Chapter 8, Problem 1P

Kelson Sporting Equipment, Inc., makes two types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 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:

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

1. a. What is the linear programming model for this problem?
2. b. Develop a spreadsheet model and find the optimal solution using Excel Solver. How many of each model should Kelson manufacture?
3. c. What is the total profit contribution Kelson can earn with the optimal production quantities?
4. d. How many hours of production time will be scheduled in each department?
5. e. What is the slack time in each department?

To determine

To form the linear programming model for the given problem.

Explanation of Solution

Through linear programming, the best outcome is achieved by using a mathematical model. The model in this case is shown below:

Let,

$\begin{array}{c}\text{R}=\text{number}\text{of}\text{units}\text{of}\text{regular}\text{model}\\ \text{C}=\text{number}\text{of}\text{units}\text{of}\text{catcher's}\text{model}\end{array}$

Max $5R+8C$

s.t.

$\begin{array}{c}R+\frac{3}{2}C\le 900\\ \frac{1}{2}R+\frac{1}{3}C\le 300\\ \frac{1}{8}R+\frac{1}{4}C\le 100\end{array}$

$R,C\ge 0$

To determine

The spreadsheet model, optimal solution, and number of units of each model manufactured by person K.

Explanation of Solution

On solving the above formulation in excel spreadsheet using solver,

Formula:

Solver:

Output:

The optimal solution will be to produce 500 regular gloves and 150 catchers.

The profit in this case will be $3,700.

To determine

To find the total profit contribution earned by person K by producing optimal quantities.

Explanation of Solution

Total profit that person K can earn by producing optimal quantities will be:

The optimal solution to produce regular gloves and catchers is substituted in the objective solution, then the profit obtained is,

$5\left(500\right)+ 8\left(150\right)=\$3,700$

The profit is $3,700

To determine

To find the production time to be scheduled in each department.

Explanation of Solution

The production time for every department can be calculated as follows:

For C&F

$\begin{array}{c}\mathrm{Pr}oductiontime=1\left(500\right)+\frac{3}{2}\left(150\right)\\ =725\end{array}$

For F

$\begin{array}{c}\mathrm{Pr}oductiontime=\frac{1}{2}\left(500\right)+\frac{1}{3}\left(150\right)\\ =300\end{array}$

For P&S

$\begin{array}{l}\begin{array}{l}\mathrm{Pr}oductiontime=\frac{1}{8}\left(500\right)+\frac{1}{4}\left(150\right)\\ =100 \end{array}\hfill \\ \hfill \end{array}$

To determine

To find the Slack time in each department.

Explanation of Solution

The slack time for each department is calculated by subtracting capacity of the department with the total usage.

The slack time in hours for every department is shown in the above table.