Graphical-4.doc

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.

PRACTICE PROBLEMS WITH SOLUTIONS

2-13.

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 Y

X : 5 . 0 0 X + 2 . 0 0 Y = 4 0 . 0 0

: 3 . 0 0 X + 6 . 0 0 Y = 4 8 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y = 7 . 0 0

: 2 . 0 0 X - 1 . 0 0 Y = 3 . 0 0

P a y o f f : 5 . 0 0 X + 3 . 0 0 Y = 4 5 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 6 . 0 0 , 5 . 0 0 ) : 5 . 0 0 X + 2 . 0 0 Y < = 4 0 . 0 0

: 3 . 0 0 X + 6 . 0 0 Y < = 4 8 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y < = 7 . 0 0 : 2 . 0 0 X - 1 . 0 0 Y > = 3 . 0 0

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 2

3

6

48.00



48

Constraint 3

1

6.00



7

Constraint 4

2

-1

7.00



3

2-14.

0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8 7 2 7 6 8 0 8 4 8 8 0

3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 0 6 3 6 6 6 9 7 2 7 5 Y

X : 1 . 0 0 X + 3 . 0 0 Y = 9 0 . 0 0

: 8 . 0 0 X + 2 . 0 0 Y = 1 6 0 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y = 7 0 . 0 0

: 3 . 0 0 X + 2 . 0 0 Y = 1 2 0 . 0 0

P a y o f f : 1 . 0 0 X + 2 . 0 0 Y = 6 8 . 5 7

O p t i m a l D e c i s i o n s ( X , Y ) : ( 2 5 . 7 1 , 2 1 . 4 3 ) : 1 . 0 0 X + 3 . 0 0 Y > = 9 0 . 0 0

: 8 . 0 0 X + 2 . 0 0 Y > = 1 6 0 . 0 0 : 0 . 0 0 X + 1 . 0 0 Y < = 7 0 . 0 0 : 3 . 0 0 X + 2 . 0 0 Y > = 1 2 0 . 0 0

See file P2-14.XLS.

X

Y

Solution

25.71 21.43

Obj coeff

1

2

68.57

Constraints:

Constraint 1

1

3

90.00



90

Constraint 2

8

2

248.57



160

Constraint 3

3

2

120.00



120

Constraint 4

1

21.43



70

2-15.

0 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8 7 2 7 6 8 0 8 4 8 8 0

3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 0 6 3 6 6 6 9 7 2 7 5 Y

X : 3 . 0 0 X + 7 . 0 0 Y = 2 3 1 . 0 0

: 1 0 . 0 0 X + 2 . 0 0 Y = 2 0 0 . 0 0

: 0 . 0 0 X + 2 . 0 0 Y = 4 5 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y = 7 5 . 0 0 P a y o f f : 4 . 0 0 X + 7 . 0 0 Y = 2 4 5 . 6 5

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 4 . 6 6 , 2 6 . 7 2 ) : 3 . 0 0 X + 7 . 0 0 Y > = 2 3 1 . 0 0

: 1 0 . 0 0 X + 2 . 0 0 Y > = 2 0 0 . 0 0 : 0 . 0 0 X + 2 . 0 0 Y > = 4 5 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y < = 7 5 . 0 0

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

Constraint 3

2

53.44



45

Constraint 4

2

29.31



75

2-16.

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 Y

X : 3 . 0 0 X + 6 . 0 0 Y = 2 9 . 0 0

: 7 . 0 0 X + 1 . 0 0 Y = 2 0 . 0 0

: 3 . 0 0 X - 1 . 0 0 Y = 1 . 0 0

P a y o f f : 1 . 0 0 X + 1 . 0 0 Y = 6 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 2 . 3 3 , 3 . 6 7 ) : 3 . 0 0 X + 6 . 0 0 Y < = 2 9 . 0 0

: 7 . 0 0 X + 1 . 0 0 Y < = 2 0 . 0 0 : 3 . 0 0 X - 1 . 0 0 Y > = 1 . 0 0

See file P2-16.XLS.

