Solving energy-efficient distributed job shop scheduling via multi-objective evolutionary algorithm with decomposition
Introduction
The rapid development of globalization makes many manufacturing industries to shift from the traditional single-factory mode to the distributed multiple-factory mode. The distributed manufacturing mode is much closer to market with the advantages in terms of lower labor cost and higher production efficiency. However, the planning and scheduling processes in the distributed systems are more complex than those in the single-factory systems. When solving the distributed scheduling problem, it needs to consider the sub-problems both in each factory and among different factories. Since these sub-problems are strongly coupled, it is full of great challenge in designing effective approaches to obtain satisfactory solutions efficiently.
Currently, the research of the distributed scheduling has attracted increasing attention. For the distributed permutation flow shop scheduling problem, Naderi and Ruiz presented six mathematical models, several heuristics [1] and a scatter search method with restarts and local search [2] to minimize makespan. Estimation of distribution algorithm (EDA) [3] and iterated greedy algorithm [4] were also proposed to solve the same problem. In Refs. [5], three heuristics and four metaheuristics were designed to minimize total flowtime. In Ref. [6], a competitive memetic algorithm was proposed to minimize both makespan and total tardiness simultaneously. In Ref. [7], a knowledge based cooperative algorithm was proposed to minimize both makespan and total energy consumption. For other generalized problems with makespan minimization, Zhang et al. [8] presented a discrete differential evolution algorithm for the distributed blocking flow shop scheduling problem. Wang and Wang [9] proposed an EDA based memetic algorithm for the distributed assembly permutation flow shop scheduling problem. Meng et al. [10] proposed three metaheuristics for the distributed permutation flow shop scheduling problem with the customer order constraint. Recently, Chen et al. [11] proposed a collaborative optimization algorithm for the energy-efficient multi-objective distributed no-idle flow-shop scheduling with the criteria of minimizing makespan and total energy consumption.
For the distributed job shop scheduling problem, genetic algorithm (GA) [12], greedy heuristics [13], simulated annealing algorithm [14], agent-based fuzzy constraint-directed negotiation mechanism [15] and hybrid ant colony algorithm with dynamic assignment rule [16] were proposed to solve the problem with makespan minimization. For the distributed flexible job shop scheduling problem, an improved GA [17] and a hybrid GA [18] were designed to minimize makespan. Wu et al. compared the effects of five chromosome representations and proposed a GA to minimize makespan [19]. Recently, Meng et al. [20] presented four mixed integer linear programming (MILP) models and a constraint programming model for the same problem. In addition to the above research of single objective optimization, Wu et al. [21] proposed an improved differential evolution (DE) algorithm to minimize the tardiness and the total cost. Xie et al. [22] presented a multi-objective artificial bee colony algorithm to minimize makespan and total energy consumption.
Environmental protection is a serious issue in modern society. Manufacturing industries should stress sustainable development. Both economic benefits and environmental criteria should be considered simultaneously. Thus, the research of green scheduling is becoming a new focus, aiming at reducing energy consumption [23,24], decreasing carbon footprint [25] and reducing noise [26]. For the unrelated parallel machine scheduling problem, Zheng and Wang [27] presented a collaborative multi-objective fruit fly optimization algorithm to minimize makespan and carbon emissions. To minimize total electricity cost, Che et al. [28] presented a MILP model and proposed a two-stage heuristic. For the permutation flow shop scheduling with criteria of makespan and energy consumption, Jiang and Wang [29] proposed an improved multi-objective evolutionary algorithm based on decomposition for the problem with sequence-dependent setup time. For the energy-efficient job shop scheduling problem, Zhang and Chiong [30] proposed a multi-objective GA to minimize total weighted tardiness and total energy consumption. For the flexible job shop scheduling problem, Mokhtari and Hasani [31] proposed a multi-objective handling technique with the criteria of minimizing both makespan and energy consumption while maximizing the total availability of the system. For the flexible job shop scheduling problem with total energy consumption threshold, Lei et al. [32] proposed a two-phase metaheuristic to minimize makespan and total tardiness. Recently, Wang et al. [33] proposed a multi-objective whale swarm algorithm to minimize makespan and energy consumption for the distributed permutation flow shop scheduling problem.
