A Tutorial on Stochastic Programming

This tutorial is aimed at introducing some basic ideas of stochastic programming. The intended audience of the tutorial is optimization practitioners and researchers who wish to acquaint themselves with the fundamental issues that arise when modeling optimization problems as stochastic programs. The emphasis of the paper is on motivation and intuition rather than technical completeness (although we could not avoid giving some technical details). Since it is not intended to be a historical overview of the subject, relevant references are given in the “Notes” section at the end of the paper, rather than in the text. Stochastic programming is an approach for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include parameters which are unknown at the time a decision should be made. When the parameters are uncertain, but assumed to lie in some given set of possible values, one might seek a solution that is feasible for all possible parameter choices and optimizes a given objective function. Such an approach might make sense for example when designing a least-weight bridge with steel having a tensile strength that is known only to within some tolerance. Stochastic programming models are similar in style but try to take advantage of the fact that probability distributions governing the data are known or can be estimated. Often these models apply to settings in which decisions are made repeatedly in essentially the same circumstances, and the objective is to come up with a decision that will perform well on average. An example would be designing truck routes for daily milk delivery to customers with random demand. Here probability distributions (e.g., of demand) could be estimated from data that have been collected over time. The goal is to find some policy that is feasible for all (or almost all) the possible parameter realizations and optimizes the expectation of some function of the decisions and the random variables.