ch02 solutions
DESCRIPTION
aaaaTRANSCRIPT
CHAPTER 2
LINEAR PROGRAMMING MODELS: GRAPHICAL AND COMPUTER METHODS
Note: Permission to use the computer program GLP for all LP graphical solution screenshots in this chapter granted by its author, Jeffrey H. Moore, Graduate School of Business, Stanford University. Software copyrighted by Board of Trustees of the Leland Stanford Junior University. All rights reserved.
SOLUTIONS TO DISCUSSION QUESTIONS
2-1. The requirements for an LP problem are listed in Section 2.2. It is also assumed that conditions of certainty exist; that is, coefficients in the objective function and constraints are known with certainty and do not change during the period being studied. Another basic assumption that mathematically sophisticated students should be made aware of is proportionality in the objective function and constraints. For example, if one product uses 5 hours of a machine resource, then making 10 of that product uses 50 hours of machine time.
LP also assumes additivity. This means that the total of all activities equals the sum of each individual activity. For example, if the objective function is to maximize Profit = 6X1 + 4X2, and if X1 = X2 = 1, the profit contributions of 6 and 4 must add up to produce a sum of 10.
2-2. If we consider the feasible region of an LP problem to be continuous (i.e., we accept non-integer solutions as valid), there will be an infinite number of feasible combinations of decision variable values (unless of course, only a single solution satisfies all the constraints). In most cases, only one of these feasible solutions yields the optimal solution.
2-3. A problem can have alternative optimal solutions if the level profit or level cost line runs parallel to one of the problem’s binding constraints (refer to Section 2.6 in the chapter).
2-4. A problem can be unbounded if one or more constraints are missing, such that the objective value can be made infinitely larger or smaller without violating any constraints (refer to Section 2.6 in the chapter).
2-5. This question involves the student using a little originality to develop his or her own LP constraints that fit the three conditions of (1) unbounded solution, (2) infeasibility, and (3) redundant constraints. These conditions are discussed in Section 2.6, but each student’s graphical displays should be different.
2-6. The manager’s statement indeed has merit if he/she understood the deterministic nature of LP input data. LP assumes that data pertaining to demand, supply, materials, costs, and resources are known with certainty and are constant during the time period being analyzed. If the firm operates in a very unstable environment (for example, prices and availability of raw materials change daily, or even hourly), the LP model’s results may be too sensitive and volatile to be trusted. The application of sensitivity analysis might, however, be useful to determine whether LP would still be a good approximating tool in decision making in this environment.
2-7. The objective function is not linear because it contains the product of X1 and X2, making it a second-degree term. The first, second, and fourth constraints are okay as is. The third and fifth constraints are
nonlinear because they contain terms to the second degree and one-half degree, respectively.
2-8. The computer is valuable in (1) solving LP problems quickly and accurately; (2) solving large problems that might take days or months by hand; (3) performing extensive sensitivity analysis automatically; and (4) allowing a manager to try several ideas, models, or data sets.
2-9. Most managers probably have Excel (or another spreadsheet software) available in their companies, and use it regularly as part of their regular activities. As such, they are likely to be familiar with its usage. In addition, a lot of the data (such as parameter values) required for developing LP models is likely to be available either in some Excel file or in a database file (such as Microsoft Access) from which it is easy to import to Excel. For these reasons, a manager may find the ability to use Excel to set up and solve LP problems very beneficial.
2-10. The three components are: target cell (objective function), changing cells (decision variables), and constraints.
2-11. Slack is defined as the RHS minus the LHS value for a constraint. It may be interpreted as the amount of unused resource described by the constraint. Surplus is defined as the LHS minus the RHS value for a ≥ constraint. It may be interpreted as the amount of over satisfaction of the constraint.
2-12. An unbounded solution occurs when the objective of an LP problem can go to infinity (negative infinity for a minimization problem) while satisfying all constraints. Solver indicates an unbounded solution by the message “The Set Cell values do not converge”.
SOLUTIONS TO PROBLEMS
2-13.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
01234567891011121314151617181920Y
X
: 5.00 X + 2.00 Y = 40.00
: 3.00 X + 6.00 Y = 48.00
: 1.00 X + 0.00 Y = 7.00
: 2.00 X - 1.00 Y = 3.00
Payoff: 5.00 X + 3.00 Y = 45.00
Optimal Decisions(X ,Y): (6.00, 5.00)
: 5.00X + 2.00Y <= 40.00
: 3.00X + 6.00Y <= 48.00
: 1.00X + 0.00Y <= 7.00
: 2.00X - 1.00Y >= 3.00
See file P2-13.XLS.X Y
Solution 6.00 5.00Obj coeff 5 3 45.00Constraints:Constraint 1 5 2 40.00 40Constraint 2 3 6 48.00 48Constraint 3 1 6.00 7Constraint 4 2 -1 7.00 3
LHS Sign RHS
2-14.
