Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher's model. The firm has $700$ hours of production time available in its cutting and sewing department, $300$ hours available in its finishing department, and $200$ hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table:

Production Time $($Hours$)$

Model Cutting and

Sewing

Finishing Packaging and

Shipping

Profit$/$Glove

Regular model $13/21/6$ $$4$

Catcher's model $3/21/23/4$ $$8$

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? If required, round your answers to $3$ decimal places or enter your answers as a fraction. If the constant is $"1"$ it must be entered in the box. Do not round intermediate calculation. If an amount is zero, enter $"0".$

Let R $=$ number of units of regular model.

C $=$ number of units of catcher's model.

Max R $+$ C

s$.$t$.$

R $+$ C Cutting and Sewing

R $+$ C Finishing

R $+$ C Packing and Shipping

R$,$ C

$($b$)$ Develop a spreadsheet model and find the optimal solution using Solver. How many gloves of each model should Kelson manufacture?

Regular Model $=$ units

Catcher's Model $=$ units

$($c$)$ What is the total profit contribution Kelson can earn with the given production quantities?

$

$($d$)$ How many hours of production time will be scheduled in each department?

Department Production Time $($Hours$)$

Cutting and Sewing

Finishing

Packing and Shipping

$($e$)$ What is the slack time in each department? If your answer is zero, enter $0.$

Department Slack Time $($Hours$)$

Cutting and Sewing

Finishing

Packing and Shipping