A genetic programming hyper-heuristic for the distributed assembly permutation flow-shop scheduling problem with sequence dependent setup times

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Abstract

In this paper, a genetic programming hyper heuristic (GP-HH) algorithm is proposed to solve the distributed assembly permutation flow-shop scheduling problem with sequence dependent setup times (DAPFSP-SDST) and the objective of makespan minimization. The main idea is to use genetic programming (GP) as the high level strategy to generate heuristic sequences from a pre-designed low-level heuristics (LLHs) set. In each generation, the heuristic sequences are evolved by GP and then successively operated on the solution space for better solutions. Additionally, simulated annealing is embedded into each LLH to improve the local search ability. An effective encoding and decoding pair is also presented for the algorithm to obtain feasible schedules. Finally, computational simulation and comparison are both carried out on a benchmark set and the results demonstrate the effectiveness of the proposed GP-HH. The best-known solutions are updated for 333 out of the 540 benchmark instances.

Introduction

As one of the important parts of modern manufacturing systems, production scheduling has received increasing attention in the past decades [[1], [2], [3]–4]. The permutation flow-shop scheduling problem (PFSP) is among the most studied topics of production scheduling and fruitful results can be found in the existing literature [[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]–16]. The assembly PFSP (APFSP) is an extension of PFSP while final products are assembled by the jobs completed in the flow-shop stage. Thus APFSP is widely recognized as a combination of PFSP and assembly scheduling problem. Typical applications of APFSP include fire engine assembly and personal computer manufacturing and the main benefit is that product varieties can be achieved by modular structure at a controlled cost [[17], [18]–19].

Nowadays, distributed production can be more commonly seen than the centralized ones in modern enterprises [3]. The economic globalization and increasingly intensified market competition are key factors that result in this trend. Since distributed production has some advantages that enterprises require, such as higher quality, lower cost and reduced risk, they would make the transformation to become competitive [20, 21]. Actually, the structure of combining assembly with distributed production has been widely used in real world applications and well-known examples can be found in the industry of automobiles, electronics and so on [21]. Nevertheless, it makes new challenges for the scheduling of this system structure because decisions of job allocation to factories, job scheduling at each factory and assembly order scheduling have to be made. Such scheduling problem is called distributed APFSP (DAPFSP) and is becoming a hot research topic in recent years.

The first effort was made by Hatami et al. [22], in which several heuristics and a variable neighborhood descent (VND) were presented to solve DAPFSP. In [23], an effective estimation of distribution algorithm-based memetic algorithm was proposed for DAPFSP. A hybrid biogeography-based optimization and a backtracking search hyper-heuristic algorithm were developed for DAPFSP in [24] and [25], respectively, and both of them were shown to be very effective. In [26] and [27], a genetic algorithm and a bounded-search iterated greedy algorithm were presented for DAPFSP, respectively. In [28], a biased-randomized iterated local search metaheuristic was presented for DAPFSP and the advantages of this algorithm are high efficiency and less parameters. The above results were concerned with standard DAPFSP with the objective of minimizing makespan. Many researchers consider DAPFSP with other objectives or certain constraints [21, [29], [30], [31]–32]. For example, DAFPSP with total flow-time criterion was considered in [21] and three discrete invasive weed optimization based metaheuristics were presented to solve it. In [29], DAPFSP with parallel machines was investigated and a competitive memetic algorithm was developed for it.

On the other hand, cleaning, adjustment and configuration are always required between two successive jobs and the time used for these procedures is called setup time [33]. Generally speaking, setup time can be categorized into sequence independent and sequence dependent cases. The later one, which is called SDST for short, is more realistic to model practical situations. Production scheduling problem with setup times is also an important research area and results can be found in [14, [34], [35], [36]–37] and the references therein. Moreover, both SDST between jobs and SDST between products exist in DAPFSP, and the former one depends on the finished job by a machine and the job to be processed next and the latter one depends on the assembled product and the product to be assembled next. DAPFSP with SDST (DAPFSP-SDST) is more close to real-word production scheduling problem and also more complicated since it considers the characteristics of distributed, assembly and setup time at the same time and covers all the complexities of them. Therefore, the DAPFSP-SDST is a practical yet complicated problem that deserves investigation. DAPFSP-SDST was firstly studied in [33], and several heuristics, VNDs and iterated greedy (IG) algorithms were proposed and some interesting results were obtained.

