# MAT 540 Week 8 Chapter 4 Homework

**Chapter**

4

4

- Betty Malloy, owner of the Eagle Tavern in Pittsburgh,

is preparing for Super Bowl Sunday, and she must determine how much beer

to stock. Betty stocks three brands of beer- Yodel, Shotz, and Rainwater.

The cost per gallon (to the tavern owner) of each brand is as follows:

**Brand**

**Cost/Gallon**

**Yodel**

$1.50

**Shotz**

0.90

**Rainwater**

0.50

The tavern has a budget of $2,000

for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per

gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on

past football games, Betty has determined the maximum customer demand to be 400

gallons of Yodel, 500 gallons of shotz, and 300 gallons of Rainwater. The

tavern has the capacity to stock 1,000 gallons of beer; Betty wants to stock up

completely. Betty wants to determine the number of gallons of each brand of

beer to order so as to maximize profit.

- Formulate a linear

programming model for this problem.

- Solve the model by using the

computer.

- As result of a recently passed bill, a congressman’s

district has been allocated $3 million for programs and projects. It is up

to the congressman to decide how to distribute the money. The congressman

has decide 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**

0.03

**Parks**

0.08

**Sanitation**

0.05

**Mobile library**

0.03

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 30% 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. Thee congressman wants to know the amount to

allocate to each program to maximize his votes.

- Formulate a linear

programming model for this problem.

- Solve the model by using the

computer.

- 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 1,300 and 2,100

calories - At least 4 mg of iron
- At least 15 but no more than

55g of fat - At least 30g of protein
- At least 60g of carbohydrates
- No more than 35 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.)**

**Carbohydrates**

**(g/lb.)**

**Fat**

**(g/lb.)**

**Cholesterol**

**(mg/lb)**

**Cost**

**($/lb.)**

Chicken

500

4.2

17

0

30

180

0.85

Fish

480

3.1

85

0

5

90

3.35

Ground beef

840

0.25

82

0

75

350

2.45

Dried beans

590

3.2

10

30

3

0

0.85

Lettuce

40

0.4

6

0

0

0

0.70

Potatoes

450

2.25

10

70

0

0

0.45

Milk (2%)

220

0.2

16

22

10

20

0.82

The dietician wants to select a menu

to meet the nutritional guidelines while minimizing the total cost per serving.

- Formulate a linear programming model for this problem

and solve.

- 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?

- Dr. Maureen Becker, the head administrator at Jefferson

County Regional Hospital, must determine a schedule for nurses to make

sure there are enough of them on duty throughout the day. During the day,

the demand for nurses varies. Maureen has broken the day in to twelve

2hour periods. The slowest time of the day encompasses the three periods

from 12:00 A.M. to 6:00 A.M., which beginning at midnight; require a

minimum of 30, 20, and 40 nurses, respectively. The demand for nurses

steadily increases during the next four daytime periods. Beginning with

the 6:00 A.M.- 8:00 A.M. period, a minimum of 50, 60, 80, and 80 nurses

are required for these four periods, respectively. After 2:00 P.M. the

demand for nurses decreases during the afternoon and evening hours. For

the five 2-hour periods beginning at 2:00 P.M. and ending midnight, 70,

70, 60, 50, and 50 nurses are required, respectively. A nurse reports for

duty at the beginning of one of the 2-hour periods and works 8 consecutive

hours (which is required in the nurses’ contract). Dr. Becker wants to

determine a nursing schedule that will meet the hospital’s minimum

requirement throughout the day while using the minimum number of nurses.

- Formulate a linear

programming model for this problem.

- Solve the model by using the

computer.

- The production manager of Videotechnics Company is attempting

to determine the upcoming 5-month production schedule for video recorders.

Past production records indicate that 2,000 recorders can be produced per

month. An additional 600 recorders can be produced monthly on an overtime

basis. Unit cost is $10 for recorders produced during regular working

hours and $15 for those produced on an overtime basis. Contracted sales

per month are as follows:

**Month**

**Contracted Sales (units)**

**1**

1200

**2**

2100

**3**

**4**

**5**

2400

3000

4000

Inventory carrying costs are $2 per recorder

per month. The manager does not want any inventory carried over past the fifth

month. The manager wants to know the monthly production that will minimize

total production and inventory costs.

- Formulate a linear programming model for this problem.

- Solve the model by using the computer.

You'll get 1 file (21.1KB)