31.26 P2 − 0.07 P22 − 1040 23. In the GRG algorithm the initial solution is called the a. originating point. b. insertion point. c. zero point. d. starting point. 24. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D4 of the spreadsheet?
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ch 7 1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 163.800 100 90 210 180
Y-Coordinate 101.000 130 60 80 110
Distance: 70.082 84.424 50.749 18.532
Total Distance:
223.787
a. =(B4-$B$3)^2+(C4-$C$3)^2 b. =SQRT((B4-$B$3)+(C4-$C$3)) c. =SQRT((B4-$B$3)^2+(C4-$C$3)^2) d. =(B4-$B$3)+(C4-$C$3) Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − .10*P1
B 2 2 18 30 − .07*P2
Available resources 150 200
Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs
A 1 2 3 Price 4 Marginal Cost 5 Profit Margin 6 7 Demand 8 9 Total Profit 10 11 Constraints: 12 Labor 13 Material
B
C
A 112.50 25.00 87.50
B 223.29 18.00 205.29
8.75
14.37
D
E
Used 54.99 37.49
Available 150.00 200.00
3715.58
3.00 1.00
2.00 2.00
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ch 7 25. Refer to Exhibit 8.1. What formula is used in cell D12 of the spreadsheet for this problem? a. =SUMPRODUCT(B3:C3, B12:C12) b. =SUMPRODUCT(B4:C4, B12:C12) c. =SUMPRODUCT(B5:C5, B12:C12) d. =SUMPRODUCT(B7:C7, B12:C12) 26. If the "Standard LP/Quadratic Engine" option is chosen in the Analytic Solver Platform (ASP) task pane, ASP a. will test for non-linearity in the model and apply GRG. b. will test for non-linearity in the model and apply Simplex. c. uses the B & B algorithm to solve the problem. d. will apply Simplex if the linearity test passes. 27. The optimal trade-off between risk and return for a given portfolio problem can be summarized by the a. efficient frontier. b. investment frontier. c. portfolio boundary. d. variance boundary. 28. The global optimum solution to a nonlinear programming problems (NLP), in which the objective function must be minimized, is: a. always larger than the local optimum solution b. always smaller than the local optimum solution c. is smaller or equal to the local optimum solution d. is larger or equal to the local optimum solution 29. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. Which cell(s) in the spreadsheet represent the decision variables in the problem?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 163.800 100 90 210 180
Y-Coordinate 101.000 130 60 80 110
Distance: 70.082 84.424 50.749 18.532
Total Distance:
223.787
a. B3 b. B3:C3 c. B4:C7 d. D9 30. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are Copyright Cengage Learning. Powered by Cognero.
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ch 7 sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What formula goes in cell B11 in this problem?
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
Total Cost:
$502,738.61
a. =B3*B7+B9/B3*B6+B9*2+B5*B7 b. =B3*B5+B3/B9*B6+B9/2*B5*B7 c. =SQRT(B3*B5+B3/B9*B6+B9/2*B5*B7) d. =B3*B3+B3/B9+B6*B9/2*B5*B7 Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − .10*P1
B 2 2 18 30 − .07*P2
Available resources 150 200
Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs
A 1 2 3 Price 4 Marginal Cost 5 Profit Margin 6
B
C
A 112.50 25.00 87.50
B 223.29 18.00 205.29
D
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E
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ch 7 7 Demand 8 9 Total Profit 10 11 Constraints: 12 Labor 13 Material
8.75
14.37
3715.58
3.00 1.00
2.00 2.00
Used 54.99 37.49
Available 150.00 200.00
31. Refer to Exhibit 8.1. What formula is used in cell B7 of the spreadsheet for this problem? a. = 30 − .07 * C3 b. = 20 − .1 * B3 c. = B3 − B4 d. = B5 * B7 + C5 * C7 32. The optimal solution to a NLP problem can occur at a(n) I. II.
corner point. interior point. a. I is true b. II is true c. I and II are true d. neither I or II is true
33. NLP problems which have slack in all the constraints a. are infeasible. b. are only at local optimal solutions. c. have zeros for all shadow prices. d. are at an interior point of the feasible region. 34. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the variable cell in this problem?
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
Total Cost:
$502,738.61
a. B3 Copyright Cengage Learning. Powered by Cognero.
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ch 7 b. B7 c. B9 d. B11 35. The main difference between LP and NLP problems is that NLPs will have a a. linear objective function and nonlinear or linear constraints. b. minimum of one nonlinear constraint or a nonlinear objective function. c. multilevel objective functions. d. multilevel objective function and nonlinear constraints. 36. In NLP a local optimum is best described as a. an interior point. b. a corner point. c. a point yielding no improving direction. d. a good starting point for subsequent searches. 37. An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell H11 of the Excel spreadsheet?
A
B
C
D
E
1
>
2 3 4 5 6 7 8 9 10 11 12 13 14 15 < < < < < < < < <
>
1 2 3 4 5 6 7 8
Year 1 2 3 4 5 6 7 8 9 10
A 3.98% −1.51% 5.36% 4.98% 3.12% 5.58% 3.49% −2.37% 5.92% 5.69%
Annual Return B 7.38% 10.45% 5.15% 5.51% 6.93% −4.32% 9.78% 5.62% 6.28% 5.75%
C 8.76% 6.66% 6.55% −1.58% 8.43% 9.07% 9.65% 8.96% −7.10% 10.96%
Average
3.42%
5.85%
6.03%
> > > > > > > > > > > > > >
F
G
H
I
A B C
Covariance Matrix A 0.00080 −0.00044 −0.00045
B −0.00044 0.00144 −0.00012
C −0.00045 −0.00012 0.00300
A
B
C
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ch 7 < < < < < < <
9 10 11 12 13 14 15
Portfolio
36.8%
Expected Return Required Return
5.00% 5.00%
Portfolio Variance
0.000274
40.6%
22.6%
100.0%
a. =SUMPRODUCT(B15:D15,G9:I9) b. =SUMPRODUCT(B15:D15,B4:D13) c. =SUMPRODUCT(B15:D15,I9) d. =SUMPRODUCT(B15:D15,G4:I6) 38. An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell G4 of the Excel spreadsheet?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Average
< < < < < < < < < < < < < < <
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
C
D
Year 1 2 3 4 5 6 7 8 9 10
A 3.98% −1.51% 5.36% 4.98% 3.12% 5.58% 3.49% −2.37% 5.92% 5.69%
Annual Return B 7.38% 10.45% 5.15% 5.51% 6.93% −4.32% 9.78% 5.62% 6.28% 5.75%
C 8.76% 6.66% 6.55% −1.58% 8.43% 9.07% 9.65% 8.96% −7.10% 10.96%
3.42%
5.85%
E
> > > > > > > > > > > > > > >
6.03%
>
F
G
H
I
A B C
Covariance Matrix A 0.00080 −0.00044 −0.00045
B −0.00044 0.00144 −0.00012
C −0.00045 −0.00012 0.00300
Portfolio
A 36.8%
B 40.6%
C 22.6%
Expected Return Required Return
5.00% 5.00%
Portfolio Variance
0.000274
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ch 7 <
15
a. =COVAR(B4:B13) b. =VARP(B4:B13) c. =COVAR(B4:D13,$D$4:$B$13) d. =COVAR(B4:B13,$B$4:$B$13) 39. The main difference between shadow prices and Lagrange multipliers is a. the range of RHS values over which the Lagrange multipliers or the shadow prices remain valid b. that shadow prices overestimate the impact of a small change in the RHS of the constraints on the objective function c. that shadow prices underestimate the impact of a small change in the RHS of the constraints on the objective function d. they are equivalent 40. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D9 of the spreadsheet?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 163.800 100 90 210 180
Y-Coordinate 101.000 130 60 80 110
Distance: 70.082 84.424 50.749 18.532
Total Distance:
223.787
a. =MIN(D4:D7)^2 b. =SUM(D4:D7) c. =SQRT(D4^2+D5^2+D6^2+D7^2) d. =SQRT(MMULT(B3:C3,D4:D7)) 41. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the objective cell in this problem?
1 2 3 4 5 6 7 8
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
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ch 7 9 10 11
Order Quantity:
483.73
Total Cost:
$502,738.61
a. B3 b. B7 c. B9 d. B11 42. The main difference between linear (LP) and nonlinear programming problems (NLP) is that a. NLP must have a nonlinear objective function b. Some constraints in NLP may be nonlinear c. No interaction terms are allowed in NLP d. Only one constraint in NLP can be nonlinear 43. Which of the following is not an assumption of an EOQ problem. a. Demand for a product is fairly constant. b. Inventory depletion rates vary non-linearly. c. Each new order is delivered in full when the inventory level reaches 0. d. All are valid assumptions. 44. A company has collected the following inventory data for an item. What is the total annual cost for this item? Annual demand for the item Unit purchase cost for the item Fixed cost of placing an order Cost of holding one unit in inventory for a year Order quantity a. 275 b. 450 c. 5,275 d. 5,450
D = 500 C = 10 S = 20 i = 30% Q = 50
45. The GRG and Simplex algorithms are similar in that a. each algorithm process continues until there is no further improvement in the objective function. b. both algorithms take their starting solution from the spreadsheet. c. both return the globally optimal solution. d. both algorithms return a solution that satisfies at least one constraint at equality. Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200. Copyright Cengage Learning. Powered by Cognero.
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ch 7 Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − .10*P1
B 2 2 18 30 − .07*P2
Available resources 150 200
Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs
A 1 2 3 Price 4 Marginal Cost 5 Profit Margin 6 7 Demand 8 9 Total Profit 10 11 Constraints: 12 Labor 13 Material
B
C
A 112.50 25.00 87.50
B 223.29 18.00 205.29
8.75
14.37
D
E
Used 54.99 37.49
Available 150.00 200.00
3715.58
3.00 1.00
2.00 2.00
46. Refer to Exhibit 8.1. What formula is used in cell B9 of the spreadsheet for this problem? a. =B3*B7+C3*C7 b. =B5*B7+C5*C7 c. =(B5-B4)*B7+(C5-C4)*B7 d. =B3*B7+C3*B7+B4*B7+C4*B7 47. In a maximization problem, the GRG algorithm's search for a feasible direction of improvement equates to ____ in the simplex algorithm for LP problems. a. an adjacent corner point b. a zero reduced cost c. a positive shadow price d. a positive reduced cost 48. The Lagrange multipliers can be used to a. estimate the impact of a small change in the RHS of the constraints on the objective function b. estimate the impact of an arbitrary change in the RHS of the constraints on the objective function c. estimate the impact of a small change in the RHS of the constraints on the shadow prices d. estimate the impact of a large change in the RHS of the constraints on the shadow prices 49. What is the straight line (Euclidean) distance between the points (5,7) and (1, 11)? a. 2.83 Copyright Cengage Learning. Powered by Cognero.
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ch 7 b. 5.65 c. 8 d. 16 50. When using the GRG algorithm to solve NLPs one should try multiple starting points because a. If two different starting points return the same solution, that solution is optimal. b. the solution returned depends upon the starting point. c. the solution returned is always near the starting point. d. a random element of GRG requires multiple starting points. 51. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What formula goes in cell C7 of the spreadsheet?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Amount Invested: Annual Return:
$9,000 10%
Year 1 2 3 4 5 6 7 8 9 10
Beginning Balance $9,000 $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207
C
D
E
F
Investment Earnings $971 $993 $1,012 $1,030 $1,046 $1,061 $1,075 $1,086 $1,096 $1,101
Earnings After Taxes $699 $715 $729 $741 $753 $764 $774 $782 $789 $793
Premium Due $500 $534 $566 $593 $607 $635 $672 $695 $737 $793
Ending Balance $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207 $10,207
a. =B7*(1+$C$3/4)^4-B7 b. =B7*(1+$C$3)-B7 c. =B7*($C$3/4)^4-B7 d. =B7*(1+$C$3/4)^4 52. A nonzero reduced gradient value indicates the approximate impact a. on the objective function of very small changes in the value of a given variable b. on the RHS of a constraint of a unitary change in the objective function value c. on the objective function of very large changes in the value of a given variable d. on the objective function of unitary change in the RHS of associated constraint 53. The global optimum solution to a nonlinear programming problems (NLP), in which the objective function must be maximized, is: a. always larger than the local optimum solution Copyright Cengage Learning. Powered by Cognero.
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ch 7 b. always smaller than the local optimum solution c. is smaller or equal to the local optimum solution d. is larger or equal to the local optimum solution 54. The optimal solution to a LP problem is always at I. II. III.
a corner point. an interior point the origin. a. I is true b. II is true c. III is true d. I and III are true
55. Which point or points are global optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function.
a. B b. D c. E d. F 56. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What constraint must be entered in the Analytic Solver Platform (ASP) task pane?
1 2 3 4 5 6 7 8 9
A
B
Amount Invested: Annual Return:
$9,000 10%
Year 1 2 3
Beginning Balance $9,000 $9,199 $9,380
C
D
E
F
Investment Earnings $971 $993 $1,012
Earnings After Taxes $699 $715 $729
Premium Due $500 $534 $566
Ending Balance $9,199 $9,380 $9,543
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ch 7 10 11 12 13 14 15 16
4 5 6 7 8 9 10
$9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207
$1,030 $1,046 $1,061 $1,075 $1,086 $1,096 $1,101
$741 $753 $764 $774 $782 $789 $793
$593 $607 $635 $672 $695 $737 $793
$9,691 $9,837 $9,966 $10,068 $10,155 $10,207 $10,207
a. $D$7:$D$16 ≤ $E$7:$E$16 b. $C$7:$D$16 ≥ $E$7:$E$16 c. $C$7:$C$16 ≥ $E$7:$E$16 d. $D$7:$D$16 ≥ $E$7:$E$16 57. The Analytic Solver Platform solution strategy for NLP problems is the GRG method. What does GRG stand for? a. Graphical Residual Gradient b. Generalized Reduced Gradient c. Goal Restricted Gradient d. Gradually Reduced Gradient 58. The GRG algorithm terminates when it a. has completed 100 iterations. b. has reached the global optimal solution. c. detects no feasible direction for improvement. d. reaches the steepest gradient. 59. The Lagrange Multiplier is similar to which of these terms from linear programming? a. Shadow Price b. Allowable Increase c. Allowable Decrease d. Reduced Cost 60. The straight line (Euclidean) distance between two points (X1, Y1) and (X2, Y2) is calculated as a. X1 − Y1 + X2 − Y2 b. (X1 − X2)2 + (Y1 − Y2)2 c. d. 61. The main difference between linear (LP) and nonlinear programming problems (NLP) is that a. NLP may have a nonlinear objective function b. All constraints in NLP must be nonlinear c. No interaction terms are allowed in NLP d. Only one constraint in NLP can be nonlinear 62. In solving the NLP problem, Solver found a degenerate model. This means that: a. Solver is cycling and redundant constraints must be removed Copyright Cengage Learning. Powered by Cognero.
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ch 7 b. a global optimal solution was found c. the objective function changed very slowly for the last few iterations d. a local optimum was found
63. A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It has four stores and their coordinates are listed in the below.
X-Coordinate Y-Coordinate Store 1 70 160 Store 2 60 90 Store 3 180 90 Store 4 150 120 Let X and Y represent the X, Y coordinates of the new warehouse. The NLP for this problem and solution is the following.
MIN: Subject to:
Q
Solution is (X, Y) = (120.0, 117.4). What values should go in cells B2:D9 of the spreadsheet for this problem?
A 1 2 3 4 5 6 7 8 9 10
B
C
D
X
Y
Distance
Warehouse Store 1 Store 2 Store 3 Store 4 Total Max
64. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk. The average return for the stocks is:
Annual Return Copyright Cengage Learning. Powered by Cognero.
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ch 7 A B Average 10.72% 10.68% The covariance matrix for the stocks is:
C 11.87%
A B C A 0.00009 −0.00009 −0.00011 B −0.00009 0.00032 −0.00007 C −0.00011 −0.00007 0.00122 Let: Pi = proportion of total funds invested in i, i = A, B, C Formulate the NLP for this problem. Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area
60,000 90,000 65,000
≥ = ≥
60,000 90,000 0
Copyright Cengage Learning. Powered by Cognero.
E Parking 80,000
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ch 7 8 9 10 11 12 13
Max. Bld Height
60
≤
65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
millions $
65. Refer to Exhibit 8.2. What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? Objective Cell: Variables Cells: Constraints Cells: 66. Find the maximum solution on this graph of a function starting from X = 12. Mark its location on the graph.
67. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk. Let: Pi = proportion of total funds invested in i, i = A, B, C The NLP for this problem is: 0.00009 P12 + 0.00032 P22 + 0.00122 P32 + 2 (−0.00009 P1P2 − 0.00011 P1P3 − 0.00007 P2P3) Subject to: P1 + P2 + P3 = 1 0.1072 P1 + 0.1068 P2 + 0.1187 P3 ≥ 0.11 P1, P2, P3 ≥ 0 P1, P2, P3 ≤ 1 What formulas should go in cells G4:J14 of the spreadsheet for this problem? NOTE: Formulas are not required in all of these cells. MIN:
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ch 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 < < < < < < < < < < < < < < < <
A
B
C
D
Year 1 2 3 4 5 6 7 8 9 10
A 9.73% 10.83% 12.14% 9.14% 11.16% 11.60% 11.06% 11.22% 9.25% 11.11%
Annual Return B 12.54% 9.52% 11.47% 13.72% 8.89% 10.72% 12.21% 8.56% 11.09% 8.10%
C 10.23% 16.10% 4.07% 12.93% 11.97% 12.03% 14.00% 16.28% 12.99% 8.06%
Average
10.72%
10.68%
11.87%
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
E
> > > > > > > > > > > > > > > >
F
G
H
I
A B C
A 0.00009 −0.00009 −0.00011
Covariance Matrix B −0.00009 0.00032 −0.00007
C −0.00011 −0.00007 0.00122
Portfolio
A 63.7%
B 27.2%
C 9.1%
Expected Return Required Return
11.00% 11.00%
Portfolio Variance
2.5312E-05
J
Total 100.0%
68. How much must the objective function coefficient of the variable X2 increase before any X2s are produced based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4
Name Number to make: X1 Number to make: X2
Final Value 9.428 0
Reduced Gradient 0 −1.96
Name
Final Value
Lagrange Multiplier
Constraints Cell
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ch 7 $D$8 $D$9 $D$10
Used Used Used
42 132 24
0 0.214 1.214
69. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. If the Analytic Solver Platform is used, which are the Objective, Variables and Constraint cells in the spreadsheet for this problem?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Amount Invested: Annual Return:
$12,000 18%
Year 1 2 3 4 5 6 7 8 9 10
Beginning Balance $12,000 $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994
C
D
E
F
Investment Earnings $2,346 $2,376 $2,403 $2,428 $2,452 $2,476 $2,497 $2,515 $2,531 $2,540
Earnings After Taxes $1,689 $1,711 $1,730 $1,748 $1,765 $1,782 $1,798 $1,811 $1,822 $1,829
Premium Due $1,536 $1,570 $1,602 $1,629 $1,643 $1,671 $1,708 $1,731 $1,773 $1,829
Ending Balance $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994 $12,994
70. The Sweet Water beverage company is designing a new soft drink can. The designers wish to minimize the manufacturing cost of the can, a cost that is directly related to the amount of aluminum used in the can. The can must hold at least 350 ml (or cm3) of beverage, have a diameter between 3 and 7 cm, and have a height between 7 and 19 cm. Formulate the NLP for Sweet Water. 71. How many local minimum solutions are there on this graph of a function Mark their locations on the graph.
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ch 7
72. How much are additional units of labor worth based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4
Name Number to make: X1 Number to make: X2
Final Value 4 8.571
Reduced Gradient 0 0
Name Wood Labor Hinges
Final Value 84 4 8.571
Lagrange Multiplier 0.714 3.714 0
Constraints Cell $D$8 $D$9 $D$10 Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
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ch 7 MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
E Parking 80,000
millions $
73. Refer to Exhibit 8.2. What formula would you place in cell D13 to calculate total cost? 74. A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formulas should go in cells D4:D9 of the spreadsheet for this problem?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 138.526 70 60 180 150
Y-Coordinate 116.227 160 90 90 120
Distance: 81.314 82.790 49.071 12.079
Total Distance:
225.253
75. An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item. Formulate the objective function for this problem. Let Q indicate the order quantity. Copyright Cengage Learning. Powered by Cognero.
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ch 7 76. An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item. What values should go in cells B3:B11 of the spreadsheet for this problem if Q = 223.61?
A 1 2 3 4 5 6 7 8 9 10 11
B
Annual Demand: Cost per Unit: Ordering Cost: Carrying Cost: Order Quantity: Total Cost:
77. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − .09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − .14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 2 2 30 50 − 0.09*P1 Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs.
B 4 8 20 30 − 0.14*P2
Available resources 150 220
Formulate the NLP for this company Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length
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ch 7 W = building width S = number of stories P = total parking area MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
E Parking 80,000
millions $
78. Refer to Exhibit 8.2. What formula would you place in cell B6 to calculate Total Lot Area? 79. How many local maximum solutions are there on this graph of a function? Mark their locations on the graph.
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ch 7 80. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − 0.09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − 0.14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 2 2 30 50 − 0.09*P1 Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs.
B 4 8 20 30 −14*P2
Available resources 150 220
The NLP for the problem is: 52.70 P1 − 0.09 P12 + 32.80 P2 − 0.14 P22 − 2100 X1 − 50 + 0.09P 1 = 0 X2 − 30 + 0.14P2 = 0 2 X1 + 4 X2 ≤ 150 2 X1 + 8 X2 ≤ 220 90 ≤ P1, P2 ≤ 140 X1, X2 ≥ 0 and the solution (P1, P2) = (140.0, 117.14) What values should go in cells B3:E18 of the spreadsheet for this problem? MAX: Subject to:
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
A
B
D
E
Used
Available
Price Min Price Max Price Marginal Cost Profit Margin Demand Total Profit Constraints: Labor Material
81. A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It Copyright Cengage Learning. Powered by Cognero.
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ch 7 has four stores and their coordinates are listed in the below.
X-Coordinate Y-Coordinate Store 1 70 160 Store 2 60 90 Store 3 180 90 Store 4 150 120 Formulate the NLP for this problem. Let X and Y represent the X, Y coordinates of the new warehouse. Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
E Parking 80,000
Cost
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ch 7 11 12 13
Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
millions $
82. Refer to Exhibit 8.2. What formula would you place into cell B5 to calculate Total Floor Area? 83. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What formulas should go in cells B7:F7 of the spreadsheet for this problem?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Amount Invested: Annual Return:
$12,000 18%
Year 1 2 3 4 5 6 7 8 9 10
Beginning Balance $12,000 $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994
C
D
E
F
Investment Earnings $2,346 $2,376 $2,403 $2,428 $2,452 $2,476 $2,497 $2,515 $2,531 $2,540
Earnings After Taxes $1,689 $1,711 $1,730 $1,748 $1,765 $1,782 $1,798 $1,811 $1,822 $1,829
Premium Due $1,536 $1,570 $1,602 $1,629 $1,643 $1,671 $1,708 $1,731 $1,773 $1,829
Ending Balance $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994 $12,994
84. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). Formulate the NLP for the problem. 85. Calculate the annual inventory costs for the following data. Order quantity Annual demand Unit purchase cost Fixed cost of placing an order Percentage cost of holding one unit in inventory for a year
= 400 units = 12,500 units = 50 = 75 = 20%
86. A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks. It has four stores and their coordinates are listed in the below.
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ch 7 X-Coordinate Y-Coordinate Store 1 70 160 Store 2 60 90 Store 3 180 90 Store 4 150 120 Formulate the objective function for this problem. Let X and Y represent the X, Y coordinates of the new warehouse. Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
Copyright Cengage Learning. Powered by Cognero.
E Parking 80,000
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ch 7 87. Refer to Exhibit 8.2. The company wishes to have a relatively square building. Thus, they wish neither the building length nor the building width exceed the other by more than 25%. Add constraint(s) to enforce this design constraint.
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ch 7 Answer Key 1. True 2. True 3. True 4. False 5. True 6. False 7. False 8. False 9. a 10. d 11. c 12. d 13. b 14. c 15. c 16. c 17. b 18. b 19. a 20. a 21. d 22. d 23. d 24. c 25. d Copyright Cengage Learning. Powered by Cognero.
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ch 7 26. d 27. a 28. c 29. b 30. b 31. b 32. c 33. d 34. c 35. b 36. c 37. a 38. d 39. a 40. b 41. d 42. b 43. b 44. c 45. a 46. b 47. d 48. a 49. b 50. b 51. a Copyright Cengage Learning. Powered by Cognero.
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ch 7 52. a 53. d 54. a 55. b 56. d 57. b 58. c 59. a 60. d 61. a 62. a 63. 1 2 3 4 5 6 7 8 9 10
A
B
C
D
Warehouse
120.0 X 70 60 180 150
117.4 Y 160 90 90 120 Total Max
Distance 65.85 65.85 65.85 30.14 227.698 65.854
Store 1 Store 2 Store 3 Store 4
64. MIN: Subject to:
0.00009 P12 + 0.00032 P22 + 0.00122 P32 + 2 (−0.00009 P1P2 − 0.00011 P1P3 − 0.00007 P2 P3) P1 + P2 + P3 = 1 0.1072 P1 + 0.1068 P2 + 0.1187 P3 ≥ 0.11 P1, P2, P3 ≥ 0 P1, P2, P3 ≤ 1
65. Objective Cell: D13 Variables Cells: Copyright Cengage Learning. Powered by Cognero.
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ch 7 B2:E2 Constraints Cells: B5 ≥ D5 B6 = D6 B7 ≥ D7 B8 ≤ D8 66. The maximum solution is at X = 19. 67. Cell B15 G4 G5 G6 H11 H14 J9
Formula =AVERAGE(B4:B13) =COVAR(B4:B13,$B$4:$B$13) =COVAR(B4:B13,$C$4:$C$13) =COVAR(B4:B13,$D$4:$D$13) =SUMPRODUCT(B15:D15,G9:I9) =SUMPRODUCT(MMULT(G9:I9,G4:I6),G9:I9) =SUM(G9:I9)
Copied to C15:D15 H4:I4 H5:I5 H6:I6
68. 1.96 69. Objective: Variables: Constraint: 70. Let
C3 C3 D7:D16≥E7:E16
D be the diameter of the can, H be the height of the can.
πDH + (π/2)D2 (π/4)D2H ≥ 350 3≤D≤7 7 ≤ H ≤ 19 Note: πDH is the cylindrical surface area while each of the two ends has area (π/4)D2 MIN Subject to:
71. There are three local minima, X = 0, X = 20, and X = 30. 72. 3.714 73. =SUMPRODUCT(B11:B12,D11:D12) 74. Cell D4 D9
Formula =SQRT((B4-$B$3)^2+(C4-$C$3)^2) =SUM(D4:D7)
Copied to D5:D7
75. MIN: 5000 (50) + 5000/Q (75) + Q/2 (50)(.30) 76. Copyright Cengage Learning. Powered by Cognero.
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ch 7
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
5,000
Cost per Unit: Ordering Cost: Carrying Cost:
$50 $75 30%
Order Quantity:
223.61
Total Cost:
$253,354
77. MAX: Subject to:
52.70 P1 − 0.09 P12 + 32.80 P2 − 0.14 P22 − 2100 X1 − 50 + 0.09P1 = 0 X2 − 30 + 0.14P2 = 0 2 X1 + 4 X2 ≤ 150 2 X1 + 8 X2 ≤ 220 90 ≤ P1, P2 ≤ 140 X1, X2 ≥ 0
78. =B2*C2+E2 79. There two local maximums, one at X = 13 and a second at X = 25. 80.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
C
Price Min Price Max Price Marginal Cost Profit Margin
A 140.00 90 140 30.00 110.00
B 117.14 90 140 20.00 97.14
Demand
37.40
13.60
Total Profit
5435.14
Constraints: Labor Material
2.00 2.00
4.00 8.00
D
E
Used 129.20 183.60
Available 150.00 220.00
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Date:
ch 7 81. MIN: Subject to:
Q
82. =B2*C2*D2 83. Cell B7 B8 C7 D7 F7 84. Let
MIN Subject to:
Formula =C2 =F7 =B7*(1+$C$3/4)^4-B7 =(1-.28)*C7 =B7+D7-E7
Copied to B9:B16 C8:C16 D8:D16 F8:F16
L = building length W = building width S = number of stories P = total parking area 0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
85. $629,343.75 86. MIN:
87. −L + 0.75W ≤ 0 L − 1.25W ≤ 0
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ch 8
Indicate whether the statement is true or false. 1. A NLP problem can have nonlinear objective and a nonlinear constraints a. True b. False 2. The solution procedure Solver uses to solve NLP problems is called the generalized reduced gradient (GRG) algorithm a. True b. False 3. A given local optimal solution is also a global optimal solution to a NLP problem a. True b. False 4. This selection of the initial feasible solution, often referred to as starting point, has no impact on the determination of the optimal solution to the NLP problem a. True b. False 5. The main difference between shadow prices and Lagrange multipliers involves the range of RHS values over which the shadow price or Lagrange multiplier remains valid a. True b. False 6. The process of formulating an NLP problem is virtually the same as formulating an LP problem a. True b. False 7. If Solver produces the completion message: "Solver found a solution. All constraints and optimality conditions are satisfied,” a global optimal solution has been found a. True b. False 8. Nonzero reduced gradient values indicate the exact impact on the objective function value of large changes in the value of a given variable a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. How much are additional units of Labor worth based on the following sensitivity report?
Changing Cells Cell $B$4
Name Number to make: X1
Final Value 9.42
Reduced Gradient 0
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ch 8 $C$4
Number to make: X2
1.71
0
Name Wood Labor Plywood
Final Value 42 132 24
Lagrange Multiplier 0 1.21 2.57
Constraints Cell $D$8 $D$9 $D$10 a. 0 b. 1.21 c. 2.57 d. 9.42 10. NLP problems which have slack in all the constraints a. are infeasible. b. are only at local optimal solutions. c. have zeros for all shadow prices. d. are at an interior point of the feasible region. 11. If the "Standard LP/Quadratic Engine" option is chosen in the Analytic Solver Platform (ASP) task pane, ASP a. will test for non-linearity in the model and apply GRG. b. will test for non-linearity in the model and apply Simplex. c. uses the B & B algorithm to solve the problem. d. will apply Simplex if the linearity test passes. 12. The main difference between linear (LP) and nonlinear programming problems (NLP) is that a. NLP may have a nonlinear objective function b. All constraints in NLP must be nonlinear c. No interaction terms are allowed in NLP d. Only one constraint in NLP can be nonlinear 13. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What constraint would you impose on this problem to ensure that at least one order is placed per year?
1 2 3 4 5 6 7 8 9
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
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ch 8 10 11
Total Cost:
$502,738.61
a. MAX B9 b. B9 = 1 c. B9 ≥ 1 d. MIN B11 14. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 5% return but with minimum risk.
Let:
Pi = proportion of total funds invested in stock i σi2 = variance of stock i σij = covariance between stocks i and j. Ri = average return on stock i What is the objective function for the NLP formulation of this problem? a. b. c. d. 15. Which point or points are global optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function.
a. B b. D c. E d. F 16. When using the GRG algorithm to solve NLPs one should try multiple starting points because a. If two different starting points return the same solution, that solution is optimal. b. the solution returned depends upon the starting point. c. the solution returned is always near the starting point. d. a random element of GRG requires multiple starting points.
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ch 8 17. A company has collected the following inventory data for an item. What is the total annual cost for this item? Annual demand for the item Unit purchase cost for the item Fixed cost of placing an order Cost of holding one unit in inventory for a year Order quantity a. 275 b. 450 c. 5,275 d. 5,450
D = 500 C = 10 S = 20 i = 30% Q = 50
18. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D9 of the spreadsheet?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 163.800 100 90 210 180
Y-Coordinate 101.000 130 60 80 110
Distance: 70.082 84.424 50.749 18.532
Total Distance:
223.787
a. =MIN(D4:D7)^2 b. =SUM(D4:D7) c. =SQRT(D4^2+D5^2+D6^2+D7^2) d. =SQRT(MMULT(B3:C3,D4:D7)) 19. The main difference between shadow prices and Lagrange multipliers is a. the range of RHS values over which the Lagrange multipliers or the shadow prices remain valid b. that shadow prices overestimate the impact of a small change in the RHS of the constraints on the objective function c. that shadow prices underestimate the impact of a small change in the RHS of the constraints on the objective function d. they are equivalent 20. In solving the NLP problem, Solver found a degenerate model. This means that: a. Solver is cycling and redundant constraints must be removed b. a global optimal solution was found c. the objective function changed very slowly for the last few iterations d. a local optimum was found 21. The optimal solution to a NLP problem can occur at a(n) I.
corner point.
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ch 8 II.
interior point. a. I is true b. II is true c. I and II are true d. neither I or II is true
22. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What formula goes in cell B11 in this problem?
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
Total Cost:
$502,738.61
a. =B3*B7+B9/B3*B6+B9*2+B5*B7 b. =B3*B5+B3/B9*B6+B9/2*B5*B7 c. =SQRT(B3*B5+B3/B9*B6+B9/2*B5*B7) d. =B3*B3+B3/B9+B6*B9/2*B5*B7 23. The Analytic Solver Platform solution strategy for NLP problems is the GRG method. What does GRG stand for? a. Graphical Residual Gradient b. Generalized Reduced Gradient c. Goal Restricted Gradient d. Gradually Reduced Gradient 24. A nonzero reduced gradient value indicates the approximate impact a. on the objective function of very small changes in the value of a given variable b. on the RHS of a constraint of a unitary change in the objective function value c. on the objective function of very large changes in the value of a given variable d. on the objective function of unitary change in the RHS of associated constraint 25. In solving the NLP problem, Solver produced a message "Solver cannot improve the current solution. All constraints are satisfied." This means that Solver found: a. a local optimal solution b. a global optimal solution c. the objective function changed very slowly for the last few iterations Copyright Cengage Learning. Powered by Cognero.
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ch 8 d. a degenerate model 26. The main difference between linear (LP) and nonlinear programming problems (NLP) is that a. NLP must have a nonlinear objective function b. Some constraints in NLP may be nonlinear c. No interaction terms are allowed in NLP d. Only one constraint in NLP can be nonlinear 27. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What constraint must be entered in the Analytic Solver Platform (ASP) task pane?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Amount Invested: Annual Return:
$9,000 10%
Year 1 2 3 4 5 6 7 8 9 10
Beginning Balance $9,000 $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207
C
D
E
F
Investment Earnings $971 $993 $1,012 $1,030 $1,046 $1,061 $1,075 $1,086 $1,096 $1,101
Earnings After Taxes $699 $715 $729 $741 $753 $764 $774 $782 $789 $793
Premium Due $500 $534 $566 $593 $607 $635 $672 $695 $737 $793
Ending Balance $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155 $10,207 $10,207
a. $D$7:$D$16 ≤ $E$7:$E$16 b. $C$7:$D$16 ≥ $E$7:$E$16 c. $C$7:$C$16 ≥ $E$7:$E$16 d. $D$7:$D$16 ≥ $E$7:$E$16 28. An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell H11 of the Excel spreadsheet?
A 1 2 3 4 5
Year 1 2
B
A 3.98% −1.51%
C Annual Return B 7.38% 10.45%
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D
C 8.76% 6.66%
E
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ch 8 6 7 8 9 10 11 12 13 14 15 < < < < < < < < < < < < < < < <
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3 4 5 6 7 8 9 10
5.36% 4.98% 3.12% 5.58% 3.49% −2.37% 5.92% 5.69%
5.15% 5.51% 6.93% −4.32% 9.78% 5.62% 6.28% 5.75%
6.55% −1.58% 8.43% 9.07% 9.65% 8.96% −7.10% 10.96%
Average
3.42%
5.85%
6.03%
> > > > > > > > > >
F
G
H
I
A B C
Covariance Matrix A 0.00080 −0.00044 −0.00045
B −0.00044 0.00144 −0.00012
C −0.00045 −0.00012 0.00300
Portfolio
A 36.8%
B 40.6%
C 22.6%
Expected Return Required Return
5.00% 5.00%
Portfolio Variance
0.000274
J
Total 100.0%
a. =SUMPRODUCT(B15:D15,G9:I9) b. =SUMPRODUCT(B15:D15,B4:D13) c. =SUMPRODUCT(B15:D15,I9) d. =SUMPRODUCT(B15:D15,G4:I6) 29. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. Which cell(s) in the spreadsheet represent the decision variables in the problem?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 163.800 100 90 210 180
Y-Coordinate 101.000 130 60 80 110
Distance: 70.082 84.424 50.749 18.532
Total Distance:
223.787
a. B3 b. B3:C3 Copyright Cengage Learning. Powered by Cognero.
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ch 8 c. B4:C7 d. D9 30. What is the search path for the following feasible solution space? The dashed line represents the objective function and the objective is to maximize the value of the objective function.
a. A, D b. A, F, E, D c. A, B, C, D d. A, F, E 31. The GRG algorithm operates by a. moving in the direction of most rapid improvement in the objective function. b. choosing a search direction at random. c. searching directly for the optimum solution. d. moving in a clockwise direction. 32. Why does the GRG algorithm not provide allowable increase or allowable decrease information with the Reduced Gradient and Lagrange multiplier information? a. Because all constraints have slack. b. Because the constraints may not be linear. c. Because the solution is not necessarily a global optimum. d. Because the solutions are not necessarily corner points. 33. The optimal trade-off between risk and return for a given portfolio problem can be summarized by the a. efficient frontier. b. investment frontier. c. portfolio boundary. d. variance boundary. 34. The main difference between LP and NLP problems is that NLPs will have a a. linear objective function and nonlinear or linear constraints. b. minimum of one nonlinear constraint or a nonlinear objective function. c. multilevel objective functions. d. multilevel objective function and nonlinear constraints. 35. A company wants to locate a new warehouse to minimize the distance traveled by its delivery trucks. It has four stores Copyright Cengage Learning. Powered by Cognero.
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ch 8 and their coordinates are listed in the accompanying spreadsheet. What formula goes in cell D4 of the spreadsheet?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 163.800 100 90 210 180
Y-Coordinate 101.000 130 60 80 110
Distance: 70.082 84.424 50.749 18.532
Total Distance:
223.787
a. =(B4-$B$3)^2+(C4-$C$3)^2 b. =SQRT((B4-$B$3)+(C4-$C$3)) c. =SQRT((B4-$B$3)^2+(C4-$C$3)^2) d. =(B4-$B$3)+(C4-$C$3) 36. The straight line (Euclidean) distance between two points (X1, Y1) and (X2, Y2) is calculated as a. X1 − Y1 + X2 − Y2 b. (X1 − X2)2 + (Y1 − Y2)2 c. d. 37. The Lagrange multipliers can be used to a. estimate the impact of a small change in the RHS of the constraints on the objective function b. estimate the impact of an arbitrary change in the RHS of the constraints on the objective function c. estimate the impact of a small change in the RHS of the constraints on the shadow prices d. estimate the impact of a large change in the RHS of the constraints on the shadow prices 38. An investor is developing a portfolio of stocks. She has identified 3 stocks to invest in. She wants to earn at least 5% return but with minimum risk. The problem data is given in the following Excel spreadsheet. What formula should be entered in cell G4 of the Excel spreadsheet?
A 1 2 3 4 5 6 7 8 9 10
Year 1 2 3 4 5 6 7
B
C
D
A 3.98% −1.51% 5.36% 4.98% 3.12% 5.58% 3.49%
Annual Return B 7.38% 10.45% 5.15% 5.51% 6.93% −4.32% 9.78%
C 8.76% 6.66% 6.55% −1.58% 8.43% 9.07% 9.65%
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E
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ch 8 11 12 13 14 15
Average
< < < < < < < < < < < < < < < <
−2.37% 5.92% 5.69%
8 9 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
3.42%
5.62% 6.28% 5.75% 5.85%
8.96% −7.10% 10.96%
> > > >
6.03%
>
F
G
H
I
A B C
Covariance Matrix A 0.00080 −0.00044 −0.00045
B −0.00044 0.00144 −0.00012
C −0.00045 −0.00012 0.00300
Portfolio
A 36.8%
B 40.6%
C 22.6%
Expected Return Required Return
5.00% 5.00%
Portfolio Variance
0.000274
J
Total 100.0%
a. =COVAR(B4:B13) b. =VARP(B4:B13) c. =COVAR(B4:D13,$D$4:$B$13) d. =COVAR(B4:B13,$B$4:$B$13) 39. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $9,000 and hopes he can earn 12%, compounded quarterly, on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What formula goes in cell C7 of the spreadsheet?
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Amount Invested: Annual Return:
$9,000 10%
Year 1 2 3 4 5 6 7 8
Beginning Balance $9,000 $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068
C
D
E
F
Investment Earnings $971 $993 $1,012 $1,030 $1,046 $1,061 $1,075 $1,086
Earnings After Taxes $699 $715 $729 $741 $753 $764 $774 $782
Premium Due $500 $534 $566 $593 $607 $635 $672 $695
Ending Balance $9,199 $9,380 $9,543 $9,691 $9,837 $9,966 $10,068 $10,155
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ch 8 15 16
9 10
$10,155 $10,207
$1,096 $1,101
$789 $793
$737 $793
$10,207 $10,207
a. =B7*(1+$C$3/4)^4-B7 b. =B7*(1+$C$3)-B7 c. =B7*($C$3/4)^4-B7 d. =B7*(1+$C$3/4)^4 40. Which point or points are local optima in this diagram? The dashed line represents the objective function and the objective is to maximize the value of the objective function.
a. D b. E c. F d. D, F 41. The global optimum solution to a nonlinear programming problems (NLP), in which the objective function must be minimized, is: a. always larger than the local optimum solution b. always smaller than the local optimum solution c. is smaller or equal to the local optimum solution d. is larger or equal to the local optimum solution Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − .10*P1
B 2 2 18 30 − .07*P2
Available resources 150 200
Let X1 = demand for As and X2 = demand for Bs. Copyright Cengage Learning. Powered by Cognero.
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ch 8 Let P1 = price for As and P2 = price for Bs
A 1 2 3 Price 4 Marginal Cost 5 Profit Margin 6 7 Demand 8 9 Total Profit 10 11 Constraints: 12 Labor 13 Material
B
C
A 112.50 25.00 87.50
B 223.29 18.00 205.29
8.75
14.37
D
E
Used 54.99 37.49
Available 150.00 200.00
3715.58
3.00 1.00
2.00 2.00
42. Refer to Exhibit 8.1. What formula is used in cell B7 of the spreadsheet for this problem? a. = 30 − .07 * C3 b. = 20 − .1 * B3 c. = B3 − B4 d. = B5 * B7 + C5 * C7 43. Refer to Exhibit 8.1. What formula is used in cell D12 of the spreadsheet for this problem? a. =SUMPRODUCT(B3:C3, B12:C12) b. =SUMPRODUCT(B4:C4, B12:C12) c. =SUMPRODUCT(B5:C5, B12:C12) d. =SUMPRODUCT(B7:C7, B12:C12) 44. The total annual cost for the economic order quantity model is a. purchase cost + ordering cost + holding cost. b. fixed ordering cost + holding cost. c. purchase cost + fixed ordering cost. d. unit cost + variable ordering cost + holding cost. 45. The GRG and Simplex algorithms are similar in that a. each algorithm process continues until there is no further improvement in the objective function. b. both algorithms take their starting solution from the spreadsheet. c. both return the globally optimal solution. d. both algorithms return a solution that satisfies at least one constraint at equality. 46. In a maximization problem, the GRG algorithm's search for a feasible direction of improvement equates to ____ in the simplex algorithm for LP problems. a. an adjacent corner point b. a zero reduced cost Copyright Cengage Learning. Powered by Cognero.
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ch 8 c. a positive shadow price d. a positive reduced cost 47. In the GRG algorithm the initial solution is called the a. originating point. b. insertion point. c. zero point. d. starting point. 48. The Reduced Gradient is similar to which of these terms from linear programming? a. Shadow Price b. Allowable Decrease c. Allowable Increase d. Reduced Cost 49. How much must the objective function coefficient of the variable Pumpkin change before any Pumpkins are produced based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4 $D$4
Final Value 9.52 0 10.79
Reduced Gradient 0 −499.99 0
Final Value 200000.00 99.00 37777.78 29.84 8000.00
Lagrange Multiplier 0.016 12.000 0 0 3.490
Name Corn Pumpkin Beans
Constraints Cell Name $E$8 Corn $E$9 Pumpkin $E$10 Beans $E$11 Water $E$12 Fertilizer a. increase by 12 b. decrease by 12 c. increase by 499.99 d. decrease by 499.99
50. The GRG algorithm terminates when it a. has completed 100 iterations. b. has reached the global optimal solution. c. detects no feasible direction for improvement. d. reaches the steepest gradient. 51. Which of the following is not an assumption of an EOQ problem. a. Demand for a product is fairly constant. Copyright Cengage Learning. Powered by Cognero.
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ch 8 b. Inventory depletion rates vary non-linearly. c. Each new order is delivered in full when the inventory level reaches 0. d. All are valid assumptions. 52. In solving the NLP problem, Solver produced a message "Solver has converged to the current solution. All constraints are satisfied." This means that: a. Solver found a local optimal solution b. Solver found a global optimal solution c. the objective function changed very slowly for the last few iterations d. Your model is degenerate 53. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of product A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of product B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − 0.10*P1 Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs.
B 2 2 18 30 − 0.07*P2
Available resources 150 200
The objective function for this problem is? a. MAX 25 X1 + 18 X2 b. MAX (20 − 0.10*P1) X1 + (30 − 0.07*P2) X2 c. MAX 20 P1 − 0.1 P12 + 30 P2 − 0.07 P22 − 1040 d. MAX 22.5 P1 − 0.1 P12 + 31.26 P2 − 0.07 P22 − 1040 54. What is the straight line (Euclidean) distance between the points (5,7) and (1, 11)? a. 2.83 b. 5.65 c. 8 d. 16 55. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the objective cell in this problem?
A
B
1 Copyright Cengage Learning. Powered by Cognero.
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Date:
ch 8 2 3 4 5 6 7 8 9 10 11
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
Total Cost:
$502,738.61
a. B3 b. B7 c. B9 d. B11 56. In NLP a local optimum is best described as a. an interior point. b. a corner point. c. a point yielding no improving direction. d. a good starting point for subsequent searches. 57. The global optimum solution to a nonlinear programming problems (NLP), in which the objective function must be maximized, is: a. always larger than the local optimum solution b. always smaller than the local optimum solution c. is smaller or equal to the local optimum solution d. is larger or equal to the local optimum solution 58. An office supply company is attempting to determine the order quantity for laser printer toner cartridges which are sold to local businesses. Annual demand is 20,000 units and each cartridge costs the store $25. It costs $30 to place an order and the inventory carrying cost rate is 25% of the value of the item. The following spreadsheet has been set up to solve the problem. What cell is the variable cell in this problem?
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
20,000
Cost per Unit: Ordering Cost: Carrying Cost:
$25 $30 25%
Order Quantity:
483.73
Total Cost:
$502,738.61
a. B3 b. B7 Copyright Cengage Learning. Powered by Cognero.
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ch 8 c. B9 d. B11 Exhibit 8.1 The following questions pertain to the problem and spreadsheet below. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 25 to make and demand is estimated to be 20 − .10 * Price of A. A unit of B costs 18 to make and demand is estimated to be 30 − .07 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 100 and 200.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 3 1 25 20 − .10*P1
B 2 2 18 30 − .07*P2
Available resources 150 200
Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs
A 1 2 3 Price 4 Marginal Cost 5 Profit Margin 6 7 Demand 8 9 Total Profit 10 11 Constraints: 12 Labor 13 Material
B
C
A 112.50 25.00 87.50
B 223.29 18.00 205.29
8.75
14.37
D
E
Used 54.99 37.49
Available 150.00 200.00
3715.58
3.00 1.00
2.00 2.00
59. Refer to Exhibit 8.1. What formula is used in cell B9 of the spreadsheet for this problem? a. =B3*B7+C3*C7 b. =B5*B7+C5*C7 c. =(B5-B4)*B7+(C5-C4)*B7 d. =B3*B7+C3*B7+B4*B7+C4*B7 60. The optimal solution to a LP problem is always at I. II. III.
a corner point. an interior point the origin.
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ch 8 a. I is true b. II is true c. III is true d. I and III are true 61. The Lagrange Multiplier is similar to which of these terms from linear programming? a. Shadow Price b. Allowable Increase c. Allowable Decrease d. Reduced Cost 62. In solving the NLP problem, Solver produced a message "Solver found a solution. All constraints and optimality conditions are satisfied." This means that Solver found: a. a local optimal solution b. a global optimal solution c. the objective function changed very slowly for the last few iterations d. a degenerate model
63. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. If the Analytic Solver Platform is used, which are the Objective, Variables and Constraint cells in the spreadsheet for this problem?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Amount Invested: Annual Return:
$12,000 18%
Year 1 2 3 4 5 6 7 8 9 10
Beginning Balance $12,000 $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994
C
D
E
F
Investment Earnings $2,346 $2,376 $2,403 $2,428 $2,452 $2,476 $2,497 $2,515 $2,531 $2,540
Earnings After Taxes $1,689 $1,711 $1,730 $1,748 $1,765 $1,782 $1,798 $1,811 $1,822 $1,829
Premium Due $1,536 $1,570 $1,602 $1,629 $1,643 $1,671 $1,708 $1,731 $1,773 $1,829
Ending Balance $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994 $12,994
64. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk. Copyright Cengage Learning. Powered by Cognero.
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ch 8 Let: Pi = proportion of total funds invested in i, i = A, B, C The NLP for this problem is: 0.00009 P12 + 0.00032 P22 + 0.00122 P32 + 2 (−0.00009 P1P2 − 0.00011 P1P3 − 0.00007 P2P3) Subject to: P1 + P2 + P3 = 1 0.1072 P1 + 0.1068 P2 + 0.1187 P3 ≥ 0.11 P1, P2, P3 ≥ 0 P1, P2, P3 ≤ 1 What formulas should go in cells G4:J14 of the spreadsheet for this problem? NOTE: Formulas are not required in all of these cells. MIN:
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 < < < < < < < < < < < < < < <
B
C
D
Year 1 2 3 4 5 6 7 8 9 10
A 9.73% 10.83% 12.14% 9.14% 11.16% 11.60% 11.06% 11.22% 9.25% 11.11%
Annual Return B 12.54% 9.52% 11.47% 13.72% 8.89% 10.72% 12.21% 8.56% 11.09% 8.10%
C 10.23% 16.10% 4.07% 12.93% 11.97% 12.03% 14.00% 16.28% 12.99% 8.06%
Average
10.72%
10.68%
11.87%
F 1 2 3 4 5 6 7 8 9 10 11 12 13 14
E
> > > > > > > > > > > > > > > >
G
H
I
A B C
A 0.00009 −0.00009 −0.00011
Covariance Matrix B −0.00009 0.00032 −0.00007
C −0.00011 −0.00007 0.00122
Portfolio
A 63.7%
B 27.2%
C 9.1%
Expected Return Required Return
11.00% 11.00%
Portfolio Variance
2.5312E-05
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J
Total 100.0%
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ch 8 <
15
65. An investor wants to determine how much interest he must earn to be able to make the payments on a 10-year mortgage which has increasing annual payments. The problem is summarized in the accompanying spreadsheet. The investor has enough money to make an initial investment of $12,000 and hopes he can earn 18% on his investments. He would like to know how low his annual return can be and still allow him to make his payments from interest income. What formulas should go in cells B7:F7 of the spreadsheet for this problem?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Amount Invested: Annual Return:
$12,000 18%
Year 1 2 3 4 5 6 7 8 9 10
Beginning Balance $12,000 $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994
C
D
E
F
Investment Earnings $2,346 $2,376 $2,403 $2,428 $2,452 $2,476 $2,497 $2,515 $2,531 $2,540
Earnings After Taxes $1,689 $1,711 $1,730 $1,748 $1,765 $1,782 $1,798 $1,811 $1,822 $1,829
Premium Due $1,536 $1,570 $1,602 $1,629 $1,643 $1,671 $1,708 $1,731 $1,773 $1,829
Ending Balance $12,153 $12,294 $12,422 $12,541 $12,664 $12,775 $12,865 $12,945 $12,994 $12,994
66. A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It has four stores and their coordinates are listed in the below.
X-Coordinate Y-Coordinate Store 1 70 160 Store 2 60 90 Store 3 180 90 Store 4 150 120 Formulate the NLP for this problem. Let X and Y represent the X, Y coordinates of the new warehouse. 67. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − .09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − .14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 2 2 30 50 − 0.09*P1 Let X1 = demand for As and X2 = demand for Bs.
B 4 8 20 30 − 0.14*P2
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Available resources 150 220
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ch 8 Let P1 = price for As and P2 = price for Bs. Formulate the NLP for this company Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
E Parking 80,000
millions $
68. Refer to Exhibit 8.2. What formula would you place into cell B5 to calculate Total Floor Area? Copyright Cengage Learning. Powered by Cognero.
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ch 8 69. A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks. It has four stores and their coordinates are listed in the below.
