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Senior Lecturer(MATHS)

vahidalli@ucsi.edu.my

allivahid@gmail.com

DR. VAHID ALLI

mailto:vahidalli@ucsi.edu.mymailto:allivahid@gmail.commailto:allivahid@gmail.commailto:vahidalli@ucsi.edu.my

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Chapter 8Linear Programming: Sensitivity Analysis

and Interpretation of Solution

Introduction to Sensitivity Analysis

Graphical Sensitivity Analysis

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Standard Computer Output

Software packages such as The Management Scientist and

Microsoft Excel provide the following LP information: Information about the objective function:

its optimal value

coefficient ranges (ranges of optimality)

Information about the decision variables: their optimal values

their reduced costs

Information about the constraints:

the amount of slack or surplus

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Standard Computer Output

In the previous chapter we discussed:

objective function value

values of the decision variables

reduced costs

slack/surplus

In this chapter we will discuss:

changes in the coefficients of the objective function

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Sensitivity Analysis

Sensitivity analysis (or post-optimality analysis) is

used to determine how the optimal solution isaffected by changes, within specified ranges, in:

the objective function coefficients

Sensitivity analysis is important to the manager who

must operate in a dynamic environment withimprecise estimates of the coefficients.

Sensitivity analysis allows him to ask certain what-ifquestions about the problem.

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Example 1

LP Formulation

Max 5x1 + 7x2

s.t. x1 < 6

2x1 + 3x2 < 19x1 + x2 < 8

x1, x2 > 0

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Example 1

Graphical Solution

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10

2x1 + 3x2 < 19

x2

x1

x1 + x2 < 8Max 5x1 + 7x2

x1 < 6

Optimal:x1 = 5, x2 = 3, z = 46

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Objective Function Coefficients

Let us consider how changes in the objective function

coefficients might affect the optimal solution.

The range of optimality for each coefficient providesthe range of values over which the current solutionwill remain optimal.

Managers should focus on those objective coefficientsthat have a narrow range of optimality andcoefficients near the endpoints of the range.

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Example 1

Changing Slope of Objective Function

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10x1

Feasible

Region

1 2

3

4

5

x2

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Range of Optimality

Graphically, the limits of a range of optimality are

found by changing the slope of the objective functionline within the limits of the slopes of the bindingconstraint lines.

The slope of an objective function line, Max c1x1 +

c2x2, is -c1/c2, and the slope of a constraint, a1x1 + a2x2= b, is -a1/a2.

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Example

Range of Optimality for c1

The slope of the objective function line is -c1/c2.The slope of the first binding constraint, x1 + x2 = 8, is

-1 and the slope of the second binding constraint,

2x1

+ 3x2

= 19, is -2/3.

Find the range of values for c1 (with c2 staying 7)such that the objective function line slope lies betweenthat of the two binding constraints:

-1 < -c1/7 < -2/3

Multiplying through by -7 (and reversing theinequalities):

14/3 < c1 < 7

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Example

Range of Optimality for c2

Find the range of values for c2 ( with c1 staying 5)such that the objective function line slope lies betweenthat of the two binding constraints:

-1 < -5/c2 < -2/3

Multiplying by -1: 1 > 5/c2 > 2/3

Inverting, 1 < c2/5 < 3/2

Multiplying by 5: 5 < c2

< 15/2

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End of Chapter 8