A matrix-cube-based estimation of distribution algorithm for blocking flow-shop scheduling problem with sequence-dependent setup times

https://doi.org/10.1016/j.eswa.2022.117602Get rights and content

Highlights

  • A fast Insert-based neighbor evaluation is proposed to accelerate the local search.

  • A diversity controlling mechanism is used to avoid stagnation of the global search.

  • A multidimensional probabilistic model is designed to learn promising patterns.

  • A local search controlled by the probabilistic model in global search is devised.

  • Two heuristics are presented to generate high-quality initial individuals.

Abstract

The blocking flow-shop scheduling problem with sequence-dependent setup times (BFSP_SDST) is a strong NP-hard problem that exists widely in practice. However, research on this issue is still quite limited. Hence, this paper presents a novel matrix-cube-based estimation of distribution algorithm (MCEDA) to minimize the makespan criterion of the BFSP_SDST. In MCEDA's global search, a matrix cube is devised to reasonably learn the promising patterns in excellent solutions or individuals, and then a matrix-cube-based probabilistic model is developed to quickly guide global search toward the potential promising regions in solution space. A diversity controlling mechanism is also added to avoid the stagnation of global search. In MCEDA's local search, an iterated multi-neighborhood local search controlled by the probabilistic model in global search is designed to execute deeper exploitation from those promising regions. Additionally, two constructive heuristics for generating high-quality initial individuals and one fast Insert-based neighbor evaluation method for accelerating the efficiency of local search are presented based on an analysis of the problem's features. MCEDA's efficacy and superiority in solving the BFSP SDST are demonstrated through comprehensive comparisons with 22 state-of-the-art algorithms.

Introduction

Production scheduling has been recognized as a realistic and reliable decision-making approach for allocating restricted resources within a certain time period in order to achieve one or more decision-maker-defined objectives (Pinedo, 2015). As a hot research topic in the field of production scheduling, the flow-shop scheduling problem (FSP) has a wide range of applications in numerous manufacturing systems, production and assembly lines, and information service facilities. For the typical FSP, it is commonly assumed that there are infinitely storage facilities or buffer units between any two adjacent machines, where finished jobs can be stored in these buffer units for an unlimited amount of time. However, in many real-world manufacturing situations, due to production characteristics and technical constraints, there are usually no intermediate storage units between machines (Grabowski & Pempera, 2007). In this sense, the traditional FSP is converted into the blocking FSP (BFSP), which is a typical NP-hard problem in the strong sense (Hall and Sriskandarajah, 1996, Ronconi and Henriques, 2009, Wang et al., 2010). As a significant subfield of FSP, BFSP has attracted the considerable attention and interest from both researchers and practitioners in recent decades. A wide variety of real-world industrial processes and manufacturing systems can be modeled as the BFSP, such as chemical and pharmaceutical manufacturing (Ronconi, 2004), iron and steel manufacturing (Gong, et al., 2010), robotic cells (Elmi & Topaloglu, 2013), serial manufacturing processes (Koren, et al., 2017), and waste treatment (Riahi, et al., 2017). Nowadays, the BFSP has garnered the tremendous attention and interest of both researchers and practitioners (see Section 2).

Setup time is prevalent in a variety of real-life manufacturing systems. In many factories, setup time is frequently derived from non-productive activities such as cleaning devices, adjusting equipment, switching machines, repairing or releasing jobs, especially in chemical or pharmaceutical plants. Although in almost all the existing research works related to BFSPs, it is usually assumed that the setup time is negligible or included in processing time, however, substantial setup times should be separable (Shao, et al., 2018b). Nevertheless, the improper handling of setup operations may result in the consumption of more than 20% of the available machine capacity (Pinedo, 2015). To the best of our knowledge, there are still very few works on BFSP that involve setup time, especially for sequence-dependent setup times (SDST) (Shao, et al., 2018b). Therefore, this paper investigates an extension of the BFSP, namely the BFSP with SDST (BFSP_SDST), whose criterion is to minimize makespan (i.e., Cmax). The SDST indicates that the setup time of each job on each machine depends not only on the job itself but also on its immediately preceding job. According to the widely used three-field notation α|β|γ proposed by Graham, et al. (1979), the BFSP under the makespan criterion and the studied problem herein can be denoted as Fm|blocking|Cmax and Fm|blocking,STsd|Cmax, respectively. Since Fm|blocking|Cmax is already recognized as strongly NP-hard, and it is obviously reduced to Fm|blocking,STsd|Cmax, it can be concluded that Fm|blocking,STsd|Cmax is also NP-hard in the strong sense.

