A hybrid method integrating an elite genetic algorithm with tabu search for the quadratic assignment problem

doi:10.1016/j.ins.2020.06.036

Information Sciences

Volume 539, October 2020, Pages 347-374

https://doi.org/10.1016/j.ins.2020.06.036 Get rights and content

Abstract

The Quadratic Assignment Problem (QAP) is one of the most studied classical combinatorial optimization problems. QAP has many practical applications. Designing enhanced meta-heuristic approaches for the QAP is an active research area. In this work, we propose a hybrid algorithm (EGATS) that combines an elite genetic algorithm and tabu search to solve the QAP. In the optimization process, EGATS employs two kinds of elite crossovers, repeated 2-exchange mutation, and tabu search to strike a balance between exploitation and exploration. We evaluated the performance of EGATS through computational experiments on 135 well-known benchmark instances from the quadratic assignment problem library, QAPLIB. EGATS obtained the best-known solution for 131 instances. Compared to other popular meta-heuristic algorithms in the literature, EGATS is a competitive method for the QAP.

Introduction

The Quadratic Assignment Problem (QAP), which was introduced by Koopmans and Beckmann in 1957 as a mathematical model for the location of a set of indivisible economic activities, is one of the most difficult combinatorial optimization problems. From the theoretical point of view, the QAP is NP-hard; even finding an $ε$ -approximate solution is difficult [49]. Because the QAP generalizes a large number of theoretical problems such as graph partitioning, maximal clique, and traveling salesman, it is one of the most fundamentally important subjects of operations research [12]. From the practical point of view, the QAP has a wide range of applications in diverse fields such as backboard wiring, typewriter keyboards and control panel design, scheduling, numerical analysis, and optimal placement of letters on touchscreen devices. Therefore, the study on the QAP has not only theoretical significance, but also practical value.

In recent years, a variety of meta-heuristic algorithms have been used to solve difficult optimization problems. Examples are genetic algorithm (GA) [4], [21], [19], tabu search (TS) [41], [47], gravitational search algorithm [19], ant colony optimization [48], greedy randomized adaptive search procedure [27], migrating birds optimization [17], population-based memetic algorithm [9], whale algorithm [2], global neighborhood algorithm [37], neighborhood decomposition-based search [7], tournament selection based antlion optimization algorithm [24], and artificial bee colony [40]. Although most of them do not ensure that the optimal solution is obtained, they nonetheless provide good results within a reasonable time [12]. Obtaining a good result within an acceptable computing time is often more desirable than spending a very long time on seeking the optimal solution. Among the meta-heuristics, GA and TS have been found to be highly efficient algorithms to solve a great diversity of optimization problems. Numerous variants of GA and TS have been developed to solve the QAP (see Section 3 for details). All these works show the popularity of GA and TS in solving the QAP.

However, we still need to develop and improve techniques for the QAP. As a very difficult and challenging optimization problem, QAP is a suitable testing platform for innovative intelligent optimization techniques. The design of enhanced meta-heuristic approaches for the QAP still remains an active research area [32]. An optimization method that is greedy and only looks in the neighborhood of the best solution can easily mislead the search process and make it fall into a local solution [39]. Therefore, it is important to develop a good optimization method that has a mechanism to strike a balance between local and global search. For this purpose, we propose a hybrid algorithm for solving the QAP that combines the diversified global search ability of GA with the intensified local search capability of TS.

GA is a bio-inspired meta-heuristic approach based upon the concepts of biological evolution and survival of the fittest individuals. There are basically three operators in GA: selection, crossover and mutation, out of which crossover is the most important operator [4]. Since crossover plays an essential role in GA search, various crossover operators have been proposed for different problems that fit to one of the many representations for a chromosome [4]. Unfortunately, the offspring produced by most of them do not inherit enough information from the elite individuals (i.e., individuals with better fitness).To overcome this issue, several variants of GA based on the concept of random key have been proposed in recent years. In this approach, when applying the crossover operator, one parent is selected from the set of elite individuals, and the other parent is selected from the rest of the population [25]. The crossover that uses elite individuals is called an elite crossover, and a GA that uses elite crossover is called an elite GA (EGA). In order to exploit the promising regions found in the search space, an EGA employs an improvement method that is applied to the best individuals in the population [25].

The search performance of GA can be improved through the choice of the method used to select, cross, and mutate individuals. Furthermore, GA performance can also be enhanced by a local optimization procedure. Among the various improvement procedures implemented within local search, TS seems to be the best candidate owing to its robustness, efficacy, and quickness [30]. Numerous hybrid genetic algorithms (HGAs) obtained by combining with TS are available in the literature [15], however, to our knowledge, an integration of an EGA with TS for solving the QAP and other combinatorial optimization problems is absent. In this paper, we propose a hybrid algorithm (EGATS) that integrates an elite genetic algorithm with tabu search to solve the QAP. EGATS employs two different kinds of elite crossovers along with TS to strengthen the exploitation of the promising regions. We evaluated EGATS on some instances of the QAPLIB, which is the benchmark problem library of the QAP. We performed many comparisons and statistical analyses, such as best deviation, average deviation, and computing time.

