Optimal control of a production-inventory system with both backorders and lost sales

We consider the optimal control of a production-inventory system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. We must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. Using numerical results, we compare the performance of the optimal policy against several heuristics.