Learning Objectives

Understand how deviational variables are used to formulate a linear programming model.

Understand the weighted and pre$-$emptive methods for solving goal programming problems.

Question $1$

A company manufactures two types of products. Let $x_1$ and $x_2,$ represent the number of units of each product that is to be produced. $10$ hours of labour are required to produce the first product and $15$ hours for the second. The first product generates $R50$ profit and the second generates R$70$ profit. The company has the following two goals:

the number of hours labour must not exceed $50$ hours

a profit target of R$1000$

Given the deviational variables and the constraints below.

$10x_1+15x_2+y_1^{-}-y_1^{+}=50$

$50x_1+70x_2+y_2^{-}-y_2^{+}=1000$

$($Labour Goal$)$

$($Profit Goal$)$

a$)$ If lahour goal is twice as important as the profit goal, using goal programming, formulate the objective function.

b$)$ If the optimum value of $z$ is not equal to zero, what can we say about the goals?