Multistage Stochastic Planning Models Using Piecewise Linear Response Functions

Abstract

Abstract : Random outcomes can often produce significant effects on planning decisions that consider several time periods. Multistage stochastic programs can model these decisions but implementations are generally restricted to a limited number of scenarios in each period. We present an alternative approximation scheme that can obtain lower and upper bounds on the optimal objective value in these stochastic programs. The method is based on building response functions to future outcomes that depend separably on the variation of random parameters around the limited set of scenarios that is initially provided. For stochastic linear programs, the resulting optimization problem involves an objective with a limited number of nonlinear terms subject to linear constraints. The method can be incorporated into various alternative procedures for solving multistage stochastic linear programs with finite numbers of scenarios. Section 2 discusses the basic model and alternative approaches. Section 3 then discusses the basic properties of piecewise linear response functions. The fourth section presents a basic model for a single scenario and randomness restricted to constraint levels. The fifth section extends this to multiple scenarios with varying scenario ranges and to possibilities for randomness among the constraint vectors.