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 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 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 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 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 ðÞ1 - 1 Chebyshev xiþ1 cos icos 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

4 Iterative ap (-1,1) xiþ ¼ sin , a = 0.7 1 xi 5 Logistic xiþ1 ¼ axiðÞ1 xi , a = 4 (0,1) 6 Piecewise (0,1) 8 > xi \ > 0 xi P > P > x P > i P x \0:5 < 0:5 P i xiþ1 ¼ 1 P xi > 0:5 xi\1 P > 0:5 P > > 1 xi > 1 P xi\1 : P P ¼ 0:4

¼ a ðp Þ 7 Sine xiþ1 4 sin xi , a = 4 (0,1) 8 Singer (0,1) ¼ l : : 2 þ : 3 : 4 ; l ¼ : xiþ1 7 86xi 23 31xi 28 75xi 13 302875xi 1 07

¼ 2 ðp Þ 9 Sinusoidal xiþ1 axi sin xi , a = 2.3 (0,1) 10 Tent (0,1) 8 > xi < xi\0:7 ¼ 0:7 xiþ1 > 10 : ðÞ1 x x 0:7 3 i i

The effect of light intensity is expressed in Equation (1). represent two fireflies. Equation (3) calculates the new tþ1 I(r) represents the amount of light to be emitted to a r position of Xi .Thet used as the index shows the iteration distance, I0, denotes the light intensity of the source. In the value in the operations. t?1 refers to the next iteration. i is equation, c is expressed fixed light absorption coefficient the vector of random variables and a is the coefficient value. parameter of randomness. Algorithm tries to find the most appropriate solution within each new position calculated cr2 iteratively. IrðÞ¼I0e ð1Þ The amount of absorption of the light source varies tþ1 t cr2 t t X ¼ X þ B e i;j X X þ a ð3Þ according to the properties of the medium. Accordingly, in i i 0 j i i Equation (2), the attractiveness of the fireflies B is expressed and the c value is taken as the absorptive coef- ficient. B0 refers to the amount of attractiveness that firefly 3.2 Particle swarm optimization algorithm has at distance 0. The PSO algorithm is proposed by Kennedy and Eberhart 2 BrðÞ¼B ecr ð2Þ in [45]. Like the firefly algorithm, it converges based on 0 swarm principles. The method mimics birds around food or At first, fireflies randomly distributed are identified in movements of fish. In the algorithm, each particle updates t t their initial positions as shown in Equation (3). Xi and Xj its position in every step of the way to reach this food. Sådhanå (2021) 46:65 Page 5 of 22 65

Start

Initilize Parameters (c1, c2, wi, wf, Xmin, Xmax, Vmin,Vmax, Pop, D,

iterationmax) and Calculate Chaotic Maps

Randomly Inıtilize Positions and Velocities

Calculate fitness, pbest and gbest

Improvement done in its fitness value in the last iteration (Eq. 6) True False

Set new value of i Set new value of rand1 and rand2 with Chaotic Maps with Chaotic Maps

Firefly Algorithm Particle Swarm Algorithm

Calculate position and Calculate position values velocity values

Is MaxFES False reached?

True

Figure 1. Chaotic Maps Values [29]. Output gbest and fitness

Stop When these updates are made, operations are carried out Figure 2. Flow diagram of proposed Chaotic HFPSO. considering the values of personal best (pbest) and global best (gbest). pbest stores the best results for each particle. gbest records the best result among the all particles. kþ1 ¼ : k þ : k: k k vi w vi c1 rand1 pbesti xi þ c :randk: gbestk xk ð4Þ 2 2 i i 3.3 Hybrid firefly and particle swarm optimization

In Equation (4), xi denotes the position of the particles, w algorithm is the inertia weight, and v denotes the velocity of the i The hybrid firefly and particle swarm optimization particles. c and c are acceleration parameters. 1 2 (HFPSO) algorithm proposed by Aydilek [2], were devel- rand1andrand2 are random values that vary in the range [0, kþ1 oped by combining strong points of the firefly and PSO 1]. xi expresses the new position of the particle and is algorithms. More successful results are obtained by com- k calculated as shown in Equation (5). xi refers to the current bining the strong points of both methods. kþ1 position of the particle and vi to its velocity. The PSO algorithm is quite successful and performs fast searching due to the convergence speed of velocity kþ1 ¼ k þ kþ1 ð Þ xi xi vi 5 parameter. It has taken place in the literature as an effective 65 Page 6 of 22 Sådhanå (2021) 46:65

Table 2. CEC 2015 learning-based functions and optimum values.

No. Functions Fi*=Fi(x*) Unimodal Functions 1 Rotated High Conditioned Elliptic Function 100 2 Rotated Cigar Function 200 Simple Multimodal Functions 3 Shifted and Rotated Ackley’s Function 300 4 Shifted and Rotated Rastrigin’s Function 400 5 Shifted and Rotated Schwefel’s Function 500 Hybrid Functions 6 Hybrid Function 1 (N = 3) 600 7 Hybrid Function 2 (N = 4) 700 8 Hybrid Function 3(N = 5) 800 Composition Functions 9 Composition Function 1 (N = 3) 900 10 Composition Function 2 (N = 3) 1000 11 Composition Function 3 (N = 5) 1100 12 Composition Function 4 (N = 5) 1200 13 Composition Function 5 (N = 5) 1300 14 Composition Function 6 (N = 7) 1400 15 Composition Function 7 (N = 10) 1500

method for global search. However, sometimes algorithm functions, the initial point that determines the rest of dis- suffers to reach the optimal results due to oscillating in tribution values, is randomly selected among [0, 1]. local searches. Thus, exploitation ability of algorithm is It is shown in figure 1, different dynamic number adversely affected. sequences are generated by the chaotic maps expressed in The firefly algorithm does not have the ability to hold the table 1. These dynamic values can cause a much better personal best values and to approach with the velocity search ability of the related . For example, parameter. Therefore, it can determine the most appropriate scanning the search space can be done better and diversity solution according to local search space conditions. Thus, of the particles are increased. successful exploitation can be realized. However, the firefly method reaches global optimums slower than PSO. The HFPSO combines the local search capabilities of firefly and global search capabilities of PSO algorithm. In 3.5 Proposed chaotic hybrid firefly and particle HFPSO, the improvement to be achieved for the fitness swarm optimization algorithm function is calculated according to Equation (6). If a par- Current optimization algorithms use randomness to scan ticle has more successful value than the current gbest,it solution space. Random variables used in these methods are uses firefly algorithm to make a local search. Otherwise, it usually in the form of uniform distribution. The chaotic continues to search with PSO equations. maps can be used instead of this randomness, small varia- true; if fitness particlet gbestt1 tion of the initial value completely changes the rest of fiðÞ¼; t i ð6Þ false; if fitness particlet [ gbestt1 whole distribution. In recent years, it has been observed that i the performances of standard and hybrid optimization The new position (XiðÞt þ 1 ) and velocity (ViðÞt þ 1 ) algorithms are enhanced using chaotic maps instead of values of the particles are calculated according to Equa- random functions. tions (7) and Equation (8). In this study, it is suggested to use the numbers that are generated from chaotic maps to determine parameter values 2 cr ; t1 XiðÞ¼t þ 1 XiðÞþt B0e i j XiðÞt gbest þ ai ð7Þ that take random values in the existing HFPSO algorithm. In Equation (4), rand and rand parameter values are used ðÞ¼þ ðÞþ ð Þ 1 2 Vi t 1 Xi t 1 Xi temp 8 in the particle swarm algorithm and in Equation (7), the parameter value [i in the firefly algorithm is randomly generated among [0, 1] values. These three parameters are assigned values generated by chaotic maps instead of ran- 3.4 Chaotic maps domly generated values at each iteration. Thus, a more In this section, chaotic maps to be used in the proposed successful scan will be performed to determine the opti- CHFPSO algorithm are mentioned. 10 different chaotic mum points. map functions have been used in this study. The list of The flow diagram of the proposed method is given in chaotic maps in the study is given in table 1. In these figure 2. The new values of the random variables that are S å dhan å

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Table 3. CEC 2015 10D Mean and Std. results.

