Discrete differential evolution algorithm for distributed blocking flowshop scheduling with makespan criterion

https://doi.org/10.1016/j.engappai.2018.09.005Get rights and content

Highlights

  • A distributed blocking flowshop scheduling problem is solved by meta-heuristics.

  • A discrete differential evolution is developed by discreteness and distribution.

  • New evolutionary operators are designed to perform DDE directly on discrete domain.

  • An efficient local search with speed-up evaluation is integrated into DDE framework.

  • The algorithmic performance is validated from both search quality and efficiency.

Abstract

This paper deals with a distributed blocking flowshop scheduling problem, which tries to solve the blocking flowshop scheduling in distributed manufacturing environment. The optimization objective is to find a suitable schedule, consisting of assigning jobs to at least two factories and sequencing the jobs assigned to each factory, to make the maximum completion time or makespan minimization. Two different mathematical models are proposed, and in view of the NP-hardness of the problem, a novel hybrid discrete differential evolution (DDE) algorithm is established. First, the problem solution is represented as several job permutations, each of which denotes the partial schedule at a certain factory. Second, four widely applied heuristics are generalized to the distributed environment for providing better initial solutions. Third, both operators of mutation and crossover are redesigned to perform the DDE directly based on the discrete permutations, and a biased section operator is used to increase the diversity of the searching information. Meanwhile, an effective local search based on distributed characteristics and an elitist retain strategy are integrated into the DDE framework to stress both local exploitation and global exploration. Taking into account the time cost, an effective speed-up technique is designed to enhance the algorithmic efficiency. In the experimental section, the parameters of DDE are calibrated by the Taguchi method. Experimental results derived from a wealth of test instances have demonstrated the algorithmic effectiveness, which further concludes that the proposed DDE algorithm is a suitable alternative approach for solving the problem under consideration.

Introduction

Production scheduling addresses the assignment of resources, typically machines, to tasks or jobs over time for optimizing a certain objective. The type of a production scheduling problem is determined by several factors: the layout of machines, the flow of jobs on the machines as well as some other production constraints (Pinedo, 2012). Among different types of scheduling problems, the flowshop scheduling is one of the most extensively studied scheduling problems with a strong engineering application. Its production prototypes can be found in a rather wide range of industries Garey et al. (1976), Gupta and Stafford Jr (2006), Li et al. (2015), Nawaz et al. (1983), Hejazi and Saghafian (2005), Ruiz and Maroto (2005), such as various manufacturing systems, assembly lines and information service facilities. When there is no storage capacity between any two adjoining machines, the classical flowshop scheduling evolves into the blocking flowshop scheduling problem Grabowski and Pempera (2000), Hall and Sriskandarajsh (1996). In this situation, a job, having completed on a machine, has to be blocked on current machine until next machine becomes idle. The applications on such a scheduling problem abound in the production environments, where the buffers amongst machines either do not deploy due to insufficient investment or have to be prohibited to utilize since some special technological requirements.

For the scheduling problems mentioned above, it is assumed that the processing of all jobs is conducted in the same factory, that is to say, in a single production center. Nevertheless, the real-life problems, concurrent or mass production emerged recently, have necessitated the patterns of distributed or multi-factory production. This machine environment is able to allocate the total tasks among some independent production units, which enables enterprises to harvest a lot of potential benefits, including higher product quality, better corporate reputation, lower production costs and manufacturing period (Behnamian and Fatemi Ghomi, 2016). At present, several different types of distributed manufacturing systems have been proposed and surveyed. Among them, there are distributed flowshop system (Naderi and Ruiz, 2010), distributed jobshop system Hsu et al. (2016), Naderi and Azab (2014), distributed flexible manufacturing system Chan et al. (2006b), Chan et al. (2006a), and distributed two-stage assembly system (Xiong et al., 2014).

In this paper, we have tackled a new distributed blocking flowshop scheduling problem (DBFSP) with makespan criterion, which focused on solving the blocking flowshop scheduling problem in a distributed environment. To the authors’ knowledge, at present there is only one journal paper (Ying and Lin, 2017) having been published for such a problem. In addition, it can be proved that it is a NP-hard problem since the single factory case with more than two machines is strongly NP-complete (Hall and Sriskandarajsh, 1996). Therefore, it is unrealistic to use the traditional mathematical methods to solve the DBFSP, especially for solving the large-scale problems. In order to obtain the solutions with both quality and efficiency, this paper establishes a novel and effective discrete differential evolution (DDE) algorithm.

Other components of this paper are arranged as follows. Section 2 systemically reviews the relevant literature. Section 3 devotes to illustrate the DBFSP. The details of DDE algorithm are presented in Section 4. Numerical experiments and statistical analyses are provided in Section 5. Finally, conclusions and future research topics are showed in Section 6.

Section snippets

Literature review

For several decades now, the typical blocking flowshop scheduling problem has been extensively focused on, and a myriad of solution methods, mainly approximation approaches such as the heuristics McCormick et al. (1989), Pan and Wang (2012), Ronconi (2004), Ronconi and Armentano (2001), Ronconi and Henriques (2009) and the meta-heuristics Caraffa et al. (2001), Grabowski and Pempera (2007), Liang et al. (2010), Ribas et al. (2011), Tasgetiren et al. (2017), Wang et al. (2010), Wang and Tang

Problem description

The blocking flowshop scheduling problem (BFSP) is first described as follows. There is a set J={Jj|(j=1,2,,n)} of n jobs to be processed on a set M={Mi|(i=1,2,,m)} of m machines disposed in a certain order. The processing of job Jj requires a set of m operations Oj={Oj,i|(i=1,2,,m)}, where Oj,i is executed on machine Mi with a processing time Pj,i. Without intermediate buffer between machines, job Jj, having completed operation Oj,i, has to remain on Mi until Mi+1 becomes idle. Both

Basic DE algorithm

Basic DE algorithm is a stochastic metaheuristic method. Based on floating-point representation, it utilizes the differential information among individuals to conduct the global optimization in the continuous domain. Starting from the target population at each generation, it orderly adopts a mutation operator to generate mutant population, a crossover operator to construct trial population, and a greedy selection operator to determine new target population at next generation. Let Xi(t)=[xi1(t),x

Experimental setup

In order to evaluate the algorithmic performance, the numerical experiments are conducted based on two sets of testing instances originally proposed for dealing with the infinite-buffer distributed flowshop scheduling problems (Naderi and Ruiz, 2010). The first set consists of the small-scale instances, in which the number of factories f{2,3,4}, the number of jobs n{4,6,8,10,12,14,16}, and the number of machines at each factory m{2,3,4,5}. Therefore, there are 84 combinations of f×n×m.

Conclusions and future researches

This paper addresses a novel distributed blocking flowshop scheduling problem (DBFSP) with makespan criterion, in which the authors tries to schedule the blocking flowshops in the distributed environment. It is a complicated scheduling problem, and to deal with it, two mathematical models are first proposed. Then, an effective hybrid discrete differential evolution (DDE) algorithm is established based on the integration of several advanced techniques, which include a solution representation

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under Grants 61573278.

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