Abstract
In this paper, we study open shop scheduling problems with synchronization. This model has the same features as the classical open shop model, where each of the n jobs has to be processed by each of the m machines in an arbitrary order. Unlike the classical model, jobs are processed in synchronous cycles, which means that the m operations of the same cycle start at the same time. Within one cycle, machines which process operations with smaller processing times have to wait until the longest operation of the cycle is finished before the next cycle can start. Thus, the length of a cycle is equal to the maximum processing time of its operations. In this paper, we continue the line of research started by Weiß et al. (Discrete Appl Math 211:183–203, 2016). We establish new structural results for the two-machine problem with the makespan objective and use them to formulate an easier solution algorithm. Other versions of the problem, with the total completion time objective and those which involve due dates or deadlines, turn out to be NP-hard in the strong sense, even for m=2 machines. We also show that relaxed models, in which cycles are allowed to contain less than m jobs, have the same complexity status.
Highlights
Scheduling problems with synchronization arise in applications where job processing includes several stages, performed by different processing machines, and all movements of jobs between machines have to be done simultaneously
In the context of shop scheduling models, synchronization aspects were initially studied for flow shops (Soylu et al 2007; Huang 2008; Waldherr and Knust 2015), and later for open shops (Weiß et al 2016)
Similar to the observation of Kouvelis and Karabati (1999) that introducing idle times in a synchronous flow shop may be beneficial, we show that a schedule for the relaxed open shop problem O|synmv, r el|Cmax consisting of more than n cycles may outperform a schedule for the nonrelaxed problem O|synmv|Cmax with n cycles
Summary
Scheduling problems with synchronization arise in applications where job processing includes several stages, performed by different processing machines, and all movements of jobs between machines have to be done simultaneously. Using the results from Demange et al (2002), Escoffier et al (2006), Kesselman and Kogan (2007), de Werra et al (2009), and Mestre and Raman (2013), formulated for MEC on cubic bipartite graphs, we conclude that these two open shop problems remain strongly NP-hard even if each job is processed by at most three machines and if there are only three different values for nonzero processing times. As observed in Weiß et al (2016), the number of dummy jobs can be bounded by (m − 1)n Both problems, Om|synmv|Cmax and Om|synmv, r el|Cmax, are solvable in O(n) time, after operations are sorted in nonincreasing (or nondecreasing) order of processing times on all machines.
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