A Selection Process for Genetic Algorithm Using Clustering Analysis
Abstract
:1. Introduction
2. Literature Review
3. Problem Definition
Algorithm 1: Given the function $f(\overrightarrow{\mathbf{x}})$, $\overrightarrow{\mathrm{x}}={\left({\mathrm{x}}_{1},\dots ,{\mathrm{x}}_{\mathrm{d}}\right)}^{\mathrm{T}}$ to minimize 

4. The Proposed Algorithm
4.1. KGA_{f}
 1.
 Choose an initial partition with K clusters.
 2.
 Generate a new partition by assigning each pattern to its nearest cluster centroid.
 3.
 Compute new cluster centroids.
 4.
 If a convergence criterion is not met, repeat steps 2 and 3.
 5.
 Clustering the population by the kmeans algorithm
 6.
 Computing the membership probability (MP) vector (Equations (2)–(4))
 7.
 Fitness scaling of MP
 8.
 Selection of the parents for recombination.
 The sum of the membership probability scores of a given cluster j of size m_{j} is equal to $\frac{{m}_{j}}{P}$. Consequently, clusters with more individuals will be attributed a larger probability sum.
 An individual with a lower fitness value $f\left({x}_{i}\right)$ inside a cluster of size m_{j} is awarded a higher MP score. This is translated in the $\frac{{S}_{j}f\left({x}_{i}\right)}{{S}_{j}}$ term, thus allocating fitter solutions a higher probability of selection.
 In order to reduce the probability of recombination between individuals from the same cluster, thus avoiding local optimal traps, fitter individuals in smaller clusters are awarded a higher MP score. This is the direct effect of $\frac{{m}_{j}}{{m}_{j}1}$ term.
 The sum of all membership probability scores is equal to one.
4.2. KGA_{o}
 External criteria: evaluation of the clustering algorithm results is based on previous knowledge about data.
 Internal criteria: clustering results are evaluated using a mechanism that takes into account the vectors of the data set themselves and prior information from the data set is not required.
 Relative criteria: aim to evaluate a clustering structure by comparing it to other clustering schemes.
 In how many clusters can the population be partitioned to?
 Is there a better “optimal” partitioning for our evolving population of chromosomes?
 Compatible Cluster Merging (CCM): starting with a large number of clusters, and successively reducing the number by merging clusters which are similar (compatible) with respect to a similarity criterion.
 Validity Indices (VI’s): clustering the data for different values of K, and using validity measures to assess the obtained partitions.
5. Numerical Simulations
 ${f}_{1}\left(x\right)={{\displaystyle \sum}}_{i=1}^{D}{x}_{i}^{2}$ ${F}_{1}\left(x\right)={f}_{1}\left(x{o}_{1}\right)$
 ${f}_{2}\left(x\right)={{\displaystyle \sum}}_{i=1}^{D}i{x}_{i}^{2}$ ${F}_{2}\left(x\right)={f}_{2}\left(x{o}_{2}\right)$
 ${F}_{3}\left(x\right)={f}_{2}\left({M}_{3}\left[x{o}_{3}\right]\right)$
 ${f}_{3}\left(x\right)={{\displaystyle \sum}}_{i=1}^{D}{\left{x}_{i}+0.5\right}^{2};{F}_{3}\left(x\right)={f}_{3}\left(x{o}_{4}\right)$
