Part $1-$ Graphical Solution

Problem $2.$ Sports Products, Inc. wants to determine the number of All$-$Pro $($A$)$ and College $($C$)$ footballs to produce in order to maximize profit over the next four$-$week planning horizon.

Constraints affecting the production quantities are the production capacities in three departments: cutting and dyeing $($CD$)$; sewing $($S$)$; and inspection and packaging $($IP$).$ For the four$-$week planning period, $340$ hours of cutting and dyeing time, $420$ hours of sewing time, and $200$ hours of inspection and packaging time are available. All$-$Pro footballs provide a profit of $$5$ per unit and College footballs provide a profit of $$4$ per unit. The linear programming model with production times expressed in minutes is as follows:

Maximize $Z=5A+4C$