Computational Intelligence and Neuroscience

Volume 2017 (2017), Article ID 5232518, 20 pages

https://doi.org/10.1155/2017/5232518

## A Bee Evolutionary Guiding Nondominated Sorting Genetic Algorithm II for Multiobjective Flexible Job-Shop Scheduling

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China

Correspondence should be addressed to Guiliang Gong; moc.361@gnailiug_gnog

Received 11 November 2016; Accepted 20 February 2017; Published 28 March 2017

Academic Editor: J. Alfredo Hernández-Pérez

Copyright © 2017 Qianwang Deng 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

Flexible job-shop scheduling problem (FJSP) is an NP-hard puzzle which inherits the job-shop scheduling problem (JSP) characteristics. This paper presents a bee evolutionary guiding nondominated sorting genetic algorithm II (BEG-NSGA-II) for multiobjective FJSP (MO-FJSP) with the objectives to minimize the maximal completion time, the workload of the most loaded machine, and the total workload of all machines. It adopts a two-stage optimization mechanism during the optimizing process. In the first stage, the NSGA-II algorithm with iteration times is first used to obtain the initial population , in which a bee evolutionary guiding scheme is presented to exploit the solution space extensively. In the second stage, the NSGA-II algorithm with* GEN* iteration times is used again to obtain the Pareto-optimal solutions. In order to enhance the searching ability and avoid the premature convergence, an updating mechanism is employed in this stage. More specifically, its population consists of three parts, and each of them changes with the iteration times. What is more, numerical simulations are carried out which are based on some published benchmark instances. Finally, the effectiveness of the proposed BEG-NSGA-II algorithm is shown by comparing the experimental results and the results of some well-known algorithms already existed.

#### 1. Introduction

As a part of production scheduling and combinatorial optimization problems, job-shop scheduling problem (JSP) attracts more and more researchers from all walks of life (e.g., mechanical engineering, mathematics, and computer software engineering) in the recent decades [1–5].

Flexible job-shop scheduling problem (FJSP) inherits the characteristics of the JSP, in which each operation is allowed to be processed by any machine in a given set rather than one specified machine, and it has been proved that the FJSP is strong NP-hard [6]. FJSP consists of two subproblems: one is the routing subproblem that each operation is assigned to one machine of a set of machines, and the other is the scheduling subproblem that a feasible schedule is obtained by sequencing the assigned operations on all machines. Therefore the FJSP is more difficult to be solved than the classical JSP because of its need to determine the assignment of operations in related machines [7].

The FJSP is firstly addressed by Brucker and Schlie [8]. And they presented a polynomial algorithm with only two jobs and identical machines. Brandimarte [9] proposed a hybrid tabu search (TS) algorithm which was based on decomposition to solve the FJSP. Dauzère-Pérès and Paulli [10] provided a TS algorithm which was based on the integrated approach and developed a new neighbourhood function [11] for the FJSP in terms of solution quality and computation time. Gao et al. [12] proposed a hybrid genetic algorithm to solve the FJSP with nonfixed availability constraints. And in order to enhance the inheritability, this genetic algorithm uses an innovative representation method and applies genetic operations to phenotype space. Saidi-Mehrabad and Fattahi [13] developed a TS algorithm that took the operation sequences and sequence-dependent setups into consideration to solve the FJSP. A genetic algorithm (GA) combined with a variable neighbourhood search (VNS) was presented by Gao et al. [14], and a GA with different strategies was proposed by Pezzella et al. [15]. Yazdani et al. [16] developed a parallel VNS algorithm based on the application of multiple independent searches which increased the exploration of search space. Xing et al. [17] put forth a knowledge-based ant colony optimization algorithm. Recently, a novel artificial bee colony (ABC) algorithm [18] and a discrete harmony search (DHS) algorithm [19] were brought forward to solve the FJSP.

As is shown above, the single-objective optimization of FJSP (SO-FJSP) has been extensively studied, which generally minimizes the makespan that is the time required to complete all jobs. However, many industries (e.g., aircraft, semiconductors manufacturing, and electronics) have trade-offs in their scheduling problems in which multiple objectives need to be considered to optimize the overall performance of the system. Therefore, the MO-FJSP may be closer to the realistic production environments and needs to be further studied. In recent years, the MO-FJSP has captured more and more interests of numerous domain researchers, and a great many of algorithms have been presented [20]. Compared with SO-FJSP, MO-FJSP has two problems to be dealt with: incommensurability between objectives and contradiction between objectives (i.e., optimizing a single-objective generally results in deterioration of another objective) [13].

