Sådhanå Ó (2021) 46:65 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