Inventory rationing on a one-for-one inventory model for two priority customer classes with backorders and lost sales

In this study, we are primarily motivated by the research problem of recognizing heterogeneous customer behavior towards waiting for order fulfillment under the threshold rationing policy (also known as the critical level policy), and aim to find its effect on system stock levels and performance measures. We assume a continuous review one-for-one ordering policy with generally distributed lead times. In the first model, we consider the case in which the low-priority customer class exhibits zero patience for waiting if the demand is not satisfied immediately (a lost sale), whereas the demand of the high-priority customer class can be backordered. This is the first study in the literature to consider this model. We provide an exact analysis for the derivation of the steady-state probability distribution and the average infinite horizon cost per unit time. We then develop an efficient optimization procedure to minimize the average expected cost rate. We also determine the forms of the optimal solutions for the two service level optimization models that are common in practice. In the second model, we study the opposite case in which the high-priority customer class exhibits zero patience for waiting. We establish a theoretical basis for the rationale of using the Continuous-Time Markov Chain (CTMC) approach as an approximation. We show that under certain assumptions, the steady-state probabilities of the system with generally distributed lead times are identical to the steady-state probabilities of the CTMC system with the same mean. This result enables us to link the dynamics of the studied model to the CTMC model, which may open new doors for future research.