Func. Chebyshev Circle Gauss/mouse Iterative Logistic Piecewise F1 3.7043E?05 std. 3.1758E?05 8.1494E?05 std. 5.5919E?05 5.7876E?05 std. 4.2305E?05 3.1969E105 std. 2.2066E105 3.4596E?05 std. 2.8592E?05 3.2109E?05 std. 2.1535E?05 F2 5.5429E103 std. 6.4540E103 6.4834E?03 std. 6.8841E?03 7.1763E?03 std. 8.3685E?03 6.9436E?03 std. 9.0182E?03 7.1825E?03 std. 8.8995E?03 6.7802E?03 std. 7.4654E?03 F3 3.2000E102 std. 1.6110E-02 3.2000E?02 std. 8.8937E-03 3.2000E?02 std. 1.4674E-03 3.2001E?02 std. 2.8347E-02 3.2000E?02 std. 2.7721E-03 3.2000E?02 std. 1.9744E-04 F4 4.3345E?02 std. 1.0700E?01 4.3734E?02 std. 1.6495E?01 4.3422E?02 std. 1.1476E?01 4.3613E?02 std. 1.5750E?01 4.3264E?02 std. 1.3388E?01 4.3393E?02 std. 1.3208E?01 F5 1.1713E?03 std. 2.4942E?02 1.3618E?03 std. 3.1116E?02 1.2507E?03 std. 2.9047E?02 1.3030E?03 std. 2.9727E?02 1.2009E?03 std. 2.4945E?02 1.3053E?03 std. 3.0584E?02 F6 3.2255E?03 std. 2.5036E?03 3.4426E?03 std. 3.1223E?03 3.5891E?03 std. 3.0192E?03 3.5199E?03 std. 2.8863E?03 3.6676E?03 std. 3.3282E?03 3.5035E?03 std. 3.0768E?03 F7 7.0279E?02 std. 1.1789E?00 7.0405E?02 std. 1.2123E?00 7.0338E?02 std. 1.3650E?00 7.0331E?02 std. 1.2146E?00 7.0255E102 std. 1.0959E100 7.0289E?02 std. 1.2657E?00 F8 4.4430E?03 std. 3.7455E?03 5.9386E?03 std. 5.2171E?03 4.7580E?03 std. 4.0643E?03 4.1227E?03 std. 4.1197E?03 3.6219E?03 std. 3.4166E?03 3.5895E103 std. 3.1532E103 F9 1.0031E103 std. 1.9066E101 1.0132E?03 std. 4.4230E?01 1.0091E?03 std. 3.4964E?01 1.0210E?03 std. 5.7238E?01 1.0192E?03 std. 4.7558E?01 1.0170E?03 std. 4.6305E?01 F10 4.3147E?03 std. 3.2905E?03 9.3770E?03 std. 8.9371E?03 4.9793E?03 std. 4.0003E?03 5.5036E?03 std. 4.1238E?03 5.8720E?03 std. 5.3363E?03 5.8443E?03 std. 4.4819E?03 F11 1.4411E?03 std. 1.0406E?02 1.4132E?03 std. 1.4734E?02 1.4156E?03 std. 1.2816E?02 1.4010E103 std. 9.7446E101 1.4047E?03 std. 1.3732E?02 1.4154E?03 std. 1.4749E?02 F12 1.3038E?03 std. 1.2733E?00 1.3058E?03 std. 2.3278E?00 1.3054E?03 std. 1.3609E?01 1.3049E?03 std. 1.9757E?00 1.3035E103 std. 1.5429E100 1.3044E?03 std. 1.7312E?00 F13 1.3000E?03 std. 2.2569E-02 1.3001E?03 std. 2.3570E-01 1.3000E?03 std. 2.0537E-02 1.3001E?03 std. 2.5624E-01 1.3001E?03 std. 6.8071E-02 1.3001E?03 std. 1.3360E-01 F14 5.3249E?03 std. 3.8512E?03 6.8027E?03 std. 4.3750E?03 4.9726E103 std. 3.7823E103 7.0428E?03 std. 4.2779E?03 6.2685E?03 std. 3.8390E?03 5.8003E?03 std. 3.7667E?03 F15 1.6000E?03 std. 2.5895E-06 1.6000E?03 std. 3.6188E-07 1.6000E?03 std. 5.4593E-06 1.6000E?03 std. 1.0590E-07 1.6000E?03 std. 4.0739E-06 1.6000E?03 std. 2.6888E-07 Func. Sine Singer Sinusoidal Tent HFPSO F1 3.7631E?05 std. 3.3254E?05 3.2706E?05 std. 1.8668E?05 3.9829E?08 std. 4.8201E?08 4.1789E?05 std. 2.6888E?05 3.6250E?05 std. 2.4943E?05 F2 6.1859E?03 std. 5.7689E?03 1.0675E?04 std. 8.9516E?03 1.3489E?10 std. 1.6188E?10 8.6635E?03 std. 1.0049E?04 6.7238E?03 std. 7.7135E?03 F3 3.2000E?02 std. 3.2804E-03 3.2002E?02 std. 1.2441E-01 3.2010E?02 std. 2.7214E-01 3.2000E?02 std. 2.1764E-03 3.2000E?02 std. 2.8434E-02 F4 4.2844E102 std. 1.2620E101 4.3982E?02 std. 1.2373E?01 4.8339E?02 std. 4.4692E?01 4.3935E?02 std. 1.5810E?01 4.3904E?02 std. 1.5278E?01 F5 1.1322E103 std. 3.0624E102 1.3323E?03 std. 3.1250E?02 2.0994E?03 std. 6.7932E?02 1.3351E?03 std. 3.0400E?02 1.3499E?03 std. 3.4549E?02 F6 2.7960E103 std. 2.2556E103 3.3423E?03 std. 3.2601E?03 4.6890E?07 std. 1.0476E?08 3.6839E?03 std. 3.2689E?03 3.8033E?03 std. 3.2807E?03 F7 7.0272E?02 std. 1.0536E?00 7.1861E?02 std. 1.0964E?02 7.6943E?02 std. 1.0752E?02 7.0344E?02 std. 1.2585E?00 7.0325E?02 std. 1.3506E?00 F8 4.4535E?03 std. 4.6627E?03 4.8488E?03 std. 5.5264E?03 1.3342E?07 std. 5.2942E?07 5.1208E?03 std. 4.2459E?03 4.8337E?03 std. 4.0549E?03 22 of 7 Page F9 1.0119E?03 std. 3.9973E?01 1.0131E?03 std. 4.3650E?01 1.1080E?03 std. 1.2368E?02 1.0151E?03 std. 4.4810E?01 1.0322E?03 std. 6.5207E?01 F10 4.3914E?03 std. 4.0614E?03 3.7711E103 std. 2.9422E103 2.0918E?07 std. 6.4220E?07 7.7244E?03 std. 4.9292E?03 3.9363E?03 std. 2.3681E?03 F11 1.4154E?03 std. 1.2610E?02 1.4427E?03 std. 1.2651E?02 1.6671E?03 std. 2.6696E?02 1.4169E?03 std. 1.3102E?02 1.4335E?03 std. 1.3211E?02 F12 1.3056E?03 std. 1.3585E?01 1.3060E?03 std. 1.3581E?01 1.3582E?03 std. 4.2714E?01 1.3080E?03 std. 1.3492E?01 1.3050E?03 std. 2.6571E?00 F13 1.3000E103 std. 1.4063E-02 1.3001E?03 std. 1.8683E-01 1.6212E?03 std. 9.4928E?02 1.3001E?03 std. 3.8868E-02 1.3001E?03 std. 5.7390E-02 F14 5.1167E?03 std. 4.1835E?03 6.8519E?03 std. 4.9327E?03 1.6287E?04 std. 1.0565E?04 5.8034E?03 std. 4.2787E?03 5.8597E?03 std. 4.5701E?03 F15 1.6000E?03 std. 2.4321E-06 1.6005E?03 std. 3.8970E?00 8.2030E?03 std. 1.0302E?04 1.6000E103 std. 6.3048E-08 1.6000E?03 std. 7.3191E-07 65

65 Page 8 of 22 Sådhanå (2021) 46:65 found in standard optimization algorithm are determined by outperforms the standard HFPSO algorithm. Standard chaotic maps. deviation values indicate that the obtained results are rea- sonably consistent. According to the results in table 3, Sine and Chebyshev 3.6 Experimental set-up chaotic map based HFPSO algorithms are better in terms of average success rank in CEC 2015 10D functions. Sinu- In experimental studies, CEC 2015 [49, 51] learning-based soidal chaotic map-based algorithm is the worst in all real-parameter test functions and computationally expen- problems and original HFPSO is better than only Tent, sive numerical optimization problems were used. As Circle, Singer and Sinusoidal chaotic algorithms. described in the proposed method for determining the In figure 3, the best solutions’ averaged results are shown results on the performance of chaotic effects, Firefly algo- as convergence graphs. When the graph is examined the rithm random (ei) and Particle swarm algorithm (rand1, sinusoidal chaos map has clearly the worst convergence rand2) parameters are generated by chaotic maps. The curve rather than the other maps and the standard HFPSO in initial values of each chaos map are initialized from a all functions. random real value between [0, 1]. Chaotic number The CEC 2015 functions are rerun as 30D to see how the sequences in the size of [1000 9 D 9 Swarm size] are results of the experiments would be in higher dimensional generated. The generated chaotic values are normalized problems. In this way, when we look at the results in between 0 and 1. table 4, it is seen that the original HFPSO algorithm is In experiments, a computer that has Windows 10 oper- generally unsuccessful compared to chaos map-based ating system, Intel Core i7 processor and 16 GB ram was algorithms. HFPSO is worse on all 10D, 30D 15 problems used. Experimental program source codes are written by against chaotic maps. use of MATLAB software version 2018a. Figure 4 shows that the Sinusoidal chaos map exhibits To comprehend the effectiveness of the proposed different convergence curves. When the CEC 2015 30D method, the ability to solve benchmark functions that are convergence graphs are examined, it is seen that the F3, F5, accepted in the literature are examined. Initially to F9, F11, F12 functions are more sensitive to the chaos- demonstrate the success of the proposed method in our based maps and the convergence graphs can be separated study, we perform tests with CEC2015 test suite II func- more clearly. tions as 10-dimensional (10D) and 30-dimensional (30D), In tables 5, 6 and figures 5, 6, algorithms are ranked and given in table 2. Finally, CEC 2015 test suite I and con- shown according to their performance. According to the straint problems are used for demonstrating performance of values in the tables, rank value 1 is the best and rank value the proposed method. 11 indicates the worst performance. Average rank success The function set given in table 2 has different charac- values were shown in the bottom row of the tables. When it teristics. In this way, the effectiveness of the proposed is sorted by these average values; Sine, Chebyshev and method can be evaluated in different problem groups. The Piecewise chaotic based HFPSO algorithms are better in desired solution being found is the x values of terms of average success in CEC 2015 10D functions. The T min FxðÞ; x ¼ ½x1; x2; x3; ...xD : The Fi value given in original HFPSO is only better than the Tent, Circle, Singer, table 2 represents the global optimum values of the func- Sinusoidal chaotic map algorithms. Rest six chaotic maps tions. The search range value is between [-100, ?100]. outperform original HFPSO algorithm. In table 7, for 30 dimensional problems, the success of the ranks of Piecewise, Sine and Chebyshev chaotic maps 4. Results and discussions are seen to be better. In 10 dimensional problems; Sine [ Chebyshev [ [ [ [ [ 4.1 Unconstrained benchmark problems Piecewise Logistic Gauss/mouse Iterative HFPSO [Tent[Circle[Singer[Sinusoidal rank is performed. 6 In experimental studies, each function was run 51 times and of the 10 maps are better than the original algorithm and thus, the general results were obtained. In experiments, a significantly improve the original algorithm. For 30 function is allowed to call maximum only 1000 9 D times. dimensional problems, Piecewise [ Sine [ Chebyshev [ The number of particles and firefly used in the algorithms Gauss/mouse [ Iterative [ Tent [ Logistic [ HFPSO[ was chosen to be the same as the problem size (Swarm Singer[Circle[Sinusoidal rank is performed. 7 of the 10 size = D). All remaining parameter values were set as same maps produce better results than the original standard with original HFPSO algorithm. Table 3 shows the mean algorithm and significantly improve the original algorithm. and standard deviation results for CEC 2015 10D functions. Sinusoidal map is the worst. The reason for this is, if the Briefly, CEC 2015 functions’ global optimums are showed initial value is greater than 0.5, the values generated by map in table 2. Thus, a lower-valued result indicates a more are mostly greater than 0.5, and if the initial value is less successful convergence. When we look at the results in the than 0.5, the values produced by the map are mostly 0. This table, clearly, HFPSO that employs chaos-based map limits the algorithm’s sensitive search capability, especially Sådhanå (2021) 46:65 Page 9 of 22 65

Figure 3. CEC 2015 10D Convergence graphs. 65 Page 10 of 22 Sådhanå (2021) 46:65

Figure 3. continued S å dhan å

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Table 4. CEC 2015 30D Mean and Std. results.

