Taylor Approximation of Inventory Policies for One-Warehouse, Multi-Retailer Systems with Demand Feature Information

Abstract

Motivated by Fresh Hema, we consider a distribution system in which retailers replenish perishable goods from a warehouse, which, in turn, replenishes from an outside source. Demand at each retailer depends on exogenous features and a random shock, and unfulfilled demand is lost. The objective is to obtain a data-driven replenishment and allocation policy that minimizes the average inventory cost per time period. The extant data-driven methods either cannot guarantee a feasible solution for out-of-sample feature observations or generate one with excessive computational time. We propose a data-driven policy that resolves these issues in two steps. In the first step, we assume that the distributions of features and random shocks are known. We develop an effective heuristic policy by using Taylor expansion to approximate the retailer's inventory cost and solve a constrained optimization problem. The resulting solution is closed-form, referred to as Taylor Approximation (TA) policy. We show that the TA policy is asymptotically optimal in the number of retailers. In the second step, we apply the linear quantile regression, kernel density estimation, and sample average approximation to the TA solution to obtain the data-driven policy called Data-Driven Taylor Approximation (DDTA) Policy. We prove that the DDTA policy is consistent with the TA policy. A numerical study shows that the DDTA policy is as effective as the best extant data-driven solution with a much shorter computational time. Using a real data set provided by Fresh Hema, we show that the DDTA policy reduces the average cost by 10.9% compared to Hema's policy. Finally, we show that the main results still hold in the cases of correlated demand features, positive lead times, and censored demand.

Keywords: distribution system, Taylor approximation, data-driven, inventory allocation