Chapter 5.3, Problem 51E

In Problems 49-64, construct a mathematical model in the form of a linear programming problem. (The answers in the back of the book for these application problems include the model.) Then solve by the geometric method.

Production scheduling. A furniture company has two plants that produce the lumber used in manufacturing tables and chairs. In $1$ day of operation, plant $A$ can produce the lumber required to manufacture $20$ tables and $60$ chairs, and plant $B$ can produce the lumber required to manufacture $25$ tables and $50$ chairs. The company needs enough lumber to manufacture at least $200$ tables and $500$ chairs.

(A) If it costs $\text{\$}1,000$ to operate plant $A$ for $1$ day and $\text{\$}900$ to operate plant $B$ for $1$ day, how many days should each plant be operated to produce a sufficient amount of lumber at a minimum cost? What is the minimum cost?

(B) Discuss the effect on the operating schedule and the minimum cost if the daily cost of operating plant $A$ is reduced to $\text{\$}600$ and all other data in part (A) remain the same.

(C) Discuss the effect on the operating schedule and the minimum cost if the daily cost of operating plant $B$ is reduced to $\text{\$}800$ and all other data in part (A) remain the same.