Func. Chebyshev Circle Gauss/mouse Iterative Logistic Piecewise F1 2.0527E?06 std. 9.2947E?05 2.0444E?06 std. 1.1295E?06 1.5933E?06 std. 7.6659E?05 1.4214E?06 std. 8.0904E?05 7.0090E?07 std. 3.6244E?08 1.3106E?06 std. 6.1635E?05 F2 3.8855E?03 std. 4.9363E?03 3.4702E?03 std. 4.7423E?03 3.1582E?03 std. 3.9203E?03 3.4134E?03 std. 3.7864E?03 3.9929E?03 std. 4.8071E?03 4.7928E?03 std. 5.6305E?03 F3 3.2011E?02 std. 4.9355E-02 3.2009E?02 std. 9.7101E-02 3.2008E?02 std. 6.0489E-02 3.2000E102 std. 5.1080E-03 3.2012E?02 std. 6.3883E-02 3.2001E?02 std. 1.4451E-02 F4 5.4100E?02 std. 3.3139E?01 5.5723E?02 std. 3.6582E?01 5.4803E?02 std. 2.7978E?01 5.5563E102 std. 3.4504E101 5.3923E?02 std. 4.0200E?01 5.4307E?02 std. 3.6203E?01 F5 4.0146E?03 std. 7.7753E?02 4.0753E?03 std. 7.0388E?02 4.1090E?03 std. 6.5670E?02 4.2562E?03 std. 7.3790E?02 4.0965E?03 std. 6.2201E?02 4.0082E?03 std. 6.9025E?02 F6 3.3712E?05 std. 2.0815E?05 1.5026E?05 std. 9.3297E?04 1.8228E?05 std. 1.1751E?05 1.5188E?05 std. 1.0938E?05 2.9099E?05 std. 1.6986E?05 1.5600E?05 std. 9.9918E?04 F7 7.1609E?02 std. 1.1510E?01 7.2315E?02 std. 2.0427E?01 7.2097E?02 std. 1.7670E?01 7.1912E?02 std. 1.6450E?01 7.1372E?02 std. 3.4459E?00 7.1896E?02 std. 1.4603E?01 F8 8.5964E?04 std. 7.8794E?04 9.8641E?04 std. 7.6697E?04 9.2152E?04 std. 7.4844E?04 7.1984E?04 std. 5.7511E?04 1.3868E?05 std. 1.7322E?05 5.0216E104 std. 4.4617E104 F9 1.0530E?03 std. 9.6087E?01 1.0492E?03 std. 1.0375E?02 1.0360E?03 std. 8.1992E?01 1.0592E?03 std. 1.1414E?02 1.0418E?03 std. 8.5209E?01 1.0193E103 std. 6.0735E101 F10 1.4900E?05 std. 9.5806E?04 2.4940E?05 std. 1.4714E?05 1.4029E?05 std. 9.0428E?04 1.2015E?05 std. 7.2647E?04 1.5593E?05 std. 8.5312E?04 1.0624E?05 std. 5.5553E?04 F11 1.7716E?03 std. 3.0392E?02 1.9876E?03 std. 3.6941E?02 1.7691E?03 std. 3.7755E?02 1.7900E?03 std. 3.5974E?02 1.7409E?03 std. 3.1268E?02 1.7416E?03 std. 3.4995E?02 F12 1.3178E103 std. 2.7510E101 1.3209E?03 std. 2.9369E?01 1.3219E?03 std. 3.1660E?01 1.3200E?03 std. 2.9691E?01 1.3231E?03 std. 3.3697E?01 1.3183E?03 std. 2.7349E?01 F13 1.3000E?03 std. 1.9565E-02 1.3001E?03 std. 1.2861E-01 1.3001E?03 std. 5.3551E-02 1.3001E?03 std. 3.6584E-02 1.3000E103 std. 9.5077E-03 1.3000E?03 std. 2.1698E-02 F14 3.4196E?04 std. 5.0666E?03 3.6390E?04 std. 2.4206E?03 3.3403E104 std. 8.3112E103 3.5265E?04 std. 2.0961E?03 3.6172E?04 std. 1.3846E?04 3.4250E?04 std. 5.0145E?03 F15 1.6011E?03 std. 3.8596E?00 1.6098E?03 std. 6.6554E?00 1.6012E?03 std. 4.1079E?00 1.6024E?03 std. 5.3098E?00 1.6010E?03 std. 6.7948E?00 1.6026E?03 std. 5.6532E?00 Func. Sine Singer Sinusoidal Tent HFPSO F1 2.2701E?06 std. 8.7609E?05 1.5424E?06 std. 7.2639E?05 3.6148E?09 std. 3.0474E?09 1.2223E106 std. 6.2568E105 1.4588E?06 std. 6.7985E?05 F2 3.0460E103 std. 3.2346E103 3.9039E?03 std. 4.4505E?03 5.9631E?10 std. 7.8129E?10 5.4046E?03 std. 4.9451E?03 4.9486E?03 std. 5.3078E?03 F3 3.2010E?02 std. 4.8000E-02 3.2013E?02 std. 7.2706E-02 3.2032E?02 std. 5.0412E-01 3.2000E?02 std. 1.7258E-02 3.2000E?02 std. 1.0646E-02 F4 5.2278E?02 std. 3.2246E?01 5.6723E?02 std. 3.0896E?01 8.3763E?02 std. 2.1933E?02 5.6181E?02 std. 3.2425E?01 5.5381E?02 std. 3.0032E?01 F5 3.9257E103 std. 7.6502E102 4.2724E?03 std. 7.7217E?02 6.7389E?03 std. 2.0005E?03 4.1195E?03 std. 6.4685E?02 4.0831E?03 std. 6.5717E?02 F6 3.1309E?05 std. 2.8605E?05 1.5805E?05 std. 9.8417E?04 2.7162E?08 std. 3.0719E?08 1.2348E105 std. 8.6142E104 1.6764E?05 std. 1.0761E?05 1 1 ? ? ? ? ? ? ? ?

