5-41
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
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 ∞
c) Percentage of allowable increase for 6am–8am = (49 – 48) / 6 = 16.7%
Percentage of allowable increase for 8am–10am = (80 – 79) / ∞ = 0%
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%
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.
5-42
Cases
5.1 a) Original Solution: 4 units of television advertising and 3 units of print media
advertising, with a total cost of $10 million.
Increasing the required minimum increase in sales for Stain Remover by 1% changes
5-43
Increasing the required minimum increase in sales for Liquid Detergent by 1% changes
Increasing the required minimmum increase in sales for Powder Detergent by 1% has
no impact on the solution nor the total cost.
5-44
b) Original Solution:
3
4
5
6
7
8
9
10
11
12
13
14
B C D E F G
Television Print Media
Unit Cost ($millions) 1 2
Increased Minimum
Sales Increase
Stain Remover 0% 1% 3% >= 3%
Liquid Detergent 3% 2% 18% >= 18%
Powder Detergent -1% 4% 8% >= 4%
Total Cost
Television Print Media ($millions)
Advertising Units 4 3 10
Increasing the required minimum increase in sales for Stain Remover by 1% increases
the total cost by $1.333 million.
3
4
5
6
7
8
9
10
11
12
13
14
B C D E F G
Television Print Media
Unit Cost ($millions) 1 2
Increased Minimum
Sales Increase
Stain Remover 0% 1% 4% >= 4%
Liquid Detergent 3% 2% 18% >= 18%
Powder Detergent -1% 4% 13% >= 4%
Total Cost
Television Print Media ($millions)
Advertising Units 3.333 4 11.333
Increasing the required minimum increase in sales for Liquid Detergent by 1%
3
4
5
6
7
8
9
10
11
12
13
14
B C D E F G
Television Print Media
Unit Cost ($millions) 1 2
Increased Minimum
Sales Increase
Stain Remover 0% 1% 3% >= 3%
Liquid Detergent 3% 2% 19% >= 19%
Powder Detergent -1% 4% 8% >= 4%
Total Cost
Television Print Media ($millions)
Advertising Units 4.333 3 10.333
5-45
Increasing the required minimmum increase in sales for Powder Detergent by 1% has
no impact on the total cost.
3
4
5
6
7
8
9
10
11
12
13
14
B C D E F G
Television Print Media
Unit Cost ($millions) 1 2
Increased Minimum
Sales Increase
Stain Remover 0% 1% 3% >= 3%
Liquid Detergent 3% 2% 18% >= 18%
Powder Detergent -1% 4% 8% >= 5%
Total Cost
Television Print Media ($millions)
Advertising Units 4 3 10
c)
17
18
19
20
21
22
23
24
25
B C D E F
Minimum Increase Total Cost Incremental
Stain Remover Television Print Media ($millions) Cost ($million)
4 3 10
0% 4.571 2.143 8.857
1% 4.571 2.143 8.857 0.000
2% 4.571 2.143 8.857 0.000
3% 4 3 10.000 1.143
4% 3.333 4 11.333 1.333
5% 2.667 5 12.667 1.333
Advertising Units
54
55
56
57
58
59
60
62
63
65
66
67
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
B C D E F
Minimum Increase Total Cost Incremental
Liquid Detergent Television Print Media ($millions) Cost ($million)
4 3 10.000
0% 0 3 6.000
1% 0 3 6.000 0.000
2% 0 3 6.000 0.000
3% 0 3 6.000 0.000
4% 0 3 6.000 0.000
5% 0 3 6.000 0.000
6% 0 3 6.000 0.000
7% 0.333 3 6.333 0.333
8% 0.667 3 6.667 0.333
9% 1 3 7.000 0.333
10% 1.333 3 7.333 0.333
11% 1.667 3 7.667 0.333
12% 2 3 8.000 0.333
13% 2.333 3 8.333 0.333
14% 2.667 3 8.667 0.333
15% 3 3 9.000 0.333
16% 3.333 3 9.333 0.333
17% 3.667 3 9.667 0.333
18% 4 3 10.000 0.333
19% 4.333 3 10.333 0.333
20% 4.667 3 10.667 0.333
21% 5 3 11.000 0.333
Advertising Units
Chapter 05 – What-If Analysis for Linear Programming
71
72
73
74
75
76
77
78
79
80
81