X

Y

Solution

2.33

3.67

Obj coeff

1

1

6.00

Constraints:

Constraint 1

3

6

29.00



29

Constraint 2

7

1

20.00



20

Constraint 3

3

-1

3.33



1

2-17.

0 1 2 3 4 5 6 7 8 9 1 0

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 Y

X : 9 . 0 0 X + 8 . 0 0 Y = 7 2 . 0 0

: 3 . 0 0 X + 9 . 0 0 Y = 2 7 . 0 0

: 9 . 0 0 X - 1 5 . 0 0 Y = 0 . 0 0 P a y o f f : 7 .0 0 X + 4 . 0 0 Y = 5 4 . 9 4

O p t i m a l D e c i s i o n s ( X , Y ) : ( 7 . 5 8 , 0 . 4 7 ) : 9 . 0 0 X + 8 . 0 0 Y < = 7 2 . 0 0

: 3 . 0 0 X + 9 . 0 0 Y > = 2 7 . 0 0 : 9 . 0 0 X - 1 5 . 0 0 Y > = 0 . 0 0

See file P2-17.XLS.

X

Y

Solution

7.58

0.47

Obj coeff

7

4

54.95

Constraints:

Constraint 1

9

8

72.00



72

Constraint 2

3

9

27.00



27

Constraint 3

9

-15

61.11



0

2-18.

0 1 2 3 4 5 6 7 8 9 1 0

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 Y

X : 9 . 0 0 X + 3 . 0 0 Y = 3 6 . 0 0 _{: 4 . 0 0 X + 5 . 0 0 Y = 4 0 . 0 0}

: 1 . 0 0 X - 1 . 0 0 Y = 0 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y = 1 3 . 0 0

P a y o f f : 3 . 0 0 X + 7 . 0 0 Y = 4 4 . 4 4

O p t i m a l D e c i s i o n s ( X , Y ) : ( 4 . 4 4 , 4 . 4 4 ) : 9 . 0 0 X + 3 . 0 0 Y > = 3 6 . 0 0

: 4 . 0 0 X + 5 . 0 0 Y > = 4 0 . 0 0 : 1 . 0 0 X - 1 . 0 0 Y < = 0 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y < = 1 3 . 0 0

See file P2-18.XLS.

X

Y

Solution

4.44

4.44

Obj coeff

3

7

44.44

Constraints:

Constraint 1

9

3

53.33



36

Constraint 2

4

5

40.00



40

Constraint 3

1

-1

0.00



0

Constraint 3

2

8.89



13

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 1 0

0 1 2 3 4 5 6 7 8 9 1 0 Y

X : 2 . 0 0 X + 1 . 0 0 Y = 6 . 0 0

: 4 . 0 0 X + 5 . 0 0 Y = 2 0 . 0 0 : 0 . 0 0 X + 2 . 0 0 Y = 7 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y = 7 . 0 0

P a y o f f : 3 . 0 0 X + 7 . 0 0 Y = 1 2 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y < = 6 . 0 0 : 4 . 0 0 X + 5 . 0 0 Y < = 2 0 . 0 0 : 0 . 0 0 X + 2 . 0 0 Y < = 7 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y > = 7 . 0 0

Formulation 2 (Multiple Optimal Solutions)

0 1 2 3 4 5 6 7 8 9 1 0

0 1 2 3 4 5 6 7 8 9 1 0 Y

X : 7 . 0 0 X + 6 . 0 0 Y = 4 2 . 0 0

: 1 . 0 0 X + 2 . 0 0 Y = 1 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y = 4 . 0 0

: 0 . 0 0 X + 2 . 0 0 Y = 9 . 0 0 P a y o f f : 3 . 0 0 X + 6 . 0 0 Y = 3 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 3 . 0 0 , 3 . 5 0 ) ( 1 . 0 0 , 4 . 5 0 ) : 7 . 0 0 X + 6 . 0 0 Y < = 4 2 . 0 0

2-19 (continued).

