Multi-Cohort Intelligence Algorithm: An Intra- and Inter-group Learning Behavior based Socio-inspired Optimization Methodology
11 Multi-Cohort Intelligence Algorithm: An Intra- and Inter-group Learning Behavior based Socio-inspired Optimization Methodology
Apoorva S Shastri , Anand J Kulkarni* Odette School of Business, University of Windsor, 401 Sunset Avenue, Windsor ON N9B3P4 Canada, Email: [email protected], Ph: 1 519 253 3000 x4939 Symbiosis Institute of Technology, Symbiosis International (Deemed University), Pune MH 412 115 India Email: [email protected]; [email protected] Ph: 91 20 39116468
Abstract
A Multi-Cohort Intelligence (Multi-CI) metaheuristic algorithm in emerging socio-inspired optimization domain is proposed. The algorithm implements intra-group and inter-group learning mechanisms. It focusses on the interaction amongst different cohorts. The performance of the algorithm is validated by solving 75 unconstrained test problems with dimensions up to 30. The solutions were comparing with several recent algorithms such as Particle Swarm Optimization, Covariance Matrix Adaptation Evolution Strategy, Artificial Bee Colony, Self-adaptive differential evolution algorithm, Comprehensive Learning Particle Swarm Optimization, Backtracking Search Optimization Algorithm and Ideology Algorithm. The Wilcoxon signed rank test was carried out for the statistical analysis and verification of the performance. The proposed Multi-CI outperformed these algorithms in terms of the solution quality including objective function value and computational cost, i.e. computational time and functional evaluations. The prominent feature of the Multi-CI algorithm along with the limitations are discussed as well. In addition, an illustrative example is also solved and every detail is provided.
Keywords:
Multi-Cohort Intelligence Algorithm, Socio-inspired optimization, Intra- and Inter-group Learning, Unconstrained Optimization, Metaheuristic Introduction
Several nature-inspired optimization algorithms have been developed so far. The notable algorithms are Evolutionary Algorithms (EAs), Genetic Algorithms (GAs), Swarm Optimization (SO) techniques, etc. These methods have proven their superiority in terms of solution quality and computational time over the traditional (exact) methods for solving a wide variety of problem classes. In agreement with the no-free-lunch theorem, certain modifications and supportive techniques are required to be incorporated into these methods when applying for solving a variety of class of problems. This motivated the researchers to resort to development of new optimization methods. An Artificial Intelligence (AI) based socio-inspired optimization methodology referred to as Cohort Intelligence (CI) was proposed by Kulkarni et al. in 2013. It is inspired from the interactive and competitive social behaviour of individual candidates in a cohort. Every candidate exhibits self-interested behaviour and tries to improve it by learning from the other candidates in the cohort. The learning refers to following/adopting the qualities associated with the behaviour of the other candidates. The candidates iteratively follow one another based on certain probability and the cohort is considered saturated/converged when no further improvement in the behaviour of any of the candidates is possible for considerable number of attempts.
The CI methodology was validated by solving several unconstrained test problems (Kulkarni et al. 2013). The algorithm performed better as compared to several versions of the Particle Swarm Optimization (PSO) such as Chaos-PSO (CPSO) and Linearly Decreasing Weight PSO (LDWPSO) (Liu et al. 2010) as well as Robust Hybrid PSO (RHPSO) (Xu et al. 2013). Then was applied for solving a combinatorial problem such as Knapsack problem (Kulkarni and Shabir2016). The algorithm yielded comparable solutions as compared to the Integer programming (IP), Harmony Search (HS) (Zou et al. 2011;Geem et al. 2001), Improved HS (IHS) (Zou et al. 2011;Mahdavi et al. 2007), Novel Global HS (NGHS) (Zou et al. 2011;Layeb 2011, 2013), Quantum Inspired HS Algorithm (QIHSA) (Layeb 2013) and Quantum Inspired Cuckoo Search Algorithm (QICSA) (Layeb 2011).The combinatorial problems from healthcare domain as well as complex large sized Supply Chain problems such as Sea-Cargo problem and Selection of Cross-Border Shippers were also solved (Kulkarni et al. 2016).
Furthermore, CI contributed in design of fractional PID controller (Shah and Kulkarni, 2017). CI was applied for solving mechanical engineering problems such as discrete and mixed variable engineering problems (Kale and Kulkarni, 2017) and cup forming design problems (Kulkarni, Kulkarni, Kulkarni, Kakandikar 2016). Recently several variations of CI were proposed by Patankar and Kulkarni (2018). In addition, CI with Cognitive Computing (CICC) was applied for solving steganography problems by (Sarmah and Kulkarni, 2017, 2018). The CI performance was better as compared to the IP solutions as well as specially developed Multi Random Start Local Search (MRSLS) method. In these problems, constraints were handled using a specially developed probability based constraint handling approach. In addition, Traveling Salesman Problem (TSP) was also solved (Kulkarni et al. 2017). The approach was further adopted for solving continuous constrained test problems (Shastri et al. 2016, Kulkarni et al 2016). In addition, complex problem of heat exchanger was also solved using CI method (Dhavale et al. 2016). The solutions were comparable to the techniques such as Differential Evolution (DE) (Price et al. 2005) and GA (Deb et al. 2000).Furthermore, a modified version of CI referred to as MCI as well as its hybridized version with K-means performed better as compared to K-means, K-means++ as well as Genetic Algorithm (GA) (Maulik and Bandyopadhyay 2000), Simulated Annealing (SA) (Niknam and Amiri 2010; Selim and Alsultan 1991), Tabu Search (TS) (Niknam and Amiri 2010), Ant Colony Optimization (ACO) (Shelokar et al. 2004), Honeybee Mating Optimization (HBMO) (Fathian and Amiri 2008) and Particle Swarm Optimization (PSO) (Kao et al. 2008). It is important to mention here that in the current version of CI (including MCI in which a mutation approach was used for sampling) the candidates learn from the candidates of the same cohort. As the selection is based on roulette wheel approach it is not necessary that the candidate will follow the best candidate in every learning attempt. Even though this helps the candidates jump out of local minima, learning options are limited as only intra-group learning exists. In the society several cohorts exist which interact and compete with one another which could be referred to as inter-group learning. This makes the candidates learn from the candidates within the cohort as well as the candidates from other cohorts. In the proposed Multi-Cohort Intelligence (Multi-CI) approach intra-group learning and inter-group learning mechanisms were implemented. In the intra-group learning mechanism, every candidate based on roulette wheel approach chooses a behaviour from within its own cohort. Then it samples certain behaviours from within the close neighbourhood of the chosen behaviour. In the inter-group learning mechanism, every candidate based on roulette wheel approach chooses a behaviour from within a pool of best behaviours associated with every cohort. Then it chooses the best behaviour by sampling certain number of behaviours from within the close neighbourhood of both behaviours chosen using the intra-group learning and inter-group learning mechanisms. This manuscript is organized as follows: Section 2 describes the Multi-Cohort Intelligence (Multi-CI) procedure. The performance analysis of the Multi-CI along with the Wilcoxon Signed Rank Test and comparison with the other algorithms is provided in Section 3. The conclusions and future directions are discussed in Section 4. A detailed illustration of the Multi-CI algorithm is provided in an Appendix at the end of the manuscript. Multi-Cohort Intelligence (Multi-CI)
Consider a general unconstrained optimization problem (in minimization sense) as follows: Minimize ๐(๐) = ๐(๐ฅ , โฆ ๐ฅ ๐ , โฆ ๐ฅ ๐ ) (1) Subject to ๐ ๐๐๐๐ค๐๐ โค ๐ฅ ๐ โค ๐ ๐๐ข๐๐๐๐ , ๐ = 1, โฆ , ๐ In the context of Multi-CI the objective function ๐(๐) is considered as the behavior of an individual candidate in each cohort with associated set of qualities
๐ = (๐ฅ , โฆ ๐ฅ ๐ , โฆ , ๐ฅ ๐ ) . The procedure begins with initialization of learning attempt counter ๐ = 1 , and ๐พ cohorts with number of candidates ๐ถ ๐ associated with every cohort ๐, (๐ = 1, โฆ , ๐พ) . Every candidate ๐ (๐ = 1, โฆ , ๐ถ ๐ ) , ๐ = 1, โฆ , ๐พ randomly generates qualities ๐ฟ ๐๐ = (๐ฅ , โฆ ๐ฅ ๐,๐๐ , โฆ , ๐ฅ ๐,๐๐ ) from within its associated sampling interval [ ๐ ๐๐๐๐ค๐๐ ,๐ ๐๐ข๐๐๐๐ ] , ๐ = 1, โฆ , ๐ . The parameters such as convergence parameter ิ , sampling interval reduction factor ๐ , behavior variations ๐ and ๐ ๐ are chosen. The algorithm steps are discussed below and the Multi-CI algorithm flowchart is presented in Figure 1. Step 1 (Evaluation of Behaviors) : The pool of objective functions/behaviors of every candidate ๐ (๐ = 1, โฆ , ๐ถ ๐ ) associated with every cohort ๐(๐ = 1, โฆ , ๐พ ) could be represented as follows: ๐ญ = [ ๐(๐ฟ ) , โฆ , ๐(๐ฟ ๐1 ) , โฆ , ๐(๐ฟ ๐พ1 )โฎ . โฎ . โฎ๐(๐ฟ ) , โฆ , ๐(๐ฟ ๐๐ ) , โฆ , ๐(๐ฟ ๐พ๐ )โฎ . โฎ . โฎ๐(๐ฟ ), โฆ ,๐(๐ฟ ๐ ), โฆ ,๐(๐ฟ ๐พ๐ถ ๐พ )] = [๐ , โฆ ,๐ ๐ , โฆ ,๐ ๐พ ] (2) Step 2 (Pool ๐ Formation) : The best behavior (objective functions with minimum value) candidate ๐ฬ ๐ , ๐(๐ =1, โฆ , ๐พ ) in each cohort are chosen and kept in separated pool ๐ and the associated set of behaviors ๐ญ ๐ is represented as follows: ๐ญ ๐ = [min (๐ ) , โฆ , min (๐ ๐ ) , โฆ , min (๐ ๐พ )] = [๐(๐ฟ ), โฆ ,๐ (๐ฟ ๐๐ฬ ๐ ), โฆ ,๐(๐ฟ ๐พ๐ฬ ๐พ )] (3) Step 3 (Probability Evaluation 1) : The probabilities associated with each candidate except pool ๐ candidates ๐ (๐ = 1, โฆ , ๐ถ ๐ โ 1) , in every cohort ๐(๐ = 1, โฆ , ๐พ ) are calculated as follows: ๐ ๐๐ = ๐๐ )โโ 1 ๐(๐ฟ ๐๐ )โ ๐ถ๐โ1 ๐=1 (4)
Step 4 (Formation of T behaviors) : Using roulette wheel approach every candidate selects a behaviour from within its corresponding cohort (except pool ๐ behaviors) and forms ๐ new behaviours by sampling in close neighbourhood of the qualities associated with the selected behaviour qualities. The neighbourhood of a quality ๐ฅ ๐,๐๐ associated with the sampling interval [๐ ๐๐,๐๐๐ค๐๐ , ๐ ๐๐,๐ข๐๐๐๐ ] , ๐ = 1, โฆ , ๐ of the follower candidate ๐ (๐ = 1, โฆ , ๐ถ ๐ โ 1) , ๐(๐ = 1, โฆ , ๐พ ) is as follows: [๐ ๐๐,๐๐๐ค๐๐ , ๐ ๐๐,๐ข๐๐๐๐ ] = [๐ฅ ๐,๐๐โ โ (โ ๐ ๐๐โ,๐ข๐๐๐๐ โ๐ ๐๐โ,๐๐๐ค๐๐ โ) ร ๐, ๐ฅ ๐,๐๐โ + (โ ๐ ๐๐โ,๐ข๐๐๐๐ โ๐ ๐๐โ,๐๐๐ค๐๐ โ) ร ๐] (5) where ๐โ represents the candidate being followed. The quality matrix ๐ ๐ associated with every candidate ๐ (๐ = 1, โฆ , ๐ถ ๐ โ 1) and corresponding cohort ๐(๐ =1, โฆ , ๐พ) is represented as follows: ๐ ๐ = [ ๐ โฆ ๐ ๐1,๐ โฆ ๐ ๐พ1,๐ โฎ โฑ โฎ โฎ๐ โฆ ๐ ๐๐,๐ โฆ ๐
๐พ๐,๐ โฎ โฎ โฑ โฎ๐ โ1,๐ โฆ ๐ ๐๐ถ ๐ โ1,๐ โฆ ๐ ๐พ๐ถ ๐พ โ1,๐ ] (6) where ๐ ๐๐,๐ = [ ๐ฟ ๐๐,1 โฎ๐ฟ ๐๐,๐ก โฎ๐ฟ ๐๐,๐ ] = [ ๐ฅ โฆ ๐ฅ ๐,๐๐,1 โฆ ๐ฅ ๐,๐๐,1 โฎ โฑ โฎ โฎ๐ฅ โฆ ๐ฅ ๐,๐๐,๐ก โฆ ๐ฅ
๐,๐๐,๐ก โฎ โฎ โฑ โฎ๐ฅ โฆ ๐ฅ ๐,๐๐,๐ โฆ ๐ฅ
๐,๐๐,๐ ] The behavior matrix ๐ญ ๐ associated with ๐ ๐ could be represented as follows: ๐ญ ๐ = [ ๐(๐ ) โฆ ๐(๐ ๐1,๐ ) โฆ ๐(๐ ๐พ1,๐ )โฎ โฑ โฎ โฎ๐(๐ ) โฆ ๐(๐ ๐๐,๐ ) โฆ ๐(๐
