库存约束下的多阶段多客户产品组合优化
Multi-stage and Multi-customer Assortment Optimization with Inventory Constraints
摘要：
我们考虑一个分类优化问题，客户选择一个单一的项目，从一系列的集合显示给她，而有限的库存约束的项目提供给客户随着时间的推移。在特殊情况下，所有的映射大小为1，我们的问题捕获的在线随机匹配超时问题。对于这个问题，我们推导出一个多项式时间的近似算法，它至少获得1-ln（2-1/e），或0.51，最佳。这提高了以前的最佳近似比0.46，而且，我们表明，它是紧的。对于一般的分类问题，对于不同类型的客户对商品的评价不同的情况，我们建立了第一常数因子近似比0.09，对于不同客户对每个商品的评价相同的情况，我们建立了近似比0.15。我们的算法是基于舍入的LP松弛多阶段的分类优化，并改进以前的随机舍入方案，以获得1-ln（2-1/e）的紧比。
英文摘要：
We consider an assortment optimization problem where a customer chooses a single item from a sequence of sets shown to her, while limited inventories constrain the items offered to customers over time. In the special case where all of the assortments have size one, our problem captures the online stochastic matching with timeouts problem. For this problem, we derive a polynomial-time approximation algorithm which earns at least 1-ln(2-1/e), or 0.51, of the optimum. This improves upon the previous-best approximation ratio of 0.46, and furthermore, we show that it is tight. For the general assortment problem, we establish the first constant-factor approximation ratio of 0.09 for the case that different types of customers value items differently, and an approximation ratio of 0.15 for the case that different customers value each item the same. Our algorithms are based on rounding an LP relaxation for multi-stage assortment optimization, and improve upon previous randomized rounding schemes to derive the tight ratio of 1-ln(2-1/e).
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