See file P2-19.XLS.

Formulation 3 (Unbounded Solution)

0 1 2 3 4 5 6 7 8 9 1 0

0 1 2 3 4 5 6 7 8 9 1 0 Y

X : 1 . 0 0 X + 2 . 0 0 Y = 1 2 . 0 0

: 8 . 0 0 X + 7 . 0 0 Y = 5 6 . 0 0

: 0 . 0 0 X + 2 . 0 0 Y = 5 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y = 9 . 0 0

P a y o f f : 2 . 0 0 X + 3 . 0 0 Y = 1 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 9 . 0 0 , 2 7 . 2 4 ) : 1 . 0 0 X + 2 . 0 0 Y > = 1 2 . 0 0

: 8 . 0 0 X + 7 . 0 0 Y > = 5 6 . 0 0 : 0 . 0 0 X + 2 . 0 0 Y > = 5 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y < = 9 . 0 0

Formulation 4 (Unique Optimal Solution)

0 1 2 3 4 5 6 7 8 9 1 0

0 1 2 3 4 5 6 7 8 9 1 0 Y

X : 3 . 0 0 X + 7 . 0 0 Y = 2 1 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y = 6 . 0 0

: 1 . 0 0 X + 1 . 0 0 Y = 2 . 0 0 : 2 . 0 0 X + 0 . 0 0 Y = 2 . 0 0

P a y o f f : 3 . 0 0 X + 4 . 0 0 Y = 1 4 . 4 5

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 . 9 1 , 2 . 1 8 ) : 3 . 0 0 X + 7 . 0 0 Y < = 2 1 . 0 0

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

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 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 0

9 1 8 2 7 3 6 4 5 5 4 6 3 7 2 8 1 9 0 9 9 1 0 8 1 1 7 1 2 6 1 3 5 1 4 4 1 5 3 1 6 2 1 7 1 1 8 0 Y

X : 1 . 0 0 X + 1 . 0 0 Y = 1 0 0 . 0 0

: 1 5 0 . 0 0 X + 5 0 . 0 0 Y = 8 0 0 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y = 4 0 . 0 0

P a y o f f : 5 0 . 0 0 X + 2 0 . 0 0 Y = 2 9 0 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 3 0 . 0 0 , 7 0 . 0 0 ) : 1 . 0 0 X + 1 . 0 0 Y < = 1 0 0 . 0 0

: 1 5 0 . 0 0 X + 5 0 . 0 0 Y < = 8 0 0 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y < = 4 0 . 0 0

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

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

X,

Y



0

Non-negativity

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0 6 0 0 6 5 0 7 0 0 7 5 0 8 0 0 8 5 0 9 0 0 9 5 0 1 0 0 0 0

6 0 1 2 0 1 8 0 2 4 0 3 0 0 3 6 0 4 2 0 4 8 0 5 4 0 6 0 0 6 6 0 7 2 0 7 8 0 8 4 0 9 0 0 9 6 0 1 0 2 0 1 0 8 0 1 1 4 0 1 2 0 0 Y

X : 2 . 5 0 X + 2 . 0 0 Y = 2 1 6 0 . 0 0

: 1 . 8 0 X + 2 . 0 0 Y = 1 8 0 0 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y = 4 0 0 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y = 3 0 0 . 0 0 P a y o f f : 3 . 5 0 X + 4 . 5 0 Y = 3 8 3 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 4 0 0 . 0 0 , 5 4 0 . 0 0 ) : 2 . 5 0 X + 2 . 0 0 Y < = 2 1 6 0 . 0 0

: 1 . 8 0 X + 2 . 0 0 Y < = 1 8 0 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y > = 4 0 0 . 0 0 : 0 . 0 0 X + 1 . 0 0 Y > = 3 0 0 . 0 0

