Non-crossing vs. independent lead times in a lost-sales inventory system with compound Poisson demand

In this paper we provide an analysis of an inventory system with compound Poisson demand and a sequential supply system, specifically focusing on non-crossing lead times. The purpose is to facilitate a comparison with order crossing caused by independent lead times. We apply a new approach for modeling exogenous, non-crossing Erlang distributed lead times. It is assumed that the inventory is controlled by continuous review and a base-stock level. Any part of a customer demand which cannot be satisfied immediately from inventory is lost. Set-up costs are negligible and the relevant cost parameters for choosing the best base stock consist of the holding cost rate and a shortage cost per unit lost. We provide a proof that the long-run average total relevant cost is a convex function of the base stock. Hence, the base stock is easy to compute and it is optimal in the case of geometrically distributed customer demand sizes. For demand sizes which are not geometric, we suggest an approximation to specify the base stock. An approximation is also suggested for the case when the lead time is only specified by its mean and standard deviation (SD). Our numerical study shows that the average cost is very sensitive to the SD. This in sharp contrast to the complete insensitivity of SD in case of geometric demand sizes and independent lead times.