MAT540 Homework Week 8

MAT540 Homework Week 8

Page 1 of 4

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MAT540

Week 8 Homework

Chapter 4

 

1. 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.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer.

2. 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.

 

 

 

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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.

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 1,300 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:

 

 

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Calories

(per lb.)

Iron

(mg/lb.)

Protein

(g/lb.)

Carbo-

hydrates

(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.

a. Formulate a linear programming model for this problem and solve.

b. 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. 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 2-

hour 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.

a. Formulate a linear programming model for this problem.

 

 

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b. Solve the model by using the computer.

5. 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.

a. Formulate a linear programming model for this problem.

b. Solve the model by using the computer.