(PDF) Ideology algorithm: a socio-inspired optimization methodology

Neural Comput & Applic DOI 10.1007/s00521-016-2379-4 ORIGINAL ARTICLE Ideology algorithm: a socio-inspired optimization methodology Teo Ting Huan1 • Anand J. Kulkarni2,3 • Jeevan Kanesan1 • Chuah Joon Huang1 Ajith Abraham4 • Received: 7 January 2016 / Accepted: 21 May 2016 The Natural Computing Applications Forum 2016 Abstract This paper introduces a new socio-inspired metaheuristic technique referred to as ideology algorithm (IA). It is inspired by the self-interested and competitive behaviour of political party individuals which makes them improve their ranking. IA demonstrated superior performance as compared to other well-known techniques in solving unconstrained test problems. Wilcoxon signed-rank test is applied to verify the performance of IA in solving optimization problems. The results are compared with seven well-known and some recently proposed optimization algorithms (PSO, CLPSO, CMAES, ABC, JDE, SADE and BSA). A total of 75 unconstrained benchmark problems are used to test the performance of IA up to 30 & Anand J. Kulkarni

;

Teo Ting Huan

Jeevan Kanesan

Chuah Joon Huang

Ajith Abraham

1 Department of Electrical Engineering, Faculty of Engineering, University Malaya, Kuala Lumpur, Malaysia 2 Odette School of Business, University of Windsor, 401 Sunset Avenue, Windsor, ON N9B3P4, Canada 3 Department of Mechanical Engineering, Symbiosis Institute of Technology, Symbiosis International University, Pune, MH 412 115, India 4 Machine Intelligence Research Labs (MIR Labs), Scientific Network for Innovation and Research Excellence, Auburn, WA 98071, USA dimensions. The results from this study highlighted that the IA outperforms the other algorithms in terms of number function evaluations and computational time. The eminent observed features of the algorithm are also discussed. Keywords Metaheuristic Ideology algorithm Socioinspired optimization Unconstrained test problems 1 Introduction and motivation Over the last two decades, metaheuristic optimization techniques have become increasingly popular and essential in applied mathematics [1–3]. Optimization algorithms are functioning as to find the best values for system variables under various conditions. Some well-known metaheuristics such as particle swarm optimization (PSO) [4], genetic algorithm (GA) [5], ant colony optimization (ACO) [6] are fairly well known, and they are applied in various fields. With regard to some drawbacks of classical optimization strategies as well as to achieve simplicity, flexibility and derivation-free mechanism, several metaheuristics have been designed [7–49]. Metaheuristics are inspired by simple concepts. They are usually related to physical phenomenon, animal’s behaviour and evolutionary concepts. The simplicity allows researchers to simulate different natural concepts and propose new metaheuristics, their hybridization and improved versions. Second, the applicability of metaheuristics to a variety of problems without significant changes in the structure/framework of algorithms makes them flexible. In other words, little problem-specific information is required. Also, different techniques could be deployed to support the algorithm solving a variety of problem classes. Third, the majority of metaheuristics have 123 Neural Comput & Applic derivative-free mechanisms. Metaheuristics find the solutions stochastically in contrast with gradient-based optimization techniques. The process of optimization starts with random solution(s), and the calculation of derivative for deciding the search direction is not required. This makes metaheuristics appropriate to apply for real-world problems. Finally, the ability of metaheuristics to avoid local optima makes them reach quickly in the close neighbourhood of the region where the global optimum could be potentially located. Generally, metaheuristics can be classified into three main classes: evolutionary, physics based and swarm based. Evolutionary algorithms (EA) are inspired by the concepts of evolution in nature. When the objective function for an optimization problem is nonlinear and nondifferentiable, EA techniques are typically used to find the global optimum [13–15]. EAs have been applied for various real-world engineering problems such as reverse engineer causal networks [16], commercial computer-automated exterior lighting design [17], nanoscale crossbar architectures [18], dynamic stochastic districting and routing problem [19], neural network classifier [20], assembly line configurations [21], configurations of mobile applications [22], electric power distribution networks [23], word sense disambiguation problem [24], surgery scheduling problems [25], image processing [26] and speech recognition [27]. The recent popular optimization techniques are from swarm intelligence (SI) domain. SI is characterized by its unique mechanism which mimics the behaviour of swarms of social insects, flocks of birds and schools of fish [15, 28]. The benefits of these approaches as compared with conventional techniques are the flexibility and robustness. These properties make SI a successful design paradigm for algorithms to deal with increasingly complex problems [29]. PSO simulates the social behaviour as a representation of the movement of organisms in the school of fish [30]. The comprehensive learning PSO (CLPSO) [32] and PSO2011 [33] are the recent versions of the standard PSO [4]. The ACO algorithm is proposed based on strategies of ants in accessing food sources [6]. In artificial bee colony (ABC) algorithm, the natural behaviour of honey bees in discovering food sources is imitated [15]. The cuckoo search (CS) algorithm is based on the behaviour of cuckoo species by laying their eggs in other nests of host birds [31]. A recently proposed algorithm, named covariance matrix adaptation evolution strategy (CMAES) [36], is based on basic genetic rules. The differential evolution (DE) algorithm [34, 35] is a population-based stochastic function minimizer. The adaptive differential evolution algorithm (JDE) [37], the parameter adaptive differential evolution algorithm (JADE) [13] and the self-adaptive differential evolution algorithm (SADE) [35] are recent versions of DE. Another recently 123 proposed algorithm referred to as backtracking search algorithm (BSA) generates a trial individual using basic genetic operators (selection, mutation and crossover). A nonuniform crossover strategy which is more complex than the crossover strategies used in many genetic algorithms is used in the BSA [38]. The work presented in this paper is motivated from ideologies which exist in human society for ages. In the context of politics, there are numerous kinds of ideologies such as conservative, socialism, left-wing, right-wing, democratic, republic, and communism. There are several political parties exist in the world which follow these ideologies in different forms, for example Conservative Party and Labour Party (UK), Republican Party and Democratic Party (USA), Communist Party (China) and Bharatiya Janata Party (India). This paper introduces a novel socio-inspired algorithm referred to as ideology algorithm (IA). The society individuals support or follow certain ideologies. These ideologies become the guide or way for the individuals to achieve their long-term goals. The IA is motivated from the competition within the members of a political party as well as competition amongst the leaders of different parties. Every local party follows certain ideology which motivates certain individuals stay associated with that party. Once associated with a party, every individual exhibits a self-interested behaviour and competes with its party members to improve and promote its rank. Every individual looks at its own local party leader as a benchmark and tries to reach as close as possible to it. Also, the individual watches other party leaders and compares itself with that leader. This may motivate it to choose different ideology associated with another party. Furthermore, every local party leader always desires to be a global leader. In other words, it competes with the other party leaders to be a global leader. Moreover, the local party leader desires to remain at least its own party leader. Thus, it also competes with the second best in the party which always desires to catch the local party leader position. In addition, the lowest rank individual following the party ideology desires to climb up in the party, however, for prolonged time if it understands that following its current party ideology is not improving its rank. Such deserted individual may change the ideology and resort to another party ideology. This competitive behaviour of individuals following certain ideology to improve and climb up in the party as well as compete with other party members is modelled. The mechanism enabled the IA to solve several numerical optimization problems with superior performance in terms of solution quality and computational cost as compared with other existing algorithms. The remainder parts in this work are organized as follows: Sect. 2 describes the mechanism of IA. Section 3 provides the detailed results of the computational Neural Comput & Applic experiments conducted to validate the algorithm. In Sect. 4, conclusions and future directions are provided. 2 Ideology algorithm (IA) In the context of the IA, every member or individual associated with a party is a possible solution. Its position in the party depends on the quality or fitness of its solution (objective function value). The individual with the best solution in a party is considered as local party leader, and the individual best amongst all the party leaders is considered as the global leader. The local party leader competes with every other party leader with a desire to be a global leader. It also competes with the second best individual in its own party as it is challenged by the second best in the party which desires to be the local party leader. The earlier makes the party leaders explore and locate promising search space. The later forces the party leader to look for a better solution in its own local neighbourhood as well as the second best in the party. This may increase its chances of remaining as the local party leader and improve. The individual in the party with worst solution checks the difference between its own solution and the penultimate worst in the same party. If the difference is greater than a pre-specified value, then such deserted individual understands that following the current party ideology is not worth for it. This makes him switchover to another party in a hope to be better off and climb up in that party. The framework makes the individual in every party to directly and indirectly compete with the same party individuals as well as other party individuals. This essentially makes every party to remain in competition and grow which motivates the individuals search for better solutions. The IA procedure is explained below in detail. Consider a general unconstrained problem (in minimization sense) as follows: f ðxÞ ¼ f ðx1 ; . . .; xi ; . . .; xN Þ Subject to Wli xi Wui ; i ¼ 1; . . .; N From within every party p ¼ 1; . . .; P subspace Wpi ¼ h i Wil;p ; Wiu;p associated with every variable xi ; i ¼ 1; . . .; N, Mp values are randomly sampled and associated objective functions are evaluated. Then, the individuals associated with every party p; p ¼ 1; . . .; P could be represented as follows: 8pf x1;mp ; . . .; xi;mp ; . . .; xN;mp ; mp ¼ 1; . . .; Mp or 3 f x1;1 ; . . .; xi;1 ; . . .; xN;1 7 6 7 .. 6 7 6 . 7 6 7 6 p ¼ 6 f x1;mp ; . . .; xi;mp ; . . .; xN;mp 7; p ¼ 1; . . .; P 7 6 7 6 .. 7 6 . 