Stochastic Programming
- Stochastic Programming

When we recognize the possibility of uncertainty or risk within the context of mathematical programming, the decision problem might be written as below. The model is shown in the typical matrix format for an LP except tildes appear over all the parameter matrices. This implies that all may be affected by a vector of random variables. In practical instances the model could be complicated by nonlinear terms and integrality restrictions. In any event, the model as stated, is not well defined because: (1) the timings of the decisions and observations of the randomness are ambiguous and (2) the meanings of optimal solution and feasible solution are unclear.

Stochastic programming addresses the first issue by explicitly defining the sequence of decisions in relation to the realization of the random variables. Given the sequence, an objective function is defined that reflects a rational criterion for evaluating the decisions at the time they must be made. Feasibility conditions must be adapted to the fact that decisions made before the realization of randomness may have feasibility consequences after the realization. How the issues are resolved leads to the several different problems considered in this section. No single problem formulation is sufficient.

For this section we assume that the probability distribution of is known. More properly we should say that stochastic programming is decision making under risk, reserving the phrase decision making under uncertainty for those situations for which probability distributions are unavailable. We will, however, use the more popular term, uncertainty, to refer to situations in which distributions are known.

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by Paul A. Jensen