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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
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