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. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state-dependent thresholds: a production base-stock level and two order-admission levels, one for each class. The production base-stock level determines when production takes place and how to allocate items that are produced. This base-stock level also determines when orders from the class with the lower shortage costs (class 2) are backordered and not fulfilled from inventory. The order-admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either non-increasing or non-decreasing) in the backorder level of class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.