MAT 540 Week 7 Homework Chapter 3 Problems P8, P10, P11, P12, P14, P15

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 (ft2)

Basketball 3 4

Football 2 5

Total resources

available

500 lb. 800 ft2

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?

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.