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

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 available

X

+ Y



350

Min required, total

Y



80

Min required, large

X,

Y



0

Non-negativity

0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0 2 7 5 3 0 0 3 2 5 3 5 0 3 7 5 4 0 0 4 2 5 4 5 0 4 7 5 5 0 0 0

2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 2 8 0 3 0 0 3 2 0 3 4 0 3 6 0 3 8 0 4 0 0 Y

X : 0 . 5 0 X + 0 . 8 5 Y = 2 4 0 . 0 0 : 1 . 0 0 X + 1 . 0 0 Y = 3 5 0 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y = 8 0 . 0 0 P a y o f f : 3 0 . 0 0 X + 4 0 . 0 0 Y = 1 3 5 2 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 3 4 4 . 0 0 , 8 0 . 0 0 ) : 0 . 5 0 X + 0 . 8 5 Y < = 2 4 0 . 0 0

: 1 . 0 0 X + 1 . 0 0 Y > = 3 5 0 . 0 0 : 0 . 0 0 X + 1 . 0 0 Y > = 8 0 . 0 0

See file P2-23.XLS.

Small

Large

Number of boxes

344.00

80.00

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

X,

Y



0

Non-negativity

0 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 0 0

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 Y

X : 1 . 0 0 X + 1 . 0 0 Y = 6 0 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y = 3 5 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y = 4 0 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y = 1 0 0 . 0 0 P a y o f f : 0 . 0 5 X + 0 . 0 4 Y = 2 . 8 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 4 0 . 0 0 , 2 0 . 0 0 ) : 1 . 0 0 X + 1 . 0 0 Y > = 6 0 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y > = 3 5 . 0 0 : 0 . 0 0 X + 1 . 0 0 Y < = 4 0 . 0 0 : 2 . 0 0 X + 1 . 0 0 Y > = 1 0 0 . 0 0

See file P2-24.XLS.

Compost Sewage

Number of pounds

40.00

20.00

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

X, Y



0

Non-negativity

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 5 1 2 0 1 3 5 1 5 0 1 6 5 1 8 0 1 9 5 2 1 0 2 2 5 2 4 0 2 5 5 2 7 0 2 8 5 3 0 0 0

7 1 4 2 1 2 8 3 5 4 2 4 9 5 6 6 3 7 0 7 7 8 4 9 1 9 8 1 0 5 1 1 2 1 1 9 1 2 6 1 3 3 1 4 0 1 4 7 Y

X : 1 . 0 0 X + 1 . 0 0 Y = 2 5 0 . 0 0

: 0 . 5 0 X - 0 . 5 0 Y = 0 . 0 0 : 0 . 3 0 X - 0 . 7 0 Y = 0 . 0 0 : - 0 . 4 2 X + 0 . 5 8 Y = 0 . 0 0

P a y o f f : 8 . 0 0 X + 9 . 0 0 Y = 2 1 0 5 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 4 5 . 0 0 , 1 0 5 . 0 0 ) : 1 . 0 0 X + 1 . 0 0 Y < = 2 5 0 . 0 0

: 0 . 5 0 X - 0 . 5 0 Y > = 0 . 0 0 : 0 . 3 0 X - 0 . 7 0 Y < = 0 . 0 0 : - 0 . 4 2 X + 0 . 5 8 Y < = 0 . 0 0

See file P2-25.XLS.

T-notes M-bonds

Amount invested

$145,000 $105,000

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

X,

Y



0

Non-negativity

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 0

5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 Y

X : 3 2 0 0 . 0 0 X + 1 3 0 0 . 0 0 Y = 9 5 2 0 0 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y = 5 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y = 1 0 . 0 0 : - 8 . 0 0 X + 1 . 0 0 Y = 0 . 0 0

P a y o f f : 3 0 0 0 0 . 0 0 X + 2 0 0 0 0 . 0 0 Y = 1 3 3 0 0 0 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 7 . 0 0 , 5 6 . 0 0 ) : 3 2 0 0 . 0 0 X + 1 3 0 0 . 0 0 Y < = 9 5 2 0 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y > = 5 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y < = 1 0 . 0 0 : - 8 . 0 0 X + 1 . 0 0 Y < = 0 . 0 0

See file P2-26.XLS.

