LINEAR PROGRAMMING UNDER UNCERTAINTY

One of the key assumptions of linear programming described in Sec. 3.3 is the certainty assumption, which says that the value assigned to each parameter of a linear programming model is assumed to be a known constant. This is a convenient assumption, but it seldom is satisfied precisely. These models typically are formulated to select some future course of action, so the parameter values need to be based on a prediction of future conditions. This sometimes results in having a significant amount of uncertainty about what the parameter values actually will turn to be when the optimal solution from the model is implemented. We now turn our attention to introducing some techniques for dealing with this uncertainty.

The most important of these techniques is sensitivity analysis. As previously mentioned in Secs. 2.3, 3.3, and 4.7, sensitivity analysis is an important part of most linear programming studies. One purpose is to determine the effect on the optimal solution from the model if some of the estimates of the parameter values turn out to be wrong. This analysis often will identify some parameters that need to be estimated more carefully be- fore applying the model. It may also identify a new solution that performs better for most plausible values of the parameters. Furthermore, certain parameter values (such as resource amounts) may represent managerial decisions, in which case the choice of the parameter values may be the main issue to be studied, which can be done through sensitivity analysis.

The basic procedure for sensitivity analysis (which is based on the fundamental in- sight of Sec. 5.3) is summarized in Sec. 7.1 and illustrated in Sec. 7.2. Section 7.3 focuses on how to use spreadsheets to perform sensitivity analysis in a straightforward way. (If you don’t have much time to devote to this chapter, it is feasible to read only Sec. 7.3 to obtain a relatively brief introduction to sensitivity analysis.)

The remainder of the chapter introduces some other important techniques for dealing with linear programming under uncertainty. For problems where there is no latitude at all for violating the constraints even a little bit, the robust optimization approach described in Sec. 7.4 provides a way of obtaining a solution that is virtually guaranteed to be feasible and nearly optimal regardless of reasonable deviations of the parameter values from their estimated values. When there is latitude for violating some constraints a little bit without very serious complications, chance constraints introduced in Sec. 7.5 can be used. A chance constraint modifies an original constraint by only requiring that there be some very high probability that the original constraint will be satin two (or more) stages, so the decisions in stage 2 can help compensate for any stage 1 decisions that do not turn out as well as hoped because of errors in estimating some parameter values. Section 7.6 describes stochastic programming with recourse for dealing with such problems.