Optimal control of production-inventory systems with constant and compound poisson demand

In this paper, we study a production-inventory systems with finite production capacity and fixed setup costs. The demand process is modeled as a mixture of a compound Poisson process and a constant demand rate. For the backlog model we establish conditions on the holding and backlogging costs such that the average-cost optimal policy is of (s, S)-type. The method of proof is based on the reduction of the production-inventory problem to an appropriate optimal stopping problem and the analysis of the associated free-boundary problem. We show that our approach can also be applied to lost-sales models and that inventory models with unconstrained order capacity can be obtained as a limiting case of our model. This allows us to analyze a large class of single-item inventory models, including many of the classical cases, and compute in a numerical efficient way optimal policies for these models, whether these optimal policies are of (s, S)-type or not.