Chapter 13, Problem 4P

The profit function for two products is:

$\text{Profit}--3x_1^2+42x_1-3x_2^2+48x_2+700,$

where x₁ represents units of production of product 1, and x₂ represents units of production of product 2. Producing one unit of product 1 requires 4 labor-hours, and producing one unit of product 2 requires 6 labor-hours. Currently, 24 labor-hours are available. The cost of labor-hours is already factored into the profit function, but it is possible to schedule overtime at a premium of $5 per hour.

1. a. Formulate an optimization problem that can be used to find the optimal production quantity of products 1 and 2 and the optimal number of overtime hours to schedule.
2. b. Solve the optimization model you formulated in part (a). How much should be produced and how many overtime hours should be scheduled?