Problem: 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
s.t. 12A + 10C + 8H ≤ 18,000 Cutting and sewing
15A + 15C + 12H ≤ 18,000 Sewing
3A + 4C + 2H ≤ 9,000 Inspection and packaging
A ≥ 1,000 All-Pro model
A, C, H ≥ 0
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?
e. How many footballs of each type should Supersport produce to maximize the total profit contribution?
f. Which constraints are binding?
g. Interpret the slack and/or surplus in each constraint.
h. Interpret the ranges of optimality for the profit contributions of the three footballs.