# A Selection Process for Genetic Algorithm Using Clustering Analysis
## 3 Problem Definition
\(f_{4}(x)=-20\exp\left[-0.2\sqrt{\frac{1}{D}\sum_{i=1}^{D}x_{i}^{2}}\right]-\exp \left[\frac{1}{D}\sum_{i=1}^{D}\cos(2\pi x_{i})\right]+20+e\); \(F_{5}(x)=f_{4}(x-o_{5})\)
6. \(f_{5}(x)=\sum_{i=1}^{D}\frac{x_{i}^{2}}{4000}-\prod_{i=1}^{D}\cos\left(\frac{x _{i}}{\sqrt{i}}\right)+1\); \(F_{6}(x)=f_{5}(x-o_{6})\)
7. \(f_{6}(x)=\sum_{i=1}^{D-1}\left[100\left(x_{i}^{2}-i_{i+1}\right)^{2}+\left(x_{ i}-1\right)^{2}\right]\); \(F_{6}(x)=f_{6}\left(M_{7}\left[\frac{2048(x-o_{7})}{20}\right]+1\right)\)

Results of the KGA techniques (KGA\({}_{o}\)-S, KGA\({}_{o}\)-DB and KGA\({}_{f}\)) were taken for \(D\) = 10 and 20 and are compared to those of the standard genetic algorithm GA and the Group Counseling Optimizer (GCO) [61] presented at the IEEE Congress on Evolutionary computation (CEC 2014). In all experiments, common parameters such as population number, maximum generation number and stopping criterion were the same for all algorithms. Population sizes of 50 and 100 were selected for dimensions 10 and 20 respectively. Each experiment is repeated 50 times to obtain the statistical features of the algorithms. A system with an Intel core i7 2.9 GHz processor and 4.096 GB RAM is used for implementing the MATLAB code for the proposed KGA techniques. All algorithms run the same number of fitness evaluations equal to 15,000 for \(D\) = 10 and 20,000 for \(D\) = 20, to ensure a fair comparison. The statistical results of the test problems are shown in Tables 2 and 3. The GCO outperformed the proposed techniques in only one data set (test = 4, \(D\) = 20). In all other cases, the best solution was obtained with either the KGA\({}_{o}\)-S or the KGA\({}_{o}\)-DB. This demonstrates the significant feasibility and efficiency of the proposed techniques over the standard GA. Although the average runtime of each experiment was increased by 10-15%, the KGA techniques ensured a broader and more exhaustive search and prevent premature death of potential solutions. The KGA methods implement an efficient partitioning of the population. They extend the diversity by intensifying the scope of the search process and reducing less favourable solutions. The recombination of two similar solutions will more likely generate a descendant with homogenous chromosomes. The evaluation of the membership probability vector inside the proposed selection process guarantees a more fitting parent selection. In addition, the elitism strategy that results from partitioning the population into a number of clusters ensures that best solutions are always carried forward to the next generation. In fact, rather than obtaining one elite solution, \(K\)-strong optimal solutions are generated in each generation. In the long run, this enhances the exploration of future generations and reduces the possibility of premature convergence at local minima. The latter was recorded with the standard GA in problems 4-7 especially. Unlike the KGA\({}_{f}\), the KGA\({}_{o}\)-S and KGA\({}_{o}\)-DB are designed in such a way that there are no additional parameters to be fine-tuned. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Problem**} & \multicolumn{1}{c}{\( \begin{array}{c}\text{KGA}_{o}\text{ (S Index)}\end{array} \)} & \multicolumn{1}{c}{\(\