On‐line algorithms for minimizing makespan on batch processing machines

Abstract

Abstract:We consider the problem of scheduling jobs on-line on batch processing machines with dynamic job arrivals to minimize makespan. A batch processing machine can handle up to B jobs simultaneously. The jobs that are processed together form a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is given by the longest processing time of any job in the batch. Each job becomes available at its arrival time, which is unknown in advance, and its processing time becomes known upon its arrival. In the first part of this paper, we address the single batch processing machine scheduling problem. First we deal with two variants: the unbounded model where B is sufficiently large and the bounded model where jobs have two distinct arrival times. For both variants, we provide on-line algorithms with worst-case ratio ( √ 5 + 1)/2 (the inverse of the Golden ratio) and prove that these results are the best possible. Furthermore, we generalize our algorithms to the general case and show a worst-case ratio of 2. We then consider the unbounded case for parallel batch processing machine scheduling. Lower bounds are given, and two on-line algorithms are presented.

An improved on-line algorithm for scheduling on two unrestrictive parallel batch processing machines

“…The algorithm performs better than the one given in [13], which has a competitive ratio of 3 2 . It would be interesting to find out whether for the problem under study there is an on-line algorithm with a competitive ratio matching the lower bound 1 + γ 2 ≈ 1.325 provided in [14], or whether the lower bound can be improved.…”

“…According to the results of [14], there does not exist any on-line algorithm for the problem with a competitive ratio smaller than 1 + γ 2 , where γ 2 ≈ 0.325 is a solution of the equation (1 + γ 2 ) 3 = γ 2 + 2. We provide an on-line algorithm for the problem that is better than the one given in [13]. In fact, our algorithm improves the competitive ratio from …”

“…Liu and Yu [8] proved that the problem is NP -hard even if there are only two release dates and derived a pseudo-polynomial time algorithm for the case where the number of release dates is fixed. Zhang et al [13] considered the on-line version of the problem. They dealt with both the unrestrictive and restrictive models.…”

A coordination mechanism for a scheduling game with parallel-batching machines

“…The results are summarized in Table 1. A number of researchers have studied the traditional parallel-batching scheduling problems that are under centralized situation (Deng et al 2003;Lee et al 1992;Lee and Uzsoy 1999;Ng et al 2003;Potts and Kovalyov 2000;Uzsoy 1994;Zhang et al 2001). Scheduling problem P|b < n|C max is N P−hard in the strong sense even for b = 1 (Lageweg et al 1981).…”

On-line integrated production and outbound distribution scheduling to minimize the maximum delivery completion time

“…When c ≥ 2, since the number of jobs n is not known in advance, we cannot assign all the jobs into n c delivery batches as in Step 2 of algorithm A 1 . In fact, by considering the vehicle as a batch-processing machine with the capacity c, the remaining job delivery problem is equivalent to an on-line batch-processing scheduling problem considered by Zhang et al (2001) as follows: All jobs arrive over time, i.e., each job and its processing time become available at its arrival time. There is a batch-processing machine which can handle up to c ≥ 2 jobs simultaneously.…”