For the NP-hard scheduling problems, existing mathematical algorithms are often of limited use due to their excessive computation time or poor performance under reasonable runtime. Hence, numerous hybrid intelligent optimization algorithms (HIOAs) have been developed to tackle this issue, aiming to achieve satisfactory solutions for a wide variety of traditional scheduling problems within several seconds or tens of seconds. Among these algorithms, the hybrid estimation of distribution algorithm (HEDA) is a unique one. Unlike the crossover and mutation operators in most existing HIOAs (e.g., hybrid genetic algorithm, hybrid particle swarm optimization algorithm, hybrid differential evolution algorithm), HEDA generates the offspring population by sampling an EDA-based probability model, which can learn and accumulate valuable information about excellent individuals from a macro perspective, as well as establish explicit probability models to effectively estimate the distribution of superior solutions and to predict promising regions in the feasible solution space. To a certain extent, such novel population generation mechanism can avoid the destruction of the blocks (i.e., the partial ordered patterns) in excellent individuals or solutions to a certain extent (Larranga & Lozano, 2001). Due to its stronger global exploration, simpler framework, and faster convergence speed, HEDA has been widely utilized to solve various scheduling problems (Faraji Amiri and Behnamian, 2020, Jarboui et al., 2009, Pan and Ruiz, 2012, Qian et al., 2017, Wang et al., 2014, Wang et al., 2013, Wu et al., 2021). These successful applications have indicated that HEDA has considerable competitive advantage against other algorithms. Therefore, HEDA is selected as the main framework of our proposed algorithm for Fm|blocking,STsd|Cmax.

Unfortunately, the majority of currently available HEDAs have two drawbacks. The first drawback is that most existing HEDAs commonly use one or more two-dimensional probabilistic models or matrices to learn the characteristic information of excellent individuals. The structure of two-dimensional matrix directly determines that only the matrix's elements and the subscripts of these elements can be utilized to store information. For the two-dimensional matrix Mn×n, its element Mn×n(x,y) is used to record the occurrence frequency of the block [x,y] in excellent individuals, while the subscript (x,y) is only enough to save the information of one block's structure or pattern. There is no extra space to record the position of this block [x,y] in each corresponding excellent individual. This makes it difficult for two-dimensional probabilistic models to correctly guide the search direction, so that the practical performance of the existing HEDAs is relatively limited (see Subsection 4.2.1). The second drawback is that almost all existing HEDAs and other HIOAs lack substantive interaction between their global and local searches. In each of these algorithms, the local search can only execute the neighborhood exploitation by using a very limited number of pre-defined common neighborhood operators (e.g., Insert, Swap, and Interchange). The lack of global exploration information to assist the local search undoubtedly limits the depth of the local search, resulting in the algorithm's overall practical performance being constrained. To overcome the aforementioned defects, a novel matrix-cube-based HEDA, namely MCEDA, is proposed to address the considered problem.

The main characteristics of our MCEDA are summarized as follows.

  • A three-dimensional matrix (i.e., matrix cube) is devised to reasonably record and reserve the valuable patterns in excellent individuals or solutions. For a three-dimensional matrix, the z in its subscript (x,y,z) is used to record the position of job block [x,y] in the corresponding excellent solutions. Meanwhile, a matrix-cube-based probabilistic model with a sampling strategy is developed to estimate the distribution of excellent solutions in solution space and correctly guide global search to promising regions. Moreover, a simple diversity controlling mechanism is designed to avoid the stagnation of global search.

  • Different from most existing HIOA's local searches that execute local search independently, a new iterated multi-neighborhood local search controlled by the matrix-cube-based probabilistic model in global search is presented to undertake deeper exploitation from those promising regions. This novel local search utilizes the block patterns saved in the probability model to approximately evaluate neighbors and dynamically create promising neighborhoods for performing fast and rich search.

  • Based on the problem's characteristics, two effective constructive heuristics are designed to ensure the quality and diversity of the initial population. Meanwhile, a fast Insert-based neighbor evaluation method is presented to improve search efficiency.

  • The proposed MCEDA is compared against twenty-two state-of-the-art algorithms on different instances. The statistical results demonstrate the efficacy and superiority of MCEDA.

The remainder of this paper is organized as follows. Section 2 briefly reviews the related literature. Section 3 describes the model of the problem. Section 4 presents MCEDA after explaining two effective heuristics for initialization, the matrix cube based global search, and the probabilistic model controlled local search. The comparison results and statistical analysis are provided in Section 5. Finally, Section 6 gives some concluding remarks and suggestions for future research.

Section snippets

Literature review of BFSP and BFSP_SDST

The comprehensive review of the BFSP can be found in (Miyata & Nagano, 2019). Since 2010, there have been mainly three types of algorithms for the BFSP.

The first is the HIOA. The existing studies have mainly concentrated on minimizing the makespan criterion. Wang, et al. (2010) presented a hybrid discrete differential evolution (HDDE), in which a speedup method was utilized to evaluate the Insert-based neighbor solutions. The test results showed that HDDE outperformed the famous tabu search

permutation-based model

The BFSP_SDST can be briefly described as follows. There are n jobs and m machines in a flow shop without intermediate buffers. Each job JiJ has a sequence of operations {Oi,1,Oi,2,...,Oi,m} to be processed sequentially on machine M1, M2, and so on until machine Mm. The operation Oi,j of job Ji should be executed on machine Mj with a period of processing time pi,j. Since there are no buffers between consecutive machines and each machine has to take some time to prepare before processing, jobs

MCEDA for BFSP_SDST

In this section, the matrix-cube-based estimation of distribution algorithm (MCEDA) is proposed to address the BFSP_SDST with makespan criterion. In the following subsections, the heuristic and initialization, the multi-dimensional probabilistic model, the diversity controlling mechanism, the multi-neighborhood based local search are firstly described in detail, and then the MCEDA’s framework is outlined. Meanwhile, the analysis of MCEDA’s computational complexity is provided.

