Complexity of late work minimization in flow shop systems and a particle swarm optimization algorithm for learning effect
Introduction
Scheduling problems with job due date constraints (Wang et al., 2017a, Yin et al., 2013, Yin et al., 2015) are motivated by many scenarios appearing in real production environments, e.g., minimizing the total delay time in a project, minimizing the storage cost in a storehouse, or minimizing the amount of hysteretic customer requests (Błażewicz et al., 2007a, Pinedo, 2015). For such models, several scheduling criteria such as lateness (McMahon & Florian, 1975), tardiness (Emmons, 1969), earliness (Sidney, 1977), or the number of tardy jobs (Moore, 1968) were proposed to reflect different objectives.
Among the criteria related to due dates, late work is one of maturely explored objective functions, with respect to its practical significance. It was first proposed by Błażewicz in 1984 to model the information collection in control systems (Błażewicz, 1984), then expanded to agriculture domain to model land cultivation (Błażewicz, Pesch, Sterna, & Werner, 2004), software development to model bugs detecting (Sterna, 2011), flexible manufacturing to model production planning (Sterna, 2007b), chip manufacturing to model burn-in operations (Ren, Zhang, & Sun, 2009), and so on. In three-field notation, late work criterion was often denoted as Y, and a number of papers were devoted to it. This criterion was investigated in various machine environments, including single-machine case (Lin and Hsu, 2005, Potts and Van Wassenhove, 1992), parallel machines environment (Błażewicz and Finke, 1987, Leung, 2004), and shop systems (Błażewicz et al., 2000, Lin et al., 2006, Pesch and Sterna, 2009, Sterna, 2007a). Recently, the late work criterion was widely studied in two agents scheduling problems by Wang et al. and Yin et al., mostly for single machine (Wang et al., 2017b, Wang et al., 2016, Wang et al., 2015, Yin et al., 2016b), as well as for parallel machine cases (Yin, Cheng, Cheng, Wang, & Wu, 2016a).
Similarly, learning effect (Wright, 1936) (LE for short) arises in many producing situations. For example, a worker improves his ability continuously over time, which implies that the processing time could be shorter if the job was processed later. Biskup (1999) was the first one who introduced learning effect to scheduling problems, in a single machine environment. Then, lots of literature is devoted to this subject, e.g., (Bachman and Janiak, 2004, Biskup, 2008, Cheng and Wang, 2000, Wang and Wang, 2013, Yin et al., 2012, Yin et al., 2009, Yin et al., 2010). Most recently, Wu, Yin, Wu, Chen, and Cheng (2016) considered learning effect on single machine with the goal of minimizing the total late work.
In this paper, we firstly revisit the complexity of the late work minimization problem in the flow shop system with a common due date (Błażewicz, Pesch, Sterna, & Werner, 2005b), and prove that problem is NP-hard in the strong sense, which implies that is also strongly NP-hard. Then, we study the problem , i.e., the late work minimization problem in the permutation flow shop system with learning effect, and propose a Particle Swarm Optimization (PSO) algorithm for it. Finally, we analyze the results of computational experiments to show the influence of different problem parameters on the efficiency of our PSO.
The rest of this paper is organized as follows. The formulation of the problem and the related work are presented in Section 2. Section 3 is devoted to NP-hardness proof for problem , based on a reduction from , which is one of the famous strongly NP-hard problems. Then, the extended model, problem , is studied in Section 4, where a PSO is designed and the computational results are analyzed. The final conclusions and future work directions are discussed in Section 5.
Section snippets
Problem formulation and related work
In a flow shop system, there are m machines and n jobs . Each job has m tasks, which have to be processed consecutively on , …, and finally on . We use to denote the task of job , with processing time . Therefore, the processing time of is equal to . For each job, a due date is defined.
