Optimal and approximate ( Q, r) inventory policies with lost sales and gamma-distributed lead time

Abstract

We consider the continuous review inventory control system with fixed reorder point r and constant order quantity Q. Demands are assumed to be generated by a Poisson process with one unit demanded at a time. Demands not covered immediately from inventory are lost. For the case of at most one order outstanding we derive and implement a model to obtain exact solutions for the reorder point and the order quantity. The model is formulated as a semi-Markov decision model and we show that if it is profitable to issue orders then a ( Q, r) policy is average-cost optimal. In general neither a ( Q, r) policy nor an ( policy is optimal if demand for more than one unit at a time is allowed in our model. A policy-iteration algorithm is developed for finding the optimal policy. We focus on the shape of the lead-time distribution by studying the optimal policy when the lead times are gamma distributed with different shape parameters. The results are compared to those obtained when applying approximate methods to the reorder-point inventory system.