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MORE THAN TWO DECISION VARIABLES 105

the resource. When the cost of a resource used is relevant, the dual price can be
interpreted as the amount by which the value of the resource exceeds its cost. So,
when the resource cost is relevant, the dual price can be interpreted as the maximum
Only relevant costs
should be included in the premium over the normal cost that the company should be willing to pay for one unit
objective function. of the resource.

NOTES AND COMMENTS

1 Computer software packages for solving linear optimality. However, when degeneracy is
programmes are readily available. Most of these present, changes beyond the end points of the
provide as a minimum the optimal solution, dual range do not necessarily mean a different
or shadow price information, the range of solution will be optimal. From a practical point of
optimality for the objective function coefficients view, changes beyond the end points of the
and the range of feasibility for the right-hand range of optimality necessitate re-solving the
sides. The labels used for the ranges of problem.
optimality and feasibility may vary, but the 4 The 100 per cent rule permits an analysis of
meaning is the same as what we have described multiple changes in the right-hand sides or
here. multiple changes in the objective function
2 Whenever one of the right-hand sides is at an end coefficients. But the 100 per cent rule cannot be
point of its range of feasibility, the dual and applied to changes in both objective function
shadow prices only provide one-sided coefficients and right-hand sides at the same
information. In this case, they only predict the time. In order to consider simultaneous changes
change in the optimal value of the objective for both right-hand side values and objective
function for changes toward the interior of the function coefficients, the problem must be
range. re-solved.
3 A condition called degeneracy can cause a 5 Managers are frequently called on to provide an
subtle difference in how we interpret changes in economic justification for new technology. Often
the objective function coefficients beyond the the new technology is developed, or purchased,
end points of the range of optimality. in order to conserve resources. The dual price
Degeneracy occurs when the dual price equals can be helpful in such cases because it can be
zero for one of the binding constraints. used to determine the savings attributable to the
Degeneracy does not affect the interpretation of new technology by showing the savings per unit
changes toward the interior of the range of of resource conserved.

3.4 More than Two Decision Variables

The graphical solution procedure is useful only for linear programmes involving two
decision variables. Computer software packages are designed to handle linear pro-
grammes involving large numbers of variables and constraints. In this section we
discuss the formulation and computer solution for two linear programmes with three
decision variables. In doing so, we will show how to interpret the reduced-cost
portion of the computer output and will also illustrate the interpretation of dual
prices for constraints that involve percentages.

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