Chapter 05 - What-If Analysis for Linear Programming
INTRODUCTION TO MANAGEMENT
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STUDIES APPROACH WITH
SPREADSHEETS 5TH EDITION HILLIER
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CHAPTER 5
WHAT-IF ANALYSIS FOR LINEAR PROGRAMMING
Review Questions
5.1-1
The parameters of a linear programming model are the constants (coefficients or right-hand sides)
in the functional constraints and the objective function.
5.1-2
Many of the parameters of a linear programming model are only estimates of quantities that
cannot be determined precisely and thus result in inaccuracies.
5.1-3
What-if analysis reveals how close each of these estimates needs to be to avoid obtaining an
erroneous optimal solution, and therefore pinpoints the sensitive parameters where extra care is
needed to refine their estimates.
5.1-4
No, if the optimal solution will remain the same over a wide range of values for a particular
coefficient, then it may be appropriate to make only a fairly rough estimate for a parameter of a
model.
5-1
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Chapter 05 - What-If Analysis for Linear Programming
5.1-5
Conditions that impact the parameters of a model, such as unit profit, may change over time and
render them inaccurate.
5.1-6
If conditions change, what-if analysis leaves signposts that indicate whether a resulting change in a
parameter of the model changes the optimal solution.
5.1-7
Sensitivity analysis is studying how changes in the parameters of a linear programming model
affect the optimal solution.
5.1-8
What-if analysis provides guidance about what the impact would be of altering policy decisions
that are represented by parameters of a model.
5.2-1
The estimates of the unit profits for the two products are most questionable.
5.2-2
The number of hours of production time that is being made available per week in the three plants
might change after analysis.
5.3-1
The allowable range for a coefficient in the objective function is the range of values over which the
optimal solution for the original model remains optimal.
5.3-2
If the true value for a coefficient in the objective function lies outside its allowable range then the
optimal solution would change and the problem would need to be resolved.
5.3-3
The Objective Coefficient column gives the current value of each coefficient. The Allowable
Increase column and the Allowable Decrease Column give the amount that each coefficient may
differ from these values to remain within the allowable range for which the optimal solution for
the original model remains optimal.
5.4-1
The 100% rule considers the percentage of the allowable change (increase or decrease) for each
coefficient in the objective function.
5.4-2
If the sum of the percentage changes do not exceed 100% then the original optimal solution
definitely will still be optimal.
5-2
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5.4-3
No, exceeding 100% may or may not change the optimal solution depending on the directions of
the changes in the coefficients.
5.5-1
The parameters in the constraints may only be estimates, or, especially for the right-hand-sides,
may well represent managerial policy decisions.
5.5-2
The right-hand sides of the functional constraints may well represent managerial policy decisions
rather than quantities that are largely outside the control of management.
5.5-3
The shadow price for a functional constraint is the rate at which the value of the objective function
can be increased by increasing the right-hand side of the constraint by a small amount.
5.5-4
The shadow price can be found with the spreadsheet by increasing the right-hand side by one, and
then re-solving to determine the increase in the objective function value. It can be found similarly
with a parameter analysis report by creating a report that shows the increase in profit for a unit
increase in the right-hand side. The shadow price is given directly in the sensitivity report.
5.5-5
The shadow price for a functional constraint informs management about how much the total
profit will increase for each extra unit of a resource (right-hand-side of a constraint).
5.5-6
Yes. The shadow price also indicates how much the value of the objective function will decrease if
the right-hand side were to be decreased by 1.
5.5-7
A shadow price of 0 tells a manager that a small change in the right-hand side of the constraint will
not change the objective function value at all.
5.5-8
The allowable range for the right-hand side of a functional constraint is found in the Solver’s
sensitivity report by using the columns labeled “Constraint R.H. Side”, “Allowable increase”, and
“Allowable decrease”.
5.5-9
The allowable ranges for the right-hand sides are of interest to managers because they tell them
how large changes in the right-hand sides can be before the shadow prices are no longer
applicable.
5.6-1
There may be uncertainty about the estimates for a number of the parameters in the functional
constraints. Also, the right-hand sides of the constraints often represent managerial policy
decisions. These decisions are frequently interrelated and so need to be considered
simultaneously.
5.6-2
The spreadsheet can be used to directly determine the impact of several simultaneous changes.
Simply change the paremeters and re-solve.
5.6-3
Using a parameter analysis report, up to two parameter cells can be varied simultaneously.
5.6-4
The right-hand sides of the constraints often represent managerial policy decisions. These
decisions are frequently interrelated and so need to be considered simultaneously.
5-3
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5.6-5
The 100 percent rule basically says that we can safely use the shadow prices to predict the effect
of simultaneous changes in the right-hand sides if the sum of the percentages of the changes does
not exceed 100 percent.
5.6-6
The data needed to apply the 100% rule for simultaneous changes in right-hand sides are given by
the Sensitivity Report (Constraint R.H. Side, Allowable Increase, and Allowable Decrease).
5.6-7
If the sum of the percentage changes does not exceed 100%, the shadow prices definitely will still
be valid.
5.6-8
If the sum of the percentages of allowable changes in the right-hand sides does exceed 100%, then
we cannot be sure if the shadow prices will still be valid.
Problems
5.1
a)
A
1
2
3
4
5
6
7
8
9
Unit Profit
Subassembly A
Subassembly B
Production
B
Toys
$3.00
2
1
C
Subassemblies
-$2.50
Resource Usage
-1
-1
Toys
2,000
D
E
F
Used
3,000
1,000
<=
<=
Available
3,000
1,000
Subassemblies
1,000
Total Profit
$3,500
b)
Unit Profit
for Toys
$2.00
$2.50
$3.00
$3.50
$4.00
Optimal
Production Rates
Toys
Subassemblies
1000
0
1000
0
2000
1000
2000
1000
2000
1000
Total
Profit
$2000
$2500
$3500
$4500
$5500
The estimate of the unit profit for toys can decrease by somewhere between $0 and $0.50
before the optimal solution will change. There is no change in the solution for an increase in
the unit profit for toys (at least for increase up to $1).
5-4
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
c)
Unit Profit
for Subassemblies
-$3.50
-$3.00
-$2.50
-$2.00
-$1.50
Optimal
Production Rates
Toys
Subassemblies
1000
0
1000
0
2000
1000
2000
1000
2000
1000
Total
Profit
$3000
$3000
$3500
$4000
$4500
The estimate of the unit profit for subassemblies can decrease by somewhere between $0 and
$0.50 before the optimal solution will change. There is no change in the solution for an
increase in the unit profit for subassemblies (at least for increases up to $1).
d) Parameter analysis report for change in unit profit for toys (part b):
Parameter analysis report for change in unit profit for subassemblies (part c):
e) The allowable range for the unit profit for toys is $2.50 to $5.00.
The allowable range for the unit profit for subassemblies as (–$3.00) to (–$1.50).
Variable Cells
Cell
Name
$B$9 Production Toys
$C$9 Production Subassemblies
Final
Value
2,000
1,000
Reduced
Cost
0
0
Objective
Coefficient
3
-2.5
Allowable
Increase
2
1
Allowable
Decrease
0.5
0.5
5-5
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
f)
g) So long as the sum of the percentage change of the unit profit for the subassemblies does not
exceed 100% (where the allowable increase and decrease are given in part f), then the
solution will not change.
5.2
a)
A
1
2
3
4
5
6
7
8
9
Unit Profit
B
Activity 1
$2
Resource Usage
1
2
1
3
Resource 1
Resource 2
Solution
C
Activity 2
$5
Activity 1
6
D
E
F
Used
10
12
<=
<=
Available
10
12
Activity 2
2
Total Profit
$22
Variable Cells
Cell
$B$9
$C$9
Name
Solution Activity 1
Solution Activity 2
Final
Value
6
2
Reduced
Cost
0
0
Objective
Coefficient
2
5
Allowable
Increase
0.5
1
Allowable
Decrease
0.33333
1
Name
Resource 1 Used
Resource 2 Used
Final
Value
10
12
Shadow
Price
1
1
Constraint
R.H. Side
10
12
Allowable
Increase
2
3
Allowable
Decrease
2
2
Constraints
Cell
$D$5
$D$6
b) The optimal solution changes to (0, 4) if the unit profit for Activity 1 changes to $1.
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$1
C
Activity 2
$5
Resource Usage
1
2
1
3
Activity 1
0
Activity 2
4
5-6
D
E
F
Used
8
12
<=
<=
Available
10
12
Total Profit
$20
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5-7
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
The optimal solution changes to (10, 0) if the unit profit for Activity 1 changes to $3.
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$3
C
Activity 2
$5
Resource Usage
1
2
1
3
Activity 1
10
D
E
F
Used
10
10
<=
<=
Available
10
12
Activity 2
0
Total Profit
$30
c) The optimal solution changes to (10, 0) if the unit profit for Activity 2 changes to $2.50.
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$2
C
Activity 2
$2.50
Resource Usage
1
2
1
3
Activity 1
10
D
E
F
Used
10
10
<=
<=
Available
10
12
Activity 2
0
Total Profit
$20
The optimal solution changes to (0, 4) if the unit profit for Activity 2 changes to $7.50.
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$2
C
Activity 2
$7.50
Resource Usage
1
2
1
3
Activity 1
0
Activity 2
4
D
E
F
Used
8
12
<=
<=
Available
10
12
Total Profit
$30
5-8
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
d)
The allowable range for the unit profit of activity 1 is approximately between $1.60 and $1.80
up to between $2.40 and $2.60.
The allowable range for the unit profit of activity 2 is between $3.50 and $4.00 up to between
$5.50 and $6.00.
e) The allowable range for the unit profit of activity 1 is approximately between $1.67 and $2.50.
The allowable range for the unit profit of activity 2 is between $4 and $6.
f)
The allowable range for the unit profit of activity 1 is approximately between $1.67 and $2.50.
The allowable range for the unit profit of activity 2 is between $4 and $6.
5-9
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
g)
5.3
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
B
C
D
E
F
G
H
Total
Shipped
Out
12
15
=
=
Output
12
15
Big M Company Distribution Problem
Shipping Cost
(per Lathe)
Factory 1
Factory 2
Units Shipped
Factory 1
Factory 2
Total To Customer
Order Size
Customer 1
$700
$800
Customer 1
10
0
10
=
10
Customer 2
$900
$900
Customer 2
2
6
8
=
8
Customer 3
$800
$700
Customer 3
0
9
9
=
9
Total Cost
$20,500
Variable Cells
Cell
$C$11
$D$11
$E$11
$C$12
$D$12
$E$12
Name
Factory 1 Customer 1
Factory 1 Customer 2
Factory 1 Customer 3
Factory 2 Customer 1
Factory 2 Customer 2
Factory 2 Customer 3
Final
Value
10
2
0
0
6
9
Reduced
Cost
0
0
100
100
0
0
Objective
Coefficient
700
900
800
800
900
700
Allowable
Increase
100
100
1E+30
1E+30
100
100
Allowable
Decrease
1E+30
100
100
100
100
1E+30
Final
Value
12
15
10
8
9
Shadow
Price
0
0
700
900
700
Constraint
R.H. Side
12
15
10
8
9
Allowable
Increase
0
2
0
0
0
Allowable
Decrease
1E+30
0
10
2
2
Constraints
Cell
$F$11
$F$12
$C$13
$D$13
$E$13
Name
Factory 1 Out
Factory 2 Out
Total To Customer Customer 1
Total To Customer Customer 2
Total To Customer Customer 3
a) All of the unit costs have a margin of error of 100 in at least one direction (increase or
decrease). Factory 1 to Customer 2 and Factory 2 to Customer 2 have the smallest margins for
error since it is 100 in both directions.
