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220 CHAPTER 5 LINEAR PROGRAMMING: THE SIMPLEX METHOD

profitability). So, to calculate the change in the objective function, if we increase x 1
by 1 we would have:
z 1 ¼ 0ð3Þþ 0ð0Þþ 0ð8Þ¼ 0
Here we multiply the objective function coefficient of the relevant basic variable by
the change in the value of that variable. This increases x 1 by 1. Clearly the net effect
on the objective function is zero – reducing the value of s 1 , s 2 and s 3 will have a zero
effect on the objective function value. The comparable calculations for the other z j
values are then:
z 2 ¼ 0ð5Þþ 0ð1Þþ 0ð5Þ¼ 0
z 3 ¼ 0ð1Þþ 0ð0Þþ 0ð0Þ¼ 0
z 4 ¼ 0ð0Þþ 0ð1Þþ 0ð0Þ¼ 0
z 5 ¼ 0ð0Þþ 0ð0Þþ 0ð1Þ¼ 0

This then gives a tableau:

Value
x 1 x 2 s 1 s 2 s 3
Basis 50 40 0 0 0 0
3 5 1 0 0 150
s 1
0 1 0 1 0 20
s 2
8 5 0 0 1 300
s 3
0 0 0 0 0 0
z j
c j –z j

In this tableau we also see a boldfaced 0 in the z j row in the last column. This
zero is the value of the objective function for the current basic feasible solution.
It is calculated in the same way as the other z j values by multiplying the objective
function coefficients of the current basic variables by the values in the last
column.

Value
x 1 x 2 s 1 s 2 s 3
Basis C B 50 40 0 0 0 0

s 1 0 3 5 1 0 0 150
s 2 0 0 1 0 1 0 20
s 3 0 8 5 0 0 1 300
0 0 0 0 0 0
z j
c j –z j

The net evaluation row, c j – z j is then simply the difference between the objective
function coefficients shown in the c row and the z values we have just calculated.
Adding these to the tableau we then have:

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