TV

Paper

Number used

7.00

56.00

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 time

2X

+ Y



140

Drilling time

1.5X + 0.5Y



100

Assembly time

X,

Y



0

Non-negativity

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 0

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 Y

X : 3 . 0 0 X + 2 . 0 0 Y = 2 4 0 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y = 1 3 9 . 2 5 : 1 . 5 0 X + 0 . 5 0 Y = 1 0 0 . 0 0

P a y o f f : 2 5 . 0 0 X + 1 5 . 0 0 Y = 1 8 9 6 . 2 5

O p t i m a l D e c i s i o n s ( X , Y ) : ( 3 8 . 5 0 , 6 2 . 2 5 ) : 3 . 0 0 X + 2 . 0 0 Y < = 2 4 0 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y < = 1 3 9 . 2 5 : 1 . 5 0 X + 0 . 5 0 Y < = 1 0 0 . 0 0

See file P2-27.XLS.

A/C

Fan

Number of units 40.00 60.00

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 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8 5 2 5 6 6 0 6 4 6 8 7 2 7 6 8 0 0

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 Y

X : 3 . 0 0 X + 2 . 0 0 Y = 2 4 0 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y = 1 4 0 . 0 0

: 1 . 5 0 X + 0 . 5 0 Y = 1 0 0 . 0 0

: 1 . 0 0 X + 0 . 0 0 Y = 5 0 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y = 3 0 . 0 0 P a y o f f : 2 5 . 0 0 X + 1 5 . 0 0 Y = 1 8 2 5 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 5 5 . 0 0 , 3 0 . 0 0 ) : 3 . 0 0 X + 2 . 0 0 Y < = 2 4 0 . 0 0

: 2 . 0 0 X + 1 . 0 0 Y < = 1 4 0 . 0 0 : 1 . 5 0 X + 0 . 5 0 Y < = 1 0 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y > = 5 0 . 0 0 : 0 . 0 0 X + 1 . 0 0 Y < = 3 0 . 0 0

See file P2-28.XLS.

A/C

Fan

Number of units 55.00 30.00

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

X,

Y



0

Non-negativity

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 5 1 2 0 1 3 5 1 5 0 1 6 5 1 8 0 1 9 5 2 1 0 2 2 5 2 4 0 2 5 5 2 7 0 2 8 5 3 0 0 0

1 5 3 0 4 5 6 0 7 5 9 0 1 0 5 1 2 0 1 3 5 1 5 0 1 6 5 1 8 0 1 9 5 2 1 0 2 2 5 2 4 0 2 5 5 2 7 0 2 8 5 3 0 0 Y

X : 1 2 0 . 0 0 X + 1 0 0 . 0 0 Y = 2 4 5 0 0 . 0 0

: 2 0 . 0 0 X + 3 0 . 0 0 Y = 6 0 0 0 . 0 0

: 1 . 0 0 X - 5 . 0 0 Y = 0 . 0 0 P a y o f f : 9 0 . 0 0 X + 7 0 . 0 0 Y = 1 8 2 0 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 7 5 . 0 0 , 3 5 . 0 0 ) : 1 2 0 . 0 0 X + 1 0 0 . 0 0 Y < = 2 4 5 0 0 . 0 0 : 2 0 . 0 0 X + 3 0 . 0 0 Y < = 6 0 0 0 . 0 0 : 1 . 0 0 X - 5 . 0 0 Y < = 0 . 0 0

See file P2-29.XLS.

