A manufacturing firm has discontinued production of a certain unprofitable product line.

Considerable excess production capacity was created as a result. Management is consid$-$

ering devoting this excess capacity to one or more of three products: $x_1,x_2,$ and $x_3.$

Machine hours required per unit are:

The available time in machine hours per week is:

The salespeople estimate they can sell all the units of $x_1$ and $x_2$ that can be made. But

the sales potential of $x_3$ is $80$ units per week maximum.

Unit profits for the three products are:

a$.$ Set up the equations that can be solved to maximize the profit per week.

$b.$ Solve these equations using the Excel Solver.

c$.$ What is the optimal solution? How many of each product should be made, and what

should the resultant profit be$?$

d$.$ What is this situation with respect to the machine groups? Would they work at capacity,

or would there be unused available time? Will $x_3$ be at maximum sales capacity?

e$.$ Suppose an additional $200$ hours per week can be obtained from the milling machines

by using staff overtime. The incremental cost would be $$1\mathrm{.}50$ per hour. Would you

recommend doing this? Explain how you arrived at your answer.