An ensemble discrete differential evolution for the distributed blocking flowshop scheduling with minimizing makespan criterion

Highlights

• An ensemble discrete differential evolution (EDE) algorithm is proposed.

• Two different heuristic methods and one random strategy are introduced into EDE.

• The mutation, crossover and selection operators are redesigned to assist the EDE algorithm.

• An elitist retain strategy is introduced into the framework of EDE algorithm.

• The parameters of the EDE algorithm are calibrated by the design of experiments (DOE) method.

Abstract

The distributed blocking flowshop scheduling problem (DBFSP) plays an essential role in the manufacturing industry and has been proven to be as a strongly NP-hard problem. In this paper, an ensemble discrete differential evolution (EDE) algorithm is proposed to solve the blocking flowshop scheduling problem with the minimization of the makespan in the distributed manufacturing environment. In the EDE algorithm, the candidates are represented as discrete job permutations. Two heuristics method and one random strategy are integrated to provide a set of desirable initial solution for the distributed environment. The front delay, blocking time and idle time are considered in these heuristics methods. The mutation, crossover and selection operators are redesigned to assist the EDE algorithm to execute in the discrete domain. Meanwhile, an elitist retain strategy is introduced into the framework of EDE algorithm to balance the exploitation and exploration ability of the EDE algorithm. The parameters of the EDE algorithm are calibrated by the design of experiments (DOE) method. The computational results and comparisons demonstrated the efficiency and effectiveness of the EDE algorithm for the distributed blocking flowshop scheduling problem.

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.

• 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.

• The two competitive discrete mutation strategies, called $\mathit{DE}/rand/1$ and $\mathit{DE}/current-to-pbest/1$, 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.