Base-stock policies for the lost sales inventory system with Poisson demand and Erlangian lead times

Abstract

The base-stock policies for the studied inventory system can easily be evaluated through Erlang's loss formula when the lead times are mutually independent. This is often the case only if the base-stock S is one. If S is larger than one, the Erlangian lead times become stochastically dependent under the realistic assumption that the replenishment orders do not cross in time. We make this assumption and show for any positive S that the number of replenishment orders outstanding has an equilibrium distribution which is a slightly modified truncated version of a negative binomial distribution. It turns out to be easy to compute the stock-out frequency recursively for S=1,2,…. For each S, the average stock can be specified in terms of this frequency. We prove that the frequency is convex in S. It is therefore straightforward to compute the base-stock for which the average cost is minimized and to compute the minimum average cost. Our numerical study illustrates that the minimum average cost is very sensitive to the shape parameter describing the Erlangian lead times, which is in sharp contrast to the complete insensitivity when lead times are independent.