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88
036912151821242730333639424548515457606366697275Y
X
: 1.00 X + 3.00 Y = 90.00
: 8.00 X + 2.00 Y = 160.00
: 0.00 X + 1.00 Y = 70.00
: 3.00 X + 2.00 Y = 120.00
Payoff: 1.00 X + 2.00 Y = 68.57
Optimal Decisions(X ,Y ): (25.71, 21.43)
: 1.00X + 3.00Y >= 90.00
: 8.00X + 2.00Y >= 160.00
: 0.00X + 1.00Y <= 70.00
: 3.00X + 2.00Y >= 120.00
See file P2-14.XLS.X Y
Solution 25.71 21.43Obj coeff 1 2 68.57Constraints:Constraint 1 1 3 90.00 90Constraint 2 8 2 248.57 160Constraint 3 3 2 120.00 120Constraint 4 1 21.43 70
LHS Sign RHS
2-15.
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88
0
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
72
75Y
X
: 3.00 X + 7.00 Y = 231.00
: 10.00 X + 2.00 Y = 200.00
: 0.00 X + 2.00 Y = 45.00
: 2.00 X + 0.00 Y = 75.00Payoff: 4.00 X + 7.00 Y = 245.65
Optimal Decisions(X ,Y): (14.66, 26.72)
: 3.00X + 7.00Y >= 231.00
: 10.00X + 2.00Y >= 200.00
: 0.00X + 2.00Y >= 45.00
: 2.00X + 0.00Y <= 75.00
See file P2-15.XLS.X Y
Solution 14.66 26.72Obj coeff 4 7 245.66Constraints:Constraint 1 3 7 231.00 231Constraint 2 10 2 200.00 200Constraint 3 2 53.44 45Constraint 4 2 29.31 75
LHS Sign RHS
2-16.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25Y
X
: 3.00 X + 6.00 Y = 29.00
: 7.00 X + 1.00 Y = 20.00
: 3.00 X - 1.00 Y = 1.00
Payoff: 1.00 X + 1.00 Y = 6.00
Optimal Decisions(X ,Y ): (2.33, 3.67)
: 3.00X + 6.00Y <= 29.00
: 7.00X + 1.00Y <= 20.00
: 3.00X - 1.00Y >= 1.00
See file P2-16.XLS.X Y
Solution 2.33 3.67Obj coeff 1 1 6.00Constraints:Constraint 1 3 6 29.00 29Constraint 2 7 1 20.00 20Constraint 3 3 -1 3.33 1
LHS Sign RHS
2-17.
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15Y
X
: 9.00 X + 8.00 Y = 72.00
: 3.00 X + 9.00 Y = 27.00
: 9.00 X - 15.00 Y = 0.00
Payoff: 7.00 X + 4.00 Y = 54.94
Optimal Decisions(X ,Y ): (7.58, 0.47)
: 9.00X + 8.00Y <= 72.00
: 3.00X + 9.00Y >= 27.00
: 9.00X - 15.00Y >= 0.00
See file P2-17.XLS.X Y
Solution 7.58 0.47Obj coeff 7 4 54.95Constraints:Constraint 1 9 8 72.00 72Constraint 2 3 9 27.00 27Constraint 3 9 -15 61.11 0
LHS Sign RHS
2-18.
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15Y
X
: 9.00 X + 3.00 Y = 36.00: 4.00 X + 5.00 Y = 40.00
: 1.00 X - 1.00 Y = 0.00
: 2.00 X + 0.00 Y = 13.00
Payoff: 3.00 X + 7.00 Y = 44.44
Optimal Decisions(X ,Y ): (4.44, 4.44)
: 9.00X + 3.00Y >= 36.00
: 4.00X + 5.00Y >= 40.00
: 1.00X - 1.00Y <= 0.00
: 2.00X + 0.00Y <= 13.00
See file P2-18.XLS.X Y
Solution 4.44 4.44Obj coeff 3 7 44.44Constraints:Constraint 1 9 3 53.33 36Constraint 2 4 5 40.00 40Constraint 3 1 -1 0.00 0Constraint 3 2 8.89 13
LHS Sign RHS
2-19. See file P2-19.XLS.
