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248 CHAPTER 5 LINEAR PROGRAMMING: THE SIMPLEX METHOD

We now have a solution for FJC where we are producing 25 litres of Sweet Grape, and 62.5 litres of Dry Grape
to generate a profit contribution of 150R. There are 125 kilos of Grape A unused, 30 kilos of the natural
flavourings and 112.5 hours of labour. We are using all the available supply of Grape B. Are further improve-
ments possible? We have one positive value in the c j –z j row for x 2 so this is set to enter the basis and we
calculate that s 3 will leave. Transforming the tableau we have:

Value
x 1 x 2 x 3 s 1 s 2 s 3 s 4 s 5
Basis c B 1 1.2 2 0 0 0 0 0

s 1 0 0 0 0 1 0 0 0 3 65
x 3 2 0 0 1 0 0.5 0 0 0.5 62.5
x 2 12 0 1 0 0 0 1 0 2 30
s 4 0 0 0 0 0 0.5 3 1 4.5 22.5
x 1 1 1 0 0 0 0 0 0 1 25
1 1.2 2 0 1 1.2 0 2.4 186
z j
0 0 0 0 1 1.2 0 2.4
c j – z j
This time we have the optimal solution given; there are no positive values in the c j –z j column. To generate a profit
contribution of 186R from tomorrow’s production FJC should produce 25 litres of Sweet Grape, 30 litres of Regular
Grape and 62.5 litres of Dry Grape. It will still have 65 kilos of Grape A unused and 22.5 hours of labour.

Problems

1 Consider the following system of linear equations:
3x 1 þ x 2 ¼ 6
2x 1 þ 4x 2 þ x 3 ¼ 12
a. Find the basic solution with x 1 ¼ 0.
b. Find the basic solution with x 2 ¼ 0.
c. Find the basic solution with x 3 ¼ 0.
d. Which of the preceding solutions would be basic feasible solutions for a linear
programme?
2 Consider the following linear programme:

Max x 1 þ 2x 2
s:t:
x 1 þ 5x 2 10
2x 1 þ 6x 2 16
x 1 ; x 2 0

a. Write the problem in standard form.
b. How many variables will be set equal to zero in a basic solution for this
problem?
c. Find all the basic solutions, and indicate which are also feasible.
d. Find the optimal solution by computing the value of each basic feasible
solution.

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