 ${f}_{4}\left(x\right)=20\mathrm{exp}\left[0.2\sqrt{\frac{1}{D}{{\displaystyle \sum}}_{i=1}^{D}{x}_{i}^{2}}\right]\mathrm{exp}\left[\frac{1}{D}{{\displaystyle \sum}}_{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)\right]+20+e$; ${F}_{5}\left(x\right)={f}_{4}\left(x{o}_{5}\right)$
 ${f}_{5}\left(x\right)={{\displaystyle \sum}}_{i=1}^{D}\frac{{x}_{i}^{2}}{4000}{{\displaystyle \prod}}_{i=1}^{D}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$; ${F}_{6}\left(x\right)={f}_{5}\left(x{o}_{6}\right)$
 ${f}_{6}\left(x\right)={{\displaystyle \sum}}_{i=1}^{D1}\left[100{\left({x}_{i}^{2}{x}_{i+1}\right)}^{2}+{\left({x}_{i}1\right)}^{2}\right];{F}_{6}\left(x\right)={f}_{6}\left({M}_{7}\left[\frac{2.048\left(x{o}_{7}\right)}{20}\right]+1\right)$
6. Conclusions and Future Work
Acknowledgments
Author Contributions
Conflicts of Interest
References
 Zhang, M.X.; Zhang, B.; Zheng, Y.J. BioInspired MetaHeuristics for Emergency Transportation Problems. Algorithms 2014, 7, 15–31. [Google Scholar] [CrossRef]
 Fister, I.; Yang, X.S.; Fister, I.; Brest, J.; Fister, D. A brief review of natureinspired algorithms for optimization. arXiv, 2013; arXiv:1307.4186. [Google Scholar]
 Yang, X.S. NatureInspired Optimization Algorithms; Elsevier: Amsterdam, The Netherlands, 2014. [Google Scholar]
 Nelder, J.A.; Mead, R. A simplex method for function minimization. Comput. J. 1965, 7, 308–313. [Google Scholar] [CrossRef]
 Hooke, R.; Jeeves, T.A. “Direct Search” Solution of Numerical and Statistical Problems. J. ACM 1961, 8, 212–229. [Google Scholar] [CrossRef]
 Li, Z.Y.; Yi, J.H.; Wang, G.G. A new swarm intelligence approach for clustering based on krill herd with elitism strategy. Algorithms 2015, 8, 951–964. [Google Scholar] [CrossRef]
 Krishna, K.; Murty, M.N. Genetic Kmeans algorithm. IEEE Trans. Syst. Man Cybern. B Cybern. 1999, 29, 433–439. [Google Scholar] [CrossRef] [PubMed]
 Wang, G.; Liu, Y.; Xiong, C. An optimization clustering algorithm based on texture feature fusion for color image segmentation. Algorithms 2015, 8, 234–247. [Google Scholar] [CrossRef]
 Sarkar, M.; Yegnanarayana, B.; Khemani, D. A clustering algorithm using an evolutionary programmingbased approach. Pattern Recognit. Lett. 1997, 18, 975–986. [Google Scholar] [CrossRef]
 Cura, T. A particle swarm optimization approach to clustering. Expert Syst. Appl. 2012, 39, 1582–1588. [Google Scholar] [CrossRef]
 Das, S.; Abraham, A.; Konar, A. Automatic kernel clustering with a multielitist particle swarm optimization algorithm. Pattern Recognit. Lett. 2008, 29, 688–699. [Google Scholar] [CrossRef]
 Yang, F.; Sun, T.; Zhang, C. An efficient hybrid data clustering method based on Kharmonic means and Particle Swarm Optimization. Expert Syst. Appl. 2009, 36, 9847–9852. [Google Scholar] [CrossRef]
 Jiang, H.; Yi, S.; Li, J.; Yang, F.; Hu, X. Ant clustering algorithm with Kharmonic means clustering. Expert Syst. Appl. 2010, 37, 8679–8684. [Google Scholar] [CrossRef]
