Model 20: Production Capacity Adjustment

A manufacturing firm has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one or more of three products; call them products 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table:

Machine type Available time
(in machine hours per week)
Milling machine 500
Lathe 350
Grinder 150

The number of machine hours required for each unit of the respective products is

Productivity coefficient (in machine hours per unit)
Machine type Product 1 Product 2 Product 3
Milling machine 9 3 5
Lathe 5 4 0
Grinder 3 0 2

The sales department indicates that the sales potential for products 1 and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $30, $12, and $15, respectively, on products 1, 2, and 3. The objective is to determine how much of each product the firm should produce to maximize profit.

Formulate the linear programming model for this problem.