Essentially, green scheduling problems are multi-objective problems (MOPs) since several objectives should be considered simultaneously. Most scheduling problems with even a single objective have been proved to be NP-hard. Obviously, it is more difficult to solve the green scheduling problems with multiple conflicting objectives. Those methods based on mathematical programming are not suitable to solve the large scale problems efficiently. Evolutionary computation and swarm intelligence provide some kinds of novel mechanisms to design effective meta-heuristic approaches for obtaining satisfactory solutions. Moreover, the population-based search framework is very suitable for solving multi-objective optimization problems to obtain a set of non-dominated solutions. So far, many algorithms based on swarm and evolutionary computation have been designed for a variety of multi-objective scheduling problems, including GA [25], DE [21], and EDA [34]. Please refer to the survey [35] about the research of multi-objective evolutionary algorithm (MOEA). This paper addresses the energy-efficient distributed job shop scheduling problem (EEDJSP) to minimize makespan and total energy consumption simultaneously. As we know, multi-objective evolutionary algorithm based on decomposition (MOEA/D) [36] has been recognized as an effective optimization framework for multi-objective optimization, which has been successfully applied to solve some scheduling problems [29,37]. Different from the original MOEA/D, in this paper we design some special search components in the framework of the MOEA/D according to the characteristics of the EEDJSP. With the problem-specific encoding and decoding schemes, four-heuristic fused initialization, collaborative search, adaptive local search and energy adjustment strategy, our algorithm is well designed for solving the EEDJSP to obtain high quality Pareto solutions. Numerical tests and statistical comparisons demonstrate the effectiveness of our designs. Better results can be achieved than other algorithms.
The remaining contents are organized as follows. Section 2 presents the problem formulation. The detail design of the algorithms is presented in Section 3. Section 4 provides the numerical results and comparisons. Finally, we end the paper with some conclusions and future work in Section 5.
Section snippets
Description of EEDJSP
The EEDJSP is described as follows. There are n jobs to be processed in f identical factories. Each factory consists of m machines, and each machine can run at s different speeds. Each job consists of m operations with a processing route given in advance that includes all machines. Once a job is assigned to a factory, it cannot be transferred to other factories. Each machine can process at most one job at a time and its running speed cannot be changed during processing any operations. There is
Procedure of MMOEA/D
The procedure of our modified multi-objective evolutionary algorithm with decomposition (MMOEA/D) is given as Algorithm 1.
The MMOEA/D includes several stages. In initialization stage, the reference point is determined by the initialized population with several heuristics. Moreover, the weight vectors are initialized randomly, and then the neighbours of each solution are determined accordingly. In every generation, collaborative search procedure, local
Experimental settings
To demonstrate the effectiveness of the MMOEA/D, we use the set of testing instances in Ref. [13], which includes 32 combinations of (f, n, m). And each combination includes 10 instances. So, there are 320 testing instances in total. The processing speed is selected from . For all the following experiments, the stopping criterion is set with the running time of (seconds) and the population size is set as 200. All the algorithms are coded with C++ on
Conclusion and future work
This is the first research work to solve the energy-efficient distributed job shop scheduling problem minimizing both makespan and total energy consumption simultaneously via multi-objective evolutionary algorithm with decomposition. By comparing to other algorithms based on extensive numerical tests, it is demonstrated that the proposed MMOEA/D is more effective in solving the EEDJSP. Through this research work, it shows that the specially designed collaborative search and local
Credit author statement
En-da Jiang: Data curation, Methodology, Formal analysis, Writing-original draft. Ling Wang: Conceptualization, Methodology, Funding acquisition, Writing-review. Zhi-ping Peng: Data curation, Funding acquisition.
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.
Acknowledgments
This research is supported by the National Natural Science Fund for Distinguished Young Scholars of China (No. 61525304), the National Natural Science Foundation of China [grant number 61873328 and 61772145] and Tsinghua University Tutor Research Fund.
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