Ch02 Solutions

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


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

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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


: 3.00X + 7.00Y >= 231.00

: 10.00X + 2.00Y >= 200.00

: 0.00X + 2.00Y >= 45.00

: 2.00X + 0.00Y <= 75.00


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

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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


: 3.00X + 6.00Y <= 29.00

: 7.00X + 1.00Y <= 20.00

: 3.00X - 1.00Y >= 1.00


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

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5

6

7

8

9

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11

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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


: 9.00X + 8.00Y <= 72.00

: 3.00X + 9.00Y >= 27.00

: 9.00X - 15.00Y >= 0.00


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

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8

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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


: 9.00X + 3.00Y >= 36.00

: 4.00X + 5.00Y >= 40.00

: 1.00X - 1.00Y <= 0.00

: 2.00X + 0.00Y <= 13.00


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


: 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


: 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

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126

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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


: 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

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1020

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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


: 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.


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

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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


: 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


: 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

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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


: 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

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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


: 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.


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

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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


: 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

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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


: 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


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

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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


: 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


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

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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


: 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


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


: 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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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


: 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


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

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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


: 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


: 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


: 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


: -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


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


: -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


: 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

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