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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

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20Y

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 5.00

Obj coeff 5 3 45.00

Constraints:

Constraint 1 5 2 40.00 40


Constraint 3 1 6.00 7

Constraint 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

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

: 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


: 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 21.43

Obj coeff 1 2 68.57

Constraints:


Constraint 2 8 2 248.57 160

Constraint 3 3 2 120.00 120


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


: 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 26.72

Obj coeff 4 7 245.66

Constraints:

Constraint 1 3 7 231.00 231

Constraint 2 10 2 200.00 200



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


: 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 3.67

Obj coeff 1 1 6.00

Constraints:




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


: 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 0.47

Obj coeff 7 4 54.95

Constraints:



Constraint 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


: 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 4.44

Obj coeff 3 7 44.44

Constraints:





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 30.00

Obj coeff 28 41 38 3,490.00

Constraints:

Constraint 1 10 15 -8 610.00 610.00

Constraint 2 0.4 0.4 0.4 40.00 40.00

Constraint 3 1 40.00 90.00

Constraint 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. Budget

150X + 50Y 8,000 Sq feet required

X 40 Rental limit

X, 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


: 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 70.00

Rent $50 $20 $2,900.00

Constraints:

Advt. budget $1 $1 $100.00 $100

Sq feet required 150 50 8,000.00 8,000

Rental 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, minutes

1.8X + 2Y 1,800 Collate time, minutes


0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000

0

60

120

180

240

300

360

420

480

540

600

660

720

780

840

900

960

1020

1080

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


: 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 400.00 540.00

Revenue $3.50 $4.50 $3,830.0

Constraints:

Print time, minutes 2.5 2.0 2,080.0 <= 2,160

Collate time, minutes 1.8 2.0 1,800.0 <= 1,800

Min Backyard to print 1 400.0 >= 400

Min 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 available

X + Y 350 Min required, total

Y 80 Min required, large


0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

320

340

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


: 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 80.00

Rent $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 bag

2X + Y 100 Fertilizer rating

X 35 Min compost, pounds

Y 40 Max sewage, pounds


0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60

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

50Y

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 20.00

Cost $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 available

X 0.7(X+Y) Max T-notes

2X + 3Y 2.42(X+Y) Max risk score

X 0.3(X+Y) Min T-notes


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


: 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 $105,000

Return 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 available

X 10 Max TV

Y 8X Paper vs TV

X 5 Min TV


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


: 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 56.00

Exposure 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 time

2X + Y 140 Drilling time

1.5X + 0.5Y 100 Assembly time


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

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 = 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 60.00

Profit $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 fans

X 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


: 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


Profit $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 available

20X + 30Y 6,000 Zinc available

X 5Y Models A vs B


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


: 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


Profit $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 hours

10X + 35Y 3,500 Redwood

X 2Y Bench vs Table


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

140

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 = 0.00

Payoff: 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


Profit $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 courses

3X + 4Y 205 Credit hours

X 20 Min core

Y 20 Min elective


0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69

0

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

69

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 25.00

Wages $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 hours

X + Y 35 Total routers

Y X Alpha 4 vs Beta 5


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

: 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


Profit $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 hours

X + 2Y 150 Acres available

X 40 Max pruned


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


: 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 62.50

Revenue $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 appr

1.67X + 1.50Y 5,000 3-year appreciation

0.04X + 0.08Y 200 Dividend income


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 $1,820.65

Investment 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 limit

0.125X + Y 15 Volume limit


0 2 4 6 8 101214161820222426283032343638404244464850

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

: 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 12.00

Profit (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 limit

3.2X + 2.4Y 1,120 Labor hours

Y 0.3(X+Y) Min Girls' bikes


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


Profit $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


0 900 1800 2700 3600 4500 5400 6300 7200 8100 9000 9900 108001170012600135001440015300162001710018000

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 5,778.61

Profit $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 Beef

0.36X + 0.40Y 13 Beans

3X + 2Y 95 Homemade salsa

+ 5Y 125 Hot sauce

X, + Y 0 Non-negativity

0 2 4 6 8 101214161820222426283032343638404244464850

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

50Y

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


Profit $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 time

1.4R + 2.1O + 1.7A 2,000 Testing time

O 0.15(R + O + A) Min Omega

R + O 0.40(R + O + A) Min Rocket & Omega

R, O, A 0 Non-negativity

See file P2-39.XLS.