5-10
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
b) The allowable range for Factory 1 to Customer 1 is Unit Cost≤ $800.
The allowable range for Factory 1 to Customer 2 is $800 ≤ Unit Cost ≤ $1,000.
The allowable range for Factory 1 to Customer 3 is Unit Cost ≥ $700.
The allowable range for Factory 2 to Customer 1 is Unit Cost ≥ $700
The allowable range for Factory 2 to Customer 2 is $800 ≤ Unit Cost ≤ $900.
The allowable range for Factory 2 to Customer 3 is Unit Cost ≤ $800.
c) The allowable range for each unit shipping cost indicates how much that shipping cost can
change before you would want to change the shipping quantities used in the optimal solution.
d) Use the 100% rule for simultaneous changes in objective function coefficients. If the sum of
the percentage changes does not exceed 100%, the optimal solution definitely will still be
optimal. If the sum does exceed 100%, then we cannot be sure.
5.4
a) Optimal solution does not change.
b) Optimal solution does change to:
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Cost per Shift
Time Period
6am-8am
8am-10am
10am- 12pm
12pm-2pm
2pm-4pm
4pm-6pm
6pm-8pm
8pm-10pm
10pm-12am
12am-6am
Number Working
C
6am-2pm
Shift
$170
1
1
1
1
0
0
0
0
0
0
6am-2pm
Shift
48
D
8am-4pm
Shift
$160
E
Noon-8pm
Shift
$175
F
4pm-midnight
Shift
$170
Shift Works Time Period? (1=yes, 0=no)
0
0
0
1
0
0
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
0
1
0
0
1
0
0
0
G
10pm-6am
Shift
$195
0
0
0
0
0
0
0
0
1
1
8am-4pm
Shift
31
Noon-8pm
Shift
33
4pm-midnight
Shift
49
10pm-6am
Shift
15
D
8am-4pm
Shift
$165
E
Noon-8pm
Shift
$175
F
4pm-midnight
Shift
$170
G
10pm-6am
Shift
$195
H
Total
Working
48
79
79
112
64
82
82
49
64
15
I
J
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
Minimum
Needed
48
79
65
87
64
73
82
43
52
15
Total Cost
$30,150
c) Optimal Solution changes to:
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Cost per Shift
Time Period
6am-8am
8am-10am
10am- 12pm
12pm-2pm
2pm-4pm
4pm-6pm
6pm-8pm
8pm-10pm
10pm-12am
12am-6am
Number Working
C
6am-2pm
Shift
$170
1
1
1
1
0
0
0
0
0
0
6am-2pm
Shift
48
Shift Works Time Period? (1=yes, 0=no)
0
0
0
1
0
0
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
0
1
0
0
1
0
0
0
8am-4pm
Shift
31
Noon-8pm
Shift
33
4pm-midnight
Shift
49
0
0
0
0
0
0
0
0
1
1
10pm-6am
Shift
15
H
Total
Working
48
79
79
112
64
82
82
49
64
15
I
J
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
Minimum
Needed
48
79
65
87
64
73
82
43
52
15
Total Cost
$30,305
d) The optimal solution does not change.
5-11
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5-12
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
e) The optimal solution does not change.
f)
Variable Cells
Cell
$C$21
$D$21
$E$21
$F$21
$G$21
Name
Number Working Shift
Number Working Shift
Number Working Shift
Number Working Shift
Number Working Shift
Final
Value
48
31
39
43
15
Reduced
Cost
0
0
0
0
0
Objective
Coefficient
170
160
175
180
195
Allowable
Increase
1E+30
10
5
1E+30
1E+30
Allowable
Decrease
10
160
175
5
195
Part a) Optimal solution does not change (within allowable increase of $10).
Part b) Optimal solution does change (outside of allowable decrease of $5).
Part c)
Percent of allowable increase for shift 2 is (165 – 160) / 10 = 50%
Percent of allowable decrease for shift 4 is (180 – 170) / 5 = 200%
Sum = 250%, so the optimal solution may or may not change.
Part d)
Percent of allowable decrease for shift 1 is (170 – 166) / 10 = 40%
Percent of allowable increase for shift 2 is (164 – 160) / 10 = 40%
Percent of allowable decrease for shift 3 is (175 – 171) / 175 = 2%
Percent of allowable increase for shift 4 is (184 – 180) / ∞ = 0%
Percent of allowable increase fo shift 5 is (199 – 195) / ∞ = 0%
The sum is 84%, so the optimal solution does not change.
Part e)
Percent of allowable increase for shift 1 is (173.40 – 170) / ∞ = 0%
Percent of allowable increase for shift 2 is (163.20 – 160) / 10 = 32%
Percent of allowable increase for shift 3 is (178.50 – 175) / 5 = 70%
Percent of allowable increase for shift 4 is (183.60 – 180) / ∞ = 0%
Percent of allowable increase for shift 5 is (198.90 – 195) / ∞ = 0%
The sum is 102%, so the optimal solution may or may not change.
5-13
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
g)
5-14
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5-15
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Chapter 05 - What-If Analysis for Linear Programming
5.5
a) The optimal solution changes to
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Net Present Value
($millions)
Now
End of Year 1
End of Year 2
End of Year 3
Participation Share
C
Office
Building
45.2
D
Hotel
70
E
Shopping
Center
50
Cumulative Capital Required ($millions)
40
80
90
100
160
140
190
240
160
200
310
220
Office
Building
13.31%
Hotel
6.12%
Shopping
Center
15.65%
F
Cumulative
Capital
Spent
24.299
45.000
65.000
80
G
H
<=
<=
<=
<=
Cumulative
Capital
Available
25
45
65
80
Total NPV
($millions)
18.12
b) The optimal solution does not change.
c) The optimal solution does not change.
d) The optimal solution does not change.
5-16
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Chapter 05 - What-If Analysis for Linear Programming
e) The optimal solution changes to
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
f)
C
Office
Building
40
Net Present Value
($millions)
Now
End of Year 1
End of Year 2
End of Year 3
D
Hotel
70.2
E
Shopping
Center
49.8
Cumulative
Capital
Spent
20.645
41.290
61.935
80
Cumulative Capital Required ($millions)
40
80
90
100
160
140
190
240
160
200
310
220
Participation Share
Office
Building
0.00%
Hotel
25.81%
F
G
H
<=
<=
<=
<=
Cumulative
Capital
Available
25
45
65
80
Shopping
Center
0.00%
Total NPV
($millions)
18.12
The optimal solution changes to
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Net Present Value
($millions)
Now
End of Year 1
End of Year 2
End of Year 3
Participation Share
C
Office
Building
46
D
Hotel
69
E
Shopping
Center
49
Cumulative Capital Required ($millions)
40
80
90
100
160
140
190
240
160
200
310
220
Office
Building
13.31%
Hotel
6.12%
F
Cumulative
Capital
Spent
24.299
45.000
65.000
80
G
H
<=
<=
<=
<=
Cumulative
Capital
Available
25
45
65
80
Shopping
Center
15.65%
Total NPV
($millions)
18.01
g) The optimal solution does not change.
h)
Variable Cells
Cell
$C$16
$D$16
$E$16
Name
Participation Share Building
Participation Share Hotel
Participation Share Center
Final
Value
0.00%
16.50%
13.11%
Reduced
Cost
-4.85%
0.00%
0.00%
Objective
Coefficient
45
70
50
Allowable
Increase
0.0485
0.4545
0.1389
Allowable
Decrease
1E+30
0.0543
0.3226
Final
Value
25
44.757
60.583
80
Shadow
Price
0.0097
0.0000
0.0000
0.2233
Constraint
R.H. Side
25
45
65
80
Allowable
Increase
0.3049
1E+30
1E+30
0.7812
Allowable
Decrease
4.3548
0.2427
4.4175
18.8889
Constraints
Cell
$F$9
$F$10
$F$11
$F$12
Name
Now Spent
End of Year 1 Spent
End of Year 2 Spent
End of Year 3 Spent
Part a) Optimal solution changes (not within allowable increase of $48,500).
Part b) Optimal solution does not change (within allowable increase of $454,500).
Part c) Optimal solution does not change (within allowable decrease of ∞).
5-17
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Chapter 05 - What-If Analysis for Linear Programming
Part d) Optimal solution does not change (within allowable decrease of $322,600).
Part e)
Percentage of allowable decrease for project 1 = (45 – 40) / ∞ = 0%
Percentage of allowable increase for project 2 = (70.2 – 70) / 0.4545 = 44%
Percentage of allowable decrease for project 3 = (50 – 49.8) / 0.3226 = 62%
Sum = 106%, so the solution may or may not change.
Part f)
Percentage of allowable increase for project 1 = (46 – 45) / 0.0485 = 2,062%
Percentage of allowable decrease for project 2 = (70 – 69) / 0.0543 = 1,842%
Percentage of allowable decrease for project 3 = (50 – 49) / 0.3226 = 310%
Sum = 4,214%, so the solution may or may not change.
Part g)
Percentage of allowable increase for project 1 = (54 – 45) / 0.0485 = 18,557%
Percentage of allowable increase for project 2 = (84 – 70) / 0.4545 = 3,080%
Percentage of allowable increase for project 3 = (60 – 50) / 0.1389 = 7,199%
Sum = 28,836%, so the solution may or may not change.
i)
5-18
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Chapter 05 - What-If Analysis for Linear Programming
5.6
The model Ep(x) is developed to identify a long-term management plan that satisfies the legal
requirements and optimizes PALCO's operations and profitability. The model consists of a linear
program with the objective of maximizing present net worth subject to harvest-flow constraints,
political and environmental constraints. Detailed sensitivity analysis is performed to "determine
the optimal mix of habitat types within each of individual watersheds" [p. 93]. Many instances of
the LP problem are run with varying parameters.
The financial benefits of this study include an increase of over $398 million in present net worth
and of over $29 million in average yearly net revenues. Sustained-yield annual- harvest levels have
increased. The habitat mix is improved in accordance with political and environmental regulations.
A more profitable long-term plan paved the way for improved short- and mid-term plans.
Sensitivity analysis enabled PALCO to improve its knowledge base of the ecosystem and to adjust
its plans quickly when a change in costs or in regulations occurs. Since its decisions are now
justified through a systematic approach, PALCO is able to obtain better terms from banks. The
study did not only affect PALCO and the habitat controlled by PALCO. It has also "shown that the
forest product industries can coexist with wildlife and contribute to their habitats” increased
quality of life for future generations" [p. 105].
5.7
a) The decrease is within the allowable decrease, so the optimal production quantities stay the
same. Total profit will decrease by ($0.30)(300) = $90 to $2440.
b) $0.30 is 0.30/0.65 = 46.2% of the allowable increase for steins.
$0.25 is 0.25/0.37 = 67.5% of the allowable decrease for plates.
46.2% + 67.5% > 100%, so the optimal production quantities may or may not change.
