Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2857564, 20 pages

https://doi.org/10.1155/2017/2857564

## A Novel Memetic Algorithm Based on Decomposition for Multiobjective Flexible Job Shop Scheduling Problem

Engineering Research Center of IoT Technology Applications, Ministry of Education, Jiangnan University, Wuxi 214122, China

Correspondence should be addressed to Yan Wang

Received 12 May 2017; Revised 1 November 2017; Accepted 9 November 2017; Published 29 November 2017

Academic Editor: Josefa Mula

Copyright © 2017 Chun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A novel multiobjective memetic algorithm based on decomposition (MOMAD) is proposed to solve multiobjective flexible job shop scheduling problem (MOFJSP), which simultaneously minimizes makespan, total workload, and critical workload. Firstly, a population is initialized by employing an integration of different machine assignment and operation sequencing strategies. Secondly, multiobjective memetic algorithm based on decomposition is presented by introducing a local search to MOEA/D. The Tchebycheff approach of MOEA/D converts the three-objective optimization problem to several single-objective optimization subproblems, and the weight vectors are grouped by* K*-means clustering. Some good individuals corresponding to different weight vectors are selected by the tournament mechanism of a local search. In the experiments, the influence of three different aggregation functions is first studied. Moreover, the effect of the proposed local search is investigated. Finally, MOMAD is compared with eight state-of-the-art algorithms on a series of well-known benchmark instances and the experimental results show that the proposed algorithm outperforms or at least has comparative performance to the other algorithms.

#### 1. Introduction

The job shop scheduling problem (JSP) is one of the most important and difficult problems in the field of manufacturing which processes a set of jobs on a set of machines. Each job consists of a sequence of successive operations, and each operation is allowed to process on a unique machine. Different from JSP which one operation is merely allowed to process on a specific machine, the flexible job shop scheduling problem (FJSP) permits one operation processed by any machine from its available machine set. Since FJSP needs to assign operations to their suited machine as well as sequence those operations assigned on the same machine, it is a complex NP-hard optimization problem [1].

The existing literatures [2–5] about solving single-objective FJSP (SOFJSP) over the past decades mainly concentrated on minimizing one specific objective such as makespan. However, in practical manufacturing process, single-objective optimization cannot fully satisfy the production requirements since many optimized objectives are usually in conflict with each other. In recent years, multiobjective flexible job shop scheduling problem (MOFJSP) has received much attention, and, until now, many algorithms have been developed to solve this kind of problem. These methods can be classified into two groups: one is a priori approach and the other is Pareto approach.

Multiple objectives are usually linearly combined into a single one by weighted sum approach in the a priori method, which can be illustrated as , where , . However, we can get only one or several Pareto solutions by using this approach, which may not well reflect the tradeoffs among different objectives, and it would be difficult to assign an appropriate weight for each problem. Even more important, the performance of the algorithm deteriorates when solving the problems contains nonconcave Pareto front (PF). The Pareto approach mainly focuses on searching the Pareto set (PS) of optimization problems by comparing two solutions based on Pareto dominance relation [6]. A solution is said to dominate solution iff is not worse than in all objectives and there exists at least one objective in which is better than . is called a Pareto optimal solution iff there is no solution that dominates . All the Pareto optimal solutions constitute the PS, and PF is the mapped vector of PS in the objective space. Since Pareto approach can achieve a set of Pareto solutions rather than a specific one, it has received much more attention than a priori approach and is recognized to be more suitable to solve MOFJSP.

Because the three objectives, makespan, total workload, and critical workload, are conflicted with each other, it is better to handle this model with knowledge about their PF. Multiobjective evolutionary algorithm (MOEA) is a kind of mature global optimization method with high robustness and wide applicability. Due to the fact that MOEAs have low requirements on the optimization problem itself and high ability to obtain multiple Pareto solutions during each run, they are suitable for solving multiobjective optimization problems (MOPs). The multiobjective evolutionary algorithm based on decomposition (MOEA/D) that integrates mathematical programming with evolutionary algorithm (EA) can obtain a set of Pareto solutions by aggregating multiple objectives into different single-objectives with many predefined weight vectors [7]. MOEA/D has shown great superiority on continuous optimization problems [8–12]; thus it is necessary to investigate its performance on multiobjective combinatorial optimization problems (MO-COPs) such as MOFJSP. To the best of our knowledge, in the literature reported, although MOEA/D has been applied to different kinds of multiobjective scheduling problems such as multiobjective flow shop scheduling problem (MOFSP) [13], multiobjective permutation flow shop scheduling problem (MOPFSP) [14], multiobjective stochastic flexible job shop scheduling problem (MOSFJSP) [15], and multiobjective job shop scheduling problem (MOJSP) [16], there is seldom corresponding application on MOFJSP.

The primary aim of this paper is to solve MOFJSP in a decomposition manner by proposing a multiobjective memetic algorithm based on decomposition (MOMAD) hybridizing MOEA/D with local search. With the purpose of making the proposed algorithm more applicable, four aspects are studied: integration of different machine assignment and operation sequencing strategies are presented to construct the initial population; objective normalization is incorporated into Tchebycheff approach to convert an MOP into a number of single-objective optimization subproblems; all weight vectors are divided into a few groups based on* K*-means clustering: some superior individuals corresponding to different weight vector groups are selected by using a selection mechanism; local search based on moving critical operations is applied on selected individuals. To evaluate the effectiveness of the proposed algorithm, some benchmark instances are tested with three purposes: investigating the effects of different aggregation functions and validating the effectiveness of local search; analyzing the influence of the key parameters on the performance of the algorithm; comparing the performance of MOMAD with other state-of-the-art algorithms for solving MOFJSP.

The rest of the paper is organized as follows. Next section presents a short overview of the existing related work. In Section 3, the background knowledge of MOFJSP is introduced. Section 4 introduces the framework of MOMAD. The implementation details of the proposed MOMAD including genetic global search and problem specific local search are described in Section 5. Afterwards, experimental studies are provided in Section 6. Finally, Section 7 concludes this paper and outlines some avenues for future research.

#### 2. Related Work

As mentioned above, there are two main methods to solve MOFJSP: a priori approach and Pareto approach. As for a priori approach, Xia and Wu [17] discussed a hybrid algorithm, where particle swarm optimization (PSO) and simulated annealing (SA) were employed in global search and local search, respectively. A bottleneck shifting-based local search was incorporated into genetic global search by Gao et al. [18]. Zhang et al. [19] introduced an effective hybrid PSO algorithm combined with tabu search (TS). Xing et al. [20] used ten different weight vectors to collect effective solution sets. A hybrid TS (HTS) algorithm was structured by combining adaptive rules with two neighborhoods. In this algorithm, three weight coefficients , , and with different settings were given to test different problems [21]. An effective estimation of distribution algorithm was proposed by Wang et al. [22], in which the new individuals were generated by sampling a probability model.

Contrary to the a priori approach, a PS can be obtained by using Pareto approach, and the tradeoffs among different objectives can be presented. The integration of fuzzy logic and EA was proposed by Kacem et al. [23]. A guide local search was incorporated into EA to enhance the convergence [24]. With the aim of keeping population diversity, immune and entropy principle were adopted in multiobjective genetic algorithm (MOGA) [25]. Two memetic algorithms (MAs) were, respectively, proposed, both of which integrate nondominated sorting genetic algorithm II (NSGA-II) [6] with effective local search techniques [26, 27]. Several effective neighborhood approaches were used in variable neighborhood search to enhance the convergence ability in a hybrid Pareto-based local search (PLS) [28]. Chiang and Lin [29] proposed a simple and effective evolutionary algorithm which only requires two parameters. Both the neighborhoods of machine assignment and operation sequence are considered in Xiong et al. [30]. An effective Pareto-based EDA was proposed by Wang et al. [31]. A novel path-relinking multiobjective TS algorithm was proposed in [32], in which a routing solution is identified by problem-specific neighborhood search and is then further refined by the TS with back-jump tracking for a sequencing decision. In addition to the successful use of EA, several swarm intelligence algorithms have also been widely used for global search. PSOs were used as global search algorithms in [33–36]. Besides, shuffled frog leaping [37] and artificial bee colony [38] were integrated with local search in related hybrid algorithms.

Besides the successful using in many scheduling problems, MOEA/Ds have also been widely dedicated to other MO-COPs. A novel NBI-style Tchebycheff approach was used in MOEA/D to solve portfolio management MOP with disparately scaled objectives [39]. Mei et al. [40] developed an effective MA by combining MOEA/D with extended neighborhood search to solve capacitated arc routing problem. Hill climbing, SA, and evolutionary gradient search were, respectively, embedded into EDA for solving multiple traveling salesmen problem (MTSP) in a decomposition manner [41]. A hybrid MOEA was established by combining ant colony optimization with MOEA/D [42], and then it was adopted to solve multiobjective 0-1 knapsack problem (MOKP) and MTSP, respectively. Then, aiming at the same two problems, Ke and Zhang proposed a hybridization of decomposition and PLS [43].

As mentioned before, MOEA/D is a kind of popular MOEA which is very suitable for solving MO-COPs such as scheduling problem. In this paper, a MOMAD is proposed that integrates MOEA/D algorithm with local search to enrich the tool-kit for solving MOFJSP.

#### 3. Related Background Knowledge

##### 3.1. Problem and Objective of MOFJSP

The MOFJSP can be formulated as follows. There are a set of jobs and machines ; each job contains one or more operations to be processed in accordance with the predetermined sequence. Each operation can be processed on any machine among its corresponding operable machine set . The problem is defined as T-FJSP iff ; otherwise, it is called P-FJSP [44]. MOFJSP not only assigns suitable processing machines for each operation but also determines the most reasonable processing sequence of operations assigned on the same machine in order to simultaneously optimize several objectives.

The following constraints should be satisfied in the process:(1)At a certain time, a machine can process one operation at most, and one operation can be processed by only one machine at a certain moment.(2)Each operation cannot be interrupted once processed.(3)All jobs and machines are available at time 0.(4)Different jobs share the same priority.(5)There exists no precedence constraint among the operations of different jobs, but there exist precedence constraints among the operations belonging to the same job.

An instance of P-FJSP with three jobs and three machines is illustrated in Table 1. Let and be the completion time of operation and its processing time on machine , respectively. denotes the completion time of job . Three considered objectives are makespan, total workload, and critical workload which are formulated as follows: