Chapter 5.3, Problem 1MP

A manufacturing plant makes two types of inflatable boats––a two-person boat and a four-person boat. Each two-person boat requires $0.9$ labor-hour from the cutting department and $0.8$ labor-hour from the assembly department. Each four-person boat requires $1.8$ labor-hours from the cutting department and $1.2$ labor-hours from the assembly department. The maximum labor-hours available per month in the cutting department and the assembly department are $864$ and $672$, respectively. The company makes a profit of $\text{\$}25$ on each two-person boat and $\text{\$}40$ on each four-person boat.

(A) Identify the decision variables.

(B) Summarize the relevant material in a table similar to Table 1 in Example 1.

(C) Write the objective function $P$.

(D) Write the problem constraints and nonnegative constraints.

(E) Graph the feasible region. Include graphs of the objective function for $P=\text{\$}5,000,P=\text{\$}10,000,P=\text{\$}15,000$, and $P=\text{\$}21,600$.

(F) From the graph and constant-profit lines, determine how many boats should be manufactured each month to maximize the profit. What is the maximum profit?