On stochastic programming I. Static linear programming under risk

Abstract

The theory of mathematical programming deals with the problem of maximizing or minimizing a function subject to constraints on the variables involved. In applications, however, these variables are often not the only ones to be considered, for certain parameters of the constraints or of the function to be optimized may be assumed to vary randomly. Such a situation gives rise to the concept of stochastic programming-a concept closely allied to parametric programming; i.e., the investigation of the behavior of the optimal value of a program as certain parameters are changed. In practice the distribution of the random variables involved in a stochastic program is almost never known, but rather must be estimated from available data. However as a first approximation, or in abstract applications, it is often expedient to assume that the required distributions are completely known and given. Following the distinction made by economists, the first case may be said to be programming under uncertainty, while the second may be called programming under risk. It is the purpose of this paper to develop a way of looking at stochastic programming problems which is natural in statistical decision theory, to relate this approach to the previous research on linear programming under risk (in which it is implicit), and to make a detailed investigation of one type of stochastic linear programming problem within this framework. In much of the paper, all parameters of the linear program considered are allowed to be jointly distributed random variables with an arbitrary covariance structure. Furthermore, although computational details will not be considered, computational feasibility has been kept in mind. Even in the simplest cases of stochastic programming, the problem involved is not immediately clear, for one has a whole class of programs depending on which values of the random variables are realized. Thus temporal con