Inventory rationing in a make-to-stock system with batch production and lost sales

We address an inventory rationing problem in a lost sales make-to-stock (MTS) production system with batch ordering and multiple demand classes. Each production order contains a single batch of a fixed lot size and the processing time of each batch is random. Assuming that there is at most one order outstanding at any point in time, we first address the case with the general production time distribution. We show that the optimal order policy is characterized by a reorder point and the optimal rationing policy is characterized by time-dependent rationing levels. We then approximate the production time distribution with a phase-type distribution and show that the optimal policy can be characterized by a reorder point and state-dependent rationing levels. Using the Erlang production time distribution, we generalize the model to a tandem MTS system in which there may be multiple outstanding orders. We introduce a state-transformation approach to perform the structural analysis and show that both the reorder point and rationing levels are state dependent. We show the monotonicity of the optimal reorder point and rationing levels for the outstanding orders, and generate new theoretical and managerial insights from the research findings.