F7 7.1352E 02 std. 5.5700E 00 7.1644E 02 std. 5.1297E 00 1.4096E 03 std. 9.2494E 02 7.1947E 02 std. 1.6085E 01 7.2130E 02 std. 1.6847E 01 22 of 11 Page F8 8.3680E?04 std. 6.7229E?04 6.6474E?04 std. 5.3182E?04 8.5585E?07 std. 1.1876E?08 6.8730E?04 std. 4.7931E?04 6.5570E?04 std. 4.9072E?04 F9 1.0323E?03 std. 8.3265E?01 1.0659E?03 std. 1.2971E?02 1.4419E?03 std. 3.3246E?02 1.0883E?03 std. 1.3414E?02 1.0501E?03 std. 1.0078E?02 F10 1.7717E?05 std. 8.8484E?04 9.3765E?04 std. 5.3446E?04 1.8896E?08 std. 2.9434E?08 9.0262E104 std. 5.0804E104 1.0955E?05 std. 6.8090E?04 F11 1.7632E?03 std. 2.9851E?02 1.8206E?03 std. 3.8933E?02 2.8722E?03 std. 1.0748E?03 1.7264E103 std. 3.6095E102 1.8012E?03 std. 3.8386E?02 F12 1.3232E?03 std. 3.3635E?01 1.3248E?03 std. 3.5355E?01 1.3897E?03 std. 6.2209E?01 1.3275E?03 std. 3.6377E?01 1.3274E?03 std. 3.6424E?01 F13 1.3000E?03 std. 1.4303E-02 1.3001E?03 std. 6.0234E-02 2.2671E?03 std. 1.2392E?03 1.3001E?03 std. 4.7977E-02 1.3001E?03 std. 7.2299E-02 F14 3.4797E?04 std. 1.7806E?03 3.4677E?04 std. 5.0526E?03 1.1576E?05 std. 7.9448E?04 3.4934E?04 std. 5.1801E?03 3.4610E?04 std. 5.1987E?03 F15 1.6002E103 std. 1.4541E100 1.6030E?03 std. 5.9144E?00 3.5400E?05 std. 4.2202E?05 1.6045E?03 std. 6.6177E?00 1.6022E?03 std. 5.3164E?00 65

65 Page 12 of 22 Sådhanå (2021) 46:65

Figure 4. CEC 2015 30D Convergence graphs. Sådhanå (2021) 46:65 Page 13 of 22 65

Figure 4. continued 65 Page 14 of 22 Sådhanå (2021) 46:65

Table 5. CEC 2015 10D Rank results.

Func. Chebyshev Circle Gauss/mouse Iterative Logistic Piecewise Sine Singer Sinusoidal Tent HFPSO F1 6 10 9 1 4 2 7 3 11 8 5 F2 1 3 7 6 8 5 2 10 11 9 4 F3 7 6 4 9 5 1 3 10 11 2 8 F4 3 7 5 6 2 4 1 10 11 9 8 F5 2 10 4 5 3 6 1 7 11 8 9 F6 2 4 7 6 8 5 1 3 11 9 10 F7 3 9 7 6 1 4 2 10 11 8 5 F8 4 10 6 3 2 1 5 8 11 9 7 F9 1 5 2 9 8 7 3 4 11 6 10 F10 3 10 5 6 8 7 4 1 11 9 2 F11 9 3 6 1 2 5 4 10 11 7 8 F12 2 8 6 4 1 3 7 9 11 10 5 F13 3 10 2 9 4 7 1 8 11 5 6 F14 3 8 1 10 7 4 2 9 11 5 6 F15 7 3 9 2 8 4 6 10 11 1 5 Avg. 3.73 7.07 5.33 5.53 4.73 4.33 3.27 7.47 11 7 6.53

Table 6. CEC 2015 30D Rank results.

Func. Chebyshev Circle Gauss/mouse Iterative Logistic Piecewise Sine Singer Sinusoidal Tent HFPSO F1 8 7 6 3 10 2 9 5 11 1 4 F2 5 4 2 3 7 8 1 6 11 10 9 F3 8 6 5 1 9 4 7 10 11 2 3 F4 3 8 5 7 2 4 1 10 11 9 6 F5 3 4 7 9 6 2 1 10 11 8 5 F6 10 2 7 3 8 4 9 5 11 1 6 F7 3 10 8 6 2 5 1 4 11 7 9 F8 7 9 8 5 10 1 6 3 11 4 2 F9 7 5 3 8 4 1 2 9 11 10 6 F10 7 10 6 5 8 3 9 2 11 1 4 F11 6 10 5 7 2 3 4 9 11 1 8 F12 1 4 5 3 6 2 7 8 11 10 9 F13 3 10 7 6 1 4 2 9 11 5 8 F14 2 10 1 8 9 3 6 5 11 7 4 F15 3 10 4 6 2 7 1 8 11 9 5 Avg. 5.07 7.27 5.27 5.33 5.73 3.53 4.40 6.87 11 5.67 5.87

Figure 5. CEC 2015 10D Rank bar graph. Sådhanå (2021) 46:65 Page 15 of 22 65

Figure 6. CEC 2015 30D Rank bar graph

Table 7. Constrained benchmark problems [50].

Problem no. Dimension Constrained function(s) Variables bounds

12 1-1 B xi B 1, i = 1, 2 22 20B xi B 10, i = 1, 2 32 213B x1 B 100, 0 B x2 B 100 47 4-10 B xi B 10, i = 1, 2, 3, 4, 5, 6, 7 55 678B x1 B 102, 33 B x2 B 45, 27 B xi B 45, i = 3, 4, 5

Table 8. Constrained problem no. 1. Table 9. Constrained problem no. 2. Algorithm Mean Std. MaxFES Algorithm Mean Std. MaxFES HM 0.7500 N/A 1400,000 ASCHEA N/A N/A 1500,000 HM -0.0892 N/A 1,400,000 CRGA 0.7570 2.50E-03 3000 ASCHEA -0.0958 N/A 1,500,000 - PSO -0.0945 9.40E-03 10,600 SAPF 0.7570 2.00E 03 500,000 - - - PSO-DE 0.0958 1.30E 12 10,600 CULDE 0.7964 1.71E 02 100,100 SR -0.0958 2.60E-17 76,200 SMES 0.7500 1.52E-04 75,000 CAEP -0.0958 0.00E?00 50,020 PSO 0.9988 8.40E-02 70,100 DE -0.0958 N/A 10,000 - - PSO-DE 0.7500 2.50E-07 70,100 HPSO 0.0958 1.20E 10 81,000 NM-PSO -0.0958 3.50E-08 2103 DE 0.7490 N/A 30,000 CRGA -0.0958 4.40E-06 64,900 SR 0.7500 8.00E-05 350,000 SAPF -0.0956 1.06E-03 500,000 DEDS 0.7499 0.00E?00 225,000 GA -0.0958 2.70E-09 4486 HEAA 0.7500 3.40E-16 200,000 SMES -0.0958 0.00E?00 240,000 - CULDE -0.0958 1.00E-07 100,100 ISR 0.7500 1.10E 16 137,200 - - a - DEDS 0.0958 4.00E 17 225,000 -Simplex 0.7499 4.90E 16 308,125 HEAA -0.0958 2.80E-17 200,000 ABC 0.7500 0.00E?00 240,000 ISR -0.0958 2.70E-17 160,000 EPSO 0.7508 1.62E-04 50,000 a-Simplex -0.0958 3.80E-13 306,248 - ABC -0.0958 0.00E?00 240,000 Chebyshev 0.7500 6,87E 05 25,000 - - - EPSO 0.0958 1.26E 17 5,000 Circle 0.7503 5,56E 04 Chebyshev -0.0958 5.53E-13 2,500 Gauss/mouse 0.7501 1,93E-04 Circle -0.0958 5.34E-17 Iterative 0.7501 2,40E-04 Gauss/mouse -0.0958 2.35E-14 Logistic 0.7500 5,45E-05 Iterative -0.0958 1.66E-16 - Logistic -0.0958 2.94E-14 Piecewise 0.7504 1,22E 03 - - - Piecewise 0.0958 8.59E 16 Sine 0.7500 3,40E 05 Sine -0.0958 2.19E-13 Singer 0.7500 4,91E-05 Singer -0.0958 1.29E-13 Sinusoidal 0.8020 9,56E-02 Sinusoidal N/A N/A - - Tent 0.7510 1,77E-03 Tent 0.0958 3.01E 17 HFPSO -0.0936 1.22E-02 HFPSO 0.7503 7,55E-04 65 Page 16 of 22 Sådhanå (2021) 46:65

Table 10. Constrained problem no. 3. Table 11. Constrained problem no. 4.