For the MO-FJSP, Schaffer [21] provided a genetic algorithm with vector evaluated. Jurisch [22] presented a branch-and-bound algorithm and some heuristic algorithms. By combining the VNS with particle swarm optimization (PSO), Liu et al. [23] presented a hybrid metaheuristic to solve the MO-FJSP. A new genetic algorithm (GA) which hybridized with a bottleneck shifting procedure was developed by Gao et al. [24] to solve the MO-FJSP. Zhang et al. [25] embedded tabu search (TS) in PSO as a local search to deal with the MO-FJSP. Xing et al. [26] advanced an efficient search method for the MO-FJSP. González-Rodríguez et al. [27] proposed a generic multiobjective model which was based on lexicographical minimization of expected values for FJSP. By introducing several metrics of the multiobjective evaluation in the MO-FJSP literature, Rahmati et al. [28] adopted two multiobjective evolutionary algorithms for the MO-FJSP.

Recently, some studies based on the Pareto dominance relation have been used to solve the MO-FJSP and they are more desirable when compared with the prior linear weighted summation ones [29]. The nondominated sorting genetic algorithm (NSGA) [30] was one of the first methods used to solve the multiobjective problem. Kacem et al. [31] brought up a Pareto approach based on the hybridization of fuzzy logic (FL) and evolutionary algorithms to solve the MO-FJSP. Ho and Tay [32] integrated a guiding local search procedure and an elitism memory mechanism into the evolutionary algorithm to solve the MO-FJSP.

Deb et al. [33] came up with a nondominated sorting-based multiobjective evolutionary algorithm (MOEA), called nondominated sorting genetic algorithm II (NSGA-II). Wang et al. [34] proposed a multiobjective GA based on immune and entropy principle for the MO-FJSP. By employing simulated annealing (SA) algorithm as a local search process, Frutos et al. [35] introduced a memetic algorithm (MA) based on the NSGA-II. Wang et al. [36] presented an effective Pareto-based estimation of distribution algorithm (P-EDA), in which various strategies are integrated to maintain quality and diversity of solutions. Rohaninejad et al. [37] advanced an MO-FJSP problem with machines capacity constraints to minimize the makespan and overtime costs of machines. Kaplanoğlu [38] put forth an object-oriented (OO) approach combined with SA optimization algorithm to solve the MO-FJSP, in which a two-string encoding scheme was used to express this problem.

Our review of the above literatures reveals that the NSGA-II algorithm has been widely used to solve the MO-FJSP for its advantages such as high efficiency to optimize the complex problems and the ability to gain widespread Pareto-optimal solutions. And the algorithms with a two-stage optimization scheme have been also widely studied to solve the MO-FJSP for it could fully tap the optimization potentials of various metaheuristic algorithms. However, we found that the NSGA-II algorithm has the disadvantage of premature convergence to local solution and the algorithms with a two-stage optimization scheme have the disadvantages of being unable to gain stable and high quality initial population in the first stage. Hence, in this paper, we propose a bee evolutionary guiding NSGA-II (BEG-NSGA-II) with a two-stage optimization scheme to solve the MO-FJSP, which aims to fully play the respective advantages of NSGA-II algorithm and the algorithms with a two-stage optimization scheme and to overcome the disadvantages of them. In the first stage, the NSGA-II algorithm with iteration times is first used to obtain the initial population which consists of three parts changing with the iteration times. In order to extensively exploit the solution space, an effective local search operator is invented in this stage. In the second stage, the NSGA-II algorithm with* GEN* iteration times is used to obtain the Pareto-optimal solutions. In order to enhance the searching ability and avoid the premature convergence, an updating mechanism and some useful genetic operators were employed in this stage. Four famous benchmarks that include 53 open problems of FJSP are chosen to estimate the performance of the proposed algorithm. Moreover, by comparing the results of our algorithm and some existing well-known algorithms, the virtues of our algorithm can be clearly demonstrated.

The rest of this paper is organized as follows. The definition and formalization of the MO-FJSP are given in the next section. In Section 3, NSGA-II is briefly introduced, and then the overview and implementation details of the proposed BEG-NSGA-II are presented, respectively. Afterwards, Experimental Studies and Discussions are made in Section 4. Finally, Conclusions and Future Studies are described in Section 5.

#### 2. Problem Definition

The FJSP is commonly defined as follows. There is a set of jobs (, ) and a set of machines (, ). One or more operation(s) (, , is the total number of operations for job ) is/are allowed to be processed by one machine of , which consists of a set of machines of the th operations for job . denotes the processing time of the th operation for job , which is processed by machine . Generally, the FJSP consists of two subproblems: the routing subproblem of assigning each operation to a machine among a set of machines available and the scheduling subproblem of sequencing the assigned operations on all machines to obtain a feasible schedule for optimizing a certain objective function [39, 40].

One classical FJSP is shown in Table 1. In this paper, we aim to minimize the following three objectives:(i)Maximal completion time of machines (makespan).(ii)Workload of the most loaded machine (MW).(iii)Total workload of all machines (TW).