5 4 f x1;Mp ; . . .; xi;Mp ; . . .; xN;Mp 2 From within every party p; p ¼ 1; . . .; P the evaluated individuals are ranked from the best individual referred to as local party leader Lp;b to the local worst individual Lp;w as shown in Fig. 1. Step 4 (Competition and Improvement for local party leader Lp;b ) Every local party leader Lp;b ; ðp ¼ 1; . . .; PÞ seeks to improve itself through introspection, local competition and global competition. The introspection refers to searching the close neighbourhood of its own current solution by modifying the current sampling space associated with its L every variable xi p;b ; i ¼ 1; . . .; N as follows: Party, ð1Þ The IA procedure starts with equally dividing the entire population into P parties. In the beginning, the number of individuals Mp in every party pðp ¼ 1; . . .; PÞ is equal, i.e. M1 ¼; . . .; Mp ; . . .; ¼ MP . Also the desertion parameter T associated with the worst individual, convergence parameter e, maximum number of iterations Imax and reduction factor R 2 ½0; 1Š are chosen. Step 1 (Party Formation) Equally divide the sampling space associated with every variable xi ; i ¼ 1; . . .; N into P party subspaces, i.e. h i 8xi Wpi ¼ Wil;p ; Wiu;p ; p ¼ 1; . . .; P; i ¼ 1; . . .; N. ð2Þ Step 3 (Local Party Ranking) Local party leader Number of members, Minimize Step 2 (Evaluation) Second best individual Other individuals Local second worst individual Local worst individual Fig. 1 The arrangement of individuals in a party 123 Neural Comput & Applic h L Lp;b 2 xi p;b R kWiu;p k Wi;insp i þR kWiu;p k Wil;p Wil;p , i.e. L ; xi p;b ð3Þ ð7Þ The local competition refers to competing with the second best Lp;2b in its own party by searching in the close neighbourhood of its current solution. In other words, the current sampling space of every variable i; i ¼ 1; . . .; N associated with every local party leader Lp;b ; ðp ¼ 1; . . .; PÞ is updated to the close neighbourhood of the local best solution Lp;2b as follows: where Lp;o ðo ¼ 1; 2; . . .; OÞ is referred to as ordinary individual for party ðp ¼ 1; 2; . . .; PÞ. Step 5(Updating party individuals) Every ordinary individual Lp;o ðo ¼ 1; 2; . . .; OÞ in every party ðp ¼ 1; 2; . . .; PÞ searches for a single solution in its L h L 2 xi p;2b R kWiu;p k i þ R kWiu;p k Wil;p Lp;b Wi;lcmp Wil;p own neighbourhood Wi;op;o as well as every local party L ; xi p;2b L ð4Þ Local competition is necessary as the local party leader will always try to remain the best in its own party which may further lead the algorithm to efficiently search for better solution. The global competition refers to searching in the close neighbourhood of the global leader. In other words, the current sampling space of every variable i; i ¼ 1; . . .; N associated with every local party leader Lp;b ðp ¼ 1; . . .; PÞ is updated to the close neighbourhood of the global best solution Lp;gb as follows: h L Lp;b 2 xi p;gb R kWiu;p k Wil;p ; Wi;gcomp i ð5Þ L xi p;gb þ R kWiu;p k Wil;p where Lp;gb ¼ min Lp;b ; p ¼ 1; . . .; P or Lp;gb ¼ min L1;b ; . . .; Lp;b ; . . .; LP;b Þ. Then, the local party leader Lp;b samples variable values L L p;b p;b from within the updated sampling intervals Wi;insp , Wi;lcmp L p;b and Wi;gcomp formed using introspection, local competition and global competition, respectively, and calculates corresponding objective functions. Then, one solution from within the three choices is selected based on the roulette wheel selection approach [42]. It is important to mention that the introspection will not make the leader ignore its recent local neighbourhood as its current solution could be far better than other individuals. For each local worst individual Lp;w ðp ¼ 1; 2; . . .; PÞ, the distance d between itself and the second worst individual Lp;2w is evaluated as follows: d ¼ Lp;w Lp;2w ð6Þ If the difference is higher than a pre-specified value T, then the individual understands that it is deserted (worst off) and switches over to another randomly selected party 123 p;o , ðp ¼ 1; 2; . . .; PÞ. The best leader’s neighbourhood Wi;lopl O solutions are chosen from within this pool, and the party individuals are updated. In other words, the current sampling space of every variable ði; i ¼ 1; . . .; N Þ associated with every individual Lp;o ðo ¼ 1; 2; . . .; OÞ; ðp ¼ 1; . . .; PÞ other than the local party leader Lp;b and deserted individual Lp;w updates its sampling space in the close neighbourhood of itself [refer to Eqs. (8) and (9)], and every local party leader Lp;b ðp ¼ 1; . . .; PÞ is as follows: h L L L Wil;p ; xi p;o þ R Wi;op;o 2 xi p;o R kWiu;p k i ð8Þ kWiu;p k Wl;p i h L L Lp;o ; xi p;b þ R Wi;lopl Wl;p 2 xi p;b R kWiu;p k i i ð9Þ kWiu;p k Wl;p i In this way, each individual of every party is updated. Step 6 (Convergence) The parties are considered converged if any of the following conditions satisfied, else continue to Step 1: (a) (b) There is no significant improvement in the local party leader solutions for a significant number of iterations and/or The maximum number of iterations Imax is reached. It is important to mention that the number of party members may change in every iteration as some of the individuals may leave a party and join any other in hope to improve. Figure 2 shows the general structure of IA, where Fig. 3 shows the flow chart of IA. The working mechanism of the IA is illustrated in Fig. 4, where the dots represent individuals or party members and the centralized dot represents leader in a party. Figure 5 illustrates the movement of individuals during a run solving multimodal Ackley function. It exhibits the ability of the algorithm to quickly jump out of the local minima and reach the global minimum solution. Neural Comput & Applic 1: Initialization 2: Party formation 3: Generation of party individuals 4: repeat 5: Evaluation 6: Local ranking 7: Competition and improvement 8: Updating party individuals 9: until convergence Fig. 2 General structure of IA 3 Results and discussion In this section, the tests and benchmark problems, statistical analysis, control parameters and convergence conditions used for the IA in the tests are presented. The performance of IA is investigated in detail. For experiments, each algorithm (PSO, CMAES, ABC, JDE, CLPSO, SADE, BSA and IA) is coded in MATLAB R2013a on Windows Platform with a T6400@4 GHz Intel Core 2 Duo processor with 4 GB RAM. 3.1 Benchmark problems Two tests are conducted to examine the performance of IA and the comparison algorithms in solving the numerical optimization problems. Test 1 involved 50 widely used benchmark problems [15, 41]. Table 1 summarizes several features of the benchmark problems used in Test 1. Test 2 involved 25 benchmark problems used in CEC2005 [45]. Table 2 summarizes several features of the benchmark problems used in Test 2. 3.2 Control parameters The values of the control parameters used in the experiments for IA are listed as shown in Table 3. 3.3 Stopping criterion The predetermined stopping criterion is set to terminate the algorithms. • • • Stop when the absolute value of the objective function evaluations is less than 10 16 . Stop when the maximum number of function evaluations reaches 200000. Stop when the maximum number of iterations ðImax Þ is reached. Parametric tests have been commonly used in the analysis of experiments. For example, a common way to test whether the difference between the results of two algorithms is nonrandom is to apply a paired t test, which checks whether the average difference in their performance over the problems is significantly different from zero. Nonparametric tests, besides their original definition for dealing with nominal or ordinal data, can be also applied to continuous data by conducting ranking-based transformations, adjusting the input data to the test requirements. They can perform two classes of analysis: pairwise comparisons and multiple comparisons. Pairwise statistical procedures perform individual comparisons between two algorithms, obtaining in each application a p value independent from another one [40]. Pairwise comparisons are the simplest kind of statistical tests that a researcher can apply within the framework of an experimental study. Such tests are directed to compare the performance of two algorithms when applied to a common set of problems. In multi-problem analysis, a value for each pair of algorithm is required (often an average value from several runs). In this section, we focus on the sign test, which is a quick and easy procedure that can provide a clearer view about the comparison. Then, the Wilcoxon signed-rank test is introduced as an example of a simple nonparametric test for pairwise statistical comparisons. 3.4 Statistical analysis The Wilcoxon signed-rank test is a nonparametric procedure employed in hypothesis testing situations, involving a design with two samples. It is commonly used for answering the following question: Do two samples represent two different populations? This is analogous to the paired t test in nonparametric statistical procedures. Thus, it is a pairwise test that aims to detect significant differences between two sample means, i.e. the behaviour of two algorithms. Table 4 shows the mean runtimes and simple statistical values for the results obtained in Test 1, whereas Table 6 lists the algorithms that obtained statistically better solutions compared with the other algorithms in Test 1, based on the Wilcoxon signed-rank test. Table 5 shows the mean runtimes and simple statistical values for the results obtained in Test 2, whereas Table 7 lists the algorithms that provided statistically better solutions compared with the other algorithms in Test 2, based on the Wilcoxon signed-rank test. Table 8 presents the multi-problem-based pairwise statistical comparison results using the averages of the global minimum values obtained through 30 runs of IA and the comparison algorithms to solve the benchmark problems in 123 Neural Comput & Applic START Generate initial population P with N solutions and sampling interval reduction factor R Form the parties according to distance between individuals Evaluate population P Determine the local best vector for every party, the global best vector amongst all the leaders, and worst vector in every party Local best vector Local worst Other vector Search in its close neighborhood, the close neighborhood of the best amongst all leaders, and the close neighborhood of the second best in its own party Evaluate the distance between itself and every other individual. If the difference is higher than a pre-specified value T, then it is switched to another party Search in its close neighborhood, the close neighborhood of the party leader, and the close neighborhood of the nearest party leader Update the parties No Convergence? Yes END Fig. 3 Flow chart of ideology algorithm (IA) Test 1 and Test 2. The results indicate that IA was statistically more successful than most of the comparison algorithms with a statistical significance value a ¼ 0:05. In Tables 6 and 7, a ‘?’ sign indicates cases in which the null hypothesis is rejected and IA displays a statistically superior performance in the problem-based statistical comparison tests at the 95 % significance level ða ¼ 0:05Þ. The ‘-’ sign indicates cases in which the null hypothesis was rejected and IA displayed an inferior performance; ‘=’ indicates cases in which there was no statistical difference between the two algorithms’ success in solving the problems. The last rows of Tables 6 and 7 depict the total 123 counts in a format of ‘þ= ¼ = ’ for the three statistical significance cases (marked with ‘?’, ‘=’ or ‘-’) in the pairwise comparison. When the ðþ= ¼ = Þ values are examined, it can be said that IA is statistically more successful than most of the other algorithms in solving the problems in Tests 1 and 2. Although, the successes IA and BSA have had are statistically identical; IA has provided statistically better solutions than other algorithms. For pairwise comparison of the problem-solving success of EAs, a problem-based or multi-problem-based statistical comparison method can be used [40]. A problem-based Neural Comput & Applic Fig. 4 a Initialization of the individuals. b Associate individuals with p parties ðp ¼ 1; 2; . . .; PÞ. c Introspection. d Global competition. e Local competition. f Search by local worst vector. f Search by ordinary individual Lp;o (a) (b) 0 0 (c) 0 (d) (e) 0 (f) 0 (g) 0 comparison can use the global minimum values obtained for the problem as the result of several runs. Problem-based pairwise comparisons are widely used to determine which of two algorithms solves a specific numerical optimization problem with greater statistical success. The global minimum values obtained are used in this paper as the result of 30 runs for its problem-based 0 pairwise comparison of the algorithms. A multi-problembased pairwise comparison can use the average of the global minimum values obtained as the result of several runs. Multi-problem-based pairwise comparisons determine which algorithm is statistically more successful in a test that includes several benchmark problems [40]. The average of global minimum values obtained is used in this 123 Neural Comput & Applic Fig. 5 Search pattern for benchmark function F5 (Ackley), different symbols denote different parties. a Initialization of population. b Iteration 1. c Iteration 3. d Iteration 5. e Iteration 10. f Convergence (a) Leader/ best individual Worst individual Every other individual (c) (d) (e) (f) paper as the result of 30 runs for its multi-problem-based comparison of the algorithms. The Wilcoxon signed-rank test was used for pairwise comparisons, with the statistical significance value a ¼ 0:05. The null hypothesis H0 for this test is: ‘There is no difference between the median of the solutions achieved by algorithm A and the median of the solutions obtained by algorithm B for the same benchmark problem’. In other words, we assume that median (A) = median (B). To determine whether algorithm A reached a statistically better solution than algorithm B, or whether the alternative hypothesis was valid, the sizes of the ranks provided by the Wilcoxon signed-rank test (T? and T- as defined in [40]) are examined thoroughly. 123 (b) 3.5 PSO versus IA In PSO, the individual particles of a swarm symbolize potential solutions. They ‘fly’ through the search space of the problem, trying to seek an optimal solution. The current positions of the particles are broadcasted to other neighbouring particles. Previously identified ‘good position’ is then used as a starting point by the swarm for further search. On the other hand, the individual particles adjust their current positions and velocities. A distinct characteristic of PSO is its fast convergent behaviour and inherent adaptability, especially when compared to conventional EAs [47]. Theoretical analysis of PSO [4, 30] proves that particles in a swarm can switch between an exploratory Neural Comput & Applic Table 1 The benchmark problems used in Test 1 (Dim dimension, low and up limitations of search space, U unimodal, M multimodal, S separable, N nonseparable) Problem Name Type Low Up Dimension F1 Foxholes MS -65.536 65.536 2 F2 Goldstein-Price MN -2 2 2 F3 Penalized MN -50 50 30 F4 Penalized2 MN -50 50 30 F5 Ackley MN -32 32 30 F6 Beale UN -4.5 4.5 5 F7 Bohachevsky1 MS -100 100 2 F8 Bohachevsky2 MN -100 100 2 F9 Bohachevsky3 MN -100 100 2 F10 Booth MS -10 10 2 F11 Branin MS -5 10 2 F12 Colville UN -10 10 4 F13 Dixon-Price UN -10 10 30 F14 Easom UN -100 100 2 F15 F16 Fletcher Fletcher MN MN -3.1416 -3.1416 3.1416 3.1416 2 5 F17 Fletcher MN -3.1416 3.1416 10 F18 Griewank MN -600 600 30 F19 Hartman3 MN 0 1 F20 Hartman6 MN 0 1 6 F21 Kowalik MN -5 5 4 F22 Langermann2 MN 0 10 2 F23 Langermann5 MN 0 10 5 F24 Langermann10 MN 0 10 10 F25 Matyas UN -10 10 2 F26 Michalewics2 MS 0 3.1416 2 F27 Michalewics5 MS 0 3.1416 5 F28 Michalewics10 MS 0 3.1416 10 F29 Perm MN -4 4 4 F30 Powell UN -4 5 24 F31 F32 Powersum Quartic MN US 0 -1.28 4 1.28 4 30 F33 Rastrigin MS -5.12 5.12 30 F34 Rosenbrock UN -30 30 30 F35 Schaffer MN -100 100 2 F36 Schwefel MS -500 500 30 F37 Schwefel_1_2 UN -100 100 30 F38 Schwefel_2_22 UN -10 10 30 F39 Shekel10 MN 0 10 4 F40 Shekel5 MN 0 10 4 F41 Shekel7 MN 0 10 4 F42 Shubert MN -10 10 2 F43 Six-hump camelback MN -5 5 F44 Sphere2 US -100 100 30 F45 Step2 US -100 100 30 F46 F47 Stepint Sumsquares US US -5.12 -10 5.12 10 5 30 F48 Trid6 UN -36 36 6 F49 Trid10 UN -100 100 10 F50 Zakharov UN -5 10 10 3 2 123 Neural Comput & Applic Table 2 The benchmark problems used in Test 2 (Dim dimension, low and up limitations of search space, U unimodal, M multimodal, E expanded, H hybrid) Problem Name Type Low Up Dimension F51 Shifted sphere U -100 100 10 F52 Shifted Schwefel U -100 100 10 F53 Shifted rotated high conditioned elliptic function U -100 100 10 F54 Shifted Schwefels problem 1.2 with noise U -100 100 10 F55 Schwefels problem 2.6 U -100 100 10 F56 Shifted Rosenbrock’s M -100 100 10 F57 F58 Shifted rotated Griewank’s Shifted rotated Ackley’s M M 0 -32 600 32 10 10 F59 Shifted Rastrigin’s M -5 5 10 F60 Shifted rotated Rastrigin’s M -5 5 10 F61 Shifted rotated Weierstrass M -0.5 0.5 10 F62 Schwefels problem 2.13 M -100 100 10 F63 Expanded extended Griewank’s ? Rosenbrock’s E -3 1 10 F64 Expanded rotated extended Scaffes E -100 100 10 F65 Hybrid composition function HC -5 5 10 F66 Rotated hybrid comp. Fn 1 HC -5 5 10 F67 Rotated hybrid comp. Fn 1 with noise HC -5 5 10 F68 Rotated hybrid comp. Fn 2 HC -5 5 10 F69 Rotated hybrid comp. Fn 2 with narrow global optimal HC -5 5 10 F70 Rotated hybrid comp. Fn 2 with the global optimum HC -5 5 10 F71 Rotated hybrid comp. Fn 3 HC -5 5 10 F72 F73 Rotated hybrid comp. Fn 3 with high condition number matrix Noncontinuous rotated hybrid comp. Fn 3 HC HC -5 -5 5 5 10 10 F74 Rotated hybrid comp. Fn 4 HC -5 5 10 F75 Rotated hybrid comp. Fn 4 HC -2 5 10 Table 3 The relevant control parameters used in the experiments for IA Control parameter Numerical values Maximum number of iteration (Imax ) 30 Number of parties presented (p) 5 Initial population size (x) 150 Reduction factor (R) 0.000001 mode with large search step sizes, as well as an exploitative mode with smaller search step sizes. Each particle in PSO is determined by its current position as shown in Eq. (10), as well as its current velocity as shown in Eq. (11) [48]. In each iteration, the particle’s velocity is modified by its personal best position, which is the position giving the best fitness value. Also, they are determined by the global best position, which is the position of the best-fit particle from the swarm [47]. As a result, each particle searches around a region defined by its personal best position and global best position. ~ x ð t þ 1Þ ¼ ~ xðt Þ þ ~ vðtÞ þ 1 123 ð10Þ ~ ~ðtÞ þ ;1 randð0; 1Þð~ vðt þ 1Þ ¼ xv pð t Þ ~ x ðt ÞÞ þ ;2 randð0; 1Þð~ g ðt Þ ~ xðtÞÞ ð11Þ The parameter x is known as inertia weight, and it controls the magnitude of the old velocity, ~ vðtÞ to calculate the new velocity, ~ vðt þ 1Þ. The parameters ;1 and ;2 determine the significance of ~ pðtÞ and ~ gðtÞ, respectively. The procedure of PSO is as shown in Fig. 6. The drawback of the basic PSO algorithm is that it easily suffers from the partial optimism, which might lead to reduced precision in speed and the regulation of direction. PSO is unable to solve the problems of scattering and optimization, as well as the problems of noncoordinate system, such as the solution to the energy field and the moving rules of the particles in the energy field [47–49]. In this paper, the proposed IA is being compared with PSO and its variants, CLPSO. The results proved that IA outperforms PSO and CLPSO in terms of runtime in most of the benchmark functions of Test 1 and Test 2. The statistical results of Test 1 as shown in Table 6 indicate that IA is equally good as compared with PSO and CLPSO. However, statistical results of Test 2 as shown in Table 7 Neural Comput & Applic Table 4 Statistical solutions obtained by PSO, CMAES, ABC, CLPSO, SADE, BSA and proposed IA in Test 1 (mean mean solution, SD standard deviation of mean solution, best best solution, runtime mean runtime in seconds) Problem Statistics PSO2011 CMAES ABC JDE F1 Mean 1.3316029264876300 10.0748846367972000 0.9980038377944500 1.0641405484285200 SD 0.9455237994690700 8.0277365400340800 0.0000000000000001 0.3622456829347420 Best 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038377944500 Runtime 72.527 44.788 64.976 51.101 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 Mean 2.9999999999999200 21.8999999999995000 3.0000000465423000 2.9999999999999200 SD 0.0000000000000013 32.6088098948516000 0.0000002350442161 0.0000000000000013 Best 2.9999999999999200 2.9999999999999200 2.9999999999999200 2.9999999999999200 Runtime 17.892 24.361 16.624 7.224 Mean 0.1278728062391630 0.0241892995662904 0.0000000000000004 0.0034556340083499 SD 0.2772792346028400 0.0802240262581864 0.0000000000000001 0.0189272869685522 Best Runtime 0.0000000000000000 139.555 0.0000000000000000 5.851 0.0000000000000003 84.416 0.0000000000000000 9.492 Mean 0.0043949463343535 0.0003662455278628 0.0000000000000004 0.0007324910557256 SD 0.0054747064090174 0.0020060093719584 0.0000000000000001 0.0027875840585535 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 Runtime 126.507 6.158 113.937 14.367 Mean 1.5214322973725000 11.7040011684582000 0.0000000000000340 0.0811017056422860 SD 0.6617570384662600 9.7201961540865200 0.0000000000000035 0.3176012689149320 Best 0.0000000000000080 0.0000000000000080 0.0000000000000293 0.0000000000000044 Runtime 63.039 3.144 23.293 11.016 Mean 0.0000000041922968 0.2540232169641050 0.0000000000000028 0.0000000000000000 SD 0.0000000139615552 0.3653844307786430 0.0000000000000030 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 Runtime 32.409 4.455 22.367 1.279 Mean 0.0000000000000000 0.0622354533647150 0.0000000000000000 0.0000000000000000 SD Best 0.0000000000000000 0.0000000000000000 0.1345061339146580 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 16.956 6.845 1.832 1.141 Mean 0.0000000000000000 0.0072771062590204 0.0000000000000000 0.0000000000000000 SD 0.0000000000000000 0.0398583525142753 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 17.039 2.174 1.804 1.139 Mean 0.0000000000000000 0.0001048363065820 0.0000000000000006 0.0000000000000000 SD 0.0000000000000000 0.0005742120996051 0.0000000000000003 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 Runtime 17.136 2.127 21.713 1.129 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 17.072 1.375 22.395 1.099 Mean 0.3978873577297380 0.6372170283279430 0.3978873577297380 0.3978873577297380 SD Best 0.0000000000000000 0.3978873577297380 0.7302632173480510 0.3978873577297380 0.0000000000000000 0.3978873577297380 0.0000000000000000 0.3978873577297380 Runtime 17.049 24.643 10.941 6.814 Mean 0.0000000000000000 0.0000000000000000 0.0715675060725970 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0579425013417103 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0013425253994745 0.0000000000000000 Runtime 44.065 1.548 21.487 1.251 123 Neural Comput & Applic Table 4 continued Problem F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 123 Statistics PSO2011 CMAES ABC JDE Mean 0.6666666666666750 0.6666666666666670 0.0000000000000038 0.6666666666666670 SD 0.0000000000000022 0.0000000000000000 0.0000000000000012 0.0000000000000002 Best 0.6666666666666720 0.6666666666666670 0.0000000000000021 0.6666666666666670 Runtime 167.094 3.719 37.604 18.689 Mean -1.0000000000000000 -0.1000000000000000 -1.0000000000000000 -1.0000000000000000 SD 0.0000000000000000 0.3051285766293650 0.0000000000000000 0.0000000000000000 Best -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 Runtime Mean 16.633 0.0000000000000000 3.606 1028.3930784026900000 13.629 0.0000000000000000 6.918 0.0000000000000000 SD 0.0000000000000000 1298.1521820113500000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 