X-Coordinate Y-Coordinate Store 1 70 160 Store 2 60 90 Store 3 180 90 Store 4 150 120 Formulate the objective function for this problem. Let X and Y represent the X, Y coordinates of the new warehouse. 70. The Sweet Water beverage company is designing a new soft drink can. The designers wish to minimize the manufacturing cost of the can, a cost that is directly related to the amount of aluminum used in the can. The can must hold at least 350 ml (or cm3) of beverage, have a diameter between 3 and 7 cm, and have a height between 7 and 19 cm. Formulate the NLP for Sweet Water. 71. How much are additional units of labor worth based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4
Name Number to make: X1 Number to make: X2
Final Value 4 8.571
Reduced Gradient 0 0
Name Wood Labor Hinges
Final Value 84 4 8.571
Lagrange Multiplier 0.714 3.714 0
Constraints Cell $D$8 $D$9 $D$10
72. An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item. What values should go in cells B3:B11 of the spreadsheet for this problem if Q = 223.61?
A 1 2 3 4 5 6 7 8 9
B
Annual Demand: Cost per Unit: Ordering Cost: Carrying Cost: Order Quantity:
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ch 8 10 11
Total Cost:
73. How many local maximum solutions are there on this graph of a function? Mark their locations on the graph.
74. A company makes products A and B from 2 resources, labor and material. The company wants to determine the selling price which will maximize profits. A unit of A costs 30 to make and demand is estimated to be 50 − 0.09 * Price of A. A unit of B costs 20 to make and demand is estimated to be 30 − 0.14 * Price of B. The utilization of labor and materials and the available quantity of resources is shown in the table. A reasonable price for the products is between 90 and 140.
Product Labor (hr/unit) Material (ounces/unit) Manufacturing cost($/unit) Demand (units)
A 2 2 30 50 − 0.09*P1 Let X1 = demand for As and X2 = demand for Bs. Let P1 = price for As and P2 = price for Bs.
B 4 8 20 30 −14*P2
Available resources 150 220
The NLP for the problem is: 52.70 P1 − 0.09 P12 + 32.80 P2 − 0.14 P22 − 2100 X1 − 50 + 0.09P 1 = 0 X2 − 30 + 0.14P2 = 0 2 X1 + 4 X2 ≤ 150 2 X1 + 8 X2 ≤ 220 90 ≤ P1, P2 ≤ 140 X1, X2 ≥ 0 and the solution (P1, P2) = (140.0, 117.14) What values should go in cells B3:E18 of the spreadsheet for this problem? MAX: Subject to:
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ch 8
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
A
B
D
E
Used
Available
Price Min Price Max Price Marginal Cost Profit Margin Demand Total Profit Constraints: Labor Material
Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1
B Length
C Width
D Stories
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ch 8 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
108.39492 300
92.255243 300
6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
80,000
millions $
75. Refer to Exhibit 8.2. What formula would you place in cell D13 to calculate total cost? 76. An investor is developing a portfolio of stocks. She has identified 3 stocks in which to invest. She wants to earn at least 11% return but with minimum risk. The average return for the stocks is:
Annual Return A B Average 10.72% 10.68% The covariance matrix for the stocks is:
C 11.87%
A B C A 0.00009 −0.00009 −0.00011 B −0.00009 0.00032 −0.00007 C −0.00011 −0.00007 0.00122 Let: Pi = proportion of total funds invested in i, i = A, B, C Formulate the NLP for this problem. 77. An office supply company is attempting to determine the order quantity for Mt. White fountain pens which are sold to local executives. Annual demand is 5,000 units and each pen costs the store $50. It costs $75 to place an order and the inventory carrying cost rate is 30% of the value of the item. Formulate the objective function for this problem. Let Q indicate the order quantity. 78. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). Formulate the NLP for the problem. 79. A company wants to locate a new warehouse to minimize the distance travelled by its delivery trucks. It has four stores and their coordinates are listed in the accompanying spreadsheet. What formulas should go in cells D4:D9 of the Copyright Cengage Learning. Powered by Cognero.
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ch 8 spreadsheet for this problem?
1 2 3 4 5 6 7 8 9
A
B
C
D
Warehouse Store 1 Store 2 Store 3 Store 4
X-Coordinate 138.526 70 60 180 150
Y-Coordinate 116.227 160 90 90 120
Distance: 81.314 82.790 49.071 12.079
Total Distance:
225.253
80. A company wants to locate a new warehouse to minimize the longest distance travelled by any of its delivery trucks. It has four stores and their coordinates are listed in the below.
X-Coordinate Y-Coordinate Store 1 70 160 Store 2 60 90 Store 3 180 90 Store 4 150 120 Let X and Y represent the X, Y coordinates of the new warehouse. The NLP for this problem and solution is the following.
MIN: Subject to:
Q
Solution is (X, Y) = (120.0, 117.4). What values should go in cells B2:D9 of the spreadsheet for this problem?
A 1 2 3 4 5 6 7 8 9 10
B
C
D
X
Y
Distance
Warehouse Store 1 Store 2 Store 3 Store 4 Total Max
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ch 8 81. Find the maximum solution on this graph of a function starting from X = 12. Mark its location on the graph.
82. How much must the objective function coefficient of the variable X2 increase before any X2s are produced based on the following sensitivity report?
Changing Cells Cell $B$4 $C$4
Name Number to make: X1 Number to make: X2
Final Value 9.428 0
Reduced Gradient 0 −1.96
Name Used Used Used
Final Value 42 132 24
Lagrange Multiplier 0 0.214 1.214
Constraints Cell $D$8 $D$9 $D$10 Exhibit 8.2 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories
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ch 8 P = total parking area MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
E Parking 80,000
millions $
83. Refer to Exhibit 8.2. The company wishes to have a relatively square building. Thus, they wish neither the building length nor the building width exceed the other by more than 25%. Add constraint(s) to enforce this design constraint. 84. How many local minimum solutions are there on this graph of a function Mark their locations on the graph.
Exhibit 8.2 Copyright Cengage Learning. Powered by Cognero.
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ch 8 The following questions pertain to the problem and spreadsheet below. A construction company just purchased a 300 × 300 foot lot upon which they plan to build an office building. They need at least 60,000 ft2 of office floor space. Zoning regulations require each floor be 10 feet high and the building not exceed 65 ft in height. Further, parking space must equal at least 30% of the total floor space available. The company's cost accountant uses a 60% factor of the building height and a 1% factor of any story's floor area to calculate the total building cost (in millions of dollars). The following is the NLP formulation for the problem.
Let
L = building length W = building width S = number of stories P = total parking area
MIN Subject to:
0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
The spreadsheet implementation of this formulation applies to the following questions.
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Max Values
B Length 108.39492 300
C Width 92.255243 300
D Stories 6 6
Total Floor Area Total Lot Area Min. Parking Area Max. Bld Height
60,000 90,000 65,000 60
≥ = ≥ ≤
60,000 90,000 0 65
Cost Height Factor Area Factor
0.6 0.01
Total Height Floor Area Cost
60 10,000 $136.0
E Parking 80,000
millions $
85. Refer to Exhibit 8.2. What values would you enter in the Analytic Solver Platform task pane for the above Excel spreadsheet? Objective Cell: Variables Cells: Constraints Cells: Copyright Cengage Learning. Powered by Cognero.
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ch 8 86. Refer to Exhibit 8.2. What formula would you place in cell B6 to calculate Total Lot Area? 87. Calculate the annual inventory costs for the following data. Order quantity Annual demand Unit purchase cost Fixed cost of placing an order Percentage cost of holding one unit in inventory for a year
= 400 units = 12,500 units = 50 = 75 = 20%
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ch 8 Answer Key 1. True 2. True 3. False 4. False 5. True 6. True 7. False 8. False 9. b 10. d 11. d 12. a 13. c 14. b 15. b 16. b 17. c 18. b 19. a 20. a 21. c 22. b 23. b 24. a 25. d Copyright Cengage Learning. Powered by Cognero.
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ch 8 26. b 27. d 28. a 29. b 30. c 31. a 32. b 33. a 34. b 35. c 36. d 37. a 38. d 39. a 40. d 41. c 42. b 43. d 44. a 45. a 46. d 47. d 48. d 49. c 50. c 51. b Copyright Cengage Learning. Powered by Cognero.
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ch 8 52. c 53. d 54. b 55. d 56. c 57. d 58. c 59. b 60. a 61. a 62. a 63. Objective: Variables: Constraint:
C3 C3 D7:D16≥E7:E16
64. Cell B15 G4 G5 G6 H11 H14 J9
Formula =AVERAGE(B4:B13) =COVAR(B4:B13,$B$4:$B$13) =COVAR(B4:B13,$C$4:$C$13) =COVAR(B4:B13,$D$4:$D$13) =SUMPRODUCT(B15:D15,G9:I9) =SUMPRODUCT(MMULT(G9:I9,G4:I6),G9:I9) =SUM(G9:I9)
Copied to C15:D15 H4:I4 H5:I5 H6:I6
65. Cell B7 B8 C7 D7 F7
Formula =C2 =F7 =B7*(1+$C$3/4)^4-B7 =(1-.28)*C7 =B7+D7-E7
Copied to
66. MIN: Subject to:
B9:B16 C8:C16 D8:D16 F8:F16
Q
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ch 8
67. MAX: Subject to:
52.70 P1 − 0.09 P12 + 32.80 P2 − 0.14 P22 − 2100 X1 − 50 + 0.09P1 = 0 X2 − 30 + 0.14P2 = 0 2 X1 + 4 X2 ≤ 150 2 X1 + 8 X2 ≤ 220 90 ≤ P1, P2 ≤ 140 X1, X2 ≥ 0
68. =B2*C2*D2 69. MIN:
70. Let
D be the diameter of the can, H be the height of the can.
πDH + (π/2)D2 (π/4)D2H ≥ 350 3≤D≤7 7 ≤ H ≤ 19 Note: πDH is the cylindrical surface area while each of the two ends has area (π/4)D2 MIN Subject to:
71. 3.714 72.
1 2 3 4 5 6 7 8 9 10 11
A
B
Annual Demand:
5,000
Cost per Unit: Ordering Cost: Carrying Cost:
$50 $75 30%
Order Quantity:
223.61
Total Cost:
$253,354
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ch 8 73. There two local maximums, one at X = 13 and a second at X = 25. 74.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
C
Price Min Price Max Price Marginal Cost Profit Margin
A 140.00 90 140 30.00 110.00
B 117.14 90 140 20.00 97.14
Demand
37.40
13.60
Total Profit
5435.14
Constraints: Labor Material
2.00 2.00
4.00 8.00
D
E
Used 129.20 183.60
Available 150.00 220.00
75. =SUMPRODUCT(B11:B12,D11:D12) 76. MIN: Subject to:
0.00009 P12 + 0.00032 P22 + 0.00122 P32 + 2 (−0.00009 P1P2 − 0.00011 P1P3 − 0.00007 P2 P3) P1 + P2 + P3 = 1 0.1072 P1 + 0.1068 P2 + 0.1187 P3 ≥ 0.11 P1, P2, P3 ≥ 0 P1, P2, P3 ≤ 1
77. MIN: 5000 (50) + 5000/Q (75) + Q/2 (50)(.30) 78. Let
MIN Subject to:
L = building length W = building width S = number of stories P = total parking area 0.60*10*X + 0.01*L*W S * L * W ≥ 60,000 L * W + P = 90,000 P ≥ 0.30 * S * L * W S * 10 ≤ 65 0 ≤ L, W ≤ 300 0≤S≤6 P≥0
{total floor area} {total lot area} {min. parking area} {max. building height}
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ch 8 79. Cell D4 D9
Formula =SQRT((B4-$B$3)^2+(C4-$C$3)^2) =SUM(D4:D7)
Copied to D5:D7
80. 1 2 3 4 5 6 7 8 9 10
A
B
C
D
Warehouse
120.0 X 70 60 180 150
117.4 Y 160 90 90 120 Total Max
Distance 65.85 65.85 65.85 30.14 227.698 65.854
Store 1 Store 2 Store 3 Store 4
81. The maximum solution is at X = 19. 82. 1.96 83. −L + 0.75W ≤ 0 L − 1.25W ≤ 0 84. There are three local minima, X = 0, X = 20, and X = 30. 85. Objective Cell: D13 Variables Cells: B2:E2 Constraints Cells: B5 ≥ D5 B6 = D6 B7 ≥ D7 B8 ≤ D8 86. =B2*C2+E2 87. $629,343.75
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ch 9
Indicate whether the statement is true or false. 1. In regression modeling, the objective is to determines the values of model coefficients that minimize the sum of squared estimation errors, or error sum of squares (ESS). a. True b. False 2. The term multicollinearity is used to describe the situation when the independent variables in a regression model are correlated among themselves. a. True b. False 3. In regression analysis, we consider models of the form: Y = f(X1, X2, ..., Xk) + a. True b. False 4. The R2 statistic (also referred to as the coefficient of determination) ranges in value from 0 to 1 (0 R2 1) and indicates the proportion of the total variation in the dependent variable Y around its mean (average) that is accounted for by the independent variable(s) in the estimated regression function. a. True b. False 5. A residual is defined as the difference between the fitted value based on a model and a corresponding actual value. a. True b. False 6. A simple linear regression model is of the form: Yi = 0 + 1X1i + i a. True b. False 7. In a model: Yi = 0 + 1X1i + i , the terms 0 and 1 are referred to as sample statistics. a. True b. False 8. The value of adjusted R2 can be negative. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. What is the formula for total sum of squares (TSS) a. b.
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ch 9 c. d.
10. On average, the differences between the actual and predicted values of Y a. are equal to b0. b. sum to an unknown value. c. are distributed uniformly. d. sum to zero. 11. An analyst has identified 3 independent variables (X1, X2, X3) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Which combination of variables provides the best regression results?
Independent Variable in the Model X1 X2 X1 and X2 X3 X1 and X3 X2 and X3 X1, X2 and X3
2
R2 Adjusted R
Se
Parameter Estimates
−0.1240 0.3104 0.2170 0.8214 0.7960 0.9824 0.9807
23.5480 18.4480 19.6540 9.3858 10.0330 2.9480 3.0850
b0 = 93.7174, b1 = 0.922 b0 = 57.0803, b2 = 1.545 b0 = 50.2927, b1 = 1.952, b2 = 1.554 b0 = 31.6238, b3 = 1.132 b0 = 31.133, b1 = 0.148, b3 = 1.132 b0 = 14.169, b2 = 0.985, b3 = 0.995 b0 = 11.113, b1 = 0.899, b2 = 0.990, b3 = 0.993
0.00089 0.38700 0.39100 0.84130 0.84130 0.98630 0.98710
a. X1 b. X1, X2 and X3 c. X1 and X2 d. X2 and X3 12. The adjusted R2 statistic a. is equal to the value of unadjusted R2 b. adjusts R2 for the degrees of freedom in the multiple regression model c. accounts for the parameters in the multiple regression model d. is always greater than R2 unadjusted 13. The estimated value of Y1 is given by a. b. c. d. Copyright Cengage Learning. Powered by Cognero.
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ch 9 14. The actual value of a dependent variable will generally differ from the regression equation estimate due to a. unaccounted for random variation. b. the inability of the nonlinear Solver to find optimal values. c. not building the regression model with enough data. d. the model R2 not equal to 1. 15. Based on the following regression output, what conclusion can you reach about β0?
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA Regression Residual Total
df 1 8 9
SS 3735.3060 704.7117 4440.0170
MS 3735.30600 88.08896
F 42.40379
Significance F 0.000186
Intercept X Variable 1
Coefficients 31.62378 1.131661
Standard Error 10.442970 0.173786
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
a. β0 = 0, with P-value = 0.016353 b. β0 ≠ 0, with P-value = 0.016353 c. β0 = 0, with P-value = 0.000186 d. β0 ≠ 0, with P-value = 0.000186 16. The regression residuals are computed as a. b. c. d. 17. The standard prediction error is a. always smaller than the standard error. b. used to construct confidence intervals for predicted values. c. measures the variability in the predicted values. d. all of these. 18. Which of the following represents a regression model? a. Copyright Cengage Learning. Powered by Cognero.
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ch 9 b. c. Y = f(X1, X2, ..., Xk) d. Y = f(X1, X2, ..., Xk) + ε 19. Polynomial regression is used when a. the independent variables are non-linear. b. there is a non-linear relationship between the dependent and independent variables. c. there is a non-linear relationship between the independent variables. d. there is a curvilinear change in the dependent variables. 20. A pattern resulting from random variation or unexplained causes is called a. noise b. trend c. seasonality d. time series 21. An analyst has identified 3 independent variables (X1, X2, X3) which might be used to predict Y. He has computed the regression equations using all combinations of the variables and the results are summarized in the following table. Why is the R2 value for the X3 model the same as the R2 value for the X1 and X3 model, but the Adjusted R2 values differ? Independent Variable in the Model X1 X2 X1 and X2 X3 X1 and X3 X2 and X3 All three
2
R2 Adjusted R
Se
Parameter Estimates
−0.1240 0.3104 0.2170 0.8214 0.7960 0.9824 0.9807
23.5480 18.4480 19.6540 9.3858 10.0330 2.9480 3.0850
b0 = 93.7174, b1 = 0.922 b0 = 57.0803, b2 = 1.545 b0 = 50.2927, b1 = 1.952, b2 = 1.554 b0 = 31.6238, b3 = 1.132 b0 = 31.133, b1 = 0.148, b3 = 1.132 b0 = 14.169, b2 = 0.985, b3 = 0.995 b0 = 11.113, b1 = 0.899, b2 = 0.990, b3 = 0.993
0.00089 0.38700 0.39100 0.84130 0.84130 0.98630 0.98710
a. The standard error for X1 is greater than the standard error for X3. b. X1 does not reduce ESS enough to compensate for its addition to the model. c. X1 does not reduce TSS enough to compensate for its addition to the model. d. X1 and X3 represent similar factors so multicollinearity exists. 22. Regression analysis is a modeling technique a. that assumes all data is normally distributed. b. for analyzing the relationship between dependent and independent variables. c. for examining linear trend data only. d. for capturing uncertainty in predicted values of Y. 23. The method of least squares finds estimates of parameter values that minimize: Copyright Cengage Learning. Powered by Cognero.
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ch 9 a. TSS. b. RSS. c. ESS. d. ESS + RSS. 24. The terms b0 and b1 are referred to as a. population variables. b. population parameters. c. estimated population variables. d. estimated population parameters. 25. Based on the following regression output, what proportion of the total variation in Y is explained by X?
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA Regression Residual Total
df 1 8 9
SS 3735.3060 704.7117 4440.0170
MS 3735.30600 88.08896
F 42.40379
Significance F 0.000186
Intercept X Variable 1
Coefficients 31.623780 1.131661
Standard Error 10.442970 0.173786
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
a. 0.917214 b. 0.841282 c. 0.821442 d. 9.385572 26. What goodness-of-fit measure is commonly used to evaluate a multiple regression function? a. R2 b. adjusted R2 c. partial R2 d. total R2 27. How many independent variables are there in simple regression analysis? a. 1 b. 2 c. 3 Copyright Cengage Learning. Powered by Cognero.
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ch 9 d. 4 28. What is the correct range for R2 values? a. (−1 ≤ R2 ≤ 0) b. (−1 ≤ R2 ≤ 1) c. (0 ≤ R2 ≤ 1) d. (0 ≤ R2 ≤ .5) 29. When using the Regression tool in Excel the dependent variable is entered as the a. X-range. b. Y-range. c. dependent-range. d. independent-range. 30. The error sum of squares term is used as a criterion for determining b0 and b1 because a. the sum of errors will always equal zero. b. the term can be solved for exact values of b0 and b1. c. both b0 and b1 can be easily calculated using the sum of squares term. d. all of these. 31. The problem of finding the optimal values of b0 and b1 is a. a linear programming problem. b. an unconstrained nonlinear optimization problem. c. a goal programming problem. d. a constrained nonlinear optimization problem. 32. What is a clear indicator of non-constant variance in a plot of regression model residuals? a. A non-linear trend in the residual plot. b. An intercept standard error larger that the estimated intercept coefficient. c. A funnel shaped trend in the residual plot. d. The standard errors from each independent variable differ. 33. A persistent upward or downward movement of data is called a. trend b. seasonality c. irregular variation d. dampening signal 34. Estimation errors are often referred to as a. mistakes. b. constant errors. c. residuals. Copyright Cengage Learning. Powered by Cognero.
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ch 9 d. squared errors. 35. The terms β0 and β1 are referred to as a. sample statistics b. random variables c. population variables d. population parameters 36. The regression function indicates the a. average value the dependent variable assumes for a given value of the independent variable. b. actual value the independent variable assumes for a given value of the dependent variable c. average value the dependent variable assumes for a given value of the dependent variable d. actual value the dependent variable assumes for a given value of the independent variable 37. In regression terms what does "best fit" mean? a. The estimated parameters, b0 and b1, are minimized. b. The estimated parameters, b0 and b1, are linear. c. The error terms are as small as possible. d. The largest error term is as small as possible. 38. Based on the following regression output, what is the equation of the regression line?
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA Regression Residual Total
df 1 8 9
SS 3735.3060 704.7117 4440.0170
MS 3735.30600 88.08896
F 42.40379
Significance F 0.000186
Intercept X Variable 1
Coefficients 31.623780 1.131661
Standard Error 10.442970 0.173786
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
a. b. c. d. 39. R2 is calculated as Copyright Cengage Learning. Powered by Cognero.
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ch 9 a. ESS/TSS b. 1 − (RSS/TSS) c. RSS/ESS d. RSS/TSS 40. In regression analysis, the total variation is: a. the sum of the squared deviations of each value of y from the mean of x b. the sum of the explained variation and unexplained variation c. the standard error of the forecast d. equal to R2 41. R2 measures a. the percentage of variability in the dependent variable, Y, explained by the model b. the unexplained variability c. the ratio of RSS/ESS d. the model sophistication 42. The reason an analyst creates a regression model is a. to determine the errors in the data collected. b. to predict a dependent variable value given specific independent variable values. c. to predict an independent variable value given specific dependent variable values. d. to verify the errors are normally distributed. 43. Residuals are assumed to be a. dependent, uniformly distributed random variables. b. independent, uniformly distributed random variables. c. dependent, normally distributed random variables. d. independent, normally distributed random variables. 44. Why do we create a scatter plot of the data in regression analysis? a. To compute the error terms. b. Because Excel calculates the function from the scatter plot. c. To visually check for a relationship between X and Y. d. To estimate predicted values. 45. The objective function in regression analysis is a. b. c. d.
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ch 9 46. When using the Regression tool in Excel the independent variable is entered as the a. X-range. b. Y-range. c. dependent-range. d. independent-range. 47. For a simple linear regression model, a 100(1 − α)% prediction interval for a new value of Y when X = Xh is computed as a. b. c. d. 48. The forecasting model that makes use of the least squares method is called a. regression b. naive approach c. moving average d. exponential smoothing 49. The R2 statistic a. varies between −1 and 1. b. compares the regression sum of squares to the total sum of squares. c. accounts for the number of parameters in the regression model. d. is the ratio of the error sum of squares to the regression sum of squares. 50. The terms b0 and b1 are a. estimated population parameters. b. estimated intercept and slope values, respectively. c. random variables. d. all of these. 51. You want to conduct a hypothesis test for β1. Based on the following regression output, what conclusion can you reach about β1?
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA df
SS
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MS
F
Significance F Page 9
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ch 9 Regression Residual Total
1 8 9
3735.3060 704.7117 4440.0170
3735.30600 88.08896
42.40379
0.000186
Intercept X Variable 1
Coefficients 31.623780 1.131661
Standard Error 10.44297 0.1737860
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
a. β1 = 0, with P-value = 0.016353 b. β1 ≠ 0, with P-value = 0.016353 c. β1 = 0, with P-value = 0.000186 d. β1 ≠ 0, with P-value = 0.000186 52. The β1 term indicates a. the average change in Y for a unit change in X. b. the Y value for a given value of X. c. the change in observed X for a given change in Y. d. the Y value when X equals zero. 53. Which of the following cannot be negative? a. coefficient of determination b. coefficient of correlation c. coefficient of the independent variable, x, in the regression equation d. y-intercept in the regression equation 54. The standard error measures the a. variability in the X values. b. variability in the actual data around the fitted regression function. c. variability in the independent variable around the fitted regression function. d. variability in the dependent variable around the fitted regression function. 55. The term ε in the regression model represents a. the slope of the regression model. b. a random error term. c. a correction for mistakes in measuring X. d. a correction for the fact that we are taking a sample. 56. R2 is also referred to as a. coefficient of determination. b. correlation coefficient. c. total sum of squares. d. regression sum of squares. 57. The regression line denotes the ____ between the dependent and independent variables. a. unsystematic variation Copyright Cengage Learning. Powered by Cognero.
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ch 9 b. systematic variation c. random variation d. average variation 58. Based on the following regression output, what is the equation of the regression line?
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.99313 0.98630 0.98238 2.94802 10
ANOVA Regression Residual Total
df 2 7 9
SS 4379.182 60.836 4440.017
MS 2189.591 8.691
F 251.943
Significance F 0.0000
Intercept X Variable 1 X Variable 2
Coefficients 14.169 0.985 0.995
Standard Error 3.856 0.114 0.057
t Stat 3.674 8.607 17.498
P-value 0.008 0.000 0.000
Lower 95% 5.050 0.714 0.860
a. b. c. d. 59. The error term ε in a regression model represents a. a random error in the data. b. unsystematic variation in the dependent variable. c. variation not explained by the independent variables. d. all of these. 60. Error sum of squares (ESS) is computed as a. b. c. d.
61. The total sum of squares (TSS) is best defined as Copyright Cengage Learning. Powered by Cognero.
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ch 9 a. the sums of squares of the dependent variables. b. the total variation of Y around its mean. c. the sums of squares of the predicted values. d. the variation of Y around its mean plus the variation of Y around the predicted values. 62. Which of the following is an advantage of using the TREND() function versus the regression tool? a. The TREND() function provides more statistical information. b. The TREND() function handles multiple dependent variable data. c. The TREND() function is dynamically updated when input to the function changes. d. The TREND() function does not use a least squares regression line. 63. What does regression analysis attempt to establish? a. a mathematical relationship between a dependent variable, for which future values will be forecast, and one or more independent variables with known values b. linearity in the relationship between independent variables c. linearity in the relationship between a dependent variable and a set of independent variables d. multicollinearity 64. In the equation Y = β0 + β1 X1i + ε, β1 is a. the Y intercept b. the slope of the regression line c. the mean of the dependent data. d. the X intercept
65. Assume you have chosen to use all three variables in your model. Test the significance of the model and explain which values you used to reach your conclusion.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adj R2 Std Error Observations
C
D
E
F 153.5555
F
G
0.993551 0.987143 0.980714 3.08453 10
ANOVA Regression Residual Total
df 3 6 9
SS 4382.93100 57.08596 4440.01700
MS 1460.97700 9.514326
Intercept
Coefficients 11.113410
Std Error 6.322591
t Stat 1.757730
14 15
B
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Significance F 4.63E-06
P-value Lower 95% Upper 95% 0.129301 −4.357430 26.58424 Page 12
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ch 9 16 17 18
X1 X2 X3
0.899230 0.989913 0.993172
1.432383 0.119994 0.059545
0.627786 8.249709 16.679410
0.553271 0.000171 2.96E-06
−2.605690 0.696299 0.847471
4.404148 1.283527 1.138874
Exhibit 9.2 The following questions are based on the problem description and spreadsheet below. A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
MS 2843.4100 0.0814
F 34930.220
t Stat 46.9741 186.8963
P-value 0.0000 0.0000
F
G
0.9999 0.9998 0.9997 0.2853 10
ANOVA Regression Residual Total
df 1 8 9
SS 2843.4178 0.6512 2844.0690
Intercept X Variable 1
Coefficients 38.1923 1.2447
Standard Error 0.8131 0.0067
14 15 16
B
Significance F 0.0000
Lower 95% 36.3174 1.2293
Upper 95% 40.0672 1.2600
66. Refer to Exhibit 9.2. Predict the mean pressure for a temperature of 120 degrees. Exhibit 9.7 The partial regression output below applies to the following questions.
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA Regression Residual Total
df 1 8 9
SS 3735.306 ? ?
Copyright Cengage Learning. Powered by Cognero.
MS 3735.306 ?
F 42.40379
Significance F 0.000186
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ch 9 Intercept X Variable 1
Coefficients 31.623780 1.131661
Standard Error 10.442970 0.173786
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
67. Refer to Exhibit 9.7. What is the SS for Residual and MS for Residual? Exhibit 9.3 The following questions are based on the problem description and spreadsheet below. A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
1 2 3 4 5 6 7 8 9 10 11 12 13
B
C
D
E
F
G
SS MS 304456.60 304456.600 13111.77 1638.972 317568.40
F 185.7608
Standard Error 79.65893 3.97892
Lower Upper P-value 95% 95% 0.0000 3812.297 4179.6850 0.0000 −4057.000 −45.0549
0.979138 0.958712 0.953551 40.48421 10
ANOVA Regression Residual Total
df 1 8 9
Intercept X Variable 1
Coefficients 3995.9910 −54.2303
14 15 16
t Stat 50.16376 −13.62940
Significance F 8.08E-07
68. Refer to Exhibit 9.3. What is the estimated regression function for this problem? Explain what the terms in your equation mean 69. How many binary variables are required to encode a person's age group as being either young, middle-age or old? What are the variables and what are the meanings of their 0, 1 values? Exhibit 9.5 The following questions are based on the description and spreadsheet below. An analyst has identified 3 independent variables (X1, X2,X3) which might be used to predict Y. He has computed the regression equations using all of the variables and the results are summarized in the following table.
Independent Variable in the Model
R2
Adjusted -R2
Se
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Parameter Estimates Page 14
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ch 9 0.00089 0.38700 0.39100 0.84130 0.84130 0.98630 0.98710
X1 X2 X1 and X2 X3 X1 and X3 X2 and X3 X1, X2 and X3
−0.1240 0.3104 0.2170 0.8214 0.7960 0.9824 0.9807
23.548 18.448 19.654 9.3858 10.033 2.948 3.085
b0 = 93.7174, b1 = 0.922 b0 = 57.0803, b2 = 1.545 b0 = 50.2927, b1 = 1.952, b2 = 1.554 b0 = 31.6238, b3 = 1.132 b0 = 31.133, b1 = 0.148, b3 = 1.132 b0 = 14.169, b2 = 0.985, b3 = 0.995 b0 = 11.113, b1 = 0.899, b2 = 0.990, b3 = 0.993
70. Refer to Exhibit 9.5. Predict the mean value based on (X1, X2, X3) = (3, 32, 50). Use the best predictive model based on data from the table. 71. The company would like to build a prediction interval on the time for a new batch of 8 parts. What formula should be entered in cells B17:F21 of the following spreadsheet to compute this prediction interval? Partial results of the Regression analysis of the data are provided below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A
B
C
Obs 1 2 3 4 5 6 7 8 9 10
Number Parts 1 2 3 4 5 6 7 8 9 10
Actual Hours 6.1 7.3 9.7 11.2 12.2 14.1 15.1 16.5 18.5 19.3
E
F
Intercept X Variable 1
Coefficients 4.8400 1.4836
Standard Error 0.2513 0.0405
Predicted Hours
Parts 8
Prediction Se Sp t
D
95% Prediction Interval Lower limit Upper limit
0.3679
Exhibit 9.1 The following questions are based on the problem description and spreadsheet below. A company has built a regression model to predict the number of labor hours (Yi) required to process a batch of parts (Xi). It has developed the following Excel spreadsheet of the results.
A
B
C
Copyright Cengage Learning. Powered by Cognero.
D
E
F
G Page 15
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ch 9 1 2 3 4 5 6 7 8 9 10 11 12 13
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations ANOVA Regression Residual Total
df 1 8 9
SS 181.5971 1.0829 182.6800
Intercept X Variable 1
Coefficients 4.8400 1.4836
Standard Error 0.2513 0.0405
14 15 16
0.9970 0.9941 0.9933 0.3679 10
MS 181.5971 0.1354
F 1341.5500
t Stat 19.2571 36.6272
P-value 0.0000 0.0000
Significance F 0.0000
Lower 95% 4.2604 1.3902
Upper 95% 5.4196 1.5770
72. Refer to Exhibit 9.1. Provide a rough 95% confidence interval on the number of labor hours for a batch of 5 parts. 73. Refer to Exhibit 9.1. Test the significance of the model and explain which values you used to reach your conclusions. Exhibit 9.3 The following questions are based on the problem description and spreadsheet below. A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
F
G
SS MS 304456.60 304456.600 13111.77 1638.972 317568.40
F 185.7608
Standard Error 79.65893 3.97892
Lower Upper P-value 95% 95% 0.0000 3812.297 4179.6850 0.0000 −4057.000 −45.0549
0.979138 0.958712 0.953551 40.48421 10
ANOVA Regression Residual Total
df 1 8 9
Intercept X Variable 1
Coefficients 3995.9910 −54.2303
14 15 16
B
t Stat 50.16376 −13.62940
Significance F 8.08E-07
74. Refer to Exhibit 9.3. Interpret the meaning of R square in cell B3 of the spreadsheet. Copyright Cengage Learning. Powered by Cognero.
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ch 9 75. The researcher would like to build a prediction interval on the calories consumed by an 18 year old man. What formula should be entered in cells B17:F21 of the following spreadsheet to compute this prediction interval? Partial results of the Regression analysis of the data are provided below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A
B
C
Obs 1 2 3 4 5 6 7 8 9 10
Age 14.6 16.0 17.2 18.0 19.8 20.3 21.0 21.8 23.2 25.7
Calories 3194 3174 3014 3088 2882 2866 2831 2832 2742 2621
Age 18
Prediction Se Sp t
D
E
F
Intercept Age
Coefficients 3995.9910 −54.2303
Standard Error 79.65893 3.97892
Predicted Calories
95% Prediction Interval Lower limit Upper limit
40.48421
Exhibit 9.2 The following questions are based on the problem description and spreadsheet below. A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
B
C
D
E
SS 2843.4178 0.6512
MS 2843.4100 0.0814
F 34930.220
F
G
0.9999 0.9998 0.9997 0.2853 10
ANOVA Regression Residual
df 1 8
Copyright Cengage Learning. Powered by Cognero.
Significance F 0.0000
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ch 9 12 13
Total
9
2844.0690
Intercept X Variable 1
Coefficients 38.1923 1.2447
Standard Error 0.8131 0.0067
14 15 16
t Stat 46.9741 186.8963
Lower 95% 36.3174 1.2293
P-value 0.0000 0.0000
Upper 95% 40.0672 1.2600
76. Refer to Exhibit 9.2. Interpret the meaning of R Square in cell B3 of the spreadsheet. Exhibit 9.5 The following questions are based on the description and spreadsheet below. An analyst has identified 3 independent variables (X1, X2,X3) which might be used to predict Y. He has computed the regression equations using all of the variables and the results are summarized in the following table.
Independent Variable in the Model X1 X2 X1 and X2 X3 X1 and X3 X2 and X3 X1, X2 and X3
R2 0.00089 0.38700 0.39100 0.84130 0.84130 0.98630 0.98710
Adjusted -R2 −0.1240 0.3104 0.2170 0.8214 0.7960 0.9824 0.9807
Se
Parameter Estimates
23.548 18.448 19.654 9.3858 10.033 2.948 3.085
b0 = 93.7174, b1 = 0.922 b0 = 57.0803, b2 = 1.545 b0 = 50.2927, b1 = 1.952, b2 = 1.554 b0 = 31.6238, b3 = 1.132 b0 = 31.133, b1 = 0.148, b3 = 1.132 b0 = 14.169, b2 = 0.985, b3 = 0.995 b0 = 11.113, b1 = 0.899, b2 = 0.990, b3 = 0.993
77. Refer to Exhibit 9.5. Based on the data in the table which is the best model for the charity to use? Explain which values you used to reach your conclusion. Exhibit 9.3 The following questions are based on the problem description and spreadsheet below. A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
B
C
D
E
SS MS 304456.60 304456.600
F 185.7608
F
G
0.979138 0.958712 0.953551 40.48421 10
ANOVA Regression
df 1
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Significance F 8.08E-07 Page 18
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ch 9 11 12 13
Residual Total
8 9
13111.77 317568.40
Intercept X Variable 1
Coefficients 3995.9910 −54.2303
Standard Error 79.65893 3.97892
14 15 16
1638.972
Lower Upper P-value 95% 95% 0.0000 3812.297 4179.6850 0.0000 −4057.000 −45.0549
t Stat 50.16376 −13.62940
78. Refer to Exhibit 9.3. Test the significance of the model and explain which values you used to reach your conclusions. Exhibit 9.1 The following questions are based on the problem description and spreadsheet below. A company has built a regression model to predict the number of labor hours (Yi) required to process a batch of parts (Xi). It has developed the following Excel spreadsheet of the results.
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
1 2 3 4 5 6 7 8 9 10 11 12 13
B
C
D
E
MS 181.5971 0.1354
F 1341.5500
t Stat 19.2571 36.6272
P-value 0.0000 0.0000
F
G
0.9970 0.9941 0.9933 0.3679 10
ANOVA Regression Residual Total
df 1 8 9
SS 181.5971 1.0829 182.6800
Intercept X Variable 1
Coefficients 4.8400 1.4836
Standard Error 0.2513 0.0405
14 15 16
Significance F 0.0000
Lower 95% 4.2604 1.3902
Upper 95% 5.4196 1.5770
79. Refer to Exhibit 9.1. Interpret the meaning of R Square in cell B3 of the spreadsheet. 80. The company would like to build a prediction interval on the pressure for a can with a temperature of 125 degrees. What formula should be entered in cells B17:F21 of the following spreadsheet to compute this prediction interval? Partial results of the Regression analysis of the data are provided below. 1 2 3 4 5 6
A
B
C
Obs 1 2 3
Temperature 100 109.4 109.8
Internal Pressure 162.5 174.3 175.3
D
E
F
Intercept Temperature
Coefficients 38.1923 1.2447
Standard Error 0.8131 0.0067
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ch 9 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
4 5 6 7 8 9 10
112.6 114.5 122.7 127.3 136.7 139.9 140.4
Temperature 125
Prediction Se Sp t
178.1 180.5 191.3 196.6 208.6 212.0 212.9 Predicted Pressure
95% Prediction Interval Lower limit Upper limit
0.2853
Exhibit 9.2 The following questions are based on the problem description and spreadsheet below. A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
MS 2843.4100 0.0814
F 34930.220
t Stat 46.9741 186.8963
P-value 0.0000 0.0000
F
G
0.9999 0.9998 0.9997 0.2853 10
ANOVA Regression Residual Total
df 1 8 9
SS 2843.4178 0.6512 2844.0690
Intercept X Variable 1
Coefficients 38.1923 1.2447
Standard Error 0.8131 0.0067
14 15 16
B
Significance F 0.0000
Lower 95% 36.3174 1.2293
Upper 95% 40.0672 1.2600
81. Refer to Exhibit 9.2. What is the estimated regression function for this problem? Explain what the terms in your equation mean. Exhibit 9.6 The partial regression output below applies to the following questions. Copyright Cengage Learning. Powered by Cognero.
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ch 9 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA Regression Residual Total
df 1 8 9
SS 3735.3060 704.7117 4440.017
MS 3735.306 ?
F ?
Significance F 0.000186
Intercept X Variable 1
Coefficients 31.623780 1.131661
Standard Error 10.442970 0.173786
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
82. Refer to Exhibit 9.6. What is the MS for Residual? 83. Refer to Exhibit 9.6. What is the F-statistic value? Exhibit 9.2 The following questions are based on the problem description and spreadsheet below. A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
MS 2843.4100 0.0814
F 34930.220
t Stat 46.9741 186.8963
P-value 0.0000 0.0000
F
G
0.9999 0.9998 0.9997 0.2853 10
ANOVA Regression Residual Total
df 1 8 9
SS 2843.4178 0.6512 2844.0690
Intercept X Variable 1
Coefficients 38.1923 1.2447
Standard Error 0.8131 0.0067
14 15 16
B
Significance F 0.0000
Lower 95% 36.3174 1.2293
Upper 95% 40.0672 1.2600
84. Refer to Exhibit 9.2. Test the significance of the model and explain which values you used to reach your conclusions. Copyright Cengage Learning. Powered by Cognero.
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ch 9 Exhibit 9.3 The following questions are based on the problem description and spreadsheet below. A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
F
G
SS MS 304456.60 304456.600 13111.77 1638.972 317568.40
F 185.7608
Standard Error 79.65893 3.97892
Lower Upper P-value 95% 95% 0.0000 3812.297 4179.6850 0.0000 −4057.000 −45.0549
0.979138 0.958712 0.953551 40.48421 10
ANOVA Regression Residual Total
df 1 8 9
Intercept X Variable 1
Coefficients 3995.9910 −54.2303
14 15 16
B
t Stat 50.16376 −13.62940
Significance F 8.08E-07
85. Refer to Exhibit 9.3. Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet. Exhibit 9.1 The following questions are based on the problem description and spreadsheet below. A company has built a regression model to predict the number of labor hours (Yi) required to process a batch of parts (Xi). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
B
C
D
E
SS 181.5971
MS 181.5971
F 1341.5500
F
G
0.9970 0.9941 0.9933 0.3679 10
ANOVA Regression
df 1
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ch 9 11 12 13
Residual Total
8 9
1.0829 182.6800
Intercept X Variable 1
Coefficients 4.8400 1.4836
Standard Error 0.2513 0.0405
14 15 16
0.1354
t Stat 19.2571 36.6272
P-value 0.0000 0.0000
Lower 95% 4.2604 1.3902
Upper 95% 5.4196 1.5770
86. Refer to Exhibit 9.1. Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet. Exhibit 9.2 The following questions are based on the problem description and spreadsheet below. A paint manufacturer is interested in knowing how much pressure (in pounds per square inch, PSI) builds up inside aerosol cans at various temperatures (degrees Fahrenheit). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
MS 2843.4100 0.0814
F 34930.220
t Stat 46.9741 186.8963
P-value 0.0000 0.0000
F
G
0.9999 0.9998 0.9997 0.2853 10
ANOVA Regression Residual Total
df 1 8 9
SS 2843.4178 0.6512 2844.0690
Intercept X Variable 1
Coefficients 38.1923 1.2447
Standard Error 0.8131 0.0067
14 15 16
B
Significance F 0.0000
Lower 95% 36.3174 1.2293
Upper 95% 40.0672 1.2600
87. Refer to Exhibit 9.2. Interpret the meaning of the "Lower 95%" and "Upper 95%" terms in cells F16:G16 of the spreadsheet. Exhibit 9.4 The following questions are based on the problem description and spreadsheet below. A charitable organization wants to determine what type of people donate to charities like itself. The charity felt that a person's education (in years), annual income, ($1,000) and the number of children the person had were important variables to consider. The charity developed regression models for all of the possible combinations of these three variables but does not know what to do with the results.
Independent Copyright Cengage Learning. Powered by Cognero.
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ch 9 Variables
Variables in the Model
Edu. Inc. Kids
X1 X2 X3
R2 0.817147 0.815521 0.711710
Edu. and Inc.
X1 and X2
Edu. and Kids
Adjusted R2 0.813338 0.811678 0.705766
Se 61.96262 62.23757 77.79432
0.817232
0.809455
62.60378
X1 and X3
0.825059
0.817615
61.24865
Inc. and Kids
X2 and X3
0.824023
0.816535
61.42967
Edu., Inc. and Kids
X1, X2 and X3
0.825119
0.813714
61.90020
Parameter Estimates b0 = 468.4985, b1 = 48.37577 b0 = −24.1245, b2 = 0.024119 b0 = 821.0877, b3 = 199.5228 b0 = 378.2853, b1 = 39.57258, b2 = 0.0044 b0 = 520.9494, b1 = 39.18062, b3 = 45.7573 b0 = 128.687, b1 = 0.019386, b2 = 47.24377 b0 = 445.0583, b1 = 31.79566, b2 = 0.003698, b3 = 45.69131
88. Refer to Exhibit 9.4. Predict the mean donation by a person with 16 years of education, $90,000 annual income and 2 children. Use a full model based on data from the table. Exhibit 9.1 The following questions are based on the problem description and spreadsheet below. A company has built a regression model to predict the number of labor hours (Yi) required to process a batch of parts (Xi). It has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6 7 8 9 10 11 12 13
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
C
D
E
MS 181.5971 0.1354
F 1341.5500
t Stat 19.2571 36.6272
P-value 0.0000 0.0000
F
G
0.9970 0.9941 0.9933 0.3679 10
ANOVA Regression Residual Total
df 1 8 9
SS 181.5971 1.0829 182.6800
Intercept X Variable 1
Coefficients 4.8400 1.4836
Standard Error 0.2513 0.0405
14 15 16
B
Significance F 0.0000
Lower 95% 4.2604 1.3902
Upper 95% 5.4196 1.5770
89. Refer to Exhibit 9.1. Predict the mean number of labor hours for a batch of 5 parts. Copyright Cengage Learning. Powered by Cognero.
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ch 9 90. Refer to Exhibit 9.1. What is the estimated regression function for this problem? Explain what the terms in your equation mean. Exhibit 9.4 The following questions are based on the problem description and spreadsheet below. A charitable organization wants to determine what type of people donate to charities like itself. The charity felt that a person's education (in years), annual income, ($1,000) and the number of children the person had were important variables to consider. The charity developed regression models for all of the possible combinations of these three variables but does not know what to do with the results.
Edu. Inc. Kids
Independent Variables in the Model X1 X2 X3
0.817147 0.815521 0.711710
Edu. and Inc.
X1 and X2
Edu. and Kids
Variables
2
R2 Adjusted R
Se
Parameter Estimates
0.813338 0.811678 0.705766
61.96262 62.23757 77.79432
0.817232
0.809455
62.60378
X1 and X3
0.825059
0.817615
61.24865
Inc. and Kids
X2 and X3
0.824023
0.816535
61.42967
Edu., Inc. and Kids
X1, X2 and X3
0.825119
0.813714
61.90020
b0 = 468.4985, b1 = 48.37577 b0 = −24.1245, b2 = 0.024119 b0 = 821.0877, b3 = 199.5228 b0 = 378.2853, b1 = 39.57258, b2 = 0.0044 b0 = 520.9494, b1 = 39.18062, b3 = 45.7573 b0 = 128.687, b1 = 0.019386, b2 = 47.24377 b0 = 445.0583, b1 = 31.79566, b2 = 0.003698, b3 = 45.69131
91. Refer to Exhibit 9.4. Based on the data in the table which is the best model for the charity to use? Explain which values you used to reach your conclusion. Exhibit 9.3 The following questions are based on the problem description and spreadsheet below. A researcher is interested in determining how many calories young men consume. She measured the age of the individuals and recorded how much food they ate each day for a month. The average daily consumption was recorded as the dependent variable. She has developed the following Excel spreadsheet of the results.
1 2 3 4 5 6
A Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
B
C
D
E
F
G
0.979138 0.958712 0.953551 40.48421 10
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ch 9 7 8 9 10 11 12 13
ANOVA Regression Residual Total
df 1 8 9
Intercept X Variable 1
Coefficients 3995.9910 −54.2303
14 15 16
SS MS 304456.60 304456.600 13111.77 1638.972 317568.40
F 185.7608
Standard Error 79.65893 3.97892
Lower Upper P-value 95% 95% 0.0000 3812.297 4179.6850 0.0000 −4057.000 −45.0549
t Stat 50.16376 −13.62940
Significance F 8.08E-07
92. Refer to Exhibit 9.3. Predict the mean number of calories consumed by a 19 year old man. Exhibit 9.7 The partial regression output below applies to the following questions.
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.917214 0.841282 0.821442 9.385572 10
ANOVA Regression Residual Total
df 1 8 9
SS 3735.306 ? ?
MS 3735.306 ?
F 42.40379
Significance F 0.000186
Intercept X Variable 1
Coefficients 31.623780 1.131661
Standard Error 10.442970 0.173786
t Stat 3.028236 6.511819
P-value 0.016353 0.000186
Lower 95% 7.542233 0.730910
93. Refer to Exhibit 9.7. What is the SS for Total?
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ch 9 Answer Key 1. True 2. True 3. True 4. True 5. False 6. True 7. False 8. True 9. c 10. d 11. d 12. b 13. a 14. a 15. b 16. c 17. b 18. d 19. b 20. a 21. b 22. b 23. c 24. d 25. b Copyright Cengage Learning. Powered by Cognero.
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ch 9 26. b 27. a 28. c 29. b 30. d 31. b 32. c 33. a 34. c 35. d 36. a 37. c 38. b 39. d 40. b 41. a 42. b 43. d 44. c 45. d 46. a 47. a 48. a 49. b 50. d 51. d Copyright Cengage Learning. Powered by Cognero.
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ch 9 52. a 53. a 54. b 55. b 56. a 57. b 58. a 59. d 60. d 61. b 62. c 63. a 64. b 65. The model is significant based on the Significance of the F value in the ANOVA table. The individual t-statistics for each of the regression coefficients indicate that the coefficients associated with X2 and X3 are statistically significantly different from zero. The regression coefficient associated with X1 is not statistically different from zero. 66. 67. SS for Residual is 704.71 and MS for Residual is 88.09 68. The average calories consumed decreases by 54.2303 calories for each additional year of age. The Y-intercept term of 3995.991 has no real meaning because we don't usually think of newborn infants (age = 0) as consuming the same level of calories as men in the 14-25 year age range. 69. Two binary variables are needed. There meanings are: Xai = 1 if person is young, 0 otherwise Xa+1I = 1 if person is middle-age, 0 otherwise If Xai and Xa+1i are both 0 then the person is old. 70.
71. Copyright Cengage Learning. Powered by Cognero.
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ch 9 Cell D17 B20 B21 E17 F17
Formula =4.8400+1.4836*8 =B19(SQRT(1+1/10)+(B17-AVERAGE(B4:B13))^2/(10*VARP(B4:B13))) =TINV(1-0.95,8) =D17-B21*B20 =D17+B21*B20
72. 73. The regression model is significant based on the P-value in the ANOVA section. The P-Value is 0.0000. 74. The R Square value of 0.958712 means that 95.87% of the variation in the Y values is explained by the X values. 75. Cell D17 B20 B21 E17 F17
Formula =3995.991−54.2303*18 =B19(SQRT(1+1/10)+(B17-AVERAGE(B4:B13))^2/(10*VARP(B4:B13))) =TINV(1-0.95,8) =D17-B21*B20 =D17+B21*B20
76. The R Square value of 0.9998 means that 99.98% of the variation in the Y values is explained by the X values. 77. The model with X2 and X3 has the largest adjusted R2 and smallest Se, so it is the best predictor. 78. The regression model is significant based on the P-value in the ANOVA section. The P-Value is 0.0000. 79. The R Square value of 0.9941 means that 99.41% of the variation in the Y values is explained by the X values. 80. Cell D17 B20 B21 E17 F17
Formula =38.19232+1.2447*125 =B19(SQRT(1+1/10)+(B17-AVERAGE(B4:B13))^2/(10*VARP(B4:B13))) =TINV(1-0.95,8) =D17-B21*B20 =D17+B21*B20
81. The average pressure in a can increases by 1.2447 hours for each additional degree of temperature. A can at 0 degrees will have a pressure of 38.1923 pounds. 82. 88.08896 83. 42.40379 84. The regression model is significant based on the P-value in the ANOVA section. The P-Value is 0.0000. 85. We are 95% confident that −63.4057 ≤ β1 ≤ −45.0549. 86. We are 95% confident that 1.3902 ≤ β1 ≤ 1.5770. Copyright Cengage Learning. Powered by Cognero.
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ch 9 87. We are 95% confident that 1.2293 ≤ β1 ≤ 1.2600. 88.
89. 90. The average labor hours for a batch of parts increases by 1.4836 hours for each additional part. A batch of 0 parts will require 4.8400 hours which can be viewed as the setup time for the machine. 91. Education and Kids has the largest adjusted R2 and smallest Se, so it is the best predictor. 92. 93. 4440.017
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ch 10
Indicate whether the statement is true or false. 1. A classification tree is a graphical representation of a set of rules for classifying observations into one group. a. True b. False 2. Classification refers to a type of data mining problem that uses the information available in a set of independent variables to predict the value of a discrete, or categorical, dependent variable. a. True b. False 3. The Mahalanobis distance measure accounts for differences in the covariances between all possible pairings of the independent variables. a. True b. False 4. The Get Data command is part of the XLMiner Platform in Excel add-on. a. True b. False 5. The data might be normalized so that each variable is expressed on a common scale. a. True b. False 6. Data mining is the process of finding and extracting useful information and insights from large data sets. a. True b. False 7. The k-nearest neighbor (k-NN) technique identifies the k observations in the training data that are most similar (or nearest) to a new observation we want to classify. a. True b. False 8. Data mining tasks fall into three potential categories: Classification, Prediction and Association/Segmentation. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. In classification techniques the dependent variable is a. discrete b. continuous c. bounded from above d. bounded from below 10. Logistic regression is a classification technique that Copyright Cengage Learning. Powered by Cognero.
