Optimizing Inventory Levels in a Two-Echelon Retailer System with Partial Lost Sales

This paper considers a retailer inventory system with N first-echelon stores and a single second-echelon distribution center (DC). Customer demands at the stores are assumed to be random. When a stockout occurs, customers are willing to wait for their order to be filled with a known probability. Customers who are unwilling to wait result in lost sales. The first and second echelons are both restocked at fixed, equally spaced time points, where the store restocking frequency is an integer multiple of the DC restocking frequency. We also assume that replenishment quantities at both echelons can be adjusted up to the time of delivery, resulting in replenishment lead times equal to zero. This simplification allows us to determine optimal solutions for the partial lost sales case, which has proven intractable for two-echelon formulations with lead times. Computational results are given for illustrative examples.