The Joint Replenishment Problem (JRP) is a classical inventory management problem, that aims to model the trade-off between coordinating orders for multiple commodities (and their cost) with holding costs incurred by meeting demand in advance. Moseley, Niaparast and Ravi introduced a natural online generalization of the JRP in which inventory corresponding to demands may be replenished late, for a delay cost, or early, for a holding cost. They established that when the holding and delay costs are monotone and uniform across demands, there is a 30-competitive algorithm that employs a greedy strategy and a dual-fitting based analysis. We develop a 5-competitive algorithm that handles arbitrary monotone demand-specific holding and delay cost functions, thus simultaneously improving upon the competitive ratio and relaxing the uniformity assumption. Our primal-dual algorithm is in the spirit of the work Buchbinder, Kimbrel, Levi, Makarychev, and Sviridenko, which maintains a wavefront dual solution to decide when to place an order and which items to order. The main twist is in deciding which requests to serve early. In contrast to the work of Moseley et al., which ranks early requests in ascending order of desired service time and serves them until their total holding cost matches the ordering cost incurred for that item, we extend to the non-uniform case by instead ranking in ascending order of when the delay cost of a demand would reach its current holding cost. An important special case of the JRP is the single-item lot-sizing problem. Here, Moseley et al. gave a 3-competitive algorithm when the holding and delay costs are uniform across demands. We provide a new algorithm for which the competitive ratio is $φ+1 \approx 2.681$, where $φ$ is the golden ratio, which again holds for arbitrary monotone holding-delay costs.

翻译：联合补货问题（JRP）是一个经典的库存管理问题，旨在建模协调多种商品订单（及其成本）与提前满足需求产生的持有成本之间的权衡关系。Moseley、Niaparast和Ravi引入了一种自然的在线化JRP推广模型，其中对应需求的库存可以延迟补货（产生延迟成本）或提前补货（产生持有成本）。他们证明当持有成本和延迟成本在需求间具有单调且均匀的特性时，存在一种基于贪婪策略和对偶拟合分析的30-竞争算法。我们提出了一种5-竞争算法，能够处理任意单调且需求特定的持有与延迟成本函数，从而在提升竞争比的同时放宽了均匀性假设。我们的原始-对偶算法沿袭了Buchbinder、Kimbrel、Levi、Makarychev和Sviridenko的研究思想，通过维护波动前沿对偶解来决定何时下订单以及订购哪些商品。算法的核心创新在于决定提前满足哪些需求。与Moseley等人按期望服务时间升序排列提前需求、直至其总持有成本匹配该商品订购成本的方法不同，我们通过按需求的延迟成本达到当前持有成本的时间点进行升序排序，将算法推广至非均匀情形。JRP的一个重要特例是单物品批量问题。在此问题上，Moseley等人在持有与延迟成本均匀的情况下给出了3-竞争算法。我们提出了一种新算法，其竞争比为$φ+1 \approx 2.681$（其中$φ$为黄金分割率），该结果同样适用于任意单调的持有-延迟成本函数。