Linear programming is a fundamental planning tool. It is often difficult to precisely estimate or forecast certain critical data elements of the linear program. In such cases, it is necessary to address the impact of uncertainty during the planning process. We discuss a variety of LP-based models that can be used for planning under uncertainty. In all cases, we begin with ade- terministicLP model and show how it can be adapted to in- cludetheimpactof uncertainty. We present models that range from simple recourse policies to more general two-stage and multistageSLP formulations. We also include a discussion of probabilistic constraints. We illustrate the various models us- ingexamplestaken from the literature. The examples involve models developed for airline yield management, telecommunications, flood control, and production planning. O verthepast several decades, linear programming (LP) has becomea fundamental planning tool. It is routinely applied in engineering, business, economics, environmental studies, and other disciplines. This widespread acceptance may be due to (1) good algorithms, (2) practitioners’understanding of the power and scope of LP, and (3) widely available and reliable software. Furthermore, research on specialized problems, such as assignment, transportation, and network problems, has made LP methodology indis- pensableinmany industries, including airlines, energy, manufacturing, and telecommunications. Notwithstanding its successes, however, the assumption that all model parameters are known with cer- taintylimitsits usefulness in planning un- deruncertainty. When one or more of the data elements inalinearprogram is repre- sentedbyarandom variable, a stochastic linear program (SLP) results. In deterministic activity analysis, plan- ningconsistsof choosing activity levels that satisfy resource constraints while maximizing total profit (or minimizing to- talcost). All the information necessary for decision making is assumed to be available at the time of planning. Under uncertainty, not all the information is available, and some parameters should be modeled as random variables. We discuss here models that can include random variables within optimization problems…

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