Strayer MAT540 week 8 homework

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 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:
·
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.

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:
·
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
(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 – 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.
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?

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 “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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