The change in total profit can not be definitively determined since it is not certain whether or
not the production quantities change.
c) 8 hours, or 480 minutes, is within the allowable decrease for molding, so the shadow price is
valid. The change in total profit is therefore ∆Profit = ($0.22)(–480) = –$105.60. The optimal
production quantities will change.
5-19
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
d) The shadow price for finishing ($0.28) is higher than the shadow price for molding ($0.22), so
shifting minutes from molding to finishing would be beneficial, and would add $0.06 to total
profit per minute shifted. This rate will remain valid at least until the 100% rule is violated. If x
is the number of minutes shifted, the 100% rule will be violated when x/600 + x/2400 > 100%,
or when x > 480 minutes.
e) 300. The shadow price is 0 because there is slack in this constraint. The shadow price will
remain 0 so long as there is slack. There will remain slack so long as the right-hand side
decreases no more than 300 minutes.
5.8
a) Optimal solution: produce no chocolate ice cream, 300 gallons of vanilla ice cream, and 75
gallons of banana ice cream. Total profit will be $341.25.
b) The optimal solution will change since $1.00 (an increase of $0.05) is outside the allowable
increase of $0.0214. The profit will go up, but how much can’t be determined without resolving.
c) The optimal solution will not change since $0.92 (a decrease of $0.03) is within the allowable
decrease ($0.05). Total profit will decrease by $2.25 ($0.03 x 75) to $339.
d) The optimal solution will change. Since the change is within the allowable range, we can
calculate the change in profit using the shadow price: ∆Z = (Shadow Price)(∆RHS) = ($1) x (–3)
= –$3. The new profit will be $338.25.
e) This increase is outside of the allowable increase so the total increase in profit with the extra
sugar can not be determined without re-solving. However, we know that the shadow price is
valid for the first increase of 10 pounds of sugar. For just this 10 pounds, the increase in profit
is ∆Z = (Shadow Price)(∆RHS) = ($1.875)(+10) = $18.75, so even just 10 pounds of sugar would
be worth the $15 price for 15 pounds.
f)
5.9
The final value is 180 as shown in the E5 in the spreadsheet. The shadow price is 0 since we
are using less milk than we have available (there is slack in the constraint). The R.H.Side value
is 200 as given in cell G5. The allowable increase is infinity since the shadow price will stay
zero no matter how much we add to the right-hand side (since this would merely add to the
slack). The allowable decrease is 20 since the solution will change (and the shadow price will
change from zero) once the right-hand side drops below 180 (the amount currently being
used).
a) The decrease is within the allowable decrease, so the optimal production quantities stay the
same. Total profit will decrease by ($50)(15) = $750 to $15,450.
b) $60 is 60/120 = 50% of the allowable decrease for tables.
$90 is 90/120 = 75% of the allowable increase for armoires.
50%+75% > 100%, so the optimal production quantities may or may not change.
The change in total profit can not be definitively determined since it is not certain whether or
not the production quantities change.
5-20
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Chapter 05 - What-If Analysis for Linear Programming
5-21
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Chapter 05 - What-If Analysis for Linear Programming
c) 4 hours, or 240 minutes, is within the allowable decrease for assembly, so the shadow price is
valid. The change in total profit is therefore ∆Profit = ($2)(–240) = –$480. The optimal
production quantities will change.
d) The shadow price for finishing ($4.50) is higher than the shadow price for assembly ($2), so
shifting minutes from assembly to finishing would be beneficial, and would add $2.50 to total
profit per minute shifted. This rate will remain valid at least until the 100% rule is violated. If x
is the number of minutes shifted, the 100% rule will be violated when x/600 + x/400 > 100%,
or when x > 240 minutes.
e) The shadow price is 0, and the allowable increase and decrease are 1E+30 (∞) and 400,
respectively. The shadow price is 0 because there is slack in this constraint. The shadow price
will remain 0 so long as there is slack. There will remain slack no matter how much the righthand side is increased (hence the allowable increase of ∞) and so long as the right-hand side
decreases no more than 400 pounds.
5.10
a) Let
G = number of grandfather clocks produced
W = number of wall clocks produced
Maximize Profit = $300G + $200W
subject to
6G + 4W ≤ 40 hours
8G + 4W ≤ 40 hours
3G + 3W ≤ 20 hours
and G ≥ 0, W ≥ 0.
b) 3.33 grandfather clocks and 3.33 wall clocks should be produced per week. If the unit profit for
grandfather clocks is changed from $300 to $375, the optimal solution does not change. If, in
addition, the estimated unit profit for wall clocks changes from $200 to $175, then the optimal
solution does change to 5 grandfather clocks and 0 wall clocks per week.
c)
A
1
2
3
4
5
6
7
8
9
10
11
12
Unit Profit
B
Grandfather
Clock
$300
C
Wall
Clock
$200
Time Required
Assembly (David)
Carving (LaDeana)
Shipping (Lydia)
Production
6
8
3
4
4
3
Grandfather
Clock
3.33
Wall
Clock
3.33
D
Hours
Used
33
40
20
E
F
<=
<=
<=
Hours
Available
40
40
20
Total Profit
$1,667
5-22
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Chapter 05 - What-If Analysis for Linear Programming
d) If the unit profit for grandfather clocks changes to $375, then the solution does not change.
A
1
2
3
4
5
6
7
8
9
10
11
12
Unit Profit
B
Grandfather
Clock
$375
C
Wall
Clock
$200
Time Required
Assembly (David)
Carving (LaDeana)
Shipping (Lydia)
Production
6
8
3
4
4
3
Grandfather
Clock
3.33
Wall
Clock
3.33
D
Hours
Used
33
40
20
E
F
<=
<=
<=
Hours
Available
40
40
20
Total Profit
$1,917
However, if the unit profit for wall clocks changes to $175 as well, then the optimal solution
does change (produce 5 grandfather clocks and 0 wall clocks).
A
1
2
3
4
5
6
7
8
9
10
11
12
Unit Profit
B
Grandfather
Clock
$375
C
Wall
Clock
$175
Time Required
Assembly (David)
Carving (LaDeana)
Shipping (Lydia)
Production
6
8
3
4
4
3
Grandfather
Clock
5
Wall
Clock
0
D
Hours
Used
30
40
15
E
F
<=
<=
<=
Hours
Available
40
40
20
Total Profit
$1,875
5-23
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Chapter 05 - What-If Analysis for Linear Programming
e)
5-24
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Chapter 05 - What-If Analysis for Linear Programming
f)
g) If David increases his hours to 45 per week, the optimal solution does not change.
A
1
2
3
4
5
6
7
8
9
10
11
12
Unit Profit
B
Grandfather
Clock
$300
C
Wall
Clock
$200
Time Required
Assembly (David)
Carving (LaDeana)
Shipping (Lydia)
Production
6
8
3
4
4
3
Grandfather
Clock
3.33
Wall
Clock
3.33
D
Hours
Used
33
40
20
E
F
<=
<=
<=
Hours
Available
45
40
20
Total Profit
$1,667
If LaDeana increases her hours to 45 per week, the optimal solution changes to
A
1
2
3
4
5
6
7
8
9
10
11
12
Unit Profit
B
Grandfather
Clock
$300
C
Wall
Clock
$200
Time Required
Assembly (David)
Carving (LaDeana)
Shipping (Lydia)
Production
6
8
3
4
4
3
Grandfather
Clock
4.58
Wall
Clock
2.08
D
Hours
Used
36
45
20
E
F
<=
<=
<=
Hours
Available
40
45
20
Total Profit
$1,792
5-25
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Chapter 05 - What-If Analysis for Linear Programming
If Lydia increases her hours to 25 per week, the optimal solution changes to
A
1
2
3
4
5
6
7
8
9
10
11
12
Unit Profit
B
Grandfather
Clock
$300
C
Wall
Clock
$200
Time Required
Assembly (David)
Carving (LaDeana)
Shipping (Lydia)
Production
6
8
3
4
4
3
Grandfather
Clock
1.67
Wall
Clock
6.67
D
Hours
Used
37
40
25
E
F
<=
<=
<=
Hours
Available
40
40
25
Total Profit
$1,833
h)
5-26
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Chapter 05 - What-If Analysis for Linear Programming
i)
The allowable range for the unit profit for the grandfather clock is $200 to $400.
The allowable range for the unit profit for the wall clock is $150 to $300.
The allowable range for David’s available hours is 33.33 and above.
The allowable range for LaDeana’s available hours is 26.67 to 53.33 hours.
The allowavle range for Lydia’s available hours is 15 to 30 hours.
Variable Cells
Cell
$B$12
$C$12
Name
Production Clock
Production Clock
Final
Value
3.33
3.33
Reduced
Cost
0.00
0.00
Objective
Coefficient
300
200
Allowable
Increase
100
100
Allowable
Decrease
100
50
Name
Assembly (David) Used
Carving (LaDeana) Used
Shipping (Lydia) Used
Final
Value
33
40
20
Shadow
Price
0
25
33.33
Constraint
R.H. Side
40
40
20
Allowable
Increase
1E+30
13.333
10
Allowable
Decrease
6.667
13.333
5
Constraints
Cell
$D$6
$D$7
$D$8
j)
Lydia should increase her hours slightly since her hours have the highest shadow price.
k) The shadow price for David is zero because all of his available hours are not being used
anyway, so an increase in his hours would not impact total profit.
l)
Yes, this increase (5 hours) is within the allowable increase (10 hours). The increase in total
profit will be ∆Z = (Shadow Price)(∆RHS) = ($33.33)(+5) = $166.65.
m) Percentage of Lydia’s available increase used = (25 – 20)/10 = 50%.
Percentage of David’s allowable decrease used = (40 – 35) / 6.667 = 75%.
The sum is 125%, so by the 100% rule, the shadow prices may or may not be valid and hence
should not be used to determine the effect on total profit.
n) The revised graph is shown below. The optimal solution changes from (3.333,3.333) with a
profit of $1666.70 to (2.5,5), (.833,7.5), and all points on the connecting line segment, with a
profit of $1750.
5-27
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Chapter 05 - What-If Analysis for Linear Programming
5-28
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Chapter 05 - What-If Analysis for Linear Programming
5.11
a)
A
1
2
3
4
5
6
7
8
9
10
11
Unit Profit
Subassembly A
Subassembly B
Production
B
Toys
$3.00
2
1
C
Subassemblies
-$2.50
Resource Usage
-1
-1
Toys
2,000
<=
2,500
Subassemblies
1,000
B
Toys
$3.00
C
Subassemblies
-$2.50
D
E
F
Used
3,000
1,000
<=
<=
Available
3,000
1,000
Total Profit
$3,500.00
b)
A
1
2
3
4
5
6
7
8
9
10
11
Unit Profit
Subassembly A
Subassembly B
Production
2
1
Resource Usage
-1
-1
Toys
2,001
<=
2,500
D
E
F
Used
3,001
1,000
<=
<=
Available
3,001
1,000
Subassemblies
1,001
Total Profit
$3,500.50
The shadow price for subassembly A is $0.50, which is the maximum premium that the
company should be willing to pay.
c)
A
1
2
3
4
5
6
7
8
9
10
11
Unit Profit
Subassembly A
Subassembly B
Production
B
Toys
$3.00
2
1
C
Subassemblies
-$2.50
Resource Usage
-1
-1
Toys
1,999
<=
2,500
Subassemblies
998
D
E
F
Used
3,000
1,001
<=
<=
Available
3,000
1,001
Total Profit
$3,502.00
The shadow price for subassembly B is $2.00, which is the maximum premium that the
company should be willing to pay.