Algorithm Mean Std. MaxFES Algorithm Mean Std. MaxFES HM -6342.6 N/A 1,400,000 HM 681.16 4.11E-02 1,400,000 ASCHEA -6961.81 N/A 1,500,000 ASCHEA 680.641 N/A 1,500,000 CULDE -6961.814 1.00E-07 100,100 IGA 680.63 1.00E-05 N/A DE -6961.814 N/A 15,000 GA 680.638 6.61E-03 320,000 CRGA -6740.288 2.70E?02 3700 GA1 680.642 N/A 350,070 PSO-DE -6961.814 2.30E-09 140,100 GA2 N/A N/A 350,070 PSO -6961.814 6.50E-06 140,100 CRGA 681.347 5.70E-01 50,000 SR -6875.94 160E?00 118,000 SAPF 681.246 3.22E-01 500,000 SMES -6961.284 1.85E?00 240,000 SR 680.656 3.40E-02 350,000 SAPF -6953.061 5.87E?00 500,000 HS N/A N/A 160,000 DEDS -6961.814 0.00E?00 225,000 DE 680.503 6.70E-01 240,000 ABC -6961.813 2.00E-03 240,000 CULDE 680.63 1.00E-07 100,100 HEAA -6961.814 4.60E-12 200,000 PSO 680.971 5.10E-01 140,100 ISR -6961.814 1.90E-12 168,800 CPSO- 680.781 1.48E-02 N/A a-Simplex -6961.814 1.30E-10 293,367 SMES 680.643 1.55E-02 240,000 EPSO -6961.813 6.20E-04 20,000 DEDS 680.63 2.90E-13 225,000 Chebyshev -6960.944 6.79E-01 10,000 HEAA 680.63 5.80E-13 200,000 Circle N/A N/A ISR 680.63 3.20E-13 271,200 Gauss/mouse -6961.354 2.52E-01 a-Simplex 680.63 2.90E-10 323,426 Iterative -6961.663 1.57E-01 PESO 680.63 N/A 350,000 Logistic -6960.557 1.03E?00 CoDE 681.503 N/A 248,000 Piecewise -6961.686 1.07E-01 ABC 680.64 4.00E-03 240,000 Sine -6960.500 1.07E?00 EPSO 680.649 8.14E-03 125,000 Singer -6961.552 1.58E-01 Chebyshev 680.679 5.53E-02 62,500 Sinusoidal N/A N/A Circle 680.746 8.94E-02 Tent -6961.795 1.93E-02 Gauss/mouse 680.685 4.99E-02 HFPSO -6961.661 2.17E-01 Iterative 680.690 4.60E-02 Logistic 680.665 2.92E-02 Piecewise 680.694 3.80E-02 Sine 680.665 1.94E-02 Singer 680.700 4.77E-02 for local search. On the contrary, Sine, Piecewise or Che- Sinusoidal N/A N/A byshev chaotic maps generate numbers more dynamically. Tent 680.700 4.89E-02 This provides diversity of particles and fireflies, and HFPSO 680.677 2.76E-02 searching space is explored and exploited much better. It can be said that by using these maps, the exploration and exploitation capabilities of the algorithm can be better improved. appears to be highly successful against most of other methods. As seen, reference of the results, EPSO algo- 4.2 Constrained benchmark problems rithm’s MaxFES sizes have halved and used. Accordingly In experiments 5 constrained benchmark problems from results, it is seen that most of our proposed chaos map- [50] are used and compared on showed table 7. The other based methods, over all problems converge to optima’s in a methods’ results are extracted from the same paper. Results competitive manner and can find a solution. are obtained after 30 independent runs. Different number of It is seen in table 8 that Chebyshev, Logistic, Sine and maximum fitness function evaluations (MaxFES) are used Singer maps are successful like the other algorithms for to compare algorithms and experimental results are shown. problem no. 1. As shown in table 9, the proposed method As shown in tables 8, 9, 10, 11 and 12 during the tests, can find the best solution like most of the other algorithms, the proposed algorithm uses lower maximum number of with all maps except Sinusoidal may. As seen in table 10, fitness evaluations (MaxFES) against to most of the rest the proposed method cannot find a solution with Circle and methods. The convergence speed of the proposed method Sinusoidal maps, under the constraints of benchmark Sådhanå (2021) 46:65 Page 17 of 22 65

Table 12. Constrained problem no. 5. Similarly, as shown in table 12, competitive solutions are obtained with chaotic maps except a few maps in problem Algorithm Mean Std. MaxFES no. 5. According to the results, it is concluded that sinu- HM -30665.3 N/A 1,400,000 soidal chaos map is not suitable for constrained optimiza- ASCHEA -30665.5 N/A 1,500,000 tion problems, on the other hand, it is determined that SR -30665.539 2.00E-05 88,200 logistic, piecewise, sine maps are suitable for constrained CAEP -30662.5 9.30E?00 50,020 optimization problems. PSO -30570.929 8.10E?01 70,100 HPSO -30665.539 1.70E-06 81,000 PSO-DE -30665.538 8.30E-10 70,100 4.3 Comparison with other optimization - - CULDE 30665.538 1.00E 07 100,100 algorithms DE -30665.536 5.07E-03 240,000 HS N/A N/A 65,000 To compare proposed method against other hybrid and CRGA -30664.398 1.60E?00 54,400 counterpart optimization algorithms, CEC 2015 test suite I, - ? SAPF 30655.922 2.04E 00 500,000 expensive numerical optimization test functions [51] are - ? SMES 30665.539 0.00E 00 240,000 used as 30 dimensional. 20 independent runs are made and ABC -30665.539 0.00E?00 240,000 maximum number of fitness function evaluations DELC -30665.539 1.00E-11 50,000 DEDS -30665.539 2.70E-11 225,000 (MaxFES) is used as 1500 and swarm size (population) are HEAA -30665.539 7.40E-12 200,000 preferred as 30. Proposed method is run with 10 different ISR -30665.539 1.10E-11 192,000 chaos maps on related problem set, on table 13 the results a-Simplex -30665.539 4.20E-11 305,343 are shown. It is indicated that, iterative map is better than EPSO -30665.538 1.07E-11 50,000 others on this problem set. It outperforms 6 out of 15 Chebyshev -30665.539 3.69E-05 25,000 problems against rest chaos maps. Circle -30653.237 4.37E?01 Arunachalam et al [52] committed hybrid particle swarm Gauss/mouse -30665.538 1.29E-03 optimization and firefly algorithm (HPSOFF). Kora et al - - Iterative 30665.539 7.88E 06 [53] introduced hybrid firefly and particle swarm opti- - - Logistic 30665.539 9.14E 12 mization (FFPSO) algorithm and Ngo et al [50] designed Piecewise -30665.539 8.22E-08 the Extraordinariness Particle Swarm Optimizer (EPSO) Sine -30665.539 1.49E-11 Singer -30665.539 2.31E-06 algorithm. Results of the Differential Evolution (DE) [54], Sinusoidal -29817.089 6.53E?02 Evolutionary Strategy (ES), Covariance Matrix Adaptation Tent -30664.946 3.05E?00 (CMAES-S, CMAES-G) [55], EPSO, HFPSO -30663.484 1.13E?01 ISRPSO [56] algorithms are directly obtained from the previous papers [2, 50, 56]. In table 14. Our proposed CHFPSO are compared to canonical PSO and FA and other hybrid algorithms. CHPSO with iterative map outperforms others on all 15 problems with comparable standard deviation values. In table 15, again, we tested our proposed method against functions in problem no. 3. The best chaotic map is some improved optimization methods in the literature. On 7 Piecewise map for this problem. According to the problem, out of 15 problems chaos based proposed method outper- the successful map can be different. The results for problem forms other improved algorithms. According to table, while no.4 are seen in table 11. The proposed method can find CMAES-S outperforms on only 2 problems, HFPSO out- very good solutions with Logistic and Sine map in problem performs the rests on 3 out of 15 problems. no. 4. 65

