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MORE THAN TWO DECISION VARIABLES 109

D 0:3S
or
0:3S þ D 0

Adding this new constraint to the modified linear programme and re-solving the
problem, we obtain the optimal solution shown in Figure 3.11.
Let us consider the interpretation of the dual price for constraint 5, the require-
ment that the number of deluxe bags produced must be at least 30 per cent of the
number of standard bags produced. The dual price of 1.38 indicates that a one-unit
increase in the right-hand side of the constraint will lower profits by $1.38. So, what
the dual price of 1.38 is really telling us is what will happen to the value of the
optimal solution if the constraint is changed to:
D 0:3S þ 1

The correct interpretation of the dual price of 1.38 can now be stated as follows: If we
are forced to produce one deluxe bag over and above the minimum 30 per cent require-
ment, total profits will decrease by $1.38. Conversely, if we relax the minimum 30 per cent
requirement by one bag (D 0.3S 1), total profits will increase by $1.38. So, we are
able to tell management that their insistence on having the number of deluxe bags at least
30 per cent of the number of standard bags is costing the company money – profit is
adversely affected.
The dual price for similar percentage (or ratio) constraints will not directly
provide answers to questions concerning a percentage increase or decrease in the
right-hand side of the constraint. For example, we might wonder what would happen
to the value of the optimal solution if the number of deluxe bags has to be at least
31 per cent of the number of standard bags. To answer such a question, we would re-
solve the problem using the constraint 0.31S + D 0.
Because percentage (or ratio) constraints frequently occur in linear programming
models, let us consider another example. For instance, suppose that management states
that the number of lightweight bags produced may not exceed 20 per cent of the total golf
bag production. If the total production of golf bags is S + D + L,wecanwritethis
constraint as:
L 0:2ðS þ D þ LÞ

L 0:2S þ 0:2D þ 0:2L
0:2S 0:2D þ 0:8L 0
The solution obtained for the model that incorporates both the effects of this new
percentage requirement and the previous requirement ( 0.3S + D 0) is shown in
Figure 3.12. After rounding, the dual price corresponding to the new constraint
(constraint 6) is 0.89. So, every additional lightweight bag we are allowed to produce
over the current 20 per cent limit will increase the value of the objective function by
$0.89; moreover, the right-hand side range for this constraint shows that this
interpretation is valid for increases of up to 156 units.

The Kenya Cattle Company Problem
To provide additional practise in formulating and interpreting the computer solution
for linear programmes involving more than two decision variables, we consider a
minimization problem involving three decision variables. Kenya Cattle Company
(KCC), located in Kenya, East Africa, has been experimenting with a special diet for
its cattle. The feed components available for the diet are a standard feed product, a
vitamin-enriched product and a new vitamin and mineral feed additive. The

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