5-29
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Chapter 05 - What-If Analysis for Linear Programming
d)
The shadow price is still valid until the maximum supply of subassembly A is at least 3,500.
5-30
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
e)
The shadow price is still valid until the maximum supply of subassembly B is at least 1,500.
f)
Variable Cells
Cell
Name
$B$9 Production Toys
$C$9 Production Subassemblies
Final
Value
2,000
1,000
Reduced
Cost
0
0
Objective
Coefficient
3
-2.5
Allowable
Increase
2
1
Allowable
Decrease
0.5
0.5
Final
Value
3,000
1,000
Shadow
Price
0.5
2
Constraint
R.H. Side
3000
1000
Allowable
Increase
500
500
Allowable
Decrease
1000
500
Constraints
Cell
Name
$D$5 Subassembly A Used
$D$6 Subassembly B Used
As shown in the sensitivity report, the shadow price is $0.50 for subassembly A is $2.00 for
subassembly B. According to the allowable increase and allowable decrease, the allowable
range for the right-hand side of the subassembly A constraint is 2,000 to 3,500. The allowable
range for the right-hand side of the subassembly B constraint is 500 to 1,500.
5-31
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Chapter 05 - What-If Analysis for Linear Programming
5.12
a) The original model:
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$2
C
Activity 2
$5
Resource Usage
1
2
1
3
Activity 1
6
D
E
F
Used
10
12
<=
<=
Available
10
12
Activity 2
2
Total Profit
$22.00
With 1 additional unit of resource 1:
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$2
C
Activity 2
$5
Resource Usage
1
2
1
3
Activity 1
9
Activity 2
1
D
E
F
Used
11
12
<=
<=
Available
11
12
Total Profit
$23.00
The shadow price is $1 (the increase in total profit).
5-32
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Chapter 05 - What-If Analysis for Linear Programming
b)
The shadow price of $1 is valid in the range of 8 to 12.
c) With 1 additional unit of resource 2:
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$2
C
Activity 2
$5
Resource Usage
1
2
1
3
Activity 1
4
Activity 2
3
D
E
F
Used
10
13
<=
<=
Available
10
13
Total Profit
$23.00
The shadow price is $1 (the increase in total profit).
d)
The shadow price of $1 is valid in the range of 10 to 15.
5-33
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Chapter 05 - What-If Analysis for Linear Programming
5-34
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Chapter 05 - What-If Analysis for Linear Programming
e) As shown in the sensitivity report, the shadow prices for both constraints are $1. According to
the allowable increase and allowable decrease, the allowable range for the right-hand side of
the first constraint is 8 to 12. Similarly, the allowable range for the right-hand side of the
second constraint is 10 to 15.
Variable Cells
Cell
Name
$B$9 Solution Activity 1
$C$9 Solution Activity 2
Final
Value
6
2
Reduced
Cost
0
0
Objective
Coefficient
2
5
Allowable
Increase
0.5
1
Allowable
Decrease
0.333
1
Final
Value
10
12
Shadow
Price
1
1
Constraint
R.H. Side
10
12
Allowable
Increase
2
3
Allowable
Decrease
2
2
Constraints
Cell
Name
$D$5 Resource 1 Used
$D$6 Resource 2 Used
5.13
a) Optimal solution: (x1, x2) = (2, 2) and Profit = $6.
5-35
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
b) When the right-hand-side of the first constraint is increased to 9, the new optimal solution
becomes (x1, x2) = (1.5, 2.5) and Profit = $6.50. Hence, the shadow price for the first constraint
is $6.50 – $6.00 = $0.50.
When the right-hand-side of the second constraint is increased to 5, the new optimal solution
becomes (x1, x2) = (3.5, 1.5) and Profit = $6.50. Hence, the shadow price for the second
constraint is $6.50 – $6.00 = $0.50.
5-36
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
c) Original model:
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$1
C
Activity 2
$2
Resource Usage
1
3
1
1
Activity 1
2
D
E
F
Used
8
4
<=
<=
Available
8
4
Activity 2
2
Total Profit
$6.00
The shadow price for resource 1 is $0.50.
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$1
C
Activity 2
$2
Resource Usage
1
3
1
1
Activity 1
1.5
D
E
F
Used
9
4
<=
<=
Available
9
4
Activity 2
2.5
Total Profit
$6.50
The shadow price for resource 2 is $0.50.
A
1
2
3
4
5
6
7
8
9
Unit Profit
Resource 1
Resource 2
Solution
B
Activity 1
$1
C
Activity 2
$2
Resource Usage
1
3
1
1
Activity 1
3.5
Activity 2
1.5
D
E
F
Used
8
5
<=
<=
Available
8
5
Total Profit
$6.50
5-37
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
d) The allowable range for the right-hand side of the resource 1 constraint is approximately from
4 (or less) to 12.
The allowable range for the right-hand side of the resource 2 constraint is approximately from
3 to 8.
e) The shadow prices for both resources are $0.50.
The allowable range for the right-hand side of the first resource is 4 to 12.
The allowable range for the right-hand side of the second resource is 2.667 to 8.
Variable Cells
Cell
Name
$B$9 Solution Activity 1
$C$9 Solution Activity 2
Final
Value
2
2
Reduced
Cost
0
0
Objective
Coefficient
1
2
Allowable
Increase
1
1
Allowable
Decrease
0.333
1
Final
Value
8
4
Shadow
Price
0.5
0.5
Constraint
R.H. Side
8
4
Allowable
Increase
4
4
Allowable
Decrease
4
1.333
Constraints
Cell
Name
$D$5 Resource 1 Used
$D$6 Resource 2 Used
5-38
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Chapter 05 - What-If Analysis for Linear Programming
5-39
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
f)
5.14
These shadow prices tell management that for each additional unit of the resource, profit will
increase by $.50 (for small changes). Management is then able to evaluate whether or not to
change the amounts of resources being made available.
a) Optimal solution: (x1, x2) = (3, 4) and Profit = $17.
b) When the right-hand-side of the first constraint is increased to 5, the optimal solution remains
the same. Hence, the shadow price for the first constraint is 0.
5-40
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
When the right-hand-side of the second constraint is increased to 16, the new optimal
solution becomes (x1, x2) = (2.8, 4.4) and P =17.2. Hence, the shadow price for the second
constraint is 17.2-17=0.2.
When the right-hand-side of the third constraint is increased to 11, the new optimal solution
becomes (x1, x2) = (3.6, 3.8) and P = 18.4. Hence, the shadow price for the third constraint is
18.4-17=1.4.
5-41
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
c) Original model:
A
1
2
3
4
5
6
7
8
9
10
Unit Profit
Resource 1
Resource 2
Resource 3
Solution
B
Activity 1
$3
C
Activity 2
$2
Resource Usage
1
0
1
3
2
1
Activity 1
3
D
E
F
Used
3
15
10
<=
<=
<=
Available
4
15
10
Activity 2
4
Total Profit
$17.00
The shadow price for resource 1 is $0.
A
1
2
3
4
5
6
7
8
9
10
Unit Profit
Resource 1
Resource 2
Resource 3
Solution
B
Activity 1
$3
C
Activity 2
$2
Resource Usage
1
0
1
3
2
1
Activity 1
3
D
E
F
Used
3
15
10
<=
<=
<=
Available
5
15
10
Activity 2
4
Total Profit
$17.00
The shadow price for resource 2 is $0.20.
A
1
2
3
4
5
6
7
8
9
10
Unit Profit
Resource 1
Resource 2
Resource 3
Solution
B
Activity 1
$3
C
Activity 2
$2
Resource Usage
1
0
1
3
2
1
Activity 1
2.8
Activity 2
4.4
D
E
F
Used
2.8
16
10
<=
<=
<=
Available
4
16
10
Total Profit
$17.20
5-42
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
The shadow price for resource 3 is $1.40.
A
1
2
3
4
5
6
7
8
9
10
Unit Profit
Resource 1
Resource 2
Resource 3
Solution
B
Activity 1
$3
C
Activity 2
$2
Resource Usage
1
0
1
3
2
1
Activity 1
3.6
Activity 2
3.8
D
E
F
Used
3.6
15
11
<=
<=
<=
Available
4
15
11
Total Profit
$18.40
d) The allowable range for the right-hand side of the resource 1 constraint is approximately from
3 to at least 10.
The allowable range for the right-hand side of the resource 2 constraint is approximately from
less than 11 to more than 21.
5-43
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
The allowable range for the right-hand side of the resource 3 constraint is approximately from
less than 6 to more than 11.
e) The shadow prices for the three resources are $0, $0.20, and $1.40, respectively.
The allowable range for the right-hand side of the first resource is 3 to ∞.
The allowable range for the right-hand side of the second resource is 10 to 30.
The allowable range for the right-hand side of the third resource is 5 to 11.667.
Variable Cells
Cell
$B$10
$C$10
Name
Solution Activity 1
Solution Activity 2
Final Reduced Objective Allowable
Value
Cost
Coefficient Increase
3
0
3
1
4
0
2
7
Allowable
Decrease
2.333
0.5
Name
Resource 1 Used
Resource 2 Used
Resource 3 Used
Final
Value
3
15
10
Allowable
Decrease
1
5
5
Constraints
Cell
$D$5
$D$6
$D$7
f)
Shadow
Price
0
0.2
1.4
Constraint Allowable
R.H. Side Increase
4
1E+30
15
15
10
1.667
These shadow prices tell management that for each additional unit of the resource, profit will
increase by $0, or $0.20, or $1.40 for the three resources, respectively (for small changes).
Management is then able to evaluate whether or not to change the amounts of resources
being made available.
5-44
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5.15
B
3
4
5
6
7
8
9
10
11
12
13
14
15
Exposures per Ad
(thousands)
Ad Budget
Planning Budget
Number of Ads
Max TV Spots
C
TV Spots
1,300
D
Magazine Ads
600
E
SS Ads
500
Cost per Ad ($thousands)
300
150
100
90
30
40
TV Spots
0
<=
5
Magazine Ads
20
SS Ads
10
F
Budget
Spent
4,000
1,000
G
H
<=
<=
Budget
Available
4,000
1,000
Total Exposures
(thousands)
17,000
Variable Cells
Cell
$C$13
$D$13
$E$13
Name
TVSpots
Number of Ads Magazine Ads
Number of Ads SS Ads
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
0
-50
1300
50
1E+30
20
0
600
150
50
10
0
500
300
33.333
Name
Ad Budget Spent
Planning Budget Spent
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side Increase Decrease
4,000
3
4000
1000
1500
1,000
5
1000
600
200
Constraints
Cell
$F$8
$F$9
a) The total number of expected exposures could be increased by 3,000 for each additional
$1,000 added to the advertising budget.
b) This remains valid for increases of up to $1,000,000.
c) The total number of expected exposure units could be increased by 5,000 for each additional
$1,000 added to the planning budget.
d) This remains valid for increases of up to $600,000.
e) Percentage of allowable increase for ad budget = (4,100 – 4,000) / 1,000 = 10%
Percentage of allowable increase for planning budget = (1,100 – 1,000) / 600 = 16.7%
The sum is 26.7% ≤ 100%, so the shadow prices are still valid.
f)
The $100,000 should be added to the planning budget since this will add 500,000 expected
exposures rather than 300,000 for the advertising budget.
g) Either shadow price would still be valid (the allowable decreases are $1,500,000 and
$200,000, respectively). The $100,000 should be removed from the advertising budget, since
this will decrease the expected number of exposures by 300,000 rather than the 500,000 for
the planning budget.