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Table 13. CHFPSO CEC 2015 computationally expensive results.

CHFPSO (Proposed) Mean

Chebyshev Circle Gauss/mouse Iterative Logistic Piecewise Sine Singer Sinusoidal Tent F1 5,2490E108 3,4382E?09 1,0338E?09 1,2042E?09 7,3808E?08 1,1816E?09 8,8591E?08 9,3722E?08 4,2800E?10 2,2773E?09 F2 8,1656E?04 8,4497E?04 7,8970E?04 8,7531E?04 7,9472E?04 8,4217E?04 7,5334E104 7,6228E?04 1,0565E?05 8,6387E?04 F3 3,2926E?02 3,3056E?02 3,2821E?02 3,2892E?02 3,2891E?02 3,2807E102 3,2905E?02 3,3037E?02 3,3862E?02 3,2894E?02 F4 4,8737E?03 4,5669E?03 4,6433E?03 4,0473E103 4,9032E?03 4,3875E?03 5,3237E?03 4,7199E?03 6,6839E?03 4,4131E?03 F5 5,0338E?02 5,0367E?02 5,0338E?02 5,0319E?02 5,0326E?02 5,0352E?02 5,0313E102 5,0329E?02 5,0337E?02 5,0340E?02 F6 6,0070E?02 6,0065E?02 6,0062E?02 6,0061E102 6,0062E?02 6,0066E?02 6,0063E?02 6,0066E?02 6,0383E?02 6,0064E?02 F7 7,0054E?02 7,0323E?02 7,0051E?02 7,0072E?02 7,0079E?02 7,0044E102 7,0097E?02 7,0048E?02 8,7587E?02 7,0112E?02 F8 1,1521E?03 6,8747E?03 1,3056E?03 1,0370E103 1,0589E?03 1,1057E?03 1,7255E?03 1,1381E?03 1,7750E?08 2,4983E?03 F9 9,1318E?02 9,1337E?02 9,1340E?02 9,1330E?02 9,1333E?02 9,1327E?02 9,1342E?02 9,1312E102 9,1375E?02 9,1335E?02 F10 5,8641E?06 5,2482E?06 4,2073E?06 3,8178E?06 6,2479E?06 3,3275E106 7,2318E?06 4,1064E?06 3,1050E?08 4,8109E?06 F11 1,1388E?03 1,1442E?03 1,1502E?03 1,1316E103 1,1398E?03 1,1461E?03 1,1375E?03 1,1379E?03 1,7313E?03 1,1360E?03 F12 1,6973E?03 1,8322E?03 1,7489E?03 1,6959E103 1,7466E?03 1,7844E?03 1,7615E?03 1,8256E?03 1,6696E?05 1,8062E?03 F13 1,6849E?03 1,7185E?03 1,6898E?03 1,6878E?03 1,6822E103 1,6828E?03 1,6913E?03 1,6895E?03 3,2402E?03 1,6981E?03 F14 1,6468E103 1,6677E?03 1,6556E?03 1,6622E?03 1,6538E?03 1,6571E?03 1,6546E?03 1,6627E?03 1,9027E?03 1,6608E?03 F15 2,5247E?03 2,5509E?03 2,5254E?03 2,4716E103 2,4852E?03 2,4906E?03 2,5507E?03 2,4978E?03 3,2912E?03 2,5598E?03 F1 3,2404E?08 1,7782E?09 6,5894E?08 7,6293E?08 2,4266E108 9,2648E?08 4,1146E?08 5,2030E?08 4,5158E?10 9,4702E?08 F2 2,1007E?04 1,8744E?04 1,8329E?04 1,8703E?04 1,7753E?04 2,0496E?04 1,5117E104 2,1103E?04 2,6225E?04 1,6531E?04 F3 2,8109E100 4,1600E?00 3,1427E?00 4,0200E?00 4,1132E?00 4,3264E?00 4,1702E?00 2,8346E?00 8,0449E?00 3,5518E?00 F4 6,3959E?02 6,1503E?02 5,8715E102 6,2383E?02 6,9882E?02 6,8540E?02 8,9169E?02 9,1693E?02 1,9310E?03 7,4048E?02 F5 6,1368E-01 8,0237E-01 7,0980E-01 6,0320E-01 6,4906E-01 6,2716E-01 6,2951E-01 7,3991E-01 1,4281E?00 6,2847E-01 F6 1,3553E-01 1,2488E-01 1,3474E-01 7,8108E-02 1,1470E-01 1,2635E-01 8,3019E-02 1,0241E-01 3,7420E?00 1,5598E-01 - ? - ? ? - ? - ? ?

F7 2,2212E 01 4,1708E 00 2,2145E 01 1,0241E 00 1,3527E 00 1,4161E 01 1,7543E 00 1,8928E 01 1,1164E 02 2,7531E 00 S

? ? ? 1 ? ? ? ? ? ? å F8 8,9162E 02 8,0292E 03 1,1044E 03 2,3469E 02 2,9093E 02 4,4279E 02 1,5715E 03 3,6387E 02 1,5356E 08 3,3516E 03 dhan F9 3,0097E-01 4,4105E-01 3,6353E-01 2,6371E-01 3,9887E-01 5,1513E-01 3,3868E-01 3,9064E-01 4,3461E-01 3,6239E-01

F10 3,4673E?06 3,8152E?06 3,3367E?06 2,2252E?06 3,8967E?06 2,5967E?06 5,2975E?06 1,9264E106 3,6324E?08 2,6184E?06 å

F11 2,6748E?01 2,8615E?01 3,5648E?01 2,0441E101 2,7565E?01 3,0978E?01 2,4258E?01 2,6345E?01 6,2206E?02 2,5300E?01 (2021) F12 2,4400E?02 2,4829E?02 2,4750E?02 2,0391E?02 2,3658E?02 2,9943E?02 2,3903E?02 2,4781E?02 4,4423E?05 1,5852E102 ? ? ? ? ? ? ? 1 ? ?