27.859 15.541 40.030 2.852 Mean 48.7465164446927000 1680.3460230073400000 0.0218688498331872 0.9443728655432830 SD 88.8658510972991000 2447.7484859066000000 0.0418409568792831 2.8815514827061600 Best 0.0000000000000000 0.0000000000000000 0.0000000000000016 0.0000000000000000 Runtime 95.352 11.947 44.572 4.719 Mean 918.9518492782850000 12340.2283326398000000 11.0681496253548000 713.7226974626920000 SD 1652.4810858411400000 22367.1698875802000000 9.8810950146557100 1710.071307430120000 Best 0.0000000000000000 0.0000000000000000 0.3274654777056860 0.0000000000000000 Runtime 271.222 7.631 43.329 16.105 Mean 0.0068943694819713 0.0011498935321349 0.0000000000000000 0.0048193578543185 SD 0.0080565201649587 0.0036449413521107 0.0000000000000001 0.0133238235582874 Best Runtime 0.0000000000000000 73.895 0.0000000000000000 2.647 0.0000000000000000 19.073 0.0000000000000000 6.914 Mean -3.8627821478207500 -3.7243887744664700 -3.8627821478207500 -3.8627821478207500 SD 0.0000000000000027 0.5407823545193820 0.0000000000000024 0.0000000000000027 Best -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 Runtime 19.280 21.881 12.613 7.509 Mean -3.3180320675402500 -3.2942534432762600 -3.3219951715842400 -3.2982165473202600 SD 0.0217068148263721 0.0511458075926848 0.0000000000000014 0.0483702518391572 Best -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 Runtime 26.209 7.333 13.562 8.008 Mean 0.0003074859878056 0.0064830287538208 0.0004414866359626 0.0003685318137604 SD 0.0000000000000000 0.0148565973286009 0.0000568392289725 0.0002323173367683 Best 0.0003074859878056 0.0003074859878056 0.0003230956007045 0.0003074859878056 Runtime 84.471 13.864 20.255 7.806 Mean -1.0809384421344400 -0.7323679641701760 -1.0809384421344400 -1.0764280762657400 SD 0.0000000000000006 0.4136688304155380 0.0000000000000008 0.0247042912888477 Best Runtime -1.0809384421344400 27.372 -1.0809384421344400 32.311 -1.0809384421344400 27.546 -1.0809384421344400 19.673 Mean -1.3891992200744600 -0.5235864386288060 -1.4999990070800800 -1.3431399432579700 SD 0.2257194403158630 0.2585330714077300 0.0000008440502079 0.2680292304904580 Best -1.4999992233524900 -0.7977041047646610 -1.4999992233524900 -1.4999992233524900 Runtime 33.809 17.940 37.986 20.333 Mean -0.9166206788680230 -0.3105071678265780 -0.8406348096500680 -0.8827152798835760 SD 0.3917752367440500 0.2080317241440800 0.2000966365984320 0.3882445165494030 Best -1.5000000000003800 -0.7976938356122860 -1.4999926800631400 -1.5000000000003800 Runtime 110.798 8.835 38.470 21.599 Neural Comput & Applic Table 4 continued Problem F25 F26 F27 F28 F29 F30 F31 F32 F33 F34 F35 F36 Statistics PSO2011 CMAES ABC JDE Mean 0.0000000000000000 0.0000000000000000 0.0000000000000004 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 Runtime 25.358 1.340 19.689 1.142 Mean -1.8210436836776800 -1.7829268228561700 -1.8210436836776800 -1.8210436836776800 SD 0.0000000000000009 0.1450583631808370 0.0000000000000009 0.0000000000000009 Best -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 Runtime Mean 19.154 -4.6565646397053900 26.249 -4.1008953007033700 17.228 -4.6934684519571100 9.663 -4.6893456932617100 SD 0.0557021530063238 0.4951250481844850 0.0000000000000009 0.0125797149251589 Best -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 Runtime 38.651 10.956 17.663 14.915 Mean -8.9717330307549300 -7.6193507368464700 -9.6601517156413500 -9.6397230986132500 SD 0.4927013165009220 0.7904830398850970 0.0000000000000008 0.0393668145094111 Best -9.5777818097208200 -9.1383975057875100 -9.6601517156413500 -9.6601517156413500 Runtime 144.093 6.959 27.051 20.803 Mean 0.0119687224560441 0.0788734736114700 0.0838440014038032 0.0154105130055856 SD 0.0385628598040034 0.1426911799629180 0.0778327303965192 0.0308963906374663 Best 0.0000044608370213 0.0000000000000000 0.0129834451730589 0.0000000000000000 Runtime 359.039 17.056 60.216 35.044 Mean 0.0000130718912008 0.0000000000000000 0.0002604330013462 0.0000000000000001 SD 0.0000014288348929 0.0000000000000000 0.0000394921919294 0.0000000000000002 Best Runtime 0.0000095067504097 567.704 0.0000000000000000 14.535 0.0001682411286088 215.722 0.0000000000000000 194.117 Mean 0.0001254882834238 0.0000000000000000 0.0077905311094958 0.0020185116261490 SD 0.0001503556280087 0.0000000000000000 0.0062425841086448 0.0077448684015362 Best 0.0000000156460198 0.0000000000000000 0.0003958766023752 0.0000000000000000 Runtime 250.248 12.062 34.665 48.692 Mean 0.0003548345513179 0.0701619169853449 0.0250163252527030 0.0013010316180679 SD 0.0001410817500914 0.0288760292572957 0.0077209314806873 0.0009952078711752 Best 0.0001014332605364 0.0299180701536354 0.0094647580732654 0.0001787238105452 Runtime 290.669 2.154 34.982 82.124 Mean 25.6367602258676000 95.9799861204982000 0.0000000000000000 1.1276202647057400 SD 8.2943512684216700 56.6919245985100000 0.0000000000000000 1.0688393637536800 Best 12.9344677422129000 29.8487565993415000 0.0000000000000000 0.0000000000000000 Runtime 76.083 2.740 4.090 7.635 Mean 2.6757043114269700 0.3986623855035210 0.2856833465904130 1.0630996944802500 SD 12.3490058210004000 1.2164328621946200 0.6247370987465170 1.7930895051734300 Best Runtime 0.0042535368984501 559.966 0.0000000000000000 9.462 0.0004266049929880 35.865 0.0000000000000000 23.278 Mean 0.0000000000000000 0.4651202457398910 0.0000000000000000 0.0038863639514140 SD 0.0000000000000000 0.0933685176073728 0.0000000000000000 0.0048411743884718 Best 0.0000000000000000 0.0097159098775144 0.0000000000000000 0.0000000000000000 Runtime 18.163 24.021 7.861 4.216 Mean -7684.6104757783800000 -6835.1836730901400000 -12569.4866181730000 -12304.9743375341000 SD 745.3954005014180000 750.7338055436110000 0.0000000000022659 221.4322514436480000 Best -8912.8855854978200000 -8340.0386911070600000 -12569.4866181730000 -12569.4866181730000 Runtime 307.427 3.174 19.225 10.315 123 Neural Comput & Applic Table 4 continued Problem F37 F38 F39 F40 F41 F42 F43 F44 F45 F46 F47 F48 123 Statistics PSO2011 CMAES ABC JDE Mean 0.0000000000000000 0.0000000000000000 14.5668734126948000 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 8.7128443012950300 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 4.0427699323673400 0.0000000000000000 Runtime 543.180 3.370 111.841 19.307 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 Runtime Mean 163.188 -10.1061873621653000 2.558 -5.2607563471326400 20.588 -10.5364098166920000 1.494 -10.3130437162426000 SD 1.6679113661236400 3.6145751818694000 0.0000000000000023 1.2234265179812200 Best -10.5364098166921000 -10.5364098166921000 -10.5364098166920000 -10.5364098166921000 Runtime 31.018 11.024 16.015 8.345 Mean -9.5373938082045500 -5.7308569926624600 -10.1531996790582000 -9.5656135761215700 SD 1.9062127067994200 3.5141202468383400 0.0000000000000055 1.8315977756329900 Best -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 Runtime 25.237 11.177 11.958 7.947 Mean -10.4029405668187000 -6.8674070870953700 -10.4029405668187000 -9.1615813354737300 SD 0.0000000000000018 3.6437803702691000 0.0000000000000006 2.8277336448396200 Best -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 Runtime 21.237 11.482 14.911 8.547 Mean -186.7309073569880000 -81.5609772893002000 -186.730908831024000 -186.730908831024000 SD 0.0000046401472660 66.4508342743478000 0.0000000000000236 0.0000000000000388 Best Runtime -186.7309088310240000 19.770 -186.7309088310240000 25.225 -186.730908831024000 13.342 -186.730908831024000 8.213 Mean -1.0316284534898800 -1.0044229658530100 -1.0316284534898800 -1.0316284534898800 SD 0.0000000000000005 0.1490105926664260 0.0000000000000005 0.0000000000000005 Best -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 Runtime 16.754 24.798 11.309 7.147 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000004 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 Runtime 159.904 2.321 21.924 1.424 Mean 2.3000000000000000 0.0666666666666667 0.0000000000000000 0.9000000000000000 SD 1.8597367258983700 0.2537081317024630 0.0000000000000000 3.0211895350832500 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 57.276 1.477 1.782 2.919 Mean 0.1333333333333330 0.2666666666666670 0.0000000000000000 0.0000000000000000 SD 0.3457459036417600 0.9444331755018490 0.0000000000000000 0.0000000000000000 Best Runtime 0.0000000000000000 20.381 0.0000000000000000 2.442 0.0000000000000000 1.700 0.0000000000000000 1.074 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 Runtime 564.178 2.565 24.172 1.870 Mean -50.0000000000002000 -50.0000000000002000 -49.9999999999997000 -50.0000000000002000 SD 0.0000000000000361 0.0000000000000268 0.0000000000001408 0.0000000000000354 Best -50.0000000000002000 -50.0000000000002000 -50.0000000000001000 -50.0000000000002000 Runtime 24.627 8.337 22.480 8.623 Neural Comput & Applic Table 4 continued Problem Statistics PSO2011 CMAES ABC JDE F49 Mean -210.0000000000010000 -210.0000000000030000 -209.999999999947000 -210.000000000003000 SD 0.0000000000009434 0.0000000000003702 0.0000000000138503 0.0000000000008251 Best -210.0000000000030000 -210.0000000000030000 -209.999999999969000 -210.000000000004000 Runtime 48.580 5.988 36.639 11.319 Mean 0.0000000000000000 0.0000000000000000 0.0000000402380424 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000002203520334 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000210 0.0000000000000000 Runtime 86.369 1.868 86.449 1.412 Problem Statistics CLPSO SADE BSA IA F1 Mean 1.8209961275956800 0.9980038377944500 0.9980038377944500 0.9980038690000000 SD 1.6979175079427900 0.0000000000000000 0.0000000000000000 0.0000000000000035 Best 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038685998520 Runtime 61.650 66.633 38.125 43.535 Mean 3.0000000000000700 2.9999999999999200 2.9999999999999200 3.0240147900000000 SD 0.0000000000007941 0.0000000000000020 0.0000000000000011 0.0787814840000000 Best Runtime 2.9999999999999200 24.784 2.9999999999999200 28.699 2.9999999999999200 7.692 3.0029461118668700 41.343 Mean 0.0000000000000000 0.0034556340083499 0.0000000000000000 0.3536752140000000 SD 0.0000000000000000 0.0189272869685522 0.0000000000000000 1.4205454130000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0014898619035614 Runtime 38.484 15.992 18.922 34.494 Mean 0.0000000000000000 0.0440448539086004 0.0000000000000000 0.0179485820000000 SD 0.0000000000000000 0.2227372747439610 0.0000000000000000 0.0526650620000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000165491 Runtime 48.667 33.019 24.309 322.808 Mean 0.1863456353861950 0.7915368220335460 0.0000000000000105 0.0000000000000009 SD 0.4389839299322230 0.7561593402959740 0.0000000000000034 0.0000000000000000 Best 0.0000000000000080 0.0000000000000044 0.0000000000000080 0.0000000000000009 Runtime 45.734 40.914 14.396 49.458 Mean 0.0000444354499943 0.0000000000000000 0.0000000000000000 0.0082236060000000 SD Best 0.0001015919507724 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0082236059357692 Runtime 125.839 4.544 0.962 50.246 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 2.926 4.409 0.825 38.506 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 2.891 4.417 0.824 39.023 Mean 0.0000193464326398 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000846531630676 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 33.307 4.303 0.829 40.896 Mean 0.0006005122443674 0.0000000000000000 0.0000000000000000 0.8346587090000000 SD Best 0.0029861918862801 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.8346587086917530 F50 F2 F3 F4 F5 F6 F7 F8 F9 F10 123 Neural Comput & Applic Table 4 continued Problem Statistics Runtime 28.508 F11 Mean 0.3978873577297390 SD 0.0000000000000049 Best Runtime F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 123 CLPSO SADE BSA IA 4.371 0.790 39.978 0.3978873577297380 0.3978873577297380 0.4156431270000000 0.0000000000000000 0.0000000000000000 0.0406451050000000 0.3978873577297380 0.3978873577297380 0.3978873577297380 0.4012748152492080 