Complexity of late work minimization in flow shop systems and a particle swarm optimization algorithm for learning effect

doi:10.1016/j.cie.2017.07.016

Computers & Industrial Engineering

Volume 111, September 2017, Pages 176-182

https://doi.org/10.1016/j.cie.2017.07.016 Get rights and content

Highlights

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3 machines flowshop late work minimization problem with common due date is NP-hard.
•
A Heuristic based on particle swarm optimization method and learning effect.
•
Experiments on the problem of flowshop with arbitrary number of machines.
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The heuristic is a reasonable solution, from performance and time-consumption views.

Abstract

Late work minimization is one of the newer branches in the scheduling theory, with the goal of minimizing the total size of late parts of all jobs in the system. In this paper, we study the scheduling problem in flow shop, which finds many practical applications. First, we prove that the problem with three machines and a common due date is NP-hard in the strong sense. Then we extend this basic model, considering the problem with the arbitrary number of machines, various due dates and learning effect, and propose a particle swarm optimization algorithm (PSO). Computational experiments show that the PSO is an efficient method for solving the problem under consideration, both from algorithm-performance and time-consumption views.

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 $(d_{j} = d)$ (Błażewicz, Pesch, Sterna, & Werner, 2005b), and prove that problem $F_{3} | d_{j} = d | Y$ is NP-hard in the strong sense, which implies that $F | d_{j} = d | Y$ is also strongly NP-hard. Then, we study the problem $F | LE | Y$ , 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 $F_{3} | d_{j} = d | Y$ , based on a reduction from $3 - PARTITION$ , which is one of the famous strongly NP-hard problems. Then, the extended model, problem $F | LE | Y$ , 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 $M = {M_{1}, M_{2}, \dots, M_{m}}$ and n jobs $J = {J_{1}, J_{2}, \dots, J_{n}}$ . Each job has m tasks, which have to be processed consecutively on $M_{1}, M_{2}$ , …, and finally on $M_{m}$ . We use $T_{ij}$ $(1 ⩽ i ⩽ m)$ to denote the $i - th$ task of job $J_{j}$ $(1 ⩽ j ⩽ n)$ , with processing time $p_{ij}$ . Therefore, the processing time of $J_{j}$ is equal to $p_{j} = \sum_{i = 1}^{m} p_{ij}$ . For each job, a due date $d_{j}$ is defined.

The late work $Y_{j}$ of job $J_{j}$ is equal to the length of the late part of this job (if any), i.e., the sum of late parts of tasks $T_{1 j},$

Strong NP-Hardness of $F_{3} | d_{j} = d | Y$

We give the strong NP-hardness proof of $F_{3} | d_{j} = d | Y$ by a reduction from $3 - PARTITION$ problem, in the following theorem.

Theorem 3.1

Problem $F_{3} | d_{j} = d | Y$ 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 $3 - PARTITION$ problem in polynomial time.

$3 - PARTITION$ $(3 - PART)$ is defined as follows: Given $3 m$ positive numbers $a_{1}, \dots, a_{3 m}$ with $\sum_{i = 1}^{3 m} a_{i} = mb$ and $\frac{b}{4} < a_{i} < \frac{b}{2}$ $(i \in [1, 3 m])$

Particle swarm optimization algorithm for $F | LE | Y$

The intractability of $F | LE | Y$ 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 $F_{3} | d_{j} = d | Y$ is NP-hard in the strong sense. Then, we considered learning effect in the late work flow shop scheduling problem, i.e., $F | LE | Y$ , 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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Complexity of late work minimization in flow shop systems and a particle swarm optimization algorithm for learning effect

Highlights

Abstract

Introduction

Section snippets

Problem formulation and related work

Strong NP-Hardness of F3|dj=d|Y

Particle swarm optimization algorithm for F|LE|Y

Conclusions

Acknowledgement

Applied Mathematical Modelling

European Journal of Operational Research

European Journal of Operational Research

Information Processing Letters

Discrete Applied Mathematics

Computers & Industrial Engineering

European Journal of Operational Research

Computers & Operations Research

Computers & Industrial Engineering

Computers & Industrial Engineering

Computers & Industrial Engineering

Omega

Applied Mathematical Modelling

International Journal of Production Economics

Omega

Omega

Computers & Industrial Engineering

Information Sciences

Applied Mathematical Modelling

Scheduling jobs with position-dependent processing times

Journal of the Operational Research Society

Scheduling preemptible tasks on parallel processors with information loss

Technique et Science Informatiques

Handbook on scheduling: From theory to applications

Total late work criteria for shop scheduling problems

A note on the two machine job shop with the weighted late work criterion

Journal of Scheduling

Single machine scheduling with learning effect considerations

Annals of Operations Research

Strong NP-Hardness of $F_{3} | d_{j} = d | Y$

Particle swarm optimization algorithm for $F | LE | Y$