A quick convergent artificial bee colony algorithm for solving quadratic assignment problems
Introduction
The quadratic assignment problem (QAP) is to allocate n facilities in n locations, taken from n available positions in such a way that minimizes the interdepartmental travel distance, cost or time for any facility. It is one of the fundamental combinatorial optimization problems from the category of facility layout optimization. This problem was first formalized by Koopmans and Beckman (1957), and can be presented as shown in Eq. (1).where Sn is the set of all n permutations of the integers 1,2,…,n; aij denotes the distance between location i and location j, brs denotes the amount of flow that occurs between location r and location s, and each individual value of the product denotes the transportation distance caused by assigning facility i to location φ(i) and facility j to location φ(j).
Sahni and Gonzalez (1976) proved that QAP could not only be solved within polynomial time, but also it would not be guaranteed to reach nearer to the optimal solution within polynomial time. Thus, QAP can be treated as an NP-hard problem. Several algorithms, for both exact and approximate solutions, have been proposed for solving this problem. The major difficulty in applying exact algorithms for QAP is that they often require high computational time for solving even small-sized QAPs. Anstreicher and Brixius (2001) used convex quadratic programming relaxation within a branch and bound algorithm for solving QAP. That algorithm provided an exact solution to the benchmark problem Nug25 (Burkard, Karisch, & Rendl, 1997). However, the entire computation process required 13 days of CPU time using sequential processing. Similarly, Hahn and Krarup (2001) solved Kra30a (Burkard et al., 1997) after 99 days of CPU time using a sequential workstation. Solving Ste36b and Ste36c (Burkard et al., 1997) took 200 days of working in a distributed environment (Nystrom, 1999), while solving Nug30 by Anstreicher, Brixius, Goux, and Linderoth (2002) required seven days with 650 processors. Clearly, with the increase in size and density of the flow, the exact solution approaches to QAPs would become computationally exhaustive even with the present day’s advanced technologies. Heuristic procedures, though do not guarantee to provide optimal solutions, are capable of reaching near optimal solutions more quickly. Hence, a lot of research works has been carried out on developing suitable heuristic-based algorithms that can reach the approximate solutions quickly. Some examples of these algorithms used for solving QAPs are genetic algorithm (GA) (Cárdenas et al., 2017, Gen and Cheng, 2000), robust Tabusearch (ROTS) (Taillard, 1991), variable neighborhood search (VNS) (Hansen and Mladenović, 2001, Mladenović and Hansen, 1997), go with the winner (GWW) (Aldous & Vazirani, 1994), simulated annealing (SA) (Battiti and Tecchiolli, 1994, Burkard and Rendl, 1984, Kirkpatrick et al., 1983), max-min ant system (MMANT) (Stützle & Hoos, 2000), greedy random adoptive search procedure (GRASP) (Fleurent and Glover, 1999, Li et al., 1994) etc.
All these metaheuristics developed by the earlier researchers can work based on two major principles: a) improvement approach and b) constructive approach. The improvement approach-based algorithms begin with a number of initial feasible solutions and try to improve the solution in each iteration by applying some search procedure. Various search procedures applied by the improvement approaches are: (a) genetic search - GA (Tale and Smith, 1995, Tosun, 2014, Wu and Ji, 2007) and (b) neighborhood search - there are mainly two kinds of neighborhood search approaches used in various algorithms - (i) SWAP - (a) random SWAP - VNS (Hansen and Mladenović, 2001, Mladenović and Hansen, 1997), GWW (Aldous & Vazirani, 1994), SA (Battiti and Tecchiolli, 1994, Burkard and Rendl, 1984, Kirkpatrick et al., 1983), and (b) best SWAP - Tabu search (TS) (Glover, 1990, Glover, 1989), ROTS (Taillard, 1991), and (ii) INSERT - GWW, VNS etc. The constructive approach, in each iteration, develops a starting solution by adding element after element using some functions. The starting solution is then used to improve while incorporating some local search heuristics for a limited number of iterations. The solution found after the local search is employed to update the benefits associated with positioning one element at a particular position for the next iteration of the algorithm. Examples of constructive approaches are MMANT and GRASP.