 Shelokar, P.; Jayaraman, V.K.; Kulkarni, B.D. An ant colony approach for clustering. Anal. Chim. Acta 2004, 509, 187–195. [Google Scholar] [CrossRef]
 Zhang, C.; Ouyang, D.; Ning, J. An artificial bee colony approach for clustering. Expert Syst. Appl. 2010, 37, 4761–4767. [Google Scholar] [CrossRef]
 Maulik, U.; Mukhopadhyay, A. Simulated annealing based automatic fuzzy clustering combined with ANN classification for analyzing microarray data. Comput. Oper. Res. 2010, 37, 1369–1380. [Google Scholar] [CrossRef]
 Selim, S.Z.; Alsultan, K. A simulated annealing algorithm for the clustering problem. Pattern Recognit. 1991, 24, 1003–1008. [Google Scholar] [CrossRef]
 Sung, C.S.; Jin, H.W. A tabusearchbased heuristic for clustering. Pattern Recognit. 2000, 33, 849–858. [Google Scholar] [CrossRef]
 Hall, L.O.; Ozyurt, I.B.; Bezdek, J.C. Clustering with a genetically optimized approach. IEEE Trans. Evolut. Comput. 1999, 3, 103–112. [Google Scholar] [CrossRef]
 Cowgill, M.C.; Harvey, R.J.; Watson, L.T. A genetic algorithm approach to cluster analysis. Comput. Math. Appl. 1999, 37, 99–108. [Google Scholar] [CrossRef]
 Maulik, U.; Bandyopadhyay, S. Genetic algorithmbased clustering technique. Pattern Recognit. 2000, 33, 1455–1465. [Google Scholar] [CrossRef]
 Tseng, L.Y.; Yang, S.B. A genetic approach to the automatic clustering problem. Pattern Recognit. 2001, 34, 415–424. [Google Scholar] [CrossRef]
 Babu, G.P.; Murty, M.N. A nearoptimal initial seed value selection in kmeans means algorithm using a genetic algorithm. Pattern Recognit. Lett. 1993, 14, 763–769. [Google Scholar] [CrossRef]
 Agustı, L.; SalcedoSanz, S.; JiménezFernández, S.; CarroCalvo, L.; Del Ser, J.; PortillaFigueras, J.A. A new grouping genetic algorithm for clustering problems. Expert Syst. Appl. 2012, 39, 9695–9703. [Google Scholar] [CrossRef]
 He, H.; Tan, Y. A twostage genetic algorithm for automatic clustering. Neurocomputing 2012, 81, 49–59. [Google Scholar] [CrossRef]
 Maulik, U.; Bandyopadhyay, S.; Mukhopadhyay, A. Multiobjective Genetic Algorithms for Clustering: Applications in Data Mining and Bioinformatics; Springer: Berlin, Germany, 2011. [Google Scholar]
 Razavi, S.H.; Ebadati, E.O.M.; Asadi, S.; Kaur, H. An efficient grouping genetic algorithm for data clustering and big data analysis. In Computational Intelligence for Big Data Analysis; Springer: Berlin, Germany, 2015; pp. 119–142. [Google Scholar]
 Krishnasamy, G.; Kulkarni, A.J.; Paramesran, R. A hybrid approach for data clustering based on modified cohort intelligence and Kmeans. Expert Syst. Appl. 2014, 41, 6009–6016. [Google Scholar] [CrossRef]
 Popat, S.K.; Emmanuel, M. Review and comparative study of clustering techniques. Int. J. Comput. Sci. Inf. Technol. 2014, 5, 805–812. [Google Scholar]
 Mann, A.K.; Kaur, N. Survey paper on clustering techniques. Int. J. Sci. Eng. Technol. Res. 2013, 2, 803–806. [Google Scholar]
