Optimizing Inventory Placement for a Downstream Online Matching Problem

We study the inventory placement problem of splitting $Q$ units of a single item across warehouses, in advance of a downstream online matching problem that represents the dynamic fulfillment decisions of an e-commerce retailer. This is a challenging problem both in theory, because the downstream matching problem itself is computationally hard, and in practice, because the fulfillment team is constantly updating its algorithm and the placement team cannot directly evaluate how a placement decision would perform. We compare the performance of three placement procedures based on optimizing surrogate functions that have been studied and applied: Offline, Myopic, and Fluid placement. On the theory side, we show that optimizing inventory placement for the Offline surrogate leads to a $(1-(1-1/d)^d)/2$-approximation for the joint placement and fulfillment problem. We assume $d$ is an upper bound on how many warehouses can serve any demand location and that stochastic arrivals satisfy either temporal or spatial independence. The crux of our theoretical contribution is to use randomized rounding to derive a tight $(1-(1-1/d)^d)$-approximation for the integer programming problem of optimizing the Offline surrogate. We use statistical learning to show that rounding after optimizing a sample-average Offline surrogate, which is necessary due to the exponentially-sized support, does indeed have vanishing loss. On the experimental side, we extract real-world sequences of customer orders from publicly-available JD.com data and evaluate different combinations of placement and fulfillment procedures. Optimizing the Offline surrogate performs best overall, even compared to simulation procedures, corroborating our theory.