Optimizing stock levels for service-differentiated demand classes with inventory rationing and demand lead times

In this paper, we study a service parts inventory management system for a single product at a parts distribution center serving two priority-demand classes: critical and non-critical. Distribution center keeps a common inventory pool to serve the two demand classes, and provides differentiated levels of service by means of inventory rationing. We assume a continuous review one-for-one ordering policy with backorders and Poisson demand arrivals. Only one demand class provides advance demand information whose orders are due after a deterministic demand lead time, whereas the orders of the other demand class need to be satisfied immediately. The problem has been studied before, but remained a challenging problem. The quality of the existing heuristic for estimating the critical class service levels can diminish significantly in some settings and the search routine for the service level optimization model relies on a brute force approach. Our contribution to the literature is twofold. For the given class of inventory replenishment and allocation policies, first we determine the form of the optimal solution to the service level optimization model, and then we derive an exact optimization routine to determine the optimal policy parameters provided the steady-state distribution is available. The computation of steady-state probabilities is needed only once. Second, we propose an alternative approach to estimate steady-state probabilities. By analyzing the limiting behavior of transition probabilities during infinitesimal time intervals, we are able to characterize the relationships between the steady-state probabilities, which satisfy nicely formed balance equations under the so-called Independence Assumption. In the numerical study section, we show that our approach provides superior performance in estimating service levels than the existing heuristic for all the examples considered. We also compare the performance of using the critical class service levels computed according to our method against the service levels computed by the existing heuristic, and show that our method can provide inventory savings up to 16.67%.