 Jain, A.K.; Maheswari, S. Survey of recent clustering techniques in data mining. Int. J. Comput. Sci. Manag. Res. 2012, 3, 72–78. [Google Scholar]
 Latter, B. The island model of population differentiation: A general solution. Genetics 1973, 73, 147–157. [Google Scholar] [PubMed]
 Qing, L.; Gang, W.; Zaiyue, Y.; Qiuping, W. Crowding clustering genetic algorithm for multimodal function optimization. Appl. Soft Comput. 2008, 8, 88–95. [Google Scholar] [CrossRef]
 Sareni, B.; Krahenbuhl, L. Fitness sharing and niching methods revisited. IEEE Trans. Evolut. Comput. 1998, 2, 97–106. [Google Scholar] [CrossRef][Green Version]
 Goldberg, D.E.; Richardson, J. Genetic algorithms with sharing for multimodal function optimization. In Proceedings of the Second International Conference on Genetic Algorithms on Genetic Algorithms and their Applications, Nagoya, Japan, 20–22 May 1996; Lawrence Erlbaum: Hillsdale, NJ, USA, 1987; pp. 41–49. [Google Scholar]
 Pétrowski, A. A clearing procedure as a niching method for genetic algorithms. In Proceedings of the IEEE International Conference on Evolutionary Computation, Nagoya, Japan, 20–22 May 1996; pp. 798–803. [Google Scholar]
 Gan, J.; Warwick, K. A genetic algorithm with dynamic niche clustering for multimodal function optimisation. In Artificial Neural Nets and Genetic Algorithms; Springer: Berlin, Germany, 1999; pp. 248–255. [Google Scholar]
 Yang, S.; Li, C. A clustering particle swarm optimizer for locating and tracking multiple optima in dynamic environments. IEEE Trans. Evolut. Comput. 2010, 14, 959–974. [Google Scholar] [CrossRef]
 Blackwell, T.; Branke, J. Multiswarm optimization in dynamic environments. In Workshops on Applications of Evolutionary Computation; Springer: Berlin, Germany, 2004; pp. 489–500. [Google Scholar]
 Li, C.; Yang, S. A general framework of multipopulation methods with clustering in undetectable dynamic environments. IEEE Trans. Evolut. Comput. 2012, 16, 556–577. [Google Scholar] [CrossRef]
 Li, C.; Yang, S. A clustering particle swarm optimizer for dynamic optimization. In Proceedings of the IEEE Congress on Evolutionary Computation, Trondheim, Norway, 18–21 May 2009; pp. 439–446. [Google Scholar]
 Kennedy, J. Stereotyping: Improving particle swarm performance with cluster analysis. In Proceedings of the IEEE Congress on Evolutionary Computation, La Jolla, CA, USA, 16–19 July 2000; pp. 1507–1512. [Google Scholar]
 Agrawal, S.; Panigrahi, B.; Tiwari, M.K. Multiobjective particle swarm algorithm with fuzzy clustering for electrical power dispatch. IEEE Trans.Evolut. Comput. 2008, 12, 529–541. [Google Scholar] [CrossRef]
 Zhang, J.; Chung, H.S.H.; Lo, W.L. Clusteringbased adaptive crossover and mutation probabilities for genetic algorithms. IEEE Trans. Evolut. Comput. 2007, 11, 326–335. [Google Scholar] [CrossRef]
 Zhang, X.; Tian, Y.; Cheng, R.; Jin, Y. A Decision Variable ClusteringBased Evolutionary Algorithm for Largescale Manyobjective Optimization. IEEE Trans. Evolut. Comput. 2016. [Google Scholar] [CrossRef]
