Answer to Question #289823 in Operations Research for Panya

Question

ABC company manufactures and sells two products P1 and P2. Each unit of P1 requires two hours of machining and one hour of skilled labour. Each unit of P2 requires one hour of machining and two hours of labour. The machine capacity is limited to 650 man hours. Only 300 unit's of product P1 can be sold in the market. The per unit contribution from product P1 is Ksh. 80 and product P2 is Ksh. 120 . Formulate a linear programming model

Answer 1

Let the number of products $P_1$ and $P_2$ produced be $x_1$ and $x_2$ respectively.

The objective function $z$ which is the contribution from each product and which is to be maximized is given as:

$$z=80x_1+120x_2$$

We have two constraints, which are; time on machining and skilled labour.

For machining, the limit is 600 hours. So, we have:

$$2x_1+x_2\leq 600$$

For skilled labour, the limit is 650 hours. So, we have:

$$x_1+2x_2 \leq 650$$

There is a constraint on the number of product $P_1$ that can be sold. So, we have:

$$x_1 \leq 300$$

Bringing everything together, we have:

$$\text{Maximize:}~~~~~~~~~~~~~~~~~~~~~~z=80x_1+120x_2\\ \text{Subject to~: }~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2x_1+x_2\leq 600\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x_1+2x_2 \leq 650\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x_1~~~~~~~~~~\leq 300~\\ x_1,x_2 \geq0$$