82
B C D E F
Minimum Increase Total Cost Incremental
Powder Detergent Television Print Media ($millions) Cost ($million)
4 3 10
0% 4 3 10
1% 4 3 10 0.000
2% 4 3 10 0.000
3% 4 3 10 0.000
4% 4 3 10 0.000
5% 4 3 10 0.000
6% 4 3 10 0.000
7% 4 3 10 0.000
8% 4 3 10 0.000
Advertising Units
d) Sensitivity Report:
Variable Cells
F in al Red uced O bjective All ow ab le All ow ab le
Cell Name Value Cost Coefficient Increase Decrease
$C$14 Advertising Units Television 4 0 1 2 1
$D$14 Advertising Units Print Media 3 0 2 1E+ 30 1.333
Constraints
F in al Sh ad ow Co nstraint All ow ab le All ow ab le
Cell Name Value Price R.H. Side Increase Decrease
$E$8 Stain Remover Sales 3% 133.33 0.03 0.06 0.008571429
$E$9 Liquid Detergent Sales 18% 33.33 0.18 0.12 0.12
$E$10 Powder Detergent Sales 8% 0 0.04 0.04 1E+ 30
The shadow price indicates the increase in total cost (in $millions) per unit increase in
the right hand side (i.e., per 100% increase). Thus, a 1% increase in the minimum
required increase in sales will only increase the total cost by one hundredth of the
The allowable range for the required minimum increase in sales constraint for Stain
Remover is 2.15% to 9%.
These allowable ranges can also be seen in the results from part (c). For Stain Remover,
5-48
e) Suppose that each of the original numbers in MinimumIncrease (G8:G10) is increased
by 1%.
Percent of allowable increase for Stain Remover used = (4% – 3%) / 6% = 16.7%.
Thus, if each of the original numbers in MinimumIncrease (G8:G10) is increased by
f) Answers will vary.
5.2 a) The decisions to be made are which types of abatement methods will be used and at
what fractions of their abatement capacities for the blast furnaces and the open-hearth
b & c)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
A B C D E F G H I J
Blast Open-Hearth Blast Open-Hearth Blast Open-Hearth
Furnaces Furnaces Furnaces Furnaces F urnaces Furnaces
Cost ($million ) 810 7 6 11 9
Total Minimum
Reduction Reduction
Pollutant (millions of lbs.) (millions of lbs.)
Particulates 12 925 20 17 13 65 >= 60
Sulfur oxides 35 42 18 31 56 49 150 >= 150
Hydrocarbons 37 53 28 24 29 20 125 >= 125
Blast Open-Hearth Blast Open-Hearth Blast O pen–Hearth Total Cost
Furnaces Furnaces Furnaces Furnaces F urnaces Furnaces ($million)
Fraction Used 100% 48.55% 57.31% 100% 7.67% 100% 32.710
<= <= <= <= <= <=
Taller Smokestacks
Filters
Better Fuels
Taller Smokestacks
Filters
Better Fuels
Reduction in Emission (Maximum F easible Use of Abatement Method)
13
14
J
Total Cost
($million)
5-50
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
Current
Value
10%
Less
Value
Solution
Changes?
10%
More
Value
Solution
Changes?
Taller Smoke (Blast)
8
7.2
No
8.8
Yes
Taller Smoke (Open H)
10
9
Yes
11
Yes
Filter (Blast)
7
6.3
No
7.7
Yes
Filter (Open H)
6
5.4
No
6.6
No
Better Fuel (Blast)
11
9.9
Yes
12.1
No
Better Fuel (Open H)
9
8.1
No
9.9
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
f)
Pollutant
Rate that
cost changes
($million)
Maximum increase
before rate changes
(million lb.)
Maximum decrease
before rate changes
(million lb.)
Particulates
0.111
14.297
7.480
Sulfur oxides
0.127
20.453
1.690
Hydrocarbons
0.069
2.042
21.692
5-51
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.
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.
Particulates and both sulfur oxides and hydrocarbons:
For each unit increase in particulate reduction, cost will increase by $0.111 million.
