# Mat 540 Week 7 Homework Solutions

MAT 540 Week 7 Homework Chapter 3

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

two resources rubber and leather. Each basketball produced results in a profit of $11 and each

football earns $15 in profit. The resource requirements for each product and the total resources

available are as follows:

Product

Total resources available 600 900

a. Find the optimal solution.

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

$11 to $12?

c. What would be the effect on optimal solution if 400 additional pounds of rubber could be

obtained? What would be the effect if 600 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:

Product

Resource Requirements per Unit

Line 1 Line 2

A 11 5

B 6 9

Total Hours 65 40

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

maximize profit.

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

c. 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.

3. 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 8 pounds of raw cotton per yard, whereas denim

requires 6 pounds of raw cotton per yard. A yard of corduroy requires 4 hours of processing time;

a yard od 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 $2.5 per yards of denim and $3.25 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. If Irwin Mills can obtain additional cotton or processing time, but not both, which should it

select? How much? Explain your answer.

4. The Bradley family owns 410 acres of farmland in North Carolina on which they grow corn and

tobacco. Each acre of corn costs $105 to plant, cultivate, and harvest; each acre of tobacco costs

$210. The Bradleys’ have a budget of $52,500 for next year. The government limits the number

of acres of tobacco that can be planted to 100. The profit from each acre of corn is $300; the

profit from each acre of tobacco is $520. The Bradleys’ want to know how many acres of each

crop to plant in order to maximize their profit.

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

b. How many acres of farmland will not be cultivated at the optimal solution? Do the Bradleys use

the entire 100-acre tobacco allotment?

c. The Bradleys’ have an opportunity to lease some extra land from a neighbor. The neighbor is

offering the land to them for $110 per acre. Should the Bradleys’ lease the land at that price?

What is the maximum price the Bradleys’ should pay their neighbor for the land, and how

much land should they lease at that price?

MAT540 Homework

d. The Bradleys’ are considering taking out a loan to increase their budget. For each dollar they

borrow, how much additional profit would they make? If they borrowed an additional $1,000,

would the number of acres of corn and tobacco they plant change?

LaTonya Meori: Matt 540 Week 7 Homework 8. Southern Sporting Goods 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 (lbs.) Leather (ft.2) Z = 12x1 + 16x2 Basketballs (X1) $12 ea. 3 4 3x1 + 4x2 ≤ 500 lbs. Footballs (X2) $16 ea. 2 5 2x1 + 5x2 ≤ 800 ft.2 Total Resources Available 500 lbs. 800 ft.2 Each basketball produced results in a profit of $12, and each football earns $16 in profit. Z = 12x1 + 16x2 a. Formulate a linear programming model to determine the number of basketballs and footballs to produce in order to maximize profit. a. Z = 12x1 + 16x2 where Z = profit, so maximize Z subject to the following assumptions: b. let x1 = the number of basketballs to produce, x2 = the number of footballs to produce. c. Rubber constraint equation: 3x1 + 4x2 ≤ 500 lbs.; Leather constraint equation: 2x1 + 5x2 ≤ 800 ft.2 d. Transform this model into standard form (with slack/surplus variables). e. Z = 12x1 + 16x2, maximize Profit (Z) subject to the following constraints: (x1, x2, s1, s2 ≥ 0) f. Rubber Resource constraint equation with slack variable s1: 3x1 + 4x2 + s1 = 500 lbs. g. Leather Resource constraint equation with slack variable s2: 2x1 + 5x2 + s2 = 800 ft.2 10. A company produces two products A and B, which have profits of$9 and $7. Each unit of product must be processed on twoassembly lines where the required production times are afollows: hr/unit Product Line1 Line2 A 12 4 B 4 8 totalhours 60 40

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