5-45
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5.16
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Exposures per Ad
(thousands)
Ad Budget
Planning Budget
Young Children
Parents of Young Children
Coupon Redemption per Ad
($thousands)
Number of Ads
Maximum TV Spots
C
TV Spots
1,300
D
Magazine Ads
600
E
SS Ads
500
F
G
H
300
90
Cost per Ad ($thousands)
150
30
100
40
Budget Spent
3,775
1,000
<=
<=
Budget Available
4,000
1,000
Number Reached per Ad (millions)
1.2
0.1
0
0.5
0.2
0.2
Total Reached
5
5.85
>=
>=
Minimum Acceptable
5
5
Total Redeemed
1,490
=
Required Amount
1,490
TV Spots
0
Magazine Ads
40
SS Ads
120
TV Spots
3
<=
5
Magazine Ads
14
SS Ads
7.75
Total Exposures
(thousands)
16,175
Variable Cells
Cell
$C$19
$D$19
$E$19
TVSpots
Number of Ads Magazine Ads
Number of Ads SS Ads
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
3
0
1300
1040
1E+30
14
0
600
1E+30
192.59
7.75
0
500
577.78
1E+30
Name
Ad Budget Budget Spent
Planning Budget Budget Spent
TotalRedeemed
Young Children Total Reached
Parents of Young Children Total Reached
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side Increase Decrease
3,775
0
4000
1E+30
225
1,000
35
1000
22.5
85
1,490
-8
1490
385
90
5
-1575.76
5
1.32
0.45
5.85
0
5
0.85
1E+30
Name
Constraints
Cell
$F$7
$F$8
$F$15
$F$11
$F$12
a) The total number of expected exposures can not be increased by adding an additional $1,000
to the advertising budget.
b) This remains valid for any increases.
c) The total number of expected exposures can be increased by 35,000 by adding an additional
$1,000 to the advertising budget.
d) This remains valid for increases of up to $22,500.
e) Percentage of allowable increase for ad budget = (4,100 – 4,000) / ∞ = 0%
Percentage of allowable increase for planning budget = (1,100 – 1,000) / 22.5 = 444%
The sum is 444% > 100%, so the shadow prices may or may not be valid.
f)
$100,000 is beyond the allowable increase for the planning budget. Therefore, the total
impact of adding $100,000 to the planning budget can not be determined without re-solving.
However, it would certainly be more worthwhile adding to the planning budget (35,000
additional exposures for each $1,000 spent up to $22,500) than adding to the advertising
budget which would not increase the expected number of exposures at all.
5-46
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5-47
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
g) The $100,000 should be removed from the advertising budget. Since the shadow price is zero
for the advertising budget (and the allowable decrease is $225,000), this will have no impact
on the total number of exposures.
5.17
B
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
C
D
6am-2pm 8am-4pm
Shift
Shift
Cost per Shift
$170
$160
Time Period
6am-8am
8am-10am
10am- 12pm
12pm-2pm
2pm-4pm
4pm-6pm
6pm-8pm
8pm-10pm
10pm-12am
12am-6am
1
1
1
1
0
0
0
0
0
0
E
Noon-8pm
Shift
$175
F
4pm-midnight
Shift
$180
Shift Works Time Period? (1=yes, 0=no)
0
0
0
1
0
0
1
0
0
1
1
0
1
1
0
0
1
1
0
1
1
0
0
1
0
0
1
0
0
0
6am-2pm 8am-4pm
Shift
Shift
Number Working
48
31
Noon-8pm
Shift
39
4pm-midnight
Shift
43
G
10pm-6am
Shift
$195
0
0
0
0
0
0
0
0
1
1
10pm-6am
Shift
15
H
Total
Working
48
79
79
118
70
82
82
43
58
15
I
J
>=
>=
>=
>=
>=
>=
>=
>=
>=
>=
Minimum
Needed
48
79
65
87
64
73
82
43
52
15
Total Cost
$30,610
Variable Cells
Cell
$C$21
$D$21
$E$21
$F$21
$G$21
Name
Number Working Shift
Number Working Shift
Number Working Shift
Number Working Shift
Number Working Shift
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
48
0
170
1E+30
10
31
0
160
10
160
39
0
175
5
175
43
0
180
1E+30
5
15
0
195
1E+30
195
Name
6am-8am Working
8am-10am Working
10am- 12pm Working
12pm-2pm Working
2pm-4pm Working
4pm-6pm Working
6pm-8pm Working
8pm-10pm Working
10pm-12am Working
12am-6am Working
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side Increase Decrease
48
10
48
6
48
79
160
79
1E+30
6
79
0
65
14
1E+30
118
0
87
31
1E+30
70
0
64
6
1E+30
82
0
73
9
1E+30
82
175
82
1E+30
6
43
5
43
6
6
58
0
52
6
1E+30
15
195
15
1E+30
6
Constraints
Cell
$H$8
$H$9
$H$10
$H$11
$H$12
$H$13
$H$14
$H$15
$H$16
$H$17
5-48
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
a) The following shifts can be increased by the indicated amounts with increasing total cost:
Serve 10–12 am
14
Serve 12–2 pm
31
Serve 2–4 pm
6
Serve 4–6 pm
9
Serve 10–12 pm
6
b) For each of the following shifts, total cost will increase by the amount indicated per unit
increase. These costs hold for the indicated increases
Shift
Increased Cost
Valid for this increase
Serve 6–8 am
$10
6
Serve 8–10 am
$160
∞
Serve 6–8 pm
$175
∞
Serve 8–10 pm
$5
6
Serve 12–6 am
$195
∞
c) Percentage of allowable increase for 6am–8am = (49 – 48) / 6 = 16.7%
Percentage of allowable increase for 8am–10am = (80 – 79) / ∞ = 0%
Percentage of allowable increase for 6pm–8pm = (83 – 82) / ∞ = 0%
Percentage of allowable increase for 8pm–10pm = (44 – 43) / 6 = 16.7%
Percentage of allowable increaes for 12am-6am = (16 – 15) / ∞ = 0%
The sum is 33.4% ≤ 100%, so the shadow prices are still valid.
d) Percentage of allowable increase for 6am–8am = (49 – 48) / 6 = 16.7%
Percentage of allowable increase for 8am–10am = (80 – 79) / ∞ = 0%
Percentage of allowable increase for 10am–12pm = (66 – 65) / 14 = 7.1%
Percentage of allowable increase for 12pm–2pm = (88 – 87) / 31 = 3.2%
Percentage of allowable increase for 2pm–4pm = (65 – 64) / 6 = 16.7%
Percentage of allowable increase for 4pm–6pm = (74 – 73) / 9 = 11.1%
Percentage of allowable increase for 6pm–8pm = (83 – 82) / ∞ = 0%
Percentage of allowable increase for 8pm–10pm = (44 – 43) / 6 = 16.7%
Percentage of allowable increase for 10pm–12am = (53 – 52) / 6 = 16.7%
Percentage of allowable increaes for 12am-6am = (16 – 15) / ∞ = 0%
The sum is 88.2% ≤ 100%, so the shadow prices are still valid.
e) All numbers can increase by (100/88.2) or 1.13 hours before it is no longer definite that the
shadow prices remain valid.
5-49
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Chapter 05 - What-If Analysis for Linear Programming
Cases
5.1
a) Original Solution: 4 units of television advertising and 3 units of print media advertising, with a
total cost of $10 million.
Increasing the required minimum increase in sales for Stain Remover by 1% changes the
solution to 3.33 units of television advertising and 4 units of print media advertising, and
increases the total cost by $1.33 million to $11.33 million.
5-50
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Increasing the required minimum increase in sales for Liquid Detergent by 1% changes the
solution to 4.33 units of television advertising and 3 units of print media advertising,, and
increases the total cost by $0.33 million to $10.33 million.
Increasing the required minimmum increase in sales for Powder Detergent by 1% has no
impact on the solution nor the total cost.
5-51
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Chapter 05 - What-If Analysis for Linear Programming
b) Original Solution:
Increasing the required minimum increase in sales for Stain Remover by 1% increases the total
cost by $1.333 million.
Increasing the required minimum increase in sales for Liquid Detergent by 1% increases the
total cost by $0.333 million.
5-52
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Increasing the required minimmum increase in sales for Powder Detergent by 1% has no
impact on the total cost.
c)
5-53
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Chapter 05 - What-If Analysis for Linear Programming
5-54
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
d) Sensitivity Report:
Variable Cells
Cell
$C$14
$D$14
Name
Advertising Units Television
Advertising Units Print Media
Final
Value
4
3
Reduced
Cost
0
0
Objective
Coefficient
1
2
Allowable
Increase
2
1E+30
Allowable
Decrease
1
1.333
Name
Stain Remover Sales
Liquid Detergent Sales
Powder Detergent Sales
Final
Value
3%
18%
8%
Shadow
Price
133.33
33.33
0
Constraint
R.H. Side
0.03
0.18
0.04
Allowable
Increase
0.06
0.12
0.04
Allowable
Decrease
0.008571429
0.12
1E+30
Constraints
Cell
$E$8
$E$9
$E$10
The shadow price indicates the increase in total cost (in $millions) per unit increase in the right
hand side (i.e., per 100% increase). Thus, a 1% increase in the minimum required increase in
sales will only increase the total cost by one hundredth of the shadow price, or $1.33 million
for the Stain Remover, $0.33 million for the Liquid Detergent, and $0 million for the Powder
Detergent.
The allowable range for the required minimum increase in sales constraint for Stain Remover is
2.15% to 9%.
The allowable range for the required minimum increase in sales constraint for Liquid Detergent
is 6% to 30%.
The allowable range for the required minimum increase in sales constraint for Powder
Detergent is -∞% to 8%.
These allowable ranges can also be seen in the results from part (c). For Stain Remover, the
incremental cost remains $1.33 million for each 1% change above 3%. Similarly, for Liquid
Detergent, the incremental cost remains $0.33 million for each 1% change above between 6%
and 30%. For Powder Detergent, the incremental cost remains $0 million for each 1% change
throughout the parameter analysis report.
e) Suppose that each of the original numbers in MinimumIncrease (G8:G10) is increased by 1%.
Percent of allowable increase for Stain Remover used = (4% – 3%) / 6% = 16.7%.
Percent of allowable increaes for Liquid Detergent used = (19% – 18%) / 12% = 8.3%.
Percent of allowable increase for Powder Detergent used = (5% – 4%) / 4% = 25%.
Sum = 50%.
Thus, if each of the original numbers in MinimumIncrease (G8:G10) is increased by 2%, the
sum will be 100%. By the 100% rule, this is the most they can be increased before the shadow
prices may no longer be valid.
5-55
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Chapter 05 - What-If Analysis for Linear Programming
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Chapter 05 - What-If Analysis for Linear Programming
f)
5.2
Answers will vary.
a) The decisions to be made are which types of abatement methods will be used and at what
fractions of their abatement capacities for the blast furnaces and the open-hearth furnaces.