17.283 27.981 5.450 40.099 Mean 0.1593872502094070 0.0000000000000000 0.0000000000000000 0.0014898620000000 SD 0.6678482786713720 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best Runtime 0.0000094069599934 166.965 0.0000000000000000 4.405 0.0000000000000000 2.460 0.0082029783984983 48.067 Mean 0.0023282133668190 0.6666666666666670 0.6444444444444440 0.2528116640000000 SD 0.0051792840882291 0.0000000000000000 0.1217161238900370 0.0000000006509080 Best 0.0000120708732167 0.6666666666666670 0.0000000000000000 0.2528116633611470 Runtime 216.261 47.833 21.192 67.463 -0.9997989620000000 Mean -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000167151 Best -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -0.9997989624626810 Runtime 16.910 28.739 5.451 39.685 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 4.030 6.020 2.067 38.867 Mean 81.7751618148164000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD Best 379.9241117377270000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 162.941 5.763 7.781 48.262 Mean 0.8530843976878610 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 2.9208253191698800 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0016957837829822 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 268.894 168.310 33.044 69.060 Mean 0.0000000000000000 0.0226359326967139 0.0004930693556077 0.0000000000000000 SD 0.0000000000000000 0.0283874287215679 0.0018764355751644 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 14.864 25.858 5.753 2.717 Mean -3.8627821478207500 -3.8627821478207500 -3.8627821478207500 -3.8596352620000000 SD 0.0000000000000027 0.0000000000000027 0.0000000000000027 0.0033967610000000 Best -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8613076574052300 Runtime 17.504 24.804 6.009 46.167 Mean -3.3219951715842400 -3.3140689634962500 -3.3219951715842400 -2.5710247593206100 SD Best 0.0000000000000013 -3.3219951715842400 0.0301641516823498 -3.3219951715842400 0.0000000000000013 -3.3219951715842400 0.0000000000000009 -2.5710247593206100 Runtime 20.099 33.719 6.822 59.083 Mean 0.0003100479704151 0.0003074859878056 0.0003074859878056 0.0016993410000000 SD 0.0000059843325073 0.0000000000000000 0.0000000000000000 0.0000013058400000 Best 0.0003074859941292 0.0003074859878056 0.0003074859878056 0.0016989914552560 Runtime 156.095 45.443 11.722 48.920 -1.4315374190000000 Mean -1.0202940450426400 -1.0809384421344400 -1.0809384421344400 SD 0.1190811583120530 0.0000000000000005 0.0000000000000005 0.0000000000000009 Best -1.0809384421344400 -1.0809384421344400 -1.0809384421344400 -1.4315374193830000 Neural Comput & Applic Table 4 continued Problem Statistics Runtime 52.853 36.659 21.421 34.714 F23 Mean -1.4765972735526500 -1.4999992233525000 -1.4821658762555300 -1.5000000000000000 SD 0.1281777579497830 0.0000000000000009 0.0976772648082733 0.0000000000000000 F24 F25 F26 F27 F28 F29 F30 F31 F32 F33 F34 CLPSO SADE BSA IA Best -1.4999992233524900 -1.4999992233524900 -1.4999992233524900 -1.5000000000000000 Runtime 42.488 36.037 18.930 41.848 -1.5000000000000000 Mean -0.9431432797743700 -1.2765515661973800 -1.3127183561646500 SD 0.3184175870987750 0.3599594108130040 0.3158807699946290 0.0000000000000000 Best Runtime -1.5000000000003800 124.609 -1.5000000000003800 47.171 -1.5000000000003800 35.358 -1.5000000000000000 54.651 Mean 0.0000041787372626 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000161643637543 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 31.632 4.090 0.813 35.662 -1.8203821100000000 Mean -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 SD 0.0000000000000009 0.0000000000000009 0.0000000000000009 0.0000000000000014 Best -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8203821095139300 Runtime 18.091 28.453 7.472 34.891 Mean -4.6920941990586400 -4.6884965299983800 -4.6934684519571100 -3.2820108350000000 SD 0.0075270931220834 0.0272323381095561 0.0000000000000008 0.0000000000000023 Best -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -3.2820108345268900 Runtime 25.843 38.446 11.971 45.085 Mean -9.6400278592589600 -9.6572038232921700 -9.6601517156413500 -6.2086254390000000 SD Best 0.0437935551332868 -9.6601517156413500 0.0105890022905617 -9.6601517156413500 0.0000000000000007 -9.6601517156413500 0.0000000000000027 -6.2086254392105500 Runtime 32.801 46.395 22.250 71.652 Mean 0.0198686590210374 0.0140272066690658 0.0007283694780796 1.3116221610000000 SD 0.0613698943155661 0.0328868042987376 0.0014793717464195 0.5590904820000000 Best 0.0000175219764526 0.0000000000000000 0.0000000000000000 1.0960146962658900 Runtime 316.817 92.412 191.881 34.697 Mean 0.0458769685199585 0.0000002733806735 0.0000000028443186 0.0000000000000000 SD 0.0620254411839524 0.0000001788830279 0.0000000033308990 0.0000000000000000 Best 0.0005277712020642 0.0000000944121661 0.0000000004769768 0.0000000000000000 Runtime 252.779 360.380 144.784 153.221 Mean 0.0002674563703837 0.0000000000000000 0.0000000111676630 0.0071082040000000 SD 0.0003044909265796 0.0000000000000000 0.0000000184322163 0.0000000000000000 Best 0.0000023064754605 0.0000000000000000 0.0000000000000000 0.0071082039505830 Runtime 227.817 220.886 149.882 43.098 Mean 0.0019635752485802 0.0016730768406953 0.0019955316015528 0.0002254250000000 SD Best 0.0043423828633839 0.0004206447422138 0.0007330246909835 0.0005630852254632 0.0009698942217908 0.0006084880639553 0.0005270410000000 0.0000023800831017 Runtime 103.283 171.637 48.237 218.722 Mean 0.6301407361590880 0.8622978494808570 0.0000000000000000 0.0000000000000000 SD 0.8046401822326410 0.9323785263847000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 18.429 23.594 5.401 2.266 Mean 5.7631786582751800 1.2137377447007000 0.3986623854300930 0.0000154715000000 SD 13.9484817304201000 1.8518519388285700 1.2164328622195200 0.0000022373400000 Best 0.0268003205820685 0.0001448955835246 0.0000000000000000 0.0000118803557196 123 Neural Comput & Applic Table 4 continued Problem Statistics Runtime 187.894 F35 Mean 0.0019431819755029 SD 0.0039528023354469 Best Runtime F36 F37 F38 F39 F40 F41 F42 F43 F44 F45 F46 123 CLPSO SADE BSA IA 268.449 34.681 7.250 0.0006477273251676 0.0000000000000000 0.0000000000000000 0.0024650053428137 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 8.304 5.902 1.779 33.155 -12569.3622100000000000 Mean -12210.8815698372000 -12549.746895737300000 -12569.486618173000000 SD 205.9313376284770000 44.8939348779747000 0.0000000000024122 0.0000000273871000 Best Runtime -12569.4866181730000 31.499 -12569.486618173000000 34.383 -12569.486618173000000 11.069 -12569.3622054081000000 2.306 Mean 6.4655746330439100 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 8.2188901353055800 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.1816624029553790 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 179.083 109.551 57.294 100.947 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 12.563 5.627 3.208 47.009 Mean -10.3130437162026000 -10.5364098166921000 -10.5364098166921000 -10.5063235800000000 SD 1.2234265179736500 0.0000000000000016 0.0000000000000018 0.0000000025211900 Best -10.5364098166920000 -10.5364098166921000 -10.5364098166920000 -10.5063235792920000 Runtime 37.275 28.031 7.045 55.666 Mean -10.1531996790582000 -9.9847854277673500 -10.1531996790582000 -10.1529842600000000 SD Best 0.0000000000000076 -10.1531996790582000 0.9224428443735560 -10.1531996790582000 0.0000000000000072 -10.1531996790582000 0.0000000000542921 -10.1529842649756000 Runtime 30.885 25.569 6.864 51.507 Mean -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.3988303400000000 SD 0.0000000000000010 0.0000000000000018 0.0000000000000017 0.0000000001978980 Best -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.3988303385534000 Runtime 31.207 27.064 8.208 53.190 -186.2926481000000000 Mean -186.730908831024000 -186.7309088310240000 -186.7309088310240000 SD 0.0000000000000279 0.0000000000000377 0.0000000000000224 0.0000000000000578 Best -186.730908831024000 -186.7309088310240000 -186.7309088310240000 -186.2926480689880000 Runtime 20.344 27.109 9.002 31.766 Mean -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0304357800000000 SD 0.0000000000000005 0.0000000000000005 0.0000000000000005 0.0014911900000000 Best -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0314500753985900 Runtime 18.564 27.650 5.691 39.897 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 14.389 5.920 3.302 174.577 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000538870000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000005399890 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000538860819891 Runtime 3.042 4.307 0.883 2.215 Mean 0.2000000000000000 0.0000000000000000 0.0000000000000000 -0.0153463301609662 SD 0.4068381021724860 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.0153463301609662 Neural Comput & Applic Table 4 continued Problem Statistics Runtime 6.142 F47 Mean 0.0000000000000000 SD 0.0000000000000000 Best Runtime F48 F49 F50 CLPSO SADE BSA IA 4.319 0.764 31.068 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 15.948 6.383 4.309 31.296 -44.7416748700000000 Mean -49.4789234062579000 -50.0000000000002000 -50.0000000000002000 SD 1.3150773145311700 0.0000000000000268 0.0000000000000361 0.0000000000000217 Best Runtime -49.9999994167392000 142.106 -50.0000000000002000 36.804 -50.0000000000002000 7.747 -44.7416748706606000 52.486 -150.5540859185450000 Mean -199.592588547503000 -210.0000000000030000 -210.0000000000030000 SD 9.6415263953591700 0.0000000000004625 0.0000000000003950 0.0000000000000000 Best -209.985867409029000 -210.0000000000040000 -210.0000000000040000 -150.5540859185450000 Runtime 187.787 54.421 11.158 70.887 Mean 0.0000000001597805 0.0000000000000000 0.0000000000000000 0.0000000000000000 SD 0.0000000006266641 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 157.838 4.930 5.702 33.573 prove that IA works better than PSO. The self-interested behaviour of every individual in IA enables them to communicate with each other in order to seek for better solutions. They respond adaptively to the shape of the fitness landscape. Thus, IA is able to achieve higher convergence rate in the iterative processes. It is because the efforts of improving the best solution depend on not only the current position of the particle itself but also the position of the global best individual ðLp;gb Þ, local best individual Lp;b and the local second best individual ðLp;2b Þ. This can prevent the problem of falling into local optimum in highdimensional space, which is the common problem faced by most of the EAs. 3.6 CMAES versus IA The CMAES algorithm stands for covariance matrix adaptation evolution strategy. It is a mathematical-based algorithm that makes use of adaptive mutation parameters through computing a covariance matrix as shown in Fig. 7 [36]. One major drawback of CMAES is the cost in calculating the covariance matrix. The cost increases rapidly with increasing dimensions. Plus, sampling using a multivariate normal distribution and factorization of the covariance matrix also becomes increasingly expensive [48]. The IA is being compared with classical CMAES in this work. The relatively simpler structure of IA as compared with CMAES leads to the successful of IA in solving unconstrained benchmark problem in terms of runtime as shown in Tables 6 and 7. Overall the convergence speed of IA is higher than CMAES. 