๐พ๐,๐ )โฎ โฎ โฑ โฎ๐(๐ โ1,๐ )โฆ ๐(๐ ๐๐ถ ๐ โ1,๐ )โฆ๐( ๐ ๐พ๐ถ ๐พ โ1,๐ )] (7) where ๐(๐ ๐๐,๐ ) = [ ๐(๐ฟ ๐๐,1 )โฎ๐(๐ฟ ๐๐,๐ก )โฎ๐(๐ฟ ๐๐,๐ )] Step 5 (Probability Evaluation 2) : The probabilities associated with each pool ๐ candidate ๐ฬ ๐ in every cohort ๐(๐ = 1, โฆ , ๐พ ) are calculated as follows: ๐ ๐ฬ ๐ = ๐๐ฬ๐ )โโ 1 ๐(๐ฟ ๐๐ฬ๐ )โ ๐พ๐=1 (8)
Step 6 (Formation of ๐ ๐ behaviors) : Also using roulette wheel approach every candidate selects a behavior from within pool ๐ and forms ๐ ๐ new behaviors by sampling in close neighbourhood of the qualities associated with the selected behaviour. The quality matrix ๐ ๐ ๐ associated with every candidate ๐ฬ ๐ , ๐(๐ = 1, โฆ , ๐พ) is represented as follows: ๐ ๐ ๐ = [๐ ๐ฬ ,๐ ๐ โฆ ๐ ๐ฬ ๐ ,๐ ๐ โฆ ๐ ๐ฬ ๐พ ,๐ ๐ ] (9) where ๐ ๐ฬ ๐ ,๐ ๐ = [ ๐ฟ ๐๐ฬ ๐ ,1 โฎ๐ฟ ๐๐ฬ ๐ ,๐ก ๐ โฎ๐ฟ ๐๐ฬ ๐ ,๐ ๐ ] = [ ๐ฅ ,1 โฆ ๐ฅ ๐,๐๐ฬ ๐ ,1 โฆ ๐ฅ ๐,๐๐ฬ ๐พ ,1 โฎ โฑ โฎ โฎ๐ฅ ,๐ก ๐ โฆ ๐ฅ ๐,๐๐ฬ ๐ ,๐ก ๐ โฆ ๐ฅ ๐,๐๐ฬ ๐พ ,๐ก ๐ โฎ โฎ โฑ โฎ๐ฅ ,๐ ๐ โฆ ๐ฅ ๐,๐๐ฬ ๐ ,๐ ๐ โฆ ๐ฅ ๐,๐๐ฬ ๐พ ,๐ ๐ ] The behavior matrix ๐ญ ๐ ๐ associated with ๐ ๐ ๐ could be represented as follows: ๐ญ ๐ ๐ = [๐(๐ ๐ฬ ,๐ ๐ )โฆ๐(๐ ๐ฬ ๐ ,๐ ๐ )โฆ๐(๐ ๐ฬ ๐พ ,๐ ๐ )] (10) where ๐(๐ ๐ฬ ๐ ,๐ ๐ ) = [ ๐(๐ฟ ๐๐ฬ ๐ ,1 )โฎ๐(๐ฟ ๐๐ฬ ๐ ,๐ก ๐ )โฎ๐(๐ฟ ๐๐ฬ ๐ ,๐ ๐ )] Step 7 (Selection) : Every candidate ๐ (๐ = 1, โฆ , ๐ถ ๐ โ 1) associated with every cohort ๐(๐ = 1, โฆ , ๐พ) selects the best behavior, i.e. minimum objective function value from within its behavior choices in ๐ญ ๐ and ๐ญ ๐ ๐ as follows: ๐ ๐,๐๐๐๐ = ๐๐๐(๐(๐ ๐๐,๐ก ), ๐(๐ ๐ฬ ๐ ,๐ก ๐ )) , (๐ = 1, โฆ , ๐ถ ๐ โ 1) , ๐(๐ = 1, โฆ , ๐พ) (11) Step 8 (Concatenation)
The pool ๐ behaviors ๐ญ ๐ (Equation (3)) are carry forwarded to the subsequent learning attempt. The modified pool of behaviors ๐ญ is given below. ๐ญ = [ ๐ , โฆ ,๐ ๐,๐๐๐1 , โฆ ,๐
๐พ,๐๐๐1 โฎ โฑ. โฎ . โฎ๐ , โฆ ,๐ ๐,๐๐๐๐ , โฆ ,๐
๐พ,๐๐๐๐ โฎ . โฎ โฑ. โฎ๐ , โฆ ,๐ ๐,๐๐๐๐ถ ๐ , โฆ ,๐ ๐พ,๐๐๐๐ถ ๐พ ] = [๐ , โฆ ,๐ ๐ , โฆ ,๐ ๐พ ] (12) Step 8 (Convergence) : The algorithm is assumed to have converged if all of the conditions listed in Equation (13) are satisfied for successive considerable number of learning attempts and accept any of the current behaviors as final solution ๐ โ from within ๐พ cohorts. โ๐๐๐ฅ(๐ญ ๐ ) โ ๐๐๐ฅ(๐ญ ๐โ1 )โ โค ๐โ๐๐๐(๐ญ ๐ ) โ ๐๐๐(๐ญ ๐โ1 )โ โค ๐โ๐๐๐ฅ(๐ญ ๐ ) โ ๐๐๐(๐ญ ๐ )โ โค ๐ } (13) where ๐ is learning attempt counter. An illustrative example (Sphere function with 2 variables) of the above discussed Multi-CI procedure is provided in Appendix A of the manuscript. It includes every details of first learning attempt followed by evaluation of every step (1 to 8) is listed in Table A.1 till convergence.
Figure 1 Multi-CI Algorithm Flowchart Results and Discussions
The Multi-CI algorithm was coded in in MATLAB R2013a onWindows Platform with a T6400@4 GHz Intel Core 2 Duoprocessor with 4 GB RAM.The algorithm was validated by solving two well studied sets of test problems. Set 1 included 50 well studied benchmark problems (Karaboga and Akay, 2009; Karaboga and Basturk, 2007). Set 2 included 25 test problems from CEC 2005 (Suganthan et al. 2005). Set 1 test problems are listed in Table 1 and Set 2 test problems are listed in Table 2. Every problem in these test cases was solved 30 times using Multi-CI. In every run, initial behaviour of every candidate was randomly initialized. The Multi-CI parameters chosen were as follows: Number of cohorts
๐พ = 3 , Number of candidates ๐ถ ๐ = 5 , Reduction factor value ๐ = 0.98 , Reduction factor value ๐ = 0.98 , Quality variation parameters ๐ = 5 and ๐ ๐ = 10 . START
Initialize ๐พ cohorts with number of candidates ๐ถ ๐ in each cohort ๐ = 1, โฆ , ๐พ . Also select quality variations ๐ , ๐ ๐ and set up interval reduction factor ๐ For every cohort the probability associated with every behavior is also calculated The best behaviors in each cohort are noted and kept in a pool ๐ Using roulette wheel approach every candidate selects a behavior from within its corresponding cohort. And forms ๐ new behaviors by sampling in close neighborhood of the qualities associated with the selected behavior Also using roulette wheel approach every candidate selects a behavior from within the pool ๐ . And forms ๐ ๐ new behaviors by sampling in close neighborhood of the qualities associated with the selected behavior. Every candidate then follows/selects the best behavior from within its
๐ + ๐ ๐ง behaviors N Y Accept the best behavior as final solution
STOP
All Cohorts saturated/converged?
The Multi-CI algorithm presented here and the other algorithms with which the results are being compared are stochastic in nature due to which in every independent run of the algorithm the converged solution may be different than one another. A pairwise comparison of Multi-CI and every other algorithm was carried out, i.e. the converged (global minimum) values of 30 independent runs solving every problem using Multi-CI are compared with every other algorithm solving 30 independent runs of these problems. The Wilcoxon Signed-Rank test was used for such pairwise comparison. Similar to (Civicioglu, 2013), t he significance value ฮฑ was chosen to be with null hypothesis H0 is: There is no difference between the median of the solutions obtained by algorithm A and the median of the solutions obtained by algorithm B for the same set of test problems, i.e. median (A) = median(B). Also, to determine whether algorithm A yielded statistically better solution than algorithm B or whether alternative hypothesis was valid, the sizes of the ranks provided by the Wilcoxon Signed-Rank test (T+ and T-) were thoroughly examined. The mean solution, best solution and standard deviation (Std. Dev.), mean run time (in seconds) over the 30 runs of the algorithms solving Test 1 and Test 2 problems are represented in Table 3 and Table 4, respectively. The algorithms with statistically better solutions for Test 1 and Test 2 problems found using Wilcoxon Signed-Rank test are presented in Table 5 and Table 6, respectively.
In these tables, โ+โ indicated that the null hypothesis H0 was rejected and Multi-CI performed better and โ - โ indicated that the null hypothesis H0 was rejected; however, Multi-CI performed worse. Th e โ=โ indicated that there is no statistical difference between the two algorithms solving the problems and none of the two algorithms being compared could be considered more successful (winner) solving that problem. The counts of statistical significant cases (+/-/=) are presented in the last row of Table 5 and 6.The multi problem based pairwise using the averages of the global solutions obtained over the 30 runs of the algorithms solving the Test 1 and Test 2 problems are presented in Table 7. The results highlighted that the Multi-CI algorithm performed significantly better than every other algorithm. The convergence plots of few unimodal and multimodal representative functions such as Ackley function, Beale function, Fletcher function, Foxhole functions, Michalewics function, Six-hump camelback function are presented in Figure 2-7. The best solutions in every learning attempt are also plotted in Figure 2(b), 3(b), 4(b), -supervised intra as well as inter cohort learning behaviour. The convergence also highlighted the significance of Multi-CI approach quickly reaching the optimum solution. 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 = Non-separable) Problem
Name
Type
Low Up Dimension F1 Foxholes MS -65.536 F2 Goldstein-Price MN -2 F3 Penalized MN -50 F4 Penalized2 MN -50 F5 Ackley MN -32 F6 Beale UN -4.5 F7 Bohachecsky1 MS -100 F8 Bohachecsky2 MN -100 F9 Bohachecsky3 MN -100 F10
Booth MS -10 F11
Branin MS -5 F12
Colville UN -10 F13
Dixon-Price UN -10 F14
Easom UN -100 F15
Fletcher MN -3.1416 F16
Fletcher MN -3.1416 F17
Fletcher MN -3.1416 F18
Griewank MN -600 F19
Hartman3 MN F20
Hartman6 MN F21
Kowalik MN -5 F22
Langermann2 MN F23
Langermann5 MN F24
Langermann10 MN F25
Matyas UN -10 F26
Michalewics2 MS F27
Michalewics5 MS F28
Michalewics10 MS F29
Perm MN -4 F30
Powell UN -4 F31
Powersum MN F32
Quartic US -1.28 F33
Rastrigin MS -5.12 F34
Rosenbrock UN -30 F35
Schaffer MN -100 F36
Schwefel MS -500 F37
Schwefel_1_2 UN -100 F38
Schwefel_2_22 UN -10 F39
Shekel10 MN F40
Shekel5 MN F41
Shekel7 MN F42
Shubert MN -10 F43
Six-hump camelback MN -5 F44
Sphere2 US -100 F45
Step2 US -100 F46
Stepint US -5.12 F47
Sumsquares US -10 F48
Trid6 UN -36 F49
Trid10 UN -100 F50
Zakharov UN -5 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 F52
Shifted Schwefel U -100 F53
Shifted rotated high conditioned elliptic function U -100 F54
Shifted Schwefels problem 1.2 with noise U -100 F55
Schwefels problem 2.6 U -100 F56
Shifted Rosenbrockโs M -100 F57
Shifted rotated Griewankโs M F58
Shifted rotated Ackleyโs M -32 F59
Shifted Rastriginโs M -5 F60
Shifted rotated Rastriginโs M -5 F61
Shifted rotated Weierstrass M -0.5 F62
Schwefels problem 2.13 M -100 F63
Expanded extended Griewankโs + Rosenbrockโs E -3 F64
Expanded rotated extended Scaffes E -100 F65
Hybrid composition function HC -5 F66
Rotated hybrid comp. Fn 1 HC -5 F67
Rotated hybrid comp. Fn 1 with noise HC -5 F68
Rotated hybrid comp. Fn 2 HC -5 F69
Rotated hybrid comp. Fn 2 with narrow global optimal HC -5 F70
Rotated hybrid comp. Fn 2 with the global optimum HC -5 F71
Rotated hybrid comp. Fn 3 HC -5 F72
Rotated hybrid comp. Fn 3 with high condition number matrix HC -5 F73
Non-continuous rotated hybrid comp. Fn 3 HC -5 F74
Rotated hybrid comp. Fn 4 HC -5 F75
Rotated hybrid comp. Fn 4 HC -2 Table 3: Statistical solutions to Test 1 Problems using PSO, CMAES, ABC, CLPSO, SADE, BSA, IA and Multi-CI (Mean = Mean solution; Std. Dev. = Standard-deviation of mean solution; Best = Best solution; Runtime = Mean runtime in seconds)
