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86 CHAPTER 3 LINEAR PROGRAMMING: SENSITIVITY ANALYSIS AND INTERPRETATION OF SOLUTION

In Chapter 2 we saw how to formulate and then solve an LP problem. This provides
management with the optimal solution to their decision problem. However, it is highly
unlikely that this will be enough by itself for management, given that the business
environment for most organizations is both constantly changing and increasingly
uncertain and unpredictable. Management are likely to say ‘OK, so that’s the optimal
solution now. But suppose things change? Suppose our costs change? Suppose demand
for the products changes? Suppose our workforce are willing to work overtime? What
should we do then to ensure an optimal solution?’. Such questions are often referred to
as what-if questions. Clearly, if some part of an LP decision problem changes then we
could re-formulate the problem taking into account such changes and then re-solve the
problem. However, this is time-consuming and costly, especially in the real world where
LP problems are large. Fortunately we don’t always need to do this. We can make use
of sensitivity analysis. Sensitivity analysis is a study of how changes in the numerical
coefficients of a linear programme affect the current, optimal solution. We conduct this
analysis from the information we already have about the optimal solution without
having to re-formulate and re-solve the problem. This is a particularly powerful feature
of LP and one which makes it attractive to many decision makers.
Because sensitivity analysis is concerned with how these changes affect the
optimal solution, the analysis does not begin until the optimal solution to the
original linear programming problem has been obtained. For that reason, sensitivity
analysis is often referred to as postoptimality analysis.
Our approach to sensitivity analysis parallels the approach used to introduce linear
programming in Chapter 2. We begin by showing how a graphical method can be used to
perform sensitivity analysis for linear programming problems with two decision variables.
Then, we show how computer packages, like Excel provide sensitivity analysis information.
Finally, we extend the discussion of problem formulation started in Chapter 2 by
formulating and solving three larger linear programming problems. In discussing the
solution for each of these problems, we focus on managerial interpretation of the
optimal solution and sensitivity analysis information.

3.1 Introduction to Sensitivity Analysis

Sensitivity analysis is important to decision makers because real-world problems
exist in a changing environment. Prices of raw materials change, product demand
changes, companies purchase new machinery, stock prices fluctuate, employee turn-
over occurs and so on. If a linear programming model has been used in such an
environment, we can expect some of the coefficients to change over time. We will
then want to determine how these changes affect the optimal solution to the original
linear programming problem. Sensitivity analysis provides us with the information
needed to respond to such changes without requiring the complete solution of a
revised linear programme.
Remember the GulfGolf problem in Chapter 2:

Max 10S þ 9D
s:t
0:7S þ 1D 630 Cutting and dyeing
0:5S þ 0:8333D 600 Sewing
1S þ 0:6667D 708 Finishing
0:1S þ 0:25D 135 Inspection and packaging
S; D 0

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