 Zhang, H.; Zhou, A.; Song, S.; Zhang, Q.; Gao, X.Z.; Zhang, J. A selforganizing multiobjective evolutionary algorithm. IEEE Trans. Evolut. Comput. 2016, 20, 792–806. [Google Scholar] [CrossRef]
 Vattani, A. The Hardness of KMeans Clustering in the Plane. 2009. Available online: https://cseweb.ucsd.edu/~avattani/papers/kmeans_hardness.pdf (accessed on 1 November 2017).
 Holland, J.H. Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence; U Michigan Press: Ann Arbor, MI, USA, 1975. [Google Scholar]
 Jain, A.K. Data clustering: 50 years beyond Kmeans. Pattern Recognit. Lett. 2010, 31, 651–666. [Google Scholar] [CrossRef]
 Aggarwal, C.C.; Reddy, C.K. Data Clustering: Algorithms and Applications; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
 Jain, A.K.; Murty, M.N.; Flynn, P.J. Data clustering: A review. ACM Comput. Surv. (CSUR) 1999, 31, 264–323. [Google Scholar] [CrossRef]
 Xu, R.; Wunsch, D. Survey of clustering algorithms. IEEE Trans. Neural Netw. 2005, 16, 645–678. [Google Scholar] [CrossRef] [PubMed]
 Steinhaus, H. Sur la division des corp materiels en parties. Bull. Acad. Pol. Sci. 1956, 1, 801. [Google Scholar]
 Drineas, P.; Frieze, A.; Kannan, R.; Vempala, S.; Vinay, V. Clustering large graphs via the singular value decomposition. Mach. Learn. 2004, 56, 9–33. [Google Scholar]
 Halkidi, M.; Batistakis, Y.; Vazirgiannis, M. Cluster validity methods: Part I. ACM SIGMM Rec. 2002, 31, 40–45. [Google Scholar] [CrossRef]
 Halkidi, M.; Batistakis, Y.; Vazirgiannis, M. On clustering validation techniques. J. Intell. Inf. Syst. 2001, 17, 107–145. [Google Scholar] [CrossRef]
 Vendramin, L.; Campello, R.J.; Hruschka, E.R. Relative clustering validity criteria: A comparative overview. Stat. Anal. Data Min. 2010, 3, 209–235. [Google Scholar]
 Halkidi, M.; Batistakis, Y.; Vazirgiannis, M. Clustering validity checking methods: Part II. ACM SIGMM Rec. 2002, 31, 19–27. [Google Scholar]
 Rousseeuw, P.J. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 1987, 20, 53–65. [Google Scholar] [CrossRef]
 Davies, D.L.; Bouldin, D.W. A cluster separation measure. IEEE Trans. Pattern Anal. Mach. Intell. 1979, 1, 224–227. [Google Scholar] [PubMed]
 Biswas, S.; Eita, M.A.; Das, S.; Vasilakos, A.V. Evaluating the performance of group counseling optimizer on CEC 2014 problems for computational expensive optimization. In Proceedings of the 2014 IEEE Congress on Evolutionary Computation, Beijing, China, 6–11 July 2014; pp. 1076–1083. [Google Scholar]
 Wilcoxon, F. Individual comparisons by ranking methods. Biom. Bull. 1945, 1, 80–83. [Google Scholar] [CrossRef]
 Chehouri, A.; Younes, R.; Perron, J.; Ilinca, A. A ConstraintHandling Technique for Genetic Algorithms using a Violation Factor. J. Comput. Sci. 2016, 12, 350–362. [Google Scholar] [CrossRef]
 Kennedy, J. Particle swarm optimization. In Encyclopedia of Machine Learning; Springer: Berlin, Germany, 2011; pp. 760–766. [Google Scholar]
