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.

翻译：我们研究了在仓库间分配单个商品的$Q$单位库存的放置问题，该问题前置了一个代表电商零售商动态履约决策的下游在线匹配问题。这是一个具有挑战性的问题，理论上由于下游匹配问题本身计算复杂，实践中则因履约团队不断更新算法而放置团队无法直接评估放置决策的效果。我们比较了三种基于优化已研究和应用的替代函数（离线、短视和流体放置）的放置程序性能。在理论方面，我们表明优化离线替代函数的库存放置，对于联合放置与履约问题可实现$(1-(1-1/d)^d)/2$逼近。我们假设$d$是可服务任何需求位置的仓库数量上限，且随机到达满足时间或空间独立性。我们理论贡献的关键在于使用随机舍入推导出整数规划问题（优化离线替代函数）的紧$(1-(1-1/d)^d)$逼近。我们利用统计学习证明，优化样本平均离线替代函数（由于指数级支持此类优化必不可少）后舍入确实具有逐渐消失的损失。在实验方面，我们从公开可用的京东数据中提取真实客户订单序列，评估放置与履约程序的不同组合。优化离线替代函数在整体上表现最佳，甚至优于仿真程序，这与我们的理论相符。