MAT540

Week

8 Homework

Chapter 4

14.

Grafton Metalworks Company produces metal alloys from six different ores it

mines. The company has an order from a

customer to produce an alloy that contains four metals according to the

following specifications: at least 21%

of metal A, no more than 12% of metal B, no more than 7% of metal C and between

30% and 65% of metal D. The proportion

of the four metals in each of the six ores and the level of impurities in each

ore are provided in the following table:

Ore

Metal (%)

Impurities (%)

Cost/Ton

A

B

C

D

1

19

15

12

14

40

27

2

43

10

25

7

15

25

3

17

0

0

53

30

32

4

20

12

0

18

50

22

5

0

24

10

31

35

20

6

12

18

16

25

29

24

When the metals are processed and refined, the

impurities are removed.

The company wants to know the amount of each ore

to use per ton of the alloy that will minimize the cost per ton of the alloy.

a.

Formulate a linear programming model

for this problem.

b.

Solve the model by using the computer.

19.

As a result of a recently passed bill, a congressmanâs district has been

allocated $4 million for programs and projects.

It is up to the congressman to decide how to distribute the money. The congressman has decided to allocate the

money to four ongoing programs because of their importance to his district â a

job training program, a parks project, a sanitation project, and a mobile

library. However, the congressman wants

to distribute the money in a manner that will please the most voters, or, in

other words, gain him the most votes in the upcoming election. His staffâs estimates of the number of votes

gained per dollar spent for the various programs are as follows.

Program

Votes/ Dollar

Job training

0.02

Parks

0.09

Sanitation

0.06

Mobile library

0.04

In order also to satisfy several local influential citizens who financed

his election, he is obligated to observe the following guidelines:

Â·

None of the programs can receive more than 40% of the total allocation.

Â·

The amount allocated to parks cannot exceed the total allocated to both the

sanitation project and the mobile library

Â·

The amount allocated to job training must at least equal the amount spent

on the sanitation project.

Any money not spent in the district will be returned to the government;

therefore, the congressman wants to spend it all. The congressman wants to know the amount to

allocate to each program to maximize his votes.

a.

Formulate a linear programming model for this problem.

b. Solve the model by using the

computer.

20.

Anna Broderick is the dietician for the State University football team, and

she is attempting to determine a nutritious lunch menu for the team. She has set the following nutritional

guidelines for each lunch serving:

Â·

Between 1,500 and 2,000 calories

Â·

At least 5 mg of iron

Â·

At least 20 but no more than 60 g of fat

Â·

At least 30 g of protein

Â·

At least 40 g of carbohydrates

Â·

No more than 30 mg of cholesterol

She selects the menu from seven basic food items, as follows, with the

nutritional contributions per pound and the cost as given:

Calories

(per lb.)

Iron

(mg/lb.)

Protein

(g/lb.)

Carbo-hydrates

(g/lb.)

Fat (g/lb.)

Chol-esterol

(mg/lb.)

Cost

$/lb.

Chicken

520

4.4

17

0

30

180

0.80

Fish

500

3.3

85

0

5

90

3.70

Ground beef

860

0.3

82

0

75

350

2.30

Dried beans

600

3.4

10

30

3

0

0.90

Lettuce

50

0.5

6

0

0

0

0.75

Potatoes

460

2.2

10

70

0

0

0.40

Milk (2%)

240

0.2

16

22

10

20

0.83

The dietician wants to select a menu to meet the nutritional guidelines

while minimizing the total cost per serving.

a.

Formulate a linear programming model for this problem.

b.

Solve the model by using the computer

c. If a serving of each of the

food items (other than milk) was limited to no more than a half pound, what

effect would this have on the solution?

22.

The Cabin Creek Coal (CCC) Company operates three mines in Kentucky and

West Virginia, and it supplies coal to four utility power plants along the East

Coast. The cost of shipping coal from

each mine to each plant, the capacity at each of the three mines and the demand

at each plant are shown in the following table:

Plant

Mine

1

2

3

4

Mine Capacity

(tons)

1

$ 7

$ 9

$10

$12

220

2

9

7

8

12

170

3

11

14

5

7

280

Demand (tons)

110

160

90

180

The cost of mining and processing coal is $62 per ton at mine 1, $67 per ton at mine 2, and $75 per ton at mine 3. The percentage of ash and sulfur content per

ton of coal at each mine is as follows:

Mine

% Ash

% Sulfur

1

9

6

2

5

4

3

4

3

Each plant has different cleaning equipment. Plant 1 requires that the coal it receives

have no more than 6% ash and 5% sulfur; plant 2 coal can have no more than 5%

ash and sulfur combined; plant 3 can have no more than 5% ash and 7% sulfur;

and plant 4 can have no more than 6% ash and sulfur combined. CCC wabts to determine the amount of coal

to produce at each mine and ship to its customers that will minimize its total

cost.

a.

Formulate a linear programming model for this problem.

b.

Solve this model by using the computer.

36.

Joe Henderson runs a small metal parts shop. The shop contains three

machines â a drill press, a lathe, and a grinder. Joe

has three operators, each certified to work on all three machines. However, each operator performs better on

some machines than on others. The shop

has contracted to do a big job that requires all three machines. The times required by the various operators

to perform the required operations on each machine are summarized as

follows:

Operator

Drill Press (min)

Lathe (min)

Grinder (min)

1

23

18

35

2

41

30

28

3

25

36

18

Joe Henderson wants to assign one operator to each machine so that the topal

operating time for all three operators is minimized.

a.

Formulate a linear programming model for this problem.

b.

Solve the model by using the computer

c.

Joeâs brother, Fred, has asked him to hire his wife, Kelly, who is a

machine operator. Kelly can perform each

of the three required machine operations in 20 minutes. Should Joe hire his sister-in-law?

43.

The Cash and Carry Building Supply Company has received the following order

for boards in three lengths:

Length

Order (quantity)

7 ft.

700

9 ft.

1,200

10 ft.

300

The company has 25-foot standard-length boards in stock. Therefore, the standard-length boards must be

cut into the lengths necessary to meet order requirements. Naturally, the company wishes to minimize the

number of standard-length boards used.

a.

Formulate a linear programming model for this problem.

b.

Solve the model by using the computer

c.

When a board is cut in a specific pattern, the amount of board left over is

referred to as âtrim-loss.â Reformulate the linear programming model for this

problem, assuming that the objective is to minimize trim loss rather than to

minimize the total number of boards used, and solve the model. How does this affect the solution?

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