满足多商品电商订单的最优相关取整
Order-optimal Correlated Rounding for Fulfilling Multi-item E-commerce Orders
AI总结：
针对电商多商品订单履约中的相关取整问题，本文提出基于膨胀时间尺度的快速新取整方案，将次优因子从q/4降至1+ln(q)（稀疏网络降至d），并证明其紧致性且在实验中稳健提升性能。
摘要：
我们研究电商中的动态履约问题，其中到来的（多商品）客户订单必须立即分派给拥有所需库存的（一个或多个组合）履约中心。解决该问题的一种主流方法由 Jasin 和 Sinha（2015）开创，即编写一个“确定性”线性规划，规定来自特定地区的多商品订单中的每件商品应以何种频率被分派至每个履约中心（FC）。然而，在满足这些频率约束的同时，又不将订单拆分至过多 FC，是极具挑战性的。Jasin 和 Sinha 将此识别为相关取整问题，并提出了一种复杂的取整方案，证明其在 q 件商品订单上最多差约 q/4 的因子。据我们所知，本文针对该相关取整问题提供了首个实质性改进的方案，其最多差 1+ln(q) 的因子。我们还为稀疏网络提供了另一种方案，若每件商品最多存储于 d 个 FC 中，则其最多差 d 的因子。我们证明，就 q 或 d 的依赖关系而言，这两种保证都是紧的。我们的方案简单快速，基于一个直观思想——商品等待 FC 在随机时间“开启”，但在“膨胀”的时间尺度上观察它们。这也蕴含了经典 Set Cover 问题的一种新随机取整方法，可能具有普遍意义。我们在与 Jasin 和 Sinha（2015）相同的现实设置下对新的取整方案进行了数值测试，发现它们改善了运行时间、缩短了代码，并稳健地提升了性能。我们的代码已公开提供。
英文摘要：
We study the dynamic fulfillment problem in e-commerce, in which incoming (multi-item) customer orders must be immediately dispatched to (a combination of) fulfillment centers that have the required inventory. A prevailing approach to this problem, pioneered by Jasin and Sinha (2015), is to write a ``deterministic'' linear program that dictates, for each item in an incoming multi-item order from a particular region, how frequently it should be dispatched to each fulfillment center (FC). However, dispatching items in a way that satisfies these frequency constraints, without splitting the order across too many FC's, is challenging. Jasin and Sinha identify this as a correlated rounding problem, and propose an intricate rounding scheme that they prove is suboptimal by a factor of at most $\approx q/4$ on a $q$-item order. This paper provides to our knowledge the first substantially improved scheme for this correlated rounding problem, which is suboptimal by a factor of at most $1+\ln(q)$. We provide another scheme for sparse networks, which is suboptimal by a factor of at most $d$ if each item is stored in at most $d$ FC's. We show both of these guarantees to be tight in terms of the dependence on $q$ or $d$. Our schemes are simple and fast, based on an intuitive idea -- items wait for FC's to ``open'' at random times, but observe them on ``dilated'' time scales. This also implies a new randomized rounding method for the classical Set Cover problem, which could be of general interest. We numerically test our new rounding schemes under the same realistic setups as Jasin and Sinha (2015) and find that they improve runtimes, shorten code, and robustly improve performance. Our code is made publicly available.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