A

B

Number of units 175.00 35.00

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

X,

Y



0

Non-negativity

0 1 5 3 0 4 5 6 0 7 5 9 0 1 0 5 1 2 0 1 3 5 1 5 0 1 6 5 1 8 0 1 9 5 2 1 0 2 2 5 2 4 0 2 5 5 2 7 0 2 8 5 3 0 0 0

1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 Y

X : 4 . 0 0 X + 6 . 0 0 Y = 1 0 0 0 . 0 0

: 1 0 . 0 0 X + 3 5 . 0 0 Y = 3 5 0 0 . 0 0 : 1 . 0 0 X - 2 . 0 0 Y = 0 . 0 0 P a y o f f : 9 . 0 0 X + 2 0 . 0 0 Y = 2 5 7 5 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 7 5 . 0 0 , 5 0 . 0 0 ) : 4 . 0 0 X + 6 . 0 0 Y < = 1 0 0 0 . 0 0

: 1 0 . 0 0 X + 3 5 . 0 0 Y < = 3 5 0 0 . 0 0 : 1 . 0 0 X - 2 . 0 0 Y > = 0 . 0 0

See file P2-30.XLS.

Bench Table

Number of units 175.00 50.00

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

X,

Y



0

Non-negativity

0 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 0 6 3 6 6 6 9 0

3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 3 3 3 6 3 9 4 2 4 5 4 8 5 1 5 4 5 7 6 0 6 3 6 6 6 9 Y

X : 1 . 0 0 X + 1 . 0 0 Y = 6 0 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y = 2 0 . 0 0

: 3 . 0 0 X + 4 . 0 0 Y = 2 0 5 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y = 2 0 . 0 0

P a y o f f : 2 6 0 0 . 0 0 X + 3 0 0 0 . 0 0 Y = 1 6 6 0 0 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 3 5 . 0 0 , 2 5 . 0 0 ) : 1 . 0 0 X + 1 . 0 0 Y > = 6 0 . 0 0

: 0 . 0 0 X + 1 . 0 0 Y > = 2 0 . 0 0 : 3 . 0 0 X + 4 . 0 0 Y > = 2 0 5 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y > = 2 0 . 0 0

See file P2-31.XLS.

Core Elective

Courses

35.00

25.00

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

X,

Y



0

Non-negativity

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 0

5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 Y

X : 2 0 . 0 0 X + 2 5 . 0 0 Y = 7 8 0 . 0 0

: 1 . 0 0 X + 1 . 0 0 Y = 3 5 . 0 0

: - 1 . 0 0 X + 1 . 0 0 Y = 0 . 0 0

: 2 0 . 0 0 X + 2 5 . 0 0 Y = 7 8 0 . 0 0

P a y o f f : 1 2 0 0 . 0 0 X + 1 8 0 0 . 0 0 Y = 5 1 6 0 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 9 . 0 0 , 1 6 . 0 0 ) : 2 0 . 0 0 X + 2 5 . 0 0 Y < = 7 8 0 . 0 0

: 1 . 0 0 X + 1 . 0 0 Y > = 3 5 . 0 0 : - 1 . 0 0 X + 1 . 0 0 Y < = 0 . 0 0 : 2 0 . 0 0 X + 2 5 . 0 0 Y > = 7 8 0 . 0 0

See file P2-32.XLS.

Alpha 4 Beta 5

Number of units

19.00

16.00

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 hours

X

+ 2Y



150 Acres available

X



40

Max pruned

X,

Y



0

Non-negativity

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 0

5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 8 5 9 0 9 5 1 0 0 Y

X : 5 . 0 0 X + 2 . 0 0 Y = 2 5 0 . 0 0

: 1 . 0 0 X + 2 . 0 0 Y = 1 5 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y = 4 0 . 0 0

P a y o f f : 2 0 . 0 0 X + 3 0 . 0 0 Y = 2 3 7 5 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 2 5 . 0 0 , 6 2 . 5 0 ) : 5 . 0 0 X + 2 . 0 0 Y < = 2 5 0 . 0 0

: 1 . 0 0 X + 2 . 0 0 Y < = 1 5 0 . 0 0 : 1 . 0 0 X + 0 . 0 0 Y < = 4 0 . 0 0

See file P2-33.XLS.