5-52
h)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
A B C D E F G H I J K
Percent Increase in Reduction
Blast Open–Hearth Blast Open-Hearth Blast Open-Hearth = 40%
Furnaces Furnaces Furnaces Furnaces F urnaces Furnaces
Cost ($millio n) 810 7 6 11 9 Original
Total Minimum Minimum
Reduction Reduction Reduction
Poll utant (millions of lbs.) (millions of lbs.) (millions of lbs.)
Particulates 12 925 20 17 13 84 >= 84 60
Sulfur oxides 35 42 18 31 56 49 210 >= 210 150
Hydrocarbons 37 53 28 24 29 20 175 >= 175 125
Blast Open-Hearth Blast Open-Hearth Blast Open-Hearth Total Cost
Furnaces Furnaces Furnaces Furnaces F urnaces Furnaces ($million)
Fractio n Used 100% 100.00% 70.53% 100% 78.16% 93% 45.898
<= <= <= <= <= <=
100% 100% 100% 100% 100% 100%
Percent Incremental
Increase Blast O pen-Hearth Blast Open-Hearth Blast O pen-Hearth Total Cost Cost
in Reduction Furnaces Furnaces F urnaces F urnaces Furnaces F urnaces ($million) ($million)
100% 100.00% 70.53% 100% 78.16% 93% 45.898
0% 100% 62.27% 34.35% 100% 4.76% 100% 32.155
10% 100% 71.88% 43.60% 100% 21.36% 100% 35.590 3.435
20% 100% 81.50% 52.85% 100% 37.96% 100% 39.025 3.435
30% 100% 91.11% 62.10% 100% 54.56% 100% 42.460 3.435
40% 100% 100% 70.53% 100% 78.16% 92.93% 45.898 3.438
Taller Smokestacks
Filters
Better Fuels
Taller Smokestacks
Filters
Better Fuels
Taller Smokestacks
Filters
Better Fuels
Reduction in Emission (Maximum Feasible Use of Abatement Method)
Each 10% reduction in pollution costs less than $3.5 million (the tax incentive) until the
i) The sensitivity report for a 40% reduction is shown below.
Variable Cells
Final Reduced Objective Allow able Allowable
Cell Name Value Cost Coefficient In crease Decrease
$B$15 Fraction Taller Smokestack (Blast) 100% -55% 8 0.553 1E+ 30
$C$15 Fraction Taller Smokestack (Open Hearth) 100.00% –42.94% 10 0. 429 1E+ 30
$D$15 Fraction Filter (Blast) 70.53% 0.00% 7 0.382 1.292
$E$15 Fraction Filter (Open Hearth) 100% -179% 6 1.789 1E+ 30
$F$15 Fraction Better Fuel (Blast) 78.16% 0.00% 11 0.384 0.045
$G$15 Fraction Better Fuel (Open Hearth) 93% 0% 9 0.044 0.372
Constraints
Final Shadow Constraint Allowable Allo wable
Cell Name Value Price R.H. Side Increase Decrease
$H$8 Particulates (millions of lbs.) 84 0.099 84 0.265 0.846
$H$9 Sulfur oxides (millions of lbs.) 210 0.124 210 1.112 6.294
$H$10 Hydrocarbons (millions of lbs.) 175 0.082 175 0.864 0.253
Pollutant
Rate that
cost changes
($million)
Maximum increase
before rate changes
(million lb.)
Maximum decrease
before rate changes
(million lb.)
Particulates
0.099
0.265
0.846
Sulfur oxides
0.124
1.112
6.294
Hydrocarbons
0.082
0.864
0.253
5-53
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.
Particulates and hydrocarbons:
For each unit increase in particulate reduction, cost will increase by $0.099 million.