The rest of this paper is organized as follows: Section 2 introduces the problem definition of QAP. A brief review of state-of-the-art meta-heuristic algorithms for solving the QAP has been carried out in Section 3. Section 4 describes the proposed EGATS for the QAP. Computational experience on some QAPLIB instances and comparison with state-of-the-art algorithms are reported in Section 5. Finally, concluding remarks are presented and some future works are suggested in Section 6.

Section snippets

Quadratic assignment problem

Consider the set $N = {1, 2, \dots, n}$ and three $n \times n$ matrices $F = (f_{ik}), D = (d_{jl})$ and $C = (c_{ij})$ , the QAP form of Koopmans & Beckmann can be stated as follows: $\min_{π \in Π} z (π) = (\sum_{i = 1}^{n} \sum_{k = 1}^{n} f_{ik} d_{π (i) π (k)} + \sum_{i = 1}^{n} c_{i π (i)})$ where $f_{ik}$ denotes the amount of flow from the facility i to facility $k, d_{jl}$ denotes the distance from location j to location $l, c_{ij}$ denotes the cost of locating facility i at location j. The assignment of facilities to locations is represented by a permutation $π$ ( $π \in Π, Π$ is the set of all possible permutations)

Related works

This section briefly reviews the literature on GAs, TS and some other meta-heuristic algorithms for solving the QAP.

Proposed EGATS algorithm for the optimization of the QAP

In this section, we introduce EGATS for the QAP whose implementation is based on robust tabu search [43], elite crossovers, and repeated 2-exchange mutation. The steps of this algorithm are stated in the following subsections, and its pseudo-code is presented in Fig. 1. Its parameters are the number of generations, G, the size of the population, P, the range of mutation rate, $[μ_{\min}, μ_{\max}]$ , the number of TS iterations, iter, the range of tabu tenure, $[h_{low}, h_{high}]$ , and the instance data, $F, D$ , and n

Computational results and discussion

In order to study the performance of EGATS, we carried out a number of computational experiments. Matlab 2010Ra was used for the implementation of the algorithm. In our experiments, all the set of instances from the QAPLIB were used. Because most of the state-of-the-art algorithms use this library, it provides a fair platform for the evaluation of new algorithms [14]. All the experiments were conducted on a computer with an Intel(R) Core(TM) i7-4790 CPU @ 3.60 GHz 3.60 GHz, with 8.0 GB of RAM.

Conclusions and future works

In this study, we proposed a hybrid method integrating an elite genetic algorithm with tabu search for solving the QAP. The results show that EGATS can obtain optimal/near-optimal results within acceptable time and outperforms most of the recent powerful algorithms reported in the literature when solving the benchmark instances of the QAPLIB. The better results of EGATS are because of three features: (1) The elite crossovers that can inherit enough information from the best individuals; (2) The

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.

CRediT authorship contribution statement

Huizhen Zhang: Conceptualization, Methodology, Software, Writing - original draft, Supervision, Writing - review & editing. Fan Liu: Validation, Formal analysis, Data curation. Yuyang Zhou: Investigation, Writing - review & editing. Ziying Zhang: Visualization.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant No. 71401106), the Shanghai Natural Science Foundation (Grant No. 14ZR1418700), the Humanities and Social Sciences Foundation from the Ministry of Education of China (Grant Nos. 16YJA630037 and 19YJAZH064), the Hujiang Foundation of China (Grant No. A14006).

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Citation Excerpt :
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A hybrid method integrating an elite genetic algorithm with tabu search for the quadratic assignment problem

Abstract

Introduction

Section snippets

Quadratic assignment problem

Related works

Proposed EGATS algorithm for the optimization of the QAP

Computational results and discussion

Conclusions and future works

Declaration of Competing Interest

CRediT authorship contribution statement

Acknowledgement

Appl. Soft Comput.

Comput. Oper. Res.

Expert Syst. Appl.

Swarm Evol. Comput.

J. Parallel Distrib. Comput.

Appl. Soft Comput.

Comput. Oper. Res.

Inf. Sci.

Comput. Ind. Eng.

Inf. Sci.

Inf. Sci.

Eur. J. Oper. Res.

Eng. Sci. Technol. Int. J.

Inf. Process. Lett.

Knowl.-Based Syst.

Knowl.-Based Syst.

Eur. J. Oper. Res.

Egypt. Inf. J.

Comput. Ind. Eng.

Parallel Comput.

Comput. Oper. Res.

Procedia Comput. Sci.

Eng. Appl. Artif. Intell.

Inf. Sci.