Rocket Omega Alpha

Number of units 296.74 178.04 712.17

Profit $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 needed

2X + 3Y + Z 80 Nutrient B needed

X + 2Z 16 Nutrient C needed

6X + 8Y + 4Z 128 Nutrient D needed

Z 5 Stock Z max

X, Y, 0 Non-negativity

See file P2-40.XLS.

Stock X Stock Y Stock Z

# of pounds 16.00 16.00 0.00

Cost $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 time

0.3J + 1.0M + 2.0T + 3.0B 17,000 Drilling time

0.2J + 4.0M + 1.0T + 2.0B 10,000 Assembly time

0.5J + 1.0M + 0.5T + 0.5B 12,000 Inspection time

J 150 Minimum XJ201

M 100 Minimum XM897

T 300 Minimum TR29

B 400 Minimum BR788

J, M, T, B 0 Non-negativity

See file P2-41.XLS.

XJ201 XM897 TR29 BR788

# of units 20,650.00 100.00 2,750.00 400.00

Profit $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 time

0.60M1 + 0.65M2 + 0.52M3 + 0.48M4 890 Milling time

1.20M1 + 0.60M2 + 0.50M3 + 0.70M4 1,200 Lathe time

0.25M1 + 0.25M2 + 0.25M3 + 0.25M4 525 Inspection time

M1 200 Minimum X409

M2 250 Minimum X3125



M1, M2, M3, M4 0 Non-negativity

See file P2-42.XLS.

X409 X3125 X4950 X2173

# of valves 332.50 250.00 600.00 450.00

Profit $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 Peanuts

0.2P + 0.3M + 0.3R 225 Cashews

+ 0.1M + 0.4R 100 Almonds

+ 0.1M + 0.4R 80 Walnuts

P 2R Plain vs Premium

P, M, R 0 Non-negativity

See file P2-43.XLS.

Plain Mixed Premium

Number of cans 375.00 400.00 100.00

Revenue $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 growth

1.59B + 1.70S + 1.55R $6,000 Intermediate growth

0.08B + 0.04S + 0.06R $250 Dividend income

B, S, R 0 Non-negativity

See file P2-44.XLS.

B & O Short Reading

$ invested $2,555.25 $1,139.50 $0.00

Investment 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 & mini

S + L + M 350 Min required, total

L 80 Min required, large

M 100 Min required, mini

S, L, M 0 Non-negativity

See file P2-45.XLS.

Small Large Mini

Number of boxes 284.00 80.00 100.00

Rent $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 0.00 600.00

Profit $34 $30 $60 $25 $59,900.00

Constraints

Drawing time 1 2 1 $2,200.00 <= 4000

Extrusion time 1 1 4 1 $1,950.00 <= 4200

Winding time 1 3 $1,850.00 <= 2000

Packaging time 1 3 2 $2,300.00 <= 2300

W75C orders 1 $1,100.00 <= 1400

W33C orders 1 $250.00 <= 250

W5X orders 1 $0.00 <= 1510

W7X orders 1 $600.00 <= 1116

Minimum W75C 1 $1,100.00 >= 150

Minimum 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 27.50

Cost $0.12 $0.09 $0.11 $0.04 $3.35

Constraints

50-lbs required 1 1 1 1 50.00 = 50.0

E-11 15% 1 27.50 7.5

C-92 & C-30 45% 1 1 22.50 22.5

D-21 & C-92 30% 1 1 15.00 15.0

LHS Sign RHS

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