The late work of job is equal to the length of the late part of this job (if any), i.e., the sum of late parts of tasks
Strong NP-Hardness of
We give the strong NP-hardness proof of by a reduction from problem, in the following theorem. Theorem 3.1 Problem is NP-hard in the strong sense. Proof Obviously, the decision counterpart of the studied problem belongs to class NP, since its solution can be verified in polynomial time. Then we show that it can be reduced from problem in polynomial time. is defined as follows: Given positive numbers with and
Particle swarm optimization algorithm for
The intractability of means that there is no polynomial time algorithm for this problem (unless P = NP), and probably no efficient exact method especially for large scale instances. In such cases the research often focuses on heuristic or meta-heuristic algorithms, which can solve the problem efficiently and effectively (cf., e.g., (Błażewicz et al., 2005a, Błażewicz et al., 2008, Lin et al., 2006, Pesch and Sterna, 2009)).
As mentioned before, learning effect is a natural outcome from real
Conclusions
In this paper, we focused on scheduling in flow shop systems with late work criterion. First, following serials of complexity results available in the literature, we proved that problem is NP-hard in the strong sense. Then, we considered learning effect in the late work flow shop scheduling problem, i.e., , and proposed a PSO algorithm for it. Computational experiments showed that this meta-heuristic is able to improve initial solutions generated by priority dispatching rules
Acknowledgement
The authors appreciate Minming Li, as well as the referees, for their helpful comments on this paper.
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2022, European Journal of Operational ResearchCitation Excerpt :Moreover, it is theoretically meaningful to consider the more general version in which the jobs of each agent have their own equal processing times. Notice that scheduling with late work criteria is a hot topic in recent years, see Chen, Yuan, & Gao (2019); Chen, Chau, Xie, Sterna, & Blazewicz (2017); Chen, Sterna, Han, & Blazewicz (2016); Gerstl, Mor, & Mosheiov (2019); He & Yuan (2020); Li, Gajpal, & Bector (2020); Mosheiov, Oron, & Shabtay (2021); Sterna & Czerniachowska (2017); Wang, Fan, Zhang, Zhang, & Leung (2017b); Yin, Xu, Cheng, Wu, & Wang (2016); Yin, Y., Wang, & Cheng (2017); Zhang & Wang (2017); Zhang & Yuan (2019), and Zhang, Geng, & Yuan (2020); Zhang, Yuan, Ng, & Cheng (2021). Among them, competing two-agent scheduling accounts for a large proportion.
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2022, European Journal of Operational ResearchCitation Excerpt :Moreover, one could consider the more general case, i.e., scheduling in multi-stage flow shop systems with arbitrary due dates. Although there were several papers devoted to the meta-heuristic methods, e.g. (Chen, Chau, Xie, Sterna, & Błażewicz, 2017; Pesch & Sterna, 2009), there is still a lack of exact algorithms for the problem. Furthermore, some factors related to practical applications, such as machine maintenance (Yin, Xu, Cheng, Wu, & Wang, 2016), deterioration and learning effect (Wang, Ji, Cheng, & Wang, 2012), job rejection (Shabtay, Gaspar, & Kaspi, 2013), scheduling with multi-agent (Wang, Yu, Qiu, Yin, & Cheng, 2020; Yin, Yang, Wang, Cheng, & Wu, 2018) or under multitasking (Xiong, Zhou, Yin, Cheng, & Li, 2019) could be further introduced into the model.
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2022, Computers and Industrial EngineeringCitation Excerpt :In the following, we just review some most related results on scheduling of two or multiple competing agents. There are other scheduling models related to our research, such as flow shop scheduling (Blazewicz, Pesch, Sterna, & Werner (2005), Chen, Chau, Xie, Sterna, & Blazewicz (2017), Chen & Li (2021)), parallel identical machines scheduling (Chen, Sterna, Han, & Blazewicz (2016)), scheduling with deadlines (Chen, Yuan, Ng, & Cheng, 2019, 2021), scheduling with job rejection (Freud & Mosheiov (2021)), scheduling with generalized due dates (Mosheiov, Oron, & Shabtay (2021)), scheduling with a rate modifying activity (Yin, Xu, Cheng, Wu, & Wang (2016), Mosheiov & Oron (2021)), and so on. From the above discussion, the lemma follows.
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