 Dorigo, M.; Birattari, M.; Stutzle, T. Ant colony optimization. IEEE Comput. Intell. Mag. 2006, 1, 28–39. [Google Scholar] [CrossRef]
 Yang, X.S. Firefly algorithm, stochastic test functions and design optimisation. Int. J. BioInspir. Comput. 2010, 2, 78–84. [Google Scholar] [CrossRef]
No.  Functions  Search Ranges  ${\mathit{f}}_{\mathit{i}}^{*}={\mathit{f}}_{\mathit{i}}\left({\mathit{x}}^{*}\right)$ 

1  shifted sphere  $\left[20,20\right]$  0 
2  shifted ellipsoid  $\left[20,20\right]$  0 
3  shifted and rotated ellipsoid  $\left[20,20\right]$  0 
4  shifted step  $\left[20,20\right]$  0 
5  shifted Ackley  $\left[32,32\right]$  0 
6  shifted Griewank  $\left[600,600\right]$  0 
7  shifted rotated Rosenbrock  $\left[20,20\right]$  0 
Problem  KGA_{o} (S Index)  KGA_{o} (DB Index)  Genetic Algorithm (GA)  KGA_{f} (K = 10)  GCO  

1  Best  8.81 × 10^{−5}  1.95 × 10^{−4}  2.76 × 10^{−4}  5.06 × 10^{−4}  3.23 
Mean  2.45 × 10^{−3}  5.33 × 10^{−3}  8.29 × 10^{−3}  2.84 × 10^{−2}  1.23 × 10^{1}  
Worst  1.13 × 10^{−2}  5.43 × 10^{−2}  1.08 × 10^{−1}  3.04 × 10^{−1}  2.96 × 10^{1}  
SD  2.63 × 10^{−3}  8.39 × 10^{−3}  1.58 × 10^{−2}  5.02 × 10^{−2}  6.37  
2  Best  2.91 × 10^{−4}  3.34 × 10^{−4}  4.60 × 10^{−4}  2.08 × 10^{−3}  8.46 
Mean  7.12 × 10^{−3}  7.12 × 10^{−3}  4.75 × 10^{−2}  2.09 × 10^{−1}  4.14 × 10^{1}  
Worst  6.27 × 10^{−2}  4.83 × 10^{−2}  7.21 × 10^{−1}  3.62  2.22 × 10^{2}  
SD  1.07 × 10^{−2}  1.06 × 10^{−2}  1.17 × 10^{−1}  6.13 × 10^{−1}  4.61 × 10^{1}  
3  Best  5.55 × 10^{−4}  2.27 × 10^{−4}  5.32 × 10^{−4}  3.23 × 10^{−3}  1.56 × 10^{1} 
Mean  1.01 × 10^{−2}  8.24 × 10^{−3}  5.01 × 10^{−2}  3.12 × 10^{−1}  8.85 × 10^{1}  
Worst  5.42 × 10^{−2}  7.49 × 10^{−2}  2.58 × 10^{−1}  2.49  2.09 × 10^{2}  
SD  1.25 × 10^{−2}  1.46 × 10^{−2}  6.73 × 10^{−2}  5.63 × 10^{−1}  5.54 × 10^{1}  
4  Best  4.00  2.00  1.50 × 10^{−1}  3.00  3.00 
Mean  9.14 × 10^{1}  6.48 × 10^{1}  1.31 × 10^{2}  6.50 × 10^{1}  1.00 × 10^{1}  
Worst  3.86 × 10^{2}  4.19 × 10^{2}  3.83 × 10^{2}  2.05 × 10^{2}  2.70 × 10^{1}  
SD  9.84 × 10^{1}  8.38 × 10^{1}  8.88 × 10^{1}  5.42 × 10^{1}  6.94  
5  Best  1.48 × 10^{−3}  8.42 × 10^{−3}  1.21 × 10^{−2}  4.01 × 10^{−2}  3.92 
Mean  1.50  6.55  5.15  5.62  6.36  
Worst  1.26 × 10^{1}  1.31 × 10^{1}  1.24 × 10^{1}  1.30 × 10^{1}  9.94  
SD  2.28  4.52  3.62  4.12  1.71  
6  Best  4.94 × 10^{−2}  4.97 × 10^{−2}  4.95 × 10^{−2}  5.04 × 10^{−2}  1.24 
Mean  6.41 × 10^{−2}  6.32 × 10^{−2}  6.38 × 10^{−2}  6.96 × 10^{−2}  2.11  
Worst  8.66 × 10^{−2}  8.56 × 10^{−2}  8.14 × 10^{−2}  1.00 × 10^{−1}  4.51  
SD  7.33 × 10^{−3}  6.73 × 10^{−3}  7.56 × 10^{−3}  1.07 × 10^{−2}  6.77 × 10^{−1}  
7  Best  2.02 × 10^{−1}  3.84 × 10^{−3}  1.28  1.48 × 10^{−1}  4.42 × 10^{1} 
Mean  3.77  3.65  3.22  4.60  9.28 × 10^{1}  
Worst  7.77  8.81  5.09  1.55 × 10^{1}  1.80 × 10^{2}  
SD  1.48  2.12  5.95 × 10^{−1}  2.78  3.22 × 10^{1} 
Problem  KGA_{o} (S Index)  KGA_{o} (DB Index)  Genetic Algorithm (GA)  KGA_{f} (K = 10)  GCO  

