Chapter 05 – What-If Analysis for Linear Programming
5-21
f)
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A B C D E F G H
Total Profit Unit Profit for W all Clocks
$1,665 $50 $100 $150 $200 $250 $300
$150 $750 $833 $1,000 $1,333 $1,667 $2,000
$200 $1,000 $1,000 $1,167 $1,333 $1,667 $2,000
Unit Profit $250 $1,250 $1,250 $1,333 $1,500 $1,667 $2,000
for Grandfather $300 $1,500 $1,500 $1,500 $1,667 $1,833 $2,000
Clocks $350 $1,750 $1,750 $1,750 $1,833 $2,000 $2,167
$400 $2,000 $2,000 $2,000 $2,000 $2,167 $2,333
$450 $2,250 $2,250 $2,250 $2,250 $2,333 $2,500
Production (G randfather Clocks, Wall Clocks) Unit Profit for W all Clocks
(3.33,3.33) $50 $100 $150 $200 $250 $300
$150 (5,0) (3.33,3.33) (3.33,3.33) (0,6.67) (0,6.67) (0,6.67)
$200 (5,0) (5,0) (3.33,3.33) (3.33,3.33) (0,6.67) (0,6.67)
Unit Profit $250 (5,0) (5,0) (3.33,3.33) (3.33,3.33) (0,6.67) (0,6.67)
for Grandfather $300 (5,0) (5,0) (5,0) (3.33,3.33) (3.33,3.33) (0,6.67)
Clocks $350 (5,0) (5,0) (5,0) (3.33,3.33) (3.33,3.33) (3.33,3.33)
$400 (5,0) (5,0) (5,0) (5,0) (3.33,3.33) (3.33,3.33)
$450 (5,0) (5,0) (5,0) (5,0) (3.33,3.33) (3.33,3.33)
g) If David increases his hours to 45 per week, the optimal solution does not change.
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A B C D E F
G randfather W all
Clock Clock
Unit Profit $300 $200
Hours Hours
Used Available
Assembly (David) 6 4 33 <= 45
Carving (LaDeana) 8 4 40 <= 40
Shipping (Lydia) 3 3 20 <= 20
G randfather W all
Clock Clock Total Profit
Production 3.33 3.33 $1,667
Time Required
If LaDeana increases her hours to 45 per week, the optimal solution changes to
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A B C D E F
G randfather W all
Clock Clock
Unit Profit $300 $200
Hours Hours
Used Available
Assembly (David) 6 4 36 <= 40
Carving (LaDeana) 8 4 45 <= 45
Shipping (Lydia) 3 3 20 <= 20
G randfather W all
Clock Clock Total Profit
Production 4.58 2.08 $1,792
Time Required
5-22
If Lydia increases her hours to 25 per week, the optimal solution changes to
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A B C D E F
G randfather W all
Clock Clock
Unit Profit $300 $200
Hours Hours
Used Available
Assembly (David) 6 4 37 <= 40
Carving (LaDeana) 8 4 40 <= 40
Shipping (Lydia) 3 3 25 <= 25
G randfather W all
Clock Clock Total Profit
Production 1.67 6.67 $1,833
Time Required
h)
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A B C D
Assembly Hours
Available Grandfather W all
(David) Clock Clock Total Profit
3.33 3.33 $1,667
35 3.33 3.33 $1,667
37 3.33 3.33 $1,667
39 3.33 3.33 $1,667
41 3.33 3.33 $1,667
43 3.33 3.33 $1,667
45 3.33 3.33 $1,667
Carving Hours
Available Grandfather W all
(LaDeana) Clock Clock Total Profit
3.33 3.33 $1,667
35 2.08 4.58 $1,542
37 2.58 4.08 $1,592
39 3.08 3.58 $1,642
41 3.58 3.08 $1,692
43 4.08 2.58 $1,742
45 4.58 2.08 $1,792
Shipping Hours
Available Grandfather W all
(Lydia) Clock Clock Total Profit
3.33 3.33 $1,667
15 5.00 0.00 $1,500
17 4.33 1.33 $1,567
19 3.67 2.67 $1,633
21 3.00 4.00 $1,700
23 2.33 5.33 $1,767
25 1.67 6.67 $1,833
5-23
i) The allowable range for the unit profit for the grandfather clock is $200 to $400.