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ch 10 a. outperforms other techniques accros a variety of data collections b. is not reliable c. is not robust d. is not feasible for most data sets 11. Suppose that two variables are found to be significantly correlated. A researcher may a. remove one variable from the data set b. replace the two variables by their product c. replace the two variables by their squared difference d. remove both variables from the data set 12. Oversampling forces a classification method to a. discriminate between groups b. classifying records correctly c. classifying records incorrectly d. perform a large number of iterations 13. Technique(s) used in classification step of data mining include a. discriminant analysis b. logistic regression c. neural networks d. all of the above Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Obs. 1 2 3 4 5 6 7 8 9 10 11
B
C
D
Group 1 1 1 1 1 1 1 2 2 2 2
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630
Verbal 633 704 570 711 645 606 684 542 736 711 622
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ch 10 15 16 17 18 19 20 21 22 23 24 25 26
12 13 14 15 16 17 18 19 20 Group Averages
2 2 2 3 3 3 3 3 3 1 2 3
668 650 633 627 613 594 577 554 561 698 650 594
597 637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Group2 41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
Group3 31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Predicted Group 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
Actual Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
Classification Matrix Copyright Cengage Learning. Powered by Cognero.
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ch 10 Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
Group3 0 0 6 6
Total 7 7 6 20
% correct 85.71% 100.00% 100.00% 95.00%
14. Refer to Exhibit 10.2. What number of observations is classified correctly? a. 19 b. 20 c. 7 d. 8 15. Two common ways of measuring impurity are ___________ and _____________ a. the Gini index and the entropy measure b. the fraction of pure data and the enthaply index c. the fraction of contaminated data and the entropy index d. the fraction of real data and the data purity coefficient Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
A
B
C
D
E
Obs.
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2
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**
** **
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ch 10 22 23 24 25 26
19 20 Group Averages
2 2 1 2
554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
655 607 654 606
2.089 2.150 1.189 1.809
2 2
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854
Predicted Group 1 1 1 1 1 1 1 2 1
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432
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ch 10 10 11 12 13 14 15 16 17 18 19 20
2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
1 2 1 1 2 2 2 2 2 2 2
1 2 2 2 2 2 2 2 2 2 2
16. Refer to Exhibit 10.1. What is the quantitative test score value of the group centroid for group 1? a. 683.8 b. 654.2 c. 610.7 d. 605.7 17. The first step in creating a classification tree involves a. recursively partitioning the independent variables using the outcome of previous partitions b. creating a subset of variables c. clustering the variables into a larger superset d. deciding the number of partitions 18. Standardization of a variable a. removes the scale factor from consideration b. requires subtracting the mean and dividing the difference by the standard deviation c. results in all variables having the mean of 0 and standard deviation of 1 d. all of the above Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
A 1 2 3 4 5 6 7 8 9
Obs. 1 2 3 4 5 6
1 1 1 1 1 1
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387
736 718 710 682 703 672
F
G
Group 1 1 1 1 1 1
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ch 10 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
1 2 1 1 2 1 1 2 2 2 2 2 2 2
**
** **
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
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ch 10 Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
Actual Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
19. Refer to Exhibit 10.1. What formula is entered in cell E4 and copied to cells E5:E25 of the spreadsheet? a. = 7.452402 + 0.00694*C4 + 0.00232*D4 b. = 7.452402 − 0.00694*C4 − 0.00232*D4 c. = 1.157926 + 0.001545*C4 + 0.01297*D4 d. = 7.452402 − 0.00694*D4 − 0.00232*C4 20. A graphical representation of a set of rules for classifying observations into 2 or more groups is called a. a classification tree b. a binary tree c. a Pareto diagram d. a branch-and-bound tree 21. Discriminant analysis (DA) differs from most other predictive statistical methods because the dependent variable is a. continuous b. random c. stochastic d. discrete Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3). Copyright Cengage Learning. Powered by Cognero.
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ch 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 1 2 3
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 698 650 594
Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3
Group1 2.112737844 2.011059054 2.39213666
Group2 41.69699282 27.71117772 21.26669295
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Group3 31.32724796 31.67722089 18.44751157
Predicted Group 1 1 1
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ch 10 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3 0 0 6 6
Total 7 7 6 20
1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
% correct 85.71% 100.00% 100.00% 95.00%
22. Refer to Exhibit 10.2. What is the verbal test score value of the group centroid for group 3? a. 697.71 b. 647.86 c. 587.67 d. 605.17 23. The dependent variable in the regression equation a. the estimated value of the dependent variable. b. the estimated value of the Group variable. c. the estimated ranking of the subject d. all of these are true.
represents
24. A way to detecting and avoiding overfitting is to a. use the validation sample to calibrate the model b. use repeated runs of the model and averaging the results c. use computer simulation d. use rigorous statistical tools Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3). Copyright Cengage Learning. Powered by Cognero.
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ch 10
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 1 2 3
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 698 650 594
Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3
Group1 2.112737844 2.011059054 2.39213666
Group2 41.69699282 27.71117772 21.26669295
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Group3 31.32724796 31.67722089 18.44751157
Predicted Group 1 1 1
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ch 10 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3 0 0 6 6
Total 7 7 6 20
1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
% correct 85.71% 100.00% 100.00% 95.00%
25. Refer to Exhibit 10.2. What is the quantitative test score value of the group centroid for group 1? a. 697.71 b. 647.86 c. 587.67 d. 650.43 26. An Excel add-in tool used for data mining is called a. XL Miner b. GS4 c. XML d. Data Miner Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
1 2 3 4 5
1 2
A
B
C
Obs.
Group
Test Scores Quantitative
1 1
736 718
D
E
Discrim. Pred. Verbal Score 633 0.873 704 0.833
F
G
Group 1 1
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ch 10 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
**
** **
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7
Verbal 654.2 605.7
Classification Matrix Copyright Cengage Learning. Powered by Cognero.
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ch 10 Actual / Predicted Group1 Group2 Total
Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
Actual Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
27. Refer to Exhibit 10.1. Based on the regression output, what is the discriminant score for a student with a quantitative score of 635 and a verbal score of 570? a. 1.72 ≤ discriminant score ≤ 1.73 b. 2.02 ≤ discriminant score ≤ 2.03 c. 3.04 ≤ discriminant score ≤ 3.05 d. 6.12 ≤ discriminant score ≤ 6.14 28. The parameters of the logistic regression model a. are derived through a nonlinear maximum likelihood estimation procedure b. are negative c. are positive d. are negative fractions 29. In the k nearest neighbor technique, a small value of k produces classifications that are a. very sensitive to the sample-specific characteristics of the training data b. not sensitive to the sample-specific characteristics of the training data c. robust d. reliable 30. Affinity analysis is a data mining technique that attempts to discover a. what goes with what Copyright Cengage Learning. Powered by Cognero.
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ch 10 b. the relationship between independent vatiables c. multicollinearity d. causality 31. A correlation coefficient a. measures the strength of a linear association between two variables b. measures the strength of statistical association between two variables c. measures the strength of physical association between two variables d. measures the strength of logical association between two variables 32. Neural networks classification methodology a. is one of the options available in XLMiner Excel add-in b. is superior to other classification schemes c. is inferior to other classification schemes d. produces results that are identical to the outcomes of the DA technique 33. Normalization of data involves a. expressing each variable on a common, standardized scale b. subtracting the grand mean from each observation c. dividing each observation by total variance d. dividing each observation by average range Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10 11 12
Group 1 1 1 1 1 1 1 2 2 2 2 2
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668
Verbal 633 704 570 711 645 606 684 542 736 711 622 597
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ch 10 16 17 18 19 20 21 22 23 24 25 26
13 14 15 16 17 18 19 20 Group Averages
2 2 3 3 3 3 3 3 1 2 3
650 633 627 613 594 577 554 561 698 650 594
637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted
Group1
Group2
Group2 41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
Group3 31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3
Total
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Predicted Group 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
Actual Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
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ch 10 Group1 Group2 Group3 Total
6 0 0 6
1 7 0 8
0 0 6 6
7 7 6 20
85.71% 100.00% 100.00% 95.00%
34. Refer to Exhibit 10.2. Based on the analysis presented in the spreadsheet, what percentage of the observations were correctly classified? a. 80% b. 85% c. 95% d. 100% 35. In the ________ step of data mining, a researcher attempts to estimate to which discrete group an observation belongs to a. classification b. prediction c. categorization d. association/segmentation 36. A ___________ algorithm is used during the training process to adjust weights in a neural network a. backpropagation b. branch-and-bound c. Simplex d. cutting plane Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
A 1 2 3 4 5 6 7 8 9 10 11 12 13
Obs. 1 2 3 4 5 6 7 8 9 10
1 1 1 1 1 1 1 1 1 1
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350
736 718 710 682 703 672 663 657 655 642
F
G
Group 1 1 1 1 1 1 1 2 1 1
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**
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ch 10 14 15 16 17 18 19 20 21 22 23 24 25 26
11 12 13 14 15 16 17 18 19 20 Group Averages
2 2 2 2 2 2 2 2 2 2 1 2
630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
622 597 637 570 526 668 622 553 655 607 654 606
1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
2 1 1 2 2 2 2 2 2 2
** **
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation
Group1
Group2
Predicted Group
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ch 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
37. Refer to Exhibit 10.1. Suppose that for a given observation, the difference between Mahalanobis distances between group 1 and 2 (G1-G2) is small. This means that a. The observation is likely to be classified incorrectly b. The observation is likely to be classified correctly c. The observation is unlikely to be classified d. The observation should be deleted from the data set 38. Refer to Exhibit 10.1. What percentage of the observations is classified correctly? a. 90% b. 80% c. 85% d. 100% 39. In preparation for mining an analyst should a. explore the relationships between variables b. verify completeness and accuracy of the data c. clean the data addressing missing values, errors, etc. d. all of the above 40. In hierarchical clustering, the measure of similarity between clusters is/are a. single linkage b. average linkage c. average group linkage d. all of the above 41. Suppose that all observations belong to the same class. The entropy measure for this situation is equal to a. 0 b. 0.25 Copyright Cengage Learning. Powered by Cognero.
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ch 10 c. 0.5 d. 1 42. Given the following confusion matrix
1 Actual 1 9 Group 2 2 Total 11 what is the correct classification rate? a. 9/13 = 69% b. 10/14 = 86% c. 19/25 = 76% d. 6/19 = 32%
Predicted Group 2 4 10 14
Total 13 12
43. Neural networks technique attempts to learn a. what relationship exists between a set of input and output variables b. whether a linear relationship exists between a set of input and output variables c. whether a nonlinear relationship exists between a set of input and output variables d. whether multicollinearity exists between a set of input variables 44. Neural networks are a. a pattern recognition technique b. a physical model representation of interrelationships c. a non-directed graph d. a binary network 45. Affinity analysis is a data mining technique used in marketing research to determine a. which products are purchased together b. product clusters c. the likelihood that a product will be purchased d. causality 46. Suppose that there are 3 variables in a data set. Approximately how many data records are required using a rule of thumb discussed in the textbook? a. 30 to 45 b. 20 to 30 c. 45 to 60 d. 50 to 100 47. Suppose that the correlation coefficient between X1 and X2 is equal to 1. This means that a. X1 and X2 are perfectly positively correlated b. X1 and X2 are perfectly negatively correlated Copyright Cengage Learning. Powered by Cognero.
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ch 10 c. X1 and X2 are weakly and positively correlated d. X1 and X2 are weakly and negatively correlated 48. The concept behind neural networks is to a. identify a function that accurately maps a set of input values to a corresponding set of output values b. simulate how the classification process works c. estimate a linear relationship between input and output values d. develop a relationship that maps input values to an arbitrary set of output values 49. If using the regression tool for two-group discriminant analysis, in the regression dialog box, the Input X-Range entry corresponds to a. the Group values. b. the independent variable values. c. the predicted variable values. d. the fitted variable values. 50. Overfitting refers to a. placing too much emphasis on the sample-specific noise b. fitting the model too tightly c. fitting the model too loosely d. underestimating model parameters 51. Suppose that all observations in partition j belong to the same group. The Gini index for this situation is equal to a. 0 b. 0.25 c. 0.5 d. 1 52. A major challenge in affinity analysis is to a. identify the most meaningful rules used to determine what goes with what b. the conditional probabilities c. the posterior probabilities d. the probabilities of states of nature 53. Technique(s) used in association step of data mining include a. affinity analysis b. binary analysis c. integer networks d. all of the above Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and Copyright Cengage Learning. Powered by Cognero.
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ch 10 quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
B
C
D
Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 1 2 3
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 698 650 594
Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation
Group1
Group2
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Predicted Group
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ch 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3 0 0 6 6
Total 7 7 6 20
1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
% correct 85.71% 100.00% 100.00% 95.00%
54. Refer to Exhibit 10.2. What is the verbal test score value of the group centroid for group 1? a. 697.71 b. 647.86 c. 587.67 d. 650.43 55. ____________ is a classification technique that estimates the probability of an observation belonging to a particular group a. logistic regression b. binary regression c. multivariate analysis d. ANCOVA 56. Two approaches to clustering discussed in the text are a. k-means clustering and hierarchical clustering b. intuitive clustering and methodical clustering c. MIPS clustering and pixel clustering d. centroid clustering and VMI 57. Steps in the data mining process include the following (in sequence) a. (identify opportunity), (collect data), (explore, understand and prepare data), (identify tasks and tools, (partition data), (build and evaluate models), (deploy models) b. (identify opportunity), (collect data), (explore, understand and prepare data), (identify tasks and tools, (build Copyright Cengage Learning. Powered by Cognero.
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ch 10 and evaluate models), (deploy models) c. (collect data), (explore, understand and prepare data), (identify tasks and tools, (partition data), (build and evaluate models), (deploy models) d. (identify opportunity), (collect data), (identify tasks and tools, (partition data), (build and evaluate models), (deploy models) Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
**
** **
0.787692 0.620459 0.575807 0.33411 20
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ch 10 ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
Actual Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
58. Refer to Exhibit 10.1. What is the verbal test score value of the group centroid for group 1? a. 683.8 Copyright Cengage Learning. Powered by Cognero.
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ch 10 b. 654.2 c. 610.7 d. 605.7 59. Suppose that the correlation coefficient between X1 and X2 is equal to -1. This means that a. X1 and X2 are perfectly positively correlated b. X1 and X2 are perfectly negatively correlated c. X1 and X2 are weakly and positively correlated d. X1 and X2 are weakly and negatively correlated 60. Prediction step in data mining is an option available in a. XLMiner Excel add-in b. Excel c. data mining software d. nonlinear multivariate regression 61. To create the training and validation data set for the model use the __________ option in the XLMiner tab a. partition with oversampling b. partition without oversampling c. partition d. sample Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 3 4 5 6 7 8 9 10 11
A
B
C
D
E
Obs.
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639
1 1 1 1 1 1 1 1 1 1 2
736 718 710 682 703 672 663 657 655 642 630
F
G
Group 1 1 1 1 1 1 1 2 1 1 2
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ch 10 15 16 17 18 19 20 21 22 23 24 25 26
12 13 14 15 16 17 18 19 20 Group Averages
2 2 2 2 2 2 2 2 2 1 2
668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
597 637 570 526 668 622 553 655 607 654 606
1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
1 1 2 2 2 2 2 2 2
** **
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992
Predicted Group 1
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1
Group1 2.780629787
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ch 10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
62. Refer to Exhibit 10.1. What is the verbal test score value of the group centroid for group 2? a. 683.8 b. 654.2 c. 610.7 d. 605.7 63. Refer to Exhibit 10.1. The university has received applications from several new students and would like to predict which group they would fall into. What is the discriminant score for a student with a Quantitative score of 686 and a Verbal score of 601. Use five (5) significant figures in your coefficients. a. 1.29 ≤ discriminant score ≤ 1.30 b. 1.69 ≤ discriminant score ≤ 1.70 c. 2.69 ≤ discriminant score ≤ 2.70 d. 6.05 ≤ discriminant score ≤ 6.06 64. In the k nearest neighbor technique, a large value of k produces classifications that a. are very sensitive to the sample-specific characteristics of the training data b. place all observations into the most frequently occurring group in the training data c. are robust d. are reliable 65. Technique(s) used in prediction step of data mining include a. regression analysis b. the k'th largest neighbor technique c. neural networks d. all of the above 66. ___________ and _________ must be chosen each time a partition is subdivided a. an independent variable and splitting value b. a dependent variable and cutoff value Copyright Cengage Learning. Powered by Cognero.
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ch 10 c. a significance level and an upper bound d. a significance level and a lower bound 67. Which of the following is not true regarding discriminant analysis? a. The classification rule translates discriminant scores into group membership. b. Discriminant analysis is based on discrete or categorical dependent variables. c. The classification rule selected perfectly classifies the data used to derive the classification rule. d. The confusion matrix summarizes classification results. 68. Suppose that a data set contains a variable EDUCATION, which has 7 discrete levels. EDUCATION is an example of a. a categorical variable b. a classification variable c. a continuous variable d. an exponential variable Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
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**
** **
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ch 10 24 Group 25 Averages 26
1 2
684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
654 606
1.189 1.809
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118
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ch 10 13 14 15 16 17 18 19 20
1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
1 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2
69. Refer to Exhibit 10.1. What is the straight line distance between (6,4) and (2,9)? a. 3.20 b. 6.40 c. 9 d. 41 70. Refer to Exhibit 10.1. The straight line distance between two points (X1, Y1) and (X2, Y2) is calculated as a. X1 − Y1 + X2 − Y2 b. (X1 − X2)2 + (Y1 − Y2)2 c. d. 71. The Fisher classification scores can be converted to a. a linear function for each of the groups in the classification problem b. probabilities of group membership c. a uniform distribution d. a half-space 72. A graphical representation of clustering outcomes showing which items should be classified to which clusters is called a(n) a. dendrogram b. hierarchical chart c. horizontal multi layer chart d. vertical bar chart 73. Suppose that a data set contains a variable EDUCATION, which has 7 discrete levels. EDUCATION can be represented by ____ binary variables a. 6 b. 7 c. 8 d. 9 74. Suppose that the observations are partitioned into m groups in equal proportion. The entropy measure for this situation is equal to a. log2(m) b. 0.25 Copyright Cengage Learning. Powered by Cognero.
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ch 10 c. 0.5 d. 1 75. One element in cleaning the data set in the mining process involves a. removing unimportant variables b. adding more variables to the data set c. calculating the adjusted R2 d. calculating the coefficient of multiple correlation Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A
B
C
D
E
Obs.
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
Regression Statistics Multiple R
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
**
** **
0.787692
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ch 10 R Square Adjusted R Square Standard Error Observations
0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733
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ch 10 19 20
17.86896792 17.7008544
2.997472384 1.72794318
2 2
2 2
76. Refer to Exhibit 10.1. How many observations are classified correctly? a. 11 b. 9 c. 17 d. 20 77. Useful data mining techniques can be found in Excel under ___________ drop menu a. Data/(Data Analysis) b. Regression c. Histogram d. Insert/Chart 78. The objective function in k-means clustering attempts to a. minimize the sum of within-cluster dispersions for all clusters b. minimize the sum of between-cluster dispersions for all clusters c. maximize the sum of within-cluster dispersions for all clusters d. maximize the sum of between-cluster dispersions for all clusters Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 1 1 1 1 1 1 1 1 2 2 2 2
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737
736 718 710 682 703 672 663 657 655 642 630 668 650 633
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2
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ch 10 18 19 20 21 22 23 24 25 26
15 16 17 18 19 20 Group Averages
2 2 2 2 2 2 1 2
627 613 594 577 554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
526 668 622 553 655 607 654 606
1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
2 2 2 2 2 2
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086
Predicted Group 1 1 1 1 1
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118
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ch 10 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
1 1 1 1 1 2 2 2 2 2 2 2 2 2 2
79. Refer to Exhibit 10.1. What percentage of the observations is classified incorrectly? a. 90% b. 80% c. 85% d. 15% Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594
Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622
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ch 10 21 22 23 24 25 26
18 19 20 Group Averages
3 3 3 1 2 3
577 554 561 698 650 594
553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
Group2 41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
Group3 31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3 0 0 6 6
Total 7 7 6 20
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Predicted Group 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
Actual Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
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ch 10 80. Refer to Exhibit 10.2. What percentage of observations is classified incorrectly? a. 5% b. 15% c. 95% d. 90% 81. The goal of discriminant analysis is a. to develop a model to predict new dependent values. b. to develop a rule for predicting to what group a new observation is most likely to belong. c. to develop a rule for predicting how independent variable values predict dependent values. d. none of these. Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
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ch 10 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554
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ch 10 16 17 18 19 20
5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
2 2 2 2 2
2 2 2 2 2
82. Refer to Exhibit 10.1. Suppose that for a given observation, the difference between Mahalanobis distances between group 1 and 2 (G1-G2) is big and negative. This means that a. The observation is likely to be classified correctly to group 2 b. The observation is likely to be classified correctly to group 1 c. The observation is likely to be classified incorrectly to group 2 d. The observation is likely to be classified incorrectly to group 1 83. In hierarchical clustering, the measure of similarity between clusters is/are a. single linkage b. complete linkage c. Ward's method d. all of the above Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
B
C
D
Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577
Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553
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ch 10 22 23 24 25 26
19 20 Group Averages
3 3 1 2 3
554 561 698 650 594
655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
Group2 41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
Group3 31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3 0 0 6 6
Total 7 7 6 20
Predicted Group 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
Actual Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
% correct 85.71% 100.00% 100.00% 95.00%
84. Refer to Exhibit 10.2. What percentage of observations is classified correctly? Copyright Cengage Learning. Powered by Cognero.
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ch 10 a. 100% b. 85.71% c. 95% d. 90% 85. Suppose that an analyst classified a new record using the following sequential steps (i) find identical records in the training sample, (ii) determine a group, to which majority of these records belong, (iii) assign the new record to the group in step (ii). This technique is called a. naive Bayes b. Bayes c. conditional probability estimation d. posterior probability estimation 86. A test sample is often used to perform ___________ of how well the model will work with new data a. an honest assessment b. an estimate c. a guess d. a belief Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2
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ch 10 20 21 22 23 24 25 26
17 18 19 20 Group Averages
2 2 2 2 1 2
594 577 554 561 684 611 Cut-off Value
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
622 553 655 607 654 606
1.889 2.167 2.089 2.150 1.189 1.809
2 2 2 2
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917
Predicted Group 1 1 1 1 1 1 1
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363
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ch 10 8 9 10 11 12 13 14 15 16 17 18 19 20
4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
2 1 1 2 1 1 2 2 2 2 2 2 2
1 1 1 2 2 2 2 2 2 2 2 2 2
87. Refer to Exhibit 10.1. What is the quantitative test score value of the group centroid for group 2? a. 683.8 b. 654.2 c. 610.7 d. 605.7 88. The discriminant score is denoted by a. b. c. Yi d. 89. The objective of classification tree algorithms is to a. minimize the average weighted impurity of the resulting partitions b. maximize the average weighted impurity of the resulting partitions c. maximize the average weighted purity of the resulting partitions d. maximize the likelihood estimator of the resulting partitions 90. Suppose the Fisher classification scores for an observation have been converted to the following probabilities: (i) 0.6 for Group 1 and (ii) 0.4 for Group 2. The observation will be classified to a. Group 1 b. Group 2 c. Group 1 or Group 2 with equal probabilities d. neither Group 1 nor Group 2 91. Cluster analysis is a data mining technique used for a. grouping together similar data b. segmentation of records within a data set c. designing effective marketing strategies d. all of the above 92. In the ________ step of data mining, a researcher attempts to form logical groupings of data in the set a. classification Copyright Cengage Learning. Powered by Cognero.
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ch 10 b. prediction c. categorization d. association/segmentation Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
B
C
D
Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 1 2 3
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 698 650 594
Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
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ch 10 Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
Group2 41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
Group3 31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Group3 0 0 6 6
Total 7 7 6 20
Predicted Group 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
Actual Group 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3
% correct 85.71% 100.00% 100.00% 95.00%
93. Refer to Exhibit 10.2. What number of observations is classified incorrectly? a. 19 b. 20 c. 7 d. 1 94. Mahalanobis distance is a measure of a. a likelihood of correctly classifying an observation to a group b. quantifying covariance between independent variables in the model c. quantifying covariance between dependent variables in the model d. quantifying correlations between independent variables in the model 95. Plots useful in data mining analysis can be accessed in Excel using the _______ add-in a. XLMiner Copyright Cengage Learning. Powered by Cognero.
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ch 10 b. Charts c. Data Analysis d. Visual Basic 96. The k-nearest neighbor classification technique a. identifies the k observations in the training data that are most similar to a new observation we want to classify b. sorts the data in an increasing order c. sorts the data in a decreasing order d. works very much like the ranking algorithm 97. A ______________ provides a visual summary of the improvements that a data mining project provides on a binary classification problem compared to a random guess a. lift chart b. Pareto chart c. Gantt chart d. histogram 98. If using the regression tool for two-group discriminant analysis, in the regression dialog box, the Input Y-Range entry corresponds to a. the Group values. b. the independent variable values. c. the predictor variable values. d. (b) and (c) are both correct Exhibit 10.2 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful (Group 1), marginally successful (Group 2) or not-successful (Group 3) in their graduate studies. The officer has data on 20 current students, 7 successful (Group 1), 6 marginally successful (Group 2) and 7 not successful (Group 3).
1 2 3 4 5 6 7 8 9 10 11 12 13
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10
Group 1 1 1 1 1 1 1 2 2 2
Test Scores Quantitative 736 718 710 682 703 672 663 657 655 642
Verbal 633 704 570 711 645 606 684 542 736 711
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ch 10 14 15 16 17 18 19 20 21 22 23 24 25 26
11 12 13 14 15 16 17 18 19 20 Group Averages
2 2 2 2 3 3 3 3 3 3 1 2 3
630 668 650 633 627 613 594 577 554 561 698 650 594
622 597 637 570 526 668 622 553 655 607 650 641 600
Discriminant Analysis Report October 1, 2013 4:22:38 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 697.7142857 2 647.8571429 3 587.6666667
Verbal 650.4285714 630.7142857 605.1666667
Group Frequencies Relative Group Frequency 1 35.00% 2 35.00% 3 30.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.112737844 2.011059054 2.39213666 1.485856258 0.044673421 2.060553332 1.892983432 8.185813378 4.452344715 5.059490343 7.700688406 2.852046055 3.649942277 10.18692176 15.83841704 10.40551791 17.20601765 28.70528187 30.65379946 30.3120197
Group2 41.69699282 27.71117772 21.26669295 7.663411634 16.39398225 3.219370309 1.831377571 1.966433516 2.503703661 1.436682922 1.733549969 2.368943616 0.033109962 1.957576354 4.609882938 6.731282922 15.60121422 28.40166911 47.27241427 40.69021299
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Group3 31.32724796 31.67722089 18.44751157 20.16628363 20.25338523 9.484373089 12.34447877 5.886859371 15.94710279 10.40940069 2.794328907 8.314403397 6.39196529 2.472196207 2.914557709 2.94864721 0.2003812 1.35907133 1.652342277 0.925000274
Predicted Group 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3
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ch 10 Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
Group1 6 0 0 6
Group2 1 7 0 8
Group3 0 0 6 6
Total 7 7 6 20
% correct 85.71% 100.00% 100.00% 95.00%
99. Refer to Exhibit 10.2. What is the quantitative test score value of the group centroid for group 2? a. 697.71 b. 647.86 c. 587.67 d. 650.43 Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A
B
C
D
E
Obs.
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
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**
** **
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ch 10 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations
0.787692 0.620459 0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492
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ch 10 17 18 19 20
9.387160758 16.72042733 17.86896792 17.7008544
0.282417171 2.503563466 2.997472384 1.72794318
2 2 2 2
2 2 2 2
100. Refer to Exhibit 10.1. What number of observations is classified incorrectly? a. 3 b. 9 c. 17 d. 20 101. In a two-group discriminant analysis problem using regression, why is the midpoint cut-off value used to determine group classification? a. Because the value minimizes the absolute misclassification error. b. Because the value minimizes the probability of misclassification error. c. Because the value represents an equal division between the groups. d. Because the value incorporates problem specific knowledge. 102. The Fisher linear discriminant function a. identifies a linear function for each of the groups in the classification problem b. fits a nonlinear function for each of the groups in the classification problem c. defines a hyperplane d. defines a half-space 103. Logistic regression in XLMiner add-in can be used for ______ groups a. 2 b. 3 c. 4 d. 5 104. Before effectively applying the k nearest neighbor classification technique, the variables need to be a. standardized b. normalized c. randomized d. trimmed 105. When purity is perfect, the Gini index is equal to a. 0 b. 0.25 c. 0.5 d. 1 106. In discriminant analysis the averages for the independent variables for a group define the a. centroid. b. median. c. mode. Copyright Cengage Learning. Powered by Cognero.
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ch 10 d. central tendency. 107. In the ________ step of data mining, a researcher attempts to predict the value of a continuous response variable based on the data set a. classification b. prediction c. categorization d. association/segmentation Exhibit 10.1 The following questions are based on the problem description and the output below. A college admissions officer wants to evaluate graduate school applicants based on their GMAT scores, verbal and quantitative. Students are classified as either successful or not-successful in their graduate studies. The officer has data on 20 current students, ten of whom are doing very well (Group 1) and ten who are not (Group 2).
B
C
D
E
Group
Test Scores Quantitative
Discrim. Verbal 633 704 570 711 645 606 684 542 736 711 622 597 637 570 526 668 622 553 655 607 654 606
Pred. Score 0.873 0.833 1.204 1.067 1.078 1.387 1.264 1.634 1.200 1.350 1.639 1.431 1.464 1.737 1.878 1.650 1.889 2.167 2.089 2.150 1.189 1.809
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 1 2
Regression Statistics Multiple R R Square
736 718 710 682 703 672 663 657 655 642 630 668 650 633 627 613 594 577 554 561 684 611 Cut-off Value
F
G
Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
**
** **
0.787692 0.620459
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ch 10 Adjusted R Square Standard Error Observations
0.575807 0.33411 20
ANOVA Regression Residual Total
df 2 17 19
SS 3.102294 1.897706 5
Intercept Quantitative Verbal
Coefficients 7.452402 −0.00694 −0.00232
Standard Error 1.157926 0.001545 0.001297
MS 1.551147 0.11163
F 13.89546
Significance F 0.000265
t Stat 6.43599 −4.49347 −1.78647
P-value 6.15E-06 0.00032 0.091869
Lower 95% 5.009388 −0.0102 −0.00505
Upper 95% 9.895417 −0.00368 0.000419
Discriminant Analysis Report October 1, 2013 3:05:44 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group Quantitative 1 683.8 2 610.7 Classification Matrix Actual / Predicted Group1 Group2 Total
Verbal 654.2 605.7 Group1 9 2 11
Group2 1 8 9
Total 10 10 20
Group2 12.01839992 14.79206851 6.963113398 10.45460384 7.402396086 2.647727028 5.717628917 3.034397466 10.91554854 6.833810307 0.450350608 2.261946848 1.793123377 0.851855838 3.152567686 1.978759442 0.282417171 2.503563466 2.997472384 1.72794318
Predicted Group 1 1 1 1 1 1 1 2 1 1 2 1 1 2 2 2 2 2 2 2
% correct 90.00% 80.00% 85.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 2.780629787 2.175986271 1.989752246 0.756823612 0.37654118 0.818322486 0.548894363 4.380783301 2.016204432 2.156062322 3.6670984 1.229779118 1.399813808 5.292489342 8.817806554 5.1735492 9.387160758 16.72042733 17.86896792 17.7008544
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ch 10 108. Refer to Exhibit 10.1. Suppose that for a given observation, the difference between Mahalanobis distances between group 1 and 2 (G1-G2) is big and positive. This means that a. The observation is likely to be classified correctly to group 2 b. The observation is likely to be classified correctly to group 1 c. The observation is likely to be classified incorrectly to group 2 d. The observation is likely to be classified incorrectly to group 1 109. Data mining tasks fall in the following categories a. classification, prediction, association b. categorization, segmentation c. prediction, association, mining d. observation, categorization, association 110. In using neural networks, an analyst must decide __________ and ___________ a. how many hidden layers to use and how many nodes to use in each of the hidden layers b. how accurate the prediction should be and how many iterations to use c. how many nodes to use and how much time to allow for simulation d. how many iterations to use and what is the limit on the budget constraint 111. The regression approach can be used in the two-group discriminant analysis problem because a. the data are not normally distributed. b. the R2 statistic is not very meaningful. c. the regression equation can generate a discriminant score. d. it scales to the k-group problem easily. 112. The k-means clustering algorithm is available a. in XLMiner Excel add-in b. as a regression option in Excel c. as ANOVA option in Excel d. as one of the options under non-parametric statistics in Excel
Exhibit 10.3 The information below is used for the following questions. A loan officer wants to determine if people will be late in making loan payments. She has information of 18 current loans including the applicants income, level of assets and whether or not the person has been late on payments. She has performed an analysis on the data and obtained the output shown below.
Regression Statistics Coefficients Intercept 3.109577 Income −0.02112 Copyright Cengage Learning. Powered by Cognero.
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ch 10 Assets A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Group Averages
−0.0212 B
C
D
E
Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2
Financial Data Income 91.00 89.00 93.00 90.00 79.00 82.00 73.00 94.00 83.00 54.60 54.60 50.40 48.30 44.10 40.60 56.70 46.90 45.50 82.86 48.39 Cut-off value
Discrim. Assets 17.00 11.50 14.00 7.00 17.50 13.00 9.00 11.00 13.00 13.00 13.50 9.00 8.00 7.50 11.00 9.00 10.50 8.00 12.60 9.56 1.488
Pred. Score 0.827 0.986 0.848 1.060 1.070 1.102 1.377 0.891 1.081 1.681 1.670 1.854 1.920 2.019 2.019 1.721 1.896 1.979 1.092 1.885
F
Group 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
G
**
Discriminant Analysis Report October 2, 2013 5:37:28 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 82.86 2 48.3875
X2 12.6 9.5625
Group Frequencies Relative Group Frequency 1 55.56% 2 44.44% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5
Group1 2.369196137 0.36901865 0.924149672 3.306331521 2.379949742
Group2 66.0278129 58.4189163 69.2027617 71.8075403 39.1364667
Predicted Group 1 1 1 1 1
Actual Group 1 1 1 1 1
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ch 10 6 7 8 9 10 11 12 13 14 15 16 17 18
0.020009681 1.970462321 1.08448764 0.015562911 5.560831725 5.603142866 8.746267347 10.57263173 13.23718625 12.77820255 6.142320172 9.53517235 11.99324528
39.2899414 23.3818835 74.7196324 41.6556521 3.47810249 4.29835367 0.29362884 0.64255268 1.35975916 3.38877281 2.90160644 0.38829888 0.72702748
1 1 1 1 2 2 2 2 2 2 2 2 2
1 1 1 1 1 2 2 2 2 2 2 2 2
Classification Matrix Actual / Predicted Group1 Group2 Total
Group1 9 0 9
Group2 1 8 9
Total 10 8 18
% correct 90.00% 100.00% 94.44%
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 2.369196137 0.36901865 0.924149672 3.306331521 2.379949742 0.020009681 1.970462321 1.08448764 0.015562911 5.560831725 5.603142866 8.746267347 10.57263173 13.23718625 12.77820255 6.142320172 9.53517235 11.99324528
Group2 66.0278129 58.4189163 69.2027617 71.8075403 39.1364667 39.2899414 23.3818835 74.7196324 41.6556521 3.47810249 4.29835367 0.29362884 0.64255268 1.35975916 3.38877281 2.90160644 0.38829888 0.72702748
Predicted Group 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
113. Refer to Exhibit 10.3. Compute the discriminant score and predicted group for someone with an income of 65 and assets of 11. 114. Refer to Exhibit 10.3. What is the percentage of observations: a. Correctly classified to a Group? b. Incorrectly classified to a Group? c. Correctly classified to Group 1? d. Correctly classified to Group 2? Exhibit 10.6 The information below is used for the following questions. Copyright Cengage Learning. Powered by Cognero.
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ch 10 An investor wants to classify companies as being either a good investment, Group 1, or a poor investment, Group 2. He has gathered Liquidity, Profitability and Activity data on 18 companies he has invested in and run a regression analysis. Discriminant Analysis output has also been generated. The data for the problem and the relevant output are shown below.
Regression Statistics Coefficients Intercept 4.690338 Liquidity −3.12192 Profitability −1.55793 Activity −0.16033 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
A
B
C
D
E
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Group Averages
Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2
Liquidity 0.95 0.93 0.97 0.94 0.83 0.86 0.77 0.98 0.87 0.83 0.83 0.77 0.74 0.68 0.63 0.86 0.72 0.70 0.89 0.74
Profitability 0.39 0.28 0.33 0.19 0.40 0.31 0.23 0.27 0.31 0.31 0.32 0.23 0.21 0.20 0.27 0.23 0.26 0.21 0.30 0.24 Cut-off Value
Activity 1.58 1.72 1.48 1.29 1.85 2.09 1.80 1.04 1.45 1.39 1.72 1.58 1.25 0.93 1.47 1.64 1.26 1.42 1.57 1.41 1.48
F
G
Discrim. Score 0.864 1.075 0.911 1.253 1.179 1.187 1.640 1.043 1.259 1.393 1.325 1.675 1.853 2.107 2.067 1.384 1.835 1.950 1.18 1.77
Pred. Group 1 1 1 1 1 1 2 1 1 1 1 2 2 2 2 1 2 2
Discriminant Analysis Report October 3, 2013 6:03:04 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 0.893 2 0.74125
X2 0.302 0.24125
X3 1.569 1.40875
Group Frequencies Relative Group Frequency 1 55.56% 2 44.44% Copyright Cengage Learning. Powered by Cognero.
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ch 10 Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 2.50887789 1.763799348 1.429746802 3.449794017 3.103590888 3.611744959 4.29341697 3.100561164 0.761088666 2.977379296 0.886799302 4.855900518 12.30578964 30.0216398 22.78198694 1.637216223 14.53793306 13.68854312
Classification Matrix Actual / Group1 Predicted Group1 10 Group2 2 Total 12
Group2 0 6 6
Group2 31.67572873 6.726769426 23.61590873 14.08711908 16.92655057 7.09261912 5.614838223 36.99856183 9.991724072 8.708624154 4.503283965 1.217350362 0.78904726 3.741434777 4.351396917 3.068990947 1.466124537 1.862371233
Predicted Group 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2
Total 10 8 18
% correct 100.00% 75.00% 88.89%
Group2 31.67572873 6.726769426 23.61590873 14.08711908 16.92655057 7.09261912 5.614838223 36.99856183 9.991724072 8.708624154 4.503283965 1.217350362 0.78904726 3.741434777 4.351396917 3.068990947 1.466124537 1.862371233
Predicted Group 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2
Actual Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 2.50887789 1.763799348 1.429746802 3.449794017 3.103590888 3.611744959 4.29341697 3.100561164 0.761088666 2.977379296 0.886799302 4.855900518 12.30578964 30.0216398 22.78198694 1.637216223 14.53793306 13.68854312
115. Refer to Exhibit 10.6. What formulas should go in cells C22:D23, E4:G24, and F24 of the spreadsheet? 116. Refer to Exhibit 10.6. Based on the 20 observations, what percentage of the observations are correctly classified? Copyright Cengage Learning. Powered by Cognero.
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ch 10 a. b. c. d.
80.00% 88.89% 75.25% 95.14%
117. Refer to Exhibit 10.6. Compute the discriminant score and predicted group for a company with Liquidity = 0.80, Profitability = 0.27 and Activity = 1.55. Exhibit 10.5 The information below is used for the following questions. A counselor wants to classify people as belonging to one of three groups based on two scores. The counselor has collected data on twenty people who are known to be in one of the three groups. The data for the problem are in the following spreadsheet. Relevant output is also included.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
Group 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 1 2 3
Score 1 245 243 243 240 242 239 239 238 238 237 236 239 238 236 236 234 233 231 230 230 241.97 237.67 232.84
Score 2 147 159 136 160 149 142 155 132 164 160 145 141 147 136 129 153 145 133 150 142 149.13 149.51 141.69
Discriminant Analysis Report October 3, 2013 5:56:34 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group 1
X1 242
X2 148.8333333
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ch 10 2 3
237.8571429 232.8571429
149.1428571 141.1428571
Group Frequencies Relative Group Frequency 1 30.00% 2 35.00% 3 35.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 1.878878546 1.485008232 1.965631332 2.050405268 0.000316616 2.619760005 2.150535081 7.234359169 5.410650963 6.148762031 7.957193516 2.815428698 3.472960993 10.17983789 13.18515704 13.34124873 17.53047429 29.9595523 30.16748987 31.5960613
Group2 44.71078439 24.59810834 23.86349594 5.240672381 15.08008682 1.463141051 1.49774236 2.336824094 1.804496269 1.481772525 3.255755002 1.572268255 0.051141495 4.687051443 6.694592753 13.02796305 21.10204534 44.66420571 54.19487001 55.30743516
Group3 28.51886788 29.27405489 16.04765018 19.14622913 17.99579731 6.685643504 12.55919407 3.975599805 16.73904474 11.17712752 2.377175692 6.440486431 6.194400342 1.467668759 2.413510551 2.816910062 0.244154325 2.148018344 1.591950492 1.317787467
Predicted Group 1 1 1 1 1 2 2 2 2 2 3 2 2 3 3 3 3 3 3 3
Actual Group 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 1.878878546 1.485008232 1.965631332 2.050405268 0.000316616 2.619760005 2.150535081 7.234359169 5.410650963 6.148762031 7.957193516 2.815428698 3.472960993 10.17983789 13.18515704 13.34124873 17.53047429 29.9595523
Group2 44.71078439 24.59810834 23.86349594 5.240672381 15.08008682 1.463141051 1.49774236 2.336824094 1.804496269 1.481772525 3.255755002 1.572268255 0.051141495 4.687051443 6.694592753 13.02796305 21.10204534 44.66420571
Group3 28.51886788 29.27405489 16.04765018 19.14622913 17.99579731 6.685643504 12.55919407 3.975599805 16.73904474 11.17712752 2.377175692 6.440486431 6.194400342 1.467668759 2.413510551 2.816910062 0.244154325 2.148018344
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ch 10 19 20
30.16748987 31.5960613
54.19487001 55.30743516
1.591950492 1.317787467
3 3
118. Refer to Exhibit 10.5. What formulas should go in cells C24:D26 of the spreadsheet? Exhibit 10.4 The information below is used for the following questions. A manager wants to classify people as belonging to one of two groups based on two scores. The manager has collected data on four current employees and has performed a regression analysis on the data. The data for the problem and the relevant output are shown below.
Regression Statistics Coefficients Intercept 31.8158 Score 1 −0.14956 Score 2 0.03825 A 1 2 3 4 5 6 7 8 9 10
Obs. 1 2 3 4 Group Averages
B
Group 1 1 2 2 1 2
C
D
Score 1 244.70 239.00 237.50 236.10 241.85 236.80 Cut-off value
Score 2 147.20 141.20 147.80 136.60 144.20 142.20 1.500
E
F
Discrim. Score 0.85 1.47 1.95 1.73 1.16 1.84
Predicted Group 1 1 2 2
Discriminant Analysis Report October 3, 2013 5:50:10 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 241.85 2 236.8
X2 144.2 142.2
Group Frequencies Relative Group Frequency 1 50.00% 2 50.00% Training Sample Classification Mahalanobis Distances Copyright Cengage Learning. Powered by Cognero.
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ch 10 Observation 1 2 3 4
Group1 0.5 0.5 2.09579E+16 7.50135E+14
Classification Matrix Actual / Group1 Predicted Group1 2 Group2 0 Total 2
Predicted Group 1 1 2 2
Group2 1.64085E+18 1.6759E+17 0.5 0.5
Group2 0 2 2
Total 2 2 4
Actual Group 1 1 2 2
% correct 100.00% 100.00% 100.00%
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4
Group1 0.5 0.5 2.09579E+16 7.50135E+14
Predicted Group 1 1 2 2
Group2 1.64085E+18 1.6759E+17 0.5 0.5
119. Refer to Exhibit 10.4. What formulas should go in cells C8:D9 and E4:G10 of the spreadsheet? 120. Refer to Exhibit 10.4. Compute the discriminant score and predicted group for someone with Score 1 of 242 and Score 2 of 142. Exhibit 10.6 The information below is used for the following questions. An investor wants to classify companies as being either a good investment, Group 1, or a poor investment, Group 2. He has gathered Liquidity, Profitability and Activity data on 18 companies he has invested in and run a regression analysis. Discriminant Analysis output has also been generated. The data for the problem and the relevant output are shown below.
Regression Statistics Coefficients Intercept 4.690338 Liquidity −3.12192 Profitability −1.55793 Activity −0.16033 1 2 3 4 5 6
A
B
C
D
E
F
G
Obs. 1 2 3
Group 1 1 1
Liquidity 0.95 0.93 0.97
Profitability 0.39 0.28 0.33
Activity 1.58 1.72 1.48
Discrim. Score 0.864 1.075 0.911
Pred. Group 1 1 1
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ch 10 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Group Averages
1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 1 2
0.94 0.83 0.86 0.77 0.98 0.87 0.83 0.83 0.77 0.74 0.68 0.63 0.86 0.72 0.70 0.89 0.74
0.19 0.40 0.31 0.23 0.27 0.31 0.31 0.32 0.23 0.21 0.20 0.27 0.23 0.26 0.21 0.30 0.24 Cut-off Value
1.29 1.85 2.09 1.80 1.04 1.45 1.39 1.72 1.58 1.25 0.93 1.47 1.64 1.26 1.42 1.57 1.41 1.48
1.253 1.179 1.187 1.640 1.043 1.259 1.393 1.325 1.675 1.853 2.107 2.067 1.384 1.835 1.950 1.18 1.77
1 1 1 2 1 1 1 1 2 2 2 2 1 2 2
Discriminant Analysis Report October 3, 2013 6:03:04 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 0.893 2 0.74125
X2 0.302 0.24125
X3 1.569 1.40875
Group Frequencies Relative Group Frequency 1 55.56% 2 44.44% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Group1 2.50887789 1.763799348 1.429746802 3.449794017 3.103590888 3.611744959 4.29341697 3.100561164 0.761088666 2.977379296 0.886799302 4.855900518 12.30578964 30.0216398 22.78198694 1.637216223 14.53793306
Group2 31.67572873 6.726769426 23.61590873 14.08711908 16.92655057 7.09261912 5.614838223 36.99856183 9.991724072 8.708624154 4.503283965 1.217350362 0.78904726 3.741434777 4.351396917 3.068990947 1.466124537
Predicted Group 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2
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Actual Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 Page 63
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ch 10 18
13.68854312
Classification Matrix Actual / Group1 Predicted Group1 10 Group2 2 Total 12
Group2 0 6 6
1.862371233
2
Total 10 8 18
% correct 100.00% 75.00% 88.89%
Group2 31.67572873 6.726769426 23.61590873 14.08711908 16.92655057 7.09261912 5.614838223 36.99856183 9.991724072 8.708624154 4.503283965 1.217350362 0.78904726 3.741434777 4.351396917 3.068990947 1.466124537 1.862371233
Predicted Group 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 2 2
2
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 2.50887789 1.763799348 1.429746802 3.449794017 3.103590888 3.611744959 4.29341697 3.100561164 0.761088666 2.977379296 0.886799302 4.855900518 12.30578964 30.0216398 22.78198694 1.637216223 14.53793306 13.68854312
121. Refer to Exhibit 10.6. Compute the discriminant score and predicted group for a company with Liquidity = 0.91, Profitability = 0.32 and Activity = 1.39. Exhibit 10.7 The information below is used for the following questions. An investor wants to classify companies as being a High Risk Investment, Group 1, a Medium Risk Investment, Group 2, or a Low Risk Investment, Group 3. He has gathered Liquidity, Profitability data on 18 companies he has invested in and produced the following spreadsheet. The following Discriminant Analysis output using Analytic Solver Platform has also been generated.
1 2 3 4 5 6 7 8 9
A
B
C
D
Obs. 1 2 3 4 5 6
Group 1 1 1 1 1 1
Liquidity 0.95 0.93 0.97 0.94 0.83 0.86
Profitability 0.39 0.28 0.33 0.19 0.40 0.31
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ch 10 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
7 8 9 10 11 12 13 14 15 16 17 18 Group Averages
2 2 2 2 2 3 3 3 3 3 3 3 1 2 3
0.77 0.98 0.87 0.83 0.83 0.77 0.74 0.68 0.63 0.86 0.72 0.70 0.91 0.86 0.73
0.23 0.27 0.31 0.31 0.32 0.23 0.21 0.20 0.27 0.23 0.26 0.21 0.32 0.29 0.23
Discriminant Analysis Report October 3, 2013 6:09:36 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 0.913333333 2 0.856 3 0.728571429
X2 0.316666667 0.288 0.23
Group Frequencies Relative Group Frequency 1 33.33% 2 27.78% 3 38.89% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Group1 1.97337755 0.247628538 1.343129567 2.684571068 2.637604687 1.11368859 11.1170375 1.495300639 0.747514185 2.652307206 2.500381176 11.1170375 16.38797225 27.28270221 32.16189652
Group2 8.018247625 1.023048854 2.978877546 8.892691791 9.412701836 0.341835695 3.16915395 3.032178576 0.350463134 0.517339085 0.930865255 3.16915395 5.742376387 9.243698571 8.405418741
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Group3 55.69209302 13.90538206 31.70006645 9.232159468 48.98843854 15.27282392 0.335282392 16.98232558 15.98445183 13.3723588 16.13581395 0.335282392 0.571428571 2.129302326 3.405581395
Predicted Group 1 1 1 1 1 2 3 1 2 2 2 3 3 3 3
Actual Group 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 Page 65
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ch 10 16 17 18
3.262216704 16.2264079 22.80611189
2.442676279 3.293838176 7.262223929
3.374352159 1.300066445 0.883986711
Group2 8.018247625 1.023048854 2.978877546 8.892691791 9.412701836 0.341835695 3.16915395 3.032178576 0.350463134 0.517339085 0.930865255 3.16915395 5.742376387 9.243698571 8.405418741 2.442676279 3.293838176 7.262223929
Group3 55.69209302 13.90538206 31.70006645 9.232159468 48.98843854 15.27282392 0.335282392 16.98232558 15.98445183 13.3723588 16.13581395 0.335282392 0.571428571 2.129302326 3.405581395 3.374352159 1.300066445 0.883986711
2 3 3
3 3 3
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 1.97337755 0.247628538 1.343129567 2.684571068 2.637604687 1.11368859 11.1170375 1.495300639 0.747514185 2.652307206 2.500381176 11.1170375 16.38797225 27.28270221 32.16189652 3.262216704 16.2264079 22.80611189
Predicted Group 1 1 1 1 1 2 3 1 2 2 2 3 3 3 3 2 3 3
122. Refer to Exhibit 10.7. Based on the 18 observations in the model complete the following confusion/classification matrix.
Actual Group 1 Group 2 Group 3 Total
Group 1
Group 2
Group 3
Total
% Correct Exhibit 10.5 The information below is used for the following questions. A counselor wants to classify people as belonging to one of three groups based on two scores. The counselor has collected data on twenty people who are known to be in one of the three groups. The data for the problem are in the following spreadsheet. Relevant output is also included.
1 2 3 4
A
B
C
D
Obs. 1
Group 1
Score 1 245
Score 2 147
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ch 10 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Group Averages
1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 1 2 3
243 243 240 242 239 239 238 238 237 236 239 238 236 236 234 233 231 230 230 241.97 237.67 232.84
159 136 160 149 142 155 132 164 160 145 141 147 136 129 153 145 133 150 142 149.13 149.51 141.69
Discriminant Analysis Report October 3, 2013 5:56:34 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group 1 2 3
X1 242 237.8571429 232.8571429
X2 148.8333333 149.1428571 141.1428571
Group Frequencies Relative Group Frequency 1 30.00% 2 35.00% 3 35.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11
Group1 1.878878546 1.485008232 1.965631332 2.050405268 0.000316616 2.619760005 2.150535081 7.234359169 5.410650963 6.148762031 7.957193516
Group2 44.71078439 24.59810834 23.86349594 5.240672381 15.08008682 1.463141051 1.49774236 2.336824094 1.804496269 1.481772525 3.255755002
Group3 28.51886788 29.27405489 16.04765018 19.14622913 17.99579731 6.685643504 12.55919407 3.975599805 16.73904474 11.17712752 2.377175692
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Predicted Group 1 1 1 1 1 2 2 2 2 2 3
Actual Group 1 1 1 1 1 1 2 2 2 2 2 Page 67
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ch 10 12 13 14 15 16 17 18 19 20
2.815428698 3.472960993 10.17983789 13.18515704 13.34124873 17.53047429 29.9595523 30.16748987 31.5960613
1.572268255 0.051141495 4.687051443 6.694592753 13.02796305 21.10204534 44.66420571 54.19487001 55.30743516
6.440486431 6.194400342 1.467668759 2.413510551 2.816910062 0.244154325 2.148018344 1.591950492 1.317787467
2 2 3 3 3 3 3 3 3
2 2 3 3 3 3 3 3 3
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Group1 1.878878546 1.485008232 1.965631332 2.050405268 0.000316616 2.619760005 2.150535081 7.234359169 5.410650963 6.148762031 7.957193516 2.815428698 3.472960993 10.17983789 13.18515704 13.34124873 17.53047429 29.9595523 30.16748987 31.5960613
Group2 44.71078439 24.59810834 23.86349594 5.240672381 15.08008682 1.463141051 1.49774236 2.336824094 1.804496269 1.481772525 3.255755002 1.572268255 0.051141495 4.687051443 6.694592753 13.02796305 21.10204534 44.66420571 54.19487001 55.30743516
Predicted Group 1 1 1 1 1 2 2 2 2 2 3 2 2 3 3 3 3 3 3 3
Group3 28.51886788 29.27405489 16.04765018 19.14622913 17.99579731 6.685643504 12.55919407 3.975599805 16.73904474 11.17712752 2.377175692 6.440486431 6.194400342 1.467668759 2.413510551 2.816910062 0.244154325 2.148018344 1.591950492 1.317787467
123. Refer to Exhibit 10.5. Based on the 20 observations in the model complete the following confusion/classification matrix. Actual Group 1 Group 2 Group 3 Total
Group 1
Group 2
Group 3
Total
% Correct Exhibit 10.4 The information below is used for the following questions. A manager wants to classify people as belonging to one of two groups based on two scores. The manager has collected data on four current employees and has performed a regression analysis on the data. The data for the problem and the relevant output are shown below.