Experimental results and statistical analysis

This section implements the extensive experiments to demonstrate the effectiveness and efficiency of the proposed MCEDA. Firstly, the experimental setup is briefly described in Section 5.1, including the testing instances, performance metrics, and experimental environment. Then, the effects of MCEDA's parameters are discussed in Section 5.2. Afterwards, the superiority of multi-dimensional probabilistic model and the advantages of improvement strategies are investigated in Section 5.3 and

Conclusion and future work

In this paper, a matrix-cube-based estimation of distribution algorithm (MCEDA) is proposed to solve a kind of important scheduling problem, i.e., the blocking flow-shop scheduling problem with sequence-dependent setup times (BFSP_SDST). To the best of our knowledge, this is the first report on the application of EDA to the BFSP problems.

From the extensive test results, it can be concluded that the use of deep and fast local search is recommended. For non-convex optimization problems such as

CRediT authorship contribution statement

Zi-Qi Zhang: Investigation, Methodology, Software, Writing – original draft. Bin Qian: Methodology, Funding acquisition, Supervision, Writing – review & editing. Rong Hu: Methodology, Funding acquisition, Investigation, Writing – review & editing. Huai-Ping Jin: . Ling Wang: Supervision, Project administration. Jian-Bo Yang: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is partially supported by the Basic Research Key Project of Yunnan Province (202201AS070030), and the National Natural Science Foundation of China (62173169, 61963022, 61873328).

References (63)

  • H.H. Miyata et al.

    The blocking flow shop scheduling problem: A comprehensive and conceptual review

    Expert Systems with Applications

    (2019)
  • G. Moslehi et al.

    A hybrid variable neighborhood search algorithm for solving the limited-buffer permutation flow shop scheduling problem with the makespan criterion

    Computers & Operations Research

    (2014)
  • M. Nawaz et al.

    A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem

    Omega

    (1983)
  • Q.-K. Pan et al.

    Effective heuristics for the blocking flowshop scheduling problem with makespan minimization

    Omega

    (2012)
  • Q.K. Pan et al.

    An estimation of distribution algorithm for lot-streaming flow shop problems with setup times

    Omega-International Journal of Management Science

    (2012)
  • B. Qian et al.

    A copula-based hybrid estimation of distribution algorithm for m-machine reentrant permutation flow-shop scheduling problem

    Applied Soft Computing

    (2017)
  • C. Rajendran et al.

    An efficient heuristic for scheduling in a flowshop to minimize total weighted flowtime of jobs

    European Journal of Operational Research

    (1997)
  • V. Riahi et al.

    Scatter search for mixed blocking flowshop scheduling

    Expert Systems with Applications

    (2017)
  • I. Ribas et al.

    An iterated greedy algorithm for the flowshop scheduling problem with blocking

    Omega-International Journal of Management Science

    (2011)
  • I. Ribas et al.

    An efficient Discrete Artificial Bee Colony algorithm for the blocking flow shop problem with total flowtime minimization

    Expert Systems with Applications

    (2015)
  • I. Ribas et al.

    An iterated greedy algorithm for the parallel blocking flow shop scheduling problem and sequence-dependent setup times

    Expert Systems with Applications

    (2021)
  • D.P. Ronconi

    A note on constructive heuristics for the flowshop problem with blocking

    International Journal of Production Economics

    (2004)
  • D.P. Ronconi et al.

    Some heuristic algorithms for total tardiness minimization in a flowshop with blocking

    Omega-International Journal of Management Science

    (2009)
  • R. Ruiz et al.

    Solving the flowshop scheduling problem with sequence dependent setup times using advanced metaheuristics - Discrete optimization

    European Journal of Operational Research

    (2005)
  • R. Ruiz et al.

    An Iterated Greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives

    European Journal of Operational Research

    (2008)
  • T. Schiavinotto et al.

    A review of metrics on permutations for search landscape analysis

    Computers & Operations Research

    (2007)
  • Z. Shao et al.

    An efficient discrete invasive weed optimization for blocking flow-shop scheduling problem

    Engineering Applications of Artificial Intelligence

    (2019)
  • Z. Shao et al.

    Effective constructive heuristic and iterated greedy algorithm for distributed mixed blocking permutation flow-shop scheduling problem

    Knowledge-Based Systems

    (2021)
  • Z.S. Shao et al.

    Self-adaptive discrete invasive weed optimization for the blocking flow-shop scheduling problem to minimize total tardiness

    Computers & Industrial Engineering

    (2017)
  • Z.S. Shao et al.

    A novel discrete water wave optimization algorithm for blocking flow-shop scheduling problem with sequence-dependent setup times

    Swarm and Evolutionary Computation

    (2018)
  • A. Sioud et al.

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

    European Journal of Operational Research

    (2018)
  • Cited by (10)

    View all citing articles on Scopus
    View full text