AI 中文总结
研究可分解次模排序成本函数族下的SJRP，提出通过新颖注水过程舍入线性规划松弛解的算法，实现O(k)近似，固定k时为该类次模排序成本提供首个常数因子保证，扩展了有保证的成本函数类别。
AI 中文摘要
本文针对广泛的可分解次模排序成本函数族下的次模联合补货问题（SJRP）开发了一种近似算法。在SJRP中，一个中央规划者协调订单以满足在有限离散规划期内对多个物品的确定性需求，同时最小化总持有和订购成本，后者被建模为每个时期订购物品子集的次模函数。本文考虑的排序成本函数基于将物品分解为k个类别来定义，其中成本是每个类别内加权总量的函数，并通过联合成本函数允许跨类别进行任意交互。所提出的算法通过使用一种新颖的注水过程根据边际成本将分数解划分为嵌套区域来对线性规划松弛的解进行舍入，然后从每个区域中选择一个订单以获得可行的整数调度。所得算法实现了O(k)近似。当类别数k固定时，这为这类广泛的次模排序成本提供了第一个常数因子保证，显著扩展了已知有此类保证的成本函数类别。
英文摘要
This paper develops an approximation algorithm for the submodular joint replenishment problem (SJRP) under a broad family of decomposable submodular ordering cost functions. In the SJRP, a central planner coordinates orders to satisfy deterministic demand for multiple items over a finite discrete planning horizon while minimizing total holding and ordering costs, with the latter modeled as a submodular function of the subset of items ordered in each period. The ordering cost functions considered in this paper are defined based on a decomposition of the items into $k$ categories, where the cost is a function of weighted aggregate quantities within each category and allows for arbitrary interactions across categories through a joint cost function. The proposed algorithm rounds the solution to a linear programming relaxation by partitioning the fractional solution into nested regions according to marginal costs using a novel water-filling procedure, and then selecting one order from each region to obtain a feasible integral schedule. The resulting algorithm achieves an $O(k)$-approximation. When the number of categories $k$ is fixed, this yields the first constant-factor guarantee for this broad class of submodular ordering costs, significantly expanding the class of cost functions for which such guarantees are known.
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