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MAT540 Homework Week 7

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MAT540

Week 7 Homework

Chapter 3

1. Southern Sporting Good Company makes basketballs and footballs. Each product is produced

from two resources rubber and leather. The resource requirements for each product and the

total resources available are as follows:

Resource Requirements per Unit

Product Rubber (lb.) Leather (ft 2 )

Basketball 3 4

Football 2 5

Total resources available

500 lb. 800 ft 2

a. State the optimal solution.

b. What would be the effect on the optimal solution if the profit for a basketball changed

from $12 to $13? What would be the effect if the profit for a football changed from $16

to $15?

c. What would be the effect on the optimal solution if 500 additional pounds of rubber

could be obtained? What would be the effect if 500 additional square feet of leather

could be obtained?

2. A company produces two products, A and B, which have profits of $9 and $7, respectively. Each

unit of product must be processed on two assembly lines, where the required production times

are as follows:

Hours/ Unit

Product Line 1 Line2

A 12 4

B 4 8

Total Hours 60 40

a. Formulate a linear programming model to determine the optimal product mix that will

maximize profit.

b. Transform this model into standard form.

3. Solve problem 2 using the computer.

a. State the optimal solution.

b. What would be the effect on the optimal solution if the production time on line 1 was

reduced to 40 hours?

MAT540 Homework Week 7

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c. What would be the effect on the optimal solution if the profit for product B was increased

from $7 to $15 to $20?

4. For the linear programming model formulated in Problem 2 and solved in Problem 3.

a. What are the sensitivity ranges for the objective function coefficients?

b. Determine the shadow prices for additional hours of production time on line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2 hours.

5. Formulate and solve the model for the following problem:

Irwin Textile Mills produces two types of cotton cloth – denim and corduroy. Corduroy is a

heavier grade of cotton cloth and, as such, requires 7.5 pounds of raw cotton per yard, whereas

denim requires 5 pounds of raw cotton per yard. A yard of corduroy requires 3.2 hours of

processing time; a yard of denim requires 3.0 hours. Although the demand for denim is

practically unlimited, the maximum demand for corduroy is 510 yards per month. The

manufacturer has 6,500 pounds of cotton and 3,000 hours of processing time available each

month. The manufacturer makes a profit of $2.25 per yard of denim and $3.10 per yard of

corduroy. The manufacturer wants to know how many yards of each type of cloth to produce to

maximize profit. Formulate the model and put it into standard form. Solve it.

. a. How much extra cotton and processing time are left over at the optimal solution? Is

the demand for corduroy met? b. What is the effect on the optimal solution if the profit per yard of denim is increased

from $2.25 to $3.00? What is the effect if the profit per yard of corduroy is increased from $3.10 to $4.00?

c. What would be the effect on the optimal solution if Irwin Mils could obtain only 6,000 pounds of cotton per month?

6. Continuing the model from Problem 5.

a. If Irwin Mills can obtain additional cotton or processing time, but not both, which should

it select? How much? Explain your answer.

b. Identify the sensitivity ranges for the objective function coefficients and for the

constraint quantity values. Then explain the sensitivity range for the demand for

corduroy.

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7. United Aluminum Company of Cincinnati produces three grades (high, medium, and low) of

aluminum at two mills. Each mill has a different production capacity (in tons per day) for each grade

as follows:

Aluminum Grade

Mill

1 2

High 6 2

Medium 2 2

Low 4 10

The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade

aluminum, and 5 tons of low-grade aluminum. It costs United $6,000 per day to operate mill 1 and

$7,000 per day to operate mill 2. The company wants to know the number of days to operate each

mill in order to meet the contract at minimum cost.

a. Formulate a linear programming model for this problem.

8. Solve the linear programming model formulated in Problem 16 for Unite Aluminum Company by

using the computer.

a. Identify and explain the shadow prices for each of the aluminum grade contract requirements.

b. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity

values.

c. Would the solution values change if the contract requirements for high-grade alumimum were

increased from 12 tons to 20 tons? If yes, what would the new solution values be?

9. Solve the linear programming model developed in Problem 22 for the Burger Doodle restaurant by

using the computer.

a. Identify and explain the shadow prices for each of the resource constraints

b. Which of the resources constrains profit the most?

c. Identify the sensitivity ranges for the profit of a sausage biscuit and the amount of sausage

available. Explain these sensitivity ranges.

Reference Problem 22. The manager of a Burger Doodle franchise wants to determine how

many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. The

two types of biscuits require the following resources:

Biscuit Labor (hr.) Sausage (lb.) Ham (lb.) Flour (lb.)

Sausage 0.010 0.10 — 0.04

Ham 0.024 — 0.15 0.04

The franchise has 6 hours of labor available each morning. The manager has a contract with a

local grocer for 30 pounds of sausage and 30 pounds of ham each morning. The manager also

MAT540 Homework Week 7

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purchases 16 pounds of flour. The profit for a sausage biscuit is $0.60; the profit for a ham

biscuit is $0.50. The manager wants to know the number of each type of biscuit to prepare each

morning in order to maximize profit. Formulate a linear programming model for this problem.