Problem Statistics PSO2011 CMAES ABC JDE CLPSO SADE BSA IA Multi CI F1 Mean 1.3316029264876300 10.0748846367972000 0.9980038377944500 1.0641405484285200 1.8209961275956800 0.9980038377944500 0.9980038377944500 0.9980038690000000 0.9980038377944500 Std. Dev. 0.9455237994690700 8.0277365400340800 0.0000000000000001 0.3622456829347420 1.6979175079427900 0.0000000000000000 0.0000000000000000 0.0000000000000035 0.0000000000000003 Best 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038377944500 0.9980038685998520 0.9980038377944500 Runtime 72.527 44.788 64.976 51.101 61.650 66.633 38.125 43.535 1.092 F2 Mean 2.9999999999999200 21.8999999999995000 3.0000000465423000 2.9999999999999200 3.0000000000000700 2.9999999999999200 2.9999999999999200 3.0240147900000000 2.9999999999999200 Std. Dev. 0.0000000000000013 32.6088098948516000 0.0000002350442161 0.0000000000000013 0.0000000000007941 0.0000000000000020 0.0000000000000011 0.0787814840000000 0.0000000000000005 Best 2.9999999999999200 2.9999999999999200 2.9999999999999200 2.9999999999999200 2.9999999999999200 2.9999999999999200 2.9999999999999200 3.0029461118668700 2.9999999999999200 Runtime 17.892 24.361 16.624 7.224 24.784 28.699 7.692 41.343 0.763 F3 Mean 0.1278728062391630 0.0241892995662904 0.0000000000000004 0.0034556340083499 0.0000000000000000 0.0034556340083499 0.0000000000000000 0.3536752140000000 0.0000000000000000 Std. Dev. 0.2772792346028400 0.0802240262581864 0.0000000000000001 0.0189272869685522 0.0000000000000000 0.0189272869685522 0.0000000000000000 1.4205454130000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0014898619035614 0.0000000000000000 Runtime 139.555 5.851 84.416 9.492 38.484 15.992 18.922 34.494 90.997 F4 Mean 0.0043949463343535 0.0003662455278628 0.0000000000000004 0.0007324910557256 0.0000000000000000 0.0440448539086004 0.0000000000000000 0.0179485820000000 0.0000000000000000 Std. Dev. 0.0054747064090174 0.0020060093719584 0.0000000000000001 0.0027875840585535 0.0000000000000000 0.2227372747439610 0.0000000000000000 0.0526650620000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000165491 0.0000000000000000 Runtime 126.507 6.158 113.937 14.367 48.667 33.019 24.309 322.808 27.589 F5 Mean 1.5214322973725000 11.7040011684582000 0.0000000000000340 0.0811017056422860 0.1863456353861950 0.7915368220335460 0.0000000000000105 0.0000000000000009 0.0000000000000000 Std. Dev. 0.6617570384662600 9.7201961540865200 0.0000000000000035 0.3176012689149320 0.4389839299322230 0.7561593402959740 0.0000000000000034 0.0000000000000000 0.0000000000000000 Best 0.0000000000000080 0.0000000000000080 0.0000000000000293 0.0000000000000044 0.0000000000000080 0.0000000000000044 0.0000000000000080 0.0000000000000009 0.0000000000000000 Runtime 63.039 3.144 23.293 11.016 45.734 40.914 14.396 49.458 5.243 F6 Mean 0.0000000041922968 0.2540232169641050 0.0000000000000028 0.0000000000000000 0.0000444354499943 0.0000000000000000 0.0000000000000000 0.0082236060000000 0.0000000000000000 Std. Dev. 0.0000000139615552 0.3653844307786430 0.0000000000000030 0.0000000000000000 0.0001015919507724 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0082236059357692 0.0000000000000000 Runtime 32.409 4.455 22.367 1.279 125.839 4.544 0.962 50.246 1.356 F7 Mean 0.0000000000000000 0.0622354533647150 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.1345061339146580 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 16.956 6.845 1.832 1.141 2.926 4.409 0.825 38.506 1.434 F8 Mean 0.0000000000000000 0.0072771062590204 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0398583525142753 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 17.039 2.174 1.804 1.139 2.891 4.417 0.824 39.023 1.542 F9 Mean 0.0000000000000000 0.0001048363065820 0.0000000000000006 0.0000000000000000 0.0000193464326398 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0005742120996051 0.0000000000000003 0.0000000000000000 0.0000846531630676 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 17.136 2.127 21.713 1.129 33.307 4.303 0.829 40.896 1.433 F10 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0006005122443674 0.0000000000000000 0.0000000000000000 0.8346587090000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0029861918862801 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.8346587086917530 0.0000000000000000 Runtime 17.072 1.375 22.395 1.099 28.508 4.371 0.790 39.978 1.260 F11 Mean 0.3978873577297380 0.6372170283279430 0.3978873577297380 0.3978873577297380 0.3978873577297390 0.3978873577297380 0.3978873577297380 0.4156431270000000 0.3978873577297380 Std. Dev. 0.0000000000000000 0.7302632173480510 0.0000000000000000 0.0000000000000000 0.0000000000000049 0.0000000000000000 0.0000000000000000 0.0406451050000000 0.0000000000000001 Best 0.3978873577297380 0.3978873577297380 0.3978873577297380 0.3978873577297380 0.3978873577297380 0.3978873577297380 0.3978873577297380 0.4012748152492080 0.3978873577297380 Runtime 17.049 24.643 10.941 6.814 17.283 27.981 5.450 40.099 0.603 F12 Mean 0.0000000000000000 0.0000000000000000 0.0715675060725970 0.0000000000000000 0.1593872502094070 0.0000000000000000 0.0000000000000000 0.0014898620000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0579425013417103 0.0000000000000000 0.6678482786713720 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0013425253994745 0.0000000000000000 0.0000094069599934 0.0000000000000000 0.0000000000000000 0.0082029783984983 0.0000000000000000 Runtime 44.065 1.548 21.487 1.251 166.965 4.405 2.460 48.067 41.69 F13 Mean 0.6666666666666750 0.6666666666666670 0.0000000000000038 0.6666666666666670 0.0023282133668190 0.6666666666666670 0.6444444444444440 0.2528116640000000 0.6728903646849310 Std. Dev. 0.0000000000000022 0.0000000000000000 0.0000000000000012 0.0000000000000002 0.0051792840882291 0.0000000000000000 0.1217161238900370 0.0000000006509080 0.2130263402454600 Best 0.6666666666666720 0.6666666666666670 0.0000000000000021 0.6666666666666670 0.0000120708732167 0.6666666666666670 0.0000000000000000 0.2528116633611470 0.0000000000000020 Runtime 167.094 3.719 37.604 18.689 216.261 47.833 21.192 67.463 11.104 F14 Mean -1.0000000000000000 -0.1000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -0.9997989620000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.3051285766293650 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000167151 0.0000000000000000 Best -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -1.0000000000000000 -0.9997989624626810 0.0000000000000000 Runtime 16.633 3.606 13.629 6.918 16.910 28.739 5.451 39.685 0.10 F15 Mean 0.0000000000000000 1028.3930784026900000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 1298.1521820113500000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 27.859 15.541 40.030 2.852 4.030 6.020 2.067 38.867 1.860 F16 Mean 48.7465164446927000 1680.3460230073400000 0.0218688498331872 0.9443728655432830 81.7751618148164000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 88.8658510972991000 2447.7484859066000000 0.0418409568792831 2.8815514827061600 379.9241117377270000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000016 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 95.352 11.947 44.572 4.719 162.941 5.763 7.781 48.262 0.459 F17 Mean 918.9518492782850000 12340.2283326398000000 11.0681496253548000 713.7226974626920000 0.8530843976878610 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 1652.4810858411400000 22367.1698875802000000 9.8810950146557100 1710.071307430120000 2.9208253191698800 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.3274654777056860 0.0000000000000000 0.0016957837829822 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 271.222 7.631 43.329 16.105 268.894 168.310 33.044 69.060 1.860 F18 Mean 0.0068943694819713 0.0011498935321349 0.0000000000000000 0.0048193578543185 0.0000000000000000 0.0226359326967139 0.0004930693556077 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0080565201649587 0.0036449413521107 0.0000000000000001 0.0133238235582874 0.0000000000000000 0.0283874287215679 0.0018764355751644 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 73.895 2.647 19.073 6.914 14.864 25.858 5.753 2.717 4.261 F19 Mean -3.8627821478207500 -3.7243887744664700 -3.8627821478207500 -3.8627821478207500 -3.8627821478207500 -3.8627821478207500 -3.8627821478207500 -3.8596352620000000 -3.8627819786235600 Std. Dev. 0.0000000000000027 0.5407823545193820 0.0000000000000024 0.0000000000000027 0.0000000000000027 0.0000000000000027 0.0000000000000027 0.0033967610000000 0.0000001322237558 Best -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8627821478207600 -3.8613076574052300 -3.8627821093820800 Runtime 19.280 21.881 12.613 7.509 17.504 24.804 6.009 46.167 1.285 F20 Mean -3.3180320675402500 -3.2942534432762600 -3.3219951715842400 -3.2982165473202600 -3.3219951715842400 -3.3140689634962500 -3.3219951715842400 -2.5710247593206100 -3.3223582775589500 Std. Dev. 0.0217068148263721 0.0511458075926848 0.0000000000000014 0.0483702518391572 0.0000000000000013 0.0301641516823498 0.0000000000000013 0.0000000000000009 0.0099173853696568 Best -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 -3.3219951715842400 -2.5710247593206100 -3.3223651489966400 Runtime 26.209 7.333 13.562 8.008 20.099 33.719 6.822 59.083 2.021 F21 Mean 0.0003074859878056 0.0064830287538208 0.0004414866359626 0.0003685318137604 0.0003100479704151 0.0003074859878056 0.0003074859878056 0.0016993410000000 0.0003516458357319 Std. Dev. 0.0000000000000000 0.0148565973286009 0.0000568392289725 0.0002323173367683 0.0000059843325073 0.0000000000000000 0.0000000000000000 0.0000013058400000 0.0000539770693216 Best 0.0003074859878056 0.0003074859878056 0.0003230956007045 0.0003074859878056 0.0003074859941292 0.0003074859878056 0.0003074859878056 0.0016989914552560 0.0003243793470953 Runtime 84.471 13.864 20.255 7.806 156.095 45.443 11.722 48.920 1.800 F22 Mean -1.0809384421344400 -0.7323679641701760 -1.0809384421344400 -1.0764280762657400 -1.0202940450426400 -1.0809384421344400 -1.0809384421344400 -1.4315374190000000 -1.0820489785202800 Std. Dev. 0.0000000000000006 0.4136688304155380 0.0000000000000008 0.0247042912888477 0.1190811583120530 0.0000000000000005 0.0000000000000005 0.0000000000000009 0.0000000000000000 Best -1.0809384421344400 -1.0809384421344400 -1.0809384421344400 -1.0809384421344400 -1.0809384421344400 -1.0809384421344400 -1.0809384421344400 -1.4315374193830000 -1.0820489785202800 Runtime 27.372 32.311 27.546 19.673 52.853 36.659 21.421 34.714 1.299 F23 Mean -1.3891992200744600 -0.5235864386288060 -1.4999990070800800 -1.3431399432579700 -1.4765972735526500 -1.4999992233525000 -1.4821658762555300 -1.5000000000000000 -1.4999999979385900 Std. Dev. 0.2257194403158630 0.2585330714077300 0.0000008440502079 0.2680292304904580 0.1281777579497830 0.0000000000000009 0.0976772648082733 0.0000000000000000 0.0000000166613427 Best -1.4999992233524900 -0.7977041047646610 -1.4999992233524900 -1.4999992233524900 -1.4999992233524900 -1.4999992233524900 -1.4999992233524900 -1.5000000000000000 -1.4999999997292700 Runtime 33.809 17.940 37.986 20.333 42.488 36.037 18.930 41.848 0.510 F24 Mean -0.9166206788680230 -0.3105071678265780 -0.8406348096500680 -0.8827152798835760 -0.9431432797743700 -1.2765515661973800 -1.3127183561646500 -1.5000000000000000 -1.4999999478041200 Std. Dev. 0.3917752367440500 0.2080317241440800 0.2000966365984320 0.3882445165494030 0.3184175870987750 0.3599594108130040 0.3158807699946290 0.0000000000000000 0.0000003682666585 Best -1.5000000000003800 -0.7976938356122860 -1.4999926800631400 -1.5000000000003800 -1.5000000000003800 -1.5000000000003800 -1.5000000000003800 -1.5000000000000000 -1.4999999976445200 Runtime 110.798 8.835 38.470 21.599 124.609 47.171 35.358 54.651 0.842 F25 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000004 0.0000000000000000 0.0000041787372626 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 0.0000161643637543 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 25.358 1.340 19.689 1.142 31.632 4.090 0.813 35.662 2.890 F26 Mean -1.8210436836776800 -1.7829268228561700 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8203821100000000 -1.8210436836776800 Std. Dev. 0.0000000000000009 0.1450583631808370 0.0000000000000009 0.0000000000000009 0.0000000000000009 0.0000000000000009 0.0000000000000009 0.0000000000000014 0.0000000000000005 Best -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8210436836776800 -1.8203821095139300 -1.8210436836776800 Runtime 19.154 26.249 17.228 9.663 18.091 28.453 7.472 34.891 0.346 F27 Mean -4.6565646397053900 -4.1008953007033700 -4.6934684519571100 -4.6893456932617100 -4.6920941990586400 -4.6884965299983800 -4.6934684519571100 -3.2820108350000000 -4.5982757883767500 Std. Dev. 0.0557021530063238 0.4951250481844850 0.0000000000000009 0.0125797149251589 0.0075270931220834 0.0272323381095561 0.0000000000000008 0.0000000000000023 0.1140777982812540 Best -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -4.6934684519571100 -3.2820108345268900 -4.6934684286288800 Runtime 38.651 10.956 17.663 14.915 25.843 38.446 11.971 45.085 0.530 F28 Mean -8.9717330307549300 -7.6193507368464700 -9.6601517156413500 -9.6397230986132500 -9.6400278592589600 -9.6572038232921700 -9.6601517156413500 -6.2086254390000000 -8.4871985036037100 Std. Dev. 0.4927013165009220 0.7904830398850970 0.0000000000000008 0.0393668145094111 0.0437935551332868 0.0105890022905617 0.0000000000000007 0.0000000000000027 0.2867921564163950 Best -9.5777818097208200 -9.1383975057875100 -9.6601517156413500 -9.6601517156413500 -9.6601517156413500 -9.6601517156413500 -9.6601517156413500 -6.2086254392105500 -8.9978275376597000 Runtime 144.093 6.959 27.051 20.803 32.801 46.395 22.250 71.652 4.784 F29 Mean 0.0119687224560441 0.0788734736114700 0.0838440014038032 0.0154105130055856 0.0198686590210374 0.0140272066690658 0.0007283694780796 1.3116221610000000 0.0049933819581781 Std. Dev. 0.0385628598040034 0.1426911799629180 0.0778327303965192 0.0308963906374663 0.0613698943155661 0.0328868042987376 0.0014793717464195 0.5590904820000000 0.0023147314691019 Best 0.0000044608370213 0.0000000000000000 0.0129834451730589 0.0000000000000000 0.0000175219764526 0.0000000000000000 0.0000000000000000 1.0960146962658900 0.0007717562336873 Runtime 359.039 17.056 60.216 35.044 316.817 92.412 191.881 34.697 0.875 F30 Mean 0.0000130718912008 0.0000000000000000 0.0002604330013462 0.0000000000000001 0.0458769685199585 0.0000002733806735 0.0000000028443186 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000014288348929 0.0000000000000000 0.0000394921919294 0.0000000000000002 0.0620254411839524 0.0000001788830279 0.0000000033308990 0.0000000000000000 0.0000000000000000 Best 0.0000095067504097 0.0000000000000000 0.0001682411286088 0.0000000000000000 0.0005277712020642 0.0000000944121661 0.0000000004769768 0.0000000000000000 0.0000000000000000 Runtime 567.704 14.535 215.722 194.117 252.779 360.380 144.784 153.221 4.297 F31 Mean 0.0001254882834238 0.0000000000000000 0.0077905311094958 0.0020185116261490 0.0002674563703837 0.0000000000000000 0.0000000111676630 0.0071082040000000 0.0003936439985429 Std. Dev. 0.0001503556280087 0.0000000000000000 0.0062425841086448 0.0077448684015362 0.0003044909265796 0.0000000000000000 0.0000000184322163 0.0000000000000000 0.0002001204487121 Best 0.0000000156460198 0.0000000000000000 0.0003958766023752 0.0000000000000000 0.0000023064754605 0.0000000000000000 0.0000000000000000 0.0071082039505830 0.0000029388885444 Runtime 250.248 12.062 34.665 48.692 227.817 220.886 149.882 43.098 8.902 F32 Mean 0.0003548345513179 0.0701619169853449 0.0250163252527030 0.0013010316180679 0.0019635752485802 0.0016730768406953 0.0019955316015528 0.0002254250000000 0.0000236661877907 Std. Dev. 0.0001410817500914 0.0288760292572957 0.0077209314806873 0.0009952078711752 0.0043423828633839 0.0007330246909835 0.0009698942217908 0.0005270410000000 0.0000236731760330 Best 0.0001014332605364 0.0299180701536354 0.0094647580732654 0.0001787238105452 0.0004206447422138 0.0005630852254632 0.0006084880639553 0.0000023800831017 0.0000050196149891 Runtime 290.669 2.154 34.982 82.124 103.283 171.637 48.237 218.722 2.860 F33 Mean 25.6367602258676000 95.9799861204982000 0.0000000000000000 1.1276202647057400 0.6301407361590880 0.8622978494808570 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 8.2943512684216700 56.6919245985100000 0.0000000000000000 1.0688393637536800 0.8046401822326410 0.9323785263847000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 12.9344677422129000 29.8487565993415000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 76.083 2.740 4.090 7.635 18.429 23.594 5.401 2.266 3.516 F34 Mean 2.6757043114269700 0.3986623855035210 0.2856833465904130 1.0630996944802500 5.7631786582751800 1.2137377447007000 0.3986623854300930 0.0000154715000000 28.8334517794009000 Std. Dev. 12.3490058210004000 1.2164328621946200 0.6247370987465170 1.7930895051734300 13.9484817304201000 1.8518519388285700 1.2164328622195200 0.0000022373400000 0.0144695690509943 Best 0.0042535368984501 0.0000000000000000 0.0004266049929880 0.0000000000000000 0.0268003205820685 0.0001448955835246 0.0000000000000000 0.0000118803557196 28.8053841187578000 Runtime 559.966 9.462 35.865 23.278 187.894 268.449 34.681 7.250 8.431 F35 Mean 0.0000000000000000 0.4651202457398910 0.0000000000000000 0.0038863639514140 0.0019431819755029 0.0006477273251676 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0933685176073728 0.0000000000000000 0.0048411743884718 0.0039528023354469 0.0024650053428137 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0097159098775144 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 18.163 24.021 7.861 4.216 8.304 5.902 1.779 33.155 2.943 F36 Mean -7684.6104757783800000 -6835.1836730901400000 -12569.4866181730000 -12304.9743375341000 -12210.8815698372000 -12549.746895737300000 -12569.486618173000000 -12569.3622100000000000 -12569.4866181730000000 Std. Dev. 745.3954005014180000 750.7338055436110000 0.0000000000022659 221.4322514436480000 205.9313376284770000 44.8939348779747000 0.0000000000024122 0.0000000273871000 0.0000000000018828 Best -8912.8855854978200000 -8340.0386911070600000 -12569.4866181730000 -12569.4866181730000 -12569.4866181730000 -12569.486618173000000 -12569.486618173000000 -12569.3622054081000000 -12569.4866181730000000 Runtime 307.427 3.174 19.225 10.315 31.499 34.383 11.069 2.306 10.825 F37 Mean 0.0000000000000000 0.0000000000000000 14.5668734126948000 0.0000000000000000 6.4655746330439100 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 8.7128443012950300 0.0000000000000000 8.2188901353055800 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 4.0427699323673400 0.0000000000000000 0.1816624029553790 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 543.180 3.370 111.841 19.307 179.083 109.551 57.294 100.947 5.112 F38 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 163.188 2.558 20.588 1.494 12.563 5.627 3.208 47.009 6.738 F39 Mean -10.1061873621653000 -5.2607563471326400 -10.5364098166920000 -10.3130437162426000 -10.3130437162026000 -10.5364098166921000 -10.5364098166921000 -10.5063235800000000 -10.5364098166920000 Std. Dev. 1.6679113661236400 3.6145751818694000 0.0000000000000023 1.2234265179812200 1.2234265179736500 0.0000000000000016 0.0000000000000018 0.0000000025211900 0.0000000000000055 Best -10.5364098166921000 -10.5364098166921000 -10.5364098166920000 -10.5364098166921000 -10.5364098166920000 -10.5364098166921000 -10.5364098166920000 -10.5063235792920000 -10.5364098166920000 Runtime 31.018 11.024 16.015 8.345 37.275 28.031 7.045 55.666 0.892 F40 Mean -9.5373938082045500 -5.7308569926624600 -10.1531996790582000 -9.5656135761215700 -10.1531996790582000 -9.9847854277673500 -10.1531996790582000 -10.1529842600000000 -10.1531996790582000 Std. Dev. 1.9062127067994200 3.5141202468383400 0.0000000000000055 1.8315977756329900 0.0000000000000076 0.9224428443735560 0.0000000000000072 0.0000000000542921 0.0000000000000000 Best -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 -10.1531996790582000 -10.1529842649756000 -10.1531996790582000 Runtime 25.237 11.177 11.958 7.947 30.885 25.569 6.864 51.507 0.860 F41 Mean -10.4029405668187000 -6.8674070870953700 -10.4029405668187000 -9.1615813354737300 -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.3988303400000000 -10.4029405668187000 Std. Dev. 0.0000000000000018 3.6437803702691000 0.0000000000000006 2.8277336448396200 0.0000000000000010 0.0000000000000018 0.0000000000000017 0.0000000001978980 0.0000000000000000 Best -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.4029405668187000 -10.3988303385534000 -10.4029405668187000 Runtime 21.237 11.482 14.911 8.547 31.207 27.064 8.208 53.190 0.395 F42 Mean -186.7309073569880000 -81.5609772893002000 -186.730908831024000 -186.730908831024000 -186.730908831024000 -186.7309088310240000 -186.7309088310240000 -186.2926481000000000 -186.7309088310240000 Std. Dev. 0.0000046401472660 66.4508342743478000 0.0000000000000236 0.0000000000000388 0.0000000000000279 0.0000000000000377 0.0000000000000224 0.0000000000000578 0.0000000000000291 Best -186.7309088310240000 -186.7309088310240000 -186.730908831024000 -186.730908831024000 -186.730908831024000 -186.7309088310240000 -186.7309088310240000 -186.2926480689880000 -186.7309088310240000 Runtime 19.770 25.225 13.342 8.213 20.344 27.109 9.002 31.766 2.466 F43 Mean -1.0316284534898800 -1.0044229658530100 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0304357800000000 -1.0316284534898800 Std. Dev. 0.0000000000000005 0.1490105926664260 0.0000000000000005 0.0000000000000005 0.0000000000000005 0.0000000000000005 0.0000000000000005 0.0014911900000000 0.0000000000000000 Best -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0316284534898800 -1.0314500753985900 -1.0316284534898800 Runtime 16.754 24.798 11.309 7.147 18.564 27.650 5.691 39.897 0.391 F44 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000004 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000000000000001 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 159.904 2.321 21.924 1.424 14.389 5.920 3.302 174.577 4.791 F45 Mean 2.3000000000000000 0.0666666666666667 0.0000000000000000 0.9000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000538870000000 0.0000000000000000 Std. Dev. 1.8597367258983700 0.2537081317024630 0.0000000000000000 3.0211895350832500 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000005399890 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000538860819891 0.0000000000000000 Runtime 57.276 1.477 1.782 2.919 3.042 4.307 0.883 2.215 14.850 F46 Mean 0.1333333333333330 0.2666666666666670 0.0000000000000000 0.0000000000000000 0.2000000000000000 0.0000000000000000 0.0000000000000000 -0.0153463301609662 0.1276607794072780 Std. Dev. 0.3457459036417600 0.9444331755018490 0.0000000000000000 0.0000000000000000 0.4068381021724860 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0737285129670879 Best 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 -0.0153463301609662 0.0148704298643416 Runtime 20.381 2.442 1.700 1.074 6.142 4.319 0.764 31.068 2.890 F47 Mean 0.0000000000000000 0.0000000000000000 0.0000000000000005 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000003 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 564.178 2.565 24.172 1.870 15.948 6.383 4.309 31.296 4.685 F48 Mean -50.0000000000002000 -50.0000000000002000 -49.9999999999997000 -50.0000000000002000 -49.4789234062579000 -50.0000000000002000 -50.0000000000002000 -44.7416748700000000 -50.0000000000000000 Std. Dev. 0.0000000000000361 0.0000000000000268 0.0000000000001408 0.0000000000000354 1.3150773145311700 0.0000000000000268 0.0000000000000361 0.0000000000000217 0.0000000000000000 Best -50.0000000000002000 -50.0000000000002000 -50.0000000000001000 -50.0000000000002000 -49.9999994167392000 -50.0000000000002000 -50.0000000000002000 -44.7416748706606000 -50.0000000000000000 Runtime 24.627 8.337 22.480 8.623 142.106 36.804 7.747 52.486 0.806 F49 Mean -210.0000000000010000 -210.0000000000030000 -209.999999999947000 -210.000000000003000 -199.592588547503000 -210.0000000000030000 -210.0000000000030000 -150.5540859185450000 -210.0000000000000000 Std. Dev. 0.0000000000009434 0.0000000000003702 0.0000000000138503 0.0000000000008251 9.6415263953591700 0.0000000000004625 0.0000000000003950 0.0000000000000000 0.0000000000000000 Best -210.0000000000030000 -210.0000000000030000 -209.999999999969000 -210.000000000004000 -209.985867409029000 -210.0000000000040000 -210.0000000000040000 -150.5540859185450000 -210.0000000000000000 Runtime 48.580 5.988 36.639 11.319 187.787 54.421 11.158 70.887 10.962 F50 Mean 0.0000000000000000 0.0000000000000000 0.0000000402380424 0.0000000000000000 0.0000000001597805 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000002203520334 0.0000000000000000 0.0000000006266641 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Best 0.0000000000000000 0.0000000000000000 0.0000000000000210 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 Runtime 86.369 1.868 86.449 1.412 157.838 4.930 5.702 33.573 2.15 Table 4: Statistical solutions to Test 2 Problems using PSO, CMAES, ABC, CLPSO, SADE, BSA, IA and Multi-CI (Mean = Mean solution; Std. Dev. = Standard-deviation of mean solution; Best = Best solution; Runtime = Mean runtime in seconds)
