Optimal control of a production-inventory system with productreturns

We consider a production-inventory system that consists of n stages. Each stage has a finite production capacity modelled by an exponential server. The downstream stage faces a Poisson demand. Each stage receives returns of products according to independent Poisson processes that can be used to serve demand. The problem is to control production to minimize discounted (or average) holding and backordering costs. For the single-stage problem (n=1), we fully characterize the optimal policy. We show that the optimal policy is base-stock and we derive an explicit formula for the optimal base-stock level. For the general n-stage problem, we show that the optimal policy is characterized by state-dependent base-stock levels. In a numerical study, we investigate three heuristic policies: the base-stock policy, the Kanban policy and the fixed buffer policy. The fixed-buffer policy obtains poor results while the relative performances of base-stock and Kanban policies depend on bottlenecks. We also show that returns have a non-monotonic effect on average costs and strongly affect the performances of heuristics. Finally, we observe that having returns at the upstream stage is preferable in some situations.