Joint replenishment meets scheduling

Abstract

In this paper we consider a combination of the joint replenishment problem (JRP) and single machine scheduling with release dates. There is a single machine and one or more item types. Each job has a release date, a positive processing time, and it requires a subset of items. A job can be started at time t only if all the required item types were replenished between the release date of the job and time point t. The ordering of item types for distinct jobs can be combined. The objective is to minimize the total ordering cost plus a scheduling criterion, such as total weighted completion time or maximum flow time, where the cost of ordering a subset of items simultaneously is the sum of a joint ordering cost, and an additional item ordering cost for each item type in the subset. We provide several complexity results for the offline problem, and competitive analysis for online variants with min-sum and min-max criteria, respectively. keywords Joint replenishment single machine scheduling complexity polynomial time algorithms online algorithms

An online joint replenishment problem combined with single machine scheduling

“…On the other hand, we show that for the p-bounded input, the competitive ratio of the algorithm tends to √ 2 as the number of jobs tends to infinity. This result generalizes the one of Györgyi et al (2021) for the regular input, and although it does not reach the √ 2-competitive ratio for short sequences of jobs, in the long run it gets arbitrarily close to it.…”

“…We mention that our online model is slightly different from that of Györgyi et al (2021). While in this paper, upon the arrival of the last job we get the information that there will be no further jobs, in Györgyi et al (2021) this information is not available at once. Therefore, the presented lower bounds cannot be directly compared with each other.…”

“…First, we present some preliminary results regarding the offline optimal solution. Although Györgyi et al (2021) already covered the offline variant of the problem, these results help in the analysis of the proposed online algorithm. We show that if the release dates of the jobs are not necessary distinct, then the problem 1|jrp, s = 1, p j = 1, distinct r j |F max + c Q is equivalent to the problem 1|jrp, s = 1, r j |F max +c Q (i.e., when the jobs can have arbitrary processing times), see Proposition 1.…”

“…In this paper, we study a combination of the classical joint replenishment problem (JRP) with machine scheduling, proposed recently by Györgyi et al (2021). The joint replenishment problem seeks an optimal replenishment policy of one or several items required to fulfill a sequence of demands over time.…”

“…Using the EDD rule whenever the machine becomes idle leads to a 2-competitive algorithm, and by introducing a clever waiting strategy, the competitive ratio reaches the lower bound of ( √ 5 + 1)/2. Györgyi et al (2021) analyzed the combination of the joint replenishment problem (JRP-W) and the single machine scheduling with different scheduling objective such as C j , w j C j , F j and F max . For the latter, the authors showed that if there are two resources, the problem is NP-hard even under very strong assumptions.…”