Integrality in Stochastic Inventory Models

Abstract

We study several dynamic, stochastic inventory control models with integer demands: the newsvendor model, its multi-period extension and a single-product, multi-echelon assembly model. Equivalent linear programs are formulated for the corresponding stochastic dynamic programs, and integrality results are derived based on the total unimodularity of the constraint matrices. Specifically, for all these models, starting with integer inventory levels, we show that there exist optimal policies that are integral. For the most general singleproduct, multi-echelon assembly system model, integrality results are also derived for a practical alternative to stochastic dynamic programming, namely rolling-horizon optimization by a similar argument. We also present a different approach to prove integrality results for stochastic inventory models. This new approach is based on a generalization we propose for the one dimensional notion of piecewise linearity with integer breakpoints to higher dimensions. The usefulness of this new approach is illustrated by showing an integrality result for the rolling-horizon optimization model of a two-product capacitated stochastic inventory control system.