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Unformatted text preview: Sensitivity Analysis and Duality Two of the most important topics in linear programming are sensitivity analysis and duality. Af- ter studying these important topics, the reader will have an appreciation of the beauty and logic of linear programming and be ready to study advanced linear programming topics such as those discussed in Chapter 10. In Section 6.1, we illustrate the concept of sensitivity analysis through a graphical example. In Section 6.2, we use our knowledge of matrices to develop some important formulas, which are used in Sections 6.3 and 6.4 to develop the mechanics of sensitivity analysis. The re- mainder of the chapter presents the important concept of duality. Duality provides many in- sights into the nature of linear programming, gives us the useful concept of shadow prices, and helps us understand sensitivity analysis. It is a necessary basis for students planning to take advanced topics in linear and nonlinear programming. 6.1 A Graphical Introduction to Sensitivity Analysis Sensitivity analysis is concerned with how changes in an LPs parameters affect the LPs optimal 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. 2 x 1 x 2 80 (Carpentry constraint) 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 would be op- timal for Giapetto to produce more soldiers ( s 3 would become nonbasic). Similarly, if the 6 . 1 A Graphical Introduction to Sensitivity Analysis 263 contribution to profit of a soldier were to decrease sufficiently, it would be optimal for Gi- apetto to produce only trains ( x 1 would now be nonbasic). We now show how to deter- mine the values of the contribution to profit for soldiers for which the current optimal ba- sis 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? At present, c 1 3, and each isoprofit line has the form 3 x 1 2 x 2 constant, or x 2 3 2 x 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 solution will change from the current optimal solution (point B ) to a new op- timal solution (point A ). If the profit for each soldier is c 1 , then the slope of each isoprofit line will be c 2 1 . Because the slope of the carpentry constraint is 1, the isoprofit lines...
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- Fall '10