优化下游在线匹配问题的库存分配
Optimizing Inventory Placement for a Downstream Online Matching Problem
AI总结：
研究电商履约前的库存分配问题，提出基于随机舍入优化Offline替代函数的方法，理论证明其能取得紧致近似比，实验验证该策略优于计算密集型模拟程序。
摘要：
我们研究库存分配问题，即在代表电子商务零售商动态履约决策的下游在线匹配问题之前，将$Q$单位的单一物品分配到各个仓库。由于下游匹配问题的计算复杂性，这在理论上是一个具有挑战性的问题；在实践中同样具有挑战性，因为履约团队不断更新其算法，而分配团队缺乏对分配决策的直接评估。我们比较了三种分配程序的性能，这些程序基于已被研究和应用的替代函数进行优化：Offline、Myopic和Fluid分配。在理论方面，我们证明在允许$\alpha$竞争履约策略的任何需求模型下，优化Offline替代函数可为联合分配与履约问题带来$\alpha(1-(1-1/d)^d)$的近似比。我们假设$d$是任何需求位置可由多少仓库服务的上界。我们理论贡献的核心是使用随机舍入法，为优化Offline替代函数的整数规划问题推导出紧致的$(1-(1-1/d)^d)$近似比。我们进一步展示了如何将此结果扩展到多SKU设置，改进了已知的最佳近似比$1/2$。我们使用统计学习表明，在优化样本平均Offline替代函数后进行舍入（由于指数级大小的支持集，这是必要的）确实具有趋于零的损失。在实验方面，我们评估了不同的分配与履约程序组合在多种合成实例上的表现。当与优秀的履约程序结合时，优化Offline替代函数表现最佳，甚至优于计算密集型的模拟程序，这证实了我们的理论。
英文摘要：
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 theoretically, due to the computational complexity of the downstream matching problem, and practically, as the fulfillment team continuously updates its algorithm while the placement team lacks direct evaluation of placement decisions. 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 an $α(1-(1-1/d)^d)$-approximation for the joint placement and fulfillment problem under any demand model that admits an $α$-competitive fulfillment policy. We assume $d$ is an upper bound on how many warehouses can serve any demand location. 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 further show how to extend this result to a multi-SKU setting, improving upon the best known approximation of $1/2$. We use statistical learning to show that rounding after optimizing a sample-average Offline surrogate, which is necessary due to the exponentially-sized support, indeed has vanishing loss. On the experimental side, we evaluate how different combinations of placement and fulfillment procedures perform on a wide array of synthetic instances. When coupled with a good fulfillment procedure, optimizing the Offline surrogate performs best even compared to computationally-intensive simulation procedures, corroborating our theory.
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