## 3 Methodology
### The Well Sequence
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sr.\#** & **Function Name** & **Objective Function** & **Search Space** & **Optimal Value** \\ \hline
01 & Sphere & \(Minf(x)=\sum\limits_{i=1}^{D}x_{i}^{2}\) & \(-5.12\leq x_{i}\leq 5.12\) & 0 \\
02 & Rastrigin & \(Minf(x)=100+\sum\limits_{i=1}^{D}\left[x_{i}^{2}-10\cos(2\pi x)\right]_{i}\) & \(-5.12\leq x_{i}\leq 5.12\) & 0 \\
03 & Axis parallel & \(Minf(x)=\sum\limits_{i=1}^{D}\left(x_{i}^{2}\right)\) & \(-5.12\leq x_{i}\leq 5.12\) & 0 \\
04 & Rotated hyper ellipsoid & \(Minf(x)=\sum\limits_{i=1}^{D}\sum\limits_{i=1}^{D}\left(x_{i}^{2}\right)\) & \(-65.536\leq x_{i}\leq 65.536\) & 0 \\
05 & Moved Axis & \(Minf(x)=\sum\limits_{i=1}^{D}\xi_{i}x_{i}^{2}\) & \(-5.12\leq x_{i}\leq 5.12\) & 0 \\
06 & Sum of different power & \(Minf(x)=\sum\limits_{i=1}^{D}\left[x_{i}\right]\) & \(-1\leq x_{i}\leq 1\) & 0 \\
07 & ChungReynolds & \(Minf(x)=\left(\sum\limits_{i=1}^{D}x_{i}^{2}\right)^{2}\) & \(-100\leq x_{i}\leq 100\) & 0 \\
08 & [PERSON] & \(Minf(x)=\sum\limits_{i=1}^{D}x_{i}^{6}\left(2+\sin\frac{1}{x}\right)\) & \(-1\leq x_{i}\leq 1\) & 0 \\
09 & [PERSON] & \(Minf(x)=05+\sin^{2}\left(x_{i}^{2}+x_{i}^{2}\right)^{2}-0.5\) & \(-100\leq x_{i}\leq 100\) & 0 \\
10 & [PERSON]\_[PERSON] & \(Minf(x)=\sum\limits_{i=1}^{D}x_{i}^{4}\) & \(-100\leq x_{i}\leq 100\) & 0 \\
11 & [PERSON] & \(Minf(x)=\left(\sum\limits_{i=1}^{D}x_{i}^{2}\right)^{2}\) & \(-100\leq x_{i}\leq 100\) & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Function table with characteristics. ## 5 Simulation Results and Discussion

### Results and Graphs on PSO Approaches

The comparative analysis can be seen from Table 4 that with a smaller dimension size, standard PSO performs well (D = 10), but as the size of dimension increases, KN-PSO outperforms in convergence significantly. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Functions**} & \multirow{2}{*}{**DIM \(\times\) Irr**} & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**SO-PSO**} & \multicolumn{1}{c}{**H-PSO**} & \multicolumn{1}{c}{**TO-PSO**} & \multicolumn{1}{c}{**WE-PSO**} & \multicolumn{1}{c}{**KN-PSO**} \\ \cline{3-8}  & & **Mean** & **Mean** & **Mean** & **Mean** & **Mean** & **Mean** & **Mean** \\ \hline \multirow{3}{*}{F1} & 10 \



### Friedman and Kruskal-Wallis Test on PSO Approaches

Mean ranks obtained by Kruskal-Wallis and [PERSON] test are given in the Table 5.

### Discussion on PSO Results

To measure the execution of proposed approaches WELL-based PSO (WE-PSO), Torus-based PSO (TO-PSO), and Knuth-based PSO (KN-PSO), on a group of 16 nonlinear benchmark test functions, have been utilized to make the comparison of WE-PSO, TO-PSO, and KN-PSO with standard PSO, SO-PSO, and H-PSO. These functions are generally applied to investigate the performance of any technique. Hence in our study, we used it to examine the optimization outcomes of the quasirandom-based approach of WE-PSO, TO-PSO, KN-PSO, SO-PSO, and H-PSO. ### Discussion

The purpose of this study continues to observe whereby the unique characteristics of experimental results rely on dimensions of these standard benchmark functions. In the experiments, three simulation experiments were performed, where the following features of the quasirandom sequence were observed:

1. Effect of using different initializing PSO approaches
2. Effect of using different dimensions for problems
3. A comparative analysis

The objective of this study to find the most suitable initializing method for the PSO, and during the first experiment, the proposed WE-PSO, TO-PSO, and KN-PSO with other approached SO-PSO, H-PSO, and standard PSO were investigated. The objective of second simulation is to find the nature of dimension regarding standard function optimization. Lastly, the simulation results of WE-PSO, TO-PSO, and KN-PSO were compared with standard PSO, SO-PSO, and H-PSO. In the rest of the paper, simulation results were discussed in detail.