0.10 Linear programming  (Page 3/4)

Writing the objective function

If the objective function is not given to you as an equation, you will need to be able to convert a word description to an equation to get the objective function.

You will need to look for words like:

• most profit
• least cost
• largest area

Writing the objective function

1. The EduFurn furniture factory manufactures school chairs and school desks. They make a profit of R50 on each chair sold and of R60 on each desk sold. Write an equation that will show how much profit they will make by selling the chairs and desks.
2. A manufacturer makes small screen GPS's and wide screen GPS's. If the profit on small screen GPS's is R500 and the profit on wide screen GPS's is R250, write an equation that will show the possible maximum profit.

Solving the problem

The numerical method involves using the points along the boundary of the feasible region, and determining which point optimises the objective function.

Investigation : numerical method

Use the objective function

to calculate Mrs Nkosi's profit for the following feasible solutions:

The question is How do you find the feasible region? We will use the graphical method of solving a system of linear equations to determine the feasible region. We draw all constraints as graphs and mark the area that satisfies all constraints. This is shown in [link] for Mrs Nkosi's farm.

Vertices (singular: vertex) are the points on the graph where two or more of the constraints overlap or cross. If the linear objective function has a minimum or maximum value, it will occur at one or more of the vertices of the feasible region.

Now we can use the methods we learnt previously to find the points at the vertices of the feasible region. In [link] , vertex A is at the intersection of $p=30$ and $m=2p$ . Therefore, the coordinates of A are (30,60). Similarly vertex B is at the intersection of $p=30$ and $m=100-p$ . Therefore the coordinates of B are (30,70). Vertex C is at the intersection of $m=100-p$ and $m=2p$ , which gives ( $33\frac{1}{3}$ , $66\frac{2}{3}$ ) for the coordinates of C.