The constraints on these decisions are the technological limits on how heavily each method
can be used and the required reductions in the annual emission rate. The overall measure of
performance is cost, which is to be minimized.
b & c)
A
1
2
3
4
Cost ($million)
5
6
7 Pollutant
8
Particulates
9
Sulfur oxides
10
Hydrocarbons
11
12
13
14
15
Fraction Used
16
17
Range Name
Cost
FractionUsed
MinimumReduction
OneHundredPercent
ReductionInEmission
TotalCost
TotalReduction
B
C
Taller Smokestacks
Blast
Open-Hearth
Furnaces
Furnaces
8
10
D
E
Filters
Blast
Open-Hearth
Furnaces
Furnaces
7
6
F
G
Better Fuels
Blast
Open-Hearth
Furnaces
Furnaces
11
9
Filters
Blast
Open-Hearth
Furnaces
Furnaces
57.31%
100%
<=
<=
100%
100%
Cells
B4:G4
B15:G15
J8:J10
B17:G17
B8:G10
J15
H8:H10
I
Total
Reduction
(millions of lbs.)
65
150
125
Reduction in Emission (Maximum Feasible Use of Abatement Method)
12
9
25
20
17
13
35
42
18
31
56
49
37
53
28
24
29
20
Taller Smokestacks
Blast
Open-Hearth
Furnaces
Furnaces
100%
48.55%
<=
<=
100%
100%
H
J
Minimum
Reduction
(millions of lbs.)
>=
60
>=
150
>=
125
Better Fuels
Blast
Open-Hearth
Furnaces
Furnaces
7.67%
100%
<=
<=
100%
100%
Total Cost
($million)
32.710
H
5
Total
6
Reduction
7
(millions of lbs.)
8 =SUMPRODUCT(B8:G8,FractionUsed)
9 =SUMPRODUCT(B9:G9,FractionUsed)
10 =SUMPRODUCT(B10:G10,FractionUsed)
J
13
Total Cost
14
($million)
15 =SUMPRODUCT(Cost,FractionUsed)
Variable Cells
Cell
$B$15
$C$15
$D$15
$E$15
$F$15
$G$15
Name
Fraction Taller Smokestack (Blast)
Fraction Taller Smokestack (Open Hearth)
Fraction Filter (Blast)
Fraction Filter (Open Hearth)
Fraction Better Fuel (Blast)
Fraction Better Fuel (Open Hearth)
Final
Value
100%
62.27%
34.35%
100%
4.76%
100%
Reduced
Cost
-34%
0.00%
0.00%
-182%
0.00%
-4%
Objective
Coefficient
8
10
7
6
11
9
Allowable
Increase
0.336
0.429
0.382
1.816
2.975
0.044
Allowable
Decrease
1E+30
0.667
2.011
1E+30
0.045
1E+30
Final
Value
60
150
125
Shadow
Price
0.111
0.127
0.069
Constraint
R.H. Side
60
150
125
Allowable
Increase
14.297
20.453
2.042
Allowable
Decrease
7.480
1.690
21.692
Constraints
Cell
$H$8
$H$9
$H$10
Name
Particulates (millions of lbs.)
Sulfur oxides (millions of lbs.)
Hydrocarbons (millions of lbs.)
5-57
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Chapter 05 - What-If Analysis for Linear Programming
d) The right-hand-side of each constraint with a non-zero shadow price is sensitive, since
changing its value will impact the total cost. All three required reductions in emission rates are
sensitive parameters. All of the objective coefficients have an allowable range to stay optimal
around them, and thus are not as sensitive. However, for some, the allowable change is
small—in particular, the cost of the two better fuel options (with an allowable increase of only
0.045 and an allowable decrease of 0.044, respectively) are fairly sensitive. Thus, all five of
these parameters should be estimated more closely, if possible.
e) The sensitivity report and, in particular, the allowable range for the objective coefficients can
be used to determine whether the solution will change. The following table shows in which
cases the optimal solution will change.
Abatement
Method
Taller Smoke (Blast)
Taller Smoke (Open H)
Filter (Blast)
Filter (Open H)
Better Fuel (Blast)
Better Fuel (Open H)
Current
Value
8
10
7
6
11
9
10% Less
Value
7.2
9
6.3
5.4
9.9
8.1
Solution
Changes?
No
Yes
No
No
Yes
No
10%
More
Value
8.8
11
7.7
6.6
12.1
9.9
Solution
Changes?
Yes
Yes
Yes
No
No
Yes
This suggests that focus should be put on estimating all of the costs except the filter for the
open hearth furnaces, since it’s optimal solution will not change with a 10% increase or
decrease. Special consideration should be given to the estimate of the cost of the taller
smokestack for the open hearth furnaces, since it affects the optimal solution for both an
increase and a decrease. Special consideration should also be given to the estimate of the cost
of the better fuel options, since the allowable decrease (for the blast furnace) or allowable
decrease (for the open hearth furnace) is so small.
f)
Pollutant
Particulates
Sulfur oxides
Hydrocarbons
Rate that
cost changes
($million)
0.111
0.127
0.069
Maximum increase
before rate changes
(million lb.)
14.297
20.453
2.042
Maximum decrease
before rate changes
(million lb.)
7.480
1.690
21.692
5-58
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Chapter 05 - What-If Analysis for Linear Programming
g) Particulates and sulfur oxides:
For each unit increase in particulate reduction, cost will increase by $0.111 million.
For each unit decrease in sulfur oxide reduction, cost will decrease by $0.127 million.
Thus, cost will remain equal if for each unit increase in particulate reduction, the sulfur oxide
reduction is reduced by $0.111 / $0.127 = 0.874 units.
Particulates and hydrocarbons:
For each unit increase in particulate reduction, cost will increase by $0.111 million.
For each unit decrease in hydrocarbon reduction, cost will decrease by $0.069 million.
Thus, cost will remain equal if for each unit increase in particulate reduction, the hydrocarbon
reduction is reduced by $0.111 / $0.069 = 1.609 units.
Particulates and both sulfur oxides and hydrocarbons:
For each unit increase in particulate reduction, cost will increase by $0.111 million.
For each simultaneous unit decrease in sulfur oxide and hydrocarbon reduction, cost will
decrease by $0.127 + $0.069 = $0.196.
Thus, cost will remain equal if for each unit increase in particulate reduction, the sulfur oxide
and hydrocarbon reduction are each reduced by $0.111 / $0.196 = 0.566 units.
5-59
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Chapter 05 - What-If Analysis for Linear Programming
h)
Each 10% reduction in pollution costs less than $3.5 million (the tax incentive) until the
reduction exceeds 40%. Since the tax incentive is $3.5 million for each 10% reduction, a 40%
reduction should be chosen to minimize the total cost of both pollution abatement and taxes.
5-60
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Chapter 05 - What-If Analysis for Linear Programming
i)
The sensitivity report for a 40% reduction is shown below.
Variable Cells
Cell
$B$15
$C$15
$D$15
$E$15
$F$15
$G$15
Name
Fraction Taller Smokestack (Blast)
Fraction Taller Smokestack (Open Hearth)
Fraction Filter (Blast)
Fraction Filter (Open Hearth)
Fraction Better Fuel (Blast)
Fraction Better Fuel (Open Hearth)
Final
Value
100%
100.00%
70.53%
100%
78.16%
93%
Reduced
Cost
-55%
-42.94%
0.00%
-179%
0.00%
0%
Objective
Coefficient
8
10
7
6
11
9
Allowable
Increase
0.553
0.429
0.382
1.789
0.384
0.044
Allowable
Decrease
1E+30
1E+30
1.292
1E+30
0.045
0.372
Final
Value
84
210
175
Shadow
Price
0.099
0.124
0.082
Constraint
R.H. Side
84
210
175
Allowable
Increase
0.265
1.112
0.864
Allowable
Decrease
0.846
6.294
0.253
Constraints
Cell
$H$8
$H$9
$H$10
Name
Particulates (millions of lbs.)
Sulfur oxides (millions of lbs.)
Hydrocarbons (millions of lbs.)
Pollutant
Particulates
Sulfur oxides
Hydrocarbons
Rate that
cost
changes
($million)
0.099
0.124
0.082
Maximum increase
before rate changes
(million lb.)
Maximum decrease
before rate changes
(million lb.)
0.265
1.112
0.864
0.846
6.294
0.253
Particulates and sulfur oxides:
For each unit increase in particulate reduction, cost will increase by $0.099 million.
For each unit decrease in sulfur oxide reduction, cost will decrease by $0.124 million.
Thus, cost will remain equal if for each unit increase in particulate reduction, the sulfur oxide
reduction is reduced by $0.099 / $0.124 = 0.798 units.
Particulates and hydrocarbons:
For each unit increase in particulate reduction, cost will increase by $0.099 million.
For each unit decrease in hydrocarbon reduction, cost will decrease by $0.082 million.
Thus, cost will remain equal if for each unit increase in particulate reduction, the hydrocarbon
reduction is reduced by $0.099 / $0.082 = 1.207 units.
Particulates and both sulfur oxides and hydrocarbons:
For each unit increase in particulate reduction, cost will increase by $0.099 million.
For each simultaneous unit decrease in sulfur oxide and hydrocarbon reduction, cost will
decrease by $0.124 + $0.082 = $0.206.
Thus, cost will remain equal if for each unit increase in particulate reduction, the sulfur oxide
and hydrocarbon reduction are each reduced by $0.099 / $0.206 = 0.481 units.
5-61
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Chapter 05 - What-If Analysis for Linear Programming
5.3
a) The decisions to be made are how much acreage should be planted in each of the crops and
how many cows and hens to have for the coming year. The constraints on these decisions are
amount of labor hours available, the investment funds available, the number of acres
available, the space available in the barn and chicken house, the minimum requirements for
feed to be planted. The overall measure of performance is monetary worth, which is to be
maximized.
b & c)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
537
736
$37,300
Plantings
Soybeans
1
1.4
$70
Corn
0.9
1.2
$60
Wheat
0.6
0.7
$40
450
30
>=
30
1
acre/cow
100
>=
100
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
400
60
$34,000
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$0
$0
30
0
30
<=
42
2,000
0
2,000
<=
5,000
Wage
W&S
$5
S&F
$5.50
Neighbor
Totals
$12,817.00
Hours Worked
1063
1364
2427
W&S Hours
S&F Hours
Acreage
Plantings
537
736
580
Livestock
2,400
2,400
60
$34,000
$35,250
$20,000
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
New Livestock
Total Livestock
Barn/House Limits
Neighboring Farm Work
Totals
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
$37,300
F
G
580
<=
Investment
Fund
$20,000
Neighbor
1,063
1,364
0
Total
4,000
4,500
640
<=
<=
<=
$12,817
$84,117
$35,250
$20,000
-$40,000
$99,367
Available
4,000
4,500
640
This model predicts that the family’s monetary worth at the end of the coming year will be
$99, 367.