1  Best  1.67 × 10^{−3}  2.36 × 10^{−3}  1.05  4.51 × 10^{−3}  3.60 × 10^{1} 
Mean  1.22 × 10^{−2}  1.53 × 10^{−2}  1.63  1.16 × 10^{−1}  1.19 × 10^{1}  
Worst  6.32 × 10^{−2}  9.89 × 10^{−2}  2.43  6.42 × 10^{−1}  2.17 × 10^{1}  
SD  1.45 × 10^{−2}  1.87 × 10^{−2}  3.13 × 10^{−1}  1.40 × 10^{−1}  5.88  
2  Best  3.76 × 10^{−3}  5.68 × 10^{−3}  8.99  5.01 × 10^{−2}  7.79 × 10^{1} 
Mean  1.17 × 10^{−1}  1.19 × 10^{−1}  1.02 × 10^{1}  4.10  9.34 × 10^{1}  
Worst  1.16  2.05  1.33 × 10^{1}  2.75 × 10^{1}  1.79 × 10^{2}  
SD  2.17 × 10^{−1}  2.94 × 10^{−1}  1.48  6.05  4.75 × 10^{1}  
3  Best  1.96 × 10^{−1}  5.50 × 10^{−3}  1.49 × 10^{1}  2.27 × 10^{−2}  3.33 
Mean  9.16 × 10^{−1}  3.34 × 10^{−1}  2.45 × 10^{1}  2.04  1.44 × 10^{2}  
Worst  3.19  4.59  3.53 × 10^{1}  1.27 × 10^{1}  2.62 × 10^{2}  
SD  4.29 × 10^{−1}  6.85 × 10^{−1}  4.19  2.55  4.76 × 10^{1}  
4  Best  7.00  7.00  3.19 × 10^{2}  1.70 × 10^{1}  3.00 
Mean  7.91 × 10^{1}  7.52 × 10^{1}  4.89 × 10^{2}  8.83 × 10^{1}  8.48  
Worst  5.37 × 10^{2}  3.32 × 10^{2}  7.23 × 10^{2}  3.17 × 10^{2}  1.40 × 10^{1}  
SD  9.23 × 10^{1}  7.44 × 10^{1}  9.99 × 10^{1}  6.41 × 10^{1}  3.01  
5  Best  1.46  1.32 × 10^{−1}  9.83  1.55  1.28 
Mean  6.75  5.59  1.16  5.18  4.58  
Worst  1.26 × 10^{1}  1.28 × 10^{1}  1.25 × 10^{1}  1.20 × 10^{1}  8.94  
SD  3.69  3.58  7.38 × 10^{−1}  2.47  1.16  
6  Best  1.46  1.32 × 10^{−1}  9.83  1.55  1.28 
Mean  6.75  5.59  1.16  5.18  4.58  
Worst  1.26 × 10^{1}  1.28 × 10^{1}  1.25 × 10^{1}  1.20 × 10^{1}  8.94  
SD  3.69  3.58  7.38 × 10^{−1}  2.47  1.16  
7  Best  1.65 × 10^{−2}  1.02 × 10^{−2}  7.99  5.04 × 10^{−2}  3.10 × 10^{1} 
Mean  1.87 × 10^{1}  1.59 × 10^{1}  2.63 × 10^{1}  1.96 × 10^{1}  1.13 × 10^{2}  
Worst  7.54 × 10^{1}  7.21 × 10^{1}  6.15 × 10^{1}  7.85 × 10^{1}  1.72 × 10^{2}  
SD  2.82 × 10^{1}  2.75 × 10^{1}  1.34 × 10^{1}  2.97 × 10^{1}  2.67 × 10^{1} 
Comparison  R^{+}  R^{−}  Alpha  zScore  pValue 

KGAo (S index)KGAo (DB index)  307  968  0.05  3.190  1.421 × 10^{−3} 
KGA_{o} (S index)GA  155  1120  0.05  4.658  3.198 × 10^{−6} 
KGA_{o} (S index)KGA_{f}  74  1201  0.05  5.440  5.339 × 10^{−8} 
KGA_{o} (DB index)GA  451  824  0.05  3.800  4.181 × 10^{−2} 
KGA_{o} (DB index)KGA_{f}  134  1141  0.05  4.860  1.170 × 10^{−6} 
GAKGA_{f}  1275  0  0.05  6.154  7.557 × 10^{−10} 
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Chehouri, A.; Younes, R.; Khoder, J.; Perron, J.; Ilinca, A. A Selection Process for Genetic Algorithm Using Clustering Analysis. Algorithms 2017, 10, 123. https://doi.org/10.3390/a10040123
Chehouri A, Younes R, Khoder J, Perron J, Ilinca A. A Selection Process for Genetic Algorithm Using Clustering Analysis. Algorithms. 2017; 10(4):123. https://doi.org/10.3390/a10040123
Chicago/Turabian StyleChehouri, Adam, Rafic Younes, Jihan Khoder, Jean Perron, and Adrian Ilinca. 2017. "A Selection Process for Genetic Algorithm Using Clustering Analysis" Algorithms 10, no. 4: 123. https://doi.org/10.3390/a10040123