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Lecture 2 Part I Mohamed A. M. A.

ISO Method • Since the objective function is Z = ax + by, draw a dotted line for the equation ax + by = k, where k is any constant.

• Sometimes it is convenient to take k as the LCM of a and b.

• Example : 6x + 5y = 30

ISO Method

Irregular Types of Linear Programming Problems For a multiple optimal solution the cj-zj (or zj-cj) value for a nonbasic variable in the final tableau equals zero.

How to find it ????????

An Infeasible Problem An infeasible problem has an artificial variable in the final simplex tableau.

An Unbounded Problem A pivot row cannot be selected for an unbounded problem.

Degeneracy Tie for the Pivot Row—Degeneracy. •Tie for the Pivot Column is broken arbitrarily.

The s3 row is selected arbitrarily

Note that • A solution to a linear programming problem is said to be degenerate if one or more of the basic variables has a value of zero. • For a two variable problem ,degeneracy can occur only when there are redundant constraints • In three variables problems we could construct four or five constraints all intersect at a common point and none of them are redundant

The primal–dual relationships

The Optimal Simplex Solution for the Primal Model

shadow prices This optimal primal tableau also contains information about the dual.

In the cj-zj row of bravaus Table, the negative values of 20 and 20/3 under the s1 and s2 columns indicate that if one unit of either s1 or s2 were entered into the solution, profit would decrease by $20 or $6.67

shadow price= marginal values The shadow price is the maximum amount that should be paid for one additional unit of a resource.

If a resource is not completely used, i.e., there is slack, its marginal value is zero

Dual

Complementary Slackness • The Theorem of Complementary Slackness is an important result that relates the optimal primal and dual solutions • we assume that the primal is a normal max problem with variables x1, x2, . . . , xn and m≤ constraints. Let s1, s2, . . . , sm be the slack variables for the primal.

• Then the dual is a normal min problem with variables y1, y2, . . . , ym and n constraints. Let e1, e2, . . . , en be the excess variables for the dual.

• Then the Theorem of Complementary Slackness follows:

• it follows that the optimal primal and dual solutions must satisfy

Using Complementary Slackness to Solve LPs If the optimal solution to the primal or dual is known, complementary slackness can sometimes be used to determine the optimal solution to the complementary problem.

For example, suppose we were told that the optimal solution of the primal problem is z =280, x1=2, x2=0, x3= 8, s1=24, s2=0, s3=0. Can we use Theorem 2 to help us find the optimal solution to the dual.

Because s1> 0, (40) tells us that the optimal dual solution must have y1=0. Because x1>0 and x3>0, (43) implies that the optimal dual solution must have e1=0, and e3=0. his means that for the optimal dual solution, the first and third constraints must be binding.

We know that y1=0, so we know that the optimal values of y2 and y3 may be found by solving the first and third dual constraints as equalities

HW#1

Sensitivity Analysis

Sensitivity Analysis • Sensitivity analysis is the analysis of the effect of parameter changes on the optimal solution

Sensitivity Analysis_G M

• the optimal solution point is: x1 = 24 x2 = 8.

Sensitivity Analysis • what if we changed the profit of a bowl, x1, from $40 to $100? How would that affect the solution ??????????????????!!!!! • Answer : the optimal solution point changes from point B to point C.

Sensitivity Analysis_S M • Changes in Objective Function Coefficients The sensitivity range for a cj value is the range of values over which the current optimal solution will remain optimal

The Optimal Simplex Tableau

The Optimal Simplex Tableau with c1 160 +∆

For the solution to remain optimal all values in the cj- zj row must be≤ 0.

The Optimal Simplex Tableau with c2=200+∆

Changes in Constraint Quantity Values The sensitivity range for a qi , value is the range of values over which the right-hand-side values can vary without changing the solution variable mix, including slack variables and the shadow prices

Example