This paper considers a combination of the joint replenishment problem with single machine scheduling. There is a single resource, which is required by all the unit-time jobs, and a job can be started at time point t on the machine if and only if the machine does not process another job at t, and the resource is replenished between its release date and t. Each replenishment has a cost, which is independent of the amount replenished. The objective is to minimize the total replenishment cost plus the maximum flow time of the jobs. We consider the online variant of the problem, where the jobs are released over time, and once a job is inserted into the schedule, its starting time cannot be changed. We propose a deterministic 2-competitive online algorithm for the general input. Moreover, we show that for a certain class of inputs (so-called p-bounded input), the competitive ratio of the algorithm tends to \(\sqrt{2}\) as the number of jobs tends to infinity. We also derive several lower bounds for the best competitive ratio of any deterministic online algorithm under various assumptions.

本文考虑联合补货问题与单机调度的结合。存在一个资源，所有单位时间作业都需要该资源，当且仅当机器在t时刻没有处理另一个作业时，才能在机器上启动作业，并且在t时刻之间补充资源它的发布日期和时间。每次补货都有成本，该成本与补货数量无关。目标是最小化总补货成本加上作业的最大流动时间。我们考虑该问题的在线变体，其中作业随着时间的推移而释放，并且一旦将作业插入到计划中，其开始时间就无法更改。我们针对一般输入提出了一种确定性的 2-竞争在线算法。此外，我们表明，对于某一类输入（所谓的p有界输入），随着工作数量趋于无穷大，算法的竞争比趋于\(\sqrt{2}\) 。我们还得出了在各种假设下任何确定性在线算法的最佳竞争比的几个下限。