Pruned Regular

Number of barrels

25.00

62.50

Revenue

$20

$30

$2,375.00

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

X,

Y



0

Non-negativity

0 2 5 0 5 0 0 7 5 0 1 0 0 0 1 2 5 0 1 5 0 0 1 7 5 0 2 0 0 0 2 2 5 0 2 5 0 0 2 7 5 0 3 0 0 0 3 2 5 0 3 5 0 0 3 7 5 0 4 0 0 0 4 2 5 0 4 5 0 0 4 7 5 0 5 0 0 0 0

1 7 5 3 5 0 5 2 5 7 0 0 8 7 5 1 0 5 0 1 2 2 5 1 4 0 0 1 5 7 5 1 7 5 0 1 9 2 5 2 1 0 0 2 2 7 5 2 4 5 0 2 6 2 5 2 8 0 0 2 9 7 5 3 1 5 0 3 3 2 5 3 5 0 0 Y

X : 0 . 3 6 X + 0 . 2 4 Y = 8 7 5 . 0 0

: 1 . 6 7 X + 1 . 5 0 Y = 5 0 0 0 . 0 0

: 0 . 0 4 X + 0 . 0 8 Y = 2 0 0 . 0 0

P a y o f f : 1 . 0 0 X + 1 . 0 0 Y = 3 1 7 9 . 3 4

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 3 5 8 . 7 0 , 1 8 2 0 . 6 5 ) : 0 . 3 6 X + 0 . 2 4 Y > = 8 7 5 . 0 0

: 1 . 6 7 X + 1 . 5 0 Y > = 5 0 0 0 . 0 0 : 0 . 0 4 X + 0 . 0 8 Y > = 2 0 0 . 0 0

See file P2-34.XLS.

Louisiana

Trimex

$ invested

$1,358.70 $1,820.65

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

X,

Y



0

Non-negativity

0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 0

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 Y

X : 5 . 0 0 0 X + 1 5 . 0 0 0 Y = 3 0 0 . 0 0 0

: 0 . 1 2 5 X + 1 . 0 0 0 Y = 1 5 . 0 0 0

P a y o f f : 6 0 . 0 0 0 X + 3 0 0 . 0 0 0 Y = 5 0 4 0 . 0 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 2 4 . 0 0 0 , 1 2 . 0 0 0 ) : 5 . 0 0 0 X + 1 5 . 0 0 0 Y < = 3 0 0 . 0 0 0

: 0 . 1 2 5 X + 1 . 0 0 0 Y < = 1 5 . 0 0 0

See file P2-35.XLS.

Coconuts

Skins

Number carried

24.00

12.00

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

X,

Y



0

Non-negativity

0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0 2 7 5 3 0 0 3 2 5 3 5 0 3 7 5 4 0 0 4 2 5 4 5 0 4 7 5 5 0 0 0

2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5 2 0 0 2 2 5 2 5 0 2 7 5 3 0 0 3 2 5 3 5 0 3 7 5 4 0 0 4 2 5 4 5 0 4 7 5 5 0 0 Y

X : - 0 . 3 0 X + 0 . 7 0 Y = 0 . 0 0

: 3 . 2 0 X + 2 . 4 0 Y = 1 1 2 0 . 0 0

: 1 . 0 0 X + 1 . 0 0 Y = 3 9 0 . 0 0 P a y o f f : 6 5 . 0 0 X + 5 5 . 0 0 Y = 2 3 7 5 0 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 2 3 0 . 0 0 , 1 6 0 . 0 0 ) : - 0 . 3 0 X + 0 . 7 0 Y > = 0 . 0 0

: 3 . 2 0 X + 2 . 4 0 Y < = 1 1 2 0 . 0 0 : 1 . 0 0 X + 1 . 0 0 Y < = 3 9 0 . 0 0

See file P2-36.XLS.

