1. 1 Chapter 5 Sensitivity Analysis: ( 敏感度分析 ) An Applied Approach

2. 2 5.1 A Graphical Introduction to Sensitivity Analysis Sensitivity analysis is concerned with how changes in an LP’s parameters affect the optimal solution. 參數改變 → LP 最佳解 Reconsider the Giapetto problem from Chapter 3. x 1 = number of soldiers produced each week x 2 = number of trains produced each week. max z = 3x 1 + 2x 2 (p.49) s.t. 2 x 1 + x 2 ≤ 100 (finishing constraint) x 1 + x 2 ≤ 80 (carpentry constraint) x 1 ≤ 40 (demand constraint) x 1,x 2 ≥ 0 (sign restriction) p.225

3. 3 The set of points satisfying the Giapetto LP is bounded by the five sided polygon DGFEH. Any point on or in the interior of this polygon (the shade area) is in the feasible region. P.57 Fig. 2

4. 4 Max z = 3x 1 + 2x 2 s.t. 2 x 1 + x 2 ≤ 100 (finishing) x 1 + x 2 ≤ 80 (carpentry) x 1 ≤ 40 (demand for soldiers) x 1, x 2 ≥ 0 BVzx1x1 x2x2 s1s1 s2s2 s3s3 RHSRatio z1-3-20000 s1s1 021100100100/2 s2s2 0110108080/1 s3s3 0100014040/1 p.226

5. 5 BVzx1x1 x2x2 s1s1 s2s2 s3s3 RHSRatio z10-2003120 s1s1 00110-22020/1 s2s2 001014040/1 x1x1 01000140none BVzx1x1 x2x2 s1s1 s2s2 s3s3 RHSRatio z10020160 x2x2 00110-220none s2s2 000112020/1 x1x1 0100014040/1 BVzx1x1 x2x2 s1s1 s2s2 s3s3 RHSRatio z100110180 x2x2 0012060 s3s3 0001120 x1x1 0101020

6. 6 The optimal solution for this LP was z = 180, x 1 =20, x 2 = 60 (point B) 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 problem’s objective function coefficients or right-hand side values change this optimal solution? 目標函數係數或等號右邊值 ( 限制條件 ) 改變 → 是否會改變 最佳值？ Figure 1 (p.228) -2 ≤ slope ≤ -1 最佳值不變

7. 7 Graphical analysis( 圖形分析 ) of the effect of a change in an objective function value for the Giapetto LP shows: By inspection, we see that making the slope of the isoprofit( 類利益 ) line more negative than the finishing constraint (slope = -2) will cause the optimal point to switch from point B to point C. Likewise, making the slope of the isoprofit line less negative than the carpentry constraint (slope = -1) will cause the optimal point to switch from point B to point A. Clearly, the slope of the isoprofit line must be between -2 and -1 for the current basis to remain optimal.

8. 8 A graphical analysis can also be used to determine whether a change in the rhs of a constraint will make the current basis no longer optimal. For example, let b 1 = number of available finishing hours. The current optimal solution (point B) is where the carpentry and finishing constraints are binding. ( 木工與修飾工限制的結合處 ) If the value of b 1 is changed, then as long as where the carpentry and finishing constraints are binding, the optimal solution will still occur where the carpentry and finishing constraints intersect.

9. 9 In the Giapetto problem to the right, we see that if b 1 > 120, x 1 will be greater than 40 and will violate the demand constraint. Also, if b 1 < 80, x 1 will be less than 0 and the nonnegativity constraint for x 1 will be violated. Therefore: 80 ≤b 1 ≤ 120 The current basis remains optimal for 80 ≤b 1 ≤ 120, but the decision variable values and z-value will change. Figure 2 (p.229) b 1 介於 80~120 ， BV 變數不會改變，但 BV 變數的值與最佳值 Z 會改變

10. 10 BVzx1x1 x2x2 s1s1 s2s2 s3s3 RHSRatio z10-2003120 s1s1 00110-22020/1 s2s2 001014040/1 x1x1 01000140none BVzx1x1 x2x2 s1s1 s2s2 s3s3 RHSRatio z1-3-20000 s1s1 021100100100/2 s2s2 0110108080/1 s3s3 0100014040/1

11. 11 It is important to determine how a change in a constraint’s rhs changes the LP’s optimal z-value. The shadow price for the ith constraint of an LP is the amount by which the optimal z-value is improved if the rhs of the ith constraint is increased by one. (p.230) 第 i 個限制式值增加 1 對最佳值 Z 的影響 ( 增加值 ) This definition applies only if the change in the rhs of constraint i leaves the current basis optimal. For the finishing constraint, 100 +  finishing hours are available. The LP’s optimal solution is then x 1 = 20 +  and x 2 = 60 –  with z = 3x 1 + 2x 2 = 3(20 + ) + 2(60 - ) = 180 + . Thus, as long as the current basis remains optimal, a one-unit increase in the number of finishing hours will increase the optimal z-value by $1. So, the shadow price for the first (finishing hours) constraint is $1.

12. 12 Sensitivity analysis is important for several reasons: Values of LP parameters might change. If a parameter changes, sensitivity analysis shows it is unnecessary to solve the problem again. In the Giapetto problem, if the profit contribution of a soldier changes to $3.50, sensitivity analysis shows the current solution remains optimal. Uncertainty about LP parameters. In the Giapetto problem, if the weekly demand for soldiers is at least 20, the optimal solution remains 20 soldiers and 60 trains. Thus, even if demand for soldiers is uncertain, the company can be fairly confident that it is still optimal to produce 20 soldiers and 60 trains.