An ensemble discrete differential evolution for the distributed blocking flowshop scheduling with minimizing makespan criterion
Graphical abstract
Introduction
Production scheduling solves the problem of allocating resources (usually machines) to a task or job at a given time to optimize the goal (Zhang, Xing, & Cao, 2018). With the rapid development of the manufacturing technology and the increasing popularity of globalized production, the traditional centralized production planning and scheduling are no longer able to respond the growing market demand (Zhang et al., 2018). Since the market disperses throughout the world, the production model of various companies has changed from single factory to a multi-factory (Li, Yang, Ruiz, Chen, & Sui, 2018). Therefore, the scheduling problem in a distributed production environment has become one of the a hot research issues (Ruiz, Pan, & Naderi, 2019). In the distributed production environment, each factory is considered a separate entity, and all factories work in parallel. Each factory has the same or different configurations and efficiencies and is subject to the same or different constraints. The goal is to assign jobs to the appropriate factory for processing. The layout of the distributed production environment is shown in Fig. 1.
The scheduling in the distributed production environment is classified as distributed scheduling problem (DFSP) (Behnamian and Ghomi, 2016, Naderi and Ruiz, 2010). The parallel flowshop scheduling problem is a special instance of DFSP (He et al., 1996, Ribas et al., 2019). The scheduling of jobs in a factory with several identical parallel production lines is considered in parallel flowshop scheduling problem. If one production line is existed in the production, the DFSP is the parallel flowshop scheduling problem (Ribas et al., 2019). Compared to the single production center, the distributed production environment has lower production costs and risks, and the quality of product was improved (Naderi & Ruiz, 2014).
In the different types of scheduling problems, the DFSP has attracted attention from researchers and practitioners in the scheduling domain (Maccarthy & Liu, 1993). The distributed permutation flow-shop scheduling problem (DPFSP) was first presented by Naderi and Ruiz (2010), which extended the tradition permutation flowshop scheduling problem (PFSP) to the distributed production environment. Following the work of DPFSP, several researchers proposed various high-performing algorithms for solving the DPFSP. A bounded-search iterated greedy (BIG) algorithm was proposed by Fernandez and Framinan (2015). A new heuristic, iterated greedy (IG) algorithm, was introduced to improve the performance of algorithm. Three different constructive heuristics (DLR, DNEH and DLR-DNEH(x)) and four metaheuristics (DABC, SS, ILS and IG) were proposed by Pan, Gao, Wang, et al. (2019) for solving the DPFSP. Ruiz et al. (2019) proposed an enhanced IG algorithm called IG2S. In IG2S, the initialization, construction and destruction procedures are the intensification mechanisms to improve the local search ability of algorithm.
In addition, other types of DFSP including the distributed no-wait flowshop scheduling problem (DNWFSP) (Lin & Ying, 2016), the distributed no-idle permutation flow-shop scheduling problem (DNIPFSP) (Ying, Lin, Cheng, & He, 2017), the distributed two-stage assembly flow-shop scheduling problem (DTSAFSP) (Zhang & Xing, 2018), and the energy-efficient scheduling of the distributed permutation flow-shop problem (EEDPFSP) (Wang & Wang, 2018), have been investigated in recent years.
If no buffers between two adjacent machines, the FSP are evolved into blocking flowshop scheduling problems (BFSP). In the BFSP, jobs completed on the previous machine are blocked only on the current machine until the next machine is idle. The BFSP is an common scheduling scenarios in modern manufacturing and production systems, e.g., the iron and steel industry (Gong, Tang, & Duin, 2010), manufacturing systems (Zhao, Qin, Yang, et al., 2019), chemical industry (Merchan & Maravelias, 2016), manufacture of metallic parts (Shao, Pi, & Shao, 2019), assembly lines (Zhao, Xue, et al., 2018), and robotic cell (Ribas, Companys, & Tort-Martorell, 2015). BFSP with more than two machines has been proved as a NP-hard problem (Hall & Sriskandarajah, 1996).