(a) Formulation 2 has multiple optimal solutions(b) Formulation 3 has an unbounded solution(c) Formulation 1 is infeasible(d) Formulation 4 has a unique optimal solution
Formulation 1 (Infeasible)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10Y
X
: 2.00 X + 1.00 Y = 6.00
: 4.00 X + 5.00 Y = 20.00
: 0.00 X + 2.00 Y = 7.00
: 2.00 X + 0.00 Y = 7.00
Payoff: 3.00 X + 7.00 Y = 12.00
: 2.00X + 1.00Y <= 6.00
: 4.00X + 5.00Y <= 20.00
: 0.00X + 2.00Y <= 7.00
: 2.00X + 0.00Y >= 7.00
Formulation 2 (Multiple Optimal Solutions)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10Y
X
: 7.00 X + 6.00 Y = 42.00
: 1.00 X + 2.00 Y = 10.00
: 1.00 X + 0.00 Y = 4.00
: 0.00 X + 2.00 Y = 9.00Payoff: 3.00 X + 6.00 Y = 30.00
Optimal Decisions(X ,Y): (3.00, 3.50) (1.00, 4.50)
: 7.00X + 6.00Y <= 42.00
: 1.00X + 2.00Y <= 10.00
: 1.00X + 0.00Y <= 4.00
: 0.00X + 2.00Y <= 9.00
2-19 (continued). See file P2-19.XLS.
Formulation 3 (Unbounded Solution)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10Y
X
: 1.00 X + 2.00 Y = 12.00
: 8.00 X + 7.00 Y = 56.00
: 0.00 X + 2.00 Y = 5.00
: 1.00 X + 0.00 Y = 9.00
Payoff: 2.00 X + 3.00 Y = 10.00
Optimal Decisions(X ,Y ): (9.00, 27.24)
: 1.00X + 2.00Y >= 12.00
: 8.00X + 7.00Y >= 56.00
: 0.00X + 2.00Y >= 5.00
: 1.00X + 0.00Y <= 9.00
Formulation 4 (Unique Optimal Solution)
0 1 2 3 4 5 6 7 8 9 10
0
1
2
3
4
5
6
7
8
9
10Y
X
: 3.00 X + 7.00 Y = 21.00
: 2.00 X + 1.00 Y = 6.00
: 1.00 X + 1.00 Y = 2.00
: 2.00 X + 0.00 Y = 2.00
Payoff: 3.00 X + 4.00 Y = 14.45
Optimal Decisions(X ,Y): (1.91, 2.18)
: 3.00X + 7.00Y <= 21.00
: 2.00X + 1.00Y <= 6.00
: 1.00X + 1.00Y >= 2.00
: 2.00X + 0.00Y >= 2.00
2-20.
See file P2-20.XLS.A B C
Solution 40.00 30.00 30.00Obj coeff 28 41 38 3,490.00Constraints:Constraint 1 10 15 -8 610.00 610.00Constraint 2 0.4 0.4 0.4 40.00 40.00Constraint 3 1 40.00 90.00Constraint 4 1 30.00 30.00
LHS Sign RHS
2-21. Let X = number of large sheds to build, Y = number of small sheds to build.
Objective: Maximize revenue = $50X + $20Y
Subject to:X + Y 100 Advertising. Budget150X + 50Y 8,000 Sq feet requiredX 40 Rental limitX, Y 0 Non-negativity
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
9
18
27
36
45
54
63
72
81
90
99
108
117
126
135
144
153
162
171
180Y
X
: 1.00 X + 1.00 Y = 100.00
: 150.00 X + 50.00 Y = 8000.00
: 1.00 X + 0.00 Y = 40.00
Payoff: 50.00 X + 20.00 Y = 2900.00
Optimal Decisions(X ,Y ): (30.00, 70.00)
: 1.00X + 1.00Y <= 100.00
: 150.00X + 50.00Y <= 8000.00
: 1.00X + 0.00Y <= 40.00
See file P2-21.XLS.Large Small
Number of sheds 30.00 70.00Rent $50 $20 $2,900.00Constraints:Advt. budget $1 $1 $100.00 $100Sq feet required 150 50 8,000.00 8,000Rental limit 1 30.00 40
LHS Sign RHS
2-22. Let X = number of copies of Backyard, Y= number of copies of Porch.
Objective: Maximize revenue = $3.50X + $4.50Y
Subject to:2.5X + 2Y 2,160 Print time, minutes1.8X + 2Y 1,800 Collate time, minutesX, Y 0 Non-negativity
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000
0
60
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300
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540
600
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720
780
840
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1020
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1140
1200Y
X
: 2.50 X + 2.00 Y = 2160.00
: 1.80 X + 2.00 Y = 1800.00
: 1.00 X + 0.00 Y = 400.00
: 0.00 X + 1.00 Y = 300.00
Payoff: 3.50 X + 4.50 Y = 3830.00
Optimal Decisions(X ,Y): (400.00, 540.00)
: 2.50X + 2.00Y <= 2160.00
: 1.80X + 2.00Y <= 1800.00
: 1.00X + 0.00Y >= 400.00
: 0.00X + 1.00Y >= 300.00
See file P2-22.XLS.Backyard Porch
Number of copies 400.00 540.00Revenue $3.50 $4.50 $3,830.0Constraints:Print time, minutes 2.5 2.0 2,080.0 <= 2,160Collate time, minutes 1.8 2.0 1,800.0 <= 1,800Min Backyard to print 1 400.0 >= 400Min Porch to print 1 540.0 >= 300
LHS Sign RHS
2-23. Let X = number of small boxes, Y = number of large boxes.
Objective: Maximize revenue = $30X + $40Y
Subject to:0.50X + 0.85Y 240 Square feet availableX + Y 350 Min required, total
Y 80 Min required, largeX, Y 0 Non-negativity
0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500
0
20
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100
120
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220
240
260
280
300
320
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360
380
400Y
X
: 0.50 X + 0.85 Y = 240.00: 1.00 X + 1.00 Y = 350.00
: 0.00 X + 1.00 Y = 80.00
Payoff: 30.00 X + 40.00 Y = 13520.00
Optimal Decisions(X ,Y ): (344.00, 80.00)
: 0.50X + 0.85Y <= 240.00
: 1.00X + 1.00Y >= 350.00
: 0.00X + 1.00Y >= 80.00
See file P2-23.XLS.Small Large
Number of boxes 344.00 80.00Rent $30 $40 $13,520.00
2-24. Let X = number of pounds of compost, Y = number of pounds of sewage in each bag.
Objective: Minimize cost = $0.05X + $0.04Y
Subject to:X + Y 60 Pounds per bag2X + Y 100 Fertilizer ratingX 35 Min compost, pounds
Y 40 Max sewage, poundsX, Y 0 Non-negativity
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
02468101214161820222426283032343638404244464850Y
X
: 1.00 X + 1.00 Y = 60.00
: 1.00 X + 0.00 Y = 35.00
: 0.00 X + 1.00 Y = 40.00
: 2.00 X + 1.00 Y = 100.00
Payoff: 0.05 X + 0.04 Y = 2.80
Optimal Decisions(X ,Y ): (40.00, 20.00)
: 1.00X + 1.00Y >= 60.00
: 1.00X + 0.00Y >= 35.00
: 0.00X + 1.00Y <= 40.00
: 2.00X + 1.00Y >= 100.00
See file P2-24.XLS.Compost Sewage
Number of pounds 40.00 20.00Cost $0.05 $0.04 $2.80
2-25. Let X = thousand of dollars to invest in Treasury notes, Y = thousand of dollars to invest in Municipal bonds.
Objective: Maximize return = 8X + 9Y
Subject to:X + Y $250,000 Amount availableX 0.7(X+Y) Max T-notes2X + 3Y 2.42(X+Y) Max risk scoreX 0.3(X+Y) Min T-notesX, Y 0 Non-negativity
0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
0
7
14
21
28
35
42
49
56
63
70
77
84
91
98
105
112
119
126
133
140
147
Y
X
: 1.00 X + 1.00 Y = 250.00
: 0.50 X - 0.50 Y = 0.00: 0.30 X - 0.70 Y = 0.00
: -0.42 X + 0.58 Y = 0.00
Payoff: 8.00 X + 9.00 Y = 2105.00
Optimal Decisions(X ,Y ): (145.00, 105.00)
: 1.00X + 1.00Y <= 250.00
: 0.50X - 0.50Y >= 0.00
: 0.30X - 0.70Y <= 0.00
: -0.42X + 0.58Y <= 0.00
See file P2-25.XLS.T-notes M-bonds
Amount invested $145,000 $105,000Return 8.00% 9.00% $21,050
2-26. Let X = number of TV spots, Y= number of newspaper ads placed.
Objective: Maximize exposure = 30,000X + 20,000Y
Subject to:$3,200X + $1,300Y $95,200 Budget availableX 10 Max TV
Y 8X Paper vs TVX 5 Min TVX, Y 0 Non-negativity
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100Y
X
: 3200.00 X + 1300.00 Y = 95200.00: 1.00 X + 0.00 Y = 5.00
: 1.00 X + 0.00 Y = 10.00
: -8.00 X + 1.00 Y = 0.00
Payoff: 30000.00 X + 20000.00 Y = 1330000.00
Optimal Decisions(X ,Y ): (7.00, 56.00)
: 3200.00X + 1300.00Y <= 95200.00
: 1.00X + 0.00Y >= 5.00
: 1.00X + 0.00Y <= 10.00
: -8.00X + 1.00Y <= 0.00
See file P2-26.XLS.TV Paper
Number used 7.00 56.00Exposure 30,000 20,000 1,330,000
2-27. Let X = number of air conditioners to produce, Y = number of fans to produce.
