Sådhanå (2021) 46:65 Ó Indian Academy of Sciences
https://doi.org/10.1007/s12046-021-01572-w Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)
Using chaos enhanced hybrid firefly particle swarm optimization algorithm for solving continuous optimization problems
I˙BRAHIM BERKAN AYDILEK1 ,I˙ZZETTIN HAKAN KARAC¸ IZMELI2 , MEHMET EMIN TENEKECI1,* , SERKAN KAYA2 and ABDU¨ LKADIR GU¨ MU¨ S¸C¸U¨ 3
1 Computer Engineering Department, Engineering Faculty, Harran University, S¸anlıurfa, Turkey 2 Industrial Engineering Department, Engineering Faculty, Harran University, S¸ anlıurfa, Turkey 3 Electrical-Electronics Engineering Department, Engineering Faculty, Harran University, S¸anlıurfa, Turkey e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]
MS received 4 May 2020; revised 19 December 2020; accepted 26 December 2020
Abstract. Optimization becomes more important and the use of optimization methods is becoming wide- spread with the developments in computer sciences. Researchers from different scientific fields are looking for better solutions to solve complex problems with optimization methods. In some complex problems, optimal results can be obtained utilizing metaheuristic algorithms. Researchers carry out different studies to improve the performance of present metaheuristic algorithms. Although the success of metaheuristic algorithms has been seen in previous studies, there are some weaknesses in these algorithms. Therefore, successful results cannot be obtained for each problem sometimes. In order to overcome this problem, more successful algorithms can be obtained by hybridizing the strong points of the different methods together. In addition, one of the important factors affecting the success of optimization algorithms is scanning ability of the solution space in order to find the optima. Exploring search space is carried out using random variables by some metaheuristic algorithms. The chaotic values that are generated by chaotic maps can be used instead of random values. Thus, search ability of algorithms performs more dynamically. In this study, hybrid firefly and particle swarm optimization algorithms are transformed to a chaotic-based algorithm by use of 10 different chaotic maps. Random valued parameters are generated by chaotic maps. In order to indicate the performances between different dimensions, CEC 2015 benchmark and constraint problems are used in experimental studies. Chaos enhanced methods are compared against canonical and hybrid optimization algorithms. It has been seen that obtained results of the proposed method were sufficiently successful and reliable.
Keywords. Chaotic maps; hybrid metaheuristic; particle swarm optimization; firefly optimization; CEC 2015.
1. Introduction most important indicators of chaotic functions. A small change that can be neglected in the value conditions may The usage of optimization methods in science, social and lead to big changes that cannot be ignored. It is thought that health sciences has increased steadily in computer tech- metaheuristic algorithms together with the coefficients nology. In the light of these developments, various resear- generated by the chaotic functions perform a better search ches have studied the run time and low-cost solution of the in the functions where local optimum or premature con- problems and steps have been taken to develop the existing vergence problems are experienced. Thus, in recent studies, methods. Many optimization methods have been proposed chaotic optimization methods have been used frequently to for this purpose until now. Some of the optimization rescue optimization from a vicious cycle. In addition to methods mentioned has a certain degree of randomness. this, many standard optimization algorithms have achieved Optimization algorithms cannot overcome the local opti- more successful results by diversifying the search space mum traps sometimes. Chaos is a randomness extremely with chaotic functions [1]. Hybrid optimization algorithms sensitive to initial value. Chaotic functions are complex and are developed by combining successful aspects of different irregular time-varying functions and they are sensitive to optimization algorithms. These algorithms could provide initial conditions. Sensitivity to initial value is one of the more successful results with chaotic functions. In this study, the random parameters of the Hybrid Firefly Particle Swarm Optimization (HFPSO) [2] algorithm employ *For correspondence 65 Page 2 of 22 Sådhanå (2021) 46:65 chaotic map functions to more effectively scan the search the capability of escaping local minima over PSO or some space. By this way, the optimum values of fitness functions other metaheuristic algorithms. The chaotic approaches can be reached more successfully. For this purpose, the give good results also with firefly algorithm. For example, HFPSO algorithm has been improved by using 10 dif- [12–15] introduce chaotic firefly algorithms that remove ferent chaotic maps. Chaotic HFPSO (CHFPSO) has problems of standard firefly algorithm, as in the chaotic been tested with CEC 2015 test suite I and II benchmark PSO algorithm. Wang et al [16] presented a comprehensive set and 5 constraint problems. In this study, experimental review of the different versions and their engineering studies are expanded compared to our previous study [3]. applications of the Krill herd algorithm which is another In [3], 2 chaos maps and a problem which is optimization nature-inspired herd-based optimization algorithm. of the FM parameters for the synthesis of the audio