We present a cutting plane algorithm for two-stage stochastic linear programs with recourse. Motivated by Benders' decomposition, our method uses randomly generated observations of random variables to construct statistical estimates of supports of the objective function. In general, the resulting piecewise linear approximations do not agree with the objective function in finite time. However, certain subsequences of the estimated supports are shown to accumulate at supports of the objective function, with probability one. From this, we establish the convergence of the algorithm under relatively mild assumptions.

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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 a deterministic LP model and show how it can be adapted to include the impact of uncertainty. We present models that range from simple recourse policies to more general two-stage and multistage SLP formulations. We also include a discussion of probabilistic constraints. We illustrate the various models using examples taken from the literature. The examples involve models developed for airline yield management, telecommunications, flood control, and production planning.

Stochastic programming (SP) was first introduced by George Dantzig in the 1950s. Since that time, tremendous progress has been made toward an understanding of properties of SP models and the design of algorithmic approaches for solving them. As a result, SP is gaining recognition as a viable approach for large-scale models of decisions under uncertainty. In this paper, we present an introduction to stochastic programming models and methodology at a level that is intended to be accessible to the breadth of members within the INFORMS community.

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We consider the problem of optimally meeting a stochastically growing demand for capacity over an infinite horizon. Under the assumption that demand for product follows either a nonlinear Brownian motion or a non-Markovian birth and death process, we show that this stochastic problem can be transformed into an equivalent deterministic problem. Consistent with earlier work by A. Manne, the equivalent problem is formed by replacing the stochastic demand by its deterministic trend and discounting all costs by a new interest rate that is smaller than the original, in approximate proportion to the uncertainty in the demand.

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