In recent years, hyper-heuristic has emerged as a powerful approach in the combinational optimization fields [[38], [39], [40]–41]. The key feature of hyper-heuristic is using a high level strategy to select or generate low-level heuristics (LLHs) to search the solution space [39]. Some effective algorithms such as particle swarm, backtracking search, Monte Carlo tree search, reinforcement learning and genetic programming (GP) have been employed as high level heuristic in hyper-heuristic to solve various problems [[42], [43], [44], [45], [46], [47], [48], [49]–50]. Among them, GP based hyper-heuristic (GP-HH) is recognized to be both effective and efficient since GP has the major advantage to automatize the assembly of the components required to create a heuristic [46]. To name a few recently reported literature, a GP-HH algorithm that can train greedy heuristics was proposed in [46] to solve large scale and combinatorial bi-level optimization problems. In [47], a dynamic job shop scheduling problem was solved by GP-HH that evolves ensembles of dispatching rules. In [48], a multi-skill resource constrained project scheduling problem was investigated by proposing a GP-HH algorithm. In [49], a GP-AM algorithm was proposed to solve some combinational optimization problems including FSP, in which GP was used as the high level strategy to evolve adaptive mechanisms. In [50], a GP-HH algorithm with cooperative co-evolution was developed to solve a dynamic flexible job shop scheduling problem.

The aforementioned insights inspire the research of using GP-HH to solve DAPFSP-SDST in this paper, which has not been investigated to the best of the authors’ knowledge. Ten efficient and easy-to-implement heuristics are used to form the LLHs set and GP is employed as the high level heuristic to manipulate them. Heuristic sequences are obtained by the operations of selection, crossover and mutation at each iterative step and operated successively to search for better solutions. Computational simulation and comparison are both given to illustrate the effectiveness of the proposed GP-HH and the superiority over the VNDs and IGs in [33] and HBBO in [24].

The contributions of this paper are summarized as follows: 1) a MILP model is presented for DAPFSP-SDST and a calculation procedure for the makespan is also given. 2) A novel GP-HH algorithm is proposed to solve DAPFSP-SDST effectively, in which GP is the high level strategy to manipulate low-level heuristics to search the solution space. 3) Modified encoding and decoding scheme is presented by taking the feature of DAPFSP-SDST and SA is embedded to the low-level heuristic to improve the local search ability. 4) The performance of the proposed algorithm is evaluated on a benchmark instance set and the obtained results are very satisfactory. Especially, new best solutions are updated for 333 out of 540 instances. The rest of the paper is organized as follows. In Section 2, the DAPFSP-SDST is formulated in detail and a MILP model is presented. The GP-HH algorithm is illustrated in Section 3. Both the computational evaluation and comparison results are presented in Section 4. Finally, a conclusion and future research directions are given in Section 5.

Section snippets

Basic of DSPFSP-SDST

The structure of the considered DAPFSP-SDST is shown in Fig. 1, where N jobs are first produced in F distributed factories and then assembled into H products by the assembly machine MA in assembly factory. The F factories are assumed to be identical and each factory has M machines. In the production stage, each one of N jobs has to be assigned to one of the factories and be processed on the machines in a predetermined order, which is always assumed to be 1,2,,M, without loss of generality. The

Genetic programming

Genetic programming is a well-known evolutionary algorithm that has found various applications in many fields. The basic idea of GP is to iteratively transform initialized populations into new ones, hopefully better quantified by a criterion called fitness, through operations of selection, crossover and mutation until accepted results are found [51].