Problem Statistics PSO2011 CMAES ABC JDE CLPSO SADE BSA IA Multi CI F51 Mean -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -447.6018854297170000 -450.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 89.3142986500000000 0.0000000000000000 Best -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 Runtime 212.862 23.146 113.623 118.477 167.675 154.232 140.736 30.282 28.930 F52 Mean -450.0000000000000000 -450.0000000000000000 -449.9999999999220000 -450.0000000000000000 -418.8551838547760000 -450.0000000000000000 -450.0000000000000000 -449.9967727000000000 -450.0000000000000000 Std. Dev. 0.0000000000000350 0.0000000000000000 0.0000000002052730 0.0000000000000615 51.0880511039985000 0.0000000000000000 0.0000000000000259 0.0176705780000000 0.0000000000000000 Best -450.0000000000000000 -450.0000000000000000 -449.9999999999970000 -450.0000000000000000 -449.4789299923810000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 -450.0000000000000000 Runtime 230.003 23.385 648.784 139.144 1462.706 185.965 243.657 48.003 74.497 F53 Mean -44.5873911956554000 -450.0000000000000000 387131.24412139700000 -197.9999999999850000 62142.8213760465000000 245.0483283713550000 -449.9999567867430000 -449.7873452000000000 67397.8550894185000000 Std. Dev. 458.5794120016290000 0.0000000000000000 166951.73365926400000 391.5169437474990000 34796.1785167236000000 790.6056596723160000 0.0001175386756044 0.0000000000001734 102453.8929479460000000 Best -443.9511286079800000 -450.0000000000000000 165173.18530956000000 -449.9999999999990000 17306.9066792474000000 -421.4054944641620000 -450.0000000000000000 -450.0000000000000000 11750.9831357565000000 Runtime 2658.937 35.464 240.094 1017.557 1789.643 1808.954 1883.713 52.463 50.447 F54 Mean -450.0000000000000000 77982.4567046980000000 140.4509447125110000 -414.0000000000000000 -178.8320689185280000 -450.0000000000000000 -450.0000000000000000 -388.7807630000000000 -450.0000000000000000 Std. Dev. 0.0000000000000460 131376.7365456010000000 217.2646715063190000 55.9309919639279000 394.8667499339530000 0.0000000000000000 0.0000000000000259 1.1928333530000000 0.0000000000000000 Best -450.0000000000000000 -450.0000000000000000 -324.3395691109350000 -450.0000000000000000 -447.9901256558030000 -450.0000000000000000 -450.0000000000000000 -389.7573633109500000 -450.0000000000000000 Runtime 247.256 32.726 209.188 143.767 1248.616 185.438 347.167 46.072 83.596 F55 Mean -310.0000000000000000 -310.0000000000000000 -291.5327549384120000 -271.0000000000000000 333.4108259915760000 -309.9999999999960000 -309.9999999999980000 -310.8207993000000000 -310.0000000000000000 Std. Dev. 0.0000000000000000 0.0000000000000000 17.6942171217937000 60.5919079609218000 512.6920837704510000 0.0000000000133965 0.0000000000023443 0.0208030240000000 0.0000000000000000 Best -310.0000000000000000 -310.0000000000000000 -307.7611364354020000 -310.0000000000000000 -309.9740055344430000 -310.0000000000000000 -310.0000000000000000 -310.8367924750510000 -310.0000000000000000 Runtime 241.517 39.293 205.568 134.078 1481.686 210.684 386.633 44.84710031 120.725 F56 Mean 393.4959999056240000 390.5315438816460000 391.2531452421960000 231.3986579112350000 405.5233436479650000 390.2657719408230000 390.1328859704120000 390.8036739982730000 392.3754700583880000 Std. Dev. 16.0224965900462000 1.3783433976378300 3.7254660805238600 247.2968415284400000 10.7480096852869000 1.0114275384776600 0.7278464357038200 0.0000000000000000 0.6527183145462900 Best 390.0000000000150000 390.0000000000000000 390.0101471658490000 -140.0000000000000000 390.5776683413440000 390.0000000000000000 390.0000000000000000 390.8036739982730000 391.2787609196740000 Runtime 1178.079 27.894 159.762 153.715 1441.859 1214.303 290.236 45.632 88.645 F57 Mean 1091.0644335162500000 1087.2645466786700000 1087.0459486286000000 1141.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.2265890000000000 1087.2402022380600000 Std. Dev. 3.4976948942723200 0.5365230018001780 0.0000000000005585 83.8964879458918000 0.0000000000004264 0.0000000000004814 0.0000000000004428 0.0019192200000000 0.2217515717485750 Best 1087.0696772583000000 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.0459486286000000 1087.2262037455100000 1087.0546352983500000 Runtime 334.064 37.047 180.472 159.922 267.342 259.760 332.132 52.621 145.678 F58 Mean -119.8190232990920000 -119.9261073509850000 -119.7446063439080000 -119.4450938018030000 -119.9300269839980000 -119.7727713703720000 -119.8356122057440000 -119.6006412865410000 -119.9717811854090000 Std. Dev. 0.0720107560874199 0.1554021446157740 0.0623866434489108 0.0927418223065644 0.0417913553101429 0.1248514853682450 0.0704515460477787 0.0000000000000434 0.0536174052618470 Best -119.9302772694110000 -120.0000000000000000 -119.8779554779730000 -119.6575717927190000 -119.9756745390830000 -119.9999999999980000 -119.9802847896350000 -119.6006412865410000 -119.9905751250590000 Runtime 602.507 49.209 265.319 160.806 1586.286 648.489 717.375 52.56165118 117.872 F59 Mean -324.6046006320200000 -306.5782069681560000 -330.0000000000000000 -329.8673387923880000 -329.4361898676470000 -329.9668346980970000 -330.0000000000000000 -327.1635938000000000 -323.0352916375860000 Std. Dev. 2.5082306041521000 21.9475396048756000 0.0000000000000000 0.3440030182812760 0.6229063711904190 0.1816538397880230 0.0000000000000000 0.0000000000001156 2.3797197171188800 Best -329.0050409429070000 -327.0151228287200000 -330.0000000000000000 -330.0000000000000000 -330.0000000000000000 -330.0000000000000000 -330.0000000000000000 -327.163593801473 -328.0100818858130000 Runtime 982.449 22.237 111.629 128.494 162.873 155.645 176.994 45.867 70.902 F60 Mean -324.3311322538170000 -314.7871102989330000 -306.7949047862760000 -319.6763749798700000 -321.7278926895280000 -322.9689591871600000 -319.2544515903510000 -335.0171647000000000 -322.5378095907100000 Std. Dev. 3.0072222933667300 8.3115989308305500 5.1787864195870400 4.9173541245304800 1.8971778613701300 2.8254645254663600 3.3091959975390800 10.6369134000000000 2.4434060592713200 Best -327.1650513120000000 -327.0151228287200000 -318.9403196374510000 -326.0201637716270000 -326.1788303102740000 -328.0100818858130000 -325.0252097523530000 -347.2509173436740000 -326.0032742097850000 Runtime 1146.013 29.860 259.258 179.039 1594.096 210.534 420.851 54.661 96.444 F61 Mean 92.5640111212146000 90.7642785704506000 94.8428485804138000 93.2972315784963000 94.6109567642977000 91.6859083842723000 92.3519494286347000 92.0170440500000000 90.0000910864369000 Std. Dev. 1.5827416781636900 26.4613831425879000 0.6869412813090850 1.8766951726453600 0.6689129174038950 0.9033073777915270 1.0901581870340800 0.000000000000014453 0.3262402308775350 Best 90.1142082473923000 -45.0054133586912000 93.1500794016147000 91.0295373630387000 92.9690673344598000 90.1363685040678000 90.2628852415150000 92.0170440535006000 90.0000665813171000 Runtime 1310.457 44.217 308.501 282.150 1421.545 506.829 1771.860 60.350 182.115 F62 Mean 18611.314225480900000 -70.0486708747625000 -337.3273080760500000 400.3240208136310000 -447.8870804905020000 -394.5206365378250000 -437.1125728026770000 -410.1361631000000000 -450.8165121271770000 Std. Dev. 12508.786612631600000 637.4585182420270000 56.5730759032367000 688.3344299264300000 11.8934815947019000 128.6353424718180000 20.3541618366546000 34.8795385900000000 4.6244223495662800 Best 4568.3350537809200000 -460.0000000000000000 -449.1707421778360000 -434.8788220982740000 -459.6890294276810000 -460.0000000000000000 -459.1772521346520000 -421.5672584975600000 -459.9993498733460000 Runtime 2381.974 34.857 232.916 202.941 1636.440 1277.975 1466.985 48.480 143.358 F63 Mean -129.2373581503910000 -128.7850616923410000 -129.8343428775830000 -129.6294851450880000 -129.8382867796110000 -129.7129164862680000 -129.8981409848090000 -122.2126680000000000 -129.3840699141840000 Std. Dev. 0.5986210944493790 0.6157633658946230 0.0408016481905455 0.1054759371085400 0.0372256921835666 0.0875456568200232 0.0682328484314248 0.0000000000000434 0.0756137409573255 Best -129.6861385930680000 -129.5105509483130000 -129.9098920058450000 -129.8125711770830000 -129.9098505660780000 -129.8717592632560000 -129.9901230990300000 -122.2126679617240000 -129.5432045245170000 Runtime 2183.218 25.496 205.194 186.347 1526.365 660.986 1064.114 46.260 170.218 F64 Mean -298.2835926212850000 -295.1290938304830000 -296.9323391084610000 -296.8839733969750000 -297.5119726691150000 -297.8403738182600000 -297.5359077431460000 -295.4721554000000000 -297.3117952188210000 Std. Dev. 0.5587676271753680 0.1634039984609270 0.2251930667702880 0.4330673614598290 0.3440115280624180 0.4536801689800720 0.4085859316264990 0.1118191570000000 0.3262896571402280 Best -299.6022022972560000 -295.7382222729600000 -297.4659619544820000 -297.8411886637500000 -298.3030560759620000 -299.2417795907860000 -298.3869295150680000 -295.6307146941910000 -297.7411404965860000 Runtime 2517.138 32.084 262.533 334.888 1615.452 1289.814 1953.289 55.118 85.817 F65 Mean 417.4613663019860000 492.5045364088000000 120.0000000000000000 326.6601114362900000 131.3550392249760000 234.2689845349590000 120.0000000000000000 120.0000000000000000 211.7934679874670000 Std. Dev. 153.9215808771580000 181.5709657779580000 0.0000000000000188 174.6877238188330000 26.1407360548431000 150.7595974059750000 0.0000000000000000 0.0000000000000000 161.3764717430970000 Best 120.0000000000000000 262.7619554120320000 120.0000000000000000 120.0000000000000000 120.0000000000000000 120.0000000000000000 120.0000000000000000 120.0000000000000000 204.1733303990720000 Runtime 3156.336 239.823 2285.787 1834.967 3210.655 1932.016 2351.478 69.052 662.766 NFE F66 Mean 221.4232628350220000 455.1151684594550000 258.8582688922670000 231.1806131539990000 231.5547154800990000 222.0256674919140000 234.4843380488580000 276.3946208000000000 223.0150462881420000 Std. Dev. 12.2450207482898000 254.3583511786970000 11.8823213189685000 13.5473380962764000 11.5441451076421000 6.1841489800660300 8.9091119100451100 19.2196655800000000 5.5838205188784200 Best 181.5746616282570000 120.0000000000000000 235.6600739998890000 210.3582705649860000 214.7661703584830000 206.4520786020840000 219.6244910167680000 259.8700033222460000 215.3853171009670000 Runtime 4242.280 202.808 2237.308 1824.388 8649.998 2970.950 8270.920 252.234 334.256 F67 Mean 217.3338617866620000 681.0349114021570000 265.0370119084380000 