F13 1,7370E 01 4,1092E 01 1,8883E 01 1,1512E 01 1,5290E 01 1,2077E 01 1,7639E 01 1,0686E 01 1,2190E 03 2,3146E 01 46:65 F14 1,5680E?01 2,4069E?01 1,4985E?01 2,6103E?01 1,3241E101 2,1360E?01 1,7157E?01 2,5673E?01 3,1953E?02 1,9668E?01 ? ? ? ? ? ? 1 ? ? ?

F15 1,2096E 02 1,7087E 02 2,0874E 02 2,1624E 02 1,6364E 02 2,0741E 02 9,8519E 01 1,5950E 02 6,2415E 02 1,5249E 02 S å dhan å

(2021) 46:65

Table 14. Comparisons of proposed CHFPSO vs. other counterpart and hybrid algorithms.

PSO FA FFPSO HPSOFF CHFPSO (Iterative)

Mean Std. Mean Std. Mean Std. Mean Std. Mean Std. F1 3,9049E?09 1,6017E?09 2,8899E?10 5,8485E?09 9,3283E?10 1,2799E?10 4,7539E?09 1,3400E?09 1,2042E109 7,6293E?08 F2 9,9760E?04 2,5812E?04 1,3418E?05 2,6344E?04 6,9430E?06 1,4507E?07 9,7376E?04 2,0814E?04 8,7531E104 1,8703E?04 F3 3,3113E?02 2,8347E?00 3,3850E?02 1,9604E?00 3,4771E?02 1,9232E?00 3,3059E?02 2,6174E?00 3,2892E102 4,0200E?00 F4 7,7928E?03 5,2384E?02 8,0330E?03 4,2608E?02 9,6696E?03 4,0262E?02 6,8199E?03 8,5746E?02 4,0473E103 6,2383E?02 F5 5,0418E?02 7,2160E-01 5,0425E?02 5,1916E-01 5,0586E?02 1,1990E?00 5,0422E?02 6,1627E-01 5,0319E102 6,0320E-01 F6 6,0096E?02 1,8792E-01 6,0410E?02 2,2901E-01 6,0755E?02 6,6978E-01 6,0090E?02 1,3068E-01 6,0061E102 7,8108E-02 F7 7,0405E?02 2,0114E?00 7,6997E?02 1,2225E?01 8,9401E?02 2,6905E?01 7,0750E?02 4,2654E?00 7,0072E102 1,0241E?00 F8 4,0013E?03 3,3477E?03 2,3491E?06 1,4471E?06 1,6032E?08 1,1620E?08 1,7191E?04 3,3261E?04 1,0370E103 2,3469E?02 F9 9,1358E?02 2,3378E-01 9,1381E?02 2,8076E-01 9,1427E?02 1,8457E-01 9,1365E?02 3,1664E-01 9,1330E102 2,6371E-01 F10 7,5602E?06 3,1209E?06 2,9938E?07 1,4513E?07 3,9391E?08 1,6830E?08 1,1337E?07 6,3462E?06 3,8178E106 2,2252E?06 F11 1,1614E?03 4,1043E?01 1,2889E?03 3,7640E?01 2,1094E?03 4,6591E?02 1,1551E?03 2,8498E?01 1,1316E103 2,0441E?01 F12 2,2417E?03 1,8647E?02 2,9655E?03 2,7960E?02 4,9876E?05 6,1606E?05 2,0617E?03 2,0746E?02 1,6959E103 2,0391E?02 F13 1,7719E?03 2,6527E?01 1,9596E?03 8,1067E?01 3,6329E?03 7,7365E?02 1,7390E?03 2,3388E?01 1,6878E103 1,1512E?01 F14 1,6644E?03 1,8615E?01 1,7522E?03 2,6072E?01 2,1683E?03 1,4580E?02 1,6711E?03 2,8498E?01 1,6622E103 2,6103E?01 22 of 19 Page F15 2,5522E?03 1,7142E?02 2,8256E?03 5,5492E?01 3,9013E?03 4,4677E?02 2,6529E?03 1,6852E?02 2,4716E103 2,1624E?02 65

65 Page 20 of 22 Sådhanå (2021) 46:65

Table 15. Comparisons of proposed CHFPSO vs. other some improved algorithms.

DE (l ? k)-ES CMAES-S CMAES-G EPSO ISRPSO HFPSO CHFPSO (Iterative) F1 2,3911E?10 3,5775E?10 6,8700E107 1,1080E?08 8,4866E?09 7,1910E?08 1,1795E?09 1,2042E?09 F2 1,8254E?05 1,6179E?05 2,3630E?05 2,9530E?05 6,3748E104 7,6860E?04 8,5653E?04 8,7531E?04 F3 3,4190E?02 3,4353E?02 6,3390E?02 6,5270E?02 3,3800E?02 3,2569E102 3,2638E?02 3,2892E?02 F4 7,9627E?03 7,0557E?03 8,6730E?03 1,2040E?04 6,6946E?03 5,8090E?03 5,1202E?03 4,0473E103 F5 5,0431E?02 5,0499E?02 1,0010E?03 1,0080E?03 5,0430E?02 5,0424E?02 5,0410E?02 5,0319E102 F6 6,0365E?02 6,0433E?02 1,2010E?03 1,2010E?03 6,0276E?02 6,0064E?02 6,0076E?02 6,0061E102 F7 7,5438E?02 7,8216E?02 1,4010E?03 1,4010E?03 7,2189E?02 7,0057E102 7,0074E?02 7,0072E?02 F8 7,9963E?05 7,3789E?06 1,7670E?03 2,3210E?03 1,2746E?05 1,4262E?03 2,6354E?03 1,0370E103 F9 9,1394E?02 9,1408E?02 1,8270E?03 1,8280E?03 9,1372E?02 9,1357E?02 9,1337E?02 9,1330E102 F10 3,8759E?07 9,5323E?07 3,6310E106 1,4730E?07 2,6363E?07 6,8320E?06 5,4690E?06 3,8178E?06 F11 1,2870E?03 1,4378E?03 2,2460E?03 2,2580E?03 1,2288E?03 1,1509E?03 1,1336E?03 1,1316E103 F12 3,0110E?03 3,8087E?03 3,4540E?03 4,0940E?03 2,4432E?03 1,9357E?03 1,7752E?03 1,6959E103 F13 1,9613E?03 2,2208E?03 3,3840E?03 3,4260E?03 1,8839E?03 1,6996E?03 1,6866E103 1,6878E?03 F14 1,7479E?03 1,8406E?03 3,2660E?03 3,3000E?03 1,7016E?03 1,6655E?03 1,6469E103 1,6622E?03 F15 2,9304E?03 2,9154E?03 4,4270E?03 4,8360E?03 2,7488E?03 2,4510E?03 2,4467E103 2,4716E?03

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