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ch 10 Regression Statistics Coefficients Intercept 31.8158 Score 1 −0.14956 Score 2 0.03825
1 2 3 4 5 6 7 8 9 10
A
B
C
D
Obs. 1 2 3 4 Group Averages
Group 1 1 2 2 1 2
Score 1 244.70 239.00 237.50 236.10 241.85 236.80 Cut-off value
Score 2 147.20 141.20 147.80 136.60 144.20 142.20 1.500
E
F
Discrim. Score 0.85 1.47 1.95 1.73 1.16 1.84
Predicted Group 1 1 2 2
Discriminant Analysis Report October 3, 2013 5:50:10 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 241.85 2 236.8
X2 144.2 142.2
Group Frequencies Relative Group Frequency 1 50.00% 2 50.00% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4
Group1 0.5 0.5 2.09579E+16 7.50135E+14
Classification Matrix Actual / Group1 Predicted Group1 2 Group2 0 Total 2
Predicted Group 1 1 2 2
Group2 1.64085E+18 1.6759E+17 0.5 0.5
Group2 0 2 2
Total 2 2 4
Actual Group 1 1 2 2
% correct 100.00% 100.00% 100.00%
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ch 10 Test Sample Classification Mahalanobis Distances Observation 1 2 3 4
Group1 0.5 0.5 2.09579E+16 7.50135E+14
Group2 1.64085E+18 1.6759E+17 0.5 0.5
Predicted Group 1 1 2 2
124. Refer to Exhibit 10.4. Compute the discriminant score and predicted group for someone with Score 1 of 238 and Score 2 of 140. Exhibit 10.3 The information below is used for the following questions. A loan officer wants to determine if people will be late in making loan payments. She has information of 18 current loans including the applicants income, level of assets and whether or not the person has been late on payments. She has performed an analysis on the data and obtained the output shown below.
Regression Statistics Coefficients Intercept 3.109577 Income −0.02112 Assets −0.0212 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
B
C
D
E
F
Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
Financial Data Income 91.00 89.00 93.00 90.00 79.00 82.00 73.00 94.00 83.00 54.60 54.60 50.40 48.30 44.10 40.60 56.70 46.90 45.50
Discrim. Assets 17.00 11.50 14.00 7.00 17.50 13.00 9.00 11.00 13.00 13.00 13.50 9.00 8.00 7.50 11.00 9.00 10.50 8.00
Pred. Score 0.827 0.986 0.848 1.060 1.070 1.102 1.377 0.891 1.081 1.681 1.670 1.854 1.920 2.019 2.019 1.721 1.896 1.979
Group 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
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G
**
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ch 10 22 23 24
Group Averages
1 2
82.86 48.39 Cut-off value
12.60 9.56 1.488
1.092 1.885
Discriminant Analysis Report October 2, 2013 5:37:28 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 82.86 2 48.3875
X2 12.6 9.5625
Group Frequencies Relative Group Frequency 1 55.56% 2 44.44% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 2.369196137 0.36901865 0.924149672 3.306331521 2.379949742 0.020009681 1.970462321 1.08448764 0.015562911 5.560831725 5.603142866 8.746267347 10.57263173 13.23718625 12.77820255 6.142320172 9.53517235 11.99324528
Group2 66.0278129 58.4189163 69.2027617 71.8075403 39.1364667 39.2899414 23.3818835 74.7196324 41.6556521 3.47810249 4.29835367 0.29362884 0.64255268 1.35975916 3.38877281 2.90160644 0.38829888 0.72702748
Predicted Group 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
Actual Group 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
Classification Matrix Actual / Predicted Group1 Group2 Total
Group1 9 0 9
Group2 1 8 9
Total 10 8 18
% correct 90.00% 100.00% 94.44%
Test Sample Classification Mahalanobis Distances Observation 1 2 3
Group1 2.369196137 0.36901865 0.924149672
Group2 66.0278129 58.4189163 69.2027617
Predicted Group 1 1 1
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ch 10 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
3.306331521 2.379949742 0.020009681 1.970462321 1.08448764 0.015562911 5.560831725 5.603142866 8.746267347 10.57263173 13.23718625 12.77820255 6.142320172 9.53517235 11.99324528
71.8075403 39.1364667 39.2899414 23.3818835 74.7196324 41.6556521 3.47810249 4.29835367 0.29362884 0.64255268 1.35975916 3.38877281 2.90160644 0.38829888 0.72702748
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2
125. Refer to Exhibit 10.3. What formulas should go in cells C22:D23 and E4:F24 of the spreadsheet? Exhibit 10.7 The information below is used for the following questions. An investor wants to classify companies as being a High Risk Investment, Group 1, a Medium Risk Investment, Group 2, or a Low Risk Investment, Group 3. He has gathered Liquidity, Profitability data on 18 companies he has invested in and produced the following spreadsheet. The following Discriminant Analysis output using Analytic Solver Platform has also been generated.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
A
B
C
D
Obs. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Group Averages
Group 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3 1 2 3
Liquidity 0.95 0.93 0.97 0.94 0.83 0.86 0.77 0.98 0.87 0.83 0.83 0.77 0.74 0.68 0.63 0.86 0.72 0.70 0.91 0.86 0.73
Profitability 0.39 0.28 0.33 0.19 0.40 0.31 0.23 0.27 0.31 0.31 0.32 0.23 0.21 0.20 0.27 0.23 0.26 0.21 0.32 0.29 0.23
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ch 10
Discriminant Analysis Report October 3, 2013 6:09:36 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 0.913333333 2 0.856 3 0.728571429
X2 0.316666667 0.288 0.23
Group Frequencies Relative Group Frequency 1 33.33% 2 27.78% 3 38.89% Training Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Group1 1.97337755 0.247628538 1.343129567 2.684571068 2.637604687 1.11368859 11.1170375 1.495300639 0.747514185 2.652307206 2.500381176 11.1170375 16.38797225 27.28270221 32.16189652 3.262216704 16.2264079 22.80611189
Group2 8.018247625 1.023048854 2.978877546 8.892691791 9.412701836 0.341835695 3.16915395 3.032178576 0.350463134 0.517339085 0.930865255 3.16915395 5.742376387 9.243698571 8.405418741 2.442676279 3.293838176 7.262223929
Group3 55.69209302 13.90538206 31.70006645 9.232159468 48.98843854 15.27282392 0.335282392 16.98232558 15.98445183 13.3723588 16.13581395 0.335282392 0.571428571 2.129302326 3.405581395 3.374352159 1.300066445 0.883986711
Predicted Group 1 1 1 1 1 2 3 1 2 2 2 3 3 3 3 2 3 3
Group2 8.018247625 1.023048854 2.978877546 8.892691791 9.412701836 0.341835695 3.16915395
Group3 55.69209302 13.90538206 31.70006645 9.232159468 48.98843854 15.27282392 0.335282392
Predicted Group 1 1 1 1 1 2 3
Actual Group 1 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 3
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4 5 6 7
Group1 1.97337755 0.247628538 1.343129567 2.684571068 2.637604687 1.11368859 11.1170375
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ch 10 8 9 10 11 12 13 14 15 16 17 18
1.495300639 0.747514185 2.652307206 2.500381176 11.1170375 16.38797225 27.28270221 32.16189652 3.262216704 16.2264079 22.80611189
3.032178576 0.350463134 0.517339085 0.930865255 3.16915395 5.742376387 9.243698571 8.405418741 2.442676279 3.293838176 7.262223929
16.98232558 15.98445183 13.3723588 16.13581395 0.335282392 0.571428571 2.129302326 3.405581395 3.374352159 1.300066445 0.883986711
1 2 2 2 3 3 3 3 2 3 3
126. Refer to Exhibit 10.7. What formulas should go in cells C22:D24 of the spreadsheet? Exhibit 10.4 The information below is used for the following questions. A manager wants to classify people as belonging to one of two groups based on two scores. The manager has collected data on four current employees and has performed a regression analysis on the data. The data for the problem and the relevant output are shown below.
Regression Statistics Coefficients Intercept 31.8158 Score 1 −0.14956 Score 2 0.03825
1 2 3 4 5 6 7 8 9 10
A
B
C
D
Obs. 1 2 3 4 Group Averages
Group 1 1 2 2 1 2
Score 1 244.70 239.00 237.50 236.10 241.85 236.80 Cut-off value
Score 2 147.20 141.20 147.80 136.60 144.20 142.20 1.500
E
F
Discrim. Score 0.85 1.47 1.95 1.73 1.16 1.84
Predicted Group 1 1 2 2
Discriminant Analysis Report October 3, 2013 5:50:10 PM Unpooled Estimates of within-group Covariance matrices are used, assuming they are different. Group Centroids Group X1 1 241.85 2 236.8
X2 144.2 142.2
Group Frequencies Copyright Cengage Learning. Powered by Cognero.
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ch 10 Group 1 2
Relative Frequency 50.00% 50.00%
Training Sample Classification Mahalanobis Distances Observation 1 2 3 4
Group1 0.5 0.5 2.09579E+16 7.50135E+14
Classification Matrix Actual / Group1 Predicted Group1 2 Group2 0 Total 2
Predicted Group 1 1 2 2
Group2 1.64085E+18 1.6759E+17 0.5 0.5
Group2 0 2 2
Total 2 2 4
Actual Group 1 1 2 2
% correct 100.00% 100.00% 100.00%
Test Sample Classification Mahalanobis Distances Observation 1 2 3 4
Group1 0.5 0.5 2.09579E+16 7.50135E+14
Group2 1.64085E+18 1.6759E+17 0.5 0.5
Predicted Group 1 1 2 2
127. Refer to Exhibit 10.4. What is the percentage of observations: a. Correctly classified to a Group? b. Incorrectly classified to a Group? c. Correctly classified to Group 1? d. Correctly classified to Group 2?
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ch 10 Answer Key 1. False 2. True 3. True 4. True 5. True 6. True 7. True 8. True 9. a 10. a 11. a 12. a 13. d 14. a 15. a 16. a 17. a 18. d 19. b 20. a 21. d 22. d 23. d 24. a 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 10 26. a 27. a 28. a 29. a 30. a 31. a 32. a 33. a 34. b 35. a 36. a 37. a 38. c 39. d 40. d 41. a 42. c 43. a 44. a 45. a 46. a 47. a 48. a 49. b 50. a 51. a Copyright Cengage Learning. Powered by Cognero.
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ch 10 52. a 53. a 54. d 55. a 56. a 57. a 58. b 59. b 60. a 61. a 62. d 63. a 64. b 65. d 66. a 67. c 68. a 69. b 70. d 71. b 72. a 73. a 74. a 75. a 76. c Copyright Cengage Learning. Powered by Cognero.
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ch 10 77. a 78. a 79. d 80. a 81. b 82. b 83. d 84. c 85. a 86. a 87. c 88. b 89. a 90. a 91. d 92. d 93. d 94. a 95. a 96. a 97. a 98. a 99. b 100. a 101. b 102. a Copyright Cengage Learning. Powered by Cognero.
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ch 10 103. a 104. a 105. a 106. a 107. b 108. a 109. a 110. a 111. c 112. a 113. DISCRIMINANT SCORE = 1.504 PREDICTED GROUP = 2 114. a. 94.44% b. 5.56% c. 90% d. 100% 115. Cell C22 F4 G4 F24
Formula =AVERAGEIF($B$4:$B$21,$B22,C$4:C$21) =4.690338+C4*(−3.12192)+D4*(−1.55793)+E4*(−0.16033) =IF(F4≤$E$24,1,2) =(F22+F23)/2
Copied to C22:F23 F5:E21 G5:G21
116. B 117. DISCRIMINANT SCORE = 1.524 PREDICTED GROUP = 2 118. Cell C24
Formula =AVERAGEIF($B$4:$B$23,$B24,C$4:C$23)
Copied to C24:D26
119. Cell C8 E4 F4
Formula =AVERAGEIF($B$4:$B$7,$B8,C$4:C$7) 31.8158+C4*(−0.14956)+D4*(0.03825) =IF(E4≤$E$10,1,2)
Copied to C8:E9 E5:E8 F5:F8
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ch 10 E10
=(E8+E9)/2
120. DISCRIMINANT SCORE = 1.054 PREDICTED GROUP = 1 121. DISCRIMINANT SCORE = 1.128 PREDICTED GROUP = 1 122. Classification Matrix Actual / Predicted Group1 Group1 5 Group2 1 Group3 0 Total 6
Group2 1 3 1 5
Group3 0 1 6 7
Total 6 5 7 18
% correct 83.33% 60.00% 85.71% 77.78%
Group1
Group2
Group3
Total
% correct
5 0 0 5
1 6 0 7
0 1 7 8
6 7 7 20
83.33% 85.71% 100.00% 90.00%
123. Classification Matrix Actual / Predicted Group1 Group2 Group3 Total
124. DISCRIMINANT SCORE = 1.58 PREDICTED GROUP = 2 125. Cell C22 E4 F4 E24
Formula =AVERAGEIF($B$4:$B$21,$B22,C$4:C$21) 3.109577+C4*(−0.02112)+D4*(−0.0212) =IF(E4≤$E$26,1,2) =(E24+E25)/2
Copied to C22:E23 E5:E23 F5:F23
126. Cell C22
Formula =AVERAGEIF($B$4:$B$21,$B22,C$4:C$21)
Copied to C22:D24
127. a. 100% b. 0% c. 100% d. 100%
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ch 11
Indicate whether the statement is true or false. 1. Exponential smoothing is applicable to non-stationary data. a. True b. False 2. Solver can be used to estimate model parameters when the time series is stationary and additive seasonal effects are present. a. True b. False 3. Seasonality is a regular, repeating pattern in the data that takes longer than 1 year to complete. a. True b. False 4. In stationary time series there is no significant upward or downward trend in the data over time. a. True b. False 5. MAD, MAPE, MSE and RMSE are measures of model accuracy. a. True b. False 6. The TREND( ) function can be used to calculate the estimated values for linear regression models. a. True b. False 7. The weighted moving average technique is a special case of the moving average technique. a. True b. False 8. A time series is a set of observations on a quantitative variable collected over time. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. Exhibit 11.5 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the multiplicative seasonal effects model.
A 1
B
C Time
D Actual
E
F Seasonal
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G
H
I
J
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ch 11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Year 1
2
3
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
Period 1 2 3 4 5 6 7 8 9 10 11 12
Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
Level 462.50 462.50 462.50 462.50 630.32 744.66 790.65 771.79 875.98 1273.96 1272.25 1280.35
Factor 0.796 0.363 1.146 1.695 1.222 0.475 1.280 1.612 1.514 0.774 1.276 1.632
Forecast alpha beta
0.332167 1
368.00 228.96 853.34 1340.26 942.82 416.43 1630.62 2050.66 85564.040
9. Refer to Exhibit 11.5. What formula should be entered in cell E3 to compute the base level when using the multiplicative seasonal effects method? a. =AVERAGE($E$3:$E$6) b. =AVERAGE(E3, E7, E11) c. =AVERAGE($D$3:$D$6) d. =AVERAGE(D3, D7, D11) 10. Refer to Exhibit 11.5. What formula should be entered in cell G13 to compute the forecast for time period 11? a. =E13+F13 b. =E12+F13 c. =E12+F8 d. =E12+F9 11. The general form of an extrapolation model for time-series analysis is a. b. c. d. Exhibit 11.4 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using the additive seasonal method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3
A
B
Year 1
Qtr 1
C Time Period 1
D Actual Sales 284
E Base Level 331.25
F Seasonal Factor −47.250
G Predicted Sales
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H
I
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alpha
0.764 Page 2
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ch 11 4 5 6 7 8 9 10 11 12 13 14 15 16
2
3
2 3 4 1 2 3 4 1 2 3 4
2 3 4 5 6 7 8 9 10 11 12
184 365 492 485 277 606 722 763 593 912 1145
331.25 331.25 331.25 484.85 438.54 540.72 556.41 714.17 745.02 822.74 942.47
−147.250 33.750 160.750 0.146 −161.541 65.279 165.591 48.827 −152.022 89.260 202.534
beta
1.0
284.00 337.60 472.29 701.47 556.56 552.63 810.30 988.33 17688.777
12. Refer to Exhibit 11.4. What are predicted sales for time period 13 using the data in the spreadsheet? a. 915 ≤ predicted sales < 916 b. 916 ≤ predicted sales < 917 c. 991 ≤ predicted sales < 992 d. 1045 ≤ predicted sales < 1046 13. As the number of periods in the forecast, k, increases a. the moving average prediction will be smoother. b. it is harder to get a good forecast. c. the forecast will respond more quickly to changes in the data. d. the moving average will increase in value. 14. How many indicator variables are required if there are p seasons in a time series and you are forecasting with a seasonal regression model? a. p − 1 b. p c. p + 1 d. p + 2 Exhibit 11.7 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Holt's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6
A
B
Year 1
Qtr 1 2 3 4
C Time Period 1 2 3 4
D Actual Sales 284 184 365 492
E Base Level
287.0 396.5
F Trend
G Predicted Sales
14.0 61.8
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H
I
J
alpha beta
0.5 0.5
209.0 301.0 Page 3
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ch 11 7 8 9 10 11 12 13 14 15 16
2
3
1 2 3 4 1 2 3 4
5 6 7 8 9 10 11 12
485 277 606 722 763 593 912 1145
471.6 408.5 508.6 641.0 747.9 720.2 834.0 1026.9
68.4 2.7 51.4 91.9 99.4 35.8 74.8 133.9
458.3 540.1 411.2 560.0 732.9 847.3 756.0 908.8 31877.0
15. Refer to Exhibit 11.7. What are predicted sales for time period 2 using the data in the spreadsheet? a. 208.5 ≤ predicted sales < 209.5 b. 233.5 ≤ predicted sales < 234.5 c. 283.5 ≤ predicted sales < 284.5 d. 300.5 ≤ predicted sales < 301.5 Exhibit 11.8 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Winter's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
alpha beta gamma
E Base Level
413.5 457.2 484.9 579.5 710.9 789.2 845.0
F Trend
G Seasonal Factor
H Predicted Sales
20.5 27.4 27.5 47.6 72.8 74.4 68.8
0.567 1.124 1.486 0.942 0.594 1.127 1.473
217.9 478.2 719.7 461.1 355.5 881.0 1283.0
MSE
28,474.73
0.2 0.3 0.1
16. Refer to Exhibit 11.8. What formula should be entered in cell E7 to compute the base level value for year 2 Quarter 1? a. =$D$17+(1-$D$17)*(E6+F6) Copyright Cengage Learning. Powered by Cognero.
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ch 11 b. =$D$17*D7/G3+(1-$D$17)*(E6+F6) c. =$D$17*D7/G3+(1-$D$17) d. =$D$17*D7/G3+($D$17)*(E6+F6) Exhibit 11.7 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Holt's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Base Level
287.0 396.5 471.6 408.5 508.6 641.0 747.9 720.2 834.0 1026.9
F Trend
G Predicted Sales
14.0 61.8 68.4 2.7 51.4 91.9 99.4 35.8 74.8 133.9
H
I
J
alpha beta
0.5 0.5
209.0 301.0 458.3 540.1 411.2 560.0 732.9 847.3 756.0 908.8 31877.0
17. Refer to Exhibit 11.7. What are predicted sales for time period 13 using the data in the spreadsheet? a. 908 ≤ predicted sales < 909 b. 1026 ≤ predicted sales < 1027 c. 1144 ≤ predicted sales < 1146 d. 1160 ≤ predicted sales < 1161 Exhibit 11.5 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the multiplicative seasonal effects model.
1 2 3 4
A
B
Year 1
Qtr 1 2
C Time Period 1 2
D Actual Sales 368 168
E Level 462.50 462.50
F Seasonal Factor 0.796 0.363
G
H
I
J
alpha beta
0.332167 1
Forecast
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ch 11 5 6 7 8 9 10 11 12 13 14 15 16
2
3
3 4 1 2 3 4 1 2 3 4
3 4 5 6 7 8 9 10 11 12
530 784 770 354 1012 1244 1326 986 1624 2090
462.50 462.50 630.32 744.66 790.65 771.79 875.98 1273.96 1272.25 1280.35
1.146 1.695 1.222 0.475 1.280 1.612 1.514 0.774 1.276 1.632
368.00 228.96 853.34 1340.26 942.82 416.43 1630.62 2050.66 85564.040
18. Refer to Exhibit 11.5. What formula should be entered in cell E7 to compute the remaining expected levels? a. =$J$3*(D7/F3) + (1-$J$3)*E6 b. =$J$3*(D7/D6) + (1-$J$3)*E6 c. =$J$4*(D7/D3) + (1-$J$4)*E6 d. =$J$4*(D7/D6) + (1-$J$4)*E6 Exhibit 11.6 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average model with k = 4.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4 1 2
2
3
4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
462.50 563.00 609.50 730.00 845.00 984.00 1142.00 1295.00 1506.50
591.25 686.88 792.13 925.25 1066.50 1231.88
868.75 1003.13 1175.88 1358.75 1523.50 1781.13
92.50 105.42 127.92 144.50 152.33 183.08
961.25 1108.54 1303.79 1503.25 1675.83
19. Refer to Exhibit 11.6. What formula should be entered in cell F9 to compute the second average when using the double moving average method with k = 4? a. =Average($E$3:$E$9) b. =Average($D$3:$D$9) Copyright Cengage Learning. Powered by Cognero.
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ch 11 c. =Average(D3:D9) d. =Average(D6:D9) Exhibit 11.12 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a seasonal regression model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is provided in the following table.
Regression Statistics Coefficients Intercept 388.88 X Variable 1 10.052 X Variable 2 4.248 X Variable 3 −79.917 X Variable 4 −296.008 X Variable 5 −84.924
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Indicator for Qtr 1
F Actual 2
G
0 0 0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 0
3
H Seasonal Sales 284 184 365 492 485 277 606 722 763 593 912 1145
I Model 130.0 372.3 497.0 465.4 306.0 582.4 741.0 743.4 618.0 928.3 1120.9
20. Refer to Exhibit 11.12. What is the Input Y Range in the Regression command settings dialog box? a. $C$3:$C$14 b. $H$3:$H$14 c. $C$3:$G$14 d. $H$3:$I$14 Exhibit 11.6 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average model with k = 4. Copyright Cengage Learning. Powered by Cognero.
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ch 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4 1 2
2
3
4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
462.50 563.00 609.50 730.00 845.00 984.00 1142.00 1295.00 1506.50
591.25 686.88 792.13 925.25 1066.50 1231.88
868.75 1003.13 1175.88 1358.75 1523.50 1781.13
92.50 105.42 127.92 144.50 152.33 183.08
961.25 1108.54 1303.79 1503.25 1675.83
21. Refer to Exhibit 11.6. What formula should be entered in cell H9 to compute the trend using the double moving average model with k = 4? a. =2*(AVERAGE(E9,F9))/4 b. =2*(E9-F9)/(4-1) c. =2*(E9-F9)/(4*2) d. =2*(D9-F9)/(4-1) Exhibit 11.11 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using seasonal indices. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8
A
B
Year 1
Qtr 1 2 3 4 1 2
2
C Time Period 1 2 3 4 5 6
D Time^2 1 4 9 16 25 36
E Actual Sales 284 184 365 492 485 277
F Quadratic Trend
G Actual as a % of Trend
H Seasonal Forecast
295.1 325.7 366.2 416.4 476.6
62% 112% 134% 116% 58%
190.0 349.1 438.5 453.3 306.9
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ch 11 9 10 11 12 13 14 15 16 17 18 19 20 21
3
3 4 1 2 3 4
7 8 9 10 11 12
Qtr 1 2 3 4
Seasonal Index 108.8% 64.4% 107.2% 119.8%
49 64 81 100 121 144
606 722 763 593 912 1145
546.6 626.4 716.1 815.6 924.9 1044.1
111% 115% 107% 73% 99% 110%
585.8 750.2 779.4 525.2 991.3 1250.5
22. Refer to Exhibit 11.11. What formula should be entered in cell F3 to compute the quadratic trend for year 1 Quarter 1? a. = 263.4545 * C3 + 5.985514 * D3 + 4.922577 b. = 263.4545 + 5.985514 * D3 + 4.922577 * C3 c. = 263.4545 + 5.985514 * C2 + 4.922577 * D2 d. = 263.4545 + 5.985514 * C3 + 4.922577 * D3 Exhibit 11.8 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Winter's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
alpha beta gamma
E Base Level
413.5 457.2 484.9 579.5 710.9 789.2 845.0
F Trend
G Seasonal Factor
H Predicted Sales
20.5 27.4 27.5 47.6 72.8 74.4 68.8
0.567 1.124 1.486 0.942 0.594 1.127 1.473
217.9 478.2 719.7 461.1 355.5 881.0 1283.0
MSE
28,474.73
0.2 0.3 0.1
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ch 11 23. Refer to Exhibit 11.8. What formula should be entered in cell G3 to compute the seasonal factor for year 1 Quarter 1? a. =D3/AVERAGE($D$3:$D$6) b. =D3/AVERAGE($D$3:$D$14) c. =D3/SUM($D$3:$D$6) d. =D3/COUNTIF($D$3:$D$6) Exhibit 11.9 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a linear trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients Intercept 114.136 X Variable 69.979
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Linear Trend 254.1 324.1 394.1 464.0 534.0 604.0 674.0 743.9 813.9 883.9 953.9
24. Refer to Exhibit 11.9. What is the Input Y Range in the Regression command settings dialog box? a. B3:B14 b. C3:C14 c. D3:D14 d. B3:D14 Exhibit 11.5 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the multiplicative seasonal effects model.
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ch 11 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Level 462.50 462.50 462.50 462.50 630.32 744.66 790.65 771.79 875.98 1273.96 1272.25 1280.35
F Seasonal Factor 0.796 0.363 1.146 1.695 1.222 0.475 1.280 1.612 1.514 0.774 1.276 1.632
G
H
I
J
alpha beta
0.332167 1
Forecast
368.00 228.96 853.34 1340.26 942.82 416.43 1630.62 2050.66 85564.040
25. Refer to Exhibit 11.5. What are predicted sales for time period 13 using the data in the spreadsheet? a. 1259 ≤ predicted sales < 1260 b. 1938 ≤ predicted sales < 1939 c. 2090 ≤ predicted sales < 2091 d. 2187 ≤ predicted sales < 2188 Exhibit 11.4 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using the additive seasonal method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Base Level 331.25 331.25 331.25 331.25 484.85 438.54 540.72 556.41 714.17 745.02 822.74 942.47
F Seasonal Factor −47.250 −147.250 33.750 160.750 0.146 −161.541 65.279 165.591 48.827 −152.022 89.260 202.534
G Predicted Sales
H
I
J
alpha beta
0.764 1.0
284.00 337.60 472.29 701.47 556.56 552.63 810.30 988.33 17688.777
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ch 11 26. Refer to Exhibit 11.4. What formula should be entered in cell G12 to compute the forecast for time period 10? a. =E10+F7 b. =E12+F12 c. =E12+F8 d. =E11+F8 Exhibit 11.3 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using exponential smoothing.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C Exp Smoothing Prediction 70.00 70.00 73.46 70.88 71.96 69.20 71.56 71.71 75.27 77.60 77.05 79.45
MSE
39.79
D
E
F
alpha
0.346
27. Refer to Exhibit 11.3. What is the exponential smoothing forecast for month 2? a. 69.75 ≤ forecast < 70.25 b. 73.00 ≤ forecast < 73.50 c. 74.75 ≤ forecast < 75.25 d. 79.75 ≤ forecast < 81.25 Exhibit 11.8 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Winter's method. The store has collected 12 quarters of data and needs your help to analyze the data.
A 1
B
C Time
D Actual
E Base
F
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G Seasonal
H Predicted Page 12
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ch 11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Year 1
2
3
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
Period 1 2 3 4 5 6 7 8 9 10 11 12
Sales 284 184 365 492 485 277 606 722 763 593 912 1145
alpha beta gamma
Level
Trend
Factor
Sales
413.5 457.2 484.9 579.5 710.9 789.2 845.0
20.5 27.4 27.5 47.6 72.8 74.4 68.8
0.567 1.124 1.486 0.942 0.594 1.127 1.473
217.9 478.2 719.7 461.1 355.5 881.0 1283.0
MSE
28,474.73
0.2 0.3 0.1
28. Refer to Exhibit 11.8. What formula should be entered in cell F7 to compute the Trend value for year 2 Quarter 1? a. =$D$18*(E6-E7)+(1-$D$18)*F6 b. =$D$18*(E7-E6)+$D$18*F6 c. =$D$18*(E7-E6)+(1-$D$18)*F6 d. =$D$18*(E7-E6)*(1-$D$18) Exhibit 11.2 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using weighted moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 2-Month Weighted Moving Avg.
D
E
F
Weights w1 w2 sum
0.244 0.756 1.000
76.59 67.95 71.56 66.93 75.02 74.44 82.00 80.54 77.95
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ch 11 15 16
MSE
26.72
29. Refer to Exhibit 11.2. What would be the forecasted value for time period 13? a. 83.024. b. 80.796. c. 79.245. d. 79.908. Exhibit 11.4 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using the additive seasonal method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Base Level 331.25 331.25 331.25 331.25 484.85 438.54 540.72 556.41 714.17 745.02 822.74 942.47
F Seasonal Factor −47.250 −147.250 33.750 160.750 0.146 −161.541 65.279 165.591 48.827 −152.022 89.260 202.534
G Predicted Sales
H
I
J
alpha beta
0.764 1.0
284.00 337.60 472.29 701.47 556.56 552.63 810.30 988.33 17688.777
30. Refer to Exhibit 11.4. What formula should be entered in cell F7 to compute the seasonal factor using the additive seasonal effects model? a. =$J$3*D4+(1-$J$3)*(E3+F3) b. =$J$4*(D7-E7)+(1-$J$4)*F3 c. =$J$4*(E7-E6)+(1-$J$4)*E6 d. =$J$3*G3+(1-$J$3)*(D3+G3) Exhibit 11.2 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using weighted moving averages. Copyright Cengage Learning. Powered by Cognero.
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ch 11 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 2-Month Weighted Moving Avg.
MSE
26.72
D
E
F
Weights w1 w2 sum
0.244 0.756 1.000
76.59 67.95 71.56 66.93 75.02 74.44 82.00 80.54 77.95
31. Refer to Exhibit 11.2. What is the 2-month weighted moving average forecast for month 3 using the weight in the spreadsheet? Associate weight w1 with sales in time period 2 and w2 with sales in time period 1. a. 72.44 b. 75.00 c. 76.59 d. 77.56 32. Why might we not be able to build a regression model to predict a dependent variable? a. We might not know the independent variables. b. There might not be any data available for the independent variables. c. The regression model might not fit the data well. d. All of these are true. Exhibit 11.9 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a linear trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients Intercept 114.136 X Variable 69.979
1 2
A
B
Year
Qtr
C Time Period
D Actual Sales
E Linear Trend
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ch 11 3 4 5 6 7 8 9 10 11 12 13 14
1
1 2 3 4 1 2 3 4 1 2 3 4
2
3
1 2 3 4 5 6 7 8 9 10 11 12
284 184 365 492 485 277 606 722 763 593 912 1145
254.1 324.1 394.1 464.0 534.0 604.0 674.0 743.9 813.9 883.9 953.9
33. Refer to Exhibit 11.9. What formula should be entered in cell E3 to compute the linear trend for year 1 Quarter 1? a. = 114.1364 + 69.7902 * 3 b. = 114.1364 + 69.7902 * 1 c. = 114.1364 + 69.7902 * 1991 d. = 114.1364 + 69.7902 * 284 34. The determination of the MSE-minimizing value of the wi is a non-linear optimization problem because a. absolute values are used. b. the wi are fractional. c. the wi sum to 1. d. MSE is a non-linear objective function. Exhibit 11.6 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average model with k = 4.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
462.50 563.00 609.50 730.00 845.00 984.00 1142.00 1295.00 1506.50
591.25 686.88 792.13 925.25 1066.50 1231.88
868.75 1003.13 1175.88 1358.75 1523.50 1781.13
92.50 105.42 127.92 144.50 152.33 183.08
961.25 1108.54 1303.79 1503.25 1675.83
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ch 11 15 16
4
1 2
13 14
35. Refer to Exhibit 11.6. What formula should be entered in cell I16 to compute the forecast for time period 14? a. =I15+2*$H$14 b. =G14+B16*H14 c. =G14+3*H14 d. =G12+H12 Exhibit 11.3 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using exponential smoothing.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C Exp Smoothing Prediction 70.00 70.00 73.46 70.88 71.96 69.20 71.56 71.71 75.27 77.60 77.05 79.45
MSE
39.79
D
E
F
alpha
0.346
36. Refer to Exhibit 11.3. Assume the forecasted value for month 13 is 79.64. What is the forecasted value for month 16? a. 85.54. b. 83.64. c. 79.64. d. 82.00. Exhibit 11.1 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
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ch 11 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 4-Month Moving Avg.
MSE
40.59
71.00 70.00 71.50 73.50 78.00 78.00 81.00
37. Refer to Exhibit 11.1. What Excel function will compute the root mean squared error in cell C18 of the spreadsheet? a. =SUMXMY2(B7:B14,C7:C14) b. =SQRT(SUMXMY2(B3:B14,C3:C14)/COUNT(C3:C14)) c. =SUMXMY2(B7:B14,C7:C14)/COUNT(C7:C14) d. =SQRT(SUMPRODUCT(B7:B14,C7:C14)/COUNT(C7:C14)) 38. Which of the following is the common approach to time-series analysis? a. Try several techniques and use the best results. b. Plot the data and count the peaks to determine a value for k. c. Plot the data and use the TREND() function. d. Use a stationary model since it is the most robust. Exhibit 11.12 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a seasonal regression model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is provided in the following table.
Regression Statistics Coefficients Intercept 388.88 X Variable 1 10.052 X Variable 2 4.248 X Variable 3 −79.917 X Variable 4 −296.008 X Variable 5 −84.924 Copyright Cengage Learning. Powered by Cognero.
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ch 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Indicator for Qtr 1
F Actual 2
G
0 0 0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 0
3
H Seasonal Sales 284 184 365 492 485 277 606 722 763 593 912 1145
I Model 130.0 372.3 497.0 465.4 306.0 582.4 741.0 743.4 618.0 928.3 1120.9
39. Refer to Exhibit 11.12. What formula should be entered in cell E3 to compute the value for the indicator variable for year 1 Quarter 1? a. =IF($B$3<>$E$2,1,0) b. =IF($B3=E$2,1) c. =IF($B$3=$E$2,0,1) d. =IF($B3=E$2,1,0) 40. Which of the following describes an additive seasonal effect in times series data? a. An occasional pattern of equal magnitude. b. An occasional pattern of unequal magnitude. c. A regular, repeating pattern of equal magnitude. d. A regular, repeating pattern of increasing magnitude. Exhibit 11.3 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using exponential smoothing.
A 1 2 3 4 5 6 7 8 9 10
Time Period 1 2 3 4 5 6 7 8
B Number of Cars Sold 70 80 66 74 64 76 72 82
C Exp Smoothing Prediction 70.00 70.00 73.46 70.88 71.96 69.20 71.56 71.71
D
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E
F
alpha
0.346
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ch 11 11 12 13 14 15 16
9 10 11 12
82 76 84 80
75.27 77.60 77.05 79.45
MSE
39.79
41. Refer to Exhibit 11.3. What is the exponential smoothing forecast for month 13? a. 79.20 ≤ forecast < 79.30 b. 79.60 ≤ forecast < 79.70 c. 80.10 ≤ forecast < 80.20 d. 81.95 ≤ forecast < 82.05 42. Which of the following is true of 2-month moving average forecasting function extrapolation? a. When forecasting ahead 4 periods, the last period forecast will be based on just forecasted values. b. The final period forecast, will equal . c. Each forecasted value will equal the mean of the last two actual data values. d. The value should not be used as only a 2-month ahead forecast is valid. Exhibit 11.1 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 4-Month Moving Avg.
MSE
40.59
71.00 70.00 71.50 73.50 78.00 78.00 81.00
43. Refer to Exhibit 11.1. What Excel function will compute the mean squared error in cell C18 of the spreadsheet? a. =SUMXMY2(B7:B14,C7:C14) Copyright Cengage Learning. Powered by Cognero.
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ch 11 b. =SUMXMY2(B3:B14,C3:C14)/COUNT(C3:C14) c. =SUMXMY2(B7:B14,C7:C14)/COUNT(C7:C14) d. =SUMPRODUCT(B7:B14,C7:C14)/COUNT(C7:C14) Exhibit 11.10 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a quadratic trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend 295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
44. Refer to Exhibit 11.10. How is a quadratic term added to the problem if we want to develop a quadratic trend model? a. Add a column containing Time2. b. Add a column containing Actual Sales2. c. Increase the Input X Range in the Regression command settings dialog box. d. Square the coefficient for Time in the regression model. 45. In an exponential smoothing method, weights are assigned a. just to the current actual and predicted values. b. just to the previous data point and its predicted value. c. to all past data points with weight α. d. to all past data points with more recent data points receiving more weight. Exhibit 11.3 Copyright Cengage Learning. Powered by Cognero.
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ch 11 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using exponential smoothing.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C Exp Smoothing Prediction 70.00 70.00 73.46 70.88 71.96 69.20 71.56 71.71 75.27 77.60 77.05 79.45
MSE
39.79
D
E
F
alpha
0.346
46. Refer to Exhibit 11.3. What formula should be entered in cell C4 to compute the exponential smoothing forecast for month 2? a. =C3-$F$3*(B3-C3) b. =C3+$F$3*(C3-B3) c. =B3+$F$3*(B3-C3) d. =C3+$F$3*(B3-C3) Exhibit 11.9 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a linear trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients Intercept 114.136 X Variable 69.979
1 2 3 4
A
B
Year 1
Qtr 1 2
C Time Period 1 2
D Actual Sales 284 184
E Linear Trend 254.1
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ch 11 5 6 7 8 9 10 11 12 13 14
2
3
3 4 1 2 3 4 1 2 3 4
3 4 5 6 7 8 9 10 11 12
365 492 485 277 606 722 763 593 912 1145
324.1 394.1 464.0 534.0 604.0 674.0 743.9 813.9 883.9 953.9
47. Refer to Exhibit 11.9. What is the Input X Range in the Regression command settings dialog box? a. B3:B14 b. C3:C14 c. D3:D14 d. B3:D14 48. In the formula for MAD, MAPE, and MSE, the Yt and a. the actual and mean values, respectively. b. the actual and forecasted values, respectively. c. the forecasted and actual values, respectively. d. the predicted and forecasted values, respectively.
terms represent
Exhibit 11.9 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a linear trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients Intercept 114.136 X Variable 69.979
1 2 3 4 5 6 7 8 9 10 11
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1
2
3
C Time Period 1 2 3 4 5 6 7 8 9
D Actual Sales 284 184 365 492 485 277 606 722 763
E Linear Trend 254.1 324.1 394.1 464.0 534.0 604.0 674.0 743.9
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ch 11 12 13 14
2 3 4
10 11 12
593 912 1145
813.9 883.9 953.9
49. Refer to Exhibit 11.9. What are predicted sales for the fourth quarter of year 4? a. 1020 ≤ predicted sales < 1025 b. 1090 ≤ predicted sales < 1095 c. 1160 ≤ predicted sales < 1165 d. 1230 ≤ predicted sales < 1235 Exhibit 11.10 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a quadratic trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend 295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
50. Refer to Exhibit 11.10. What are predicted sales for the fourth quarter of year 4? a. 1170 ≤ predicted sales < 1175 b. 1310 ≤ predicted sales < 1315 c. 1460 ≤ predicted sales < 1465 d. 1615 ≤ predicted sales < 1620 Exhibit 11.2 The following questions use the data below. Copyright Cengage Learning. Powered by Cognero.
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ch 11 Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using weighted moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 2-Month Weighted Moving Avg.
MSE
26.72
D
E
F
Weights w1 w2 sum
0.244 0.756 1.000
76.59 67.95 71.56 66.93 75.02 74.44 82.00 80.54 77.95
51. Refer to Exhibit 11.2. What formula should be entered in cell C6 to compute the 2-month weighted moving average forecast for month 3? a. =F3*B4+F4*B3 b. =$F$3*B4+$F$4*B3 c. =(B3+B4)/2 d. =SUMPRODUCT($F$3:$F$4,B3:B4) 52. Which of the following is not a quantitative technique for evaluating the accuracy of a time-series modeling technique? a. Constructing line graphs of the data. b. The mean absolute deviation. c. The mean absolute percent error. d. The root mean square error. 53. Why might a forecaster calculate MSE values on just the most recent data in the time-series data set? a. The forecaster might be interested in how well the forecasting method performs on the more recent data. b. Because the most recent data may be a better predictor of future values. c. Because the resulting forecasting function might fit the older data better that the more recent data. d. All of these. 54. As alpha increases the exponential smoothing model a. produces sluggish forecasts. b. reacts quickly to changes in the data. c. reacts slowly to changes in the data. Copyright Cengage Learning. Powered by Cognero.
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ch 11 d. does not change. 55. The correct formula for a k period moving average is a. b. c. d. 56. Which of the following describes a multiplicative seasonal effect in times series data? a. An occasional pattern of equal magnitude. b. An occasional pattern of unequal magnitude. c. A regular, repeating pattern of equal magnitude. d. A regular, repeating pattern of increasing magnitude. Exhibit 11.10 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a quadratic trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend 295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
57. Refer to Exhibit 11.10. What is the Input Y Range in the Regression command settings dialog box? a. B3:B14 Copyright Cengage Learning. Powered by Cognero.
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ch 11 b. C3:C14 c. D3:D14 d. E3:E14 Exhibit 11.1 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 4-Month Moving Avg.
MSE
40.59
71.00 70.00 71.50 73.50 78.00 78.00 81.00
58. Refer to Exhibit 11.1. What formula should be entered in cell C13 to compute the 4-month moving average forecast for month 11? a. =AVERAGE(B9:B12)/4 b. =AVERAGE(B7+B10) c. =AVERAGE(A9:A12) d. =AVERAGE(B9:B12) Exhibit 11.10 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a quadratic trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Intercept
Coefficients 263.4545
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ch 11 X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
5.985514 4.922577
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend 295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
59. Refer to Exhibit 11.10. What formula should be entered in cell F3 to compute the quadratic trend for year 1 Quarter 1? a. = 263.4545 * C3 + 5.985514 * D3 + 4.922577 b. = 263.4545 + 5.985514 * D3 + 4.922577 * C3 c. = 263.4545 + 5.985514 * C2 + 4.922577 * D2 d. = 263.4545 + 5.985514 * C3 + 4.922577 * D3 60. The correct formula for the weighted moving average extrapolation technique with different weights is a. b. c. d. Exhibit 11.9 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a linear trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients Intercept 114.136 X Variable 69.979
A 1
B
C Time
D Actual
E Linear
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ch 11 2 3 4 5 6 7 8 9 10 11 12 13 14
Year 1
2
3
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
Period 1 2 3 4 5 6 7 8 9 10 11 12
Sales 284 184 365 492 485 277 606 722 763 593 912 1145
Trend 254.1 324.1 394.1 464.0 534.0 604.0 674.0 743.9 813.9 883.9 953.9
61. Refer to Exhibit 11.9. Which column in the spreadsheet represents the independent variable for a regression model? a. A b. B c. C d. D 62. A time series which has a significant upward or downward trend is referred to as a. static. b. non-moving. c. stationary. d. non-stationary. 63. Which of the following statements are true regarding the difference between forecasts using exponential smoothing and forecasts using a weighted moving average method? a. The exponential smoothing forecasts will have a steeper trend line. b. The weighted moving average forecasts will form a level line of constant value. c. The exponential smoothing forecasts will form a level line of constant value. d. The exponential smoothing forecasts will do a better job of capturing the underlying trend in the data. Exhibit 11.12 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a seasonal regression model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is provided in the following table.
Regression Statistics Coefficients Intercept 388.88 X Variable 1 10.052 X Variable 2 4.248 X Variable 3 −79.917 X Variable 4 −296.008 Copyright Cengage Learning. Powered by Cognero.
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ch 11 −84.924
X Variable 5
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Indicator for Qtr 1
F Actual 2
G
0 0 0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 0
3
H Seasonal Sales 284 184 365 492 485 277 606 722 763 593 912 1145
I Model 130.0 372.3 497.0 465.4 306.0 582.4 741.0 743.4 618.0 928.3 1120.9
64. Refer to Exhibit 11.12. What is the Input X Range in the Regression command settings dialog box? a. $C$3:$C$14 b. $H$3:$H$14 c. $C$3:$G$14 d. $H$3:$I$14 Exhibit 11.6 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average model with k = 4.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4 1 2
2
3
4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
462.50 563.00 609.50 730.00 845.00 984.00 1142.00 1295.00 1506.50
591.25 686.88 792.13 925.25 1066.50 1231.88
868.75 1003.13 1175.88 1358.75 1523.50 1781.13
92.50 105.42 127.92 144.50 152.33 183.08
961.25 1108.54 1303.79 1503.25 1675.83
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ch 11 65. Refer to Exhibit 11.6. What formula should be entered in cell E6 (and copied to E7:E14) to compute the first average when using the double moving average method with k = 4? a. =Average($E$3:$E$6) b. =Average($D$3:$D$6) c. =Average(D3:D6) d. =Average(D5:D6) Exhibit 11.11 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using seasonal indices. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
Qtr 1 2 3 4
Seasonal Index 108.8% 64.4% 107.2% 119.8%
2
3
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend
G Actual as a % of Trend
H Seasonal Forecast
295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
62% 112% 134% 116% 58% 111% 115% 107% 73% 99% 110%
190.0 349.1 438.5 453.3 306.9 585.8 750.2 779.4 525.2 991.3 1250.5
66. Refer to Exhibit 11.11. What formula should be entered in cell C18 to compute the Seasonal Index value for quarter 1? a. =SUM($B$3:$B$14)/COUNTIF($B$3:$B$14,B18) b. =SUMIF($B$3:$B$14,B18,$G$3:$G$14)/COUNTIF($B$3:$B$14,B18) Copyright Cengage Learning. Powered by Cognero.
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ch 11 c. =SUMIF($B$3:$B$14,$G$3:$G$14)/COUNT($B$3:$B$14,B18) d. =SUMIF($B$3:$B$14,B18)/COUNTIF($B$3:$B$14,B18) Exhibit 11.7 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Holt's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Base Level
287.0 396.5 471.6 408.5 508.6 641.0 747.9 720.2 834.0 1026.9
F Trend
G Predicted Sales
14.0 61.8 68.4 2.7 51.4 91.9 99.4 35.8 74.8 133.9
H
I
J
alpha beta
0.5 0.5
209.0 301.0 458.3 540.1 411.2 560.0 732.9 847.3 756.0 908.8 31877.0
67. Refer to Exhibit 11.7. What formula should be entered in cell E4 to compute the base level when using Holt's method? a. =$J$3*D4+(1-$J$3)*(E3+F3) b. =$J$4*D4+(1-$J$4)*(E3+F3) c. =$J$3*G4+(1-$J$3)*(E3+F3) d. =$J$4*(E4-E3)+(1-$J$4)*F3 Exhibit 11.8 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Winter's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5
A
B
Year 1
Qtr 1 2 3
C Time Period 1 2 3
D Actual Sales 284 184 365
E Base Level
F Trend
Copyright Cengage Learning. Powered by Cognero.
G Seasonal Factor
H Predicted Sales
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ch 11 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
4 1 2 3 4 1 2 3 4
2
3
4 5 6 7 8 9 10 11 12
492 485 277 606 722 763 593 912 1145
alpha beta gamma
0.2 0.3 0.1
413.5 457.2 484.9 579.5 710.9 789.2 845.0
20.5 27.4 27.5 47.6 72.8 74.4 68.8
0.567 1.124 1.486 0.942 0.594 1.127 1.473
217.9 478.2 719.7 461.1 355.5 881.0 1283.0
MSE
28,474.73
68. Refer to Exhibit 11.8. What formula should be entered in cell H7 to compute the Predicted Sales value for year 2 Quarter 1? a. =SUM(E6:F6)*G3 b. =SUM(E6:F6)+G3 c. =SUM(E6:F6) d. =SUM(E6:G3) Exhibit 11.11 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using seasonal indices. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1
2
3
C Time Period 1 2 3 4 5 6 7 8 9
D Time^2 1 4 9 16 25 36 49 64 81
E Actual Sales 284 184 365 492 485 277 606 722 763
F Quadratic Trend
G Actual as a % of Trend
H Seasonal Forecast
295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1
62% 112% 134% 116% 58% 111% 115% 107%
190.0 349.1 438.5 453.3 306.9 585.8 750.2 779.4
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ch 11 12 13 14 15 16 17 18 19 20 21
2 3 4
10 11 12
Qtr 1 2 3 4
Seasonal Index 108.8% 64.4% 107.2% 119.8%
100 121 144
593 912 1145
815.6 924.9 1044.1
73% 99% 110%
525.2 991.3 1250.5
69. Refer to Exhibit 11.11. What formula should be entered in cell H3 to compute the Seasonal Forecast value for year 1 Quarter 1? a. =F3*VLOOKUP(B3,$B$18:$C$21,2) b. =F3*HLOOKUP(B3,$B$18:$B$21,2) c. =F3*VLOOKUP(B3,$B$18:$B$21,2) d. =F3*VLOOKUP(B3,$C$18:$C$21,2) Exhibit 11.7 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Holt's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Base Level
287.0 396.5 471.6 408.5 508.6 641.0 747.9 720.2 834.0 1026.9
F Trend
G Predicted Sales
14.0 61.8 68.4 2.7 51.4 91.9 99.4 35.8 74.8 133.9
H
I
J
alpha beta
0.5 0.5
209.0 301.0 458.3 540.1 411.2 560.0 732.9 847.3 756.0 908.8 31877.0
70. Refer to Exhibit 11.7. What formula should be entered in cell F4 to compute the trend when using Holt's method? a. =$J$3*D4+(1-$J$3)*(E3+F3) b. =$J$4*(E4-E3)+(1-$J$4)*F3 c. =$J$4*(E4-E3)+(1-$J$4)*G3 d. =$J$4*G3+(1-$J$4)*(D3+G3) Copyright Cengage Learning. Powered by Cognero.
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ch 11 Exhibit 11.11 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using seasonal indices. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
Qtr 1 2 3 4
Seasonal Index 108.8% 64.4% 107.2% 119.8%
2
3
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend
G Actual as a % of Trend
H Seasonal Forecast
295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
62% 112% 134% 116% 58% 111% 115% 107% 73% 99% 110%
190.0 349.1 438.5 453.3 306.9 585.8 750.2 779.4 525.2 991.3 1250.5
71. Refer to Exhibit 11.11. What formula should be entered in cell G3 to compute the "Actual as a % of Trend" value for year 1 Quarter 1? a. =E3/H3 b. =F3/H3 c. =F3/E3 d. =E3/F3 72. Seasonality in a time series is indicated by a. regular, repeating patterns in the data around a trend line. b. regular patterns in the data around a trend line. c. irregular patterns in the data around a trend line. Copyright Cengage Learning. Powered by Cognero.
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ch 11 d. random patterns in the data around a trend line. Exhibit 11.10 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a quadratic trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend 295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
73. Refer to Exhibit 11.10. What Excel command can be used in cells F4:F14 in lieu of a formula based on the regression statistics? a. =TREND($E$3:$E$14,$C$3:$C$14,C4) b. =TREND($E$3:$E$14,$C$3:$D$14,C4:D4) c. =TREND($E$3:$E$14,$C$3:$D$14,D4) d. =TREND($E$3:$E$14,$B$3:$D$14,B4:D4) 74. Forecasting into distant future periods a. is unreliable b. is expensive c. is easy d. assumes changing data patterns Exhibit 11.1 The following questions use the data below. Copyright Cengage Learning. Powered by Cognero.
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ch 11 Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 4-Month Moving Avg.
MSE
40.59
71.00 70.00 71.50 73.50 78.00 78.00 81.00
75. Refer to Exhibit 11.1. If predicting the cars sold for time period 14, what formula must be placed in cell B16? a. =AVERAGE(B12:B15) b. =AVERAGE(B12:B15)/4 c. =TREND(B12:B15) d. =B14 Exhibit 11.5 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the multiplicative seasonal effects model.
