Chapter 6.2, Problem 48E

In Problems 41-56, construct a mathematical model in the form of a linear programming problem. (The answer in the back of the book for these application problems include the model.) Then solve the problem using the simplex method. Include an interpretation of any nonzero slack variables in the optimal solution.

Bicycle manufacturing. A company manufactures three-speed, five-speed and ten-speed bicycles. Each bicycle passes through three departments: fabrication, painting $\&$ plating, and final assembly. The relevant manufacturing data are given in the table.

$\begin{array}{l}\text{ Labor-Hours per Bicycle}\\ \begin{array}{ccccc}& \text{Three-Speed}& \text{Five-Speed}& \text{Ten-Speed}& \begin{array}{l}\text{Mamimum Labor-Hours }\\ \text{ Available per day}\end{array}\\ \text{Fabrication}& 3& 4& 5& 120\\ \text{Painting \& plating}& 5& 3& 5& 130\\ \text{Final assembly}& 4& 3& 5& 120\\ \text{Profit per bicycle}\left(\text{\$}\right)& 80& 70& 100& \end{array}\end{array}$

How many bicycles of each type should the company manufacture per day in order to maximize its profit? What is the maximum profit?