MAT540 Week 8 Homework Chapter 4

| October 3, 2018

MAT540 Homework

Week 8
Page 1 of 4

MAT540

Week 8 Homework

Chapter 4

1.
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:

Metal (%)

Impurities

Ore

A

B

C

D

(%)

Cost/Ton

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.

2.
As a result of a recently passed bill, a
congressman’s 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 – 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 staff’s 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:

MAT540 Homework

Week 8
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·
None of the
programs can receive more than 40% of the total allocation.

·
The amount
allocated to parks cannot exceed the total allocated to both the sanitation
project and the mobile library

·
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.

3.
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:

·
Between
1,500 and 2,000 calories

·
At least 5
mg of iron

·
At least 20
but no more than 60 g of fat

·
At least 30
g of protein

·
At least 40
g of carbohydrates

·
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

Iron

Protein

Carbo-

Fat

Chol-

Cost

(per lb.)

(mg/lb.)

(g/lb.)

hydrates

(g/lb.)

esterol

(g/lb.)

(mg/lb.)

$/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.

MAT540 Homework

Week 8
Page 3 of 4

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?

4.
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 Capacity

Mine

1

2

3

4

(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.

5.
Joe
Henderson runs a small metal parts shop. The shop contains three machines – 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

MAT540 Homework

Week 8
Page 4 of 4

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

Lathe

Grinder

(min)

(min)

(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.
Joe’s
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?

6.
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 “trim-loss.” 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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