1 2 3 4 5 6 7 8 9 10 11
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1
2
3
C Time Period 1 2 3 4 5 6 7 8 9
D Actual Sales 368 168 530 784 770 354 1012 1244 1326
E Level 462.50 462.50 462.50 462.50 630.32 744.66 790.65 771.79 875.98
F Seasonal Factor 0.796 0.363 1.146 1.695 1.222 0.475 1.280 1.612 1.514
G
H
I
J
alpha beta
0.332167 1
Forecast
368.00 228.96 853.34 1340.26 942.82
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ch 11 12 13 14 15 16
2 3 4
10 11 12
986 1624 2090
1273.96 1272.25 1280.35
0.774 1.276 1.632
416.43 1630.62 2050.66 85564.040
76. Refer to Exhibit 11.5. What formula should be entered in cell F7 to compute the seasonal factor using the multiplicative seasonal effects model? a. =$J$3*D4+(1-$J$3)*(E3+F3) b. =$J$4*(D7/E7)+(1-$J$4)*F3 c. =$J$4*(E7/E6)+(1-$J$4)*E6 d. =$J$3*G3+(1-$J$3)*(D3+G3) 77. How is mean absolute deviation calculated? a. b. c. d.
Exhibit 11.2 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using weighted moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
C 2-Month Weighted Moving Avg.
MSE
26.72
D
E
F
Weights w1 w2 sum
0.244 0.756 1.000
76.59 67.95 71.56 66.93 75.02 74.44 82.00 80.54 77.95
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ch 11 78. Refer to Exhibit 11.2. Which cell in the spreadsheet is the objective cell in the Analytic Solver Platform (ASP) task pane area? a. F3 b. F4 c. F5 d. C16 Exhibit 11.11 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using seasonal indices. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Coefficients 263.4545 5.985514 4.922577
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
Qtr 1 2 3 4
Seasonal Index 108.8% 64.4% 107.2% 119.8%
2
3
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend
G Actual as a % of Trend
H Seasonal Forecast
295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
62% 112% 134% 116% 58% 111% 115% 107% 73% 99% 110%
190.0 349.1 438.5 453.3 306.9 585.8 750.2 779.4 525.2 991.3 1250.5
79. Refer to Exhibit 11.11. What are predicted sales for the first quarter of year 4? a. 755 ≤ predicted sales < 780 b. 1255 ≤ predicted sales < 1260 c. 1275 ≤ predicted sales < 1280 Copyright Cengage Learning. Powered by Cognero.
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ch 11 d. 1405 ≤ predicted sales < 1410 80. A technique that analyzes past behavior of a time-series variable to predict the future is referred to as a. a regression model. b. a seasonal model. c. a past performance model. d. an extrapolation model. Exhibit 11.6 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average model with k = 4.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4 1 2
2
3
4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
462.50 563.00 609.50 730.00 845.00 984.00 1142.00 1295.00 1506.50
591.25 686.88 792.13 925.25 1066.50 1231.88
868.75 1003.13 1175.88 1358.75 1523.50 1781.13
92.50 105.42 127.92 144.50 152.33 183.08
961.25 1108.54 1303.79 1503.25 1675.83
81. Refer to Exhibit 11.6. What are predicted sales for time period 16 using the data in the spreadsheet? a. 1964 ≤ predicted sales < 1965 b. 2147 ≤ predicted sales < 2148 c. 2330 ≤ predicted sales < 2331 d. 2513 ≤ predicted sales < 2513 82. A time series which has no significant upward or downward trend is referred to as a. static. b. non-moving. c. stationary. d. non-stationary. Exhibit 11.12 Copyright Cengage Learning. Powered by Cognero.
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ch 11 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a seasonal regression model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is provided in the following table.
Regression Statistics Coefficients Intercept 388.88 X Variable 1 10.052 X Variable 2 4.248 X Variable 3 −79.917 X Variable 4 −296.008 X Variable 5 −84.924
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Indicator for Qtr 1
F Actual 2
G
0 0 0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 0
3
H Seasonal Sales 284 184 365 492 485 277 606 722 763 593 912 1145
I Model 130.0 372.3 497.0 465.4 306.0 582.4 741.0 743.4 618.0 928.3 1120.9
83. Refer to Exhibit 11.12. What Excel command can be used in cells F4:F14 in lieu of a formula based on the regression statistics? a. =TREND($H$3:$H$14,$C$3:$G$14, C3:G3) b. =TREND($H$3:$H$14,$B$3:$G$14, B3:G3) c. =TREND($H$3:$H$14,$A$3:$G$14, A3:G3) d. =TREND($H$3:$H$14,$C$3:$D$14, C3:D3) Exhibit 11.10 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a quadratic trend model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is in the following table.
Regression Statistics Intercept
Coefficients 263.4545
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ch 11 X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14
5.985514 4.922577
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
F Quadratic Trend 295.1 325.7 366.2 416.4 476.6 546.6 626.4 716.1 815.6 924.9 1044.1
84. Refer to Exhibit 11.10. What is the Input X Range in the Regression command settings dialog box? a. B3:B14 b. C3:C14 c. C3:D14 d. C3:E14 Exhibit 11.12 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a seasonal regression model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is provided in the following table.
Regression Statistics Coefficients Intercept 388.88 X Variable 1 10.052 X Variable 2 4.248 X Variable 3 −79.917 X Variable 4 −296.008 X Variable 5 −84.924
1 2 3 4 5
A
B
Year 1
Qtr 1 2 3
C Time Period 1 2 3
D Time^2 1 4 9
E Indicator for Qtr 1
F Actual 2
G
0 0
1 0
0 1
Copyright Cengage Learning. Powered by Cognero.
3
H Seasonal Sales 284 184 365
I Model 130.0 372.3 Page 42
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ch 11 6 7 8 9 10 11 12 13 14
2
3
4 1 2 3 4 1 2 3 4
4 5 6 7 8 9 10 11 12
16 25 36 49 64 81 100 121 144
0 1 0 0 0 1 0 0 0
0 0 1 0 0 0 1 0 0
0 0 0 1 0 0 0 1 0
492 485 277 606 722 763 593 912 1145
497.0 465.4 306.0 582.4 741.0 743.4 618.0 928.3 1120.9
85. Refer to Exhibit 11.12. What are predicted sales for the first quarter of year 4? a. 1155 ≤ predicted sales < 1160 b. 1065 ≤ predicted sales < 1070 c. 1410 ≤ predicted sales < 1415 d. 1635 ≤ predicted sales < 1640 Exhibit 11.8 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using Winter's method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
alpha beta gamma
0.2 0.3 0.1
E Base Level
413.5 457.2 484.9 579.5 710.9 789.2 845.0
F Trend
G Seasonal Factor
H Predicted Sales
20.5 27.4 27.5 47.6 72.8 74.4 68.8
0.567 1.124 1.486 0.942 0.594 1.127 1.473
217.9 478.2 719.7 461.1 355.5 881.0 1283.0
MSE
28,474.73
86. Refer to Exhibit 11.8. What are predicted sales for the first quarter of year 4? a. 795 ≤ predicted sales < 800 b. 860 ≤ predicted sales < 865 Copyright Cengage Learning. Powered by Cognero.
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ch 11 c. 910 ≤ predicted sales < 915 d. 1280 ≤ predicted sales < 1285 87. A model or technique that uses past behavior of a time-series variable to predict the future is referred to as a. a forecasting model. b. an extrapolation model. c. a trend model. d. all of these. Exhibit 11.4 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using the additive seasonal method. The store has collected 12 quarters of data and needs your help to analyze the data.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 284 184 365 492 485 277 606 722 763 593 912 1145
E Base Level 331.25 331.25 331.25 331.25 484.85 438.54 540.72 556.41 714.17 745.02 822.74 942.47
F Seasonal Factor −47.250 −147.250 33.750 160.750 0.146 −161.541 65.279 165.591 48.827 −152.022 89.260 202.534
G Predicted Sales
H
I
J
alpha beta
0.764 1.0
284.00 337.60 472.29 701.47 556.56 552.63 810.30 988.33 17688.777
88. Refer to Exhibit 11.4. What formula should be entered in cell E3 to compute the base level when using the additive seasonal effects method? a. =AVERAGE($E$3:$E$6) b. =AVERAGE(E3, E7, E11) c. =AVERAGE($D$3:$D$6) d. =AVERAGE(D3, D7, D11) 89. Refer to Exhibit 11.4. What formula should be entered in cell E7 to compute the remaining expected levels? a. =$J$3*(D7-D3) + (1-$J$3)*E6 b. =$J$3*(D7-D6) + (1-$J$3)*E6 c. =$J$4*(D7-D3) + (1-$J$4)*E6 d. =$J$4*(D7-D6) + (1-$J$4)*E6 Copyright Cengage Learning. Powered by Cognero.
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ch 11 Exhibit 11.12 The following questions use the data below. Joe's Sporting Goods wants to forecast quarterly sales figures using a seasonal regression model. The store has collected 12 quarters of data and needs your help to analyze the data. The relevant regression output is provided in the following table.
Regression Statistics Coefficients Intercept 388.88 X Variable 1 10.052 X Variable 2 4.248 X Variable 3 −79.917 X Variable 4 −296.008 X Variable 5 −84.924
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Indicator for Qtr 1
F Actual 2
G
0 0 0 1 0 0 0 1 0 0 0
1 0 0 0 1 0 0 0 1 0 0
0 1 0 0 0 1 0 0 0 1 0
3
H Seasonal Sales 284 184 365 492 485 277 606 722 763 593 912 1145
I Model 130.0 372.3 497.0 465.4 306.0 582.4 741.0 743.4 618.0 928.3 1120.9
90. Refer to Exhibit 11.12. What formula should be entered in cell I3 to compute the Seasonal Value for year 1 Quarter 1? a. =388.88+10.052*C3+4.248*D3+79.917*E3+296.008*F3+84.924*G3 b. =10.052*C3+4.248*D3-79.917*E3-296.008*F3-84.924*G3 c. =388.88+10.052*C3+4.248*D3-79.917*E3-296.008*F3-84.924*G3 d. =I3+10.052*C3+4.248*D3-79.917*E3-296.008*F3-84.924*G3 Exhibit 11.1 The following questions use the data below. Honest Al's Used Cars wants to predict how many cars are sold each month. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1
B Number of
C 4-Month
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ch 11 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Cars Sold 70 80 66 74 64 76 72 82 82 76 84 80
Moving Avg.
MSE
40.59
71.00 70.00 71.50 73.50 78.00 78.00 81.00
91. Refer to Exhibit 11.1. What would be the forecasted values for time periods 13 and 14? a. 81.00 and 79.50. b. 78.00 and 78.00. c. 80.50 and 80.13. d. 80.50 and 80.00. 92. Refer to Exhibit 11.1. What is the 4-month moving average forecast for month 5? a. 71 b. 72.5 c. 74 d. 75 93. What is the correct form of the exponential smoothing model? a. b. c. d. 94. Seasonality is defined as a. regular variation above and below the trend line with a cycle completing itself within a year b. regular variation above and below the trend line with a cycle completing itself within a period longer than one year c. a sudden change in process level due to a known cause d. part of the error term 95. Assume that seasonality exists in the monthly product sales data. The number of seasonal indexes needed to model this is equal to Copyright Cengage Learning. Powered by Cognero.
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ch 11 a. 11 b. 10 c. 12 d. 4
Exhibit 11.25 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use a seasonal regression model to forecast sales.
Regression Statistics Coefficients Intercept 60.42 X Variable 1 1.91 X Variable 2 0.16 X Variable 3 −11.85 X Variable 4 −3.57 X Variable 5 25.98
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4 1
2
3
4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144 169
E Indicator for Qtr 1 1 0 0 0 1 0 0 0 1 0 0 0 1
F Actual 2 0 1 0 0 0 1 0 0 0 1 0 0 0
G 3 0 0 1 0 0 0 1 0 0 0 1 0 0
H Seasonal Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
I Model 50.6 61.3 93.6 70.6 62.1 74.1 107.6 85.9 78.7 92.0 126.8 106.4
96. Refer to Exhibit 11.25. Based on the regression output what formulas should go in cells D3:I14? Exhibit 11.18 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the multiplicative seasonal method. Copyright Cengage Learning. Powered by Cognero.
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ch 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Level 462.50 462.50 462.50 462.50 630.32 744.66 790.65 771.79 875.98 1273.96 1272.25 1280.35
F Seasonal Factor 0.796 0.363 1.146 1.695 1.222 0.475 1.280 1.612 1.514 0.774 1.276 1.632
G
H
I
J
alpha beta
0.332167 1
Forecast
368.00 228.96 853.34 1340.26 942.82 416.43 1630.62 2050.66 85564.040
97. Refer to Exhibit 11.18. What formulas should go in cells E3:G16 of the spreadsheet if the multiplicative seasonal method is used to forecast sales? Exhibit 11.22 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use regression and a linear trend model.
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations Intercept X Variable 1
1 2 3 4 5 6 7 8 9
A
B
Year 1
Qtr 1 2 3 4 1 2 3
2
0.895 0.791 0.773 269.406 12 Coefficients 28.273 139.958
C Time Period 1 2 3 4 5 6 7
D Actual Sales 368 168 530 784 770 354 1012
E Linear Trend 168.2 308.2 448.1 588.1 728.1 868.0 1008.0
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ch 11 10 11 12 13 14
3
4 1 2 3 4
8 9 10 11 12
1244 1326 986 1624 2090
1147.9 1287.9 1427.9 1567.8 1707.8
98. Refer to Exhibit 11.22. Based on the regression output, what formulas should go in cells E3:E14? Exhibit 11.20 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using Holt's method.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Base Level 368.0 288.0 356.0 540.2 703.6 651.2 811.6 1048.8 1270.7 1281.2 1471.8 1809.1
F Trend 0.0 −48.0 21.6 119.1 145.7 26.8 107.0 185.1 207.2 89.1 150.0 262.4
G Predicted Sales
H
368.0 240.0 377.6 659.3 849.3 678.0 918.6 1233.9 1477.9 1370.3 1621.8
MSE
I
J
alpha beta
0.4 0.6
30469.5
99. Refer to Exhibit 11.20. What is the forecast for time period 13? Exhibit 11.21 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use Winter's method to forecast sales.
1 2 3 4 5 6
A
B
Year 1
Qtr 1 2 3 4
C Time Period 1 2 3 4
D Actual Sales 55.2 60.0 86.4 74.4
E Base Level
F Trend
69.0
0.0
Copyright Cengage Learning. Powered by Cognero.
G Seasonal Factor 0.800 0.870 1.252 1.078
H Predicted Sales
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ch 11 7 8 9 10 11 12 13 14 15 16 17 18 19
2
3
1 2 3 4 1 2 3 4
5 6 7 8 9 10 11 12
62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
alpha beta gamma
0.2 0.3 0.1
70.8 72.5 77.1 79.3 83.5 92.0 97.1 100.5
0.5 0.9 2.0 2.1 2.7 4.4 4.6 4.3
0.808 0.875 1.276 1.079 0.816 0.897 1.280 1.074
55.2 62.0 91.9 85.3 65.8 75.5 123.1 109.8
MSE
174.43
100. Refer to Exhibit 11.21. What is the forecast for time period 13? Exhibit 11.20 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using Holt's method.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Base Level 368.0 288.0 356.0 540.2 703.6 651.2 811.6 1048.8 1270.7 1281.2 1471.8 1809.1
F Trend 0.0 −48.0 21.6 119.1 145.7 26.8 107.0 185.1 207.2 89.1 150.0 262.4 MSE
G Predicted Sales 368.0 240.0 377.6 659.3 849.3 678.0 918.6 1233.9 1477.9 1370.3 1621.8
H
I
J
alpha beta
0.4 0.6
30469.5
101. Refer to Exhibit 11.20. The store wishes to use Solver to find the optimal values for cell E3. Provide the following Analytic Solver Platform (ASP) settings. Objective Cell: Variables Cells: Constraints Cells: Copyright Cengage Learning. Powered by Cognero.
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ch 11 Exhibit 11.14 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs help analyzing the data using a 2-month moving average.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C 2-Month Moving Average
181.5 178.5 174.0 172.5 174.0 180.0 184.5 192.0 187.5 189.0
102. Refer to Exhibit 11.14. What is the forecast for time periods 13-15? Exhibit 11.23 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use a quadratic trend model to forecast sales.
Regression Statistics Coefficients 56.29 3.59 0.08
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6
A
B
Year 1
Qtr 1 2 3 4
C Time Period 1 2 3 4
D Time^2 1 4 9 16
E Actual Sales 55.2 60.0 86.4 74.4
F Quadratic Trend 60.0 63.8 67.8 71.9
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ch 11 7 8 9 10 11 12 13 14 15
2
3
1 2 3 4 1 2 3 4
5 6 7 8 9 10 11 12 13
25 36 49 64 81 100 121 144 169
62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
76.2 80.7 85.3 90.1 95.1 100.2 105.5 110.9 116.5
103. Refer to Exhibit 11.23. Based on the regression output, what formulas should go in cells D3:F14? Exhibit 11.13 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C 4-Month Moving Avg.
MSE
91.34
177.75 175.50 174.00 176.25 179.25 186.00 186.00 190.50
104. Refer to Exhibit 11.13. What formulas should go in cells C3:C16 of the spreadsheet if Bill uses a 4 month moving average forecasting model? Exhibit 11.21 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use Winter's method to forecast sales.
1 2
A
B
Year
Qtr
C Time Period
D Actual Sales
E Base Level
F Trend
Copyright Cengage Learning. Powered by Cognero.
G Seasonal Factor
H Predicted Sales Page 52
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ch 11 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1
2
3
1 2 3 4 1 2 3 4 1 2 3 4
1 2 3 4 5 6 7 8 9 10 11 12
55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
alpha beta gamma
0.2 0.3 0.1
69.0 70.8 72.5 77.1 79.3 83.5 92.0 97.1 100.5
0.0 0.5 0.9 2.0 2.1 2.7 4.4 4.6 4.3
0.800 0.870 1.252 1.078 0.808 0.875 1.276 1.079 0.816 0.897 1.280 1.074
55.2 62.0 91.9 85.3 65.8 75.5 123.1 109.8
MSE
174.43
105. Refer to Exhibit 11.21. What formulas should go in cells E3:H16? 106. Refer to Exhibit 11.21. The store wishes to use Solver to find the optimal values for cell E6. Provide the following Analytic Solver Platform (ASP) settings. Objective Cell: Variables Cells: Constraints Cells: Exhibit 11.20 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using Holt's method.
1 2 3 4 5 6 7 8 9 10 11 12 13
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624
E Base Level 368.0 288.0 356.0 540.2 703.6 651.2 811.6 1048.8 1270.7 1281.2 1471.8
F Trend 0.0 −48.0 21.6 119.1 145.7 26.8 107.0 185.1 207.2 89.1 150.0
G Predicted Sales
Copyright Cengage Learning. Powered by Cognero.
368.0 240.0 377.6 659.3 849.3 678.0 918.6 1233.9 1477.9 1370.3
H
I
J
alpha beta
0.4 0.6
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ch 11 14 15 16
4
12
2090
1809.1
262.4
1621.8
MSE
30469.5
107. Refer to Exhibit 11.20. What formulas should go in cells E3:G16 of the spreadsheet if Holt's method is used to forecast sales? Exhibit 11.23 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use a quadratic trend model to forecast sales.
Regression Statistics Coefficients 56.29 3.59 0.08
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13
D
E Actual Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
Time^2 1 4 9 16 25 36 49 64 81 100 121 144 169
F Quadratic Trend 60.0 63.8 67.8 71.9 76.2 80.7 85.3 90.1 95.1 100.2 105.5 110.9 116.5
108. Refer to Exhibit 11.23. What is the forecast for time period 13? Exhibit 11.17 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the additive seasonal method.
1 2
A
B
Year
Qtr
C Time Period
D Actual Sales
E Level
F Seasonal Factor
G
H
I
J
Forecast
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ch 11 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1991
1992
1993
1 2 3 4 1 2 3 4 1 2 3 4
1 2 3 4 5 6 7 8 9 10 11 12
368 168 530 784 770 354 1012 1244 1326 986 1624 2090
462.50 462.50 462.50 462.50 769.71 677.08 881.44 912.82 1228.35 1290.04 1445.48 1684.93
−94.500 −294.500 67.500 321.500 0.293 −323.081 130.558 331.182 97.653 −304.044 178.520 405.068
alpha beta
0.764197 1
368.00 475.21 744.58 1202.94 913.11 905.27 1420.60 1776.66 70755.110
109. Refer to Exhibit 11.17. What formulas should go in cells E3:G16 of the spreadsheet if the additive seasonal method is used to forecast sales? Exhibit 11.14 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs help analyzing the data using a 2-month moving average.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C 2-Month Moving Average
181.5 178.5 174.0 172.5 174.0 180.0 184.5 192.0 187.5 189.0
110. Refer to Exhibit 11.14. What formulas should go into cells B15:B18 and cells C15:C18 of the spreadsheet if a 2month moving average forecasting model is used? Exhibit 11.16 The following questions use the data below. Copyright Cengage Learning. Powered by Cognero.
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ch 11 The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using exponential smoothing.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C Exp Smoothing Prediction 174.00 174.00 180.60 175.06 177.23 171.85 176.76 176.86 183.52 187.25 185.38 189.61
MSE
91.18
D
E
F
alpha
0.440
111. Refer to Exhibit 11.16. What is the forecast for time period 13? Exhibit 11.15 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using weighted moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C 2-Month Weighted Moving Avg.
MSE
62.20
D
179.25 181.65 172.20 174.75 171.30 180.90 182.25 192.00 188.85 187.20
Copyright Cengage Learning. Powered by Cognero.
E
F
Weights w1 w2 sum
0.350 0.650 1.000
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ch 11 112. Refer to Exhibit 11.15. What is the forecast for time period 13-15? Exhibit 11.17 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the additive seasonal method.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1991
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
1992
1993
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Level 462.50 462.50 462.50 462.50 769.71 677.08 881.44 912.82 1228.35 1290.04 1445.48 1684.93
F Seasonal Factor −94.500 −294.500 67.500 321.500 0.293 −323.081 130.558 331.182 97.653 −304.044 178.520 405.068
G
H
I
J
alpha beta
0.764197 1
Forecast
368.00 475.21 744.58 1202.94 913.11 905.27 1420.60 1776.66 70755.110
113. Refer to Exhibit 11.17. What is the forecast for time period 13? Exhibit 11.22 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use regression and a linear trend model.
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations Intercept X Variable 1
A 1
B
0.895 0.791 0.773 269.406 12 Coefficients 28.273 139.958
C Time
D Actual
E Linear
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ch 11 2 3 4 5 6 7 8 9 10 11 12 13 14
Year 1
2
3
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
Period 1 2 3 4 5 6 7 8 9 10 11 12
Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
Trend 168.2 308.2 448.1 588.1 728.1 868.0 1008.0 1147.9 1287.9 1427.9 1567.8 1707.8
114. Refer to Exhibit 11.22. Interpret the R2 value for your model. Exhibit 11.25 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use a seasonal regression model to forecast sales.
Regression Statistics Coefficients Intercept 60.42 X Variable 1 1.91 X Variable 2 0.16 X Variable 3 −11.85 X Variable 4 −3.57 X Variable 5 25.98
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4 1
2
3
4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144 169
E Indicator for Qtr 1 1 0 0 0 1 0 0 0 1 0 0 0 1
Copyright Cengage Learning. Powered by Cognero.
F Actual 2 0 1 0 0 0 1 0 0 0 1 0 0 0
G 3 0 0 1 0 0 0 1 0 0 0 1 0 0
H Seasonal Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
I Model 50.6 61.3 93.6 70.6 62.1 74.1 107.6 85.9 78.7 92.0 126.8 106.4
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ch 11 115. Refer to Exhibit 11.25. What is the forecast for time period 13? Exhibit 11.22 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data. He wants to use regression and a linear trend model.
Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations Intercept X Variable 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
0.895 0.791 0.773 269.406 12 Coefficients 28.273 139.958
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Linear Trend 168.2 308.2 448.1 588.1 728.1 868.0 1008.0 1147.9 1287.9 1427.9 1567.8 1707.8
116. Refer to Exhibit 11.22. What is the forecast for time period 13? Exhibit 11.13 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1 2 3 4
Time Period 1 2
B Number of Toys Sold 174 189
C 4-Month Moving Avg.
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ch 11 5 6 7 8 9 10 11 12 13 14 15 16
3 4 5 6 7 8 9 10 11 12
168 180 165 183 177 192 192 183 195 189
177.75 175.50 174.00 176.25 179.25 186.00 186.00 190.50
MSE
91.34
117. Refer to Exhibit 11.13. What formulas go into cells B7:B9 of the following partial spreadsheet where data for periods 9-12 represent actual data? Exhibit 11.15 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using weighted moving averages.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C 2-Month Weighted Moving Avg.
MSE
62.20
D
179.25 181.65 172.20 174.75 171.30 180.90 182.25 192.00 188.85 187.20
E
F
Weights w1 w2 sum
0.350 0.650 1.000
118. Refer to Exhibit 11.15. What formulas should go in cells C5:C16 of the spreadsheet if Bill uses a 2 month weighted moving average forecasting model? Exhibit 11.19 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average method. Copyright Cengage Learning. Powered by Cognero.
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ch 11
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
69.00 70.80 72.60 79.80 82.80 85.80 94.20 97.20 101.40
73.05 76.50 80.25 85.65 90.00 94.65
86.55 89.10 91.35 102.75 104.40 108.15
4.50 4.20 3.70 5.70 4.80 4.50
91.05 93.30 95.05 108.45 109.20
119. Refer to Exhibit 11.19. What formulas should go in cells E6:G14 of the spreadsheet if the double moving average method is used to forecast sales? Exhibit 11.18 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the multiplicative seasonal method.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 368 168 530 784 770 354 1012 1244 1326 986 1624 2090
E Level 462.50 462.50 462.50 462.50 630.32 744.66 790.65 771.79 875.98 1273.96 1272.25 1280.35
F Seasonal Factor 0.796 0.363 1.146 1.695 1.222 0.475 1.280 1.612 1.514 0.774 1.276 1.632
G
H
I
J
alpha beta
0.332167 1
Forecast
368.00 228.96 853.34 1340.26 942.82 416.43 1630.62 2050.66 85564.040
120. Refer to Exhibit 11.18. What is the forecast for time period 13? Copyright Cengage Learning. Powered by Cognero.
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ch 11 Exhibit 11.24 The following questions use the data below. A store wants to predict quarterly sales. The owners have has collected 3 years of sales data and wants your help in analyzing the data. They want to use a quadratic trend model with seasonality to forecast sales.
Regression Statistics Coefficients 56.29 3.59 0.08
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
Qtr 1 2 3 4
Seasonal Index 84.1% 92.6% 127.7% 97.5%
2
3
D Time^2 1 4 9 16 25 36 49 64 81 100 121 144
E Actual Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
F Quadratic Trend 60.0 63.8 67.8 71.9 76.2 80.7 85.3 90.1 95.1 100.2 105.5 110.9
G Actual as a % of Trend 92% 94% 127% 103% 82% 83% 135% 96% 78% 101% 121% 93%
H Seasonal Forecast 50.4 59.1 86.5 70.1 64.1 74.8 109.0 87.8 79.9 92.8 134.7 108.1
121. Refer to Exhibit 11.24. Based on the regression output, what formulas should go in cells D3:H21? Exhibit 11.13 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using moving averages.
A 1 2
Time Period
B Number of Toys Sold
C 4-Month Moving Avg.
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ch 11 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 2 3 4 5 6 7 8 9 10 11 12
174 189 168 180 165 183 177 192 192 183 195 189
177.75 175.50 174.00 176.25 179.25 186.00 186.00 190.50
MSE
91.34
122. Refer to Exhibit 11.13. What is the forecast for time periods 13-15? Exhibit 11.19 The following questions use the data below. A store wants to predict quarterly sales. The owner has collected 3 years of sales data and wants your help in analyzing the data using the double moving average method.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3 4
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11 12
D Actual Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2 103.2
E Moving Ave
F Dbl Moving Ave
G
H
I
Level
Trend
Forecast
69.00 70.80 72.60 79.80 82.80 85.80 94.20 97.20 101.40
73.05 76.50 80.25 85.65 90.00 94.65
86.55 89.10 91.35 102.75 104.40 108.15
4.50 4.20 3.70 5.70 4.80 4.50
91.05 93.30 95.05 108.45 109.20
123. Refer to Exhibit 11.19. What are the forecasts for time periods 13-16? Exhibit 11.16 The following questions use the data below. The owner of Tim's Toys wants to predict monthly sales. He has collected data for 12 months. He needs your help in analyzing this data using exponential smoothing. Copyright Cengage Learning. Powered by Cognero.
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ch 11 A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Time Period 1 2 3 4 5 6 7 8 9 10 11 12
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189
C Exp Smoothing Prediction 174.00 174.00 180.60 175.06 177.23 171.85 176.76 176.86 183.52 187.25 185.38 189.61
MSE
91.18
D
E
F
alpha
0.440
124. Refer to Exhibit 11.16. What formulas should go in cells C3:C16 of the spreadsheet if Bill uses an exponential smoothing forecasting model? Exhibit 11.24 The following questions use the data below. A store wants to predict quarterly sales. The owners have has collected 3 years of sales data and wants your help in analyzing the data. They want to use a quadratic trend model with seasonality to forecast sales.
Regression Statistics Coefficients 56.29 3.59 0.08
Intercept X Variable 1 X Variable 2
1 2 3 4 5 6 7 8 9 10 11 12 13
A
B
Year 1
Qtr 1 2 3 4 1 2 3 4 1 2 3
2
3
C Time Period 1 2 3 4 5 6 7 8 9 10 11
D Time^2 1 4 9 16 25 36 49 64 81 100 121
E Actual Sales 55.2 60.0 86.4 74.4 62.4 67.2 115.2 86.4 74.4 100.8 127.2
F Quadratic Trend 60.0 63.8 67.8 71.9 76.2 80.7 85.3 90.1 95.1 100.2 105.5
Copyright Cengage Learning. Powered by Cognero.
G Actual as a % of Trend 92% 94% 127% 103% 82% 83% 135% 96% 78% 101% 121%
H Seasonal Forecast 50.4 59.1 86.5 70.1 64.1 74.8 109.0 87.8 79.9 92.8 134.7 Page 64
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ch 11 14 15 16 17 18 19 20 21
4
12
Qtr 1 2 3 4
Seasonal Index 84.1% 92.6% 127.7% 97.5%
144
103.2
110.9
93%
108.1
125. Refer to Exhibit 11.24. What is the forecast for time period 13?
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ch 11 Answer Key 1. False 2. True 3. False 4. True 5. True 6. True 7. False 8. True 9. c 10. d 11. d 12. c 13. a 14. a 15. c 16. b 17. d 18. a 19. d 20. b 21. b 22. d 23. a 24. c 25. c Copyright Cengage Learning. Powered by Cognero.
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ch 11 26. d 27. a 28. c 29. a 30. b 31. a 32. d 33. b 34. d 35. a 36. c 37. b 38. a 39. d 40. c 41. b 42. a 43. c 44. a 45. d 46. d 47. b 48. b 49. d 50. c 51. b Copyright Cengage Learning. Powered by Cognero.
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ch 11 52. a 53. d 54. b 55. b 56. d 57. d 58. d 59. d 60. a 61. c 62. d 63. c 64. c 65. c 66. b 67. a 68. a 69. a 70. b 71. d 72. a 73. c 74. a 75. a 76. b Copyright Cengage Learning. Powered by Cognero.
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ch 11 77. a 78. d 79. c 80. d 81. d 82. c 83. a 84. c 85. c 86. c 87. d 88. c 89. a 90. c 91. c 92. b 93. a 94. a 95. a 96. Cell D3 E3 I3
Formula =C3^2 =IF($B3=E$2,1,0) =60.42+1.91*C3+0.16*D3-11.85*E3-3.57*F3+25.98*G3
Copied to D4:D14 E4:E14 I4:I14
97. Cell E3 E7 F3 F7 G4
Formula =AVERAGE($D$3:$D$6) =$J$3*(D7/F3)+(1-$J$3)*E6 =D3/E3 =$J$4*(D7/F7)+(1-$J$4)*F3 =E6+F3
Copied to E4:E6 E8:E14 F4:F6 F8:F14 G8:G14
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ch 11 G16
=SUMXMY2(D7:D14,G7:G14)/COUNT(G7:G14)
98. Cell E3
Formula =28.273+139.958*C3
Copied to E4:E14
99. Year
Qtr
Time Period
1993 1994
4 1
12 13
Year
Qtr
Time Period
1994
1
13
Actual Sales
Base Level
Trend
2090
1809.1
Predicted Sales
2071.5
100. Actual Sales
Base Level
Trend
Seasonal Factor
Predicted Sales
100.5
4.3
0.816
85.547
101. Objective Cell: G16 Variables Cells: E3 Constraints Cells: 102. A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
103. Cell D3 F3
Time Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
B Number of Toys Sold 174 189 168 180 165 183 177 192 192 183 195 189 192.0 190.5 191.3 190.9
C 2-Month Moving Average
181.5 178.5 174.0 172.5 174.0 180.0 184.5 192.0 187.5 189.0 192.0 190.5 191.3 190.9
Formula =C3^2 =56.29+3.59*C3+0.08*D3
Copied to F4:F14
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ch 11 104. Cell C3:C6 C7 C16
Formula None =AVERAGE(C3:C6) =SUMXMY2(B7:B14,C7:C14)/COUNT(C7:C14)
Copied to
105. Cell G3 E6 E7 F6 F7 G7 H7 H24
Formula =D3/AVERAGE($D$3:$D$6) =D6/G6 =$D$17*D7/G3+(1-$D$17)*(E6+F6) =0 =$D$18*(E7-E6)+(1-$D$18)*F6 =$D$19*D7/E7+(1-$D$19)*G3 =SUM(E6:F6)*G3 =SUMXMY2(D7:D14,H7:H14)/COUNT(H7:H14)
Copied to G4:G6 − E8:E14 − F8:F14 G8:G14 H8:H14 −
C8:C14
106. Objective Cell: H16 Variables Cells: E6 Constraints Cells: 107. Cell E3 E4 F3 F4 G4 G16
Formula =D3 =$J$3*D4+(1-$J$3)(E3+F3) =0 =$J$4*(E4-E3)+(1-$J$4)*F3 =SUM(E3:F3) =SUMXMY2(D3:D14,G3:G14)/COUNT(G3:G14)
Copied to E5:E14 F5:F14 G5:G14
108. Year
Qtr
Time Period
Time^2
13
169
Actual Sales
Quadratic Trend
116.5
109. Cell E3 E7 F3 F7 G4 G16
Formula =AVERAGE($D$3:$D$6) =$J$3*(D7-F3)+(1-$J$3)*E6 =D3-E3 =$J$4*(D7-F7)+(1-$J$4)*F3 =E6+F3 =SUMXMY2(D7:D14,G7:G14)/COUNT(G7:G14)
Copied to E4:E6 E8:E14 F4:F6 F8:F14 G8:G14
110. Cell B15 C15
Formula =AVERAGE(B13:B14) =AVERAGE(B13:B14)
Copied to B16:B18 C16:C18
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ch 11 111. Time Period 12 13 112. Time Period 11 12 13 14 15
Number of Toys Sold 189
Exp Smoothing Prediction 189.61 189.34
Number of Toys Sold 195 189 187.20 192.90 188.37
2-Month Weighted Moving Avg. 188.85 187.20 192.90 188.37 189.20
113. Year
Qtr
Time Period
1993 1994
4 1
12 13
Actual Sales
Level
Seasonal Factor
2090
1684.93
405.068
Forecast
1776.66 1782.59
114. The R2 value of 0.791 means that 79.1% of the variability in sales is explained by the time period. 115. Year
Qtr
Time Period
4
1
13
Year
Qtr
Time Period
4
1
13
Time^2
Indicator for Qtr 1
Actual 2
Seasonal 3 Sales
Model
169
1
0
0
100.4
116. Actual Sales
Linear Trend
1847.7
117. Cell B7 C7
Formula =AVERAGE(B3:B6) =AVERAGE(B3:B6)
Copied to B8:B9 C8:C9
118. Cell C5 C16
Formula =$F$3*B3+$F$4*B4 =SUMXMY2(B5:B14,C5:C14)/COUNT(C5:C14)
Copied to C6:C14
119. Cell E3 F9 G9 H9
Formula =AVERAGE(D3:D6) =AVERAGE(E6:E9) =2*E9-F9 =3*(E9-F9)/(4-1)
Copied to E7:E14 F10:F14 G10:G14 H10:H14
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ch 11 I10
=G9+H9
I11:I14
120. Year
Qtr
Time Period
3 4
4 1
12 13
Actual Sales
Level
Seasonal Factor
2090
1280.35
1.632
Forecast
2050.66 1938.11
121. Cell
Formula
Copied to
D3 F3 G3 C26 H3
=C3^2 =56.29+3.59*C3+0.08*D3 =E3/F3 =SUMIF($B$3:$B$14,B18,$G$3:$G$14)/COUNTIF($B$3:$B$14,B18) =F3*VLOOKUP(B3,$B$18:$C$21,2)
F4:F14 F4:F14 C19:C21
122. Time Period 9 10 11 12 13 14 15
Number of Toys Sold 192 183 195 189 189.75 189.19
2-Month Weighted Moving Avg.
189.75 189.19 190.74
123. Year
Qtr
Time Period
Actual Sales
Moving Ave
Dbl Moving Ave
Level
Trend
Forecast
3 4
4 1 2 3 4
12 13 14 15 16
103.2
101.40
94.65
108.15
4.50
109.20 112.65 117.15 121.65 126.15
124. Cell C3 C4 C16
Formula =B3 =C3+$F$3*(B3-C3) =SUMXMY2(B3:B14,C3:C14)/COUNT(C3:C14)
Copied to C5:C14
125. Qtr
Time Period
Time^2
1
13
169
Actual Sales
Quadratic Trend
116.5
Actual as a % of Trend
Seasonal Forecast
97.9
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ch 12
Indicate whether the statement is true or false. 1. The historical data itself can be sampled from using Analytic Solver Platform’s PsiDisUniform( ), PsiResample( ), PsiSip( ) or PsiSlurp( ) functions. a. True b. False 2. The term “risk” also implies the potential for loss. a. True b. False 3. The educational version of Analytic Solver Platform has no limit on the number of trials per simulation. a. True b. False 4. One of Analytic Solver Platform’s amazing capabilities is its ability to perform interactive simulation. a. True b. False 5. In the face of uncertainty, some people react with paralysis, or they do exhaustive research to avoid making a decision. a. True b. False 6. To perform simulation in a spreadsheet, we must first place a random number generator (RNG) formula in each cell that represents a random, or uncertain, independent variable. a. True b. False 7. Several techniques are available to help managers analyze risk. Three of the most common are best-case/worst-case analysis, what-if analysis, and simulation. Of these methods, what-if analysis the most powerful. a. True b. False 8. If you have historical data for any of the random variables in your model, you can use this dialog to instruct Analytic Solver Platform to automatically identify and suggest appropriate probability distributions for your data. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. Risk needs to be analyzed using models in order to a. make decisions better than those based on informed guesses b. obtain a feasible solution c. obtain an optimal solution d. confound the decision-making process Copyright Cengage Learning. Powered by Cognero.
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ch 12 10. A random variable is a. a variable whose value cannot be predicted with certainty b. a parameter c. a population variable d. a sample variable Exhibit 12.2 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels the change in number of cars can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%). The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. A spreadsheet model to simulate the problem has been run 300 times. A part of the simulation statistics output from Risk Solver Platform (RSP)and a spreadsheet for computing confidence intervals follows.
Simulation Statistics Name Description Cell Minimum= Maximum= Mean= Std Deviation=
1 2 3 4 5 6 7 8 9 10 11 12
Income/Revenue Output D21 3339249.82 5086714.77 4119518.91 291116.83 A
B
Sample Size:
300
Sample Mean: Sample Standard Deviation:
4,119,519 291,117
95% LCL for the population mean: 95% UCL for the population mean:
4,086,576 4,152,462
Target Proportion 95% Lower Confidence Limit: 95% Upper Confidence Limit:
0.900 0.866 0.934 Confidence Intervals
11. Using the information in Exhibit 12.2, what formula should go in cell B8 of the Confidence Intervals spreadsheet to compute the upper limit on a 95% confidence interval for the true population mean? a. =B4+1.96*B5/SQRT(B2) b. =B4+1.645*B5/SQRT(B2) c. =B4-1.96*B5/SQRT(B2) Copyright Cengage Learning. Powered by Cognero.
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ch 12 d. =B4+1.96*B5/B2 Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
9
2
19
0
3
3
>
10
3
16
0
7
7
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7
7
>
MODEL
<
>
F
G
H
I
Copyright Cengage Learning. Powered by Cognero.
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ch 12 <
1
<
2
Performance Measures
<
3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
<
5
<
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ch 12 15 DATA
12. Using the information in Exhibit 12.3, what Analytic Solver Platform function should be used in cell I8 to determine the lead time for an order? a. =PsiDiscrete($B$6:$B$8, $C$6:$C$8) b. =PsiPoisson(Data!$B$6:$B$8,$Data!$C$6:$C$8) c. =PsiBinomial(Data!$B$6:$B$8,$Data!$C$6:$C$8) d. =PsiNormal($B$6:$C$8) 13. The PsiTarget(.) function in Analytic Solver Platform a. returns the cumulative probability of a specified distribution cell being less or equal to the specified target value b. returns the cumulative probability of a specified distribution cell being equal to or larger than the specified target value c. returns the cumulative probability of a specified distribution cell being equal to the specified target value d. returns the probability of a specified distribution cell being equal to the specified target value 14. Why would a manager be interested in analyzing risk? a. To determine a most likely outcome. b. To determine a range of outcomes. c. To determine a distribution of outcomes. d. To determine a confidence interval on most likely outcomes. 15. Some discrete distributions available in Analytic Solver Platform are a. binomial b. Poisson c. Hypergeometric d. all of the above 16. Which of the following do not help determine the magnitude of risk in a decision-making problem? a. The level of uncertainty in outcome. b. The magnitude of potential loss. c. The level of management interest in the problem. d. The level of uncertainty in input variables. 17. As the number of replicates in a simulation increases the width of a confidence interval computed from the simulation results will a. decrease. b. increase. c. remain the same. d. change depends on standard deviation. Copyright Cengage Learning. Powered by Cognero.
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ch 12 Exhibit 12.4. The following questions use the information below. The manager of a Washington, DC sightseeing tour company is concerned about overbooking for one of his bus tours. The bus has 15 seats but sometimes there are empty seats. His records show that about 20% of ticket holders do not show up for their tour. Tickets cost $10 and are non-refundable. If the manager overbooks the tour and more than 15 passengers show up, some of them will be bumped to a later tour. This bumping costs the company $25 in various expenses to keep the customer happy until the next tour. The manager wants to see what happens to profits if 18 reservations are accepted.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B Capital Tours Tour Reservation System
C
Seats Available Ticket Price per Seat Prob. of No-Show Cost of Bumping Reservations Accepted
15 $10 0.2 $25 18
Passengers to Board
12
Ticket Revenue Opp. Cost of Empty Seats Cost of Bumping Passengers Marginal Profit MODEL
$180 $30 0 $150
18. Using the information in Exhibit 12.4, what formula should go in cell C15 of the worksheet to determine the Marginal Profit? a. =C12+C13+C14 b. =C12-C13-C14 c. =C12-C14+C13 d. =C12-(C13-C14) 19. Which Analytic Solver Platform function will generate random integer numbers between 2 and 8? a. =PsiIntUniform(2, 8) b. =PsiDiscrete(2, 8) c. =PsiUniform(2, 8) d. =PsiRandom(2, 8) Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the Copyright Cengage Learning. Powered by Cognero.
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ch 12 inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
9
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G
H
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Performance Measures
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3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
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Order?
I
J
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ch 12 <
7
End. Inv.
Pos.
(0=n,1=y)
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3
18
0
0
0
<
14
13
13
0
0
0
<
15
12
12
0
0
0
<
16
11
11
0
0
0
<
17
5
5
1
2
13
<
18
4
19
0
0
0
<
19
0
15
0
0
0
<
20
13
13
0
0
0
<
21
6
6
0
0
0
A
B
C
D
E
F
G
1 2 3 4
Shipping Time
Quantity Demanded
5
Days
Prob.
Units
Prob.
6
3
0.20
0
0.01
7
4
0.60
1
0.02
8
5
0.20
2
0.04
9
Total
1.00
3
0.06
10
4
0.25
11
5
0.20
12
6
0.20
13
7
0.22
14
Total
1.00
15 DATA
20. Using the information in Exhibit 12.3, what Analytic Solver Platform function should be used in cell D8 and copied to cells D9:D21 of the MODEL sheet to compute daily demand? a. =PsiDiscrete($E$6:$E$13, $F$6:$F$13) Copyright Cengage Learning. Powered by Cognero.
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ch 12 b. =PsiHypergeo($E$6:$F138) c. =PsiBinomial($E$6, $F$13) d. =PsiCustom(Data!$E$6, $F$13) 21. Using the information in Exhibit 12.3, what formula should go in cell H8 to determine if an order should be placed? a. =IF(G8<$D$3,1,0) b. =IF(G8>$D$3,1,0) c. =IF(G8<$D$3,0,1) d. =IF(G8<$D$4,1,0) 22. The worst-case analysis approach to risk analysis a. is optimistic b. is pessimistic c. is most likely d. generally produces good outcomes 23. Which of the following distributions can be generated by Analytic Solver Platform? a. Poisson b. Normal c. Weibull d. All of these 24. If a spreadsheet simulation user has a probability distribution that may assume 1 of 5 values with nearly equal probability, this user has what type of distribution? a. A discrete distribution. b. A continuous distribution. c. A triangular distribution. d. A normal distribution. 25. In running simulations under Analytic Solver Platform it is desirable to use more simulation runs because a. confidence level regarding the decision precision is improved b. point estimates are more accurate c. simulation run length is decreased d. cost is decreased Exhibit 12.1 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels this can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%) over the previous months. The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. You have created the following spreadsheet to simulate the problem. Copyright Cengage Learning. Powered by Cognero.
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Name:
Class:
Date:
ch 12
A
B
C
D
>
1 Initial Conditions:
>
2 Number of Cars:
100
>
3 Average Damage:
$3,000
>
4
>
5
>
6
Number of
Average
Total
>
7
Month
Cars
Damage
Revenue
>
8
1
99.81
$4,006
$399,868
>
9
2
98.93
$3,188
$315,402
>
10
3
97.57
$2,579
$251,620
>
11
4
97.57
$3,804
$371,193
>
12
5
95.64
$3,045
$291,270
>
13
6
93.18
$2,537
$236,396
>
14
7
92.72
$3,182
$295,054
>
15
8
90.56
$2,945
$266,745
>
16
9
90.96
$2,038
$185,383
>
17
10
92.87
$3,786
$351,624
>
18
11
93.71
$3,037
$284,560
>
19
12
97.77
$3,404
$332,819
>
20
>
21
Income:
$3,581,933 >
FIXADENT MODEL
<
E
F
G
>
H
I
< 1 Assumptions: < 2 Max Decrease:
3%
Max Increase:
5%
Uniform
< 3 Mthly. Increase:
1%
Std Dev:
500
Normal
< 4 Constant < 5 < 6 < 7 < 8 < 9 < 10 Copyright Cengage Learning. Powered by Cognero.
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ch 12 < 11 < 12 < 13 < 14 < 15 < 16 < 17 < 18 < 19 < 20 < 21 <
26. Using the information in Exhibit 12.1, what Analytic Solver Platform function should go in cell C8 and copied to cells C9:C19 to compute the average damage per car in the month? a. =PsiNormal($C$3*$F$3^$A$8, $H$3) b. =PsiNormal($C$3*$F$3^A8, $H$3) c. =PsiNormal($C$3, $D$3*(1+$F$3)^A8) d. =PsiNormal($C$3*(1+$F$3)^A8, $H$3) Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
Copyright Cengage Learning. Powered by Cognero.
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Date:
ch 12 9
2
19
0
3
3
>
10
3
16
0
7
7
>
11
4
9
0
1
1
>
12
5
8
0
4
4
>
13
6
4
0
1
1
>
14
7
3
15
5
5
>
15
8
13
0
1
1
>
16
9
12
0
1
1
>
17
10
11
0
6
6
>
18
11
5
0
1
1
>
19
12
4
0
4
4
>
20
13
0
15
2
2
>
21
14
13
0
7
7
>
MODEL
<
>
F
G
H
<
1
<
2
Performance Measures
<
3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
<
5
<
6
<
7
<
I
J
Inv.
Order?
Lead
Arrives
End. Inv.
Pos.
(0=n,1=y)
Time
On Day
8
19
19
0
0
0
<
9
16
16
0
0
0
<
10
9
9
0
0
0
<
11
8
8
0
0
0
<
12
4
4
1
1
7
<
13
3
18
0
0
0
<
14
13
13
0
0
0
<
15
12
12
0
0
0
<
16
11
11
0
0
0
<
17
5
5
1
2
13
<
18
4
19
0
0
0
<
19
0
15
0
0
0
<
20
13
13
0
0
0
Copyright Cengage Learning. Powered by Cognero.
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Date:
ch 12 <
21
6
A
6 B
0 C
0 D
0 E
F
G
1 2 3 4
Shipping Time
Quantity Demanded
5
Days
Prob.
Units
Prob.
6
3
0.20
0
0.01
7
4
0.60
1
0.02
8
5
0.20
2
0.04
9
Total
1.00
3
0.06
10
4
0.25
11
5
0.20
12
6
0.20
13
7
0.22
14
Total
1.00
15 DATA
27. Using the information in Exhibit 12.3, what formula should go in cell G9 to compute inventory position? a. =G8+E9-IF(H8=1,$D$4,0) b. =G8-E9+IF(H8=1,$D$4,0) c. =G8-E9+IF(H8=1,$D$5,0) d. =G8-E9+IF(H8=1,0,$D$4) 28. Simulation is used to a. find possible worst case values for the dependent variable(s). b. find worst case and best case values for the dependent variable(s). c. find distribution information for the dependent variable(s). d. find median values for the dependent variable(s). 29. Uncertainty a. is the most difficult thing about decision-making b. simplifies decision-making c. is easy to capture with computers d. requires the use of quantitative models for supporting decision-making 30. Which Analytic Solver Platform function will generate random numbers between 3 and 7 from a continuous uniform distribution? Copyright Cengage Learning. Powered by Cognero.
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ch 12 a. =PsiConUniform(3, 7) b. =PsiUniDist(3, 7) c. =PsiUniform(3, 7) d. =PsiTruncate(3, 7) Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
9
2
19
0
3
3
>
10
3
16
0
7
7
>
11
4
9
0
1
1
>
12
5
8
0
4
4
>
13
6
4
0
1
1
>
14
7
3
15
5
5
>
15
8
13
0
1
1
>
16
9
12
0
1
1
>
17
10
11
0
6
6
>
18
11
5
0
1
1
>
19
12
4
0
4
4
>
20
13
0
15
2
2
>
21
14
13
0
7
7
>
MODEL
>
Copyright Cengage Learning. Powered by Cognero.
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Name:
Class:
Date:
ch 12
<
F
G
H
<
1
<
2
Performance Measures
<
3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
<
5
<
6
<
7
<
I
J
Inv.
Order?
Lead
Arrives
End. Inv.
Pos.
(0=n,1=y)
Time
On Day
8
19
19
0
0
0
<
9
16
16
0
0
0
<
10
9
9
0
0
0
<
11
8
8
0
0
0
<
12
4
4
1
1
7
<
13
3
18
0
0
0
<
14
13
13
0
0
0
<
15
12
12
0
0
0
<
16
11
11
0
0
0
<
17
5
5
1
2
13
<
18
4
19
0
0
0
<
19
0
15
0
0
0
<
20
13
13
0
0
0
<
21
6
6
0
0
0
A
B
C
D
E
F
G
1 2 3 4
Shipping Time
Quantity Demanded
5
Days
Prob.
Units
Prob.
6
3
0.20
0
0.01
7
4
0.60
1
0.02
8
5
0.20
2
0.04
9
Total
1.00
3
0.06
10
4
0.25
11
5
0.20
12
6
0.20
Copyright Cengage Learning. Powered by Cognero.
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ch 12 13
7
0.22
14
Total
1.00
15 DATA
31. Using the information in Exhibit 12.3, what formula should go in cell H3 to compute the service level? a. =SUM(G8:G21)/SUM(D8:D21) b. =SUM(D8:D21)/SUM(E8:E21) c. =SUM(E8:E21)/SUM(D8:D21) d. =SUM(F8:F21)/SUM(G8:G21) 32. If chance or uncertainty is present in a system then there is an element of ____ in the decision-making problem. a. danger b. security c. risk d. difficulty 33. Which of the following probability distributions are associated with discrete outcomes? a. Gamma b. Custom c. Normal d. Exponential 34. A variable whose value cannot be predicted or set with certainty is a a. discrete variable b. random variable c. realistic variable d. simulation variable Exhibit 12.1 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels this can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%) over the previous months. The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
1 Initial Conditions:
D
> >
Copyright Cengage Learning. Powered by Cognero.
