A Stochastic Programming Based Inventory Policy for Assemble- To-Order Systems with Application to the W Model

We consider an Assemble-To-Order inventory system, formulating an associated stochastic program (SP). We show that the solution of a relaxation of this SP provides a lower bound on total inventory cost for all feasible policies. We then focus our attention on the stylized W model, which involves three components used to produce two products. (There are two unique parts and a common part. Each product uses the common part and its own unique part.) For the W model, the solution of the SP provides base-stock levels and motivates a priority based allocation policy. We also provide a condition under which the original SP and its relaxation have the same solution and develop efficient solution procedures for the SP under both continuous and discrete demand distributions. We show that our SP based policy achieves the lower bound, and is thus optimal, in two situations: when a certain symmetry condition in the cost parameters holds and when the SP solution indicates that it is optimal to stock enough of the common part so that the two products effectively do not share it. For other cases, numerical results demonstrate that our policy works well, and outperforms alternative approaches in many circumstances.