228.7309024901770000 240.3635189964930000 221.1801916743850000 228.3769828342800000 201.0516618000000000 222.0150462881420000 Std. Dev. 20.6685850658838000 488.0618274343640000 12.4033917090208000 12.3682716268631000 14.8435137485293000 5.7037006844690500 8.7086794471239900 2.4309010810000000 4.5838205188784200 Best 120.0000000000000000 223.0782617790520000 241.9810089596350000 181.6799927773160000 221.3817133141830000 209.2509748304710000 204.6479138174220000 197.8966349103590000 215.3853171009670000 Runtime 8208.697 197.497 2159.392 5873.112 4599.027 5938.879 8189.243 254.253 294.256 F68 Mean 668.9850326105730000 926.9488078829420000 513.8925774904480000 743.9859973770210000 892.4391527217660000 845.4504613493740000 587.5732354221340000 310.0161021000000000 366.0263626038670000 Std. Dev. 275.8071370273340000 174.1027182659660000 31.0124861524005000 175.6497294240330000 79.1422224454971000 120.8505129523180000 250.0556329707140000 0.0370586450000000 231.2038781149790000 Best 310.0000000000000000 310.0000000000000000 444.4692044973030000 310.0000000000000000 738.3764781625320000 310.0000000000000000 310.0000000000000000 310.0014955442130000 310.0622831487350000 Runtime 3687.235 251.155 2445.259 1777.638 8398.690 3073.274 4554.102 253.064 310.288 F69 Mean 708.2979222913040000 831.2324139697050000 500.5478931040730000 776.5150806087790000 863.8926908090610000 809.7183195902260000 587.6511686191670000 310.0029796000000000 810.0062247333440000 Std. Dev. 256.2419561521300000 250.1848775931620000 31.2240894705539000 160.7307526692470000 96.5618989087194000 147.3158109824600000 236.1141037692630000 0.0082796490000000 223.3247693967360000 Best 310.0000000000000000 310.0000000000000000 407.3155842366960000 363.8314566805740000 493.0042540796450000 310.0000000000000000 310.0000000000000000 310.0000285440690000 310.0000000000000000 Runtime 5258.509 222.015 2341.791 1849.670 9909.479 3213.601 4764.968 291.084 281.124 F70 Mean 711.2970397614200000 876.9306188768990000 483.2984167460740000 761.2954767038960000 844.6391674419360000 810.5227124472170000 612.0906184834040000 310.0041570000000000 660.0000000106290000 Std. Dev. 258.9317052508320000 289.7296413284470000 99.3976740616107000 163.4084080635650000 113.6848457105400000 104.7139423525340000 249.5599278421970000 0.0128812140000000 202.7587641256140000 Best 310.0000000000000000 310.0000000000000000 155.5049931377980000 363.8314568648180000 489.0742585970560000 310.0000000000000000 310.0000000000000000 310.0002219576930000 310.0000000051680000 Runtime 4346.055 228.619 2250.917 1900.279 9988.261 2818.575 4945.132 268.701 440.520 F71 Mean 1117.8857079625100000 1258.1065536572400000 659.5351969346130000 959.3735119754180000 911.4640642691360000 990.8546718748010000 836.1411004458200000 577.7786170000000000 760.0000063070120000 Std. Dev. 311.0011859260640000 359.7382897536570000 98.5410511961986000 240.5568407069990000 238.3180009803040000 235.1014092849970000 128.9346234954740000 1.8288684190000000 105.4092619871220000 Best 560.0000000000000000 660.0000000000000000 560.0001912324020000 660.0000000000000000 560.0000121795840000 660.0000000000000000 560.0000000000000000 574.8590032551840000 660.0000000000000000 Runtime 3012.883 241.541 2728.060 1573.484 10891.124 1769.459 2972.618 279.0646913 530.172 NFE F72 Mean 1094.8305116977000000 -7.159E + 49 915.4958100611630000 1133.7536009808600000 1075.5292326436900000 1094.6823697304900000 984.5106541514410000 694.3706620000000000 1088.6626563226300000 Std. Dev. 121.3539576317800000 4.387E + 50 242.1993331983530000 42.1171260000361000 166.9355145236330000 87.9884000140656000 199.1563947691970000 20.9754439100000000 136.0666138798300000 Best 660.0000000000000000 -133.9585340104890000 660.0006867770510000 1088.9543269392600000 660.0000000000020000 660.0000000000000000 660.0000000000000000 644.2542524502140000 660.0000000000000000 Runtime 6363.267 290.334 2326.112 1730.723 9601.880 3854.148 10458.467 273.922 997.068 F73 Mean 1304.3661550124000000 1159.9280867973000000 830.2290165794410000 1167.9040488743800000 1070.4327462836400000 1105.2511774948600000 976.2273885425320000 559.6581705000000000 919.4683268438060000 Std. Dev. 262.1065863453340000 742.1215416320490000 60.2286903507069000 236.7325108248320000 203.0676662707430000 190.6172874229610000 160.1543461970300000 16.1193896300000000 136.2630721125810000 Best 919.4683107913200000 -460.7504508023100000 785.1725102979490000 785.1725102979490000 785.1725102979480000 919.4683107913240000 785.1725102979480000 546.1130231359180000 919.4683114379170000 Runtime 2165.640 238.261 2045.582 1580.067 7459.005 1901.540 4209.110 287.271 118.771 F74 Mean 500.0000000000000000 653.3355378428050000 460.0000000000020000 510.0000000000000000 493.3333333333340000 490.0000000000000000 460.0000000000000000 463.2262530000000000 460.0000000000000000 Std. Dev. 103.7237710925280000 302.5312999719650000 0.0000000000016493 113.7147065368360000 137.2973951415090000 91.5385729888094000 0.0000000000000000 4.9321910760000000 0.0000000000000000 Best 460.0000000000000000 460.0000000000000000 460.0000000000000000 460.0000000000000000 460.0000000000000000 460.0000000000000000 460.0000000000000000 458.5444354721460000 460.0000000000000000 Runtime 1811.980 165.962 1698.121 1366.710 3016.959 1410.399 1795.637 257.960 1278.572 F75 Mean 1107.9038127876700000 1401.6553278264300000 930.4565414149210000 1072.9924659809200000 1258.5157766524700000 1074.3695435628600000 1063.7363787709700000 471.2797518000000000 1084.7073068225200000 Std. Dev. 127.9566489362040000 253.2428066220210000 87.9959072391079000 2.2606058314671500 241.4024507676890000 2.8314182838917800 55.8479313799755000 2.2346287190000000 4.9504851126832800 Best 1069.5511765775700000 1072.4973401423200000 862.4476004191700000 1068.5560012648600000 871.8607884176050000 1069.8723890709000000 856.8214538442850000 469.3372925643150000 1078.2231646698500000 Runtime 4060.091 214.580 2113.339 2951.018 5262.210 3410.902 4280.901 263.829 711.530
Table 5: Statistical results for Test 1 Problems using two-sided Wilcoxon Signed-Rank Test ( ๐ผ = 0.05 ) Problem PSO2011 vs Multi CI CMAES vs Multi CI ABC vs Multi CI JDE vs Multi CI p-value T+ T- winner p-value T+ T- winner p-value T+ T- winner p-value T+ T- winner F1 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F2 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F3 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 1.34E-06 465 0 - F4 4.32E-08 0 465 + 6.80E-08 0 465 + 3.35E-07 0 465 + 4.32E-08 0 465 + F5 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 465 0 - 1.73E-06 0 465 + F6 1.73E-06 465 0 - 1.73E-06 0 465 + 1.73E-06 465 0 - 1.73E-06 465 0 - F7 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 465 0 - 1.73E-06 465 0 - F8 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + F9 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F10 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F11 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F12 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F13 1.66E-06 0 465 + 1.66E-06 465 0 - 1.66E-06 465 0 - 5.66E-02 140 325 + F14 1.67E-06 465 0 - 1.67E-06 465 0 - 1.55E-06 465 0 - 1.55E-06 465 0 - F15 1.01E-07 0 465 + 1.01E-07 0 465 + 1.01E-07 465 0 - 1.01E-07 465 0 - F16 6.87E-07 0 465 + 6.87E-07 0 465 + 6.87E-07 0 465 + 6.87E-07 0 465 + F17 1.10E-06 0 465 + 1.10E-06 0 465 + 1.10E-06 0 465 + 1.10E-06 0 465 + F18 1.01E-07 465 0 - 1.01E-07 465 0 - 1.01E-07 465 0 - 1.01E-07 465 0 - F19 1.20E-06 0 465 + 1.20E-06 0 465 + 1.20E-06 465 0 - 1.20E-06 465 0 - F20 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F21 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + F22 1.66E-06 0 465 + 1.66E-06 0 465 + 1.66E-06 0 465 + 1.66E-06 0 465 + F23 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + F24 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F25 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F26 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F27 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F28 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F29 5.99E-07 0 465 + 5.99E-07 0 465 + 5.99E-07 0 465 + 5.99E-07 0 465 + F30 1.70E-06 465 0 - 1.70E-06 465 0 - 1.70E-06 465 0 - 1.70E-06 465 0 - F31 1.08E-06 0 465 + 1.08E-06 465 0 - 1.08E-06 0 465 + 1.08E-06 0 465 + F32 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F33 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F34 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F35 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F36 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F37 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F38 3.32E-07 465 0 - 3.32E-07 465 0 - 4.32E-08 0 465 + 3.32E-07 465 0 - F39 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F40 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F41 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F42 1.96E-07 465 0 - 1.96E-07 465 0 - 1.96E-07 465 0 - 1.96E-07 465 0 - F43 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F44 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F45 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F46 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F47 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F48 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F49 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F50 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + +/=/- 30/0/20 29/0/21 26/0/24 26/0/24
Problem CLPSO vs Multi CI SADE vs Multi CI BSA vs Multi CI IA vs Multi CI p-value T+ T- winner p-value T+ T- winner p-value T+ T- winner p-value T+ T- winner F1 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1.67E-06 0 465 + F2 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1.01E-06 0 465 + F3 1.34E-06 465 0 - 4.32E-08 0 465 + 1 0 0 = 4.32E-08 0 465 + F4 3.35E-07 465 0 - 4.32E-08 0 465 + 1 0 0 = 1.66E-06 0 465 + F5 1.73E-06 0 465 + 1.73E-06 0 465 + 4.32E-08 0 465 + 1 0 0 = F6 1.73E-06 465 0 - 1.73E-06 465 0 - 1 0 0 = 4.32E-08 0 465 + F7 1.73E-06 465 0 - 1.73E-06 465 0 - 1 0 0 = 1 0 0 = F8 1.73E-06 0 465 + 1.73E-06 0 465 + 1 0 0 = 1 0 0 = F9 4.32E-08 0 465 + 1 0 0 = 1 0 0 = 1 0 0 = F10 4.32E-08 0 465 + 4.32E-08 465 0 - 1 0 0 = 4.32E-08 0 465 + F11 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1.72E+00 0 465 + F12 4.32E-08 0 465 + 4.32E-08 465 0 - 1 0 0 = 5.99E-07 0 465 + F13 0.0566 140 325 + 1.66E-06 0 465 + 0.0027 378 87 - 3.19E-06 459 6 - F14 1.55E-06 465 0 - 1.55E-06 465 0 - 4.88E-04 78 0 - 3.19E-06 459 6 - F15 1.01E-07 465 0 - 1.01E-07 465 0 - 1 0 0 = 1 0 0 = F16 6.87E-07 0 465 + 6.87E-07 0 465 + 1 0 0 = 1 0 0 = F17 1.10E-06 0 465 + 1.10E-06 465 0 - 1 0 0 = 1 0 0 = F18 1.01E-07 465 0 - 1.01E-07 465 0 - 4.32E-08 0 465 + 1 0 0 = F19 1.20E-06 465 0 - 1.20E-06 465 0 - 1.69E-06 465 0 - 1.73E-06 0 465 + F20 4.32E-08 465 0 - 4.32E-08 465 0 - 3.11E-06 459 6 - 1.69E-06 0 465 + F21 1.73E-06 0 465 + 1.73E-06 0 465 + 1.69E-06 465 0 - 1.70E-06 0 465 + F22 1.66E-06 0 465 + 1.66E-06 0 465 + 4.32E-08 0 465 + 1.69E-06 465 0 - F23 1.73E-06 0 465 + 1.73E-06 0 465 + 1.69E-06 0 465 + 1.69E-06 465 0 - F24 4.32E-08 465 0 - 4.32E-08 465 0 - 1.69E-06 0 465 + 1.69E-06 465 0 - F25 4.32E-08 0 465 + 1 0 0 = 1 0 0 = 1 0 0 = F26 4.32E-08 465 0 - 4.32E-08 465 0 - 1.69E-06 0 465 + 1.69E-06 0 465 + F27 4.32E-08 465 0 - 4.32E-08 465 0 - 1.66E-06 465 0 - 1.69E-06 0 465 + F28 4.32E-08 465 0 - 4.32E-08 465 0 - 