Due to the complex nature of the traditional BFSP, exact methods are used only to solve small scale instances. For large-scale and moderate size problems, the heuristic algorithm and meta-heuristic algorithm have a satisfactory performance to find a desirable solution than exact methods. In the past few decades, the heuristic algorithm and meta-heuristic algorithm have been proven as an effective way to solve the BFSP, such as the profile fitting (PF) heuristic (McCormick, Pinedo, Shenker, & Wolf, 1989), the minimax (MM) heuristic (Ronconi, 2004), the Nawaz-Enscore-Ham (NEH) heuristic (Nawaz, Enscore, Emory, & Ham, 1983) and the branch-and-bound (b&b) algorithm (Ronconi & Armentano, 2001).
With the development of technology and science, the production model of various enterprise has changed from the single factory to the multi-factory (Li et al., 2018). The multi-factory production model improves the efficiency of production, increases the dependability of system and the flexibility of process simultaneous. Therefore, the distributed scheduling problem has become as a popular research topic (Pan, Gao, Li, et al., 2019). Recently, the distributed blocking flowshop scheduling problems (DBFSP) have attracted comprehensive attention from researchers and practitioner in various fields. The solution methods including heuristics and meta-heuristics (Shao, Pi, & Shao, 2020), have been proposed to solve the DBFSP. The heuristic has been testified as an effective way to solve the flowshop scheduling problem (Fernandez-Viagas et al., 2018). An iterated greedy algorithm, which is named IGA, is applied to solve the parallel flow shop scheduling problem (Ribas et al., 2019). A novel hybrid discrete differential evolution (HDDE) algorithm, which is based on the characteristics of discreteness and distribution features, is proposed by L. Wang, Pan, Suganthan, Wang, and Wang (2010) to solve the BFSP. A revised artificial immune system (RAIS) algorithm is presented by Lin and Ying (2013) for the BFSP. A populated local search strategy is introduced in differential evolution (DE_PLS) proposed by Tasgetiren, Pan, Kizilay, and Suer (2015) to enhance the exploitation and exploration of the DE_PLS for solving the DBFSP. In the latest study, Zhang et al. (2018) proposed a discrete DE (DDE) algorithm for the DBFSP with makespan minimization. Furthermore, Zhang and Xing (2019) proposed an enhanced DDE (CDE) algorithm for the distributed limited-buffer flowshop scheduling problem.
Although various algorithms have been proposed for solving the DBFSP, the optimal solution of the DBFSP is still a difficult problem to obtain the satisfactory solution sets in desirable time, especially for the large-scale instances (Pan & Wang, 2011). As a population-based metaheuristic algorithm, DE has the advantages of strong versatility, simple and effective mechanism to solve discrete combinatorial optimization problems. The application of DE algorithm for the scheduling problems is mainly through two schemes including encoding the scheduling solutions as numerical vectors and discretizing the evolutionary operators as the permutation representation.
In this paper, an ensemble discrete differential evolution algorithm (EDE) is proposed to solve the DBFSP with minimizing the makespan criterion. The main contributions in this paper are listed as follows.
- •
The coding and decoding mechanism of a classic single-shop scheduling solution is extended as a novel representation rule for multi-shop scheduling solutions. Furthermore, unlike traditional greedy selection strategy of DE, bias selection and elitism strategies are adopted to improve the global search capability of EDE.
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The problem-related implicit properties are extracted out as a prior knowledge to construct a framework to provide a relatively reliable initial solution for the EDE.
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The two competitive discrete mutation strategies, called and , are discretized and integrated into the EDE to balance the exploration and exploitation of the algorithm.
The remainder of this paper is organized as follows. The definition and details about the DBFSP are introduced in Section 2. The basic DE algorithm and the proposed EDE algorithm are described in Section 3. The numerical experiments and comparisons are presented in Section 4. The conclusion and some future works are suggested in Section 5.
Section snippets
Problem description
The blocking flowshop scheduling problem (BFSP) is first described and analyzed with Graham, Lawler, Lenstra, Kan, and Rinnooy. (1979). jobs from set have to be sequenced through machines from set without intermediated buffers. The processing of job requires a set of operations , where is executed on machine with processing time , job having completed operation , has to remain on until becomes idle. All
EDE for DBFSP with minimizing makespan
To simplify formulation exposition, the following notations are used:Number of machines Number of jobs Number of factories Time level, Index of machines Index of jobs Time level Maximum elapsed CPU time Maximal completion time of the last job to be processed in any factory Number of current generations is an exclusive and randomly selected number from , for different Best individual in the population in a generation Randomly selected number from the top
Experiments and comparisons
In this section, to test the performance of the proposed EDE algorithm, a numerical experiment and comparison with other four state-of-the-art algorithms are presented based on the adaptation of the well-known flowshop benchmark set of Taillard (1993), as it was proposed in Naderi and Ruiz (2010). Afterwards, the parameters are calibrated with the DOE method. Finally, the comprehensive details of all the experiments are presented out in the following sections.