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Name:
Class:
Date:
ch 12 2 Number of Cars:
100
>
3 Average Damage:
$3,000
>
4
>
5
>
6
Number of
Average
Total
>
7
Month
Cars
Damage
Revenue
>
8
1
99.81
$4,006
$399,868
>
9
2
98.93
$3,188
$315,402
>
10
3
97.57
$2,579
$251,620
>
11
4
97.57
$3,804
$371,193
>
12
5
95.64
$3,045
$291,270
>
13
6
93.18
$2,537
$236,396
>
14
7
92.72
$3,182
$295,054
>
15
8
90.56
$2,945
$266,745
>
16
9
90.96
$2,038
$185,383
>
17
10
92.87
$3,786
$351,624
>
18
11
93.71
$3,037
$284,560
>
19
12
97.77
$3,404
$332,819
>
20
>
21
Income:
$3,581,933 >
FIXADENT MODEL
<
E
F
G
>
H
I
< 1 Assumptions: < 2 Max Decrease:
3%
Max Increase:
5%
Uniform
< 3 Mthly. Increase:
1%
Std Dev:
500
Normal
< 4 Constant < 5 < 6 < 7 < 8 < 9 < 10 < 11 < 12 < 13 Copyright Cengage Learning. Powered by Cognero.
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ch 12 < 14 < 15 < 16 < 17 < 18 < 19 < 20 < 21 <
35. Using the information in Exhibit 12.1, what formula should go cell G5 to calculate the 80th percentile of the empirical distribution of income? a. =IF(RANK(D21) = 80, 1, 0) b. None, use Excel Histogram on cell D21. c. =IF(COUNTIF(D8:D19 > D21) ≥ 8, 1, 0) d. =PsiPercentile(B21, .8). 36. Common continuous distributions available in Analytic Solver Platform are a. exponential b. Normal c. truncated Normal d. all of the above Exhibit 12.1 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels this can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%) over the previous months. The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
D
1 Initial Conditions:
> >
2 Number of Cars:
100
>
3 Average Damage:
$3,000
>
4
>
5
>
Copyright Cengage Learning. Powered by Cognero.
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Name:
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Date:
ch 12 6
Number of
Average
Total
>
7
Month
Cars
Damage
Revenue
>
8
1
99.81
$4,006
$399,868
>
9
2
98.93
$3,188
$315,402
>
10
3
97.57
$2,579
$251,620
>
11
4
97.57
$3,804
$371,193
>
12
5
95.64
$3,045
$291,270
>
13
6
93.18
$2,537
$236,396
>
14
7
92.72
$3,182
$295,054
>
15
8
90.56
$2,945
$266,745
>
16
9
90.96
$2,038
$185,383
>
17
10
92.87
$3,786
$351,624
>
18
11
93.71
$3,037
$284,560
>
19
12
97.77
$3,404
$332,819
>
20
>
21
Income:
$3,581,933 >
FIXADENT MODEL
<
E
F
G
>
H
I
< 1 Assumptions: < 2 Max Decrease:
3%
Max Increase:
5%
Uniform
< 3 Mthly. Increase:
1%
Std Dev:
500
Normal
< 4 Constant < 5 < 6 < 7 < 8 < 9 < 10 < 11 < 12 < 13 < 14 < 15 < 16 < 17 Copyright Cengage Learning. Powered by Cognero.
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ch 12 < 18 < 19 < 20 < 21 <
37. Using the information in Exhibit 12.1, what Analytic Solver Platform function should go in cell B8 to compute the number of cars repaired in the first month? a. =PsiUniform($C$2*(1-$F$2), $C$2*(1+$H$2)) b. =PsiUniform($C$2*1-$F$2, $C$2*1+$H$2) c. =PsiUniform($C$2-(1-$F$2), $C$2+(1+$H$2)) d. =PsiUniform($C$2+(1-$F$2), $C$2+(1+$H$2)) 38. Best case analysis is a(n) ____ view of the problem. a. pessimistic b. optimistic c. risky d. certain Exhibit 12.2 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels the change in number of cars can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%). The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. A spreadsheet model to simulate the problem has been run 300 times. A part of the simulation statistics output from Risk Solver Platform (RSP)and a spreadsheet for computing confidence intervals follows.
Simulation Statistics Name Description Cell Minimum= Maximum= Mean= Std Deviation=
1 2 3
Income/Revenue Output D21 3339249.82 5086714.77 4119518.91 291116.83 A
B
Sample Size:
300
Copyright Cengage Learning. Powered by Cognero.
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Name:
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Date:
ch 12 4 5 6 7 8 9 10 11 12
Sample Mean: Sample Standard Deviation:
4,119,519 291,117
95% LCL for the population mean: 95% UCL for the population mean:
4,086,576 4,152,462
Target Proportion 95% Lower Confidence Limit: 95% Upper Confidence Limit:
0.900 0.866 0.934 Confidence Intervals
39. Using the information in Exhibit 12.2, what is the worst case scenario for the Fix-a-dent company based on this output? a. $1,747,464.94 b. $3,339,249.82 c. $4,122,024.01 d. $4,207,301.98 40. What is the correct Analytic Solver Platform function for generating random numbers from the following distribution on the number of children in families (assume a reference to the current worksheet).
A # of children 1 2 3 4 5 a. =PsiCustom(A2:A6,B2:B6) b. =PsiRandom(A2:A6,B2:B6) c. =PsiDiscrete(A2:A6,B2:B6) d. =PsiUniform(A2:A6,B2:B6)
1 2 3 4 5 6
B P(# of children) 0.20 0.30 0.25 0.20 0.05
Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
1
D
E
> >
Copyright Cengage Learning. Powered by Cognero.
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Name:
Class:
Date:
ch 12 2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
9
2
19
0
3
3
>
10
3
16
0
7
7
>
11
4
9
0
1
1
>
12
5
8
0
4
4
>
13
6
4
0
1
1
>
14
7
3
15
5
5
>
15
8
13
0
1
1
>
16
9
12
0
1
1
>
17
10
11
0
6
6
>
18
11
5
0
1
1
>
19
12
4
0
4
4
>
20
13
0
15
2
2
>
21
14
13
0
7
7
>
MODEL
<
>
F
G
H
<
1
<
2
Performance Measures
<
3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
<
5
<
6
<
7
<
I
J
Inv.
Order?
Lead
Arrives
End. Inv.
Pos.
(0=n,1=y)
Time
On Day
8
19
19
0
0
0
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9
16
16
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<
10
9
9
0
0
0
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11
8
8
0
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0
<
12
4
4
1
1
7
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ch 12 <
13
3
18
0
0
0
<
14
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17
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1
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21
6
6
0
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0
A
B
C
D
E
F
G
1 2 3 4
Shipping Time
Quantity Demanded
5
Days
Prob.
Units
Prob.
6
3
0.20
0
0.01
7
4
0.60
1
0.02
8
5
0.20
2
0.04
9
Total
1.00
3
0.06
10
4
0.25
11
5
0.20
12
6
0.20
13
7
0.22
14
Total
1.00
15 DATA
41. Using the information in Exhibit 12.3, what formula should go in cell C9 and copied to C10:C21 of the MODEL sheet to compute units received? a. =COUNT($J$8:J8,A9)*$D$4 b. =COUNTIF($J$8:J8,A8)*$C$4 c. =COUNTIF($J$8:J8,A9)*$D$5 d. =COUNTIF($J$8:J8,A9)*$D$4 42. The Analytic Solver Platform is a good simulation tool because a. it is difficult to use b. it provides several polymorphic spreadsheet interpreter "psi" functions to facilitate Excel calculations Copyright Cengage Learning. Powered by Cognero.
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ch 12 c. it has a limited choice of statistical distributions making it difficult to approximate reality d. it requires a standalone platform operating independently of Excel 43. In a what-if analysis the decision maker a. changes the values of the uncertain input variables to see what happens to the performance measure being studied b. assumes that all inputs are random variables c. assumes that all outputs are deterministic d. corrects for underestimated risk 44. Large sample size, n, is desirable because a. upper and lower limits of the confidence interval for the true population mean are closer to one another b. upper and lower limits of the confidence interval for the true population mean are further apart c. point estimate of the population mean is larger d. point estimate of the population mean is smaller Exhibit 12.4. The following questions use the information below. The manager of a Washington, DC sightseeing tour company is concerned about overbooking for one of his bus tours. The bus has 15 seats but sometimes there are empty seats. His records show that about 20% of ticket holders do not show up for their tour. Tickets cost $10 and are non-refundable. If the manager overbooks the tour and more than 15 passengers show up, some of them will be bumped to a later tour. This bumping costs the company $25 in various expenses to keep the customer happy until the next tour. The manager wants to see what happens to profits if 18 reservations are accepted.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B Capital Tours Tour Reservation System
C
Seats Available Ticket Price per Seat Prob. of No-Show Cost of Bumping Reservations Accepted
15 $10 0.2 $25 18
Passengers to Board
12
Ticket Revenue Opp. Cost of Empty Seats Cost of Bumping Passengers Marginal Profit MODEL
$180 $30 0 $150
45. Using the information in Exhibit 12.4, what formula should go in cell C13 of the worksheet to determine the Opportunity Cost of Empty Seats? a. =C5*MAX(C4,C10) Copyright Cengage Learning. Powered by Cognero.
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ch 12 b. =C5*MAX(C4,0) c. =MAX(C4-C10,0) d. =C5*MAX(C4-C10,0) Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
9
2
19
0
3
3
>
10
3
16
0
7
7
>
11
4
9
0
1
1
>
12
5
8
0
4
4
>
13
6
4
0
1
1
>
14
7
3
15
5
5
>
15
8
13
0
1
1
>
16
9
12
0
1
1
>
17
10
11
0
6
6
>
18
11
5
0
1
1
>
19
12
4
0
4
4
>
20
13
0
15
2
2
>
21
14
13
0
7
7
>
MODEL
>
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ch 12
<
F
G
H
<
1
<
2
Performance Measures
<
3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
<
5
<
6
<
7
<
I
J
Inv.
Order?
Lead
Arrives
End. Inv.
Pos.
(0=n,1=y)
Time
On Day
8
19
19
0
0
0
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9
16
16
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0
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19
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19
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20
13
13
0
0
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21
6
6
0
0
0
A
B
C
D
E
F
G
1 2 3 4
Shipping Time
Quantity Demanded
5
Days
Prob.
Units
Prob.
6
3
0.20
0
0.01
7
4
0.60
1
0.02
8
5
0.20
2
0.04
9
Total
1.00
3
0.06
10
4
0.25
11
5
0.20
12
6
0.20
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ch 12 13
7
0.22
14
Total
1.00
15 DATA
46. The average demand is 4.45 cases per day. Using the information in Exhibit 12.3, what formula should go in cell H4 to determine the average lost sales? a. =H3*4.45*10*30 b. =(1-H3)*4.45*10 c. =(1-H3)*4.45*30 d. =(1-H3)*4.45*10*30 47. Using the information in Exhibit 12.3, what formula should go in cell J8 to determine the arrival date for an order? a. =IF(I8=0,0,A8+I8) b. =IF(I8=0,0,A8+1+I8) c. =IF(I8=1,0,A8+1+I8) d. =IF(I8=0,A8+1+I8,0) 48. Methods for analyzing risk that are discussed in the textbook include a. best-case/worst case analysis b. what-if analysis c. simulation d. all of the above 49. The benefit(s) of simulation discussed in the text is/are a. gaining better insight into a problem b. ability to make more informed decisions c. eliminating the need to build a system with fine-tuning occurring after the launch d. all of the above 50. In general there are two primary issues involved in risk. What are these two issues? a. Validation of spreadsheet model and setting up for the simulation. b. Uncertainty of the outcome and magnitude of the potential loss. c. Maximum amount of profit made and the probability of a maximum profit. d. Maximum amount of loss incurred and the likelihood of that loss. 51. The best-case analysis approach to risk analysis a. is optimistic b. is pessimistic c. is most likely d. is likely to produce good outcomes Copyright Cengage Learning. Powered by Cognero.
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ch 12 52. Worst case analysis is a(n) ____ view of the problem. a. pessimistic b. optimistic c. risky d. certain 53. A good way to fit the distribution to historical patterns when historical data is available is to a. use the Fit icon in the Tools group of the Analytic Solver Platform b. guess the distribution parameters c. assume that the data distribution is Normal d. assume that the data distribution is Weibull Exhibit 12.4. The following questions use the information below. The manager of a Washington, DC sightseeing tour company is concerned about overbooking for one of his bus tours. The bus has 15 seats but sometimes there are empty seats. His records show that about 20% of ticket holders do not show up for their tour. Tickets cost $10 and are non-refundable. If the manager overbooks the tour and more than 15 passengers show up, some of them will be bumped to a later tour. This bumping costs the company $25 in various expenses to keep the customer happy until the next tour. The manager wants to see what happens to profits if 18 reservations are accepted.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B Capital Tours Tour Reservation System
C
Seats Available Ticket Price per Seat Prob. of No-Show Cost of Bumping Reservations Accepted
15 $10 0.2 $25 18
Passengers to Board
12
Ticket Revenue Opp. Cost of Empty Seats Cost of Bumping Passengers Marginal Profit MODEL
$180 $30 0 $150
54. Using the information in Exhibit 12.4, what Analytic Solver Platform function should be used in cell C10 to determine the number of Passengers to Board? a. =PsiBinomial($C$8, 1-$C$6) b. =PsiBinomial(1-$C$6, $C$8) c. =PsiBinomial($C$8, $C$6) d. =PsiBinomial($C$6, $C$8) Copyright Cengage Learning. Powered by Cognero.
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ch 12 Exhibit 12.1 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels this can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%) over the previous months. The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
D
>
1 Initial Conditions:
>
2 Number of Cars:
100
>
3 Average Damage:
$3,000
>
4
>
5
>
6
Number of
Average
Total
>
7
Month
Cars
Damage
Revenue
>
8
1
99.81
$4,006
$399,868
>
9
2
98.93
$3,188
$315,402
>
10
3
97.57
$2,579
$251,620
>
11
4
97.57
$3,804
$371,193
>
12
5
95.64
$3,045
$291,270
>
13
6
93.18
$2,537
$236,396
>
14
7
92.72
$3,182
$295,054
>
15
8
90.56
$2,945
$266,745
>
16
9
90.96
$2,038
$185,383
>
17
10
92.87
$3,786
$351,624
>
18
11
93.71
$3,037
$284,560
>
19
12
97.77
$3,404
$332,819
>
20
>
21
Income:
$3,581,933 >
FIXADENT MODEL
<
E
F
G
>
H
I
5%
Uniform
< 1 Assumptions: < 2 Max Decrease:
3%
Max Increase:
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ch 12 < 3 Mthly. Increase:
1%
Std Dev:
500
Normal
< 4 Constant < 5 < 6 < 7 < 8 < 9 < 10 < 11 < 12 < 13 < 14 < 15 < 16 < 17 < 18 < 19 < 20 < 21 <
55. Using the information in Exhibit 12.1, what Analytic Solver Platform function should go in cell B9 and copied to B10:B19 to compute the number of cars repaired in the subsequent months? a. =PsiUniform($B$8*(1+$F$2), $B$8*(1-$H$2)) b. =PsiUniform(B8*(1-$F$2), B8*(1+$H$2)) c. =PsiUniform($C$2*(1-$F$2), $C$2*(1+$H$2)) d. =PsiIntUniform($C$2*(1-$F$2), $C$2*(1+$H$2)) 56. Which of the following probability distributions are associated with continuous outcomes? a. Poisson b. Binomial c. Custom d. Triangular 57. Inventory position is defined as a. ending inventory + outstanding orders b. ending inventory − backorders c. outstanding orders − on hand inventory d. ending inventory Copyright Cengage Learning. Powered by Cognero.
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ch 12 58. What is a weakness of manual what-if analysis? a. biased sample values of performance measures b. hard to do many what-if scenarios c. does not provide distribution information d. all of these are weaknesses 59. How should one determine which RNGs to employ in a spreadsheet simulation model? a. Use the Analytic Solver Platform gallery and pick a distribution. b. Generate thousands of samples and compare the resulting histogram to ensure the distribution is correct. c. Solve the deterministic model repeatedly and use Analytic Solver Platform distribution fitting tools. d. The distributions selected should represent the underlying pool of values expected to occur. Exhibit 12.4. The following questions use the information below. The manager of a Washington, DC sightseeing tour company is concerned about overbooking for one of his bus tours. The bus has 15 seats but sometimes there are empty seats. His records show that about 20% of ticket holders do not show up for their tour. Tickets cost $10 and are non-refundable. If the manager overbooks the tour and more than 15 passengers show up, some of them will be bumped to a later tour. This bumping costs the company $25 in various expenses to keep the customer happy until the next tour. The manager wants to see what happens to profits if 18 reservations are accepted.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B Capital Tours Tour Reservation System
C
Seats Available Ticket Price per Seat Prob. of No-Show Cost of Bumping Reservations Accepted
15 $10 0.2 $25 18
Passengers to Board
12
Ticket Revenue Opp. Cost of Empty Seats Cost of Bumping Passengers Marginal Profit MODEL
$180 $30 0 $150
60. Using the information in Exhibit 12.4, what formula should go in cell C14 of the worksheet to determine the Cost of Bumping Passengers? a. =C5*MAX(C10-C4,0) b. =C5*MAX(C10,0) c. =MAX(C10-C4,0) d. =C5*MAX(C10,C4) Copyright Cengage Learning. Powered by Cognero.
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ch 12 61. Which of the following best describes a random variable? a. A spreadsheet input cell containing a random number generator. b. The outcome of a simulation model. c. Any variable whose value cannot be predicted with certainty. d. All of these describe random variables. Exhibit 12.2 The following questions use the information below. The owner of Fix-a-dent Auto Repair wants to study the growth of his business using simulation. He is interested in simulating the number of damaged cars and the amount of damage to the cars each month. He currently repairs 100 cars per month and feels the change in number of cars can vary uniformly between a decrease of as much as 3% and an increase of up to 5% (average change of 1%). The dollar value of the damage to the cars is a normally distributed random variable with a mean of $3,000 and a standard deviation of $500. The average repair bill has been increasing steadily over the years and the owner expects the mean repair bill will increase by 1% per month. A spreadsheet model to simulate the problem has been run 300 times. A part of the simulation statistics output from Risk Solver Platform (RSP)and a spreadsheet for computing confidence intervals follows.
Simulation Statistics Name Description Cell Minimum= Maximum= Mean= Std Deviation=
1 2 3 4 5 6 7 8 9 10 11 12
Income/Revenue Output D21 3339249.82 5086714.77 4119518.91 291116.83 A
B
Sample Size:
300
Sample Mean: Sample Standard Deviation:
4,119,519 291,117
95% LCL for the population mean: 95% UCL for the population mean:
4,086,576 4,152,462
Target Proportion 95% Lower Confidence Limit: 95% Upper Confidence Limit:
0.900 0.866 0.934 Confidence Intervals
62. Using the information in Exhibit 12.2, what formula should go in cell B12 of the Confidence Intervals spreadsheet to compute the upper limit on a 95% confidence interval for the population proportion below 90%? a. =B10+1.96*B10*(1-B10)/SQRT(B2) b. =B10+1.96*SQRT(B10*(1-B10)/B2) c. =B10+1.96*SQRT(B10*(1-B10)*B2) Copyright Cengage Learning. Powered by Cognero.
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ch 12 d. =B10+1.96*B10*(1-B10)/B2 Exhibit 12.3 The following questions use the information below. An auto parts store wants to simulate its inventory system for engine oil. The company has collected data on the shipping time for oil and the daily demand for cases of oil. A case of oil generates a $10 profit. Customers can buy oil at any auto parts store so there are no backorders (the company loses the sale and profit). The company orders 30 cases whenever the inventory position falls below the reorder point of 15 cases. Orders are placed at the beginning of the day and delivered at the beginning of the day so the oil is available on the arrival day. An average service level of 99% is desired. The following spreadsheets have been developed for this problem. The company has simulated 2 weeks of operation for their inventory system. The current level of on-hand inventory is 25 units and no orders are pending.
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
6
>
4
Order Quantity:
15
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
6
6
>
9
2
19
0
3
3
>
10
3
16
0
7
7
>
11
4
9
0
1
1
>
12
5
8
0
4
4
>
13
6
4
0
1
1
>
14
7
3
15
5
5
>
15
8
13
0
1
1
>
16
9
12
0
1
1
>
17
10
11
0
6
6
>
18
11
5
0
1
1
>
19
12
4
0
4
4
>
20
13
0
15
2
2
>
21
14
13
0
7
7
>
MODEL
<
>
F
G
H
I
Copyright Cengage Learning. Powered by Cognero.
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ch 12 <
1
<
2
Performance Measures
<
3
Service Level:
100%
<
4
Avg. Lost Profit:
$229.16
<
5
<
6
<
7
<
Inv.
Order?
Lead
Arrives
End. Inv.
Pos.
(0=n,1=y)
Time
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8
19
19
0
0
0
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16
16
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0
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0
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13
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21
6
6
0
0
0
A
B
C
D
E
F
G
1 2 3 4
Shipping Time
Quantity Demanded
5
Days
Prob.
Units
Prob.
6
3
0.20
0
0.01
7
4
0.60
1
0.02
8
5
0.20
2
0.04
9
Total
1.00
3
0.06
10
4
0.25
11
5
0.20
12
6
0.20
13
7
0.22
14
Total
1.00
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ch 12 15 DATA
63. Using the information in Exhibit 12.3, what Analytic Solver Platform function should be used for generating a random shipping time based on the Data spreadsheet distribution for shipping time? a. =PsiCumul($B$6:$C$8) b. =PsiCustom($B$6:$B$8, $C$8:$C$8) c. =PsiGeneral(Data!$B$6, probability $C$8) d. =PsiDiscrete($B$6:$B$8, $C$8:$C$8) 64. What method is used to generate observations from a distribution? a. random number generator b. sample generator c. problem generator d. steady state generator
Exhibit 12.5 The following questions use the information below. The owner of Sal's Italian Restaurant wants to study the growth of his business using simulation. He is interested in simulating the number of customers and the amount ordered by customers each month. He currently serves 1000 customers per month and feels this can vary uniformly between a decrease of as much as 5% and an increase of up to 9%. The bill for each customer is a normally distributed random variable with a mean of $20 and a standard deviation of $5. The average order has been increasing steadily over the years and the owner expects the mean order will increase by 2% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
D
1 Initial Conditions:
> >
2 Number of Customers:
1000
>
3 Average Bill:
$20
>
4
>
5
>
6
Number of
Avg. Bill
Total
>
Revenue
>
7
Month
Customers
8
1
1014.8
$17
$17,703
>
9
2
1064.54
$23
$24,645
>
10
3
1054.23
$24
$24,904
>
11
4
1114.73
$31
$35,001
>
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ch 12 12
5
1208.94
$25
$29,947
>
13
6
1310.42
$16
$21,202
>
14
7
1352.23
$24
$32,983
>
15
8
1336.12
$31
$40,851
>
16
9
1412.07
$22
$30,548
>
17
10
1444.66
$28
$40,292
>
18
11
1462.64
$31
$44,899
>
19
12
1539.91
$26
$39,695
>
20
>
21
Income:
$382,670
>
SAL'S MODEL
<
E
>
F
G
H
I
<
1
Assumptions:
<
2
Max Decrease:
5%
Max Increase:
9%
Uniform
<
3
Mthl. Increase:
2%
Std Dev:
5
Normal
<
4
Constant
<
5
<
6
<
7
<
8
<
9
<
10
<
11
<
12
<
13
<
14
<
15
<
16
<
17
<
18
<
19
<
20
<
21
<
65. Using the information in Exhibit 12.5 and the Analytic Solver Platform, what formulas should go in cells B8:D21 of Copyright Cengage Learning. Powered by Cognero.
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ch 12 the spreadsheet? Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 66. University Florists makes bouquets from a variety of materials. The Daily Special Bouquet is priced at $20. The florist assembles this bouquet each day from a variety of low cost flowers he buys from his flower supplier. The actual cost of flowers ranges uniformly from $2 to $7, with all intermediate values being equally likely. The florist (who studied management science many years ago) knows that the time to assemble a bouquet is normally distributed with a mean time of 5 minutes and standard deviation of 1 minute. This will be the time required for all of the Daily Special Bouquets for that day. The florist values his labor time at $10 per hour. Sales are normally distributed with a mean of 10 bouquets per day with a standard deviation of 1 bouquet. What formulas should go in cells B8:B14 to simulate daily profits for the store?
1 2 3 4 5 6 7 8 9 10 11 12 13 14
A Initial Conditions: Sales Price Number of Bouquets: Cost of Flowers Labor Time Labor Cost Cost of Flowers Labor Time (Min) Labor Cost Total Cost Profit/Bouquet # Bouquets Sold Total Profit
B
C
D
E
F
Average Lo Average
$20 10 $2 5 $10
Std dev Hi Std dev
1 $7 1
Constant Normal Cont. Uniform Normal Constant
$4.41 2.55 $0.43 4.83 $15.17 12.20 $184.99
Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 67. What function should be used for generating random numbers from a normal distribution with mean μ and standard deviation σ between the values of a and b only? 68. What function should be used for generating random numbers from a normal distribution with mean μ and standard deviation σ? 69. What is the expected number of phone calls per hour based on the following distribution on the number of phone calls per hour? 1 2 3 4 5 6
A # of phone calls 1 2 3 4 5
B P(# of phone calls) 0.10 0.40 0.30 0.15 0.05
C # of phone calls 1 2 3 4 5
D P(# of phone calls) 0.10 0.50 0.80 0.95 1.00
70. A simulation model was replicated 100 times yielding a mean of 82.59 with variance of 17.66. Of the 100 replications, 11 replications yielded an outcome over a value of 100. What is the 95% confidence interval on the proportion of simulations whose outcomes exceeded a value of 100? Copyright Cengage Learning. Powered by Cognero.
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ch 12 Exhibit 12.5 The following questions use the information below. The owner of Sal's Italian Restaurant wants to study the growth of his business using simulation. He is interested in simulating the number of customers and the amount ordered by customers each month. He currently serves 1000 customers per month and feels this can vary uniformly between a decrease of as much as 5% and an increase of up to 9%. The bill for each customer is a normally distributed random variable with a mean of $20 and a standard deviation of $5. The average order has been increasing steadily over the years and the owner expects the mean order will increase by 2% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
D
>
1 Initial Conditions:
>
2 Number of Customers:
1000
>
3 Average Bill:
$20
>
4
>
5
>
6
Number of
Avg. Bill
Total
>
Revenue
>
7
Month
Customers
8
1
1014.8
$17
$17,703
>
9
2
1064.54
$23
$24,645
>
10
3
1054.23
$24
$24,904
>
11
4
1114.73
$31
$35,001
>
12
5
1208.94
$25
$29,947
>
13
6
1310.42
$16
$21,202
>
14
7
1352.23
$24
$32,983
>
15
8
1336.12
$31
$40,851
>
16
9
1412.07
$22
$30,548
>
17
10
1444.66
$28
$40,292
>
18
11
1462.64
$31
$44,899
>
19
12
1539.91
$26
$39,695
>
20
>
21
Income:
$382,670
>
SAL'S MODEL
<
E
<
1
Assumptions:
<
2
Max Decrease:
F
5%
>
G
Max Increase:
H
I
9%
Uniform
Copyright Cengage Learning. Powered by Cognero.
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ch 12 <
3
Mthl. Increase:
<
4
Constant
<
5
<
6
<
7
<
8
<
9
<
10
<
11
<
12
<
13
<
14
<
15
<
16
<
17
<
18
<
19
<
20
<
21
2%
Std Dev:
5
Normal
<
71. The spreadsheet model for Sal, from Exhibit 12.5, has been run 300 times to produce the following output. What is the best case scenario for Sal based on this output?
Simulation Statistics Name Description Cell Minimum= Maximum= Mean= Std Deviation=
Income/Revenue Output D21 211786.92 435002.74 4119518.91 33831.44
Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 72. What function should be used for generating random numbers between a and b from a symmetric triangular distribution when the most likely value is (a + b)/2? Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 73. An office supply store wants to simulate its inventory system for notebooks. The company has collected data on the shipping time and daily demand for notebooks. Each notebook generates a $2 profit. Customers can buy notebooks at any Copyright Cengage Learning. Powered by Cognero.
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ch 12 office supply store so there are no backorders (the company loses the sale and profit). The company orders 200 notebooks whenever the inventory position falls below the reorder point of 100 notebooks. Orders are placed at the beginning of the day and delivered at the beginning of the day so the notebooks are available on the arrival day. The average daily demand is 23.5 notebooks. An average service level of 99% is desired. There are currently 25 notebooks on hand and no orders are pending. The following spreadsheets have been developed for this problem. The company has simulated 1 month of operation for its inventory system. (HINT: This covers Rows 8 to 37 of the spreadsheet.) What formulas go in cells A1:J37 of the "Model" worksheet for this simulation?
A
B
C
D
E
F
1 2
Data For Random Number Generators
3 4
Shipping Time
Quantity Demanded
5
Days
Prob
Units
Prob
6
1
0.5
20
0.05
7
2
0.3
21
0.1
8
3
0.2
22
0.15
9
23
0.25
10
24
0.15
11
25
0.15
12
26
0.1
13
27
0.05
DATA
A
B
C
D
E
1
> >
2
Model Parameters
>
3
Reorder Point:
100
>
4
Order Quantity:
200
>
5
>
6
Units
Quantity
Demand
>
7
Day
Beg. Inv.
Received
Demanded
Satisfied
>
8
1
25
0
23
23
>
9
2
2
0
26
2
>
10
3
0
0
22
0
>
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
Copyright Cengage Learning. Powered by Cognero.
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ch 12 37
30
0
0
22
0
>
MODEL
<
F
>
G
H
<
1
<
2
Performance Measures
<
3
Service Level:
78.9%
<
4
Avg. Lost Sales:
$297.49
<
5
<
6
<
7
<
I
J
Inv.
Order?
Lead
Arrives
End. Inv.
Pos.
(0=n,1=y)
Time
On Day
8
2
2
1
3
5
<
9
0
200
0
0
0
<
10
0
200
0
0
0
<
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
∙ ∙ ∙
<
37
0
200
0
0
0
< <
74. Jim Johnson operates a bus service to take college students to "The Big City" on Friday night and bring them back to school on Sunday night. The bus has 45 seats but sometimes there are empty seats. His records show that about 5% of ticket holders do not show up for their ride. Tickets cost $20 and are non-refundable. If Jim overbooks the bus and more than 45 passengers show up, some of them will be bumped and have to miss the trip. This bumping costs the company $40 because Jim has a double-your-money back policy for bumped passengers. Jim plans to accept 48 reservations (overbook 3 seats).
A 1 2 3 4 5 6 7 8 9 10 11 12 13
B Jim's Big City Bus Reservation System
C
Seats Available Ticket Price per Seat Prob. of No-Show Cost of Bumping Reservations Accepted
45 $20 0.05 $40 48
Passengers to Board
47
Ticket Revenue Opp. Cost of Empty Seats
$960 0
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ch 12 14 15
Cost of Bumping Passengers Marginal Profit MODEL
$80 $880
What is Jim Johnson's expected marginal profit? Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 75. What function should be used for generating random numbers from the following distribution on the number of phone calls per hour? A # of phone calls 1 2 3 4 5
1 2 3 4 5 6
B P(# of phone calls) 0.10 0.40 0.30 0.15 0.05
C # of phone calls 1 2 3 4 5
D P(# of phone calls) 0.10 0.50 0.80 0.95 1.00
Exhibit 12.5 The following questions use the information below. The owner of Sal's Italian Restaurant wants to study the growth of his business using simulation. He is interested in simulating the number of customers and the amount ordered by customers each month. He currently serves 1000 customers per month and feels this can vary uniformly between a decrease of as much as 5% and an increase of up to 9%. The bill for each customer is a normally distributed random variable with a mean of $20 and a standard deviation of $5. The average order has been increasing steadily over the years and the owner expects the mean order will increase by 2% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
D
1 Initial Conditions:
> >
2 Number of Customers:
1000
>
3 Average Bill:
$20
>
4
>
5
>
6
Number of
Avg. Bill
Total
>
Revenue
>
7
Month
Customers
8
1
1014.8
$17
$17,703
>
9
2
1064.54
$23
$24,645
>
10
3
1054.23
$24
$24,904
>
11
4
1114.73
$31
$35,001
>
12
5
1208.94
$25
$29,947
>
13
6
1310.42
$16
$21,202
>
14
7
1352.23
$24
$32,983
>
15
8
1336.12
$31
$40,851
>
Copyright Cengage Learning. Powered by Cognero.
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ch 12 16
9
1412.07
$22
$30,548
>
17
10
1444.66
$28
$40,292
>
18
11
1462.64
$31
$44,899
>
19
12
1539.91
$26
$39,695
>
20
>
21
Income:
$382,670
>
SAL'S MODEL
<
E
>
F
G
H
I
<
1
Assumptions:
<
2
Max Decrease:
5%
Max Increase:
9%
Uniform
<
3
Mthl. Increase:
2%
Std Dev:
5
Normal
<
4
Constant
<
5
<
6
<
7
<
8
<
9
<
10
<
11
<
12
<
13
<
14
<
15
<
16
<
17
<
18
<
19
<
20
<
21
<
76. Sal, from Exhibit 12.5, has produced the following spreadsheet to compute confidence intervals on his income. What formula should go in cell B8 to compute the upper limit on a 95% confidence interval for the true population mean?
A
B
1 Copyright Cengage Learning. Powered by Cognero.
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ch 12 2 3 4 5 6 7 8 9 10 11 12
Sample Size:
300
Sample Mean: Sample Standard Deviation:
309,773.56 33,831.44
95% LCL for the population mean: 95% UCL for the population mean:
305,945.17 313,601.95
Target Proportion 95% Lower Confidence Limit: 95% Upper Confidence Limit:
0.900 0.866 0.934
Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 77. Jim Johnson operates a bus service to take college students to "The Big City" on Friday night and bring them back to school on Sunday night. The bus has 45 seats but sometimes there are empty seats. His records show that about 5% of ticket holders do not show up for their ride. Tickets cost $20 and are non-refundable. If Jim overbooks the bus and more than 45 passengers show up, some of them will be bumped and have to miss the trip. This bumping costs the company $40 because Jim has a double-your-money back policy for bumped passengers. Jim wants to see what happens to profits if 48 reservations are accepted.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B Jim's Big City Bus Reservation System
C
Seats Available Ticket Price per Seat Prob. of No-Show Cost of Bumping Reservations Accepted
45 $20 0.05 $40 48
Passengers to Board
47
Ticket Revenue Opp. Cost of Empty Seats Cost of Bumping Passengers Marginal Profit MODEL
$960 0 $80 $880
What formulas should go in cell C10 − C15 of the worksheet? Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 78. What function should be used for generating random numbers between 1 and 12 from a continuous uniform distribution? 79. What gallery distribution should be used for generating the number of times "tails" come up over 15 flips of a "fair" coin? 80. A simulation model was replicated 100 times yielding a mean of 82.59 with variance of 17.66. Of the 100 replications, 11 replications yielded an outcome over a value of 100. What is the 90% confidence interval on the mean? Copyright Cengage Learning. Powered by Cognero.
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ch 12 Exhibit 12.5 The following questions use the information below. The owner of Sal's Italian Restaurant wants to study the growth of his business using simulation. He is interested in simulating the number of customers and the amount ordered by customers each month. He currently serves 1000 customers per month and feels this can vary uniformly between a decrease of as much as 5% and an increase of up to 9%. The bill for each customer is a normally distributed random variable with a mean of $20 and a standard deviation of $5. The average order has been increasing steadily over the years and the owner expects the mean order will increase by 2% per month. You have created the following spreadsheet to simulate the problem.
A
B
C
D
>
1 Initial Conditions:
>
2 Number of Customers:
1000
>
3 Average Bill:
$20
>
4
>
5
>
6
Number of
Avg. Bill
Total
>
Revenue
>
7
Month
Customers
8
1
1014.8
$17
$17,703
>
9
2
1064.54
$23
$24,645
>
10
3
1054.23
$24
$24,904
>
11
4
1114.73
$31
$35,001
>
12
5
1208.94
$25
$29,947
>
13
6
1310.42
$16
$21,202
>
14
7
1352.23
$24
$32,983
>
15
8
1336.12
$31
$40,851
>
16
9
1412.07
$22
$30,548
>
17
10
1444.66
$28
$40,292
>
18
11
1462.64
$31
$44,899
>
19
12
1539.91
$26
$39,695
>
20
>
21
Income:
$382,670
>
SAL'S MODEL
<
E
<
1
Assumptions:
<
2
Max Decrease:
F
5%
>
G
Max Increase:
H
I
9%
Uniform
Copyright Cengage Learning. Powered by Cognero.
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ch 12 <
3
Mthl. Increase:
<
4
Constant
<
5
<
6
<
7
<
8
<
9
<
10
<
11
<
12
<
13
<
14
<
15
<
16
<
17
<
18
<
19
<
20
<
21
2%
Std Dev:
5
Normal
<
81. Sal, from Exhibit 12.5, has produced the following spreadsheet to compute confidence intervals on his income. What formula should go in cell B12 to compute the upper limit on a 95% confidence interval for the population proportion below 90%?
1 2 3 4 5 6 7 8 9 10 11 12
A
B
Sample Size:
300
Sample Mean: Sample Standard Deviation:
4,119,519 291,117
95% LCL for the population mean: 95% UCL for the population mean:
4,086,576 4,152,462
Target Proportion 95% Lower Confidence Limit: 95% Upper Confidence Limit:
0.900 0.866 0.934
Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 82. What is the probability that 3 or more phone calls are received in any hour of operation? Copyright Cengage Learning. Powered by Cognero.
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ch 12 A # of phone calls 1 2 3 4 5
1 2 3 4 5 6
B P(# of phone calls) 0.10 0.40 0.30 0.15 0.05
C # of phone calls 1 2 3 4 5
D P(# of phone calls) 0.10 0.50 0.80 0.95 1.00
83. A simulation model was replicated 100 times yielding a mean of 82.59 with variance of 17.66. Of the 100 replications, 11 replications yielded an outcome over a value of 100. The 95% confidence interval of the mean is the interval (81.77, 83.41). Of the 100 simulation outcomes, 65 outcomes failed to fall within this interval. What is wrong with the confidence interval? Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 84. A machine produces an average of 500 parts per day with a standard deviation of 10 parts. This is a normally distributed variable. The percent of defective parts ranges from 8-12%. Parts which need minor repair comprise 75% of the defective parts and cost $5 to repair. The rest of the defective parts cost $20 to repair. What formulas should go in cells B8:B15 to compute the daily cost of defective parts?
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A Initial Conditions: Number of Parts: % Defective % Minor Defects Cost of Minor Defect Cost of Major Defect Units Produced % Defective Total # Defective # With Minor Defects Cost of Minor Defects # With Major Defects Cost of Major Defects Total Cost
B
C
D
E
F
Average Lo
500 8% 75% $5 $20
Std dev Hi
10 12%
Normal Cont. Uniform Constant Constant Constant
500 8.2% 41 31 $155 10 $200 $355
Instructions: Answer the following questions using the Analytic Solver Platform Excel add-in. 85. What function should be used for generating random integer numbers between 2 and 8?
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ch 12 Answer Key 1. True 2. True 3. False 4. True 5. True 6. True 7. False 8. True 9. a 10. a 11. a 12. a 13. a 14. c 15. d 16. c 17. a 18. b 19. a 20. a 21. a 22. b 23. b 24. a 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 12 26. d 27. b 28. c 29. a 30. c 31. c 32. c 33. b 34. b 35. d 36. d 37. a 38. b 39. b 40. c 41. d 42. b 43. a 44. a 45. d 46. d 47. b 48. d 49. d 50. b 51. a Copyright Cengage Learning. Powered by Cognero.
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ch 12 52. a 53. a 54. b 55. b 56. d 57. a 58. d 59. d 60. a 61. d 62. b 63. d 64. a 65. Cell B8 B9 C8 D8 D21
Formula =PsiUniform ($D$2*(1-$F$2), $D$2*(1+$H$2)) =PsiUniform ($B$8*(1-$F$2), $B$8*(1+$H$2)) =PsiNormal ($D$3*(1+$F$3)^A8, $H$3) =B8*C8 =SUM(D8:D19)+PsiOutput()
66. Cell B8 B9 B10 B11 B12 B13 B14
Formula =PsiUniform($C$4, $E$4) =PsiNormal($C$5, $E$5) =B9/60*C6 =B8+B10 =C2-B11 =PsiNormal($C$3, $E$3) =B12*B13+PsiOutput()
Copied to − B10:B19 C9:C19 D9:D19
67. =PsiNormal(μ, σ, PsiTruncate(a,b)) 68. =PsiNormal (Mean = μ, Stdev = σ) 69. (1*0.1)+(2*0.4)+(3*0.3)+(4*0.15)+(5*0.05) = 2.65 Copyright Cengage Learning. Powered by Cognero.
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ch 12 70. 0.11 ± 1.96*SQRT((0.11 * 0.89)/100) = (0.0487, 0.1713) 71. $435,002.74 72. =PsiTriangular(a, (a+b)/2, b) 73. Cell H3 H4 B8 C8 B9 C9 D8 E8 F8 G8 G9 H8 I8 J8
Formula =SUM(E8:E21)/SUM(D8:D21)+PsiOutput() =(1-H3)*23.5*2*30+PsiOutput() 25 0 =F8 =COUNTIF($J$8:J8,A9)*$D$4 =PsiDiscrete($E$6:$E$16, $F$6:$F$16) =MIN(D8,B8+C8) =B8+C8-E8 =F8 =G8-E9+IF(H8=1,$D$4,0) =IF(G3<$D$3,1,0) =PsiDiscrete($B$6:$B$8, $C$6:$C$8) =IF(H8=0,0,A8+1+I8)
Copied to − − − − B10:B37 C10:C37 D9:D37 E9:E37 F9:F37 − G10:G37 H9:H37 I9:I37 J8:J37
74. Revenue: 48 seats * $20 = $960 No shows: 48 seats * 0.05 = 2.4 no shows Seats bumped: 48 seats − 2.4 no shows = 45.6 passengers or 0.6 bumps Cost of bumping: 0.6 bumps * $40 bumping cost = $24 Marginal profit: $960 − $24 = $936 Expected marginal profit is $936.00 75. =PsiDiscrete($A$2:$A$6, $B$2:$B$6) 76. =B4+1.96*B5/SQRT(B2) 77. Cell C10 C12 C13 C14 C15
Formula =PsiBinomial($C$8, 1-$C$6) =C8 * C5 =C5*MAX(C4-C10,0) =C7*MAX(C10-C4,0) =C12-C13-C14+PsiOutput()
78. =PsiUniform(1, 12) 79. =PsiBinomial(15, 0.5) 80. 82.59 ± 1.645*SQRT(17.66/100) = (81.90, 83.28) 81. =B10+1.96*SQRT(B10*(1-B10)/B2) Copyright Cengage Learning. Powered by Cognero.
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ch 12 82. 1 − (P ≤ 2) = 1 − 0.50 = 0.50 83. Nothing. The confidence interval is on the mean not on the outcomes. 84. Cell B8 B9 B10 B11 B12 B13 B14 B15
Formula =PsiNormal($C$2, $E$2) =PsiUniform($C$3, $E$3) =B9*B8 =B9*B8*C45 =B11*C5 =B9*B8*(1-C4) =B13*C6 =B12+B14+PsiOutput()
85. =PsiIntUniform(2, 8)
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ch 13
Indicate whether the statement is true or false. 1. The number of arrivals that occurs in a given time period represents a random variable in a queuing system. a. True b. False 2. In a queuing problem, Wq > W. a. True b. False 3. The objective in queuing problems is to find the service level that achieves an acceptable balance between the cost of providing service and customer satisfaction. a. True b. False 4. An arrival process is memoryless if the time until the next arrival occurs is inversely proportional to the time elapsed since the last arrival. a. True b. False 5. One way to improve performance of a queuing system from the customer perspective is to reduce the number of servers. a. True b. False 6. If the number of arrivals in a given period of time follows a Poisson distribution with mean , the interarrival times follow an exponential probability distribution with mean 1/λ a. True b. False 7. The term queuing theory refers to the body of knowledge dealing with waiting lines. a. True b. False 8. A situation where cars arrive at an intersection can be modeled as an M/D/s queue with finite capacity. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. In the Kendall notation M/G/4, the number 4 indicates a. the number of servers b. deterministic departure distribution c. memoryless arrival and departure distributions d. queue capacity Copyright Cengage Learning. Powered by Cognero.
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ch 13 10. A balk refers to a. a customer who refuses to join the queue. b. a customer who refuses service by a specific server. c. a customer who joins the queue but leaves before service is complete. d. a customer who requires extra service time. 11. The service times for a grocery store with one checkout line have a mean of 3 minutes and a standard deviation of 20 seconds. Customer arrivals at the checkout stand follow a Poisson distribution. What type of system is it? a. M/G/1 b. M/D/1 c. G/M/1 d. M/M/1 12. The M/M/s model with finite population can be used to model a. a machine breakdown process in a shop with 10 identical machines b. a process of patient arrival to a dentist's office c. a process of patient departure from a dentist's office d. a process of installing new machines in a shop 13. The M in M/G/1 stands for a. Markovian inter-arrival times. b. Mendelian inter-arrival times. c. Mean inter-arrival times. d. Mathematical inter-arrival times. 14. If a service system has a constant service time, Poisson arrival rates and 2 servers its Kendall notation is a. P/D/2 b. M/D/2 c. M/D/1 d. G/D/2 15. A jockey refers to a. a customer who refuses to join the queue. b. a customer who refuses service by a specific server. c. a customer who joins the queue but leaves before service is complete. d. a customer who switches between queues in the system. 16. What is the service policy in the queuing systems presented in this chapter that is considered "fair" by the customers? a. FIFO b. LIFO c. FILO d. Priority 17. In the Kendall notation M/G/4, G stands for a. memoryless arrival distribution Copyright Cengage Learning. Powered by Cognero.
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ch 13 b. general departure distribution c. memoryless arrival and departure distributions d. none of the above 18. Which of the following best describes queuing theory? a. The study of arrival rates. b. The study of service times. c. The study of waiting lines. d. The evaluation of service time costs. 19. A renege refers to a. a customer who refuses to join the queue. b. a customer who refuses service by a specific server. c. a customer who joins the queue but leaves before service is complete. d. a customer who requires extra service time. 20. For a Poisson random variable, λ represents the ____ number of arrivals per time period a. maximum b. minimum c. average d. standard deviation of 21. If a company adds an additional identical server to its M/M/1 system, making an M/M/2 system, what happens to a customer's average service time? a. increases b. decreases c. it is unchanged d. depends on the arrival rate Exhibit 13.1 The following questions are based on the output below. A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system. M/M/s queuing computations Arrival rate Service rate Number of servers
6 8 1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system
75.00% 0.2500 2.2500 3.0000 0.3750 0.5000
Copyright Cengage Learning. Powered by Cognero.
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ch 13 Probability that a customer waits
0.7500
22. Refer to Exhibit 13.1. What is the probability that a customer must wait in queue before being served? a. 0.00 b. 0.25 c. 0.75 d. 1.00 23. To find steady-state values for the M/M/S queuing system, which of the following statements must be true about the arrival rate? a. λ < s μ b. λ − s = μ c. λ > s μ d. λ = s μ 24. The standardized queuing system notation such as M/M/1 or M/G/2 is referred to as a. Kendall notation. b. Erlang notation. c. Poisson notation. d. Queuing notation. 25. What is the formula for P(t ≤ T) under the exponential distribution with rate μ? a. 1 − eμT b. eμT c. 1 − e−μT d. 1 − eT 26. An arrival process is memoryless if a. the time until the next arrival depends on the time elapsed since the last arrival. b. the time until the next arrival is based on the time elapsed since the last arrival. c. the time until the next arrival does not depend on the time elapsed since the last arrival. d. the time until the next arrival is based on the arrival rate. 27. Which of the following is the typical operating characteristic for average time a unit spends waiting for service? a. W b. Wq c. L d. Lq 28. The memoryless property is also referred to as the ____ property. a. Markov b. Erlang c. Poisson Copyright Cengage Learning. Powered by Cognero.
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ch 13 d. Normal 29. In the Kendall notation M/G/4, M stands for a. memoryless arrival distribution b. memoryless departure distribution c. memoryless arrival and departure distributions d. none of the above 30. Which of the following is a reason to employ queuing theory? a. To reduce customer wait time in line. b. To reduce service times. c. To generate more arrivals to the system. d. To reduce worker idle time in line. 31. Joe's Copy Center has 10 copiers. They break down at a rate of 0.02 copiers per hour and are sent to the service facility. What is the average arrival rate of broken copiers to the service facility? a. 0.02 b. 0.2 c. 10 d. It cannot be determined from the information provided. 32. The M/D/1 model results can be derived from which of the following systems? a. M/M/1 with λ = 0 b. M/G/1 with μ = 0 c. M/G/1 with σ = 0 d. M/M/2 with finite queue length. 33. If the arrival process is modeled as a Poisson random variable with arrival rate λ, then the average time between arrivals is a. 1/μ b. 1/λ c. 1/λ2 d. σ 34. Joe's Copy Center has 10 copiers. They break down and require service quite often. Time between breakdowns follows an exponential distribution for each copier. The repair person services machines as quickly as possible, but the service time follows an exponential distribution. What type of system is it? a. M/M/1 with Finite Population b. M/M/1 with Finite Queue c. M/M/1 d. M/M/10 35. A doctor's office only has 8 chairs. The doctor's service times and customer inter-arrival times are exponentially distributed. What type of system is it? a. M/M/1 Copyright Cengage Learning. Powered by Cognero.
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ch 13 b. M/M/8 c. M/M/1 with Finite Queue d. M/M/1 with Finite Population 36. A company has recorded the following list of service rates (customers/hour) for one of its servers. What is the mean service time for this server? Customers / hour: 4, 4, 5, 6, 5, 4, 3, 4, 3, 5, 5, 6 a. 0.22 min b. 1.11 min c. 4.5 min d. 13.3 min Exhibit 13.1 The following questions are based on the output below. A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system. M/M/s queuing computations Arrival rate Service rate Number of servers
6 8 1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
75.00% 0.2500 2.2500 3.0000 0.3750 0.5000 0.7500
(max of 40)
37. Refer to Exhibit 13.1. How many customers will be in the store on average at any one time? a. 0.375 b. 0.50 c. 2.25 d. 3.00 38. The Kendall notation for a queuing system with Poisson arrivals, exponential service and 3 service providers is a. M/M/3 b. M/G/1 c. G/G/3 d. G/G/1 39. If the number of arrivals in a time period follow a Poisson distribution with mean λ then the inter-arrival times follow a(n) ____ distribution with mean ____. a. normal; μ Copyright Cengage Learning. Powered by Cognero.
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ch 13 b. constant; λ c. exponential; λ d. exponential; 1/λ 40. Which type of queuing system are you likely to encounter at an ATM? a. Single waiting line, single service station. b. Multiple waiting lines, single service station. c. Single waiting line, multiple service stations. d. Multiple waiting lines, multiple service stations. 41. If cell B2 contains the value for μ and cell A5 contains the value for T, what formula should go in cell B5 to compute the P(Service time) ≤ T for this exponential distribution? a. =1-EXP($B$2*A5) b. =EXP(-$B$2*A5) c. =1-EXP(-$B$2) d. =1-EXP(-$B$2*A5) 42. If the service rate decreases as the arrival rate remains constant, then, in general a. customer waiting time increases. b. customer waiting time decreases. c. service costs increase. d. customer dissatisfaction decreases. 43. The M/M/s model with finite capacity queue can be used to model a. a machine breakdown process in a shop with 10 identical machines b. traffic in a dentist's office c. a process of patient departure from a dentist's office d. a process of installing new machines in a shop 44. The amount of time a customer spends with the server is referred to as a. system time. b. queue time. c. service time. d. served time. 45. A barber shop has one barber, a Poisson arrival rate and exponentially distributed service times. What is the Kendall notation for this system? a. M/M/E b. M/M/1 c. M/E/1 d. P/M/1 46. What is the formula for the probability of x arrivals, p(x), under a Poisson distribution with arrival rate λ? a. Copyright Cengage Learning. Powered by Cognero.
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ch 13 b. c. d.
47. Which type of queuing system are you likely to encounter at a grocery store? a. Single waiting line, single service station. b. Multiple waiting lines, single service station. c. Single waiting line, multiple service stations. d. Multiple waiting lines, multiple service stations. 48. Which of the following is the typical operating characteristic for the probability an arriving unit has to wait for service? a. Wp b. P0 c. Pw d. Pn 49. The number of arrivals to a store follows a Poisson distribution with mean λ = 10/hour. What is the mean inter-arrival time? a. 6 seconds b. 6 minutes c. 10 minutes d. 10 hours 50. A store is considering adding a second clerk. The customer arrival rate at this new server will be a. twice the old rate. b. half the old rate. c. the same as the old rate. d. unpredictable. 51. The M/D/1 model with infinite capacity queue can be used to model a. a machine breakdown process in a shop with 10 identical machines b. a process of washing a car in an automatic car wash c. a process of patient departure from a dentist's office d. a process of installing new machines in a shop 52. Which type of queuing system are you likely to encounter at a Wendy's restaurant? a. Single waiting line, single service station. b. Multiple waiting lines, single service station. c. Single waiting line, multiple service stations. d. Multiple waiting lines, multiple service stations. Copyright Cengage Learning. Powered by Cognero.