5-62
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Chapter 05 - What-If Analysis for Linear Programming
5-63
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Chapter 05 - What-If Analysis for Linear Programming
A
B
C
1 Plantings
2
Soybeans
3
W&S Hours Required 1
4
S&F Hours Required 1.4
5
Net Value 70
6
7
Acres Planted 450
8
9
10
11
D
0.9
1.2
60
0.6
0.7
40
30
100
=SUM(AcresPlanted)
>=
=C10*B28
1
acre/cow
>=
=D10*C28
0.05
acre/hen
Corn
A
Wheat
B
13 Livestock
14
15
Hours Required per Month
16
Grazing Land Required
17
Net Annual Cash Income
18
19
Beginning Value (Current Livestock)
20
Decrease in Value per Year
21
End Value (Current Livestock)
22
23
Cost of New Livestock
24
End Value (New Livestock)
25
26
Current Livestock
27
New Livestock
28
Total Livestock
29
30
Barn/House Limits
C
D
E
10
2
850
0.05
0
4.25
Livestock
Totals
=SUMPRODUCT(B15:C15,TotalLivestock)
=SUMPRODUCT(B16:C16,TotalLivestock)
=SUMPRODUCT(B17:C17,TotalLivestock)
35000
0.1
=(1-B20)*B19
5000
0.25
=(1-C20)*C19
=SUM(B21:C21)
1500
=(1-B20)*B23
3
=(1-C20)*C23
=SUMPRODUCT(B23:C23,NewLivestock)
=SUMPRODUCT(B24:C24,NewLivestock)
30
0
=CurrentLivestock+NewLivestock
<=
42
2000
0
=CurrentLivestock+NewLivestock
<=
5000
Cows
A
B
32 Neighboring Farm Work
33
W&S
34
Wage 5
35
36
Hours Worked 1063
A
38 Totals
39
W&S Hours
40
S&F Hours
41
Acreage
42
43
Net Income
44
End of Year Value
45
Leftover Investment Fund
46
Living Expenses
47
Total Monetary Worth
Range Name
AcresPlanted
Available
BarnHouseLimits
CurrentLivestock
HoursWorked
InvestmentFund
MonetaryWorth
NewLivestock
Total
TotalLivestock
Wage
E
Planting
Totals
=SUMPRODUCT(B3:D3,AcresPlanted)
=SUMPRODUCT(B4:D4,AcresPlanted)
=SUMPRODUCT(B5:D5,AcresPlanted)
Hens
C
S&F
5.5
D
Neighbor
Totals
=SUMPRODUCT(Wage,HoursWorked)
1364
=SUM(HoursWorked)
B
C
D
E
Plantings
=E3
=E4
=E7
Livestock
=6*D15
=6*D15
=D16
Neighbor
Total
=B36
=SUM(B39:D39)
=C36
=SUM(B40:D40)
0
=SUM(B41:D41)
=E5
=D17
=D21+D24
=InvestmentFund-D23
=D34
F
F
Investment
Fund
<= 20000
G
Available
<= 4000
<= 4500
<= 640
=SUM(B43:D43)
=SUM(B44:D44)
=SUM(B45:D45)
-40000
=SUM(E43:E46)
Cells
B7:D7
G39:G41
B30:C30
B26:C26
B36:C36
F23
E47
B27:C27
E39:E41
B28:C28
B34:C34
5-64
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Chapter 05 - What-If Analysis for Linear Programming
Variable Cells
Cell
$B$7
$C$7
$D$7
$B$27
$C$27
$B$36
$C$36
Name
Acres Planted Soybeans
Acres Planted Corn
Acres Planted Wheat
New Livestock Cows
New Livestock Hens
Hours Worked W&S
Hours Worked S&F
Final Reduced Objective Allowable Allowable
Value
Cost
Coefficient Increase Decrease
450
0
70
1E+30
8.4
30
0
60
8.4
1E+30
100
0
40
17.15
1E+30
0
-53
700
53
1E+30
0
-0.857
3.5
0.857
1E+30
1063
0
5
57.3
0.915
1364
0
5.5
34.5
0.930
Name
Acres Planted Corn
Acres Planted Wheat
Cost of New Livestock Totals
Total Livestock Cows
Total Livestock Hens
W&S Hours Total
S&F Hours Total
Acreage Total
Final Shadow Constraint Allowable Allowable
Value
Price
R.H. Side Increase Decrease
30
-8.4
0
450
30
100
-24.15
0
450
100
$0
$0
20000
1E+30
20000
30
0
42
1E+30
12
2000
0
5000
1E+30
3000
4000
5
4000
1E+30
1063
4500
5.5
4500
1E+30
1364
640
57.3
640
974.29
450
Constraints
Cell
$C$7
$D$7
$D$23
$B$28
$C$28
$E$39
$E$40
$E$41
d) The allowable range for the value per acre planted of soybeans is 61.6 to ∞.
The allowable range for the value per acre planted of corn is –∞ to 68.4.
The allowable range for the value per acre planted of wheat is –∞ to 57.15.
5-65
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Chapter 05 - What-If Analysis for Linear Programming
e) Drought
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
117.8
143.733
-$630
Plantings
Soybeans
1
1.4
-$10
Corn
0.9
1.2
-$15
Wheat
0.6
0.7
$0
0
42
>=
42
1
acre/cow
133.33
>=
133.33
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
553.333333
84
$47,033
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$20,000
$17,700
30
12
42
<=
42
2,000
667
2,667
<=
5,000
Wage
W&S
$5
S&F
$5.50
Hours Worked
562.2
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
New Livestock
Total Livestock
Barn/House Limits
Neighboring Farm Work
Totals
-$630
G
175.333
<=
Investment
Fund
$20,000
Neighbor
562
1,036
0
Total
4,000
4,500
259.333
<=
<=
<=
$8,510
$54,914
$52,950
$0
-$40,000
$67,864
Neighbor
Totals
$8,510.47
1036.267 1598.46665
Plantings Livestock
W&S Hours
117.8
3,320
S&F Hours 143.7333
3,320
Acreage 175.3333
84
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
F
$47,033
$52,950
$0
Available
4,000
4,500
640
In a drought, the model predicts (under the optimal solution) that the family’s monetary worth
at the end of the year will be $67,864.
5-66
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Flood
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
460.4
600.533
$9,787
Plantings
Soybeans
1
1.4
$15
Corn
0.9
1.2
$20
Wheat
0.6
0.7
$10
0
422.6667
>=
42
1
acre/cow
133.33
>=
133.33
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
553.333333
84
$47,033
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$20,000
$17,700
30
12
42
<=
42
2,000
667
2,667
<=
5,000
Wage
W&S
$5
S&F
$5.50
Neighbor
Totals
$4,285.07
Hours Worked
219.6
579.4667
799.067
Plantings Livestock
W&S Hours
460.4
3,320
S&F Hours 600.5333
3,320
Acreage
556
84
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
New Livestock
Total Livestock
Barn/House Limits
Neighboring Farm Work
Totals
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
$9,787
$47,033
$52,950
$0
F
G
556
<=
Investment
Fund
$20,000
Neighbor
220
579
0
Total
4,000
4,500
640
<=
<=
<=
$4,285
$61,105
$52,950
$0
-$40,000
$74,055
Available
4,000
4,500
640
In a flood, the model predicts (under the optimal solution) that the family’s monetary worth at
the end of the year will be $74,055.
5-67
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Early Frost
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
537
736
$26,700
Plantings
Soybeans
1
1.4
$50
Corn
0.9
1.2
$40
Wheat
0.6
0.7
$30
450
30
>=
30
1
acre/cow
100.00
>=
100.00
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
400
60
$34,000
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$0
$0
30
0
30
<=
42
2,000
0
2,000
<=
5,000
Wage
W&S
$5
S&F
$5.50
Neighbor
Totals
$12,817.00
Hours Worked
1063
1364
2427.000
Plantings
537
736
580
Livestock
2,400
2,400
60
$26,700
$34,000
$35,250
$20,000
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
New Livestock
Total Livestock
Barn/House Limits
Neighboring Farm Work
Totals
W&S Hours
S&F Hours
Acreage
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
F
G
580
<=
Investment
Fund
$20,000
Neighbor
1,063
1,364
0
Total
4,000
4,500
640
<=
<=
<=
$12,817
$73,517
$35,250
$20,000
-$40,000
$88,767
Available
4,000
4,500
640
In an early frost, the model predicts (under the optimal solution) that the family’s monetary
worth at the end of the year will be $88,767.
5-68
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Drought and Early Frost
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
97.8
120.4
-$1,840
Plantings
Soybeans
1
1.4
-$15
Corn
0.9
1.2
-$20
Wheat
0.6
0.7
-$10
0
42
>=
42
1
acre/cow
100.00
>=
100.00
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
520
84
$44,200
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$18,000
$16,200
30
12
42
<=
42
2,000
0
2,000
<=
5,000
Wage
W&S
$5
S&F
$5.50
Neighbor
Totals
$10,838.80
Hours Worked
782.2
1259.6
2041.800
Plantings
97.8
120.4
142
Livestock
3,120
3,120
84
-$1,840
$44,200
$51,450
$2,000
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
New Livestock
Total Livestock
Barn/House Limits
Neighboring Farm Work
Totals
W&S Hours
S&F Hours
Acreage
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
F
G
142
<=
Investment
Fund
$20,000
Neighbor
782
1,260
0
Total
4,000
4,500
226
<=
<=
<=
$10,839
$53,199
$51,450
$2,000
-$40,000
$66,649
Available
4,000
4,500
640
In a drought and early frost, the model predicts (under the optimal solution) that the family’s
monetary worth at the end of the year will be $66,649.
5-69
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Flood and Early Frost
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
183.6
219.8
$1,623
Plantings
Soybeans
1
1.4
$10
Corn
0.9
1.2
$10
Wheat
0.6
0.7
$5
0
37.33333
>=
37.33333
1
acre/cow
250.00
>=
250.00
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
623.333333
74.6666667
$52,983
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$20,000
$16,650
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
30
New Livestock 7.333333
Total Livestock 37.33333
<=
Barn/House Limits
42
Wage
Hours Worked 76.39999
Totals
$1,623
287.333
<=
Investment
Fund
$20,000
Neighbor
76
540
0
Total
4,000
4,500
362
<=
<=
<=
$3,353
$57,960
$51,900
$0
-$40,000
$69,860
S&F
$5.50
Neighbor
Totals
$3,353.10
540.2
616.600
Plantings Livestock
W&S Hours
183.6
3,740
S&F Hours
219.8
3,740
Acreage 287.3333 74.66667
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
G
2,000
3,000
5,000
<=
5,000
Neighboring Farm Work
W&S
$5
F
$52,983
$51,900
$0
Available
4,000
4,500
640
In a flood and early frost, the model predicts (under the optimal solution) that the family’s
monetary worth at the end of the year will be $69,860.
5-70
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
f)
Opt. Sol. Used
Good Weather
Drought
Flood
Early Frost
Drought & E.F.
Flood & E.F.
Good
$99,367
$76,348
$94,962
$99,367
$75,009
$80,476
Family’s monetary worth at year’s end if the scenario is actually:
Drought
Flood
Early Frost
Drought&EF
Flood&EF
$57,117
$70,417
$88,767
$53,717
$67,367
$67,864
$70,668
$74,174
$66,321
$69,581
$57,929
$74,055
$85,175
$54,482
$69,162
$57,117
$70,417
$88,767
$53,717
$67,367
$67,859
$70,329
$73,169
$66,649
$69,409
$67,676
$71,483
$77,230
$64,990
$69,860
Answers will vary. No solution is clearly best. The Good Weather solution is the riskiest, with
the highest upside and downside. The Flood solution appears to be a good middle ground. The
Drought, Drought&EF, and Flood&EF solutions are the most conservative.
g and h)
The expected net value for each of the crops is calculated as follows:
Soybeans: ($70)(0.4) + (–$10)(0.2) + ($15)(0.1) + ($50)(0.15) + (–$15)(0.1) +
($10)(0.05) = $34,
Corn: ($60)(0.4) + (–$15)(0.2) + ($20)(0.1) + ($40)(0.15) + (–$20)(0.1) +
($10)(0.05) = $27.5,
Wheat: ($40)(0.4) + ($0)(0.2) + ($10)(0.1) + ($30)(0.15) + (–$10)(0.1) +
($5)(0.05) = $20.75.
