Chapter 6, Problem 39RE

In problems 38-41, 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 by the simplex, dual problem, or big $M$ methods.

Manufacturing. A company manufactures outdoor furniture consisting of regular chairs, rocking chairs, and chaise lounges. Each piece of furniture passes through three different production departments: fabrication, assembly, and finishing. Each regular chair takes $1$ hour to fabricate, $2$ hours to assemble, and $3$ hours to finish. Each rocking chair takes $2$ hours to fabricate, $2$ hours to assemble, and $3$ to finish. Each chaise lounge takes $3$ hours to fabricate, $4$ hours to assemble, and $2$ hours to finish. There are $2,500$ labor-hours available in the fabrication department, $3,000$ in the assembly department, and $3,500$ in the finishing department. The company makes a profit of $\text{\$}17$ on each regular chair, $\text{\$}24$ on each rocking chair, and $\text{\$}31$ on each chaise lounge.

(A) How many chairs of each type should the company produce in order to maximize profit? What is the maximum profit?

(B) Discuss the effect on the optimal solution in part (A) if the profit on a regular chair is increased to $\text{\$}25$ and all other data remain the same.

(C) Discuss the effect on the optimal solution in part (A) if the available hours on the finishing department are reduced to $3,000$ and all other data remain the same.