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274 CHAPTER 6 SIMPLEX-BASED SENSITIVITY ANALYSIS AND DUALITY

we must see this as a negative value (Table 6.1). This indicates that if s 5 were to enter the solution, forcing
production of x 1 above the minimum of 25 litres, then the optimal profit contribution would suffer and fall
by R2.4. This implies that the current requirement to produce 25 litres of x 1 is adversely affecting profit
contribution – if we didn’t have this constraint the company would actually be able to increase total profit
contribution. Finally, let us look at s 3 relating to the available supply of Natural Flavourings. This has a shadow
price of 1.2 indicating that if we can increase supplies above the current limit of 80 kilos, profit contribution will
increase by R1.2 for each extra kilo obtained. Examination of the s 3 column tells us that this will be achieved
through increased production of Regular Grape, x 2 . The analysis is valid up to an extra 30 kilos of Natural
Flavouring. After that point we would need to re-formulate and re-solve the problem to assess the impact on
the current optimal solution.

Problems

1 Consider the following linear programming problem.
Max 5x 1 þ 6x 2 þ 4x 3
s:t:
3x 1 þ 4x 2 þ 2x 3 120
x 1 þ 2x 2 þ x 3 50
x 1 þ 2x 2 þ 3x 3 30
x 1 ; x 2 ; x 3 0
The optimal simplex tableau is:

x 1 x 2 x 3 s 1 s 2 s 3
Basis c B 5 6 4 0 0 0

0 0 4 0 2 7 1 80
s 3
4 0 2 1 1 3 0 30
x 3
5 1 0 0 1 2 0 20
x 1
5 8 4 1 2 0 220
z j
c j – z j 0 2 0 1 2 0

a. Calculate the range of optimality for c 1 .
b. Calculate the range of optimality for c 2 .
.
c. Calculate the range of optimality for c s 1
2 For the HighTech problem, we found the range of optimality for c 1 , the profit contribu-
tion per unit of the Deskpro. The final simplex tableau is given in Section 6.1. Find the
following:
a. The range of optimality for c 2 .
.
b. The range of optimality for c s 2
.
c. The range of optimality for c s 3
d. Suppose the per-unit profit contribution of the UltraPortable (c 2 ) dropped to E35. How
would the optimal solution change? What is the new value for total profit?

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