Variable Cells
F in al Red uced O bjective Allow ab le Al low able
Cell Name Valu e Cost Coefficient Increase Decrease
$B$12 Production Clock 3.33 0.00 300 100 100
$C$12 Production Clock 3.33 0.00 200 100 50
Constraints
F in al Sh ad ow Co nstrai nt Allo w ab le Allow ab le
Cell Name Valu e Price R.H. Side Increase Decrease
$D$6 Assembly (David) Used 33 040 1E+ 30 6.667
$D$7 Carving (LaDeana) Used 40 25 40 13.333 13.333
$D$8 Shipping (Lydia) Used 20 33.33 20 10 5
j) Lydia should increase her hours slightly since her hours have the highest shadow price.
m) Percentage of Lydia’s available increase used = (25 – 20)/10 = 50%.
n) The revised graph is shown below. The optimal solution changes from (3.333,3.333)
5-24
5.11 a)
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A B C D E F
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,000 <= 3,000
Subassembly B 1 -1 1,000 <= 1,000
Toys Subassemblies Total Profit
Production 2,000 1,000 $3,500.00
<=
2,500
Resource Usage
b)
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A B C D E F
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,001 <= 3,001
Subassembly B 1 -1 1,000 <= 1,000
Toys Subassemblies Total Profit
Production 2,001 1,001 $3,500.50
<=
2,500
Resource Usage
c)
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A B C D E F
Toys Subassemblies
Unit Profit $3.00 -$2.50
Used Available
Subassembly A 2 -1 3,000 <= 3,000
Subassembly B 1 -1 1,001 <= 1,001
Toys Subassemblies Total Profit
Production 1,999 998 $3,502.00
<=
2,500
Resource Usage
5-25
d)
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A B C D E
Available Total Incremental
Subassembly A Toys Subassemblies Profit Profit
2,000 1,000 $3,500.00
3,000 2,000 1,000 $3,500.00
3,100 2,100 1,100 $3,550.00 $50.00
3,200 2,200 1,200 $3,600.00 $50.00
3,300 2,300 1,300 $3,650.00 $50.00
3,400 2,400 1,400 $3,700.00 $50.00
3,500 2,500 1,500 $3,750.00 $50.00
3,600 2,500 1,500 $3,750.00 $0.00
3,700 2,500 1,500 $3,750.00 $0.00
3,800 2,500 1,500 $3,750.00 $0.00
3,900 2,500 1,500 $3,750.00 $0.00
4,000 2,500 1,500 $3,750.00 $0.00
Production
e)
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A B C D E
Available Total Incremental
Subassembly B Toys Subassemblies Profit Profit
2,000 1,000 $3,500.00
1,000 2,000 1,000 $3,500.00
1,100 1,900 800 $3,700.00 $200.00
1,200 1,800 600 $3,900.00 $200.00
1,300 1,700 400 $4,100.00 $200.00
1,400 1,600 200 $4,300.00 $200.00
1,500 1,500 0 $4,500.00 $200.00
1,600 1,500 0 $4,500.00 $0.00
1,700 1,500 0 $4,500.00 $0.00
1,800 1,500 0 $4,500.00 $0.00
1,900 1,500 0 $4,500.00 $0.00
2,000 1,500 0 $4,500.00 $0.00
Production
f)
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$B$9 Production Toys 2,000 0 3 2 0.5
$C$9 Production Subassemblies 1,000 0 -2.5 1 0.5
Constraints
Final Shadow Constraint Allowable Allo wable
Cell Name Value Price R.H. Side Increase Decrease
$D$5 Subassembly A Used 3,000 0.5 3000 500 1000
$D$6 Subassembly B Used 1,000 2 1000 500 500
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,
5.12 a) The original model:
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A B C D E F
Activity 1 Activity 2
Unit Profit $2 $5
Used Available
Resource 1 1 2 10 <= 10
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 6 2 $22.00
Resource Usage
With 1 additional unit of resource 1:
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A B C D E F
Activity 1 Activity 2
Unit Profit $2 $5
Used Available
Resource 1 1 2 11 <= 11
Resource 2 1 3 12 <= 12
Activity 1 Activity 2 Total Profit
Solution 9 1 $23.00
Resource Usage
5-27
b)
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A B C D E
Available Total Incremental
Resource 1 Activity 1 Activity 2 Profit Profit
6 2 $22.00
5 0 2.5 $12.50
6 0 3 $15.00 $2.50
7 0 3.5 $17.50 $2.50
8 0 4 $20.00 $2.50
9 3 3 $21.00 $1.00
10 6 2 $22.00 $1.00
11 9 1 $23.00 $1.00
12 12 0 $24.00 $1.00
13 12 0 $24.00 $0.00
14 12 0 $24.00 $0.00
15 12 0 $24.00 $0.00
Solution
The shadow price of $1 is valid in the range of 8 to 12.