signal Similarly, good results are obtained with the chaotic are used. approach in other algorithms. Guvenc et al [17] embed 10 The rest of the article is organized as follows. Section 2 chaotic maps into moth swarm algorithm (MSA) for elim- summarizes previous works in the literature. Utilized inating the slow convergence problem. The sinusoidal map methods and algorithms are discussed in section 3.In is the best map among the other nine chaotic maps. Liang section 4, the results obtained from the proposed methods et al [18] hybridize random black hole model into bat are given and the performance of the study is evaluated by algorithm (BA) and chaotic maps for solving economic examining the results. In the last section, the contributions dispatch problems in power systems. Gandomi and Yang of the proposed method are indicated. [19] propose a chaotic BA. The results of their study indicate that chaotic BA are superior to BA in some cases. Alatas [20] proposes 7 chaotic artificial bee colony variants 2. Literature review with different chaotic maps. Metlicka and Davendra [21] present a chaotic artificial bee colony algorithm for solving According to the literature review made within the scope of quadratic assignment problems and vehicle routing prob- this study, chaotic optimization algorithms are usually more lems. In both sets of problems, Tinkerbell’s functional successful than standard optimization algorithms in terms chaotic algorithm achieves the best performance. of the capability of avoiding local minima and better con- Differential evolution algorithm (DEA) is a well-known vergence time performance. Therefore, the use of chaotic optimization algorithm with fast convergence ability. Sen- maps among researchers in optimization and stochastic kerik et al [22] hybridize the chaos theory with the DEA. search algorithms have been increasing steadily. The The authors perform an experimental study to determine chaotic optimization algorithms use chaotic maps to gen- mutation, crossover rates and chaotic system parameters. In erate random sequences for optimization problems. When another study, Senkerik et al [23] focused on the impact of the statistical properties such as probability density function chaotic sequences on the population, unlike many studies of chaotic sequences are considered, chaotic optimization on the combination of chaotic and meta-heuristic methods. algorithms can reach to optimal results with global searches The authors researched different randomization schemes by avoiding local searches according to standard opti- for the selection of individuals in the Differential Devel- mization algorithms. opment Algorithm and tested them with 15 test functions Particle swarm optimization (PSO) and genetic algo- from CEC 2015. Multimodal optimization problems that rithms (GA) are optimization algorithms that yield effective have various local and global optimization points, are results in the class of metaheuristic algorithms. However, among the most difficult problem types in the field of these algorithms often cause to premature convergence in optimization. Damanahi et al [24] introduce a chaotic DEA complex optimization problems. The chaotic algorithms for solving high dimensional multimodal problems. The use a more dynamic number range to overcome the con- authors compared the algorithms with the problem sets in vergence problem. Hosseinpourfard and Javidi [4] propose the literature and obtained successful results. a new chaotic PSO algorithm that employ Lorenz system, Although the gravitational search algorithm (GSA) is Tent map and Henon map for producing random numbers. used in complex optimization problems, it has disadvan- The authors prove that the performance of the proposed tages such as slow convergence and falling into local algorithm is better than PSO, GA and chaotic GA. Simi- optima. Shen et al [25] propose a chaotic GSA that uses 4 larly, Pluhacek et al [5] propose a chaotic PSO algorithm different chaotic maps to remove these disadvantages. that uses 6 different chaotic functions simultaneously. The Talatahari et al [26] introduce a chaotic imperialist com- authors tested the performance of algorithms with the CEC petitive algorithm (ICA) for global optimization. They 2013 benchmark problem set and obtained effective results. show that logistic and sinusoidal maps give better results In other study, Pluhacek et al [6] showed that the perfor- from the other 7 different chaotic maps. Yuan et al [27] mance of PSO algorithm improved with the application of introduce a new parallel chaos optimization algorithm that chaotic sequences. In similar researches [7–11], the includes migration and merging operations. researchers show the efficiency and robustness of their Successful results are also obtained from non-linear chaotic PSO algorithms in terms of the solution speed and problems with chaotic approaches. Yang et al [28] compare Sådhanå (2021) 46:65 Page 3 of 22 65 the performance of the hybrid chaotic Broyden–Fletcher– global optimization. They test 10 different chaotic maps on Goldfarb–Shanno (BFGS) algorithm, which is a quasi- 14 unimodal and multimodal benchmark optimization Newton method for local optimization, based on 8 one- problems for enhancing the convergence rate and resulting dimensional chaotic maps with different probability density precision of SSA. The results of their tests show that the functions and Lyapunov exponents. The authors show that logistic chaotic map is the optimal map. Aziz et al [40] the probability distribution property and