16. Supersport Footballs, Inc., has to determine the best number of All-Pro (A), College (C), and High School (H) models of footballs to produce in order to maximize profits. Constraints include production capacity limitations (time available in minutes) in each of three departments (cutting and dyeing, sewing, and inspection and packaging) as well as a constraint that requires the production of at least 1000 All-Pro footballs. The linear programming model of Supersport’s problem is shown here:

Max 3A+5C+4H

St

12A+10C+8H≤18,000 Cutting and dyeing

15A+15C+12H≤18,000 Sewing

3A+4C+2H≤9,000 Inspection and packaging

1A≥1,000 All-Pro model

A, C, H ≥0

The computer solution to the Supersport problem is shown in Figure 8.21.

a. How many footballs of each type should Supersport produce to maximize the total profit contribution?

b. Which constraints are binding?

c. Interpret the slack and/or surplus in each constraint.

d. Interpret the objective coefficient ranges.

17. Refer to the computer solution of Problem 16 in Figure 8.21.

a. Overtime rates in the sewing department are $12 per hour. Would you recommend that the company consider using overtime in that department? Explain.

b. What is the dual price for the fourth constraint? Interpret its value for management.

c. Note that the reduced cost for H, the High School football, is zero, but H is not in the solution at a positive value. What is your interpretation of this value?

d. Suppose that the profit contribution of the College ball is increased by $1. How do you expect the solution to change?


FIGURE 8.21THE MANAGEMENT SCIENTIST SOLUTION FOR THE SUPERSPORT FOOTBALLS PROBLEM

Objective Function Value = 4000.000

Variable Value Reduced Costs

------

A 1000.000 0.000

C 200.000 0.000

H 0.000 0.000

Constraint Slack/Surplus Dual Prices

------

1 4000.000 0.000

2 0.000 0.333

3 5200.000 0.000

4 0.000 –2.000

OBJECTIVE COEFFICIENT RANGES

Variable Lower Limit Current Value Upper Limit

------

A No Lower Limit 3.000 5.000

C 5.000 5.000 No Upper Limit

H No Lower Limit 4.000 4.000

RIGHT HAND SIDE RANGES

Constraint Lower Limit Current Value Upper Limit

------

1 14000.000 18000.000 No Upper Limit

2 15000.000 18000.000 24000.000

3 3800.000 9000.000 No Upper Limit

4 0.000 1000.000 1200.000