3.7 ABC versus IA In ABC algorithm, the artificial bee colony is made up of employed bees, onlooker bees and scout bees. An onlooker bee waits on the dance area for making decision in choosing a food source. An employed bee goes to the previously visited food source to search for food. A scout bee carries out random search [41]. The working mechanism of ABC is described in Fig. 8. An existing challenge to all stochastic optimization methods is the balance between exploration and exploitation. A poor optimization will meet the problems of premature convergence and get trapped from local minima. Meanwhile, excessively exploitative will cause the algorithm to converge very slowly. ABC is good at exploration but poor at exploitation; its convergence speed is also an issue in some cases [50]. The results of the proposed IA are being compared with ABC. Results from Table 6 denote that IA works equally well as ABC. However, Table 7 shows that IA has a superior performance as compared with ABC. This proves that IA works better in dealing with high-performance and more complicated benchmark functions in Test 2. Table 8 also proves that IA outperforms ABC as indicated in p values as well as T- and T? values. 123 Neural Comput & Applic Table 5 Statistical solutions obtained by PSO, CMAES, ABC, CLPSO, SADE, BSA and proposed IA in Test 2 (mean mean solution, SD standard deviation of mean solution, best best solution, runtime mean runtime in seconds) Problem Statistics PSO2011 CMAES ABC JDE F51 Mean -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 F52 F53 F54 F55 F56 F57 F58 F59 F60 F61 F62 123 Runtime 212.862 23.146 113.623 118.477 Mean -450.0000000000000000 -450.0000000000000000 -449.9999999999220000 -450.0000000000000000 SD 0.0000000000000350 0.0000000000000000 0.0000000002052730 0.0000000000000615 Best -450.0000000000000000 -450.0000000000000000 -449.9999999999970000 -450.0000000000000000 Runtime 230.003 23.385 648.784 139.144 Mean -44.5873911956554000 -450.0000000000000000 387131.24412139700000 -197.9999999999850000 SD 458.5794120016290000 0.0000000000000000 166951.73365926400000 391.5169437474990000 Best Runtime -443.9511286079800000 2658.937 -450.0000000000000000 35.464 165173.18530956000000 240.094 -449.9999999999990000 1017.557 -414.0000000000000000 Mean -450.0000000000000000 77982.4567046980000000 140.4509447125110000 SD 0.0000000000000460 131376.7365456010000000 217.2646715063190000 55.9309919639279000 Best -450.0000000000000000 -450.0000000000000000 -324.3395691109350000 -450.0000000000000000 Runtime 247.256 32.726 209.188 143.767 Mean -310.0000000000000000 -310.0000000000000000 -291.5327549384120000 -271.0000000000000000 SD 0.0000000000000000 0.0000000000000000 17.6942171217937000 60.5919079609218000 Best -310.0000000000000000 -310.0000000000000000 -307.7611364354020000 -310.0000000000000000 Runtime 241.517 39.293 205.568 134.078 Mean 393.4959999056240000 390.5315438816460000 391.2531452421960000 231.3986579112350000 SD 16.0224965900462000 1.3783433976378300 3.7254660805238600 247.2968415284400000 Best 390.0000000000150000 390.0000000000000000 390.0101471658490000 -140.0000000000000000 Runtime 1178.079 27.894 159.762 153.715 Mean 1091.0644335162500000 1087.2645466786700000 1087.0459486286000000 1141.0459486286000000 SD Best 3.4976948942723200 1087.0696772583000000 0.5365230018001780 1087.0459486286000000 0.0000000000005585 1087.0459486286000000 83.8964879458918000 1087.0459486286000000 Runtime 334.064 37.047 180.472 159.922 Mean -119.8190232990920000 -119.9261073509850000 -119.7446063439080000 -119.4450938018030000 SD 0.0720107560874199 0.1554021446157740 0.0623866434489108 0.0927418223065644 Best -119.9302772694110000 -120.0000000000000000 -119.8779554779730000 -119.6575717927190000 Runtime 602.507 49.209 265.319 160.806 Mean -324.6046006320200000 -306.5782069681560000 -330.0000000000000000 -329.8673387923880000 SD 2.5082306041521000 21.9475396048756000 0.0000000000000000 0.3440030182812760 Best -329.0050409429070000 -327.0151228287200000 -330.0000000000000000 -330.0000000000000000 Runtime 982.449 22.237 111.629 128.494 Mean -324.3311322538170000 -314.7871102989330000 -306.7949047862760000 -319.6763749798700000 SD 3.0072222933667300 8.3115989308305500 5.1787864195870400 4.9173541245304800 Best -327.1650513120000000 -327.0151228287200000 -318.9403196374510000 -326.0201637716270000 Runtime 1146.013 29.860 259.258 179.039 Mean 92.5640111212146000 90.7642785704506000 94.8428485804138000 93.2972315784963000 SD Best 1.5827416781636900 90.1142082473923000 26.4613831425879000 -45.0054133586912000 0.6869412813090850 93.1500794016147000 1.8766951726453600 91.0295373630387000 Runtime 1310.457 44.217 308.501 282.150 Mean 18611.314225480900000 -70.0486708747625000 -337.3273080760500000 400.3240208136310000 SD 12508.786612631600000 637.4585182420270000 56.5730759032367000 688.3344299264300000 Best 4568.3350537809200000 -460.0000000000000000 -449.1707421778360000 -434.8788220982740000 Runtime 2381.974 34.857 232.916 202.941 Neural Comput & Applic Table 5 continued Problem F63 F64 F65 Statistics PSO2011 CMAES ABC JDE -129.6294851450880000 Mean -129.2373581503910000 -128.7850616923410000 -129.8343428775830000 SD 0.5986210944493790 0.6157633658946230 0.0408016481905455 0.1054759371085400 Best -129.6861385930680000 -129.5105509483130000 -129.9098920058450000 -129.8125711770830000 Runtime 2183.218 25.496 205.194 186.347 Mean -298.2835926212850000 -295.1290938304830000 -296.9323391084610000 -296.8839733969750000 SD 0.5587676271753680 0.1634039984609270 0.2251930667702880 0.4330673614598290 Best -299.6022022972560000 -295.7382222729600000 -297.4659619544820000 -297.8411886637500000 Runtime Mean 2517.138 417.4613663019860000 32.084 492.5045364088000000 262.533 120.0000000000000000 334.888 326.6601114362900000 SD 153.9215808771580000 181.5709657779580000 0.0000000000000188 174.6877238188330000 Best 120.0000000000000000 262.7619554120320000 120.0000000000000000 120.0000000000000000 Runtime 3156.336 239.823 2285.787 1834.967 NFE F66 F67 F68 F69 F70 F71 Mean 221.4232628350220000 455.1151684594550000 258.8582688922670000 231.1806131539990000 SD 12.2450207482898000 254.3583511786970000 11.8823213189685000 13.5473380962764000 Best 181.5746616282570000 120.0000000000000000 235.6600739998890000 210.3582705649860000 Runtime 4242.280 202.808 2237.308 1824.388 Mean 217.3338617866620000 681.0349114021570000 265.0370119084380000 228.7309024901770000 SD 20.6685850658838000 488.0618274343640000 12.4033917090208000 12.3682716268631000 Best 120.0000000000000000 223.0782617790520000 241.9810089596350000 181.6799927773160000 Runtime 8208.697 197.497 2159.392 5873.112 Mean 668.9850326105730000 926.9488078829420000 513.8925774904480000 743.9859973770210000 SD Best 275.8071370273340000 310.0000000000000000 174.1027182659660000 310.0000000000000000 31.0124861524005000 444.4692044973030000 175.6497294240330000 310.0000000000000000 Runtime 3687.235 251.155 2445.259 1777.638 Mean 708.2979222913040000 831.2324139697050000 500.5478931040730000 776.5150806087790000 SD 256.2419561521300000 250.1848775931620000 31.2240894705539000 160.7307526692470000 Best 310.0000000000000000 310.0000000000000000 407.3155842366960000 363.8314566805740000 Runtime 5258.509 222.015 2341.791 1849.670 Mean 711.2970397614200000 876.9306188768990000 483.2984167460740000 761.2954767038960000 SD 258.9317052508320000 289.7296413284470000 99.3976740616107000 163.4084080635650000 Best 310.0000000000000000 310.0000000000000000 155.5049931377980000 363.8314568648180000 Runtime 4346.055 228.619 2250.917 1900.279 Mean 1117.8857079625100000 1258.1065536572400000 659.5351969346130000 959.3735119754180000 SD 311.0011859260640000 359.7382897536570000 98.5410511961986000 240.5568407069990000 Best 560.0000000000000000 660.0000000000000000 560.0001912324020000 660.0000000000000000 Runtime 3012.883 241.541 2728.060 1573.484 Mean SD 1094.8305116977000000 121.3539576317800000 -7.159E?49 4.387E?50 915.4958100611630000 242.1993331983530000 1133.7536009808600000 42.1171260000361000 Best 660.0000000000000000 -133.9585340104890000 660.0006867770510000 1088.9543269392600000 Runtime 6363.267 290.334 2326.112 1730.723 NFE F72 F73 Mean 1304.3661550124000000 1159.9280867973000000 830.2290165794410000 1167.9040488743800000 SD 262.1065863453340000 742.1215416320490000 60.2286903507069000 236.7325108248320000 Best 919.4683107913200000 -460.7504508023100000 785.1725102979490000 785.1725102979490000 Runtime 2165.640 238.261 2045.582 1580.067 123 Neural Comput & Applic Table 5 continued Problem Statistics PSO2011 CMAES ABC JDE F74 Mean 500.0000000000000000 653.3355378428050000 460.0000000000020000 510.0000000000000000 SD 103.7237710925280000 302.5312999719650000 0.0000000000016493 113.7147065368360000 Best 460.0000000000000000 460.0000000000000000 460.0000000000000000 460.0000000000000000 F75 Problem F51 F52 F53 F54 F55 F56 F57 F58 F59 F60 123 Runtime 1811.980 165.962 1698.121 1366.710 Mean 1107.9038127876700000 1401.6553278264300000 930.4565414149210000 1072.9924659809200000 SD 127.9566489362040000 253.2428066220210000 87.9959072391079000 2.2606058314671500 Best 1069.5511765775700000 1072.4973401423200000 862.4476004191700000 1068.5560012648600000 Runtime 4060.091 214.580 2113.339 2951.018 Statistics CLPSO SADE BSA IA Mean -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -447.6018854297170000 SD 0.0000000000000000 0.0000000000000000 0.0000000000000000 89.3142986500000000 -450.0000000000000000 Best -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 Runtime 167.675 154.232 140.736 30.282 Mean -418.8551838547760000 -450.0000000000000000 -450.0000000000000000 -449.9967727000000000 SD 51.0880511039985000 0.0000000000000000 0.0000000000000259 0.0176705780000000 Best Runtime -449.4789299923810000 1462.706 -450.0000000000000000 185.965 -450.0000000000000000 243.657 -450.0000000000000000 48.003 -449.7873452000000000 Mean 62142.8213760465000000 245.0483283713550000 -449.9999567867430000 SD 34796.1785167236000000 790.6056596723160000 0.0001175386756044 0.0000000000001734 Best 17306.9066792474000000 -421.4054944641620000 -450.0000000000000000 -450.0000000000000000 Runtime 1789.643 1808.954 1883.713 52.463 Mean -178.8320689185280000 -450.0000000000000000 -450.0000000000000000 -388.7807630000000000 SD 394.8667499339530000 0.0000000000000000 0.0000000000000259 1.1928333530000000 Best -447.9901256558030000 -450.0000000000000000 -450.0000000000000000 -389.7573633109500000 Runtime 1248.616 185.438 347.167 46.072 Mean 333.4108259915760000 -309.9999999999960000 -309.9999999999980000 -310.8207993000000000 SD 512.6920837704510000 0.0000000000133965 0.0000000000023443 0.0208030240000000 Best -309.9740055344430000 -310.0000000000000000 -310.0000000000000000 -310.8367924750510000 Runtime 1481.686 210.684 386.633 44.84710031 Mean 405.5233436479650000 390.2657719408230000 390.1328859704120000 390.8036739982730000 SD Best 10.7480096852869000 390.5776683413440000 1.0114275384776600 390.0000000000000000 0.7278464357038200 390.0000000000000000 0.0000000000000000 390.8036739982730000 Runtime 1441.859 1214.303 290.236 45.632 Mean 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.2265890000000000 SD 0.0000000000004264 0.0000000000004814 0.0000000000004428 0.0019192200000000 Best 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.2262037455100000 Runtime 267.342 259.760 332.132 52.621 Mean -119.9300269839980000 -119.7727713703720000 -119.8356122057440000 -119.6006412865410000 SD 0.0417913553101429 0.1248514853682450 0.0704515460477787 0.0000000000000434 Best -119.9756745390830000 -119.9999999999980000 -119.9802847896350000 -119.6006412865410000 Runtime 1586.286 648.489 717.375 52.56165118 Mean -329.4361898676470000 -329.9668346980970000 -330.0000000000000000 -327.1635938000000000 SD 0.6229063711904190 0.1816538397880230 0.0000000000000000 0.0000000000001156 -327.163593801473 Best -330.0000000000000000 -330.0000000000000000 -330.0000000000000000 Runtime 162.873 155.645 176.994 45.867 Mean -321.7278926895280000 -322.9689591871600000 -319.2544515903510000 -335.0171647000000000 SD Best 1.8971778613701300 -326.1788303102740000 2.8254645254663600 -328.0100818858130000 3.3091959975390800 -325.0252097523530000 10.6369134000000000 -347.2509173436740000 Neural Comput & Applic Table 5 continued Problem Statistics Runtime F61 Mean SD 0.6689129174038950 Best 92.9690673344598000 90.1363685040678000 90.2628852415150000 92.0170440535006000 Runtime 1421.545 506.829 1771.860 