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Chapter 5

Sensitivity Analysis: (敏感度分析 ) An Applied Approach

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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.

x1 = number of soldiers produced each week x2 = number of trains produced each week.

max z = 3x1 + 2x2 (p.49)

s.t. 2 x1 + x2 ≤ 100 (finishing constraint)

x1 + x2 ≤ 80 (carpentry constraint)

x1 ≤ 40 (demand constraint)

x1,x2 ≥ 0 (sign restriction)

p.225

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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

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Max z = 3x1 + 2x2

s.t. 2 x1 + x2 ≤ 100 (finishing)

x1 + x2 ≤ 80 (carpentry)

x1 ≤ 40 (demand for soldiers)

x1 , x2 ≥ 0

BV z x1 x2 s1 s2 s3 RHS Ratio

z 1 -3 -2 0 0 0 0

s1 0 2 1 1 0 0 100 100/2

s2 0 1 1 0 1 0 80 80/1

s3 0 1 0 0 0 1 40 40/1

p.226

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BV z x1 x2 s1 s2 s3 RHS Ratio

z 1 0 -2 0 0 3 120

s1 0 0 1 1 0 -2 20 20/1

s2 0 0 1 0 1 -1 40 40/1

x1 0 1 0 0 0 1 40 none

BV z x1 x2 s1 s2 s3 RHS Ratio

z 1 0 0 2 0 -1 160

x2 0 0 1 1 0 -2 20 none

s2 0 0 0 -1 1 1 20 20/1

x1 0 1 0 0 0 1 40 40/1

BV z x1 x2 s1 s2 s3 RHS Ratio

z 1 0 0 1 1 0 180

x2 0 0 1 -1 2 0 60

s3 0 0 0 -1 1 1 20

x1 0 1 0 1 -1 0 20

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The optimal solution for this LP was z = 180, x1=20, x2= 60 (point B) and it has x1, x2, and s3 (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

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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.

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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 b1 = 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 b1 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.

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In the Giapetto problem to the right, we see that if b1 > 120, x1 will be greater than 40 and will violate the demand constraint. Also, if b1 < 80, x1 will be less than 0 and the nonnegativity constraint for x1 will be violated.

Therefore: 80 ≤b1≤ 120 The current basis remains

optimal for 80 ≤b1≤ 120, but the decision variable values and z-value will change.

Figure 2 (p.229)

b1 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

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BV z x1 x2 s1 s2 s3 RHS Ratio

z 1 0 -2 0 0 3 120

s1 0 0 1 1 0 -2 20 20/1

s2 0 0 1 0 1 -1 40 40/1

x1 0 1 0 0 0 1 40 none

BV z x1 x2 s1 s2 s3 RHS Ratio

z 1 -3 -2 0 0 0 0

s1 0 2 1 1 0 0 100 100/2

s2 0 1 1 0 1 0 80 80/1

s3 0 1 0 0 0 1 40 40/1

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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 x1 = 20 + and x2 = 60 – with z = 3x1 + 2x2 = 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.

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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.