1.66E-06 465 0 - 1.69E-06 0 465 + F29 5.99E-07 0 465 + 5.99E-07 0 465 + 1.40E-06 465 0 - 1.43E-06 0 465 + F30 1.70E-06 465 0 - 1.70E-06 465 0 - 4.32E-08 0 465 + 1 0 0 = F31 1.08E-06 0 465 + 1.08E-06 0 465 + 1.66E-06 465 0 - 1.69E-06 0 465 + F32 4.32E-08 0 465 + 4.32E-08 0 465 + 1.66E-06 0 465 + 8.94E-04 71 394 + F33 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1 0 0 = F34 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1.69E-06 465 0 - F35 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1 0 0 = F36 4.32E-08 465 0 - 4.32E-08 465 0 - 1 0 0 = 1.69E-06 465 0 - F37 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1 0 0 = F38 3.32E-07 0 465 + 3.32E-07 465 0 - 1 0 0 = 1 0 0 = F39 4.32E-08 465 0 - 4.32E-08 465 0 - 1.69E-06 465 0 - 1.73E-06 0 465 + F40 4.32E-08 465 0 - 4.32E-08 465 0 - 1 0 0 = 1.73E-06 0 465 + F41 4.32E-08 465 0 - 4.32E-08 465 0 - 1 0 0 = 1.73E-06 0 465 + F42 1.96E-07 465 0 - 1.96E-08 465 0 - 1 0 0 = 1.69E-06 0 465 + F43 4.32E-08 465 0 - 4.32E-08 465 0 - 1 0 0 = 2.10E-03 382 83 - F44 4.32E-08 465 0 - 4.32E-08 465 0 - 1 0 0 = 1 0 0 = F45 1 0 0 = 1 0 0 = 1 0 0 = 1.69E-06 0 465 + F46 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 1.69E-06 0 465 - F47 4.32E-08 0 465 + 4.32E-08 0 465 + 1 0 0 = 1 0 0 = F48 4.32E-08 465 0 - 4.32E-08 0 465 + 1 0 0 = 1.69E-06 0 465 + F49 4.32E-08 465 0 - 4.32E-08 465 0 - 1 0 0 = 1.69E-06 0 465 + F50 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 - 1 0 0 = +/=/- 25/1/24 22/3/25 8/30/12 24/17/9 Table 6: Statistical results for Test 2 Problems using two-sided Wilcoxon Signed-Rank Test ( ๐ผ = 0.05 ) Problem PSO vs Multi CI CMAES vs Multi CI ABC vs Multi CI JDE vs Multi CI p-value T+ T- winner p-value T+ T- p-value T+ T- winner winner p-value T+ T- winner F51 6.91E-07 465 0 - 6.91E-07 465 0 - 6.91E-07 465 0 - 6.91E-07 465 0 - F52 1.18E-06 0 465 + 1.18E-06 465 0 - 1.18E-06 465 0 - 1.18E-06 0 465 + F53 4.32E-08 0 465 + 4.32E-08 465 0 - 4.32E-08 0 465 + 4.32E-08 0 465 + F54 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 465 0 - F55 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + F56 4.32E-08 0 465 + 4.32E-08 465 0 - 4.32E-08 0 465 + 4.32E-08 465 0 - F57 6.80E-08 0 465 + 6.80E-08 0 465 + 6.80E-08 0 465 + 6.80E-08 0 465 + F58 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 0 465 + F59 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 465 0 - 4.32E-08 465 0 - F60 3.96E-05 36 429 + 1.16E-06 0 465 + 1.16E-06 0 465 + 1.16E-06 0 465 + F61 4.32E-08 0 465 + 4.32E-08 465 0 - 4.32E-08 0 465 + 4.32E-08 0 465 + F62 1.44E-07 0 465 + 1.44E-07 0 465 + 2.99E-07 6 459 + 1.44E-07 0 465 + F63 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F64 1.51E-06 465 0 - 1.51E-06 0 465 + 1.51E-06 465 0 - 1.51E-06 465 0 - F65 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - 4.32E-08 465 0 - F66 7.86E-07 465 0 - 7.86E-07 0 465 + 7.86E-07 465 0 - 7.86E-07 465 0 - F67 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + F68 6.98E-07 0 465 + 6.98E-07 0 465 + 6.98E-07 0 465 + 6.98E-07 0 465 + F69 1.19E-06 0 465 + 1.19E-06 0 465 + 1.19E-06 0 465 + 1.19E-06 0 465 + F70 1.20E-06 0 465 + 1.20E-06 0 465 + 1.20E-06 0 465 + 1.20E-06 0 465 + F71 9.27E-07 0 465 + 9.27E-07 0 465 + 9.27E-07 0 465 + 9.27E-07 0 465 + F72 1.97E-07 465 0 - 4.32E-08 465 0 - 1.97E-07 465 0 - 4.32E-08 465 0 - F73 8.89E-07 0 465 + 8.89E-07 0 465 + 8.89E-07 0 465 + 8.89E-07 0 465 + F74 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + F75 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + 1.73E-06 0 465 + +/=/- 18/0/7 16/0/9 16/0/9 16/0/9 Problem CLPSO vs Multi CI SADE vs Multi CI BSA vs Multi CI IA vs Multi CI p-value T+ T- winner p-value T+ T- winner p-value T+ T- winner p-value T+ T- winner F51 1.18E-06 0 465 + 6.91E-07 465 0 - 1 0 0 = 6.91E+00 0 465 + F52 4.32E-08 0 465 + 1.18E-06 465 0 - 1 0 0 = 1.18E-06 0 465 + F53 1.73E-06 0 465 + 4.32E-08 0 465 + 1.69E-06 465 0 - 1.58E-06 465 0 - F54 1.73E-06 0 465 + 1.73E-06 465 0 - 1 0 0 = 1.73E-06 0 465 + F55 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 0 465 + 4.32E-08 465 0 - F56 4.32E-08 0 465 + 4.32E-08 465 0 - 1.58E-06 465 0 - 1.58E-06 465 0 - F57 6.80E-08 465 0 - 6.80E-08 465 0 - 1.58E-06 465 0 - 1 0 0 = F58 4.32E-08 465 0 - 4.32E-08 465 0 - 3.11E-06 6 459 + 1.69E-06 0 465 + F59 4.32E-08 465 0 - 4.32E-08 465 0 - 1.69E-06 465 0 - 3.13E-06 459 6 - F60 1.16E-06 0 465 + 1.16E-06 0 465 + 1.28E-05 21 444 + 3.13E-06 459 6 - F61 4.32E-08 0 465 + 4.32E-08 0 465 + 1.66E-06 0 465 + 1.69E-06 0 465 + F62 1.44E-07 465 0 - 9.96E-04 87 378 + 1.69E-06 0 465 + 1.69E-06 0 465 + F63 4.32E-08 465 0 - 4.32E-08 465 0 - 1.66E-06 465 0 - 1.69E-06 0 465 + F64 1.51E-06 465 0 - 1.51E-06 465 0 - 8.66E-05 423 42 - 1.73E-06 0 465 + F65 4.32E-08 465 0 - 4.32E-08 465 0 - 8.66E-05 465 0 - 4.32E-08 465 0 - F66 7.86E-07 465 0 - 7.86E-07 465 0 - 2.96E-06 6 459 + 1.69E-06 0 465 + F67 1.73E-06 0 465 + 1.73E-06 0 465 + 3.72E-04 60 405 + 1.73E-06 465 0 - F68 6.98E-07 0 465 + 6.98E-07 0 465 + 0.0266 126 339 + 1.73E-06 465 0 - F69 1.19E-06 0 465 + 1.19E-06 0 465 + 6.07E-04 399 66 - 1.73E-06 465 0 - F70 1.20E-06 0 465 + 1.20E-06 0 465 + 0.1646 300 165 - 1.73E-06 465 0 - F71 9.27E-07 0 465 + 9.27E-07 0 465 + 0.001 78 387 + 1.73E-06 465 0 - F72 4.32E-08 465 0 - 1.97E-07 465 0 - 1.97E-07 465 0 - 4.32E-08 465 0 - F73 8.89E-07 0 465 + 8.89E-07 0 465 + 1.13E-04 420 45 - 1.73E-06 465 0 - F74 1.73E-06 0 465 + 1.73E-06 0 465 + 1 0 0 = 1.69E-06 0 465 + F75 1.73E-06 0 465 + 1.73E-06 0 465 + 1.69E-06 465 0 - 1.73E-06 465 0 - +/=/- 16/0/9 13/0/12 9/4/12 10/1/14 Table 7: Multi-problem based statistical pairwise comparison of PSO, CMAES, ABC, JDE, CLPSO, SADE, BSA, IA and Multi-CI
Other Algorithm vs IA p-Value T+ T- Winner
PSO vs Multi CI 0.0035 368 1010 Multi CI CMAES vs Multi CI 3.3367e-07 235 1656 Multi CI ABC vs Multi CI 0.1355 615 981 Multi CI JDE vs Multi CI 4.6305e-04 320 1111 Multi CI CLPSO vs Multi CI 1.507e-04 366 1345 Multi CI SADE vs Multi CI 0.2031 424 657 Multi CI BSA vs Multi CI 0.9144 402 418 Multi CI IA vs Multi CI 0.7003 834 936 Multi CI (a) Convergence of all candidates (b)
Convergence of best candidates Figure 2 Convergence for Ackley Function (F5)
Learning attempts F un c t i on v a l ue s Learning attempts F un c t i on v a l ue s (a) Convergence of all candidates (c) Convergence of best solutions Figure 3 Convergence for Beale Function (F6)
Learning attempts F un c t i on v a l ue s Learning attempts F un c t i on v a l ue s (a) Convergence of all candidates (b) Convergence of best candidates Figure 4 Convergence for Fletcher Function (F16) Learning attempts F un c t i on v a l ue s Learning attempts F un c t i on v a l ue s (a) Convergence of all candidates (d) Convergence of best candidates Figure 5 Convergence for Foxholes Function (F1)
Learning attempts F un c t i on v a l ue s Learning attempts F un c t i on v a l ue s (a) Convergence of all candidates (e) Convergence of best candidates Figure 6 Convergence for Michalewics Function (F28)
Learning attempts F un c t i on v a l ue s Learning attempts F un c t i on v a l ue s This section provides theoretical comparison of the algorithms being compared with Multi-CI. The method of PSO a swarm of solutions modify their positions in the search space. Every particle of (a) Convergence of all candidates (b) Convergence of best candidates Figure 7 Convergence for Six-hump camelback Function (F43)
Learning attempts F un c t i on v a l ue s Learning attempts F un c t i on v a l ue s the swarm represents a solution which moves with certain velocity in the search space based on the best solution in the entire swarm as well as the best solution in certain close neighborhood. It imparts exploration as well as exploitation abilities to entire swarm. According to Teo et al. (2016), Li and Yao (2012) and Selvi and Umrani (2010) the PSO may not be efficient solving the problems with discrete search space as well as non-coordinate systems and may need supporting techniques to solve such problems. In this paper Multi-CI is compared with the advanced versions of the PSO referred to as Comprehensive Learning PSO (CLPSO) (Liang et al., 2016) and PSO2011 (Omran and Clerc2011).The technique of CMAES (Igel et al. 2007) is a mathematical-based algorithm which exploits adaptive mutation parameters through computing a covariance matrix. The computational cost of the covariance matrix calculation, sampling using multivariate normal distribution and factorization of covariance matrix may increase exponentially with increase in problem dimension (Selvi and Umrani, 2010). The algorithm of ABC (Karaboga and Akay, 2009) carries out exploration using random search by scout bees and exploitation using employed bees. Some studies highlightedthat the algorithm of ABC can perform well with exploration; however, it is not efficient in local search and exploitation (Murugan and Mohan, 2012). This may make the algorithm trap into local minima.The BSA (Civicioglu, 2013) is a populations based technique which deploys genetic operators to generate initial solutions. Then therandomly chooses individuals to find the new solutions in the search space. The non-uniform crossover makes BSA unique and powerful technique. Similar to the BSA, DE (Storn and Price 1997, Qin and Suganthan 2005) is also a population based technique which exploits genetic operators. The search process is mainly driven by the mutation and selection operation. The crossover operator is further deployed for effectively sorting the trial vectors which helps to choose and retain better solutions. Teo et al. (2016) recently proposed IA. It is inspired from competitive behavior of political party individuals. The local party leaders exploit the concepts such as introspection, local competition and global competition improving the solution quality through exploitation and exploration. In addition, the ordinary party membersmay follow the own party leader or other party leader. This changes the priority of the search lead by certain party. The algorithm performed better as compared to most of the contemporary algorithms. The Multi-CI algorithm proposed here exhibited certain prominent characteristics and limitations. These are discussed below. 1. In Multi-CI the best individual from within every cohort are moved to a separate pool ๐ . One best candidate is chosen from within each cohort. Then every candidate chooses the best behavior/objective function value from within the ๐ + ๐ ๐ choices. Thus every candidate competes with its own local best behavior as well as the best behavior chosen from the other cohorts. This gives more exploitation power to the algorithm due to which chance of avoiding the local minima increases with faster convergence. 2. The elite candidate behaviours from Pool ๐ is carry forwarded to the subsequent learning attempt. This helps not to lose the best behaviour (solution) so far and also the influence of such solutions does not diminish if not followed by any candidate. 3. Initial random walks of individuals around the cohorts ensure exploration of the search space around the candidates. 4.
The Multi-CI parameters could be easily tuned which may make it a flexible algorithm for handling variety of problems with different dimensions and complexity. The results highlighted that the algorithm is sufficiently robust with reasonable computational cost and is successful at exploring multi-modal search spaces. 6.