60.350 -410.1361631000000000 F62 F63 F64 F65 CLPSO SADE BSA IA 1594.096 210.534 420.851 54.661 94.6109567642977000 91.6859083842723000 92.3519494286347000 92.0170440500000000 0.9033073777915270 1.0901581870340800 0.000000000000014453 Mean -447.8870804905020000 -394.5206365378250000 -437.1125728026770000 SD 11.8934815947019000 128.6353424718180000 20.3541618366546000 34.8795385900000000 Best Runtime -459.6890294276810000 1636.440 -460.0000000000000000 1277.975 -459.1772521346520000 1466.985 -421.5672584975600000 48.480 -122.2126680000000000 Mean -129.8382867796110000 -129.7129164862680000 -129.8981409848090000 SD 0.0372256921835666 0.0875456568200232 0.0682328484314248 0.0000000000000434 Best -129.9098505660780000 -129.8717592632560000 -129.9901230990300000 -122.2126679617240000 Runtime 1526.365 660.986 1064.114 46.260 -295.4721554000000000 Mean -297.5119726691150000 -297.8403738182600000 -297.5359077431460000 SD 0.3440115280624180 0.4536801689800720 0.4085859316264990 0.1118191570000000 Best -298.3030560759620000 -299.2417795907860000 -298.3869295150680000 -295.6307146941910000 Runtime 1615.452 1289.814 1953.289 55.118 Mean 131.3550392249760000 234.2689845349590000 120.0000000000000000 120.0000000000000000 SD 26.1407360548431000 150.7595974059750000 0.0000000000000000 0.0000000000000000 Best 120.0000000000000000 120.0000000000000000 120.0000000000000000 120.0000000000000000 Runtime 3210.655 1932.016 2351.478 69.052 Mean SD 231.5547154800990000 11.5441451076421000 222.0256674919140000 6.1841489800660300 234.4843380488580000 8.9091119100451100 276.3946208000000000 19.2196655800000000 Best 214.7661703584830000 206.4520786020840000 219.6244910167680000 259.8700033222460000 Runtime 8649.998 2970.950 8270.920 252.234 Mean 240.3635189964930000 221.1801916743850000 228.3769828342800000 201.0516618000000000 SD 14.8435137485293000 5.7037006844690500 8.7086794471239900 2.4309010810000000 Best 221.3817133141830000 209.2509748304710000 204.6479138174220000 197.8966349103590000 NFE F66 F67 F68 F69 F70 F71 Runtime 4599.027 5938.879 8189.243 254.253 Mean 892.4391527217660000 845.4504613493740000 587.5732354221340000 310.0161021000000000 SD 79.1422224454971000 120.8505129523180000 250.0556329707140000 0.0370586450000000 Best 738.3764781625320000 310.0000000000000000 310.0000000000000000 310.0014955442130000 Runtime 8398.690 3073.274 4554.102 253.064 Mean 863.8926908090610000 809.7183195902260000 587.6511686191670000 310.0029796000000000 SD 96.5618989087194000 147.3158109824600000 236.1141037692630000 0.0082796490000000 Best 493.0042540796450000 310.0000000000000000 310.0000000000000000 310.0000285440690000 Runtime 9909.479 3213.601 4764.968 291.084 Mean SD 844.6391674419360000 113.6848457105400000 810.5227124472170000 104.7139423525340000 612.0906184834040000 249.5599278421970000 310.0041570000000000 0.0128812140000000 Best 489.0742585970560000 310.0000000000000000 310.0000000000000000 310.0002219576930000 Runtime 9988.261 2818.575 4945.132 268.701 Mean 911.4640642691360000 990.8546718748010000 836.1411004458200000 577.7786170000000000 SD 238.3180009803040000 235.1014092849970000 128.9346234954740000 1.8288684190000000 Best 560.0000121795840000 660.0000000000000000 560.0000000000000000 574.8590032551840000 Runtime 10891.124 1769.459 2972.618 279.0646913 NFE 123 Neural Comput & Applic Table 5 continued Problem Statistics CLPSO SADE BSA IA F72 Mean 1075.5292326436900000 1094.6823697304900000 984.5106541514410000 694.3706620000000000 SD 166.9355145236330000 87.9884000140656000 199.1563947691970000 20.9754439100000000 Best 660.0000000000020000 660.0000000000000000 660.0000000000000000 644.2542524502140000 F73 F74 F75 Runtime 9601.880 3854.148 10458.467 273.922 Mean 1070.4327462836400000 1105.2511774948600000 976.2273885425320000 559.6581705000000000 SD 203.0676662707430000 190.6172874229610000 160.1543461970300000 16.1193896300000000 Best 785.1725102979480000 919.4683107913240000 785.1725102979480000 546.1130231359180000 Runtime Mean 7459.005 493.3333333333340000 1901.540 490.0000000000000000 4209.110 460.0000000000000000 287.271 463.2262530000000000 SD 137.2973951415090000 91.5385729888094000 0.0000000000000000 4.9321910760000000 Best 460.0000000000000000 460.0000000000000000 460.0000000000000000 458.5444354721460000 Runtime 3016.959 1410.399 1795.637 257.960 Mean 1258.5157766524700000 1074.3695435628600000 1063.7363787709700000 471.2797518000000000 SD 241.4024507676890000 2.8314182838917800 55.8479313799755000 2.2346287190000000 Best 871.8607884176050000 1069.8723890709000000 856.8214538442850000 469.3372925643150000 Runtime 5262.210 3410.902 4280.901 263.829 3.8 DE versus IA DE is a population-based algorithm which uses the similar operators as GA: crossover, mutation and selection. The only difference is that GA relies on crossover where DE relies on mutation operation. DE algorithm uses mutation operation as a search mechanism and selection operation to direct the search in the search space as shown in Eqs. (12) and (13). By creating trial vectors using the components of existing individuals in the population, the crossover operator effectively sorts information about successful combinations, enabling better solution search space [15]. Mutation x^i ¼ xr1 þ F ðxr3 xr2 Þ; xr1 ; xr2 ; xr3 jr1 6¼ r2 6¼ r3 6¼ i j x^i ; Rj CR Crossover yij ¼ ; xij ; Rj [ CR F ¼ ½0; 1Š ð12Þ ð13Þ Rj ¼ ½0; 1Š ð14Þ During mutation, the parameter x^i is mutant solution vector, while F is scaling factor and i is an index of current solution. In the stage of crossover, CR is the crossover constant, while j represents the jth component of the corresponding array. In DE, a population of solution vectors is randomly created at the start. This population is successfully improved by applying mutation, crossover and selection operators as shown in Fig. 9. In DE algorithm, each new solution produced competes with a mutant vector and the better one wins the competition. In other words, the chance of succession is independent on their fitness values. Every new solution produced competes with its parent, and the better one wins the competition [15]. 123 In this section, IA is being compared with the variants modified based on DE, which are JDE and SADE. The IA outperforms JDE and SADE in most of the benchmark functions of Test 1 and Test 2. The statistical results in Tables 6 and 7 indicate that IA performs better than JDE and SADE. 3.9 BSA versus IA In BSA, three basic genetic operators—selection, mutation and crossover—are used to generate trial individuals. A random mutation strategy is performed such that only one direction individual is used for each target individual. BSA randomly chooses the direction individual from a randomly chosen individual from previous generation. BSA uses a nonuniform crossover strategy that is more complex as compared with other GAs [38]. The procedure of BSA is shown in Fig. 10. BSA is divided into five processes: initialization, selection I, mutation, crossover and selection II. In the selection II stage, the Ti s that have better fitness values than the corresponding Pi s are used to update the Pi s based on the concept of greedy selection. If the best individual of P (Pbest ) has better fitness value than the global minimum value, the global minimizer is updated to be Pbest . Hence, the global minimum value is updated to be the fitness value of Pbest . Mutation Mutant ¼ P þ F ðoldP PÞ ð15Þ Crossover mapn;m ¼ 1; Tn;m :¼ Pn;m n 2 f1; 2; 3; . . .; N g m 2 f1; 2; 3; . . .; Dg ð16Þ Problem PSO2011 versus IA p value T? CMAES versus IA T- Winner p value T? ABC versus IA T- Winner p value JDE versus IA T? T- Winner p value T? T- Winner F1 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F2 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F3 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F4 4.3205E-08 0 465 ? 6.7988E-08 0 465 ? 3.3465E-07 0 465 ? 4.3205E-08 0 465 ? F5 1.7289E-06 0 465 ? 1.7289E-06 0 465 ? 1.7289E-06 465 0 – 1.7289E-06 0 465 ? F6 1.7300E-06 465 0 – 1.7300E-06 0 465 ? 1.7300E-06 465 0 – 1.7300E-06 465 0 – – F7 1.7344E-06 465 0 – 1.7344E-06 0 465 ? 1.7344E-06 465 0 – 1.7344E-06 465 0 F8 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? F9 1.00E?00 0 0 = 1.7279E-06 0 465 ? 4.3205E-08 0 465 ? 1.00E?00 0 0 = F10 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F11 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F12 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 0 465 ? 4.3205E-08 465 0 – F13 1.6594E-06 0 465 ? 1.6594E-06 0 465 ? 1.6594E-06 465 0 – 1.6594E-06 0 465 ? F14 1.5450E-06 465 0 – 1.6657E-06 465 0 – 1.5450E-06 465 0 – 1.5450E-06 465 0 – F15 1.0135E-07 465 0 – 1.0135E-07 0 465 ? 1.0135E-07 465 0 – 1.0135E-07 465 0 – F16 6.8714E-07 0 465 ? 6.8714E-07 0 465 ? 6.8714E-07 0 465 ? 6.8714E-07 0 465 ? F17 1.1048E-06 0 465 ? 1.1048E-06 0 465 ? 1.1048E-06 0 465 ? 1.1048E-06 0 465 ? F18 1.0135E-07 465 0 – 1.0135E-07 465 0 – 1.0135E-07 465 0 – 1.0135E-07 465 0 – F19 1.2033E-06 465 0 – 1.2033E-06 0 465 ? 1.2033E-06 465 0 – 1.2033E-06 465 0 – F20 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F21 1.7300E-06 0 465 ? 1.7300E-06 0 465 ? 1.7300E-06 0 465 ? 1.7300E-06 0 465 ? ? F22 1.6647E-06 0 465 ? 1.6647E-06 0 465 ? 1.6647E-06 0 465 ? 1.6647E-06 0 465 F23 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? F24 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F25 1.00E?00 0 0 = 1.00E?00 0 0 = 4.3205E-08 0 465 ? 1.00E?00 0 0 = F26 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F27 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F28 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F29 5.9869E-07 0 465 ? 5.9869E-07 0 465 ? 5.9869E-07 0 465 ? 5.9869E-07 0 465 ? F30 1.6976E-06 465 0 – 1.6976E-06 465 0 – 1.6976E-06 465 0 – 1.6976E-06 465 0 – F31 1.0789E-06 0 465 ? 1.0789E-06 0 465 ? 1.0789E-06 0 465 ? 1.0789E-06 0 465 ? F32 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F33 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 1.9773E-07 465 0 – 4.3205E-08 0 465 ? 123 F34 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F35 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? Neural Comput & Applic Table 6 Statistical results for each benchmark problem in Test 1 using two-sided Wilcoxon signed-rank test (x = 0.05) 123 Table 6 continued Problem PSO2011 versus IA p value T? CMAES versus IA T- Winner p value ABC versus IA T? T- Winner p value JDE versus IA T? T- Winner p value T? T- Winner F36 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F37 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F38 3.3248E-07 465 0 – 3.3248E-07 465 0 – 4.3205E-08 0 465 ? 3.3248E-07 465 0 – F39 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F40 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F41 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – – F42 1.9618E-07 465 0 – 1.9618E-07 465 0 – 1.9618E-07 465 0 – 1.9618E-07 465 0 F43 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F44 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F45 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 1.00E?00 0 0 = 4.3205E-08 0 465 ? F46 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F47 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F48 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F49 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F50 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? ?/=/- 24/2/24 Problem 29/1/20 CLPSO versus IA p value 24/1/25 24/2/24 SADE versus IA T? T- Winner p value BSA versus IA T? T- Winner p value T? T- Winner F1 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F2 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F3 1.3422E-06 465 0 – 4.3205E-08 0 465 ? 1.3422E-06 465 0 – F4 3.3465E-07 465 0 – 4.3205E-08 0 465 ? 3.3465E-07 465 0 – F5 1.7289E-06 0 465 ? 1.7289E-06 0 465 ? 1.7289E-06 465 0 – F6 1.7300E-06 465 0 – 1.7300E-06 465 0 – 1.7300E-06 465 0 – 1.7344E-06 465 0 – 1.7344E-06 465 0 – 1.7344E-06 465 0 – 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? F9 4.3205E-08 0 465 ? 