Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. The firm has $850$ hours of production time available in its cutting and sewing department, $200$ hours available in its finishing department, and $100$ hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table.

Model Production Time

$($hours$)$

Cutting and Sewing Finishing Packaging and

Shipping Profit$/$Glove

Regular model $1$

$1$

$2$

$1$

$8$

$$4$

Catcher's model

$3$

$2$

$1$

$3$

$1$

$4$

$$7$

Assuming that the company is interested in maximizing the total profit contribution, answer the following.

$($a$)$

What is the linear programming model for this problem? $($Assume R is the number of units of regular model gloves and C is the number of units of catcher's model gloves.$)$

$($b$)$

Develop a spreadsheet model and find the optimal solution using Excel Solver. How many of each model should Kelson manufacture?

$($c$)$

What is the total profit contribution $($in dollars$)$ Kelson can earn with the optimal production quantities?

$($d$)$

How many hours of production time will be scheduled in each department?

$($e$)$

What is the slack time $($in hours$)$ in each department?