Sequential Bounding Methods for Two-Stage Stochastic Programs

Abstract

In rare situations, stochastic programs can be solved analytically. Otherwise, approximation is necessary to solve stochastic programs with a large or infinite number of scenarios to a desired level of accuracy. This involves statistical sampling or deterministic selection of a finite set of scenarios to obtain a tractable deterministic equivalent problem. Some of these approaches rely on bounds for primal and dual decision variables of the second stage. We develop new algorithms to improve these bounds and reduce the deterministic approximation error. Experiments were conducted to compare a sequential approximation approach with and without these new algorithms. Each algorithm is applied to a set of test instances for a problem of managing semiconductor inventory with downward substitutions, where random variables only appear in the right-hand side of the second stage. Experiments were also conducted using a sample average approximation (SAA) algorithm. The sequential approximation and SAA algorithm generate a feasible solution upon termination. We directly compare the quality of these solutions using a paired student t-test.