Artificial bee colony (ABC) algorithm is a swarm-based metaheuristic, and it mimics the foraging behavior of honeybees to search for optimality within the solution space (Karaboga, 2005, Karaboga et al., 2014). Because of its powerful convergence ability, it has been widely applied in various fields for solving several complex optimization problems, such as parametric optimization of non-traditional machining processes (Samanta & Chakraborty, 2011), cluster analysis (Teja & Indu, 2016), training artificial neural networks (Du et al., 2015, Sarangi et al., 2014), image processing (Banharnsakun, 2018, Xie et al., 2019), wireless sensor network (Al-Ariki et al., 2018, Saad et al., 2019, Szeto and Jiang, 2014), power transmission (Bai et al., 2017, Das et al., 2018, Ma et al., 2019), medical research (Karpagam & Raghavan, 2019) etc. The ABC algorithm has also been successfully employed for solving various combinatorial optimization problems, e.g. job shop scheduling problem (JSSP) (Li et al., 2017, Meng et al., 2018), travelling salesman problem (TSP) (Khan and Maiti, 2019, Pandiri and Singh, 2018, Zhong et al., 2017), vehicle routing problem (VRP) (Baradaran, Shafaei, & Hosseinian, 2019), container inventory management (Poo & Yip, 2019), workforce scheduling and balancing (Yurtkuran, Yagmahan, & Emel, 2018), QAP (Dokeroglu, Sevinc, & Cosar, 2019) etc.
The ABC algorithm consists of two local searches and an optional global search in each iteration. Researchers have used different or similar search approaches in the two local search phases of the ABC algorithm while solving combinatorial optimization problems. However, most of them have failed to provide a valid justification behind selecting a proper search approach which would raise an obvious question that what would happen if some different search approaches would have been used at any of the local search phases. Moreover, comparisons made by the researchers with the other state-of-the-art algorithms are mostly based on only average reachability of the algorithm, which are not statistically consistent enough to declare a particular algorithm’s superiority over the other. This paper first attempts to determine the combination of local search approaches that provides a considerably better result when used in ABC algorithm for solving QAP by comparing ten candidate alternatives. The best option, chosen with proper statistical analysis, along with some further modifications, is named as quick convergent ABC (qABC). It is important to mention here that the qABC was first formulated by Samanta, Philip, and Chakraborty (2018) for solving a real time bi-objective dependent location QAP. This paper shows how the qABC is systematically developed and how it is comparable with other state-of-the-art algorithms available for solving QAPs. The performance of this algorithm is evaluated while solving some benchmark problems from the QAP library (Burkard et al., 1997), and compared with seven other state-of-the-art algorithms for solving QAPs, i.e. (a) GA, (b) GWW, (c) ROTS, (d) SA, (e) VNS, (f) MMANT and (g) GRASP. Appropriate statistical analyses are also performed to contrast among the simulation results to make the comparisons more consistent. This paper is organized as follows: Section 2 presents the development of qABC algorithm. Section 3 provides information about different experimental conditions and test problems. The comparison of simulation runs of the developed qABC with other state-of-the-art algorithms is presented in Section 4. Finally, Section 5 concludes the paper.
Section snippets
Development of various alternatives and selection of the best alternative
The ABC algorithm uses an improvement approach in its general form. It implies that the algorithm starts by initializing with some feasible solutions. Those feasible solutions are then tried to improve in each iteration. The algorithm consists of three main phases in each iteration, i.e. (a) a compulsory local search phase named as employed bee phase, (b) a selective local search phase designated as onlooker bee phase, and (c) an optional global search phase named as scout bee phase. During the
Experiments
Experiments are now conducted to compare the computability of the developed qABC over the other state-of-the-art algorithms in solving QAPs. The other algorithms considered here are (a) GA, (b) GWW, (c) ROTS, (d) SA and (e) VNS. It is important to mention here that Dokeroglu et al. 2019) also used an ABC algorithm for solving QAP. However, the adopted algorithm was mainly a hybridization of Tabu search having exploration phase of finding out a new solution within the solution space and
Results and analysis
Table 5, Table 6, Table 7, Table 8 respectively summarize the results based on 50 runs of each of the categories of the benchmark QAP instances using the considered algorithms. It is worthwhile to mention here that all the benchmark QAPs were divided into four categories, based on their different characteristics. These tables provide the values of the percentage difference of the best value found by an algorithm with minima (if available) or LB (mGAP), number of times an algorithm reaches the
Conclusions and future work
This paper provides a new variant of ABC algorithm, named as quick convergent ABC (qABC) for solving combinatorial optimization problems. The performance of this algorithm is tested on 51 standard QAPs (divided into four categories) selected from the QAP library. The results obtained are compared with seven other state-of-the-art algorithms to show its efficacy in reliably finding out the BKS within reasonably less time.
In contrast to the general ABC algorithm, where both the employed and
Acknowledgement
The authors sincerely acknowledge those anonymous reviewers whose valuable comments have helped us to enhance the quality of this paper.
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