search speed of proposed the Simulated Kalman Filter algorithm to solve chaotic sequences from different chaotic maps have a sig- real valued optimization problems. The proposed algorithm nificant effect on the global search ability and optimization works with only one agent and has fewer parameters to set. efficiency of the chaotic optimization algorithm. El-Shor- The authors tested their algorithms with 30 functions of bagy et al [1] present a chaotic-based GA to solve non- CEC 2014 and obtained effective results compared to other linear programming problems. The authors hybridize GA algorithms. and chaotic local search algorithms to improve convergence Among the recent studies on the subject; Turgut et al speed. Naanaa [29] proposes a new algorithm for global [41] presented the Crow Search algorithm, which has optimization, using the spatiotemporal map to improve the advanced search capability among subpopulations that performance of the chaos optimization algorithm. The interact during iterations. In addition, four different author shows that the proposed algorithm has more effec- migration topologies are proposed in the algorithm. Their tive results than the various hybrid optimization algorithm effectiveness was evaluated with classical optimization in the literature for non-linear multimodal optimization problems in the literature and CEC 2015 benchmark functions. functions. Sattar and Salim [42] proposed a Smart Flower Chaotic algorithms are also used in various combinato- Optimization Algorithm (SFOA) that operates in two rial optimization problems [30]. Xusheng et al [31] propose modes, sunny and cloudy/rainy. The authors tested their a chaotic GA for solving the path planning problem of robot algorithms with 15 benchmark functions of CEC 2015 and for avoiding the obstacle on the basis of space division. with four different engineering design problems. Joshi and This chaotic GA reduces the dangerous movements of the Bansal [43] proposed a generalized strategy that found the robot and the operation is safe. Chunyan et al [32] proposed optimal parameter value of meta-heuristic algorithms and a two-step algorithm for the target tracking problem. The applied it to the Gravitational Search Algorithm. The a first step is the PSO and the second step is the chaos inertia parameter, which plays an important role in the conver- weight and chaos search strategy for better search ability. gence of the algorithm, is precisely adjusted using the Saremi et al [33] used 10 chaotic maps to improve the proposed strategy. The proposed strategy has been tested performance of the biogeography-based optimization with CEC 2015 benchmark functions and it has been shown algorithm. Chaotic maps are used to describe the possibil- that effective results are obtained. ities of selection, migration and mutation. Wang and Li As it can be seen in recent studies, chaotic approaches [34] solve the sequence dependent setup times flow shop can improve the success of optimization algorithms. scheduling problem by using their chaotic biogeography- However, the superiority of chaotic maps to each other may based optimization approach. Ryter et al [35] investigate differ according to the type of optimization problem and the effects of 9 different chaotic maps in order to improve algorithm. the quality of the results obtained by a self-organizing map to solve the traveling salesman problem. According to the experimental results, chaotic maps provide significant 3. Chaotic hybrid firefly particle swarm improvement in several cases. optimization Sayed et al [36] proposed a chaotic crow search algo- rithm. They use 5 chaotic maps that are chebyshev, circle, 3.1 Firefly optimization algorithm gauss/mouse, logistic and piecewise maps. They implement the proposed algorithm to a three-bar truss, tension/com- In recent years, the use of nature inspired optimization pression spring design, and welded beam problems. The algorithms has become widespread. Firefly algorithm is a superiority of the proposed algorithm with piecewise map swarm-based metaheuristic algorithm inspired from nature. are indicated. Rahman et al [37] proposed 4 chaotic vari- It is assumed in this algorithm that, fireflies are unisex; each ants of stochastic fractal search algorithm. They used 2 individual iteratively approaches to the other. The modeling types of chaotic maps that are chebyshev and gauss/mouse of this behavior and its use as an optimization algorithm are maps. According to their study, the best results are achieved suggested by Yang [44]. As fireflies move in swarm, each with gaussian maps. Tavazoei and Haeri [38] investigate 10 of them approaches towards the brightest one. The light one-dimensional chaotic maps with weighted gradient intensity that affects the movement of the firefly. This method. It cannot be said that these maps are superior to depends on the brightness and the distance. As the distance each other, according to the results of convergence success increases, the intensity of the light will decrease. For this rate, algorithm speed and accuracy parameters. Sayed et al reason, the far firefly will be less effective than the nearby [39] propose a chaotic salp swarm algorithm (SSA) for one. The model was developed considering these factors. 65 Page 4 of 22 Sådhanå (2021) 46:65
Table 1. Chaotic Maps [14, 19, 46–48].
No Name Chaotic Map Range ¼ ðÞ 1 - 1 Chebyshev xiþ1 cos i cos xi ( 1,1) 2 Circle ¼ þ a ðÞp ; (0,1) xiþ1 mod xi b 2p sin 2 xi 1 , a = 0.5 and b = 0.2 3 Gauss/mouse (0,1) 8 < 1 xi ¼ 0 ¼ 1 xiþ1 : otherwise MODðÞ xi; 1