Objective: Maximize revenue = $25X + $15Y
Subject to:3X + 2Y 240 Wiring time2X + Y 140 Drilling time1.5X + 0.5Y 100 Assembly timeX, Y 0 Non-negativity
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
10
20
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80
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100
110
120
130
140
150
160
170
180
190
200Y
X
: 3.00 X + 2.00 Y = 240.00
: 2.00 X + 1.00 Y = 139.25
: 1.50 X + 0.50 Y = 100.00
Payoff: 25.00 X + 15.00 Y = 1896.25
Optimal Decisions(X ,Y ): (38.50, 62.25)
: 3.00X + 2.00Y <= 240.00
: 2.00X + 1.00Y <= 139.25
: 1.50X + 0.50Y <= 100.00
See file P2-27.XLS.A/C Fan
Number of units 40.00 60.00Profit $25 $15 $1,900.00
2-28. X and Y are defined as in Problem 2-27. Objective remains the same.
Now subject to the following additional constraints:
Y 30 Max fansX 50 Min A/c
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200Y
X
: 3.00 X + 2.00 Y = 240.00
: 2.00 X + 1.00 Y = 140.00
: 1.50 X + 0.50 Y = 100.00
: 1.00 X + 0.00 Y = 50.00
: 0.00 X + 1.00 Y = 30.00
Payoff: 25.00 X + 15.00 Y = 1825.00
Optimal Decisions(X ,Y): (55.00, 30.00)
: 3.00X + 2.00Y <= 240.00
: 2.00X + 1.00Y <= 140.00
: 1.50X + 0.50Y <= 100.00
: 1.00X + 0.00Y >= 50.00
: 0.00X + 1.00Y <= 30.00
See file P2-28.XLS.A/C Fan
Number of units 55.00 30.00Profit $25 $15 $1,825.00
2-29. Let X = number of model A tubs to produce, Y= number of model B tubs to produce.
Objective: Maximize profit = $90X + $70Y
Subject to:120X + 100Y 24,500 Steel available20X + 30Y 6,000 Zinc availableX 5Y Models A vs BX, Y 0 Non-negativity
0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
0
15
30
45
60
75
90
105
120
135
150
165
180
195
210
225
240
255
270
285
300Y
X
: 120.00 X + 100.00 Y = 24500.00
: 20.00 X + 30.00 Y = 6000.00
: 1.00 X - 5.00 Y = 0.00
Payoff: 90.00 X + 70.00 Y = 18200.00
Optimal Decisions(X ,Y ): (175.00, 35.00)
: 120.00X + 100.00Y <= 24500.00
: 20.00X + 30.00Y <= 6000.00
: 1.00X - 5.00Y <= 0.00
See file P2-29.XLS.A B
Number of units 175.00 35.00Profit $90 $70 $18,200.00
2-30. Let X = number of benches to produce, Y = number of tables to produce.
Objective: Maximize profit = $9X + $20Y
Subject to:4X + 6Y 1,000 Labor hours10X + 35Y 3,500 RedwoodX 2Y Bench vs TableX, Y 0 Non-negativity
0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
0
10
20
30
40
50
60
70
80
90
100
110
120
130
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150
160
170
180
190
200Y
X
: 4.00 X + 6.00 Y = 1000.00
: 10.00 X + 35.00 Y = 3500.00
: 1.00 X - 2.00 Y = 0.00Payoff: 9.00 X + 20.00 Y = 2575.00
Optimal Decisions(X ,Y ): (175.00, 50.00)
: 4.00X + 6.00Y <= 1000.00
: 10.00X + 35.00Y <= 3500.00
: 1.00X - 2.00Y >= 0.00
See file P2-30.XLS.Bench Table
Number of units 175.00 50.00Profit $9 $20 $2,575.00
2-31. Let X = number of core courses, Y = number of elective courses.
