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Unformatted text preview: Sensitivity Analysis: An Applied Approach In this chapter, we discuss how changes in an LPs parameters affect the optimal solution. This is called sensitivity analysis. We also explain how to use the LINDO output to answer ques- tions of managerial interest such as What is the most money a company would be willing to pay for an extra hour of labor? We begin with a graphical explanation of sensitivity analysis. 5.1 A Graphical Introduction to Sensitivity Analysis Sensitivity analysis is concerned with how changes in an LPs parameters affect the op- timal solution. Reconsider the Giapetto problem of Section 3.1: max z 3 x 1 2 x 2 s.t. 2 x 1 x 2 100 (Finishing constraint) s.t. s.t. 2 x 1 x 2 80 (Carpentry constraint) s.t. s.t. x 1 x 2 40 (Demand constraint) s.t. 2 x 1 , x 2 where x 1 number of soldiers produced per week x 2 number of trains produced per week The optimal solution to this problem is z 180, x 1 20, x 2 60 (point B in Figure 1), and it has x 1 , x 2 , and s 3 (the slack variable for the demand constraint) as basic variables. How would changes in the problems objective function coefficients or right-hand sides change this optimal solution? Graphical Analysis of the Effect of a Change in an Objective Function Coefficient If the contribution to profit of a soldier were to increase sufficiently, then it seems rea- sonable that it would be optimal for Giapetto to produce more soldiers (that is, s 3 would become nonbasic). Similarly, if the contribution to profit of a soldier were to decrease suf- ficiently, then it would become optimal for Giapetto to produce only trains ( x 1 would now be nonbasic). We now show how to determine the values of the contribution to profit for soldiers for which the current optimal basis will remain optimal. Let c 1 be the contribution to profit by each soldier. For what values of c 1 does the cur- rent basis remain optimal? Currently, c 1 3, and each isoprofit line has the form 3 x 1 2 x 2 constant, or x 2 3 2 x 1 con 2 stant and each isoprofit line has a slope of 3 2 . From Figure 1, we see that if a change in c 1 causes the isoprofit lines to be flatter than the carpentry constraint, then the optimal so- lution will change from the current optimal solution (point B ) to a new optimal solution (point A ). If the profit for each soldier is c 1 , the slope of each isoprofit line will be c 2 1 . Because the slope of the carpentry constraint is 1, the isoprofit lines will be flatter than the carpentry constraint if c 2 1 1, or c 1 2, and the current basis will no longer be optimal. The new optimal solution will be (0, 80), point A in Figure 1. If the isoprofit lines are steeper than the finishing constraint, then the optimal solution will change from point B to point C. The slope of the finishing constraint is 2. If c 2 1 2, or c 1 4, then the current basis is no longer optimal and point C , (40, 20), will be optimal. In summary, we have shown that (if all other parameters remain unchanged) the current basis remains optimal for 2...
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This note was uploaded on 11/02/2010 for the course ORIE 1101 taught by Professor Trotter during the Fall '10 term at Cornell University (Engineering School).
- Fall '10