## 5 Results and Discussion
### Discussion on PSO Results
Simulation results are presented in Table 4. From these simulation results, it was found that functions having larger dimensions were tougher to optimize and it can be seen from the Table 4 when dimension size id is \(D=20\) and \(D=30\) and our proposed approach KN-PSO shows better result on higher dimensions on other approaches WE-PSO, TO-PSO, standard PSO, H-PSO, and SO-PSO. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Functions**} & \multirow{2}{*}{**DIM \(\times\) Irr**} & **PSO** & **SO\(-\)PSO** & **H\(-\)PSO** & **TO\(-\)PSO** & **WE\(-\)PSO** & **KN\(-\)PSO** \\ \cline{3-8}  & & **Mean** & **Mean** & **Mean** & **Mean** & **Mean** & **Mean** \\ \hline \multirow{4}{*}{F1} & \(10\times 1000\) & \(2.33\times 10^{-74}\) & \(2.74\times 10^{-76}\) & \(3.10\times 10^{-77}\) & \(5.57\times 10^{-78}\) & \(5  
KN-PSO is compared with the other approaches like WE-PSO, TO-PSO, SO-PSO, H-PSO, and standard PSO where each technique true value is presented for comparison with other techniques for the same nature of problems. Standard benchmark functions are presented in the Table 3 and their parameter settings are also shown in the Table 1. Table 4 shows that with dimension D-30, KN-PSO is more superior and outperforms in convergence than the WE-PSO, TO-PSO, standard PSO, SO-PSO, and H-PSO. The comparative analysis can be seen from Table 4 that with smaller dimension size, standard PSO performs well (\(D=10\)); while the size of the dimension increases, KN-PSO outperforms in convergence significantly. Hence, KN-PSO is best for higher dimensions. The experimental results from Table 4 show that KN-PSO outclassed the results of WE-PSO, TO-PSO, SO-PSO, H-PSO, and traditional PSO for all functions. It can be seen that the TO-PSO outperformed the results of the other techniques in all functions on SO-PSO, H-PSO, standard PSO; while in the other approaches, H-PSO performs better on functions f4, f1, f2 for 20D but H-PSO gives overall poor result on higher dimensions and SO-PSO gives slightly better results on functions f8, f9, f15 on 10-D but worst results on larger dimensions. Standard PSO did not provide better results. Figures 7-15 depict that WE-PSO outperforms in simulation results than other approaches for solving the standard benchmark tests functions for dim size \(D=10\), \(D=20\), \(D=30\). To validate the numerical results, mean ranks obtained by Kruskal-Wallis and Friedman tests for KN-PSO, WE-PSO, TO-PSO, SO-PSO, HA-PSO, and standard PSO are given in Table 5.  
For global optimization, the most considerable variety of benchmark problems can be used. All benchmark problems have their own individual abilities and the variety of detailed characteristics of such functions explains the level of complexity for benchmark problems. For the efficiency analysis of the above-mentioned optimization algorithms, Table 3 displays the benchmark problems that are utilized. Table 3 explains the following contents of benchmark problems: name, range, domain, and formulas. In this study, those benchmark problems are incorporated, which have been extensively utilized in the literature for conveying a deep knowledge of the performance related to the above-mentioned optimization algorithms. To measure the effectiveness and robustness of optimization algorithms, benchmark functions are applied. In this study, fifteen computationally-expensive black box functions are applied with their various abilities and traits. The purpose to utilize these benchmark functions is to examine the effectiveness of the above-mentioned proposed approaches. In this section, a comparison among the low discrepancies sequence, methods, is performed with each other with reference to capabilities and efficiency with the help of highdimensional fifteen benchmark functions. Nevertheless, the whole performance of optimization algorithms varies on the basis of setting parameters, and also with other testing criteria.