The resulting spreadsheet solution is shown below:
5-71
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
B
C
D
E
Planting
Totals
511.8
700
$17,306
Plantings
Soybeans
1
1.4
$34.00
Corn
0.9
1.2
$27.50
Wheat
0.6
0.7
$20.75
414
42
>=
42
1
acre/cow
100.00
>=
100.00
0.05
acre/hen
Cows
10
2
$850
Hens
0.05
0
$4
Livestock
Totals
520
84
$44,200
Beginning Value (Current Livestock)
Decrease in Value per Year
End Value (Current Livestock)
$35,000
10%
$31,500
$5,000
25%
$3,750
$35,250
Cost of New Livestock
End Value (New Livestock)
$1,500
$1,350
$3
$2
$18,000
$16,200
30
12
42
<=
42
2,000
0
2,000
<=
5,000
Wage
W&S
$5
S&F
$5.50
Neighbor
Totals
$5,581.00
Hours Worked
368.2
680
1048.200
Plantings
511.8
700
556
Livestock
3,120
3,120
84
$17,306
$44,200
$51,450
$2,000
W&S Hours Required
S&F Hours Required
Net Value
Acres Planted
Livestock
Hours Required per Month
Grazing Land Required
Net Annual Cash Income
Current Livestock
New Livestock
Total Livestock
Barn/House Limits
Neighboring Farm Work
Totals
W&S Hours
S&F Hours
Acreage
Net Income
End of Year Value
Leftover Investment Fund
Living Expenses
Total Monetary Worth
F
G
556
<=
Investment
Fund
$20,000
Neighbor
368
680
0
Total
4,000
4,500
640
<=
<=
<=
$5,581
$67,087
$51,450
$2,000
-$40,000
$80,537
Available
4,000
4,500
640
This model predicts that the family’s monetary worth at the end of the coming year will be (on
average) $80,537.
5-72
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
Variable Cells
Cell
$B$7
$C$7
$D$7
$B$27
$C$27
$B$36
$C$36
Name
Acres Planted Soybeans
Acres Planted Corn
Acres Planted Wheat
New Livestock Cows
New Livestock Hens
Hours Worked W&S
Hours Worked S&F
Final
Value
414
42
100
12
0
368.2
680
Reduced
Cost
0
0
0.00
0
0
0
0
Objective
Coefficient
34
27.5
20.75
700
3.5
5
5.5
Allowable
Increase
7.5
4.9
0.4
1E+30
0.02
0.389
0.395
Allowable
Decrease
0.4
22.5
1E+30
22.5
1E+30
0.071
0.075
Final
Value
42
100.00
$18,000
42
2,000
4,000
4,500
640
Shadow
Price
-4.9
-7.40
$0
22.5
0
5
5
21.3
Constraint
R.H. Side
0
0
20000
42
5000
4000
4500
640
Allowable
Increase
414
414
1E+30
1.333
1E+30
1E+30
1E+30
368.2
Allowable
Decrease
42
100
2000
12
3000
368.2
680
414
Constraints
Cell
$C$7
$D$7
$D$23
$B$28
$C$28
$E$39
$E$40
$E$41
Name
Acres Planted Corn
Acres Planted Wheat
Cost of New Livestock Totals
Total Livestock Cows
Total Livestock Hens
W&S Hours Total
S&F Hours Total
Acreage Total
i)
The shadow price for the investment constraint is zero, indicating that additional investment
funds will not increase their total monetary worth at all. Thus, it is not worthwhile to obtain a
bank loan. The shadow price would need to be at least $1.10 before a loan at 10% interest
would be worthwhile.
j)
The expected net value for soybeans can increase up to $7.50 or decrease up to $0.40; for
corn can increase up to $4.90 or decrease up to $22.50; for wheat can increase up to $0.40 or
decrease any amount without changing the optimal solution. The expected net value for
soybeans and wheat should be estimated most carefully.
The solution is sensitive to decreases in the expected value of soybeans and increases in the
expected value of wheat. If the cumulative decrease in the expected value of soybeans and
increase in the expected value of wheat exceeds $0.40, then the 100% rule will be violated,
and the solution might change.
k) Answers will vary.
5-73
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
5.4
a)
Range Name
BussingCost
Capacity
NumberOfStudents
PercentageInGrade
Solution
TotalBussingCost
TotalFromArea
TotalInSchool
Cells
E4:G9
B22:D22
G14:G19
B4:D9
B14:D19
G24
E14:E19
B20:D20
5-74
© 2014 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
b)
Variable Cells
Cell
$B$14
$C$14
$D$14
$B$15
$C$15
$D$15
$B$16
$C$16
$D$16
$B$17
$C$17
$D$17
$B$18
$C$18
$D$18
$B$19
$C$19
$D$19
Name
Area 1 School 1
Area 1 School 2
Area 1 School 3
Area 2 School 1
Area 2 School 2
Area 2 School 3
Area 3 School 1
Area 3 School 2
Area 3 School 3
Area 4 School 1
Area 4 School 2
Area 4 School 3
Area 5 School 1
Area 5 School 2
Area 5 School 3
Area 6 School 1
Area 6 School 2
Area 6 School 3
Final
Value
0
450
0
0
422.22
177.78
0
227.78
322.22
350
0
0
366.67
0
133.33
83.33
0
366.67
Reduced
Cost
177.778
0
266.667
-800.000
0
0
11.111
0
0
0
366.667
-433.333
0
233.333
0
0
200
0
Objective
Coefficient
300
0
700
0
400
500
600
300
200
200
500
0
0
0
400
500
300
0
Allowable
Increase
1E+30
177.778
1E+30
1E+30
34.211
4.545
1E+30
4.545
34.211
366.667
1E+30
1E+30
16.667
1E+30
108.333
33.333
1E+30
166.667
Allowable
Decrease
177.778
1E+30
266.667
800.000
4.545
34.211
11.111
34.211
7.692
2.08E+17
366.667
433.333
108.333
233.333
16.667
166.667
200
33.333
Name
8th Graders <=
8th Graders <=
8th Graders <=
Total In School School 1
Total In School School 2
Total In School School 3
6th Graders <=
6th Graders <=
6th Graders <=
6th Graders <=
6th Graders <=
6th Graders <=
7th Graders <=
7th Graders <=
7th Graders <=
7th Graders <=
7th Graders <=
7th Graders <=
8th Graders <=
8th Graders <=
8th Graders <=
Area 1 From Area
Area 2 From Area
Area 3 From Area
Area 4 From Area
Area 5 From Area
Area 6 From Area
Final
Value
242.67
369.33
360.00
800
1,100
1,000
269.33
368.56
339.11
269.33
368.56
339.11
288.00
362.11
300.89
288.00
362.11
300.89
242.67
369.33
360.00
450
600
550
350
500
450
Shadow
Price
0.00
0.00
-6666.67
0
-178
-144
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-2777.78
0.00
0.00
0.00
0.00
0.00
177.778
577.778
477.778
311.111
-55.556
277.778
Constraint
R.H. Side
0
0
0
900
1100
1000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
450
600
550
350
500
450
Allowable
Increase
1E+30
1E+30
5.333
1E+30
36.364
42.105
29.333
38.556
39.111
1E+30
1E+30
1E+30
48
32.111
0.889
0.258
1E+30
1E+30
2.667
39.333
60
3.774
3.774
3.774
72.727
12.903
3.226
Allowable
Decrease
45.333
26.667
0.667
100
3.774
3.883
1E+30
1E+30
1E+30
18.667
27.444
20.889
1E+30
1E+30
1E+30
2.909
33.889
59.111
1E+30
1E+30
1E+30
36.364
36.364
36.364
6.452
145.455
36.364
Constraints
Cell
$B$30
$C$30
$D$30
$B$20
$C$20
$D$20
$B$28
$C$28
$D$28
$B$28
$C$28
$D$28
$B$29
$C$29
$D$29
$B$29
$C$29
$D$29
$B$30
$C$30
$D$30
$E$14
$E$15
$E$16
$E$17
$E$18
$E$19
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Chapter 05 - What-If Analysis for Linear Programming
c) The bussing cost from area 6 to school 1 can increase $33.33 before the current optimal
solution would no longer be optimal. The new solution with a 10% increase ($50) is shown
below.
d) The bussing cost from area 6 to school 2 can increase any amount and the optimal solution
from part (a) will still be optimal.
e) If the bussing costs increase 1% from area 6 to all the schools, then:
Percentage of allowable increase for school 1 used = ($505 – $500) / $33.33 = 15%.
Percentage of allowable increase for school 2 used = ($303 – $300) / ∞ = 0%.
Percentage of allowable increase for school 3 used = ($0 – $0) / $166.67 = 0%.
Sum = 15%. Therefore, the bussing costs from area 6 can increase uniformly by
(100%/15%)(1%) = 6.67% before 100% will be reached. Beyond that, the solution might
change.
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
Chapter 05 - What-If Analysis for Linear Programming
If the bussing costs increase 10% from area 6 to all schools, the new solution is:
f)
The shadow price for school 1 is zero. Thus, adding a temporary classroom at school 1 would
not save any money, and thus would not be worthwhile.
The shadow price for school 2 is –$177.78. Thus, adding a temporary classroom at school 2
would save ($177.78)(20) = $3,555.60 in bussing cost. This is worthwhile, since it exceeds the
$2500 leasing cost.
The shadow price for school 3 is –$144.44. Thus, adding a temporary classroom at school 3
would save ($144.44)(20) = $2,888.80 in bussing cost. This is also worthwhile, since it exceeds
the $2500 leasing cost.
g) For school 2, the allowable increase for school capacity is 36. This means the shadow price is
only valid for a single additional portable classroom.
For school 3, the allowable increase for school capacity is 42. This means the shadow price is
valid for up to two additional portable classrooms.
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Chapter 05 - What-If Analysis for Linear Programming
h) The following combinations do not violate the 100% rule:
Portables to
add
to school 2
1
0
0
Portables to
add
to school 3
0
1
2
100%-rule calculation
(20/36) + (0/42) = 55.6%
(0/36) + (20/42) = 47.6%
(0/36) + (40/42) = 95.23%
Each combination yields the following total savings
Portables to
add
to school 2
1
0
0
Portables to
add
to school 3
0
1
2
Bussing Cost Savings
($177.78)(20) = $3555.60
($144.44)(20) = $2888.80
($144.44)(40) = $5777.60
Lease
Cost
$2500
$2500
$5000
Total
Savings
$1055.60
$388.80
$777.60
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Chapter 05 - What-If Analysis for Linear Programming
Of these combinations, adding one portable to school 2 is best in terms of minimizing total
cost. The spreadsheet solution is shown below.
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Chapter 05 - What-If Analysis for Linear Programming
i)
Adding two portables to school 2 yields the following solution. This is the best plan.
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.