In GP, individuals are described by syntax trees whose nodes are usually operators and functions and leaves are constants and variables. Selection

Computational results and comparison

In this section, computational simulation is carried out for the proposed GP-HH algorithm. The benchmark instances set in [33] is used for the test and comparison here. The set has 540 instances and contains all possible combinations of the number of jobs N, machines M, factories F, products H and the distributions of the SDSTs. Specifically, N and SDSTs are tested at two levels with N={100,200} and SDST={[1,50],[1,125]}, and M, F and H are tested at three levels, M={5,10,20}, F={4,6,8} and H={

Conclusions

A GP-HH algorithm was proposed to solve the DAPFSP-SDST with the objective of minimizing makespan. An encoding and decoding scheme pair was presented to apply the algorithm and obtain feasible schedules. Ten efficient and easy-to-implement heuristics were developed to form the LLH set and SA was also embedded into each LLH to improve the local search ability. GP was used as the high level heuristic to generate new heuristic sequences from the LLH set in each iteration. Then the heuristic

Author statement

Hong-Bo Song: Conceptualization, Writing-Original draft, Writing-Reviewing & Editing, Programming, Investigation.

Jian Lin: Methodology, Validation, Writing-Reviewing & Editing, Data curation, Resources.

Declaration of Competing Interest

The authors declared that they have no conflicts of interest to this work.

Acknowledgment

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. This work is part of a project supported by the National Natural Science Foundation of China (Grant Nos. 61973267, 61503331 and 71671160), the Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LY19F030007 and LY19G010004) and the Zhejiang Provincial High-Education Teaching Reform Project (Grant No. jg20180199).

References (53)

  • J. Lin et al.

    A backtracking search hyper-heuristic for the distributed assembly flow-shop scheduling problem

    Swarm Evol. Comput.

    (2017)
  • H. Ochi et al.

    Scheduling the distributed assembly flowshop problem to minimize the makespan

    Proc. Comput. Sci.

    (2019)
  • S. Hatami et al.

    Heuristics and metaheuristics for the distributed assembly permutation flowshop scheduling problem with sequence dependent setup times

    Int. J. Prod. Econ.

    (2015)
  • M. Yokoyama

    Flow-shop scheduling with setup and assembly operations

    Eur. J. Oper. Res.

    (2008)
  • A. Rossi

    Flexible job shop scheduling with sequence-dependent setup and transportation times by ant colony with reinforced pheromone relationships

    Int. J. Prod. Econ.

    (2014)
  • A. Sioud et al.

    Enhanced migrating birds optimization algorithm for the permutation flow shop problem with sequence dependent setup times

    Eur. J. Oper. Res.

    (2018)
  • S.S. Choong et al.

    An artificial bee colony algorithm with a Modified Choice Function for the traveling salesman problem

    Swarm Evol. Comput.

    (2019)
  • G. Koulinas et al.

    A particle swarm optimization based hyper-heuristic algorithm for the classic resource constrained project scheduling problem

    Inf. Sci.

    (2014)
  • N.R. Sabar et al.

    Population based Monte Carlo tree search hyper-heuristic for combinatorial optimization problems

    Inf. Sci.

    (2015)
  • J. Lin

    Backtracking search based hyper-heuristic for the flexible job-shop scheduling problem with fuzzy processing time

    Eng. Appl. Artif. Intell.

    (2019)
  • S.S. Choong et al.

    Automatic design of hyper-heuristic based on reinforcement learning

    Inf. Sci.

    (2018)
  • J. Park et al.

    An investigation of ensemble combination schemes for genetic programming based hyper-heuristic approaches to dynamic job shop scheduling

    Appl. Soft Comput.

    (2018)
  • J. Lin et al.

    A genetic programming hyper-heuristic approach for the multi-skill resource constrained project scheduling problem

    Exp. Syst. Appl.

    (2020)
  • R. Ruiz et al.

    Two new robust genetic algorithms for the flowshop scheduling problem

    Omega

    (2006)
  • P. Brucker et al.

    Job-shop scheduling with multi-purpose machines

    Computing

    (1990)
  • S.R. Hejazi et al.

    Flowshop-scheduling problems with makespan criterion: a review

    Int. J. Prod. Res.

    (2005)
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