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ch 13 53. A Poisson distribution shape can be described as a. slightly skewed to the left. b. symmetric around the parameter λ. c. skewed to the right. d. discrete so it lacks any definable shape. 54. What is the mean arrival rate based on the following 8 arrival rate observations? Number of arrivals per hour: 6, 5, 3, 4, 7, 6, 4, 5 a. 3 b. 4 c. 5 d. 6 55. In the Kendall notation M/D/4, D stands for a. memoryless arrival distribution b. deterministic departure distribution c. memoryless arrival and departure distributions d. none of the above 56. Which of the following is the typical operating characteristic for average number of units in a queue? a. W b. Wq c. L d. Lq 57. One reason to use queuing models in business is a. to trade-off the cost of providing service and the cost of customer dissatisfaction b. to maximize the number of service providers c. to minimize the cost of providing service d. all of the above Exhibit 13.1 The following questions are based on the output below. A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are exponentially distributed. The following spreadsheet has been developed for the system. M/M/s queuing computations Arrival rate Service rate Number of servers
6 8 1
Utilization
75.00%
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ch 13 P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
0.2500 2.2500 3.0000 0.3750 0.5000 0.7500
58. Refer to Exhibit 13.1. What is the probability that a customer can go directly into service without waiting in line? a. 0.00 b. 0.25 c. 0.75 d. 1.00 59. A common queue discipline used in practice is a. first-in-first-out b. random c. last-in-first-out d. group arrival 60. What is the probability that it will take less than or equal to 0.25 hours to service any call based on the following exponential probability distribution with rate μ = 5? Average Service Rate μ= 5 per hour P(service time ≤ T) 0.00 0.22 0.39 0.53 0.63 0.71 0.78 0.83 0.86 0.89 0.92 0.94 0.95 0.96
T 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 a. 0.00 b. 0.71 c. 0.92 d. 1.00 Exhibit 13.1
The following questions are based on the output below. A store currently operates its service system with 1 operator. Arrivals follow a Poisson distribution and service times are Copyright Cengage Learning. Powered by Cognero.
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ch 13 exponentially distributed. The following spreadsheet has been developed for the system. M/M/s queuing computations Arrival rate Service rate Number of servers
6 8 1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
75.00% 0.2500 2.2500 3.0000 0.3750 0.5000 0.7500
(max of 40)
61. Refer to Exhibit 13.1. What is the average amount of time spent waiting in line? a. 0.375 b. 0.50 c. 2.25 d. 3.00
Exhibit 13.7 The following questions refer to the information and output below. A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Average service time Standard dev. of service time
1 0.5 0.1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system
50.00% 0.5000 0.2600 0.7600 0.2600 0.7600
average service rate 2
62. Refer to Exhibit 13.7. What is the Kendall notation for this system? 63. The customer service desk at Joe's Discount Electronics store receives 5 customers per hour on average. On average, each customer requires 10 minutes for service. The customer service desk is staffed by a single person. What is the average number of customers in the customer service area, if modeled as an M/M/1 queuing system? Exhibit 13.3 Copyright Cengage Learning. Powered by Cognero.
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ch 13 The following questions refer to the information below. A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.
Customer 1 2 3 4 5 6 7 8 9 10
All time in minutes Inter-arrival Service 11.08 2.20 2.50 2.50 6.00 1.10 5.75 14.50 8.50 2.00 4.15 2.70 15.50 5.00 13.00 8.50 10.50 5.00 6.00 1.50
64. Refer to Exhibit 13.3. What is the mean arrival rate per hour? Exhibit 13.4 The following questions refer to the information and output below. A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers
240 90 4
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
(max of 40) 66.67% 0.0599 0.7568 3.4235 0.0032 0.0143 0.3784
65. Refer to Exhibit 13.4. Based on this report what percent of the time is a grocery clerk busy serving a customer? 66. A grocery clerk can serve 20 customers per hour on average and the service time follows an exponential distribution. What is the expected service time per customer? Exhibit 13.3 The following questions refer to the information below. Copyright Cengage Learning. Powered by Cognero.
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ch 13 A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.
Customer 1 2 3 4 5 6 7 8 9 10
All time in minutes Inter-arrival Service 11.08 2.20 2.50 2.50 6.00 1.10 5.75 14.50 8.50 2.00 4.15 2.70 15.50 5.00 13.00 8.50 10.50 5.00 6.00 1.50
67. Refer to Exhibit 13.3. What is the mean service rate per hour? Exhibit 13.7 The following questions refer to the information and output below. A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Average service time Standard dev. of service time
1 0.5 0.1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system
50.00% 0.5000 0.2600 0.7600 0.2600 0.7600
average service rate 2
68. Refer to Exhibit 13.7. Based on this report what is the probability that a customer does not have to wait for assistance with his or her taxes? Exhibit 13.5 The following questions refer to the information and output below. A computer printer in a large administrative office has a printer buffer (memory to store printing jobs) capacity of 3 jobs. If the buffer is full when a user wants to print a file the user is told that the job cannot be printed and to try again later. There are so many users in this office that we can assume that there is an infinite calling population. Jobs arrive at the printer at a Poisson rate of 55 jobs per hour and take an average of 1 minute to print. Printing times are exponentially Copyright Cengage Learning. Powered by Cognero.
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ch 13 distributed. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Maximum queue length
55 60 1 3
(max of 40) (max of 40 combined)
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits Probability that a customer balks
76.38% 0.2362 1.0628 1.8265 0.0193 0.0360 0.7638 0.1668
69. Refer to Exhibit 13.5. Based on this report what is the average number of jobs waiting to be printed? 70. Refer to Exhibit 13.5. Based on this report what is the probability that a computer user will be told to resubmit a print job at a later time? 71. Refer to Exhibit 13.5. What is the Kendall notation for this system? Exhibit 13.6 The following questions refer to the information and output below. The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Population Size Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
4 60 1 10
(max of 40) (max of 100) 58.97% 0.41034 0.56545 1.15511 0.01598 0.03265 0.58966
72. Refer to Exhibit 13.6. Based on this report how much time do students spend getting help before they can resume work on their computers? Copyright Cengage Learning. Powered by Cognero.
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ch 13 73. A company has recorded the following list of service rates (customers/hour) for one of its servers. What is the mean service time for this server? Customers 2 3 2 3 4 4 2
/ hour 3 4 3 3 2 3 4
Exhibit 13.4 The following questions refer to the information and output below. A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers
240 90 4
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
(max of 40) 66.67% 0.0599 0.7568 3.4235 0.0032 0.0143 0.3784
74. Refer to Exhibit 13.4. Based on this report what is the average total time spent in line and being checked out? 75. A grocery clerk can serve 20 customers per hour on average and the service time follows an exponential distribution. What is the probability that a customer's service time is more than 4 minutes? Exhibit 13.4 The following questions refer to the information and output below. A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers
240 90 4
Copyright Cengage Learning. Powered by Cognero.
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ch 13 Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
66.67% 0.0599 0.7568 3.4235 0.0032 0.0143 0.3784
76. Refer to Exhibit 13.4. What is the Kendall notation for this system? 77. Customers arrive at a store randomly, following a Poisson distribution at an average rate of 20 per hour. What is the probability of exactly 0, 1 2, and 3 arrivals in a 15 minute period? 78. Customers arrive at a store randomly, following a Poisson distribution at an average rate of 90 per hour. How many customers would you expect to arrive in a 20 minute period? Exhibit 13.4 The following questions refer to the information and output below. A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers
240 90 4
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
(max of 40) 66.67% 0.0599 0.7568 3.4235 0.0032 0.0143 0.3784
79. Refer to Exhibit 13.4. Based on this report what is the average number of customers waiting for a checker? 80. Customers arrive at a store randomly, following a Poisson distribution at an average rate of 90 per hour. How many customers arrive per minute, on average? Exhibit 13.2 The following questions refer to the information and output below. A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information. Copyright Cengage Learning. Powered by Cognero.
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ch 13
Arrival rate Service rate Number of servers
8 12 1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
(max of 40) 66.67% 0.3333 1.3333 2.0000 0.1667 0.2500 0.6667
81. Refer to Exhibit 13.2. Based on this report what is the average waiting time before the barber begins a customer's haircut? Exhibit 13.3 The following questions refer to the information below. A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.
Customer 1 2 3 4 5 6 7 8 9 10
All time in minutes Inter-arrival Service 11.08 2.20 2.50 2.50 6.00 1.10 5.75 14.50 8.50 2.00 4.15 2.70 15.50 5.00 13.00 8.50 10.50 5.00 6.00 1.50
82. Refer to Exhibit 13.3. What is the average number of customers in the system? 83. What is the mean arrival rate based on the following 10 arrival rate observations? Number of arrivals per hour 16 15 14 15 17 16 14 15 Copyright Cengage Learning. Powered by Cognero.
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ch 13 13 16 Exhibit 13.2 The following questions refer to the information and output below. A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers
8 12 1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
(max of 40) 66.67% 0.3333 1.3333 2.0000 0.1667 0.2500 0.6667
84. Refer to Exhibit 13.2. Based on this report what is the average total time spent waiting for a haircut and getting a haircut? 85. The customer service desk at Joe's Discount Electronics store receives 5 customers per hour on average. On average, each customer requires 10 minutes for service. The customer service desk is staffed by a single person. What is the average time a customer spends in the customer service area if modeled as an M/M/1 queuing system? 86. A grocery clerk can serve 20 customers per hour on average and the service time follows an exponential distribution. What is the probability that a customer's service time is less than 2 minutes? Exhibit 13.7 The following questions refer to the information and output below. A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Average service time Standard dev. of service time
1 0.5 0.1
Utilization P(0), probability that the system is empty Lq, expected queue length
50.00% 0.5000 0.2600
Copyright Cengage Learning. Powered by Cognero.
average service rate 2
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ch 13 L, expected number in system Wq, expected time in queue W, expected total time in system
0.7600 0.2600 0.7600
87. Refer to Exhibit 13.7. Based on this report what is the average number of customers waiting to be helped? Exhibit 13.6 The following questions refer to the information and output below. The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Population Size
4 60 1 10
(max of 40) (max of 100)
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
58.97% 0.41034 0.56545 1.15511 0.01598 0.03265 0.58966
88. Refer to Exhibit 13.6. Based on this report what is the average number of students waiting to be helped? Exhibit 13.4 The following questions refer to the information and output below. A grocery store can serve an average of 360 customers per hour. The service times are exponentially distributed. The store has 4 checkout lines each of which serves 90 customers per hour. Customers arrive at the store at a Poisson rate of 240 customers per hour. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers
240 90 4
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system
(max of 40) 66.67% 0.0599 0.7568 3.4235 0.0032 0.0143
Copyright Cengage Learning. Powered by Cognero.
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ch 13 Probability that a customer waits
0.3784
89. Refer to Exhibit 13.4. Based on this report how long does a customer wait before the checker begins serving them? Exhibit 13.6 The following questions refer to the information and output below. The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Population Size Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
4 60 1 10
(max of 40) (max of 100) 58.97% 0.41034 0.56545 1.15511 0.01598 0.03265 0.58966
90. Refer to Exhibit 13.6. Based on this report what is the probability that a student will not get instantaneous help? Exhibit 13.2 The following questions refer to the information and output below. A barber shop has one barber who can give 12 haircuts per hour. Customers arrive at a rate of 8 customers per hour. Customer inter-arrival times and service times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits Copyright Cengage Learning. Powered by Cognero.
8 12 1
(max of 40) 66.67% 0.3333 1.3333 2.0000 0.1667 0.2500 0.6667 Page 20
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ch 13 91. Refer to Exhibit 13.2. What is the Kendall notation for this system? 92. Refer to Exhibit 13.2. Based on this report what is the average number of customers waiting for a haircut? 93. Refer to Exhibit 13.2. Based on this report what percent of the time is the barber busy cutting hair? Exhibit 13.7 The following questions refer to the information and output below. A tax accountant has found that the time to serve a customer has a mean of 30 minutes (or 0.5 hours) and a standard deviation of 6 minutes (or 0.1 hours). Customer arrivals follow a Poisson distribution with an average of 60 minutes between arrivals. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Average service time Standard dev. of service time
1 0.5 0.1
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system
50.00% 0.5000 0.2600 0.7600 0.2600 0.7600
average service rate 2
94. Refer to Exhibit 13.7. Based on this report how long does a customer spend at the tax accountant's office? Exhibit 13.5 The following questions refer to the information and output below. A computer printer in a large administrative office has a printer buffer (memory to store printing jobs) capacity of 3 jobs. If the buffer is full when a user wants to print a file the user is told that the job cannot be printed and to try again later. There are so many users in this office that we can assume that there is an infinite calling population. Jobs arrive at the printer at a Poisson rate of 55 jobs per hour and take an average of 1 minute to print. Printing times are exponentially distributed. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Maximum queue length
55 60 1 3
Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system
(max of 40) (max of 40 combined) 76.38% 0.2362 1.0628 1.8265 0.0193 0.0360
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ch 13 Probability that a customer waits Probability that a customer balks
0.7638 0.1668
95. Refer to Exhibit 13.5. Based on this report how long does a computer user have to wait for his/her job to be completed? Exhibit 13.6 The following questions refer to the information and output below. The university computer lab has 10 computers which are constantly being used by students. Users need help from the one lab assistant fairly often. Students ask for help at a Poisson rate of with an average of 4 requests per hour for any one computer. The assistant answers questions as quickly as possible and the service time follows an exponential distribution with mean of 1 minute per help session. The following queuing analysis spreadsheet was developed from this information.
Arrival rate Service rate Number of servers Population Size Utilization P(0), probability that the system is empty Lq, expected queue length L, expected number in system Wq, expected time in queue W, expected total time in system Probability that a customer waits
4 60 1 10
(max of 40) (max of 100) 58.97% 0.41034 0.56545 1.15511 0.01598 0.03265 0.58966
96. Refer to Exhibit 13.6. What is the Kendall notation for this system? Exhibit 13.3 The following questions refer to the information below. A company has recorded the following customer inter-arrival times and service times for 10 customers at one of its single teller service lines. Assume the data are exponentially distributed and the 10 data points represent a reasonable sample.
Customer 1 2 3 4 5 6 7 8 9 10
All time in minutes Inter-arrival Service 11.08 2.20 2.50 2.50 6.00 1.10 5.75 14.50 8.50 2.00 4.15 2.70 15.50 5.00 13.00 8.50 10.50 5.00 6.00 1.50
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ch 13 97. Refer to Exhibit 13.3. What is the average time a customer spends in the system?
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ch 13 Answer Key 1. True 2. False 3. True 4. False 5. False 6. True 7. True 8. False 9. a 10. a 11. a 12. a 13. a 14. b 15. d 16. a 17. b 18. c 19. c 20. c 21. c 22. c 23. a 24. a 25. c Copyright Cengage Learning. Powered by Cognero.
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ch 13 26. c 27. b 28. a 29. a 30. a 31. b 32. c 33. b 34. a 35. c 36. d 37. d 38. a 39. d 40. a 41. d 42. a 43. b 44. c 45. b 46. b 47. d 48. c 49. b 50. b 51. b Copyright Cengage Learning. Powered by Cognero.
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ch 13 52. c 53. c 54. c 55. b 56. d 57. a 58. b 59. a 60. b 61. a 62. M/G/1 63. W = 1 / (μ − λ) = 1 / (6 − 5) = 1, so L = λ W = (5)(1) = 5 64. Mean inter-arrival rate is 8.30 so that 60 / 8.30 = 7.23 is mean arrival rate per hour (λ) 65. 66.67% 66. 3 minutes per customer 67. The mean service time is 4.50 minutes so the 60 / 4.5 = 13.33 is the mean service rate (μ). 68. 0.5000 69. 1.0628 70. 0.1688 71. M/M/1/ with finite queue length 72. 0.03265 73. 3 customers per hour gives a mean service time of 20 minutes per customer 74. 0.0143 75. T = 4/60 = 0.0667 hours P(t > T) = e−μT = e−20*0.0667 = 0.2636 76. M/M/4 Copyright Cengage Learning. Powered by Cognero.
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ch 13 77. λ = 5 customers/15 minute period Arrivals 0 1 2 3
Probability 0.0067 0.0336 0.0842 0.1404
78. 30 customers per 20 minute period 79. 0.7568 80. 1.5 customers per minute 81. 0.1667 82. λ = 7.23, μ = 13.33, W = 1 / (μ − λ) = 1 / (13.33 − 7.23) = 0.164, L = λW = 7.23*0.164 = 1.186 83. 15.1 arrivals per hour 84. 0.2500 85. W = 1 / (μ − λ) = 1 / (6 − 5) = 1 86. T = 2/60 = 0.033 hours P(t < T) = 1 − e−μT = 1 − e−20*0.033 = 1 − 0.5134 = 0.4866 87. 0.2600 88. 0.56545 89. 0.0032 90. 0.58966 91. M/M/1 92. 1.3333 93. 66.67% 94. 0.7600 95. 0.0360 96. M/M/1 with finite population 97. λ = 7.23, μ = 13.33, W = 1 / (μ − λ) = 1 / (13.33 − 7.23) = 0.164 Copyright Cengage Learning. Powered by Cognero.
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ch 13
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ch 14
Indicate whether the statement is true or false. 1. Under maximin rule a decision maker hedges against the worst possible outcome of a decision. a. True b. False 2. The expected value of perfect information (EVPI) is equivalent to the minimum expected opportunity loss (EOL). a. True b. False 3. The decision rules that assume that probabilities of occurrence can be assigned to the states of nature in a decision problem are called probabilistic methods. a. True b. False 4. The expected monetary value decision rule selects the decision alternative with the largest expected regret. a. True b. False 5. One of the primary advantages in decision making is that we usually know which state of nature will occur. a. True b. False 6. An analyst can apply a process known as rolling back to a decision tree to determine the decision with the largest EMV. a. True b. False 7. The criteria in a decision problem represent various factors that are important to the decision maker. a. True b. False 8. A payoff matrix is a table that summarizes the final outcome (or regret) for each decision alternative under each possible condition. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. Which decision rule optimistically assumes that nature will always be "on our side" regardless of what decision we make? a. maximax decision rule. b. maximin decision rule. c. minimax regret decision rule. d. minimin decision rule. Copyright Cengage Learning. Powered by Cognero.
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ch 14 10. The decision rule which determines the maximum payoff for each alternative and then selects the alternative associated with the largest payoff is the a. maximax decision rule. b. maximin decision rule. c. minimax regret decision rule. d. minimin decision rule. 11. The minimum EOL in a decision problem will always a. exceed the EVPI. b. be less than the EVPI. c. equal the EVPI. d. equal the EMV. 12. Which of the following summarizes the final outcome for each decision alternative? a. payoff matrix b. outcome matrix c. yield matrix d. performance matrix Exhibit 14.7 The following questions use the information below. A decision maker is faced with two alternatives. The decision maker has determined that she is indifferent between the two alternatives when p = 0.45. Alternative 1: Alternative 2:
Receive $82,000 with certainty Receive $143,000 with probability p and lose $15,000 with probability (1 − p).
13. Refer to Exhibit 14.7. What is the decision maker's risk premium for this problem? a. −$20,000 b. −$25,900 c. $70,000 d. $80,000 14. Which decision rule pessimistically assumes that nature will always be "against us" regardless of what decision we make? a. maximax decision rule. b. maximin decision rule. c. minimax regret decision rule. d. minimin decision rule. 15. The decision rule which determines the minimum payoff for each alternative and then selects the alternative associated with the largest minimum payoff is the a. maximax decision rule. b. maximin decision rule. Copyright Cengage Learning. Powered by Cognero.
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ch 14 c. minimax regret decision rule. d. minimin decision rule. 16. Business decision models can be categorized as a. decision-making under uncertainty b. decision-making under risk c. decision making under certainty d. (a) and (b) only Exhibit 14.3 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 70% and an expanding economy at 30%.
A 1 2 3 4 5 6 7 8 9 10
B
C
D
E
EMV
Investment A B C Bank
Payoff Matrix Economy Investment A B C Bank
Decline −10 20 40 15
Expand 90 50 45 20
Probability
0.7
0.3
Probability
F Regret Matrix
G
Economy Decline
EOL
0.7
0.3
H
17. Refer to Exhibit 14.3. What decision should be made according to the expected monetary value decision rule? a. A b. B c. C d. Bank 18. Decision models are applicable when a. there are multiple alternatives b. there are multiple states of nature c. there is only one alternative d. there is only one state of nature 19. Suppose that the payoffs for an alternative with three states of nature are: 10, 20, and 30. The probabilities of these states of nature are 0.2, 0.3, and 0.5, respectively. The expected payoff for the alternative is equal to a. 23 b. 30 Copyright Cengage Learning. Powered by Cognero.
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ch 14 c. 60 d. 20 Exhibit 14.8 The following questions use the information below. A company needs to buy a new insurance policy. They have three policies to choose from, A, B and C. The policies differ with respect to price, coverage and ease of billing. The company has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
D
A B C Sum
A 1.000 0.333 0.200 1.533
Pairwise Comparisons B 3.000 1.000 0.500 4.500
A B C
A 0.652 0.217 0.130
Normalized Comparisons B 0.667 0.222 0.111
E
C 5.000 2.000 1.000 8.000
C 0.625 0.250 0.125 Consistency ratio:
F
G
Price Scores 0.648 0.230 0.122
Consistency Measure 3.007 3.003 3.001 0.003
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Coverage Billing Weighted Avg. Score: Summary
A 0.648 0.213 0.120 0.215
B 0.230 0.701 0.272 0.404
C 0.122 0.085 0.608 0.381
F
G
0.123 0.320 0.557 1.000
20. Refer to Exhibit 14.8. What formula should go in cell F11 and get copied to F12:F13 of the Price worksheet to compute the Price Score? a. =AVERAGE(C4:C6) b. =AVERAGE(C11:E11) c. =AVERAGE(G11:G13) d. =AVERAGE(C7:E7) Copyright Cengage Learning. Powered by Cognero.
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ch 14 21. The expected monetary value criterion (EMV) is the decision-making approach used a. in decision-making under risk b. in decision-making under uncertainty c. in decision-making under certainty d. all of the above 22. The ____ correspond to future events that are not under the control of the decision maker. a. payoffs b. states of nature c. criteria d. alternatives Exhibit 14.3 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 70% and an expanding economy at 30%.
A 1 2 3 4 5 6 7 8 9 10
B
C
D
E
Payoff Matrix Economy Investment A B C Bank
Decline −10 20 40 15
Expand 90 50 45 20
Probability
0.7
0.3
EMV
Investment A B C Bank Probability
F Regret Matrix
G
Economy Decline
EOL
0.7
0.3
H
23. Refer to Exhibit 14.3. What is the expected monetary value of Investment A? a. 34. b. 30. c. 20. d. 15. 24. Although modeling provides valuable insight to decision makers, decision making remains a difficult task. Which of the following is not a primary cause for this difficulty discussed in the Decision Analysis chapter? a. Uncertainty regarding the future. b. Models provide decisions for the decision maker. c. Conflicting values. d. Conflicting objectives. Copyright Cengage Learning. Powered by Cognero.
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ch 14 Exhibit 14.6 The following questions use the information below. A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 69% chance that demand for their products will be high and a 31% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 63% chance that the customers will like the products and a 37% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is a 92% chance that demand will be high for the products. If the survey is unfavorable there is only a 30% chance that the demand will be high. The following decision tree has been built for this problem. The company has computed that the expected monetary value of the best decision without sample information is 154.35 million. The company has developed the following conditional probability table for their decision problem.
A B C 1 2 Joint Probabilities 3 High Demand Low Demand 4 Favorable Response 0.58 0.05 5 Unfavorable Response 0.11 0.26 6 Total 0.69 0.31 7 8 9 Conditional Probability 10 For A Given Survey Response 11 High Demand Low Demand 12 Favorable Response 0.92 0.08 13 Unfavorable Response 0.30 0.70 14 15 Conditional Probability 16 For A Given Demand Level 17 High Demand Low Demand 18 Favorable Response 0.84 0.16 19 Unfavorable Response 0.16 0.84
D
Total 0.63 0.37 1.00
25. Refer to Exhibit 14.6. What is P(F∩H), where F = favorable response and H = high demand? a. .58 b. .63 c. .84 d. .92 Exhibit 14.8 The following questions use the information below. A company needs to buy a new insurance policy. They have three policies to choose from, A, B and C. The policies differ with respect to price, coverage and ease of billing. The company has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations. Copyright Cengage Learning. Powered by Cognero.
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ch 14
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
D
E
A B C Sum
A 1.000 0.333 0.200 1.533
Pairwise Comparisons B 3.000 1.000 0.500 4.500
C 5.000 2.000 1.000 8.000
A B C
A 0.652 0.217 0.130
Normalized Comparisons B 0.667 0.222 0.111
C 0.625 0.250 0.125 Consistency ratio:
F
G
Price Scores 0.648 0.230 0.122
Consistency Measure 3.007 3.003 3.001 0.003
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Coverage Billing Weighted Avg. Score: Summary
A 0.648 0.213 0.120 0.215
B 0.230 0.701 0.272 0.404
C 0.122 0.085 0.608 0.381
F
G
0.123 0.320 0.557 1.000
26. Refer to Exhibit 14.8. What formula should go in cell G11 and get copied to G12:G13 of the Price worksheet to compute the Consistency Measure? a. =MMULT(C4:E4,$F$11:$F$13) b. =SUMPRODUCT(C4:E4,$F$11:$F$13)/F11 c. =MMULT(C4:E4,$F$11:$F$13)/F11 d. =MMULT(C7:E7,$F$11:$F$13)/F11 27. What is the formula for the weighted average score for alternative j when using a multi-criteria scoring model? a. b. c. d.
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ch 14 28. An investor is considering 2 investments, A, B, which can be made now. After these investments are made he can pursue choices C, D, E and F depending on whether he chose A or B originally. He has developed the following decision tree to aid in his selection process. What are the correct original and subsequent decisions based on an expected monetary value criterion?
a. A, C b. A, D c. B, E d. B, F 29. An alternative a. is a course of action intended to solve a problem b. is always feasible c. is never feasible d. is realistic Exhibit 14.5 The following questions are based on the information below. An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following decision tree has been developed for the problem. The investor has estimated the probability of a declining economy at 40% and an expanding economy at 60%.
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ch 14
30. Refer to Exhibit 14.5. How high can P(E) go before the investor's decision, based on expected monetary value criteria, changes? a. 0.65 b. 0.70 c. 0.75 d. 0.80 Exhibit 14.4 The following questions are based on the information below.
A
B
1 2 3 4 5 6 7 8 9 10 11
Investment A B C Bank
12
Payoff of decision made with perfect information:
C
D
Decline 0 30 50 20
Expand 80 70 35 20
EMV
0.7
0.3
E
Payoff Matrix Economy
Probability
13 Copyright Cengage Learning. Powered by Cognero.
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ch 14 14
EVPI: EVPI
31. Refer to Exhibit 14.4. What is the expected value of perfect information for the investor? a. 13.5 b. 20 c. 45.5 d. 59 Exhibit 14.1 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A 1 2 3 4 5 6 7 8
Investment A B C Bank Payoffs
B C Payoff Matrix
D
Economy Decline Expand 0 85 25 65 40 30 10 10
32. Refer to Exhibit 14.1. What decision should be made according to the maximax decision rule? a. A b. B c. C d. Bank 33. Based on the radar chart of the weighted scores provided below, which of the following interpretations is incorrect?
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ch 14
a. Site A wins on the Sales criteria but is last on the Location criteria. b. Site C wins on the Security criteria and scores high on the remaining three criteria. c. Site B scores lowest on each of the four criteria. d. No site dominates on each of the four criteria. 34. A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 69% chance that demand for their products will be high and a 31% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 63% chance that the customers will like the products and a 37% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is a 92% chance that demand will be high for the products. If the survey is unfavorable there is only a 30% chance that the demand will be high. The following decision tree has been built for this problem. The company has computed that the expected monetary value of the best decision without sample information is 154.35 million. What is the EVSI for this problem (in $ million)? Factory Size Large Small
Demand High 200 100
Low 85 95
Plant Cost ($million) 10 2
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ch 14
a. 0.07 b. 26.38 c. 109.5 d. 180.8 35. What is the formula for the exponential utility function U(x)? a. −e−x/R b. 1 + e−x/R c. 1 − ex/R d. 1 − e−x/R Exhibit 14.1 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A 1 2 3 4 5 6
Investment A B
B C Payoff Matrix
D
Economy Decline Expand 0 85 25 65
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ch 14 7 8
C Bank Payoffs
40 10
30 10
36. Refer to Exhibit 14.1. What formula should go in cell D5 to implement the maximax decision rule? a. =MAX(MAX(B5:C5)) b. =MIN(B5:C5) c. =AVERAGE(B5:C5) d. =MAX(B5:C5) 37. Expected regret is also called a. EMV. b. EOL. c. EPA. d. EOQ. 38. A "risk averse" decision maker assigns the ____ relative utility to any payoff but has a(n) ____ marginal utility for increased payoffs. a. largest; increasing b. largest; diminishing c. smallest; diminishing d. smallest; increasing 39. Which one of these is not used in decision-making under risk? a. minimax regret b. EVPI c. EMV d. decision trees 40. How are states of nature assigned probabilities? a. Use historical data. b. Use best judgement. c. Use interview results. d. All of these. Exhibit 14.6 The following questions use the information below. A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 69% chance that demand for their products will be high and a 31% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 63% chance that the customers will like the products and a 37% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is a 92% chance that demand will be high for the products. If the survey is unfavorable there is only a 30% chance that the demand will be high. The following decision tree has been built for this problem. The Copyright Cengage Learning. Powered by Cognero.
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ch 14 company has computed that the expected monetary value of the best decision without sample information is 154.35 million. The company has developed the following conditional probability table for their decision problem.
A B C 1 2 Joint Probabilities 3 High Demand Low Demand 4 Favorable Response 0.58 0.05 5 Unfavorable Response 0.11 0.26 6 Total 0.69 0.31 7 8 9 Conditional Probability 10 For A Given Survey Response 11 High Demand Low Demand 12 Favorable Response 0.92 0.08 13 Unfavorable Response 0.30 0.70 14 15 Conditional Probability 16 For A Given Demand Level 17 High Demand Low Demand 18 Favorable Response 0.84 0.16 19 Unfavorable Response 0.16 0.84
D
Total 0.63 0.37 1.00
41. Refer to Exhibit 14.6. What formula should go in cell C13 of the probability table? a. =C5/$D4 b. =C5/C$6 c. =C5/$D5 d. =C4/$D4 Exhibit 14.2 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A 1 2 3 4 5 6 7 8
Investment A B C Bank
B Payoff Matrix
C
Economy Decline Expand 0 85 25 65 40 30 10 10
D
E
F Regret Matrix
Investment A B C Bank
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G
H
Economy Decline Expand
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ch 14 42. Refer to Exhibit 14.2. What formula should go in cell F5 of the Regret Matrix above to compute the regret value? a. =B$5-MAX(B$5:B$8) b. =MAX(B$5:B$8)-MAX(B5) c. =MAX(B$5:B$8)-MIN(B$5:B$8) d. =MAX(B$5:B$8)-B5 43. A course of action intended to solve a problem is called a(n) a. alternative. b. option. c. decision. d. criteria. 44. Suppose that the regrets for an alternative with three states of nature are: 20, 10, and 0. The probabilities of these states of nature are 0.2, 0.3, and 0.5, respectively. The expected regret for the alternative is equal to a. 7 b. 20 c. 10 d. 30 45. Leaves of a decision tree are also called ____ nodes. a. end b. terminal c. decision d. payoff 46. A circular node in a decision tree is called a(n) ____ node. a. chance b. random c. decision d. event Exhibit 14.5 The following questions are based on the information below. An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following decision tree has been developed for the problem. The investor has estimated the probability of a declining economy at 40% and an expanding economy at 60%.
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ch 14
47. Refer to Exhibit 14.5. What is the expected monetary value for the investor's problem? a. 32 b. 36 c. 38 d. 42 48. Probabilistic decision rules can be used if the states of nature in a decision problem can be assigned probabilities that represent their likelihood of occurrence. Which of the following is not true regarding the probabilities employed? a. The probabilities are always obtained from historical data. b. The probabilities must always be unbiased. c. The probabilities can be assigned subjectively. d. Subjective probabilities obtained can be accurate and unbiased. 49. Sensitivity analysis is most useful in a. decision-making under risk b. decision-making under uncertainty c. decision-making under certainty d. all of the above 50. Based on the radar chart of raw scores provided below, why is this decision complex?
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ch 14
a. The chart is hard to read. b. No site wins on all four criteria. c. No site achieves a perfect score of 1.0 on a criteria. d. No sites have sufficient security. Exhibit 14.1 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A 1 2 3 4 5 6
Investment A B
B C Payoff Matrix
D
Economy Decline Expand 0 85 25 65
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ch 14 7 8
C Bank Payoffs
40 10
30 10
51. Refer to Exhibit 14.1. What decision should be made according to the maximin decision rule? a. A b. B c. C d. Bank Exhibit 14.8 The following questions use the information below. A company needs to buy a new insurance policy. They have three policies to choose from, A, B and C. The policies differ with respect to price, coverage and ease of billing. The company has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
D
E
A B C Sum
A 1.000 0.333 0.200 1.533
Pairwise Comparisons B 3.000 1.000 0.500 4.500
C 5.000 2.000 1.000 8.000
A B C
A 0.652 0.217 0.130
Normalized Comparisons B 0.667 0.222 0.111
C 0.625 0.250 0.125 Consistency ratio:
F
G
Price Scores 0.648 0.230 0.122
Consistency Measure 3.007 3.003 3.001 0.003
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Coverage Billing Weighted Avg. Score: Summary
A 0.648 0.213 0.120 0.215
B 0.230 0.701 0.272 0.404
C 0.122 0.085 0.608 0.381
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F
G
0.123 0.320 0.557 1.000
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ch 14
52. Refer to Exhibit 14.8. The Consistency Ratio indicates consistency in the pairwise comparison matrix if the ratio is a. ≤ 0.05 b. ≤ 0.10 c. ≤ 0.20 d. ≤ 0.30 53. A payoff matrix depicts ____ versus ____ with payoffs for each intersection cell. a. decision criteria; states of nature. b. decision alternatives; potential outcomes. c. decision alternatives; states of nature. d. decision criteria; potential outcomes. 54. The decision with the smallest expected opportunity loss (EOL) will also have the a. smallest EMV. b. largest EMV. c. smallest regret. d. largest regret. 55. The amount of opportunity lost in making a decision is called a. loss. b. frustration. c. negative profit. d. regret. 56. The ____ in a decision problem represent factors that are important to the decision maker. a. payoffs b. states of nature c. criteria d. alternatives 57. A state of nature a. is observed b. is under control of a decision maker c. is known with certainty d. is estimated using a decision model of choice 58. In decision-making, luck a. often plays a role in determining whether good or bad outcomes occur b. can be quantified c. cannot be quantified d. can be ignored Exhibit 14.7 Copyright Cengage Learning. Powered by Cognero.
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ch 14 The following questions use the information below. A decision maker is faced with two alternatives. The decision maker has determined that she is indifferent between the two alternatives when p = 0.45. Alternative 1: Alternative 2:
Receive $82,000 with certainty Receive $143,000 with probability p and lose $15,000 with probability (1 − p).
59. Refer to Exhibit 14.7. What is the decision maker's certainty equivalent for this problem? a. −$15,000 b. $84,000 c. $56,100 d. $82,000 60. A square node in a decision tree is called a(n) ____ node. a. chance b. random c. decision d. event 61. In decision analysis, good decisions a. always result in good outcomes b. always result in bad outcomes c. guarantee good outcomes d. may be reached when the model accounts for unforeseeable circumstances 62. The total worth, value or desirability of a decision alternative is called its a. usefulness. b. worthiness. c. utility. d. risk. 63. A fast food restaurant is considering opening a new store at one of four locations. They have developed the following multi-criteria scoring model for this problem. What location should they choose based on this information?
A 1 2 3 4 5 6 7 8
B
Criterion Sales Location Security Growth Weighted average score
C Scores Site A 0.80 0.70 0.40 0.60
D
E
F
Site B 0.75 0.80 0.50 0.60
Site C 0.70 0.88 0.60 0.80
Site D 0.65 0.95 0.50 0.85
G
H Criterion Weights 0.4 0.3 0.2 0.1 1
a. A Copyright Cengage Learning. Powered by Cognero.
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ch 14 b. B c. C d. D 64. The scores in a scoring model range from a. 0 to 1 b. −1 to +1 c. 0 to 5 d. 0 to 10 Exhibit 14.3 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 70% and an expanding economy at 30%.
A 1 2 3 4 5 6 7 8 9 10
B
C
D
E
EMV
Investment A B C Bank
Payoff Matrix Economy Investment A B C Bank
Decline −10 20 40 15
Expand 90 50 45 20
Probability
0.7
0.3
Probability
F Regret Matrix
G
Economy Decline
EOL
0.7
0.3
H
65. Refer to Exhibit 14.3. What formula should go in cell F5 and copied to F6:F8 of the spreadsheet if the expected regret decision rule is to be used? a. =B$5-MAX(B$5:B$8) b. =MAX(B$5:B$8)-MAX(B5) c. =MAX(B$5:B$8)-MIN(B$5:B$8) d. =MAX(B$5:B$8)-B5 Exhibit 14.1 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A
B
C
D
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ch 14 1 2 3 4 5 6 7 8
Payoff Matrix
Investment A B C Bank Payoffs
Economy Decline Expand 0 85 25 65 40 30 10 10
66. Refer to Exhibit 14.1. What decision should be made according to the minimax regret decision rule? a. A b. B c. C d. Bank 67. The category of decision rules that contains the maximax decision rule is the a. optimistic category. b. non-probabilistic category. c. probabilistic category. d. optimality category. 68. The maximin approach to decision-making a. maximizes the minimum return b. maximizes the maximum return c. maximizes the minimum regret d. minimizes the minimum regret 69. A(n) ____ is a course of action intended to solve a problem. a. decision b. criteria c. state of nature d. alternative 70. Which are characteristics of decision-making under uncertainty? a. the probability of possible future events is unknown b. decision-makers must rely on probabilities in evaluating outcomes c. all process parameters have known values d. some process parameters have known values Exhibit 14.8 The following questions use the information below. A company needs to buy a new insurance policy. They have three policies to choose from, A, B and C. The policies differ Copyright Cengage Learning. Powered by Cognero.
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ch 14 with respect to price, coverage and ease of billing. The company has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
B
C
D
A B C Sum
A 1.000 0.333 0.200 1.533
Pairwise Comparisons B 3.000 1.000 0.500 4.500
A B C
A 0.652 0.217 0.130
Normalized Comparisons B 0.667 0.222 0.111
E
C 5.000 2.000 1.000 8.000
C 0.625 0.250 0.125 Consistency ratio:
F
G
Price Scores 0.648 0.230 0.122
Consistency Measure 3.007 3.003 3.001 0.003
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Coverage Billing Weighted Avg. Score: Summary
A 0.648 0.213 0.120 0.215
B 0.230 0.701 0.272 0.404
C 0.122 0.085 0.608 0.381
F
G
0.123 0.320 0.557 1.000
71. Refer to Exhibit 14.8. What formula should go in cell C7 and get copied to D7:E7 of the Summary worksheet to compute the Weighted Average Score? a. =SUMPRODUCT(C4:E4,$G$4:$G$6) b. =SUMPRODUCT(C4:C6,$C$5:$C$7) c. =SUMPRODUCT($G$4,$G$6) d. =SUMPRODUCT(C4:C6,$G$4:$G$6) 72. Every nonprobabilistic method has a weakness for decision making. Which of the following is incorrect regarding a method and its weakness? a. The maximax method ignores potentially large losses. b. The maximin method ignores potentially large payoffs. c. The minimax regret method can lead to inconsistent decisions. d. All of these are correct. Copyright Cengage Learning. Powered by Cognero.
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ch 14 Exhibit 14.4 The following questions are based on the information below.
A
B
1 2 3 4 5 6 7 8 9 10 11
Investment A B C Bank
12
Payoff of decision made with perfect information:
C
D
Decline 0 30 50 20
Expand 80 70 35 20
EMV
0.7
0.3
E
Payoff Matrix Economy
Probability
13 14
EVPI: EVPI
73. Refer to Exhibit 14.4. What is the expected value with perfect information for the investor? a. 13.5 b. 45.5 c. 59 d. 80 Exhibit 14.1 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A 1 2 3 4 5 6 7 8
Investment A B C Bank Payoffs
B C Payoff Matrix
D
Economy Decline Expand 0 85 25 65 40 30 10 10
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ch 14 74. Refer to Exhibit 14.1. What formula should go in cell D5 to implement the maximin decision rule? a. =MAX(MIN(B5:C5)) b. =MIN(B5:C5) c. =AVERAGE(B5:C5) d. =MAX(B5:C5) 75. Which of the following is a goal of decision analysis? a. Help individuals make good decisions. b. Ensure decisions lead to good outcomes. c. Avoiding decisions leading to bad outcomes. d. Reduce the role of luck in a decision. Exhibit 14.2 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem.
A 1 2 3 4 5 6 7 8
Investment A B C Bank
B Payoff Matrix
C
Economy Decline Expand 0 85 25 65 40 30 10 10
D
E
F Regret Matrix
Investment A B C Bank
G
H
Economy Decline Expand
76. Refer to Exhibit 14.2. What formula should go in cell H5 and copied to H6:H8 of the Regret Table above to implement the minimax regret decision rule? a. =MAX(MAX(F5:G5)) b. =MIN(F5:G5) c. =AVERAGE(F5:G5) d. =MAX(F5:G5) 77. In a graphical representation of decision trees the decision nodes are represented by a. squares b. circles c. solid dots d. ovals 78. Suppose that EVPI=0. This means that a. the decision problem involves no risk b. the decision problem is certain Copyright Cengage Learning. Powered by Cognero.
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ch 14 c. the payoff under risk iz zero d. the decision problem is incorrectly formulated Exhibit 14.5 The following questions are based on the information below. An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following decision tree has been developed for the problem. The investor has estimated the probability of a declining economy at 40% and an expanding economy at 60%.
79. Refer to Exhibit 14.5. What is the correct decision for this investor based on an expected monetary value criterion? a. A b. B c. C d. D Exhibit 14.8 The following questions use the information below. A company needs to buy a new insurance policy. They have three policies to choose from, A, B and C. The policies differ with respect to price, coverage and ease of billing. The company has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A
B
C
D
E
F
G
1 Copyright Cengage Learning. Powered by Cognero.
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ch 14 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A B C Sum
A 1.000 0.333 0.200 1.533
Pairwise Comparisons B 3.000 1.000 0.500 4.500
C 5.000 2.000 1.000 8.000
A B C
A 0.652 0.217 0.130
Normalized Comparisons B 0.667 0.222 0.111
C 0.625 0.250 0.125
Price Scores 0.648 0.230 0.122
Consistency ratio:
Consistency Measure 3.007 3.003 3.001 0.003
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Coverage Billing Weighted Avg. Score: Summary
A 0.648 0.213 0.120 0.215
B 0.230 0.701 0.272 0.404
C 0.122 0.085 0.608 0.381
F
G
0.123 0.320 0.557 1.000
80. Refer to Exhibit 14.8. What formula should go in cell G15 of the Price worksheet to compute the Consistency Ratio? a. =AVERAGE(G11:G13)-3)/(2*0.58) b. =AVERAGE(G11:G13)-3) c. =AVERAGE(G11:G13))/(2*0.58) d. =AVERAGE(G11:G13)-3)/0.58 81. Refer to Exhibit 14.8. Which policy should the company choose based on the Summary worksheet? a. A b. B c. C d. None of these 82. The difference between expected payoff under certainty and expected payoff under risk is a. EVPI b. EMV c. expected regret value d. none of the above
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ch 14 Exhibit 14.3 The following questions are based on the information below. An investor is considering 4 investments, A, B, C and leaving his money in the bank. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 70% and an expanding economy at 30%.
A 1 2 3 4 5 6 7 8 9 10
B
C
D
E
EMV
Investment A B C Bank
Payoff Matrix Economy Investment A B C Bank
Decline −10 20 40 15
Expand 90 50 45 20
Probability
0.7
0.3
Probability
F Regret Matrix
G
Economy Decline
EOL
0.7
0.3
H
83. Refer to Exhibit 14.3. What decision should be made according to the expected regret decision rule? a. A b. B c. C d. Bank 84. In a graphical representation of decision trees the event nodes are represented by a. squares b. circles c. solid dots d. ovals Exhibit 14.7 The following questions use the information below. A decision maker is faced with two alternatives. The decision maker has determined that she is indifferent between the two alternatives when p = 0.45. Alternative 1: Alternative 2:
Receive $82,000 with certainty Receive $143,000 with probability p and lose $15,000 with probability (1 − p).
85. Refer to Exhibit 14.7. What is the expected value of Alternative 2 for this decision maker? a. $82,000 b. $56,100 c. $64,350 Copyright Cengage Learning. Powered by Cognero.
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ch 14 d. $72,600 86. The scores in a scoring model can be thought of as subjective assessments of a. usefulness. b. worthiness. c. utility. d. payoff. Exhibit 14.4 The following questions are based on the information below.
A
B
1 2 3 4 5 6 7 8 9 10 11
Investment A B C Bank
12
Payoff of decision made with perfect information:
C
D
Decline 0 30 50 20
Expand 80 70 35 20
EMV
0.7
0.3
E
Payoff Matrix Economy
Probability
13 14
EVPI: EVPI
87. Refer to Exhibit 14.4. What formula should go in cell D14 of the spreadsheet to compute the EVPI? a. MAX(D5:D8)-D12 b. D12-MIN(D5:D8) c. SUMPRODUCT(B12:C12,B10:C10)-MAX(D5:D8) d. D12-MAX(D5:D8) 88. Decision Analysis techniques provide modeling techniques to help decision makers make decisions. Which of the following is not typically a benefit of decision analysis? a. Incorporating uncertainty via probabilities. b. Incorporating risk via utility theory functions. c. Incorporating uncertainty via exponential distributions. d. Structuring decision strategies via decision trees. 89. The decision rule which selects the alternative associated with the smallest maximum opportunity loss is the a. maximax decision rule. b. maximin decision rule. Copyright Cengage Learning. Powered by Cognero.
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ch 14 c. minimax regret decision rule. d. minimin decision rule. 90. Decision analysis supports all but one of the following goals. Which goal is not supported? a. Help make good decisions. b. Ensure selection of good outcomes. c. Analyze decision problems logically. d. Incorporate problem uncertainty.
Exhibit 14.12 The following questions use the information below. A decision maker is faced with two alternatives. Alternative 1: Alternative 2:
Receive $40,000 with certainty Receive $80,000 with probability p and lose $5,000 with probability (1 − p).
The decision maker has determined that she is indifferent between the two alternatives when p = 0.7. 91. Refer to Exhibit 14.12. What is the expected value of Alternative 2 for this decision maker? Exhibit 14.13 The following questions use the information below. A student wants to buy a new car. She has three cars to choose from, A, B and C. The cars differ with respect to price, performance and looks. The student has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9
B
C
A B C Sum
A 1.000 1.001 4.000 6.001
D
E
11 12 13 14 15
A B C
G
Price
Consistency
Pairwise Comparisons B C 0.999 0.250 1.000 0.200 5.000 1.000 6.999 1.450 Normalized Comparisons
10
F
A
B
C
Scores
Measure
0.167 0.167 0.667
0.143 0.143 0.714
0.172 0.138 0.690
0.161 0.149 0.690
3.003 3.003 3.012
Consistency ratio:
0.005
Price Copyright Cengage Learning. Powered by Cognero.
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ch 14
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Performance Looks Weighted Avg. Score: Summary
A 0.557 0.161 0.115 0.184
B 0.320 0.149 0.182 0.188
C 0.123 0.690 0.703 0.628
F
G
0.123 0.320 0.557 1.000
92. Refer to Exhibit 14.13. Which car should the student choose based on the Summary worksheet? Exhibit 14.14 The following questions use the Decision Tree model and strategy table information below.
A 1 2 3 4
B
C
D
E
F
G
H
I
J
K
Sell B
0
0.1
0.2
0.3
P(G|B) 0.4
0.5
0.6
0.7
0.8
0
Trade A for C
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Strategy Table
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ch 14 5
0.1
6
0.2
7
P(G|A)
0.3
8
0.4
9
0.5
10
0.6
11
0.7
12
0.8
13
0.9
14
1
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Sell B Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Sell B
Sell B
Sell B
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Sell B
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Sell B
Sell B
Sell B
Sell B
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Sell B
Sell B
Sell B
Sell B
Trade A for C
Sell B
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C
93. Refer to Exhibit 14.14. You want to conduct a risk analysis on P(G|A) and P(G|B). What Decision Tree model changes must you make to be able to use a strategy table? Exhibit 14.9 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the investment decision problem.
A 1 2 3 4 5 6 7 8
Investment W X Y Z Payoffs
B Payoff Matrix Economy Decline 0 30 50 20
C
Expand 80 70 35 20
D
E
Choice
94. Refer to Exhibit 14.9. What decision should be made according to the maximax decision rule? Exhibit 14.14 The following questions use the Decision Tree model and strategy table information below.
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ch 14
A 1 2 3
B
Sell B
4
0
5
0.1
6
0.2
7
C
D
E
F
G
H
I
J
K
0.5
0.6
0.7
0.8
Strategy Table
P(G|A)
0.3
8
0.4
9
0.5
10
0.6
11
0.7
12
0.8
13
0.9
14
1
0 Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
0.1
0.2
0.3
P(G|B) 0.4
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
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Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
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Sell B
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Sell B
Trade A for C
Sell B
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C
95. Refer to Exhibit 14.14. Why does the strategy table, examining the risk associated with P(G|A) and P(G|B) never show Copyright Cengage Learning. Powered by Cognero.
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ch 14 "Sell A" or "Trade B for D" as selected options? Exhibit 14.10 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 80% and an expanding economy at 20%.
A 1 2 3 4 5 6 7 8 9 10
B Payoff Matrix Economy Decline 10 20 40 35
Investment W X Y Z Probability Payoffs
C
D
Expand 60 80 30 25
EMV
0.8
0.2
96. Refer to Exhibit 14.10. Complete the table using the expected monetary value decision rule and indicate which decision should be made according to that rule. 97. Refer to Exhibit 14.10. Complete the following table to determine the expected value of perfect information for the investor.
A
B
1 2 3 4 5 6 7 8 9 10 11
Investment W X Y Z
Decline 10 20 40 35
Probability
0.8
12
Payoff of decision made with perfect information:
13 14
C Payoff Matrix Economy Expand 60 80 30 25
D
E
EMV
0.2
EVPI: EVPI
98. A convenience store chain is considering opening a new store at one of four locations. They have developed the Copyright Cengage Learning. Powered by Cognero.
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ch 14 following multi-criteria scoring model for this problem. What formulas must be placed in cells C13:F16 to compute the weighted scores for use in generating a Weighted Score radar chart?
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
B
C
D
E
F
Criterion Sales Location Security Growth Weighted average score
Site A 0.65 0.95 0.50 0.85 0.6825
Site B 0.75 0.80 0.50 0.60 0.69
Scores Site C 0.70 0.88 0.60 0.80 0.716
Criterion Sales Location Security Growth
Site A ? ? ? ?
Site B ? ? ? ?
Scores Site C ? ? ? ?
G
H
Site D 0.80 0.70 0.40 0.60 0.67
Criterion Weights 0.50 0.20 0.25 0.05 1
Site D ? ? ? ?
Criterion Weights 0.50 0.20 0.25 0.05 1
99. An investor is considering 2 investments, A, B, which can be purchased now for $10. There is a 40% chance that investment A will grow rapidly in value and a 60% chance that it will grow slowly. If A grows rapidly the investor can cash it in for $80 or trade it for investment C which has a 25% chance of growing to $100 and a 75% chance of reaching $80. If A grows slowly it is sold for $50. There is a 70% chance that investment B will grow rapidly in value and a 30% chance that it will grow slowly. If B grows rapidly the investor can cash it in for $100 or trade it for investment D which has a 20% chance of growing to $95 and an 80% chance of reaching $80. If B grows slowly it is sold for $45. What is the multistage decision for this investor and what is the EMV for this decision? 100. A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is a 72% chance that demand for their products will be high and a 28% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 76% chance that the customers will like the products and a 24% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is an 87% chance that demand will be high for the products. If the survey is unfavorable there is only a 25% chance that the demand will be high. Draw the decision tree for the problem when the survey is performed before the plant size decision is made.
Demand Factory Size Large Small
High 90 55
Low 40 20
Plant Cost ($million) 5 1
Exhibit 14.9 The following questions are based on the information below. Copyright Cengage Learning. Powered by Cognero.
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ch 14 An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the investment decision problem.
A 1 2 3 4 5 6 7 8
Investment W X Y Z Payoffs
B Payoff Matrix Economy Decline 0 30 50 20
C
D
Expand 80 70 35 20
E
Choice
101. Refer to Exhibit 14.9. What formula should go in cell D5 of the following Regret Table to implement the minimax regret decision rule? Assume that cells B5:C8 contain the regret values for the problem.
A 1 2 3 4 5 6 7 8
Investment W X Y Z Regret
B Regret Matrix Economy Decline
C
D
Expand
102. Refer to Exhibit 14.9. What decision should be made according to the minimax regret decision rule? Exhibit 14.12 The following questions use the information below. A decision maker is faced with two alternatives. Alternative 1: Alternative 2:
Receive $40,000 with certainty Receive $80,000 with probability p and lose $5,000 with probability (1 − p).
