Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 3630869, 9 pages

https://doi.org/10.1155/2017/3630869

## Synergy of Genetic Algorithm with Extensive Neighborhood Search for the Permutation Flowshop Scheduling Problem

^{1}Department of Distribution Management, National Taichung University of Science and Technology, Taichung 404, Taiwan^{2}Department of Computer Science and Information Engineering, National Taichung University of Science and Technology, Taichung 404, Taiwan

Correspondence should be addressed to Chien-Che Huang; moc.liamg@8290selrahc

Received 19 August 2016; Revised 24 December 2016; Accepted 17 January 2017; Published 13 February 2017

Academic Editor: Jean J. Loiseau

Copyright © 2017 Rong-Chang Chen 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

The permutation flowshop scheduling problem (PFSP) is an important issue in the manufacturing industry. The objective of this study is to minimize the total completion time of scheduling for minimum makespan. Although the hybrid genetic algorithms are popular for resolving PFSP, their local search methods were compromised by the local optimum which has poorer solutions. This study proposed a new hybrid genetic algorithm for PFSP which makes use of the extensive neighborhood search method. For evaluating the performance, results of this study were compared against other state-of-the-art hybrid genetic algorithms. The comparisons showed that the proposed algorithm outperformed the other algorithms. A significant 50% test instances achieved the known optimal solutions. The proposed algorithm is simple and easy to implement. It can be extended easily to apply to similar combinatorial optimization problems.

#### 1. Introduction

The permutation flowshop scheduling problem (PFSP) [1–7] is the combination of jobs and machines of production scheduling problems. Each job must be processed on machines. All jobs are processed in the same sequence on each machine, but with different processing times. Each machine is identical and can only process one job at a time, and the processing cannot be interrupted or preempted. Specifically, let denote the processing time of job on Machine , and let denote a permutation of the jobs. Then the jobs are processed consecutively on Machine 1 in the order given by , with no delays between the executions of consecutive jobs. The jobs are processed in the same order on Machine 2, but each job must wait until it is finished processing on Machine 1 so that it can be started on Machine 2. Similarly, a job must wait to be processed on Machine until it has finished processing on Machine .

Subject to the foregoing stipulations, find a permutation that minimizes the maximum makespan, that is, the time to compete the processing of the last job in on the last Machine . By the preceding definition, PTSP has ! possible schedules. With an increase in the number of jobs and machines, the complexity of PTSP grows exponentially. Garey et al. [8] have proven that PFSP belonged to an NP-complete problem for . Thus, in recent years, approximation methods have been popularly used to solve PFSP. One such method is the metaheuristic algorithms [9]. In metaheuristic algorithms, studies are focused on the hybrid genetic algorithms which combine genetic algorithms and local search methods. The performance of hybrid genetic algorithms, or memetic algorithms (MAs) [10], is affected by local search methods. However, the local search methods are easily trapped in the local optima which increase the difficulty to find global optima.

This study proposes a new hybrid genetic algorithm (GA_ENS algorithm) which combines genetic algorithm and extensive neighborhood search. The diversity of the local search is thus significantly increased. When the GA_ENS algorithm explores the neighborhood of the current solution, the neighborhood will be dynamically changed to prevent redundant searches in the same region. This increases the probability of finding the global optimum.

In this study, we compared the GA_ENS algorithm with two other algorithms which belonged to state-of-the-art hybrid genetic algorithms for PFSP. One is the HGA_RMA algorithm proposed by Ruiz et al. [11] and the other is the NEGAVNS algorithm proposed by Zobolas et al. [12]. The two algorithms used different local search methods.

The remainder contents of this paper are organized as follows: Section 2 is a brief overview of the related algorithms for PFSP. Section 3 is the detailed description of this study proposing the new hybrid genetic algorithm. Section 4 is the experimental results and analyses. Finally, conclusions are presented in Section 5.

