Single-Stage Bounds for Optimal Policies in Serial Inventory Systems with Non-stationary Demand

Clark and Scarf (1960) consider a two-stage inventory system in a finite horizon. They show that echelon base-stock policies are optimal. The optimal base-stock levels can be obtained by solving two sets of recursive, functional equations. Nevertheless, finding the upstream base-stock level is more complex because the functional equation depends on the downstream base-stock level. This paper provides an approach that can generate an effective upstream base-stock level without considering the decision made by downstream stages. More specifically, we show that the optimal upstream base-stock level is bounded above and below by the optimal base-stock level obtained from a single-stage system. The construction of these single-stage solution bounds is based on three results: (1) The optimal value function of the original system is bounded above and below by that of a revised two-stage system; (2) the optimal base-stock levels of the revised systems, in turn, bound the optimal upstream solution; (3) the revised two-stage systems are essentially equivalent to a single-stage system. An effective heuristic for the upstream base-stock level is proposed by employing a weighted average of the single-stage solution bounds. Our results can be extended to general stage systems with Markov modulated demand.