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. Because of the demand features and the allocation constraint, solving the problem with the extent datadriven methods either possibly violates the constraint or is subject to the curse of dimensionality. We construct 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. The resulting solution yields a closed-form one, 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 empirical risk minimization and conditional kernel density estimation on the TA solution to obtain the data-driven policy, referred to as Data-Driven Taylor Approximation (DTA). We prove that the DTA policy is consistent with the TA policy. A numerical study shows that our DTA policy outperforms the other data-driven policies in the literature. Finally, using a real data set provided by Fresh Hema, we show that the DTA policy reduces the average cost by 11%, compared to the current policy.

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