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
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
5-54
b & c)
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
A B C D E F G
Plantings Planting
Soybeans Corn W heat T otals
W &S Hours Required 1 0.9 0.6 537
S&F Hours Required 1.4 1.2 0.7 736
Net Value $70 $60 $40 $37,300
Acres Planted 450 30 100 580
>= >=
30 100
1 0.05
acre/cow acre/hen
Liv estock Livestock
Cows Hens Totals
Hours Required per Month 10 0.05 400
Grazing Land Required 2 0 60
Net Annual Cash Income $850 $4 $34,000
Beginning Value (Current Livestock) $35,000 $5,000
Decrease in Value per Year 10% 25%
End Value (Current Livestock) $31,500 $3,750 $35,250 Investment
Fund
Cost of New Livestock $1,500 $3 $0 <= $20,000
End Value (New Livestock) $1,350 $2 $0
Current Livestock 30 2,000
New Livestock 0 0
Total Livestock 30 2,000
<= <=
Barn/House Limits 42 5,000
Neighboring Farm Work Neighbor
W &S S&F T otals
W age $5 $5.50 $12,817.00
Hours W orked 1063 1364 2427
Totals Plantings Livestock Neighbor Total Available
W &S Hours 537 2,400 1,063 4,000 <= 4,000
S&F Hours 736 2,400 1,364 4,500 <= 4,500
Acreage 580 60 0 640 <= 640
Net Income $37,300 $34,000 $12,817 $84,117
End of Year Value $35,250 $35,250
Leftover Investment Fund $20,000 $20,000
Living Expenses -$40,000
Total Monetary Worth $99,367
This model predicts that the family’s monetary worth at the end of the coming year will
5-55
1
2
3
4
5
6
7
8
9
10
A B C D E
Plantings Planting
Soybeans Corn Wheat Totals
W&S Hours Required 1 0.9 0.6 =SUMPRODUCT(B3:D3,AcresPlanted)
S&F Hours Required 1.4 1.2 0.7 = SUMPRODUCT(B4:D4,AcresPlanted)
Net Value 70 60 40 =SUMPRODUCT(B5:D5,AcresPlanted)
Acres Planted 450 30 100 =SUM(AcresPlanted)
>= >=
=C10*B28 =D10*C28
1 0.05
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
A B C D E F
Liv estock Livestock
Cows Hens Totals
Hours Required per Month 10 0.05 = SUMPRO DUCT(B 15: C15,T otalLivest ock)
Grazing Land Required 2 0 =SUMPRODUCT(B16:C16,TotalLivestock)
Net Annual Cash Income 850 4.25 = SUMPRO DUCT(B17:C17,T otalLivestock)
Beginning Value (Current Livestock) 35000 5000
Decrease in Value per Year 0.1 0.25
End Value (Current Livestock)=(1-B20)*B19 = (1-C20)*C19 = SUM(B21:C21) Investment
Fund
Cost of New Livestock1500 3 = SUMPRO DUCT(B23:C23,NewLivestock) <= 20000
End Value (New Livestock) = (1-B20)*B23 = (1-C20)*C23 = SUMPRO DUCT(B24: C24,NewLivestock)
Current Livestock 30 2000
New Livestock 0 0
Total Livestock = CurrentLivest ock+NewLivestock = CurrentLivestock+NewLivestock
<= <=
Barn/House Limits 42 5000
32
33
34
35
36
A B C D
Neighboring Farm Work Neighbor
W&S S&F Totals
Wage 5 5.5 =SUMPRODUCT(Wage,HoursW orked)
Hours Worked 1063 1364 = SUM(HoursW orked)
38
39
40
41
42
43
44
45
46
47
A B C D E F G
Totals Plantings Livestock Neighbor Total Available
W &S Hours = E3 = 6*D15 = B36 = SUM(B39:D39) <= 4000
S&F Hours = E4 = 6*D15 = C36 = SUM(B40:D40) <= 4500
Acreage = E7 = D16 0 = SUM(B41: D41) <= 640
Net Income =E5 = D17 =D34 = SUM(B43:D43)
End of Year Value = D21+ D24 = SUM(B44:D44)
Leftover Investment Fund = InvestmentFund-D23 = SUM(B45:D45)
Living Expenses -40000
Total Monetary W orth = SUM(E43:E46)
Rang e Name Cells
AcresPlanted B7:D7
Available G39:G41
BarnHouseLimits B30:C30
CurrentLivestock B26:C26
HoursW orked B36:C36
InvestmentFund F 23
MonetaryWorth E47
NewLivestock B27:C27
Total E39:E41
TotalLivestock B28:C28
Wage B34:C34