c) With 1 additional unit of resource 2:
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A B C D E F
Activity 1 Activity 2
Unit Profit $2 $5
Used Available
Resource 1 1 2 10 <= 10
Resource 2 1 3 13 <= 13
Activity 1 Activity 2 Total Profit
Solution 4 3 $23.00
Resource Usage
5-28
d)
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18 0 5 $25.00 $0.00
A B C D E
Available Total Incremental
Resource 2 Activity 1 Activity 2 Profit Profit
6 2 $22.00
6 6 0 $12.00
7 7 0 $14.00 $2.00
8 8 0 $16.00 $2.00
9 9 0 $18.00 $2.00
10 10 0 $20.00 $2.00
11 8 1 $21.00 $1.00
12 6 2 $22.00 $1.00
13 4 3 $23.00 $1.00
14 2 4 $24.00 $1.00
15 0 5 $25.00 $1.00
16 0 5 $25.00 $0.00
17 0 5 $25.00 $0.00
Solution
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
Variable Cells
Final Reduced O bjective Allo wable Allowable
Cell Name Valu e Cost Coefficient Increase Decrease
$B$9 Solution Activity 1 6 0 2 0.5 0.333
$C$9 Solution Activity 2 2 0 5 1 1
Constraints
Final Shadow Constraint Allowable Allo wable
Cell Name Valu e Price R.H. Side Increase Decrease
$D$5 Resource 1 Used 10 110 2 2
$D$6 Resource 2 Used 12 112 3 2
5-29
b) When the right-hand-side of the first constraint is increased to 9, the new optimal
5-30
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c) Original model:
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A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 8 <= 8
Resource 2 1 1 4 <= 4
Activity 1 Activity 2 Total Profit
Solution 2 2 $6.00
Resource Usage
The shadow price for resource 1 is $0.50.
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A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 9 <= 9
Resource 2 1 1 4 <= 4
Activity 1 Activity 2 Total Profit
Solution 1.5 2.5 $6.50
Resource Usage
The shadow price for resource 2 is $0.50.
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A B C D E F
Activity 1 Activity 2
Unit Profit $1 $2
Used Available
Resource 1 1 3 8 <= 8
Resource 2 1 1 5 <= 5
Activity 1 Activity 2 Total Profit
Solution 3.5 1.5 $6.50
Resource Usage
5-32
d) The allowable range for the right-hand side of the resource 1 constraint is
approximately from 4 (or less) to 12.
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A B C D E
Available Total Incremental
Resource 1 Activity 1 Activity 2 Profit Profit
2 2 $6.00
4 4 0 $4.00
5 3.5 0.5 $4.50 $0.50
6 3 1 $5.00 $0.50
7 2.5 1.5 $5.50 $0.50
8 2 2 $6.00 $0.50
9 1.5 2.5 $6.50 $0.50
10 1 3 $7.00 $0.50
11 0.5 3.5 $7.50 $0.50
12 0 4 $8.00 $0.50
13 0 4 $8.00 $0.00
14 0 4 $8.00 $0.00
Solution
The allowable range for the right-hand side of the resource 2 constraint is
approximately from 3 to 8.
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A B C D E
Available Total Incremental
Resource 2 Activity 1 Activity 2 Profit Profit
2 2 $6.00
0 0 0 $0.00
1 0 1 $2.00 $2.00
2 0 2 $4.00 $2.00
3 0.5 2.5 $5.50 $1.50
4 2 2 $6.00 $0.50
5 3.5 1.5 $6.50 $0.50
6 5 1 $7.00 $0.50
7 6.5 0.5 $7.50 $0.50
8 8 0 $8.00 $0.50
9 8 0 $8.00 $0.00
10 8 0 $8.00 $0.00
Solution
Variable Cells
Final Reduced Objective Allo w able Allo w able
Cell Name Value Cost Coefficient Increase Decrease
$B$9 Solution Activity 1 2 0 1 1 0.333
$C$9 Solution Activity 2 2 0 2 1 1
Constraints
Final Shado w Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$D$5 Resource 1 Used 8 0.5 8 4 4
$D$6 Resource 2 Used 4 0.5 4 4 1.333
5-33
5.14 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.
5-34
When the right-hand-side of the second constraint is increased to 16, the new optimal
When the right-hand-side of the third constraint is increased to 11, the new optimal
5-35
c) Original model:
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A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3 <= 4
Resource 2 1 3 15 <= 15
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 3 4 $17.00
Resource Usage
The shadow price for resource 1 is $0.
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A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3 <= 5
Resource 2 1 3 15 <= 15
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 3 4 $17.00
Resource Usage
The shadow price for resource 2 is $0.20.
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A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 2.8 <= 4
Resource 2 1 3 16 <= 16
Resource 3 2 1 10 <= 10
Activity 1 Activity 2 Total Profit
Solution 2.8 4.4 $17.20
Resource Usage
The shadow price for resource 3 is $1.40.
