$1$ Part :$-$Graphical Solution

$2$

$3$ Problem $2.$ Sports Products, Inc. wants to determine the number of All$-$Pro $($A$)$ and College $($C$)$ footballs to

$4$ 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:

$11$

$12$ Maximize $Z=5A+4C$

$13$ Subject to:

$12A+6C20,400$ Cutting and dyeing

$9A+15C25,200$ Sewing

$6A+6C12,000$ Inspection and packaging

$A,C0,$

a$.$ Graph the constraints and show the region of feasible solutions.

b$.$ Solve the problem and find the optimum solution. Give the results in the following table.

$\backslash$table$[[$Number of A footballs,$],[$Number of C footballs,$],[$Slack in CD$,],[$Slack in S$,],[$Slack in IP$,]]$

SHOW YOUR GRAPH BELOW.