Objective: Minimize wages = $2,600X + $3,000Y
Subject to:X + Y 60 Total courses3X + 4Y 205 Credit hoursX 20 Min core
Y 20 Min electiveX, Y 0 Non-negativity
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69
03691215182124273033363942454851545760636669
Y
X
: 1.00 X + 1.00 Y = 60.00
: 0.00 X + 1.00 Y = 20.00
: 3.00 X + 4.00 Y = 205.00: 1.00 X + 0.00 Y = 20.00
Payoff: 2600.00 X + 3000.00 Y = 166000.00
Optimal Decisions(X ,Y ): (35.00, 25.00)
: 1.00X + 1.00Y >= 60.00
: 0.00X + 1.00Y >= 20.00
: 3.00X + 4.00Y >= 205.00
: 1.00X + 0.00Y >= 20.00
See file P2-31.XLS.Core Elective
Courses 35.00 25.00Wages $2,600 $3,000 $166,000
2-32. Let X = number of Alpha 4 routers to produce, Y = number of Beta 5 routers to produce
Objective: Maximize profit = $1,200X + $1,800Y
Subject to:20X + 25Y = 780 Labor hoursX + Y 35 Total routers
Y X Alpha 4 vs Beta 5X, Y 0 Non-negativity
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
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80
85
90
95
100Y
X
: 20.00 X + 25.00 Y = 780.00
: 1.00 X + 1.00 Y = 35.00
: -1.00 X + 1.00 Y = 0.00
: 20.00 X + 25.00 Y = 780.00
Payoff: 1200.00 X + 1800.00 Y = 51600.00
Optimal Decisions(X ,Y ): (19.00, 16.00)
: 20.00X + 25.00Y <= 780.00
: 1.00X + 1.00Y >= 35.00
: -1.00X + 1.00Y <= 0.00
: 20.00X + 25.00Y >= 780.00
See file P2-32.XLS.Alpha 4 Beta 5
Number of units 19.00 16.00Profit $1,200 $1,800 $51,600.00
2-33. Let X = barrels of pruned olives, Y = barrels of regular olives to produce
Objective: Maximize revenue = $20X + $30Y
Subject to:5X + 2Y 250 Labor hoursX + 2Y 150 Acres availableX 40 Max prunedX, Y 0 Non-negativity
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100Y
X
: 5.00 X + 2.00 Y = 250.00
: 1.00 X + 2.00 Y = 150.00
: 1.00 X + 0.00 Y = 40.00
Payoff: 20.00 X + 30.00 Y = 2375.00
Optimal Decisions(X ,Y ): (25.00, 62.50)
: 5.00X + 2.00Y <= 250.00
: 1.00X + 2.00Y <= 150.00
: 1.00X + 0.00Y <= 40.00
See file P2-33.XLS.Pruned Regular
Number of barrels 25.00 62.50Revenue $20 $30 $2,375.00
Number of acres 25.00 125.00
2.34. Let X = dollars to invest in Louisiana Gas and Power, Y = dollars to invest in Trimex
Objective: Minimize total investment = X + Y
Subject to:0.36X + 0.24Y 875 Short term appr1.67X + 1.50Y 5,000 3-year appreciation0.04X + 0.08Y 200 Dividend incomeX, Y 0 Non-negativity
0 250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250 4500 4750 5000
0
175
350
525
700
875
1050
1225
1400
1575
1750
1925
2100
2275
2450
2625
2800
2975
3150
3325
3500Y
X
: 0.36 X + 0.24 Y = 875.00
: 1.67 X + 1.50 Y = 5000.00
: 0.04 X + 0.08 Y = 200.00
Payoff: 1.00 X + 1.00 Y = 3179.34
Optimal Decisions(X ,Y ): (1358.70, 1820.65)
: 0.36X + 0.24Y >= 875.00
: 1.67X + 1.50Y >= 5000.00
: 0.04X + 0.08Y >= 200.00
See file P2-34.XLS.Louisiana Trimex
$ invested $1,358.70 $1,820.65Investment 1 1 $3,179.35
2-35. Let X = number of coconuts to load on the boat, Y = number of skins to load on the boat.
Objective: Maximize profit = 60X + 300Y
Subject to:5X + 15Y 300 Weight limit0.125X + Y 15 Volume limitX, Y 0 Non-negativity
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
012345678910111213141516171819202122232425Y
X
: 5.000 X + 15.000 Y = 300.000
: 0.125 X + 1.000 Y = 15.000
Payoff: 60.000 X + 300.000 Y = 5040.000
Optimal Decisions(X ,Y ): (24.000, 12.000)
: 5.000X + 15.000Y <= 300.000
: 0.125X + 1.000Y <= 15.000
See file P2-35.XLS.Coconuts Skins
Number carried 24.00 12.00Profit (rupees) 60 300 5,040.00
2.36. Let X = number of boys’ bikes to produce, Y = number of girls’ bikes to produce.
Objective: Maximize profit = (225 - 101.25 - 38.75-20)X + (175 - 70 - 30-20)Y = $65X + $55Y
Subject to:X + Y 390 Production limit3.2X + 2.4Y 1,120 Labor hours
Y 0.3(X+Y) Min Girls' bikesX, Y 0 Non-negativity
0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500
0
25
50
75
100
125
150
175
200
225
250
275
300
325
350
375
400
425
450
475
500Y
X
: -0.30 X + 0.70 Y = 0.00
: 3.20 X + 2.40 Y = 1120.00
: 1.00 X + 1.00 Y = 390.00
Payoff: 65.00 X + 55.00 Y = 23750.00
Optimal Decisions(X ,Y ): (230.00, 160.00)
: -0.30X + 0.70Y >= 0.00
: 3.20X + 2.40Y <= 1120.00
: 1.00X + 1.00Y <= 390.00
See file P2-36.XLS.Boys Girls
Number of units 230.00 160.00Profit $65 $55 $23,750.00
2.37. Let X = number of regular modems to produce, Y = number of intelligent modems to produce
Objective: Maximize profits = $22.67X + $29.01Y
Subject to:0.555X + Y 15,400 Direct labor
+ Y 8,000 Microprocessor+ Y 0.25(X + Y) Min intelligent
X, Y 0 Non-negativity
0 900 1800 2700 3600 4500 5400 6300 7200 8100 9000 9900 10800 11700 12600 13500 14400 15300 16200 17100 18000
0
600
1200
1800
2400
3000
3600
4200
4800
5400
6000
6600
7200
7800
8400
9000
9600
10200
10800
11400
12000Y
X
: -0.250 X + 0.750 Y = 0.000
: 0.555 X + 1.000 Y = 15400.000
: 0.000 X + 1.000 Y = 8000.000
Payoff: 22.670 X + 29.010 Y = 560640.900
Optimal Decisions(X ,Y ): (17335.835, 5778.612)
: -0.250X + 0.750Y >= 0.000
: 0.555X + 1.000Y <= 15400.000
: 0.000X + 1.000Y <= 8000.000
See file P2-37.XLS.Regular Intelligent
Number of units 17,335.83 5,778.61Profit $22.67 $29.01 $560,640.90
2.38. Let X = number of Mild servings to make, Y = number of Spicy servings to make.
Objective: Maximize profit = $0.58X + $0.45Y
Subject to:0.15X + 0.30Y 8.5 Beef0.36X + 0.40Y 13 Beans3X + 2Y 95 Homemade salsa
+ 5Y 125 Hot sauceX, + Y 0 Non-negativity
0 2 4 6 8 101214161820222426283032343638404244464850
02468101214161820222426283032343638404244464850Y
X
: 0.15 X + 0.30 Y = 8.50
: 0.36 X + 0.40 Y = 13.00
: 3.00 X + 2.00 Y = 95.00
: 0.00 X + 5.00 Y = 125.00
Payoff: 0.58 X + 0.45 Y = 19.00
Optimal Decisions(X ,Y ): (25.00, 10.00)
: 0.15X + 0.30Y <= 8.50
: 0.36X + 0.40Y <= 13.00
: 3.00X + 2.00Y <= 95.00
: 0.00X + 5.00Y <= 125.00
See file P2-38.XLS.Mild Spicy
Number of units 25.00 10.00Profit $0.58 $0.45 $19.00
2.39. Let R = number of Rocket printers to produce, O, A defined similarly.
Objective: Maximize profit = $60R + $90O + $73A
Subject to:2.9R + 3.7O + 3.0A 4,000 Assembly time1.4R + 2.1O + 1.7A 2,000 Testing time
O 0.15(R + O + A) Min OmegaR + O 0.40(R + O + A) Min Rocket & OmegaR, O, A 0 Non-negativity
See file P2-39.XLS.Rocket Omega Alpha
Number of units 296.74 178.04 712.17Profit $60 $90 $73 $85,816.02
2-40. Let X = pounds of Stock X to mix into feed for one cow, Y, Z defined similarly.
