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Practice #6
Solutions
Linear Programming Sensitivity Analysis
BUAD311 – Operations Management
Fall 2010
1.
Solved Problem 2 on p49 of the textbook.
2.
Consider the following LP:
Max
3x
1
+ 4x
2
+ 6x
3
+ 10x
4
Subject to:
x
1
+ x
2
+ 3x
4
≤
120
x
1
+ x
2
+ 3x
3
+ x
4
≤
150
x
1
+ 2x
2
+ x
3
+ 2x
4
≤
100
x
1
≥
0, x
2
≥
0, x
3
≥
0, x
4
≥
0
After solving the problem using Excel solver, the following sensitivity repost was generated:
Microsoft Excel 10.0 Sensitivity Report
Adjustable Cells
d
d
Final
Reduced
Objective
Allowable
Allowable
Cell
Nam
e
Value
Cost
Coefficient
Increase
Decrease
$A$1
d
X1
0
2.2
3
2.2
1E+30
$B$1
d
X2
0
6
4
6
1E+30
$C$1
d
X3
40
0
6
24
1
$D$1
d
X4
30
0
10
2
5.499999999
Constraints
d
d
Final
Shadow
Constraint
Allowable
Allowable
Cell
Nam
e
Value
Price
R.H. Side
Increase
Decrease
$E$3
d
90
0
120
1E+30
30
$E$4
d
150
0.4
150
150
50
$E$5
d
100
4.8
100
16.66666667
50
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View Full Document (a) What is the value of optimal objective function?
6 * 40 + 10 * 30 = 540
(b) Suppose the first constraint instead was:
x
1
+ x
2
+ 3x
4
≤
100.
What is the new
optimal objective function value? Explain.
540.
The change (20) is less than the allowable decrease and the shadow price is 0.
(c) Suppose the second constraint instead was:
x
1
+ x
2
+ 3x
3
+ x
4
≤
190.
What is the
new optimal objective function value? Explain.
The increase is 40 and is less than the allowable increase (150). Therefore, we can use the
shadow price 0.4 to calculate the new objective value. It will be
540 + 40 * 0.4 = 556
(d) Suppose the coefficient multiplying x
3
in the objective function was 10 instead of 6.
What is the new optimal objective function value? Why?
The increase is 4 and is less than the allowable increase (24). The optimal decision
variables stay the same. The new objective’s value is 10*40+10*30=700
(e) Suppose the coefficient multiplying x
4
in the objective function was 13 instead of 10.
What is the new optimal objective function value? Why?
The increase is 3 and is more than the allowable increase (2). The optimal decision
variables will change. We need to rerun solver to find the new objective’s value.
3.
Consider the following LP
minimize 2x
1
+ 5x
2
+ 6x
3
+ 12x
4
subject to:
x
1
+ x
2
+ 3x
3
≥ 50
x
1
+ x
2
+ x
3
+ 6x
4
≥ 155
x
1
+ x
2
+ 8x
4
≥ 100
x
1
≥0,
x
2
≥0,
x
3
≥0,
x
4
≥ 0
Using Excel solver, we have the following sensitivity report
Adjustable Cells
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This note was uploaded on 10/05/2010 for the course BUAD 311 at USC.
 '07
 Vaitsos
 Management

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