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
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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