Objective: Minimize cost = $3.00X + $4.00Y + $2.25Z
Subject to:3X + 2Y + 4Z 64 Nutrient A needed2X + 3Y + Z 80 Nutrient B neededX + 2Z 16 Nutrient C needed6X + 8Y + 4Z 128 Nutrient D needed
Z 5 Stock Z maxX, Y, 0 Non-negativity
See file P2-40.XLS.Stock X Stock Y Stock Z
# of pounds 16.00 16.00 0.00Cost $3.00 $4.00 $2.25 $112.00
2.41. Let J = number of units of XJ201 to produce, M, T, B defined similarly.
Objective: Maximize profit = $9J + $12M + $15T + $11B
Subject to:0.5J + 1.5M + 1.5T + 1.0B 15,000 Wiring time0.3J + 1.0M + 2.0T + 3.0B 17,000 Drilling time0.2J + 4.0M + 1.0T + 2.0B 10,000 Assembly time0.5J + 1.0M + 0.5T + 0.5B 12,000 Inspection timeJ 150 Minimum XJ201
M 100 Minimum XM897T 300 Minimum TR29
B 400 Minimum BR788J, M, T, B 0 Non-negativity
See file P2-41.XLS.XJ201 XM897 TR29 BR788
# of units 20,650.00 100.00 2,750.00 400.00Profit $9 $12 $15 $11 $232,700.00
2-42. Let M1 = number of X409 valves to produce, M2, M3, M4 defined similarly.
Objective: Maximize profit = $16M1 + $12M2 + $13M3 + $8M4
Subject to:0.40M1 + 0.30M2 + 0.45M3 + 0.35M4 700 Drilling time0.60M1 + 0.65M2 + 0.52M3 + 0.48M4 890 Milling time1.20M1 + 0.60M2 + 0.50M3 + 0.70M4 1,200 Lathe time0.25M1 + 0.25M2 + 0.25M3 + 0.25M4 525 Inspection timeM1 200 Minimum X409
M2 250 Minimum X3125M3 600 Minimum X4950
M4 450 Minimum X2173M1, M2, M3, M4 0 Non-negativity
See file P2-42.XLS.X409 X3125 X4950 X2173
# of valves 332.50 250.00 600.00 450.00Profit $16 $12 $13 $8 $19,720.00
2.43. Let P = cans of Plain nuts to produce. M, R defined similarly
Objective: Maximize revenue = $2.25P + $3.37M + $6.49R
Subject to:0.8P + 0.5M 500 Peanuts0.2P + 0.3M + 0.3R 225 Cashews
+ 0.1M + 0.4R 100 Almonds+ 0.1M + 0.4R 80 Walnuts
P 2R Plain vs PremiumP, M, R 0 Non-negativity
See file P2-43.XLS.Plain Mixed Premium
Number of cans 375.00 400.00 100.00Revenue $2.25 $3.37 $6.49 $2,840.75
2.44. Let B = dollars invested in B&O. S, R defined similarly.
Objective: Minimize investment = B + S + R
Subject to:0.39B + 0.26S + 0.42R $1,000 Short term growth1.59B + 1.70S + 1.55R $6,000 Intermediate growth0.08B + 0.04S + 0.06R $250 Dividend incomeB, S, R 0 Non-negativity
See file P2-44.XLS.B & O Short Reading
$ invested $2,555.25 $1,139.50 $0.00Investment 1 1 1 $3,694.75
2-45. Let S = number of Small boxes to include, L, M defined similarly.
Objective: Maximize rent collected = $30S + $40L + $17M
Subject to:0.50S + 0.85L + 0.30M 240 Square feet available
+ 0.85L + 0.30M 120 Max space, large & miniS + L + M 350 Min required, total
L 80 Min required, largeM 100 Min required, mini
S, L, M 0 Non-negativity
See file P2-45.XLS.Small Large Mini
Number of boxes 284.00 80.00 100.00Rent $30 $40 $17 $13,420.00
Case: Mexicana Wire Works
See file P2-Mexicana.XLS.W75C W33C W5X W7X
Number of units 1,100.00 250.00 0.00 600.00Profit $34 $30 $60 $25 $59,900.00ConstraintsDrawing time 1 2 1 $2,200.00 <= 4000Extrusion time 1 1 4 1 $1,950.00 <= 4200Winding time 1 3 $1,850.00 <= 2000Packaging time 1 3 2 $2,300.00 <= 2300W75C orders 1 $1,100.00 <= 1400W33C orders 1 $250.00 <= 250W5X orders 1 $0.00 <= 1510W7X orders 1 $600.00 <= 1116Minimum W75C 1 $1,100.00 >= 150Minimum W7X 1 $600.00 >= 600
LHS Sign RHS
Case: Golding Landscaping and Plants, Inc.
See file P2-Golding.XLS.C-30 C-92 D-21 E-11
# of pounds 7.50 15.00 0.00 27.50Cost $0.12 $0.09 $0.11 $0.04 $3.35Constraints50-lbs required 1 1 1 1 50.00 = 50.0E-11 15% 1 27.50 7.5C-92 & C-30 45% 1 1 22.50 22.5D-21 & C-92 30% 1 1 15.00 15.0
LHS Sign RHS