# A Selection Process for Genetic Algorithm Using Clustering Analysis
## 3 Problem Definition

Table 3: Comparison of statistical results of four algorithms for test problems 1–7 of dimensions \(D\) = 20. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**Problem**} & \multicolumn{2}{c}{**KGA\({}_{\mathbf{o}}\)**} & \multicolumn{2}{c}{**KGA\({}_{\mathbf{o}}\)**} & \multicolumn{2}{c}{**Genetic**} & \multicolumn{2}{c}{**KGA\({}_{\mathbf{f}}\)**} & \multicolumn{1}{c}{**GCO**} \\ \hline \multirow{3}{*}{4} & Best & \(1.67\times 10^{-3}\) & \(2.36\times 10^{-3}\) & \(1.05\) & \(4.51\times 1 The most frequently used statistical tests to determine significant differences between two computational intelligence algorithms are the \(t\)-test and Wilcoxon signed-ranks test [62]. The later is a non-parametric counterpart of the paired \(t\)-test, which ranks the differences in performances of two algorithms over each data set. In brief, the test omits the signs, and compares the ranks for the positive and the negative differences. The differences are ranked based on their absolute values and in case of ties average ranks are calculated. A Wilcoxon test is used for pairwise comparisons between the following pairs of algorithms: KGA\({}_{0}\) (S index)-KGA\({}_{0}\) (DB index), KGA\({}_{0}\) (S index)-GA, KGA\({}_{0}\) (S index)-KGA\({}_{f}\), KGA\({}_{0}\) (DB index)-GA, KGA\({}_{f}\), GA-KGA\({}_{f}\) for test function seven (refer to Table 4). As we can see, the \(p\)-values obtained by the paired Wilcoxon test indicate that the algorithms behave differently, since all \(p\)-values are less than the level of significance \(\alpha=0.05\). ## 6 Conclusions and Future Work

A \(k\)-means Genetic Selection (KGA) is proposed to solve multimodal function optimization problems. Two different versions of the KGA technique were presented: using a fixed number of clusters \(K\) (KGA\(\beta\)) and via an optimal number \(K_{opt}\) (KGA\({}_{0}\)). In the latter, the optimal number of clusters is determined using two validity indexes: Silhouette and Davies-Bouldin. The KGA techniques are composed of four stages: clustering, membership phase, fitness scaling and selection. Clustering the population aids the search algorithm to preserve a selection pressure throughout the evolution. A membership probability number is assigned to each individual following the \(k\)-mean clustering phase. Fitness scaling converts the membership scores in a range suitable for the selection function which selects the parents of the next generation. The performance of each KGA technique (KGA\({}_{o}\)-S, KGA\({}_{o}\)-DB and KGA\({}_{f}\)) is tested on seven benchmark problems for two separate dimensions of the search spaces \(D=10\) and \(20\). Traditional GAs perform well on single optimum problems but fail to provide multiple solutions. By combining the strength of clustering analysis and genetic search, the proposed KGA techniques permit the evaluation of multimodal functions. The computational results reveal that the proposed selection process is superior to or competitive with the standard genetic algorithm for the problems considered. In the current study, combining the strengths of evolutionary computation and data mining were limited to single-objective optimization problems. Future research could test the performance of KGA techniques in solving constrained optimization problems and/or multiobjective formulations [63]. The stability of the novel selection processes should also be considered in future work. It would be compelling to integrate the KGA processes in further population-based optimization algorithms such as particle swarm optimization (PSO) [64], ant colony optimization (ACO) [65] and firefly algorithm (FA) [66]. Lastly, larger-scale examples should be tested and further research on the impact of GA parameters (such as population size, probabilities of crossover and mutation) on the KGA process will be examined.