Question

1. A tailor in Huntsville, Alabama makes wool sports coats and slacks. He is able to get a shipment of 300 square yards of wool cloth from Atlanta, Georgia each week to make coats and slacks, and he has 400 labor hours to make them each week. A coat requires 4 square yards of wool and 5 hours to make, and a pair of slacks requires 3 square yards of wool and 5 hours to make. The tailor earns $30 in profit from each coat he makes and $25 from each pair of slacks. He wants to know how many coats and pairs of slacks to make each month to maximize profit. (a) Formulate an integer linear programming model for this problem. That is, you will define the decision variables, state the objective function, and thirdly, state the constraints.

(b) Use the computer software (QM for Windows or its equivalent) to determine the integer optimal solution to this problem by determining;

(i) The optimal number of coats.

(ii) The optimal number of pairs of slacks.

(iii) The optimal profit.

Answer 1

(a) The decisions that needs to be made is how many coats and slacks to be produced. The decision variables can be denoted by X and Y where X is the number of coats and Y is the number of slacks.

The objective is to maximize profit with the resources he has. Since coats give a profit of $30 and slacks a profit of $25, the objective function can be written as

Maximize Profit (P) = 30X + 25Y

The constraint is the limitation of resources. He gets 300 sqyd of wool and 400 labor hours. This means that for X coats and Y slacks,

4X + 3Y <= 300

5X + 5Y <= 400

Also the values of X and Y cannot be negative. Hence,

X, Y >= 0

This completes the formulation of the problem.

(b) Next we shall use solver on Excel to find the optimal solution. The setup and the results are shown in the next two pictures.

(i) Optimal number of coats is 60

(ii) Optimal number of slacks is 20

(iii) The optimal profit is $2300