Chapter 27 - Stochastic programming

Abstract

This chapter discusses the stochastic programming. A problem of stochastic linear programming arises when the coefficients of the linear functions, that is, the parameters of the linear programming model are random variables. The linear programming model for such a case is not relevant and it is necessary to formulate a new model to deal with such cases. The initial approach to reduce the effect of uncertainty in the problem was to replace the random variables by their expected values or by some good estimates of them and then to solve the resulting linear program. There are two different approaches to deterministic formulation to stochastic linear programs named as “wait and see” and the ‘‘here and now” model. In the “wait and see” model, the decision maker waits for the realization of the random variables and then solves the resulting linear program. In “here and now” model, a decision has to be taken at the very beginning before the realization of the random variables. In a linear programming problem, where some or all the parameters (A,b,c) are random variables with a known joint probability distribution, the problem is to determine an X ≥ 0 satisfies the constraints with a certain preassigned probability α and minimize the expected value of the objective function.