The decision maker has determined that she is indifferent between the two alternatives when p = 0.7. 103. Refer to Exhibit 14.12. What is the decision maker's risk premium for this problem? Exhibit 14.13 The following questions use the information below. A student wants to buy a new car. She has three cars to choose from, A, B and C. The cars differ with respect to price, performance and looks. The student has developed the following AHP tables for price and summary. The other tables are Copyright Cengage Learning. Powered by Cognero.
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ch 14 not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9
B
C
D
A B C Sum
A 1.000 1.001 4.000 6.001
E
11 12 13 14 15
A B C
G
Price
Consistency
Pairwise Comparisons B C 0.999 0.250 1.000 0.200 5.000 1.000 6.999 1.450 Normalized Comparisons
10
F
A
B
C
Scores
Measure
0.167 0.167 0.667
0.143 0.143 0.714
0.172 0.138 0.690
0.161 0.149 0.690
3.003 3.003 3.012
Consistency ratio:
0.005
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Performance Looks Weighted Avg. Score: Summary
A 0.557 0.161 0.115 0.184
B 0.320 0.149 0.182 0.188
C 0.123 0.690 0.703 0.628
F
G
0.123 0.320 0.557 1.000
104. Refer to Exhibit 14.13. What formula should go in cell C7 and copied to cells D7:E7 of the Summary worksheet to compute the Weighted Average Score? 105. A convenience store chain is considering opening a new store at one of four locations. They have developed the following multi-criteria scoring model for this problem. Complete the table for this problem. What location should they choose based on this information?
A 1 2 3 4 5 6 7
B
C
D
E
F
Criterion Sales Location Security Growth
Site A 0.65 0.95 0.50 0.85
Site B 0.75 0.80 0.50 0.60
Scores Site C 0.70 0.88 0.60 0.80
Site D 0.80 0.70 0.40 0.60
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H Criterion Weights 0.50 0.20 0.25 0.05 Page 37
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ch 14 8
Weighted average score
1
Exhibit 14.10 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 80% and an expanding economy at 20%.
A 1 2 3 4 5 6 7 8 9 10
B Payoff Matrix Economy Decline 10 20 40 35
Investment W X Y Z Probability Payoffs
C
D
Expand 60 80 30 25
EMV
0.8
0.2
106. Refer to Exhibit 14.10. What formulas should go in cell D5:D14 and B12:C12 of the spreadsheet to compute the EVPI?
A
B
1 2 3 4 5 6 7 8 9 10 11
C Payoff Matrix
Investment A B C Bank
Decline 10 20 40 35
Probability
0.8
12
Payoff of decision made with perfect information:
Economy Expand 60 80 30 25
D
E
EMV
0.2
13 14
EVPI: EVPI
Exhibit 14.13 The following questions use the information below. Copyright Cengage Learning. Powered by Cognero.
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ch 14 A student wants to buy a new car. She has three cars to choose from, A, B and C. The cars differ with respect to price, performance and looks. The student has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9
B
C
D
A B C Sum
A 1.000 1.001 4.000 6.001
E
11 12 13 14 15
A B C
G
Price
Consistency
Pairwise Comparisons B C 0.999 0.250 1.000 0.200 5.000 1.000 6.999 1.450 Normalized Comparisons
10
F
A
B
C
Scores
Measure
0.167 0.167 0.667
0.143 0.143 0.714
0.172 0.138 0.690
0.161 0.149 0.690
3.003 3.003 3.012
Consistency ratio:
0.005
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Performance Looks Weighted Avg. Score: Summary
A 0.557 0.161 0.115 0.184
B 0.320 0.149 0.182 0.188
C 0.123 0.690 0.703 0.628
F
G
0.123 0.320 0.557 1.000
107. Refer to Exhibit 14.13. What formula should go in cell F11 and copied to cells F12:F13 of the Price worksheet to compute the Price Score? Exhibit 14.9 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the investment decision problem.
A 1
B Payoff Matrix
C
D
E
Copyright Cengage Learning. Powered by Cognero.
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ch 14 2 3 4 5 6 7 8
Economy Decline 0 30 50 20
Investment W X Y Z Payoffs
Expand 80 70 35 20
Choice
108. Refer to Exhibit 14.9. Assume the formula =MAX(B5:C5) was entered in cell D5 and copied to cells D6:D8. What formula should go in cell E5 and get copied to cells E6:E8 to place a "<==" to indicate the choice according to the maximax decision rule? Exhibit 14.10 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 80% and an expanding economy at 20%.
A 1 2 3 4 5 6 7 8 9 10
B Payoff Matrix Economy Decline 10 20 40 35
Investment W X Y Z Probability Payoffs
0.8
C
D
Expand 60 80 30 25
EMV
0.2
109. Refer to Exhibit 14.10. The original payoff data is in the worksheet above called "Payoffs". What formula should go in cell B5 of the spreadsheet if the expected regret decision rule is to be used?
A 1 2 3 4 5 6 7 8 9 10
Investment W X Y Z Probability
B
C Regret Matrix
Economy Decline Expand
0.8
D
EOL
0.2
Copyright Cengage Learning. Powered by Cognero.
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ch 14 Regret
110. An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following decision tree has been developed for the problem. The investor has estimated the probability of a declining economy at 25% and an expanding economy at 75%. What is the correct decision for this investor based on an expected monetary value criteria? Draw the decision tree for this problem. Exhibit 14.9 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the investment decision problem.
A 1 2 3 4 5 6 7 8
Investment W X Y Z Payoffs
B Payoff Matrix Economy Decline 0 30 50 20
C
D
Expand 80 70 35 20
E
Choice
111. Refer to Exhibit 14.9. Assume the formula =MIN(B5:C5) was entered in cell D5 and copied to cells D6:D8. What formula should go in cell E5 and get copied to cells E6:E8 to place a "<==" to indicate the choice according to the maximin decision rule? Exhibit 14.12 The following questions use the information below. A decision maker is faced with two alternatives. Alternative 1: Alternative 2:
Receive $40,000 with certainty Receive $80,000 with probability p and lose $5,000 with probability (1 − p).
The decision maker has determined that she is indifferent between the two alternatives when p = 0.7. 112. Refer to Exhibit 14.12. What is the decision maker's certainty equivalent for this problem? Exhibit 14.10 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the decision problem. The investor has estimated the probability of a declining economy at 80% and an expanding Copyright Cengage Learning. Powered by Cognero.
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ch 14 economy at 20%.
A 1 2 3 4 5 6 7 8 9 10
Investment W X Y Z
B Payoff Matrix Economy Decline 10 20 40 35
Probability Payoffs
C
D
Expand 60 80 30 25
EMV
0.8
0.2
113. Refer to Exhibit 14.10. Complete the Regret Table according to the expected regret decision rule. 114. An investor is considering 2 investments, A, B, which can be purchased now for $10. There is a 40% chance that investment A will grow rapidly in value and a 60% chance that it will grow slowly. If A grows rapidly the investor can cash it in for $80 or trade it for investment C which has a 25% chance of growing to $100 and a 75% chance of reaching $80. If A grows slowly it is sold for $50. There is a 70% chance that investment B will grow rapidly in value and a 30% chance that it will grow slowly. If B grows rapidly the investor can cash it in for $100 or trade it for investment D which has a 20% chance of growing to $95 and an 80% chance of reaching $80. If B grows slowly it is sold for $45. Draw the decision tree for this problem. Exhibit 14.9 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the investment decision problem.
A 1 2 3 4 5 6 7 8
Investment W X Y Z Payoffs
B Payoff Matrix Economy Decline 0 30 50 20
C
Expand 80 70 35 20
D
E
Choice
115. Refer to Exhibit 14.9. What formula should go in cell D5 and get copied to D6:D8 to implement the maximin decision rule? Exhibit 14.13 Copyright Cengage Learning. Powered by Cognero.
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ch 14 The following questions use the information below. A student wants to buy a new car. She has three cars to choose from, A, B and C. The cars differ with respect to price, performance and looks. The student has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9
B
C
D
A B C Sum
A 1.000 1.001 4.000 6.001
E
A B C
G
Price
Consistency
Scores 0.161 0.149 0.690
Measure 3.003 3.003 3.012
Pairwise Comparisons B C 0.999 0.250 1.000 0.200 5.000 1.000 6.999 1.450 Normalized Comparisons
10 11 12 13 14 15
F
A 0.167 0.167 0.667
B 0.143 0.143 0.714
C 0.172 0.138 0.690 Consistency ratio:
0.005
Price
A 1 2 3 4 5 6 7
B
C
D
E
Criterion Price Performance Looks Weighted Avg. Score: Summary
A 0.557 0.161 0.115 0.184
B 0.320 0.149 0.182 0.188
C 0.123 0.690 0.703 0.628
F
G
0.123 0.320 0.557 1.000
116. Refer to Exhibit 14.13. What formula should go in cell G15 of the Price worksheet to compute the Consistency Ratio? 117. A convenience store chain is considering opening a new store at one of four locations. They have developed the following multi-criteria scoring model for this problem. Complete the following table to prepare the spreadsheet for use in generating a Weighted Score radar chart?
A 1 2 3 4
B
Criterion Sales
C Site A 0.65
D
E
F
Site B 0.75
Scores Site C 0.70
Site D 0.80
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H Criterion Weights 0.50 Page 43
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ch 14 5 6 7 8 9 10 11 12 13 14 15 16 17
Location Security Growth Weighted average score
0.95 0.50 0.85 0.6825
0.80 0.50 0.60 0.69
0.88 0.60 0.80 0.716
0.70 0.40 0.60 0.67
0.20 0.25 0.05 1
Criterion Sales Location Security Growth
Site A ? ? ? ?
Site B ? ? ? ?
Scores Site C ? ? ? ?
Site D ? ? ? ?
Criterion Weights 0.50 0.20 0.25 0.05 1
Exhibit 14.13 The following questions use the information below. A student wants to buy a new car. She has three cars to choose from, A, B and C. The cars differ with respect to price, performance and looks. The student has developed the following AHP tables for price and summary. The other tables are not shown due to space limitations.
A 1 2 3 4 5 6 7 8 9
B
A B C Sum
C
D
E
A B C
G
Price
Consistency
Scores 0.161 0.149 0.690
Measure 3.003 3.003 3.012
Pairwise Comparisons B C 0.999 0.250 1.000 0.200 5.000 1.000 6.999 1.450
A 1.000 1.001 4.000 6.001
Normalized Comparisons
10 11 12 13 14 15
F
A 0.167 0.167 0.667
B 0.143 0.143 0.714
C 0.172 0.138 0.690 Consistency ratio:
0.005
Price
A 1 2 3 4 5 6
B
C
D
E
Criterion Price Performance Looks
A 0.557 0.161 0.115
B 0.320 0.149 0.182
C 0.123 0.690 0.703
Copyright Cengage Learning. Powered by Cognero.
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G
0.123 0.320 0.557 Page 44
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ch 14 7
Weighted Avg. Score: Summary
0.184
0.188
0.628
1.000
118. Refer to Exhibit 14.13. What formula should go in cell G11 and copied to cells G12:G13 of the Price worksheet to compute the Consistency Measure? Exhibit 14.11 The following questions use the information below. A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 72% chance that demand for their products will be high and a 28% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 76% chance that the customers will like the products and a 24% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is an 87% chance that demand will be high for the products. If the survey is unfavorable there is only a 25% chance that the demand will be high.
Demand Factory Size Large Small
High 90 55
Low 40 20
Plant Cost ($million) 5 1
The company has developed the following conditional probability table for their decision problem.
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
B
C
Favorable Response Unfavorable Response Total
Joint Probabilities High Demand Low Demand 0.66 0.10 0.06 0.18 0.72 0.28
Favorable Response Unfavorable Response
Conditional Probability For A Given Survey Response High Demand Low Demand 0.87 0.13 0.25 0.75
Favorable Response Unfavorable Response
Conditional Probability For A Given Demand Level High Demand Low Demand 0.92 0.36 0.08 0.64
D
Total 0.76 0.24 1.00
119. Refer to Exhibit 14.11. What is P(F∩H), where F = favorable response and H = high demand? Copyright Cengage Learning. Powered by Cognero.
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ch 14 120. An investor is considering 4 investments, A, B, C, D. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can be weak or strong. The investor has estimated the probability of a declining economy at 30% and an expanding economy at 70%. Draw the decision tree for this problem and determine the correct decision for this investor based on the expected monetary value criteria.
Investment A B C D
Payoff Matrix Economy Weak Strong −30 120 20 60 30 35 15 30
Exhibit 14.14 The following questions use the Decision Tree model and strategy table information below.
A 1 2 3 4
B
C
D
E
F
G
H
I
J
K
Sell B
0
0.1
0.2
0.3
P(G|B) 0.4
0.5
0.6
0.7
0.8
0
Trade A for C
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Strategy Table
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ch 14 5
0.1
6
0.2
7
P(G|A)
0.3
8
0.4
9
0.5
10
0.6
11
0.7
12
0.8
13
0.9
14
1
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Sell B Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Sell B
Trade A for C
Sell B
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C Trade A for C
Trade A for C Trade A for C Trade A for C
121. Refer to Exhibit 14.14. What formula is placed in cell B3 of the strategy table to complete the table as provided? 122. A convenience store chain is considering opening a new store at one of four locations. They have developed the following multi-criteria scoring model for this problem.
A
B
1 2 3 4 5 6 7 8
Criterion Sales Location Security Growth Weighted average score
C
D
E
F
Site A 0.65 0.95 0.50 0.85
Scores Site B 0.75 0.80 0.50 0.60
Site C 0.70 0.88 0.60 0.80
Site D 0.80 0.70 0.40 0.60
G
H
Criterion Weights 0.50 0.20 0.25 0.05 1
What formula should be entered into cell C8-F8 to compute the weighted average scores? Exhibit 14.9 The following questions are based on the information below. An investor is considering 4 investments, W, X, Y, and Z. The payoff from each investment is a function of the economic climate over the next 2 years. The economy can expand or decline. The following payoff matrix has been developed for the investment decision problem.
A 1 2 3 4 5
Investment W
B Payoff Matrix Economy Decline 0
C
Expand 80
D
E
Choice
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ch 14 6 7 8
X Y Z Payoffs
30 50 20
70 35 20
123. Refer to Exhibit 14.9. The original payoff data is in the worksheet called "Payoffs". What formula should go in cell B5 of this Regret Matrix to compute the regret value?
A 1 2 3 4 5 6 7 8
B Regret Matrix Economy Decline
Investment W X Y Z Regret
C
D
Expand
124. Refer to Exhibit 14.9. What formula should go in cell D5 and get copied to D6:D8 to implement the maximax decision rule? 125. Refer to Exhibit 14.9. What decision should be made according to the maximin decision rule? Exhibit 14.11 The following questions use the information below. A company is planning a plant expansion. They can build a large or small plant. The payoffs for the plant depend on the level of consumer demand for the company's products. The company believes that there is an 72% chance that demand for their products will be high and a 28% chance that it will be low. The company can pay a market research firm to survey consumer attitudes towards the company's products. There is a 76% chance that the customers will like the products and a 24% chance that they won't. The payoff matrix and costs of the two plants are listed below. The company believes that if the survey is favorable there is an 87% chance that demand will be high for the products. If the survey is unfavorable there is only a 25% chance that the demand will be high.
Demand Factory Size Large Small
High 90 55
Low 40 20
Plant Cost ($million) 5 1
The company has developed the following conditional probability table for their decision problem.
A 1 2 3 4
Favorable Response
B
C
Joint Probabilities High Demand Low Demand 0.66 0.10
D
Total 0.76
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ch 14 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Unfavorable Response Total
0.06 0.72
0.18 0.28
Favorable Response Unfavorable Response
Conditional Probability For A Given Survey Response High Demand Low Demand 0.87 0.13 0.25 0.75
Favorable Response Unfavorable Response
Conditional Probability For A Given Demand Level High Demand Low Demand 0.92 0.36 0.08 0.64
0.24 1.00
126. Refer to Exhibit 14.11. What formula should go in cell C13 of the probability table?
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ch 14 Answer Key 1. True 2. True 3. True 4. False 5. False 6. True 7. True 8. False 9. a 10. a 11. c 12. a 13. b 14. b 15. b 16. d 17. c 18. a 19. a 20. b 21. a 22. b 23. c 24. b 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 14 26. c 27. a 28. d 29. a 30. d 31. a 32. a 33. c 34. a 35. d 36. d 37. b 38. b 39. a 40. d 41. c 42. d 43. a 44. a 45. b 46. d 47. c 48. a 49. a 50. b 51. c Copyright Cengage Learning. Powered by Cognero.
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ch 14 52. b 53. c 54. b 55. d 56. c 57. a 58. a 59. d 60. c 61. d 62. c 63. c 64. a 65. d 66. b 67. b 68. a 69. d 70. a 71. d 72. d 73. c 74. b 75. a 76. d Copyright Cengage Learning. Powered by Cognero.
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ch 14 77. a 78. a 79. d 80. a 81. b 82. a 83. c 84. b 85. b 86. c 87. d 88. c 89. c 90. b 91. = 80,000 * .7 − 5,000 * .3 = 54,500 92. Car C 93. Change cell H17 to =(1-H7) and change cell H21 to =(1-H32) 94. W 95. The expected value of Trade A for C will always exceed the certain value of Sell A and the certain value of Sell B will always exceed the expected value of Trade B for D. 96. 1 2 3 4 5 6 7 8 9
A
B
C Payoff Matrix
Investment W X Y Z
Decline 10 20 40 35
Economy Expand 60 80 30 25
D
EMV 20 32 38 <== 33
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ch 14 10
Probability Payoffs
0.8
0.2
97. A
B
1 2 3 4 5 6 7 8 9 10 11
Investment W X Y Z
Decline 10 20 40 35
Probability
0.8
0.2
12
Payoff of decision made with perfect information:
40
80
48
EVPI:
10
13 14
C Payoff Matrix Economy Expand 60 80 30 25
D
E
EMV 20 32 38 33
EVPI
98. Cell C13
Formula =C4*$H13
Copied to: D13:F13, C14:F16
99. Pick B, Sell B, EMV = 83.5.
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ch 14
100. 101. =MAX(B5:C5) 102. X 103. $14,500 104. =SUMPRODUCT(C4:C6,$G$4:$G$6) 105. A 1 2 3 4 5 6 7 8
B
C
D
Criterion Sales Location Security Growth Weighted average score
Site A 0.65 0.95 0.50 0.85 0.6825
Site B 0.75 0.80 0.50 0.60 0.69
E Scores Site C 0.70 0.88 0.60 0.80 0.716
F Site D 0.80 0.70 0.40 0.60 0.67
G
H Criterion Weights 0.50 0.20 0.25 0.05 1
Site C should be selected. Copyright Cengage Learning. Powered by Cognero.
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ch 14 106. Cell D5 B5 D14
Formula =SUMPRODUCT(B5:C5,$B$10:$C$10) =MAX(B5:B8) =D12-MAX(D5:D8)
Copied to D6:D8, D12 C5
107. =AVERAGE(C11:E11) 108. =IF(D5=MAX($D$5:$D$8), "<==","") 109. =MAX(Payoffs!B$5:B$8)-Payoffs!B5
110. C, EMV = 45 111. =IF(D5=MAX($D$5:$D$8), "<==","") Copyright Cengage Learning. Powered by Cognero.
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ch 14 112. $40,000 113. 1 2 3 4 5 6 7 8 9 10
A
B
C Regret Matrix
Investment W X Y Z
Decline 30 20 5
50 55
Probability Regret
0.8
0.2
Economy Expand 20
D
EOL 30 16 10 15
114. 115. =MIN(B5:C5) 116. =AVERAGE(G11:G13)-3)/(2*0.58) 117. Copyright Cengage Learning. Powered by Cognero.
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ch 14 A 10 11 12 13 14 15 16 17
B
C
D
Criterion Sales Location Security Growth
Site A 0.3250 0.1900 0.1250 0.0425
Site B 0.375 0.160 0.125 0.030
E Scores Site C 0.350 0.176 0.150 0.040
F Site D 0.40 0.14 0.10 0.03
G
H Criterion Weights 0.50 0.20 0.25 0.05 1
118. =MMULT(C4:E4,$F$11:$F$13)/F11 119. 0.66
120. Correct decision, A with EMV = 75. 121. =IF(B20=1,CHOOSE(J9,L4,L12),CHOOSE(J34,L29,L37)) or =IF(B20=1,CHOOSE(J9,"Trade A for C","Sell A"),CHOOSE(J34,"Trade B for D","Sell B")) Copyright Cengage Learning. Powered by Cognero.
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ch 14 122. Cell C8
Formula SUMPRODUCT(C4:C7,$H$4:$H$7)
Copied to: D8:F8
123. =MAX(Payoffs!B$5:B$8)-Payoffs!B5 124. =MAX(B5:C5) 125. Y 126. C5/$D5
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ch 15
Indicate whether the statement is true or false. 1. In Project Management the early start (ES) and early finish (EF) times for each activity are determined by using a backward pass. a. True b. False 2. Two techniques that were developed to help managers plan, organize, and control projects are: the Critical Path Method (CPM) and the Program Evaluation and Review Technique (PERT). a. True b. False 3. Array formulas typically perform operations on one or more ranges of cells. a. True b. False 4. Two network design techniques used in Project Management are: the Activity-On-Node (AON) and the Activity on Arcs (AOA). a. True b. False 5. In a project network there must be a unique critical path. a. True b. False 6. Slack is the amount of time by which the start of an activity can be delayed without delaying the project. a. True b. False 7. Noncritical activities have some negative amount of slack. a. True b. False 8. A Gantt chart is a popular technique for portraying the schedule of a project’s activities over time. a. True b. False
Indicate the answer choice that best completes the statement or answers the question. 9. How many paths are there in the following project precedence network?
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ch 15
a. 6 b. 7 c. 8 d. 9 10. What is the correct formula to compute the variance of an activity using PERT? a. b. c. d.
11. What are the immediate successors of activity D in this network?
a. A, B, C b. B, C c. E, F d. E 12. Slack for activity I is computed as a. LFTi − ESTi b. LSTi − ESTi Copyright Cengage Learning. Powered by Cognero.
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ch 15 c. ESTi − LSTi d. EFTi − LFTi 13. A weakness of Gantt charts is that they do not explicitly a. indicate activity times. b. show precedence relationships. c. show activity start times. d. show activity finish times. 14. In a PERT network the time required to complete a path is a(n) a. constant b. random variable c. Poisson distributed variable d. exponentially distributed variable 15. If we are employing Activity-On-Node (AON) network design, the arcs in the network diagram represent a. significance relationships. b. precedence relationships. c. tasks to complete. d. task times. 16. Which of the following definitions correctly defines the acronym, PERT? a. Program Evaluation and Review Technique. b. Process Evaluation and Review Technique. c. Program Evaluation and Reduction Technique. d. Program Explanation and Reduction Technique. 17. Activity-on-Arc (AOA) networks do not allow multiple arcs with common start and finish nodes. If this must be represented, how is an AOA network modified? a. The network is redrawn into an Activity-on-Node network. b. Connect the arcs to the node using an IFERROR() function. c. Phantom (dummy) activity arcs are added. d. Zero duration activity arcs are added. 18. Activities with zero slack a. are dummy activities b. belong to the critical path c. must be completed first d. must be delayed 19. If we are employing Activity-On-Arc (AOA) network design, the arcs in the network diagram represent a. significance relationships. b. precedence relationships. c. tasks to complete. Copyright Cengage Learning. Powered by Cognero.
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ch 15 d. task times. 20. For an activity with duration t, slack can be calculated as a. LF-EF b. LF-t c. LS+t d. LS+ES 21. There are various similarities between the CPM and PERT methods. Which of the following is not one of those similarities? a. Any project can be broken down into component activities that require different amounts of time. b. Any project can be broken down into component activities that must be accomplished in a specific order. c. Both techniques use the same method to calculate the time element of the activities in the project. d. Both require a detailed network of the project that clearly indicates each of the main activities and the precedence relationship among the activities. 22. The term "time zero" identifies a. the start time for each activity. b. the start time for the project. c. midnight on each work day. d. days with not wasted effort. 23. Which activities are critical in the following diagram?
a. A, C, D, F b. B, D, E c. A, C, D, E d. A, B, D, E, F 24. A company wants to use PERT to manage a project. They have developed the following spreadsheet for the problem. If The Analytic Solver Platform (ASP) is used, what formula should go in cell F3 of the spreadsheet to compute the Activity Time when simulating a PERT analysis?
1 2 3
A Time Estimates Act A
B Random a 8
C Activity m 10
D
E
F
G
H
I
J
KL
b 14
Number 0.73
Time 11.46
EST 0
EFT 11.462
LST 0.00
LFT 11.46
SLACK 0.00 **
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ch 15 4 5 6 7 8 9
B C D E F G
2 3 5 1 3 1 MODEL
3 4 8 2 5 2
4 5 12 4 7 3
0.19 0.56 0.94 0.35 0.09 0.59
2.62 4.06 10.72 2.03 3.87 2.09
11.462 11.462 15.521 26.24 26.24 30.109
14.081 15.521 26.24 28.272 30.109 32.202
12.90 11.46 15.52 28.08 26.24 30.11
15.52 15.52 26.24 30.11 30.11 32.20
1.44 0.00 ** 0.00 ** 1.84 0.00 ** 0.00 **
a. =IF(E3≤(C3-E3)/(D3-B3),D3-SQRT((D3-B3)*(D3-C3)*(1-E3)),B3+SQRT((D3-E3)*(C3-B3)*E3)) b. =IF(E3≤(D3-B3)/(C3-E3),B3+SQRT((D3-E3)*(C3-B3)*E3),D3-SQRT((D3-B3)*(D3-C3)*(1-E3))) c. =IF(E3≤(C3-E3)/(D3-B3),B3-SQRT((D3-E3)*(C3-B3)*E3),D3+SQRT((D3-B3)*(D3-C3)*(1-E3))) d. =PsiTriangular(B3,C3,D3) 25. For an activity with duration t, slack can be calculated as a. LS-ES b. LF-t c. LS+t d. LS+ES 26. In CPM, the duration of the project a. is equal to the duration of the critical path b. is determined by managers c. is determined by investors d. is determined by a supervising government agency 27. In project control charts, the horizontal bars are shaded to indicate a. work completed b. work to be completed c. overcapacity d. undercapacity 28. Critical activities are distinguished by the presence of a. slack. b. idle resources. c. wasted time. d. zero slack. 29. What is the latest finish time for activity D in the following diagram?
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ch 15 a. 6 b. 13 c. 14 d. 15 Exhibit 15.1 The following questions employ the AON network and partial spreadsheet below.
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A A ? D D E,F
D Time 4 3 2 7 1 2 3
E EST 0 4 4 ? 14 14 16
F EFT 4 7 6 ? 15 16 19
G LST 0 4 5 ? 15 14 16
H LFT 4 7 7 ? 16 16 19
I Slack
30. Refer to Exhibit 15.1. What values are placed in cell C5 of the spreadsheet? a. E, F b. B c. C d. B, C 31. Non-critical activities are distinguished by the presence of a. positive slack. b. idle resources. c. wasted time. d. zero slack. Exhibit 15.2 The following questions employ the AON network and completed spreadsheet below.
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ch 15
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A A B,C D D E,F
D Time 4 3 2 7 1 2 3
E EST 0 4 4 7 14 14 16
F EFT 4 7 6 14 15 16 19
G LST 0 4 5 7 15 14 16
H LFT 4 7 7 14 16 16 19
I Slack 0 0 1 0 1 0 0
32. Refer to Exhibit 15.2. What formula is placed in cell G5 to calculate the Latest Start Time? a. H5 − G5 b. F5 − G5 c. H5 − D5 d. F5 − D5 33. Which of the following correctly describes the focus of the Critical Path Method (CPM)? a. To order activities in a project in terms of their completion time to facilitate scheduling of the activities. b. To determine when a project should be completed and to schedule when each activity in the project must begin in order to keep the project on schedule. c. To estimate the probability of completing a project by a given deadline when the time required to perform the activity is essentially a random variable. d. To structure the activities of a project in order to eliminate or reduce critical dependencies among the activities. Exhibit 15.2 The following questions employ the AON network and completed spreadsheet below.
A
B
C
D
E
F
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G
H
I Page 7
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ch 15 1 2 3 4 5 6 7 8
Activity A B C D E F G
Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
Predecessor A A B,C D D E,F
Time 4 3 2 7 1 2 3
EST 0 4 4 7 14 14 16
EFT 4 7 6 14 15 16 19
LST 0 4 5 7 15 14 16
LFT 4 7 7 14 16 16 19
Slack 0 0 1 0 1 0 0
34. Refer to Exhibit 15.2. Which of the following activities is not on the critical path for the network? a. Activity A b. Activity C c. Activity D d. Activity G 35. What are the immediate predecessors of activity D in this network?
a. A, B, C b. B, C c. E, F d. E, G, H 36. CPM and PERT differ because of a. activity time estimates b. cost calculations c. precedence relationship d. network diagram 37. An activity a. is an effort required to complete part of the project b. is a milestone c. consumes no resources d. describes precedence relationship 38. The main difference between CPM and PERT is that a. CPM is deterministic and PERT is probabilistic b. CPM is probabilisttic and PERT is deterministic c. CPM is best for longer projects while PERT is best for shorter projects Copyright Cengage Learning. Powered by Cognero.
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ch 15 d. CPM requires 3 duration estimates for each activity while PERT requires only one duration estimate for each activity 39. The critical path is the ____ path throughout the network. a. longest b. most expensive c. shortest d. least expensive 40. What formula should go in cell H3 of the following spreadsheet to compute Crash Cost Per Day?
1 2 3 4 5 6 7 8 9
A
B
Activity A B C D E F G COSTS
Description − − − − − − −
C Normal Time 4 3 2 7 2 2 3
D Cost 12000 6000 18000 14000 3500 6000 9000
E Crash Time 2 2 1 4 1 1 1
F Cost 14000 6500 19000 20000 4000 7500 12000
G Allowable Crash Days 2 1 1 3 1 1 2
H Crash Cost Per Day 1000 500 1000 2000 500 1500 1500
a. =(C3-E3)/(F3-D3) b. =(F3-D3)/(C3-E3) c. =(F3)/(C3-E3) d. =(F3-D3)/(E3) 41. The longest path through a network is comprised of the ____ activities. a. longest b. essential c. mandatory d. critical 42. The purpose of the forward pass in the Critical Path Method (CPM) technique is to a. review each of the precedence relationships in the activity network. b. calculate the slack time within each node on the activity network. c. determine the earliest time each activity can start and finish. d. determine the latest time each activity can start and finish. 43. If the following project must be completed by day 18 at minimum cost how do we change the LP model?
A 1 2 3
B
Nodes
C D LP Model For Project Crashing Normal
Start
E
F
Amount
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G
Arcs
H
I
Actual Time
J Minimum Time Page 9
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ch 15 4 5 6 7 8 9 10 11 12 13 14
Activity A B C D E F G
Time 4 3 2 7 1 2 3
Time 0 4 4 6 14 13 15
Crashed 0 1 0 0 0 0 0
Total Crash Cost: Finish Time: MODEL
From A A B C D D E F
To C B D D E F G G
Between Starts Between Starts
4 4 2 2 8 7 1 2
4 4 2 2 7 7 1 2
500 18
a. Enter 18 in cell E14. b. Add a constraint that E14 ≤ 18. c. Make cell E14 the target cell. d. Add a constraint that E14 = 18. 44. What is the formula for the earliest finish time for activity i? a. EFTi = ESTi + ti b. EFTi = EFTj + tj c. EFTi = EFTi−1 + ti d. EFTi = ESTi − ti 45. Which of the following correctly describes the focus of the Program Evaluation and Review Technique (PERT)? a. To order activities in a project in terms of their completion time to facilitate scheduling of the activities. b. To determine when a project should be completed and to schedule when each activity in the project must begin in order to keep the project on schedule. c. To estimate the probability of completing a project by a given deadline when the time required to perform the activity is essentially a random variable. d. To structure the activities of a project in order to eliminate or reduce critical dependencies among the activities. 46. The purpose of the backward pass in the Critical Path Method (CPM) technique is to a. review each of the precedence relationships in the activity network. b. calculate the slack time within each node on the activity network. c. determine the earliest time each activity can start and finish. d. determine the latest time each activity can start and finish. 47. What formula should go in cell I5 of the following spreadsheet to compute Actual Time Between Starts?
A 1 2 3
B
Nodes
C D LP Model For Project Crashing Normal
Start
E
F
Amount
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G
Arcs
H
I
J
Actual Time
Minimum Time Page 10
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ch 15 4 5 6 7 8 9 10 11 12 13 14
Activity A B C D E F G
Time 4 3 2 7 1 2 3
Time 0 4 4 6 14 13 15
Crashed 0 1 0 0 0 0 0
Total Crash Cost: Finish Time:
500 18
From A A B C D D E F
To C B D D E F G G
Between Starts Between Starts
4 4 2 2 8 7 1 2
4 4 2 2 7 7 1 2
MODEL
a. =VLOOKUP(H5,$B$5:$D$11,3)-VLOOKUP(G5,$B$5:$D$11,3) b. =VLOOKUP(H5,$B$5:$D$11,2)-VLOOKUP(G5,$B$5:$D$11,2) c. =VLOOKUP(G5,$B$5:$D$11,3)-VLOOKUP(H5,$B$5:$D$11,3) d. =VLOOKUP(H5,$B$5:$D$11,3)-VLOOKUP(G5,$B$5:$D$11,2) Exhibit 15.1 The following questions employ the AON network and partial spreadsheet below.
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A A ? D D E,F
D Time 4 3 2 7 1 2 3
E EST 0 4 4 ? 14 14 16
F EFT 4 7 6 ? 15 16 19
G LST 0 4 5 ? 15 14 16
H LFT 4 7 7 ? 16 16 19
I Slack
48. Refer to Exhibit 15.1. What array formula is placed in cell E5 to calculate the Earliest Start Time? a. E2 + D2 b. MAX(IF(ISERR(FIND($A$5:$A$17,C5)),0,$F$5:$F$17)) c. H2 − D2 d. MIN(IF(ISNUMBER(FIND(A5,$C$2:$C$8)),$G$2:$G$8)) 49. Slack represents the amount of time by which an activity can be delayed without delaying the entire project, assuming that Copyright Cengage Learning. Powered by Cognero.
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ch 15 a. all successor activities start at their earliest start times. b. all predecessor activities start at their earliest start times. c. immediate predecessor activities start at their earliest start times. d. all predecessor activities start at their latest start times. 50. Which of the following is true regarding projects? a. Projects can have a unique start activity and a unique finish activity. b. Projects can have multiple start activities and a unique finish activity. c. Projects can have multiple start and finish activities. d. All of these are true regarding projects. 51. What is the earliest start time for activity D in the following diagram?
a. 3 b. 4 c. 6 d. 7 52. What is the correct formula to compute the expected duration of an activity using PERT? a. b. c. d. 53. A project plan is established a. before project activities begin and is modified as time goes on and conditions change b. before project activities begin c. before project organization d. dynamically, as conditions change 54. Shortening the activity completion time is called a. expediting b. speeding c. accelerating d. crashing Copyright Cengage Learning. Powered by Cognero.
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ch 15 Exhibit 15.1 The following questions employ the AON network and partial spreadsheet below.
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A A ? D D E,F
D Time 4 3 2 7 1 2 3
E EST 0 4 4 ? 14 14 16
F EFT 4 7 6 ? 15 16 19
G LST 0 4 5 ? 15 14 16
H LFT 4 7 7 ? 16 16 19
I Slack
55. Refer to Exhibit 15.1. What array formula is placed in cell H5 to calculate the Latest Finish Time? a. E2 + D2 b. MAX(IF(ISNUMBER(FIND($A$2:$A$8,C5)),$F$2:$F$8)) c. H2 − D2 d. MIN(IF(ISERR(FIND(A5,$C$5:$C$17)),MAX($F$5:$F$17),$G$5:$G$17)) 56. A strength of Gantt charts is that they a. clearly portray precedence relationships. b. model uncertainty in activity duration times. c. provide a timeline-based, graphical view of when activities can begin and end. d. clearly identifies the critical path. 57. Which activities have slack in the following diagram?
a. A, C, D, F Copyright Cengage Learning. Powered by Cognero.
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ch 15 b. B, E c. A, C, D, E d. B, D, E 58. In deciding which activities to crash, select a. activities on the critical path(s) b. all activities with positive slack time c. longest activities d. lowest cost activities 59. One of PERT's bold assumptions is that a. individual activity times are independent of each other. b. individual activity times are dependent of each other. c. individual activity times are constant. d. individual activity times can be added together. 60. The critical path in PERT analysis is the path with the a. smallest variance. b. longest expected completion time. c. smallest number of tasks. d. smallest variance and longest expected completion time. 61. A way to crash an activity is to a. spend extra money b. subcontract c. use overtime d. all of the above
Exhibit 15.3 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Activity A B C D E F G
Days Required 5 3 4 7 6 5 4
Predecessor Activities − A A B, C C E D, F
62. Refer to Exhibit 15.3. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. What formulas should go into cells E2:E8 to calculate Earliest Start Times? Copyright Cengage Learning. Powered by Cognero.
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ch 15
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A A B,C C E D, F
D Time 5 3 4 7 6 5 4
E EST 0 5 5 9 9 15 20
F EFT 5 8 9 16 15 20 24
G LST 0 10 5 13 9 15 20
H LFT 5 13 9 20 15 20 24
I Slack 0 5 0 4 0 0 0
Exhibit 15.4 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Activity A B C D E F G
Days Required 1 4 3 3 2 4 2
Predecessor Activities − A B C B E D, F
63. Refer to Exhibit 15.4. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. What formulas should go into cells E2:E8 to calculate Earliest Start Times?
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A B C B E D, F
D Time 1 4 3 3 2 4 2
E EST 0 1 5 8 5 7 11
F EFT 1 5 8 11 7 11 13
G LST 0 1 5 8 5 7 11
H LFT 1 5 8 11 7 11 13
I Slack 0 0 0 0 0 0 0
64. Refer to Exhibit 15.4. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. Based on the data in the spreadsheet, which activities are on the critical path?
1 2 3 4 5
A Activity A B C D
B Description Activity A Activity B Activity C Activity D
C Predecessor A B C
D Time 1 4 3 3
E EST 0 1 5 8
F EFT 1 5 8 11
Copyright Cengage Learning. Powered by Cognero.
G LST 0 1 5 8
H LFT 1 5 8 11
I Slack 0 0 0 0 Page 15
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ch 15 6 7 8
E F G
Activity E Activity F Activity G
B E D, F
2 4 2
5 7 11
7 11 13
5 7 11
7 11 13
0 0 0
Exhibit 15.5 The following questions use the information below. A company wants to crash a project and reduce the time required to complete it. They have developed the following table of activity costs and times and a spreadsheet to compute starting time for each activity.
1 2 3 4 5 6 7 8 9
A
B
Activity A B C D E F G Costs
Description − − − − − − −
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B
C Normal Time 5 3 4 7 6 5 4
D Cost 6000 18000 14000 6000 9000 12000 3500
C D LP Model For Project Crashing
Nodes Activity A B C D E F G
Normal Time 5 3 4 7 6 5 4
E Crash Time 3 2 2 4 1 3 2
Start Time 0 3 3 7 11 11 16
Total Crash Cost: Finish Time: Model
E
F Cost 8000 19000 19000 9000 14000 14000 7500
F
Amount Crashed 2 0 0 3 1 0 2
G Allowable Crash Days 2 1 2 3 5 2 2
H Crash Cost Per Day 1000 1000 2500 1000 1000 1000 2000
G
H
Arcs From A A B C D D E F
To C B D D E F G G
I
J
Actual Time
Minimum Time
Between Starts Between Starts
3 3 4 4 4 4 5 5
3 3 3 4 4 4 5 5
1000 18
65. Refer to Exhibit 15.5. What formula should go in cell I5 of the Model spreadsheet to compute Actual Time Between Starts? Exhibit 15.3 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Days
Predecessor
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ch 15 Activity A B C D E F G
Required 5 3 4 7 6 5 4
Activities − A A B, C C E D, F
66. Refer to Exhibit 15.3. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. What formulas should go into cells F2:F8 to calculate Earliest Finish Times, G2:G8 to calculate Latest Start Times, and cells I2:I8 to calculate the Slack? A Activity A B C D E F G
1 2 3 4 5 6 7 8
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor A A B,C C E D, F
D Time 5 3 4 7 6 5 4
E EST 0 5 5 9 9 15 20
F EFT 5 8 9 16 15 20 24
G LST 0 10 5 13 9 15 20
H LFT 5 13 9 20 15 20 24
I Slack 0 5 0 4 0 0 0
67. Refer to Exhibit 15.3. Identify each path through the network and its expected length. 68. Refer to Exhibit 15.3. Manually determine the earliest and latest start and finish times for each activity and the amount of slack for each activity. What is the critical path for the project and how long should it take to complete it? Exhibit 15.4 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Activity A B C D E F G
Days Required 1 4 3 3 2 4 2
Predecessor Activities − A B C B E D, F
69. Refer to Exhibit 15.4. Identify each path through the network and its expected length. 70. Refer to Exhibit 15.4. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. What formulas should go into cells F2:F8 to calculate Earliest Finish Times, G2:G8 to calculate Latest Start Times, and cells I2:I8 to calculate the Slack?
1
A Activity
B Description
C Predecessor
D Time
E EST
F EFT
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H LFT
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ch 15 2 3 4 5 6 7 8
A B C D E F G
Activity A Activity B Activity C Activity D Activity E Activity F Activity G
1 4 3 3 2 4 2
A B C B E D, F
0 1 5 8 5 7 11
1 5 8 11 7 11 13
0 1 5 8 5 7 11
1 5 8 11 7 11 13
0 0 0 0 0 0 0
Exhibit 15.3 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Days Required 5 3 4 7 6 5 4
Activity A B C D E F G
Predecessor Activities − A A B, C C E D, F
71. Refer to Exhibit 15.3. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. Based on the data in the spreadsheet, which activities are on the critical path?
1 2 3 4 5 6 7 8
A Activity A B C D E F G
B Description Activity A Activity B Activity C Activity D Activity E Activity F Activity G
C Predecessor
D Time 5 3 4 7 6 5 4
A A B,C C E D, F
E EST 0 5 5 9 9 15 20
F EFT 5 8 9 16 15 20 24
G LST 0 10 5 13 9 15 20
H LFT 5 13 9 20 15 20 24
I Slack 0 5 0 4 0 0 0
Exhibit 15.6 The following questions use the information below. Consider the CPM network for the following set of activities: Activity A B C D E F G H
Predecessor Activities --A A,B A,B A,B,C A,B,C A,B,C,D A,B,D
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ch 15 I
A,B,C,D,F,G
72. Refer to Exhibit 15.6. Is it necessary to draw an arc from the node for activity A to the node for activity C? Exhibit 15.3 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Days Required 5 3 4 7 6 5 4
Activity A B C D E F G
Predecessor Activities − A A B, C C E D, F
73. Refer to Exhibit 15.3. Draw the CPM network for this problem. Use AON notation. Exhibit 15.4 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Days Required 1 4 3 3 2 4 2
Activity A B C D E F G
Predecessor Activities − A B C B E D, F
74. Refer to Exhibit 15.4. Manually determine the earliest and latest start and finish times for each activity and the amount of slack for each activity. What is the critical path for the project and how long should it take to complete it? Exhibit 15.5 The following questions use the information below. A company wants to crash a project and reduce the time required to complete it. They have developed the following table of activity costs and times and a spreadsheet to compute starting time for each activity.
1 2 3
A
B
Activity A
Description −
C Normal Time 5
D Cost 6000
E Crash Time 3
F Cost 8000
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G Allowable Crash Days 2
H Crash Cost Per Day 1000 Page 19
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ch 15 4 5 6 7 8 9
− − − − − −
B C D E F G Costs
A 1 2 3 4 5 6 7 8 9 10 11 12 13 14
B
3 4 7 6 5 4
18000 14000 6000 9000 12000 3500
C D LP Model For Project Crashing
Nodes Activity A B C D E F G
Normal Time 5 3 4 7 6 5 4
Start Time 0 3 3 7 11 11 16
2 2 4 1 3 2
E
19000 19000 9000 14000 14000 7500
F
G
Amount Crashed 2 0 0 3 1 0 2
Total Crash Cost: Finish Time: Model
1 2 3 5 2 2
1000 2500 1000 1000 1000 2000
H
Arcs From A A B C D D E F
To C B D D E F G G
I
J
Actual Time
Minimum Time
Between Starts Between Starts
3 3 4 4 4 4 5 5
3 3 3 4 4 4 5 5
1000 18
75. Refer to Exhibit 15.5. If the following project must be completed by day 18 at minimum cost how do we change the LP model? Exhibit 15.3 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Activity A B C D E F G
Days Required 5 3 4 7 6 5 4
Predecessor Activities − A A B, C C E D, F
76. Refer to Exhibit 15.3. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. What formulas should go into cells H2:H8 to calculate Latest Finish Times?
1 2
A Activity A
B Description Activity A
C Predecessor
D Time 5
E EST 0
F EFT 5
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ch 15 3 4 5 6 7 8
B C D E F G
Activity B Activity C Activity D Activity E Activity F Activity G
A A B,C C E D, F
3 4 7 6 5 4
5 5 9 9 15 20
8 9 16 15 20 24
10 5 13 9 15 20
13 9 20 15 20 24
5 0 4 0 0 0
Exhibit 15.4 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Days Required 1 4 3 3 2 4 2
Activity A B C D E F G
Predecessor Activities − A B C B E D, F
77. Refer to Exhibit 15.4. Draw the CPM network for this problem. Use AON notation. Exhibit 15.5 The following questions use the information below. A company wants to crash a project and reduce the time required to complete it. They have developed the following table of activity costs and times and a spreadsheet to compute starting time for each activity.
1 2 3 4 5 6 7 8 9
A 1 2 3 4 5
A
B
Activity A B C D E F G Costs
Description − − − − − − −
B
Nodes Activity A
C Normal Time 5 3 4 7 6 5 4
D Cost 6000 18000 14000 6000 9000 12000 3500
C D LP Model For Project Crashing Normal Time 5
Start Time 0
E Crash Time 3 2 2 4 1 3 2
E
F Cost 8000 19000 19000 9000 14000 14000 7500
F
Amount Crashed 2
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G Allowable Crash Days 2 1 2 3 5 2 2
H Crash Cost Per Day 1000 1000 2500 1000 1000 1000 2000
G
H
Arcs From A
To C
I
J
Actual Time
Minimum Time
Between Starts Between Starts
3
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ch 15 6 7 8 9 10 11 12 13 14
B C D E F G
3 4 7 6 5 4
3 3 7 11 11 16 Total Crash Cost: Finish Time: Model
0 0 3 1 0 2
A B C D D E F
B D D E F G G
3 4 4 4 4 5 5
3 3 4 4 4 5 5
1000 18
78. Refer to Exhibit 15.5. What formula should go in cell H3 of the Costs spreadsheet to compute Crash Cost Per Day? Exhibit 15.6 The following questions use the information below. Consider the CPM network for the following set of activities: Activity A B C D E F G H I
Predecessor Activities --A A,B A,B A,B,C A,B,C A,B,C,D A,B,D A,B,C,D,F,G
79. Refer to Exhibit 15.6. Is it necessary to draw an arc from the node for activity F to the node for activity I? 80. A company wants to use PERT to manage a project. They have developed the following spreadsheet for the problem. What formula should go in cell F3 of the spreadsheet to compute the Activity Time when simulating a PERT analysis?
1 2 3 4 5 6 7 8 9
A Time Estimates Act A B C D E F G
B Random a 4 4 5 3 2 6 2 MODEL
C Activity m 5 6 8 4 3 7 4
D
E
F
G
H
I
J
K
b 7 8 12 5 5 8 6
Number 0.756 0.284 0.855 0.726 0.043 0.183 0.941
Time 5.79 5.51 9.98 4.26 2.36 6.60 5.31
EST 0.00 5.79 5.79 15.77 20.03 20.03 26.64
EFT 5.79 11.30 15.77 20.03 22.39 26.64 31.95
LST 0.06 10.33 5.85 15.84 24.34 20.10 26.70
LFT 5.85 15.84 15.84 20.10 26.70 26.70 32.01
SLACK 0 5 0 0 4 0 0
L
** ** ** ** **
Exhibit 15.6 The following questions use the information below. Copyright Cengage Learning. Powered by Cognero.
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ch 15 Consider the CPM network for the following set of activities: Activity A B C D E F G H I
Predecessor Activities --A A,B A,B A,B,C A,B,C A,B,C,D A,B,D A,B,C,D,F,G
81. Refer to Exhibit 15.6. Modify the table above by completing the following table. For column 3, list only the immediate predecessors for each activity. Activity A B C D E F G H I
Predecessor Activities --A A,B A,B A,B,C A,B,C A,,B,C,D A,B,D A,B,C,D,F,G
Immediate Predecessors --A B B
Exhibit 15.4 The following questions are based on the information below. A company needs to manage a project which consists of the following set of activities: Activity A B C D E F G
Days Required 1 4 3 3 2 4 2
Predecessor Activities − A B C B E D, F
82. Refer to Exhibit 15.4. The following spreadsheet was developed to determine the earliest start times and latest start times for the project. What formulas should go into cells H2:H8 to calculate Latest Finish Times?
1 2 3 4 5
A Activity A B C D
B Description Activity A Activity B Activity C Activity D
C Predecessor A B C
D Time 1 4 3 3
E EST 0 1 5 8
F EFT 1 5 8 11
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H LFT 1 5 8 11
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ch 15 6 7 8
E F G
Activity E Activity F Activity G
B E D, F
2 4 2
5 7 11
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7 11 13
5 7 11
7 11 13
0 0 0
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ch 15 Answer Key 1. False 2. True 3. True 4. True 5. False 6. True 7. False 8. True 9. d 10. a 11. c 12. b 13. b 14. b 15. b 16. a 17. c 18. b 19. c 20. a 21. c 22. b 23. a 24. d 25. a Copyright Cengage Learning. Powered by Cognero.
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ch 15 26. a 27. a 28. d 29. b 30. d 31. a 32. c 33. b 34. b 35. b 36. a 37. a 38. a 39. a 40. b 41. d 42. c 43. b 44. a 45. c 46. d 47. a 48. b 49. b 50. d 51. c Copyright Cengage Learning. Powered by Cognero.
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ch 15 52. d 53. a 54. d 55. d 56. c 57. b 58. a 59. a 60. b 61. d 62. Cell E2
Formula =MAX(IF(ISNUMBER(FIND($A$2:$A$8,C2)),$F$2:$F$8))
Copied to: E3:E8
63. Cell E2
Formula =MAX(IF(ISNUMBER(FIND($A$2:$A$8,C2)),$F$2:$F$8))
Copied to: E3:E8
64. All activities are on the critical path. There are two critical paths. These are A-B-C-D-G and A-B-E-F-G. 65. =VLOOKUP(H5,$B$5:$D$11,3)-VLOOKUP(G5,$B$5:$D$11,3) 66. Cell F2 G2 I2
Formula E2 + D2 H2 − D2 H2 − F2
Copied to: F3:F8 G3:G8 I3:I8
67. A-B-D-G = 19 DAYS A-C-E-F-G = 24 DAYS A-C-D-G = 20 DAYS 68. Activity
Time
EST
EFT
LST
LFT
Slack
A B C D E
5 3 4 7 6
0 5 5 9 9
5 8 9 16 15
0 10 5 13 9
5 13 9 20 15
0 5 0 4 0
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ch 15 F G
5 4
15 20
20 24
15 20
20 24
0 0
Critical path = A-C-E-F-G 69. A-B-C-D-G = 13 DAYS A-B-E-F-G = 13 DAYS 70. Cell F2 G2 I2
Formula E2 + D2 H2 − D2 H2 − F2
Copied to: F3:F8 G3:G8 I3:I8
71. Activities A, C, E, F, and G. 72. No, B is the only immediate predecessor, so an arc from node A to node C is superfluous.
73. 74. Activity
Time
EST
EFT
LST
LFT
Slack
A B C D E F G
1 4 3 3 2 4 2
0 1 5 8 5 7 11
1 5 8 11 7 11 13
0 1 5 8 5 7 11
1 5 8 11 7 11 13
0 0 0 0 0 0 0
Critical paths = A-B-C-D-G and A-B-E-F-G 75. Add a constraint that E14 ≤ 18. 76. Cell H2
Formula =MIN(IF(ISNUMBER(FIND(A2,$C$2:$C$8)),$G$2:$G$8))
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ch 15
77. 78. =(F3-D3)/(C3-E3) or (F3-D3)/G3 79. Yes, activity F is an immediate predecessor of activity I. 80. =IF(E3≤(C3-E3)/(D3-B3),B3+SQRT((D3-E3)*(C3-B3)*E3,D3-SQRT((D3-B3)*(D3-C3)*(1-E3))) 81. Activity A B C D E F G H I
Predecessor Activities --A A,B A,B A,B,C A,B,C A,,B,C,D A,B,D A,B,C,D,F,G
82. Cell H2
Formula =MIN(IF(ISNUMBER(FIND(A2,$C$2:$C$8)),$G$2:$G$8))
Immediate Predecessors --A B B C C C,D D F,G
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