1.00E?00 0 0 = 1.00E?00 0 0 = F10 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 465 0 – F11 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F12 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 465 0 – F13 0.0566 140 325 ? 1.6594E-06 0 465 ? 1.6594E-06 0 465 ? F14 1.5450E-06 465 0 – 1.5450E-06 465 0 – 1.5450E-06 465 0 – F15 1.0135E-07 465 0 – 1.0135E-07 465 0 – 1.0135E-07 465 0 – F16 6.8714E-07 0 465 ? 6.8714E-07 0 465 ? 6.8714E-07 0 465 ? Neural Comput & Applic F7 F8 Problem CLPSO versus IA p value SADE versus IA T? T- Winner p value BSA versus IA T? T- Winner p value T? T- Winner F17 1.1048E-06 0 465 ? 1.1048E-06 465 0 – 1.1048E-06 465 0 – F18 1.0135E-07 465 0 – 1.0135E-07 465 0 – 1.0135E-07 465 0 – F19 1.2033E-06 465 0 – 1.2033E-06 465 0 – 1.2033E-06 465 0 – F20 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F21 1.7300E-06 0 465 ? 1.7300E-06 0 465 ? 1.7300E-06 0 465 ? F22 1.6647E-06 0 465 ? 1.6647E-06 0 465 ? 1.6647E-06 0 465 ? F23 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? 1.7279E-06 0 465 ? – F24 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 F25 4.3205E-08 0 465 ? 1.00E?00 0 0 = 1.00E?00 0 0 = F26 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – – F27 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 F28 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F29 5.9869E-07 0 465 ? 5.9869E-07 0 465 ? 5.9869E-07 465 0 – F30 1.6976E-06 465 0 – 1.6976E-06 465 0 – 1.6976E-06 465 0 – F31 1.0789E-06 0 465 ? 1.0789E-06 0 465 ? 1.0789E-06 0 465 ? F32 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F33 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 1.9773E-07 465 0 – F34 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F35 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F36 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F37 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F38 3.3248E-07 465 0 – 3.3248E-07 465 0 – 3.3248E-07 465 0 – F39 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F40 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F41 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F42 1.9618E-07 465 0 – 1.9618E-07 465 0 – 1.9618E-07 465 0 – F43 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – – F44 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 F45 1.00E?00 0 0 = 1.00E?00 0 0 = 1.00E?00 0 0 = F46 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F47 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F48 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 123 F49 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F50 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? ?/=/- 25/1/24 22/3/25 17/3/30 Neural Comput & Applic Table 6 continued 123 Table 7 Statistical results for each benchmark problem in Test 2 using two-sided Wilcoxon signed-rank test (x = 0.05) Problem PSO versus IA CMAES versus IA ABC versus IA JDE versus IA p value T? T- Winner p value T? T- Winner p value T? T- Winner p value T? T- Winner F51 6.9066E-07 465 0 – 6.9066E-07 465 0 – 6.9066E-07 465 0 – 6.9066E-07 465 0 – F52 1.1826E-06 465 0 – 1.1826E-06 465 0 – 1.1826E-06 465 0 – 1.1826E-06 465 0 – ? F53 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 0 465 ? 4.3205E-08 0 465 F54 1.7333E-06 465 0 – 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? 1.7333E-06 465 0 – F55 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F56 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 0 465 ? 4.3205E-08 465 0 – F57 6.7988E-08 0 465 ? 6.7988E-08 0 465 ? 6.7988E-08 465 0 – 6.7988E-08 0 465 ? F58 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 0 465 ? F59 F60 4.3205E-08 3.9575E-05 0 36 465 429 ? ? 4.3205E-08 1.1567E-06 0 0 465 465 ? ? 4.3205E-08 1.1567E-06 465 0 0 465 – ? 4.3205E-08 1.1567E-06 465 0 0 465 – ? F61 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? F62 1.4403E-07 0 465 ? 1.4403E-07 0 465 ? 2.9866E-07 6 459 ? 1.4403E-07 0 465 ? F63 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F64 1.5117E-06 465 0 – 1.5117E-06 0 465 ? 1.5117E-06 465 0 – 1.5117E-06 465 0 – F65 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 1.00E?00 0 0 = 4.3205E-08 0 465 ? F66 7.8641E-07 465 0 – 7.8641E-07 0 465 ? 7.8641E-07 465 0 – 7.8641E-07 465 0 – F67 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? F68 6.9824E-07 0 465 ? 6.9824E-07 0 465 ? 6.9824E-07 0 465 ? 6.9824E-07 0 465 ? F69 1.1881E-06 0 465 ? 1.1881E-06 0 465 ? 1.1881E-06 0 465 ? 1.1881E-06 0 465 ? F70 1.2001E-06 0 465 ? 1.2001E-06 0 465 ? 1.2001E-06 0 465 ? 1.2001E-06 0 465 ? ? F71 9.2745E-07 0 465 ? 9.2745E-07 0 465 ? 9.2745E-07 0 465 ? 9.2745E-07 0 465 F72 1.9721E-07 0 465 ? 4.3205E-08 465 0 – 1.9721E-07 0 465 ? 1.9721E-07 0 465 ? F73 8.8940E-07 0 465 ? 8.8940E-07 0 465 ? 8.8940E-07 0 465 ? 8.8940E-07 0 465 ? F74 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? 0.0752 319 146 = 1.7333E-06 0 465 ? F75 ?/=/- 1.7344E-06 18/0/7 0 465 ? 1.7344E-06 17/0/8 0 465 ? 1.7344E-06 15/2/8 0 465 ? 1.7344E-06 17/0/8 0 465 ? Problem SADE versus IA BSA versus IA p value T? T- Winner p value T? T- Winner p value T? T- Winner 6.9066E-07 1.1826E-06 465 0 0 465 – ? 6.9066E-07 1.1826E-06 465 465 0 0 – – 6.9066E-07 1.1826E-06 465 465 0 0 – – F53 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 465 0 – F54 1.7333E-06 0 465 ? 1.7333E-06 465 0 – 1.7333E-06 465 0 – F55 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? Neural Comput & Applic F51 F52 CLPSO versus IA Problem CLPSO versus IA p value SADE versus IA T? T- Winner p value BSA versus IA T? T- Winner p value T? T- Winner F56 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 465 0 – F57 6.7988E-08 465 0 – 6.7988E-08 465 0 – 6.7988E-08 465 0 – F58 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F59 F60 4.3205E-08 1.1567E-06 465 0 0 465 – ? 4.3205E-08 1.1567E-06 465 0 0 465 – ? 4.3205E-08 1.1567E-06 465 0 0 465 – ? F61 4.3205E-08 0 465 ? 4.3205E-08 465 0 – 4.3205E-08 0 465 ? F62 1.4403E-07 465 0 – 9.9562E-04 87 378 ? 1.4403E-07 465 0 – F63 4.3205E-08 465 0 – 4.3205E-08 465 0 – 4.3205E-08 465 0 – F64 1.5117E-06 465 0 – 1.5117E-06 465 0 – 1.5117E-06 465 0 – F65 4.3205E-08 0 465 ? 4.3205E-08 0 465 ? 1.00E?00 0 0 = F66 7.8641E-07 465 0 – 7.8641E-07 465 0 – 7.8641E-07 465 0 – F67 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? 1.7333E-06 0 465 ? F68 6.9824E-07 0 465 ? 6.9824E-07 0 465 ? 6.9824E-07 0 465 ? F69 1.1881E-06 0 465 ? 1.1881E-06 0 465 ? 1.1881E-06 0 465 ? F70 1.2001E-06 0 465 ? 1.2001E-06 0 465 ? 1.2001E-06 0 465 ? F71 9.2745E-07 0 465 ? 9.2745E-07 0 465 ? 9.2745E-07 0 465 ? F72 1.9721E-07 0 465 ? 1.9721E-07 0 465 ? 1.9721E-07 0 465 ? F73 8.8940E-07 0 465 ? 8.8940E-07 0 465 ? 8.8940E-07 0 465 ? F74 F75 1.7333E-06 1.7344E-06 0 0 465 465 ? ? 1.7333E-06 1.7344E-06 0 0 465 465 ? ? 0.0752 1.7344E-06 319 0 146 465 = ? ?/=/- 17/0/8 14/0/11 11/2/12 Neural Comput & Applic Table 7 continued 123 Neural Comput & Applic 1: Initialization 2: repeat 3: Calculate fitness values of particles 4: Modify the best particles in the swarm 5: Choose the best particle 6: Calculate the particles’ velocity 7: Update the particles’ position 8: until convergence Fig. 6 General structure of PSO 1: Initialization 2: repeat 3: Sample and validate offspring’s fitness value 4: Sort the offspring by fitness 5: Perform environmental selection 6: Update the evolution path for covariance matrix adaptation 7: Update the covariance matrix 8: Update the step size 9: Update the mean 10: until convergence Fig. 7 General structure of CMAES The proposed IA is compared with BSA. The results indicate that IA works equally well as BSA in Test 1 and Test 2 as shown in Tables 6 and 7. However, Table 8 shows that BSA is better than IA. The unique mutation and crossover strategies of BSA make it a powerful minimization technique. However, results of Test 1 and Test 2 denote that IA has higher convergence rate as compared with BSA because of the relatively simpler structure of IA. 4 Conclusions and future directions Table 8 Multi-problem-based statistical pairwise comparison of PSO, CMAES, ABC, JDE, CLPSO, SADE, BSA and proposed IA Other algorithm versus IA p Value PSO versus IA 0.0038 Fig. 8 General structure of ABC 123 T- Winner 55 270 IA CMAES versus IA 0.0087 65 260 IA ABC versus IA 0.0207 69 231 IA JDE versus IA 0.0058 60 265 IA CLPSO versus IA 0.0025 50 275 IA SADE versus IA 0.0264 80 245 IA BSA versus IA 0.2904 113 187 BSA exploration and exploitation. The performance of the proposed algorithm was benchmarked on 75 test functions in terms of exploration, exploitation, local optima avoidance, fitness improvement of the population and convergence rate. It can be concluded that the proposed algorithm benefits from high exploitation and convergence rate. The IA is compared to seven well-known and recent algorithms: PSO, CMAES, ABC, JDE, CLPSO, SADE and BSA. Wilcoxon statistical tests were also conducted when comparing the algorithms. The results showed that the proposed algorithm outperforms other algorithms in the majority of test functions. The statistical tests proved that the results were statistically significant for the IA. Thus, it may be concluded from the results that the proposed IA is comparable with other algorithms. Also, it is able to be applied as alternative optimizer for different optimization problems. It is concluded that the IA improves the overall fitness of random initial solutions on optimization problems from the overall individuals’ fitness. IA effectively searches and converges towards promising search space. Thus, the proposed algorithm is able to discover different regions of an optimization problem. Other remarks based on the results of this study are as follows: • In this work, a novel socio-inspired algorithm referred to as ideology algorithm (IA), which is mainly inspired from the human society individuals following certain ideology, is proposed. Several operators were proposed and mathematically modelled for equipping the IA with high T? • Initial random walks of individuals around the parties emphasize exploration of the search space around the individuals. Effective in local optima avoidance since IA employs a population of search agents to approximate the global optimum. 1: Initialization 2: repeat 3: Place the employed bees on their food sources 4: Place the onlooker bees on the food sources depending on their nectar amounts 5: Send the scouts to the search area for discovering new food sources 6: Memorize the best food source found so far 7: until convergence Neural Comput & Applic 1: Initialization 2: Evaluation 3: repeat 4: Mutation 5: Recombination 6: Evaluation 7: Selection 8: until convergence Fig. 9 General structure of DE 1: Initialization 2: repeat 3: Selection I 4: Mutation 5: Crossover 6: Selection II 7: until convergence Fig. 10 General structure of BSA • • • • • Promising search spaces are ensured since individuals relocate to the position of the best individuals during optimization. The best individual from each iteration is saved and considered as the elite, so all individuals tend towards the best solution obtained so far as well. IA has very few parameters to adjust. Thus, it is a flexible algorithm for solving diverse class of problems. The unique mechanism of IA where the local party leader competes with every other party leader and the second best individual in its own party. This motivates the party leaders to explore a greater and promising search space. Also, they continuously look for a better solution in its own local neighbourhood. Every individual in every party to directly and indirectly compete with the same party individuals as well as other party individuals. This makes every party to remain in competition and grow which motivates the individuals search for better solutions. Several research directions can be recommended for future studies with the proposed algorithm. 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