A one-warehouse/multi-retailer inventory system with lost sales, Erlangian lead times, and emergency shipments

Abstract

We consider a two-echelon inventory system with one central warehouse and a number of retailers. In case of a stock-out at the warehouse, the external supplier replenishes retailers' stocks by emergency shipments. The warehouse and retailers apply a base-stock ordering policy and unsatisfied demands at retailers will be lost. We assume that lead times of retailers are Erlangian distributed. Under the realistic assumption that the replenishment orders do not cross in time, the Erlangian lead times become stochastically dependent. This dependency will affect the optimal base-stock levels at retailers and, consequently, the optimal base-stock level at the warehouse. In this paper, we estimate the demand rate at the warehouse, and calculate the total cost function of the inventory system. We also present an algorithm to compute the optimal base-stock levels at the warehouse and retailers. Our numerical study illustrates that the sequential supply system cannot be ignored.