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A B C D E F
Activity 1 Activity 2
Unit Profit $3 $2
Used Available
Resource 1 1 0 3.6 <= 4
Resource 2 1 3 15 <= 15
Resource 3 2 1 11 <= 11
Activity 1 Activity 2 Total Profit
Solution 3.6 3.8 $18.40
Resource Usage
5-36
d) The allowable range for the right-hand side of the resource 1 constraint is
approximately from 3 to at least 10.
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A B C D E
Available Total Incremental
Resource 1 Activity 1 Activity 2 Profit Profit
3 4 $17.00
0 0 5 $10.00
1 1 4.667 $12.33 $2.33
2 2 4.333 $14.67 $2.33
3 3 4 $17.00 $2.33
4 3 4 $17.00 $0.00
5 3 4 $17.00 $0.00
6 3 4 $17.00 $0.00
7 3 4 $17.00 $0.00
8 3 4 $17.00 $0.00
9 3 4 $17.00 $0.00
10 3 4 $17.00 $0.00
Solution
The allowable range for the right-hand side of the resource 2 constraint is
approximately from less than 11 to more than 21.
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A B C D E
Available Total Incremental
Resource 2 Activity 1 Activity 2 Profit Profit
3 4 $17.00
11 3.8 2.4 $16.20
12 3.6 2.8 $16.40 $0.20
13 3.4 3.2 $16.60 $0.20
14 3.2 3.6 $16.80 $0.20
15 3 4 $17.00 $0.20
16 2.8 4.4 $17.20 $0.20
17 2.6 4.8 $17.40 $0.20
18 2.4 5.2 $17.60 $0.20
19 2.2 5.6 $17.80 $0.20
20 2 6 $18.00 $0.20
21 1.8 6.4 $18.20 $0.20
Solution
5-38
5.15
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B C D E F G H
TV Spots Magazine Ads SS Ads
Exposures per Ad 1,300 600 500
(thousands)
Budget Budget
Cost per Ad ($thousands) Spent Available
Ad Budget 300 150 100 4,000 <= 4,000
Planning Budget 90 30 40 1,000 <= 1,000
Total Exposures
TV Spots Magazine Ads SS Ads (thousands)
Number of Ads 0 20 10 17,000
<=
Max TV Spots 5
Variable Cells
Final Reduced Objective Allowable Allowable
Cell Name Valu e Cost Coefficient Increase Decrease
$C$13 TVSpots 0 -50 1300 50 1E+ 30
$D$13 Number of Ads Magazine Ads 20 0 600 150 50
$E$13 Number of Ads SS Ads 10 0 500 300 33.333
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Valu e Price R.H. Side Increase Decrease
$F$8 Ad Budget Spent 4,000 3 4000 1000 1500
$F$9 Planning Budget Spent 1,000 5 1000 600 200
a) The total number of expected exposures could be increased by 3,000 for each additional
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%
f) The $100,000 should be added to the planning budget since this will add 500,000
5-39
5.16
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B C D E F G H
TV Spots Magazine Ads SS Ads
Exposures per Ad 1,300 600 500
(thousands)
Cost per Ad ($thousands) Budget Spent Budget Available
Ad Budget 300 150 100 3,775 <= 4,000
Planning Budget 90 30 40 1,000 <= 1,000
Number Reached per Ad (millions) Total Reached Minimum Acceptable
Young Children 1.2 0.1 0 5 >= 5
Parents of Young Children 0.5 0.2 0.2 5.85 >= 5
TV Spots Magazine Ads SS Ads Total Redeemed Required Amount
Coupon Redemption per Ad 0 40 120 1,490 = 1,490
($thousands)
Total Exposures
TV Spots Magazine Ads SS Ads (thousands)
Number of Ads 3 14 7.75 16,175
<=
Maximum TV Spots 5
Variable Cells
Final Redu ced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
$C$19 TVSpots 3 0 1300 1040 1E+ 30
$D$19 Number of Ads Magazine Ads 14 0 600 1E+ 30 192.59
$E$19 Number of Ads SS Ads 7.75 0 500 577.78 1E+ 30
Constraints
Final Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease
$F$7 Ad Budget Budget Spent 3,775 0 4000 1E+ 30 225
$F$8 Planning Budget Budget Spent 1,000 35 1000 22.5 85
$F$15 TotalRedeemed 1,490 -8 1490 385 90
$F$11 Young Children Total Reached 5 -1575.76 5 1.32 0.45
$F$12 Parents of Young Children Total Reached 5.85 0 5 0.85 1E+ 30
a) The total number of expected exposures can not be increased by adding an additional
d) This remains valid for increases of up to $22,500.
f) $100,000 is beyond the allowable increase for the planning budget. Therefore, the total