Boys

Girls

Number of units 230.00 160.00

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 9 0 0 1 8 0 0 2 7 0 0 3 6 0 0 4 5 0 0 5 4 0 0 6 3 0 0 7 2 0 0 8 1 0 0 9 0 0 0 9 9 0 0 1 0 8 0 0 1 1 7 0 0 1 2 6 0 0 1 3 5 0 0 1 4 4 0 0 1 5 3 0 0 1 6 2 0 0 1 7 1 0 0 1 8 0 0 0 0

6 0 0 1 2 0 0 1 8 0 0 2 4 0 0 3 0 0 0 3 6 0 0 4 2 0 0 4 8 0 0 5 4 0 0 6 0 0 0 6 6 0 0 7 2 0 0 7 8 0 0 8 4 0 0 9 0 0 0 9 6 0 0 1 0 2 0 0 1 0 8 0 0 1 1 4 0 0 1 2 0 0 0 Y

X : - 0 . 2 5 0 X + 0 . 7 5 0 Y = 0 . 0 0 0

: 0 . 5 5 5 X + 1 . 0 0 0 Y = 1 5 4 0 0 . 0 0 0

: 0 . 0 0 0 X + 1 . 0 0 0 Y = 8 0 0 0 . 0 0 0

P a y o f f : 2 2 . 6 7 0 X + 2 9 . 0 1 0 Y = 5 6 0 6 4 0 . 9 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 1 7 3 3 5 . 8 3 5 , 5 7 7 8 . 6 1 2 ) : - 0 . 2 5 0 X + 0 . 7 5 0 Y > = 0 . 0 0 0

: 0 . 5 5 5 X + 1 . 0 0 0 Y < = 1 5 4 0 0 . 0 0 0 : 0 . 0 0 0 X + 1 . 0 0 0 Y < = 8 0 0 0 . 0 0 0

See file P2-37.XLS.

Regular Intelligent

Number of units

17,335.83 5,778.61

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 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 0

2 4 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 4 8 5 0 Y

X : 0 . 1 5 X + 0 . 3 0 Y = 8 . 5 0

: 0 . 3 6 X + 0 . 4 0 Y = 1 3 . 0 0 : 3 . 0 0 X + 2 . 0 0 Y = 9 5 . 0 0

: 0 . 0 0 X + 5 . 0 0 Y = 1 2 5 . 0 0

P a y o f f : 0 . 5 8 X + 0 . 4 5 Y = 1 9 . 0 0

O p t i m a l D e c i s i o n s ( X , Y ) : ( 2 5 . 0 0 , 1 0 . 0 0 ) : 0 . 1 5 X + 0 . 3 0 Y < = 8 . 5 0

: 0 . 3 6 X + 0 . 4 0 Y < = 1 3 . 0 0 : 3 . 0 0 X + 2 . 0 0 Y < = 9 5 . 0 0 : 0 . 0 0 X + 5 . 0 0 Y < = 1 2 5 . 0 0

See file P2-38.XLS.

Mild

Spicy

Number of units

25.00

10.00

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

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 M

= number of X409 valves to produce, M

, M

, M

defined similarly.

Objective: Maximize profit = $16M

+ $12M

+ $13M

+ $8M

Subject to:

0.40M

+ 0.30M

+ 0.45M

+ 0.35M



700

Drilling time

0.60M

+ 0.65M

+ 0.52M

+ 0.48M



890

Milling time

1.20M

+ 0.60M

+ 0.50M

+ 0.70M



1,200

Lathe time

0.25M

+ 0.25M

+ 0.25M

+ 0.25M



525

Inspection time

M



200

Minimum X409

M



250

Minimum X3125

M



600

Minimum X4950

M



450

Minimum X2173

M

,

M

,

M

,

M



0

Non-negativity

See file P2-42.XLS.

X409 X3125 X4950 X2173

# of valves

332.50 250.00 600.00 450.00

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

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

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