#### 2. Related Work

In 1954, Johnson [13] proposed Johnson’s rule to solve two-machine flowshop scheduling problem. As a result, much research had been focused on resolving the different flowshop scheduling problems. Currently, the* m*-machine () PFSP is the main trend of research.

Reza Hejazi and Saghafian [14] presented a complete review of flowshop scheduling problems with makespan objectives from 1954 to 2005. They reviewed various production scheduling problems of flowshops, including PFSP.

PFSP belongs to an NP-complete problem [8]. Therefore, approximation methods are applied to solve PFSP. However, the approximation methods cannot provide optimal solutions. They can only provide acceptable solutions within a reasonable time range. Approximate methods can generally be classified as heuristic algorithms and metaheuristic algorithms [15].

The design core of the heuristic algorithms is based on experiences. In other words, the design of the algorithms relies heavily on understanding the problems and experiences. Heuristic algorithms can solve problems quickly and obtain reasonable solutions within reasonable time. However, they cannot guarantee the same quality solution in different data. The greatest drawback is that they cannot guarantee global and consistent optimal solutions.

The heuristic algorithms can mainly be divided into two categories: constructive methods and improvement methods [15]. The constructive methods construct solutions from scratch according to some special rules and can provide solutions rather quickly. However, their solution quality is not guaranteed, examples of which include Palmer’s heuristic method [16], CDS heuristic [17], rapid access (RA) [18], and NEH heuristic [19]. NEH heuristic is the best algorithm in the constructive methods [15, 20]. NEH is a quick local search for minimum makespan. For the jobs (!), for the first-level current permutation, the* k*th job which minimizes the partial makespan is inserted. The sequence is calculated with . Taillard’s insertion moves jobs by new neighborhoods where is the best solution. Current research has been focused on improving the NEH mechanisms which also involved the hybrid designs [21, 22]. The improvement methods are also called local search methods. They can provide good solutions. However, they are time-consuming. The basic design idea of the improvement methods is to improve the initial solution by some specific rules and expect to obtain the better solutions, examples of which are RACS and RAES [18].

In the last 20 years, much work has been done on the relatively new type approximate metaheuristic algorithms which are also known as modern heuristics. The algorithms inject the probability concept into the process of solving problems. Compared with the heuristic algorithms, metaheuristic algorithms require more time for getting the solutions. However, they resulted in higher quality solutions than heuristic algorithms [15].

In order to enhance the quality of solutions in metaheuristic algorithms, algorithms must be designed with the balance between diversification and intensification. Diversification belongs to the capability of global search and maintaining the population diversity; it can explore all areas of the search space. Intensification belongs to the capability of local search; it can exploit the neighborhood of the current solution and find a local optimum.

Metaheuristic algorithms can be classified as trajectory methods and population-based methods according to the number of solutions used in the search process [9]. Trajectory methods generate a feasible solution in each iteration and attempt to find the best solution along the trajectory of the search space, such as simulated annealing (SA) [23–28], Tabu search (TS) [29–34], iterated local search (ILS) [9, 15], and variable neighborhood search (VNS) [9, 35]. Population-based methods generate a set of feasible solutions to perform parallel search in each generation and get the best solution after iterative evolution, such as genetic algorithms (GAs) [36–40], ant colony optimization (ACO), and particle swarm optimization (PSO).

#### 3. A New Hybrid Genetic Algorithm

Traditional genetic algorithms utilize the population to execute multiple points search using the genetic operators. As a result, they have the capability of global search and population diversity. In order to enhance the efficiency of local search for obtaining better quality solutions, this study proposes a new hybrid genetic algorithm: the GA_ENS algorithm (Figure 1). The difference between the GA_ENS algorithm and traditional genetic algorithms is that the GA_ENS algorithm added an operator of extensive neighborhood search, which can enhance the force of intensification; we named this operator “extensive neighborhood search operator” (ENS operator). In the following, we will describe all operators of the GA_ENS algorithm.