ZAGAZIG UNIVERSITY FACULTY OF COMPUTERS & INFORMATICS DS200 OPERATIONS RESEARCH FIRST SEMESTER SECOND YEAR Midterm EXAM: DEC. 2014
Time allowed: 60 MINUTES ANSWER ANY FOUR OF THE FOLLOWING FIVE QUESTIONS. ** INSTRUCTIONS * Verify that your copy of the exam has all 3 pages * All questions carry equal marks. * A list of useful formulae is given as an appendix. * Calculators are permitted. Q1: a. Consider the following LPP max 2x1 + x2 s.t. x1 + 2x2 ≤ 14 2x1 − x2 ≤ 10 x1 − x2 ≤ 3 x1 , x2 ≥ 0. i. Write the dual problem. ii. Given that ) is an optimal solution to this LPP, use the complementary slackness theorem, to find optimal Solution to the dual problem. b. Define: Quadratic Programming ,Infeasibility, Unboundedness, Alternate optimal, Degenerate basic feasible, Non-degenerate basic feasible, Basic infeasible, Non-basic feasible Solutions with respect to an LP solution. Q2: a. What is sensitivity analysis in LP? Which type of changes in sensitivity analysis affect the: i. feasibility ii. Optimality b. Solve the following nonlinear program: 2 Min w= x12 + 2x2 – 8x1 – 12x2 + 34 2 2 Subject to: x1 + 2x2 = 5 Q3: i. Given the following data and seasonal index:
(a) Compute the seasonal index using only year 1 data. (b) Determine the deseasonalized demand values using year 2 data and year 1's seasonal indices. 1
(c) Determine the trend line on year 2's deseasonalized data. (d) Forecast the sales for the first 3 months of year 3, adjusting for seasonality. ii. Consider the following nonlinear programming problem. 2 2 Maximize Z = 2x1 2x2 4x3 x3 , subject to 2x1 + x2 + x3 ≤ 4 and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0. Use the KKT conditions to derive an optimal solution Q4: Consider the following linear programming problem: Max 5x1 + 6x2 + 4x3 s.t. 3x1 + 4x2+ 2x3 ≤ 120 x1 +2x2 + x3≤ 50 x1 + 2x2 + 3x3 ≥ 30 x1, x2, x3 ≥0 The optimal simplex tableau is:
i. Compute the range of optimality for c1 ii. Find the dual price for the second constraint. iii. Suppose the right-hand side of the first constraint is increased from 120 to 125.Find the new optimal solution and its value. iv. If c1 changed from $5 to $7, how will the optimal solution be affected? Q5: the following sales data are available for 2007-2012.
i. Determine a 4-year weighted moving average forecast for 2013, where weights are W1 = 0.1, W2 =0.2, W3 = .02 and W4 = 0.5. 2
ii. Assume that the forecasted demand for 2011 is 15. Use the above data set and exponential smoothing with α = 0.3 to forecast for 2013. iii. Forecast for 2013 using linear trend line. iv. Determine the forecasted demand for 2013 based on adjusted exponential smoothing with α = 0.2, β = 0.3.(hint: an initial trend adjustment of 0 for 2011)
Appendix *Exponential Smoothing Ft = Ft – 1 + a(At – 1 - Ft – 1) *Exponential Smoothing with Trend Adjustment: FIT = Ft + Tt Ft = a(At - 1) + (1 - a)(Ft - 1 + Tt - 1) Tt = b(Ft - Ft - 1) + (1 - b)Tt - 1 *Linear Regression Equation Ft = y = a + bx where, Xi Yi / Xi 2 ( Xi) 2 , a Yi b Xi b XiYi n n n n
HINT: you may need to find b before you can find a
Good Luck Prof. Naser H. R.
Dr. Mohamed A. M.
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