Model $6$: The Factory Planning Problem

In this model, we will tackle a production planning problem for a factory that produces seven different products $($P$1$ to P$7)$ using five types of machines. Our objective is to find the optimal production plan that maximizes the total profit over a six$-$month period, subject to various constraints such as machine availability, sales limits$,$ and inventory levels.

The factory operates with four grinders, two vertical drills, three horizontal drills, one borer, and one planer. Each product has a known profit contribution per unit sold, and requires a certain amount of time on each machine for its manufacturing. The tables below show the profit contributions and the machine time requirements for each product. A dash indicates that the product does not need that machine.

Profit contributions $($$$)$: