Everything Maths Grade 11

15.5 CHAPTER 15. LINEARPROGRAMMING

Activity: Numerical Method

Use the objective function

(650× q) + (1 500× p)

to calculate Mrs Nkosi’sprofit for the following feasible solutions:

q p Profit

60 30

65 30

70 30

(^66233313)
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 drawall constraints as graphsand
mark the area that satisfies all constraints. This isshown in Figure 15.1 for Mrs Nkosi’s farm.

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 10 0

p

q

A

B

C

Figure 15.1: Graph of the feasible region

Vertices (singular: vertex) are the points on the graph where two or moreof the constraints overlap or
cross. If the linear objective function has a minimum or maximum value, it will occur at one ormore
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 Figure 15.1, vertex A is at the intersection of p = 30 and q = 2p. Therefore, the coordinates
of A are (30;60). Similarly vertex B is at the intersection of p = 30 and q = 100− p. Therefore the
coordinates of B are (30;70). Vertex C is at the intersection of q = 100− p and q = 2p, which gives
( 3313 ;66^23 ) for the coordinates of C.

If we now substitute these points into the objective function, we get thefollowing: