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DUALITY 273

WORKED EXAMPLE

e shall return to the Worked Example that we introduced in Chapter 5. Recollect that the Fresh Juice
W Company had identified the optimal solution in terms of quantities of its three grape juice products to
produce tomorrow. We had a formulation such that:

Max 1x 1 þ 1:2x 2 þ 2x 3
s:t:
1x 1 þ 2x 2 150
1x 1 þ 2x 3 150
2x 1 þ 1x 2 80
2x 1 þ 3x 2 þ 1x 2 225
x 1 25
x 1 ; x 2 ; x 3 0
The final simplex tableau was:

Basis x 1 x 2 x 3 s 1 s 2 s 3 s 4 s 5
C 1 1.2 2 0 0 0 0 0
Unused kilos of 0 0 0 1 0 2 0 3 65
Grape A
Litres of Dry Grape 0 1 1 0 0.5 0 0 0.5 62.5
juice produced
Litres of Regular 0 0 0 0 0 1 0 2 30
Grape juice
produced
Unused labour 0 0 0 0 0.5 3 1 4.5 22.5
hours
Litres of Sweet 1 0 0 0 0 0 0 1 25
Grape juice
produced
z j 1 1.2 2 0 1 1.2 0 2.4 186
c j z j 0 0 0 0 1 1.2 0 2.4
Let us see what other advice we can provide the company’s management team. If we carry out sensitivity
analysis on the objective function coefficients we obtain the following results:

Current value Maximum increase Maximum decrease

x 1 1 2.4 1
x 2 1.2 1 1.2
x 3 2 1 2
If we look at x 1 we see that there is an infinite allowable decrease. In other words, no matter how low the profit
contribution of x 1 it will remain in the solution. In the context of the problem this is not surprising since we have
a constraint forcing the company to produce 25 litres of x 1 no matter what. On the other hand, if the profit
contribution of x 1 rises by more than 2.4, the current optimal solution will change. For x 2 and x 3 we see that
there is an infinite increase allowed and a decrease in both cases that would take the profit contribution to
zero. In other words, as long as the profit contribution for these two products is non-zero they will continue to
be produced. If we now turn to sensitivity analysis on the constraints we see that the largest positive z j value is
associated with s 5 at 2.4. Recollect that s 5 is a surplus variable associated with a constraint of the type so

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