Conclusions and future research
In this paper, an ensemble different evolution (EDE) algorithm is presented to solve the distributed blocking flowshop scheduling problem with the objective of minimizing makespan. First, a large number of experimental results show that the hidden knowledge from knowledge specific problem is used to design efficient scheduling rules by constructing the initial solution, which is significantly better than those methods that do not use domain knowledge. Second, the integration of different
CRediT authorship contribution statement
Fuqing Zhao: Conceptualization, Funding acquisition, Supervision, Writing - review & editing. Lexi Zhao: Data curation, Formal analysis, Writing - original draft. Ling Wang: Methodology, Resources. Houbin Song: Investigation, Resources, Software, Validation.
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.
Acknowledgement
This work was financially supported by the National Key Research and Development Plan under grant number 2018YFB1703105 and National Natural Science Foundation of China under grant numbers 61663023 and 61873328, China. It is also supported by the Key Research Programs of Science and Technology Commission Foundation of Gansu Province (2017GS10817), Lanzhou Science Bureau project (2018-rc-98), Public Welfare Project of Zhejiang Natural Science Foundation (LGJ19E050001) and Wenzhou Public Welfare
References (49)
- et al.
Variable neighborhood algebraic Differential Evolution: An application to the Linear Ordering Problem with Cumulative Costs
Information Sciences
(2020) - et al.
The distributed permutation flow shop to minimise the total flowtime
Computers and Industrial Engineering
(2018) - et al.
A two-stage flow shop scheduling problem on a batching machine and a discrete machine with blocking and shared setup times
Computers and Operations Research
(2010) - et al.
Optimization and approximation in deterministic sequencing and scheduling: A survey
Annals of Discrete Mathematics
(1979) - et al.
An iterated greedy heuristic for no-wait flow shops with sequence dependent setup times, learning and forgetting effects
Information Sciences
(2018) - et al.
Minimizing makespan in a blocking flowshop using a revised artificial immune system algorithm
Omega
(2013) - et al.
Minimizing makespan for solving the distributed no-wait flowshop scheduling problem
Computers and Industrial Engineering
(2016) - et al.
Preprocessing and tightening methods for time-indexed MIP chemical production scheduling models
Computers and Chemical Engineering
(2016) - et al.
The distributed permutation flowshop scheduling problem
Computers and Operations Research
(2010) - et al.
A scatter search algorithm for the distributed permutation flowshop scheduling problem
European Journal of Operational Research
(2014)
A heuristic algorithm for the m -machine, n -job flow-shop sequencing problem
Omega
Effective heuristics and metaheuristics to minimize total flowtime for the distributed permutation flowshop problem
Expert Systems with Applications
Effective constructive heuristics and meta-heuristics for the distributed assembly permutation flowshop scheduling problem
Applied Soft Computing
An efficient Discrete Artificial Bee Colony algorithm for the blocking flow shop problem with total flowtime minimization
Expert Systems with Applications
Efficient heuristics for the parallel blocking flow shop scheduling problem
Expert Systems with Applications
An Iterated Greedy Algorithm for Solving the Total Tardiness Parallel Blocking Flow Shop Scheduling Problem
Expert Systems with Applications
A note on constructive heuristics for the flowshop problem with blocking
International Journal of Production Economics
Iterated Greedy methods for the distributed permutation flowshop scheduling problem
Omega
A novel multi-objective discrete water wave optimization for solving multi-objective blocking flow-shop scheduling problem
Knowledge-Based Systems
An efficient discrete invasive weed optimization for blocking flow-shop scheduling problem
Engineering Applications of Artificial Intelligence
Benchmarks for basic scheduling problems
European Journal of Operational Research
New hard benchmark for flowshop scheduling problems minimising makespan
European Journal of Operational Research
A knowledge-based cooperative algorithm for energy-efficient scheduling of distributed flow-shop
IEEE Transactions on Systems Man and Cybernetics Systems
A novel hybrid discrete differential evolution algorithm for blocking flow shop scheduling problems
Computers and Operations Research
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