The computational performance was essentially governed by sampling interval reduction factor ๐ . Its value was chosen based on the preliminary trials of the algorithm. Conclusions and Future Directions
A modified version of the Cohort Intelligence (CI) algorithm referred to as Multi-Cohort Intelligence (Multi-CI) was proposed. In the proposed Multi-CI approach intra-group learning and inter-group learning mechanisms were implemented. It is more realistic representation of the learning through interaction and competition of the cohort candidates. It imparted the exploitation and exploration capabilities to the algorithm. The approach was validated by solving two sets of test problems from CEC 2005. Wilcoxon statistical tests were conducted for comparing the performance of the algorithm with the existing algorithms. The performance of Multi-CI was exceedingly better as compared to PSO2011, CMAES, ABC, JDE, CLPSO and SADE in terms of objective function value (best and mean), robustness, as well as computational time. The performance of the Multi-CI was marginally better as compared to BSA and IA. The solution quality highlighted that the Multi-CI is a robust approach with reasonable computational cost and could quickly reach in the close neighborhood of the global optimum solution. A generalized constraint handling mechanism needs to be developed and incorporated into the algorithm. This can help Multi-CI to solve real world problems which are generally constrained in nature. The Multi-CI algorithm could be further modified for solving constrained test problems as well as real world problems. The constrained Multi-CI version could be further extended to solve complex structural optimization problems (Azad 2017, 2018). This work is currently underway. A self-adaptive mechanism needs to be developed for the selection of the sampling interval reduction factor ๐ . Appendix A: Illustration of Multi-CI Algorithm
An illustrative example (Sphere function with 2 variables: ๐๐๐ง๐ โ ๐ฅ ๐22๐=1 , ๐๐ข๐๐๐๐๐ก ๐ก๐ โ 5.12 โค ๐ฅ ๐ โค5.12 , ๐ = 1, 2 ) of the Multi-CI procedure discussed in Section 2 is detailed below. It includes every details of first learning attempt followed by evaluation of every step (1 to 8) is listed in Table A.1 till convergence along with the convergence plot in Figure A.1.The Multi-CI parameters chosen were as follows: number of cohorts ๐พ = 3 , number of candidates ๐ถ ๐ = 3 , reduction factor value ๐ = 0.98 , quality variation parameters ๐ = 2 and ๐ ๐ = 4 , the algorithm stopped when the objective function value is less than โ16 . Learning Attempt ๐ = 1
๐ฟ = [ 0.4426 โ2.7631 1.7060 4.5039 โ2.2525 โ1.3291, โ4.4698 1.2841โ4.4155 โ0.7989 โ2.4503 3.8907 , โ4.4839 4.8435โ4.0203 โ1.1923 0.1308 โ1.8813] ( Step 1, Eq 2 ): ๐ญ = [ 7.8304 21.6280 43.564823.1957 20.1344 17.58416.8402 21.1409 3.5564 ] ( Step 2, Eq 3 ): ๐ญ ๐ = [6.8402 20.1344 3.5564] ( Step 3, Eq 4 ): ๐ = 0.7476 ๐ = 0.2524 ๐ = 0.4943 ๐ = 0.5057 ๐ = 0.2876 ๐ = 0.7124 ( Step 4, Eq 5 ): Now every cohort ๐ (๐ = 1, โฆ , ๐พ ) is left with ๐ถ ๐ โ 1 candidates. Consider candidate ๐ถ in cohort 1 and the associated qualities ๐ฟ = [๐ฅ , ๐ฅ ] = [0.4426, ๏ญ . Sampling interval for ๐ฅ is given by [๐ , ๐ ] = [0.4426 โ (โ5.12 โ (โ5.12)2 โ) ร 0.98, 0.4426 + (โ5.12 โ (โ5.12)2 โ) ร 0.98] [๐ , ๐ ] = [โ4.575, 5.4602] = [โ4.575, 5.12] ( ๏ ๐ฅ is given by [๐ , ๐ ] = [ ๏ญ ๏ญ [๐ , ๐ ] = [ ๏ญ = [ ๏ญ ( ๏ ๏ญ ๐ถ in cohort 1 chooses to follow candidate ๐ถ . So sampling intervals for candidate ๐ถ will be the same as that of candidate ๐ถ . The ๐ = 2 sampling intervals of every candidate ๐ถ ๐ โ 1 , ๐ (๐ = 1, โฆ , ๐พ ) are as follows: [ [ [ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ๏ญ ( Step 4, Eq 6 ): Every candidate samples the qualities from within these sampling intervals and forms quality matrix ๐ ๐ as follows: ๐ ๐ = [ [ 3.7775 โ4.9854โ0.0271 2.1268 ][โ1.5062 โ0.00348โ2.2765 โ2.2535 ] [โ2.4902 4.1556โ2.8815 4.4581][โ4.7380 4.1285โ1.7774 2.8955] [โ1.6575 โ0.5544โ1.3713 โ4.2098][ โ4.4906 โ2.1964โ4.65257 โ1.0430]] Cohort 1 Cohort 2 Cohort 3 ( Step 4, Eq 7 ): ๐ญ ๐ = [[39.12424.5240 ] [23.470228.1783] [ 3.056219.6036][ 2.270011.6100] [39.494111.5436] [24.990122.4855]] ( Step 5, Eq 8 ): ๐ = 0.3065 ๐ = 0.1041 ๐ = 0.5894 ( Step 6, Eq 9 ): ๐ ๐ ๐ = [ [โ0.9714 0.1627โ4.5412 โ2.4676โ4.0294 โ0.5608โ4.2622 โ1.8622] [ 0.0057 0.6931โ4.8913 2.3370โ4.4565 โ1.9554โ1.5795 1.0522 ] [โ2.9102 โ3.4787โ2.9835 โ0.8647โ2.1960 โ4.7693โ2.2042 โ3.3672][ 1.1820 โ2.2922โ0.9737 โ3.0238โ1.7371 โ1.506450.0012 2.3825 ] [ 1.2259 โ3.5126โ0.6327 โ0.3592โ0.2333 โ2.3488โ2.5639 โ1.2459] [ 0.7412 โ2.7817โ3.0381 โ0.8921โ0.3807 2.1570โ2.7075 โ0.9127]] ( Step 6, Eq 10 ): ๐ญ ๐ ๐ = [ [ 0.970326.712216.550821.6350] [ 0.480529.38723.68423.6022 ] [20.57149.649527.569218.3430][ 9.935810.09165.28715.6763 ] [13.84160.52945.57138.1260 ] [ 8.287610.02644.79768.1636 ]] ( Step 8, Eq 12 ): ๐ญ = [0.9703 0.4805 3.05622.2700 0.5294 4.79766.8402 20.1344 3.5564]
๐๐๐๐๐๐ข๐ = 0.4805
End of Learning Attempt F un c t i o n V a l u e s Learning Attempts
Cohort 1 Cohort 2 Cohort 3 Figure A.1 Convergence for Sphere Function Table A.1 Illustration of Multi-CI Algorithm solving Sphere Function
Learning Attempt (๐) ๐ฟ ๐ญ (Step 1, Eq 2) ๐ญ ๐ (Step 2, Eq 3) ๐ = 2 [ โ0.9715 โ0.1627 โ0.9715 โ0.1627 โ2.2525 โ1.3291, 0.0057 0.6932 0.0057 0.6932 โ4.4155 โ0.7989, โ1.6579 โ0.5544โ2.9836 โ0.8647 0.1308 โ1.8813 ] [0.9703 0.4805 3.05620.9703 0.4805 9.64656.8402 20.1344 3.5564] [0.9703 0.4805 3.0562] ๐ญ ๐ (Step 4, Eq 7) ๐ญ ๐ ๐ (Step 6, Eq 10) ๐ญ (Step 8, Eq 12) Minimum [[ 6.696620.3143] [ 3.203913.4206] [17.481820.2246][4.69844.1166] [3.50173.7385] [3.57444.4140] ] [ [ 6.647517.52172.99300.7515 ] [7.79842.95761.16477.7003] [8.74003.36706.12572.4325][0.98721.07323.03780.3062] [3.91200.49514.55881.2267] [10.23980.60332.41800.6774 ]] [0.7515 1.1647 2.43250.3062 0.4951 0.60330.9703 0.4805 3.0562] ๐ = 3 ๐ฟ ๐ญ (Step 1, Eq 2) ๐ญ ๐ (Step 2, Eq 3) [ โ0.4357 โ0.7494 โ0.4357 โ0.7494 โ0.9715 0.1627 , โ0.3565 โ1.0186 โ0.3565 โ1.0186 0.0057 0.6932 , 1.0491 1.15411.0491 1.1541 โ1.6579 โ0.5544] [0.7515 1.1647 2.43250.7515 1.1647 2.43250.9703 0.4805 3.0562] [0.7515 0.4805 2.4325] ๐ญ ๐ (Step 4, Eq 7) ๐ญ ๐ ๐ (Step 6, Eq 10) ๐ญ (Step 8, Eq 12) Minimum [[0.32760.6218] [1.76951.0992] [0.08818.6904][1.27726.1061] [1.60182.0007] [ 2.588611.1223]] [ [10.47356.16701.62760.7038 ] [3.15050.68581.55010.0470] [0.93882.20836.81895.8672][4.31812.07175.45491.4087] [4.66740.43196.61151.2808] [0.16117.01182.32740.3228]] [0.3276 0.0470 0.08811.2772 0.4319 0.16110.7515 0.4805 2.4325] โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ โฎ ๐ = 48 ๐ฟ ๐ญ (Step 1, Eq 2) ๐ญ ๐ (Step 2, Eq 3) [ 4.98E โ 8 1.03E โ 7โ4.33E โ 8 4.71E โ 8 โ2.59E โ 8 6.17E โ 8, โ2.95E โ 9 โ7.71E โ 8 โ2.95E โ 9 โ7.71E โ 8 6.5E โ 8 8.76E โ 8 , โ8.29E โ 8 โ1.11E โ 7โ8.29E โ 8 โ1.11E โ 7 1.03E โ 7 โ5.9E โ 9 ] [1.31๐ธ โ 14 5.96๐ธ โ 15 1.90๐ธ โ 144.09๐ธ โ 15 5.96๐ธ โ 15 1.90๐ธ โ 144.48๐ธ โ 15 1.12๐ธ โ 14 1.06๐ธ โ 14] [4.09E โ 15 5.96E โ 15 1.06E โ 14] ๐ญ ๐ (Step 4, Eq 7) ๐ญ ๐ ๐ (Step 6, Eq 10) ๐ญ (Step 8, Eq 12) Minimum [[2.05E โ 145.64E โ 14] [3.59E โ 145.99E โ 14] [9.92E โ 155.32E โ 14][4.13E โ 142.92E โ 15] [3.54E โ 149.02E โ 15] [2.12E โ 143.58E โ 14]] [ [1.80E โ 145.62E โ 153.76E โ 141.57E โ 14] [2.54E โ 142.00E โ 141.95E โ 157.82E โ 14] [1.05E โ 134.06E โ 142.19E โ 141.70E โ 14][2.49E โ 152.38E โ 141.67E โ 152.30E โ 14] [3.56E โ 151.35E โ 147.74E โ 152.36E โ 14] [3.86E โ 146.24E โ 155.02E โ 143.03E โ 14]] [5.62E โ 15 1.95E โ 15 9.92E โ 151.67E โ 15 3.56E โ 15 6.24E โ 154.09E โ 15 5.96E โ 15 1.06E โ 14] ๐ = 49 ๐ฟ ๐ญ (Step 1, Eq 2) ๐ญ ๐ (Step 2, Eq 3) [ 5.05E โ 8 โ5.54E โ 81.28E โ 9 โ5.4E โ 8 โ4.33E โ 8 4.71E โ 8 , 1.92E โ 8 โ3.98E โ 8 โ1.92E โ 8 โ3.98E โ 8 2.95E โ 9 โ7.77E โ 8 , โ9.49E โ 8 3E โ 8โ1.28E โ 7 2.46E โ 8 1.03E โ 7 โ5.9E โ 9] [5.62๐ธ โ 15 1.95๐ธ โ 15 9.92๐ธ โ 152.92๐ธ โ 15 1.95๐ธ โ 15 1.72๐ธ โ 154.09๐ธ โ 15 5.96๐ธ โ 14 1.06๐ธ โ 14] [2.92E โ 15 1.95E โ 15 1.72E โ 15] ๐ญ ๐ (Step 4, Eq 7) ๐ญ ๐ ๐ (Step 6, Eq 10) ๐ญ (Step 8, Eq 12) Minimum [[3.34E โ 144.73E โ 15] [7.96E โ 152.11E โ 14] [1.78๐ธ โ 144.17๐ธ โ 14][2.47๐ธ โ 147.68๐ธ โ 15] [1.19E โ 148.36E โ 16] [6.41E โ 151.33E โ 14]] [ [2.38E โ 152.25E โ 141.86E โ 142.87E โ 14] [4.28E โ 141.15E โ 141.32E โ 144.69E โ 14] [2.79๐ธ โ 143.41๐ธ โ 141.09๐ธ โ 148.24๐ธ โ 16][2.65E โ 143.08E โ 152.02E โ 143.53E โ 14] [4.21E โ 143.54E โ 147.60E โ 153.37E โ 14] [1.24E โ 143.10E โ 153.37E โ 146.32E โ 16]] [2.38๐ธ โ 15 7.96๐ธ โ 15 8.24๐ธ โ 163.08๐ธ โ 15 8.36๐ธ โ 16 3.10๐ธ โ 152.92E โ 15 1.95E โ 15 1.72E โ 15] ๐ = 50 ๐ฟ ๐ญ (Step 1, Eq 2) ๐ญ ๐ (Step 2, Eq 3) [ 3E โ 8 โ3.84E โ 83E โ 8 โ3.84E โ 8 1.28E โ 9 5.4E โ 8 , 8.31E โ 8 3.18E โ 8 โ2.78E โ 8 8E โ 9 1.92E โ 8 โ3.98E โ 8, โ8.6E โ 9 2.9E โ 8โ8.6E โ 9 2.9E โ 8 9.4E โ 8 3E โ 8 ] [ 2.4 โ 15 7.96๐ธ โ 15 8.24๐ธ โ 162.4๐ธ โ 15 8.36๐ธ โ 16 8.24๐ธ โ 152.9๐ธ โ 15 1.95๐ธ โ 15 9.92๐ธ โ 15] [2.4E โ 15 8.36E โ 16 8.24E โ 16] ๐ญ ๐ (Step 4, Eq 7) ๐ญ ๐ ๐ (Step 6, Eq 10) ๐ญ (Step 8, Eq 12) Final Solution ๐ โ [[2.67E โ 159.29E โ 15] [2.77๐ธ โ 151.15๐ธ โ 14] [3.82E โ 161.29E โ 14][4.01E โ 158.98E โ 15] [1.24E โ 141.15E โ 14] [8.41E โ 158.02E โ 15]] [ [7.15E โ 156.48E โ 156.32E โ 151.33E โ 14] [1.11E โ 155.92E โ 157.86E โ 151.61E โ 14] [9.89๐ธ โ 159.98๐ธ โ 151.32๐ธ โ 143.60๐ธ โ 14][2.93๐ธ โ 151.65๐ธ โ 141.64๐ธ โ 155.38๐ธ โ 15] [2.05E โ 154.46E โ 151.69E โ 151.11E โ 14] [2.44E โ 151.51E โ 142.53E โ 154.77E โ 15]] [2.67๐ธ โ 15 1.11๐ธ โ 15 3.82๐ธ โ 161.64๐ธ โ 15 1.69๐ธ โ 15 2.44๐ธ โ 152.4E โ 15 8.36E โ 15 8.24E โ 16] References Civicioglu P (2013) Backtracking search optimization algorithm for numerical optimization problems. Appl Math Comput219:8121 โ Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186(2/4), 311 โ
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