## 3 Methodology
### The Well Sequence
\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{3}{*}{F12} & \(10\times 1000\) & \(1.13\times 10^{-01}\) & \(1.67\times 10^{-02}\) & \(2.28\times 10^{-02}\) & \(4.78\times 10^{-03}\) & \(2.89\times 10^{-03}\) & \(2.739\times 10^{-12}\) \\  & \(20\times 2000\) & \(1.39\times 10^{01}\) & \(5.03\times 10^{-00}\) & \(2.95\times 10^{-00}\) & \(1.28\times 10^{-00}\) & \(1.67\times 10^{-00}\) & \(7.819\times 10^{-00}\) \\  & \(30\times 3000\) & \(7.45\times 10^{00}\) & \(1.22\times 10^{-01}\) & \(8.74\times 10^{-00}\) & \(2.94\times 10^{-00}\) & \(4.94\times 10^{-00}\) & \(2.239\times 10^{-01}\) \\  & \(10\times 1000\) & \(8.04\times 10^{-26}\) & \(8.01\times 10^{-27}\) & \(3.59\times 10^{-27}\) & \(1.24\times 10^{-27}\) & \(1.41\times 10^{-27}\) & \(0.0000\times 10^{-00}\) \\  & \(20\times 2000\) & \(1.42\times 10^{-08}\) & \(2.64\times 10^{-11}\) & \(3.29\times 10^{-10}\) & \(2.99\times 10^{-10}\) & \(2.14\times 10^{-12}\) & \(0.0000\times 10^{-00}\) \\  & \(30\times 3000\) & \(6.20\times 10^{-03}\) & \(1.41\times 10^{-03}\) & \(9.36\times 10^{-03}\) & \(1.12\times 10^{-03}\) & \(1.41\times 10^{-03}\) & \(0.0000\times 10^{-00}\) \\  & \(10\times 1000\) & \(3.62\times 10^{-38}\) & \(3.62\times 10^{-38}\) & \(5.92\times 10^{-36}\) & \(6.92\times 10^{-39}\) & \(1.95\times 10^{-38}\) & \(7.78286\times 10^{-197}\) \\ F14 & \(20\times 2000\) & \(6.27\times 10^{-10}\) & \(1.38\times 10^{-09}\) & \(7.91\times 10^{-13}\) & \(2.49\times 10^{-12}\) & \(1.17\times 10^{-13}\) & \(6.6163\times 10^{-12}\) \\  & \(30\times 3000\) & \(2.56\times 10^{-06}\) & \(4.80\times 10^{-01}\) & \(1.34\times 10^{-06}\) & \(5.40\times 10^{-11}\) & \(4.88\times 10^{-09}\) & \(9.3032\times 10^{-06}\) \\  & \(10\times 1000\) & \(1.10\times 10^{-294}\) & \(3.19\times 10^{-301}\) & \(2.78\times 10^{-307}\) & \(1.94\times 10^{-307}\) & \(3.21\times 10^{-308}\) & \(6.26612\times 10^{-138}\) \\ F15 & \(20\times 2000\) & \(6.16\times 10^{-271}\) & \(5.09\times 10^{-26}\) & \(3.74\times 10^{-200}\) & \(1.60\times 10^{-26}\) & \(4.85\times 10^{-268}\) & \(1.29033\times 10^{-25}\) \\  & \(30\times 3000\) & \(3.08\times 10^{-207}\) & \(1.04\times 10^{-200}\) & \(8.12\times 10^{-209}\) & \(2.34\times 10^{-215}\) & \(3.06\times 10^{-212}\) & \(2.27\times 10^{-06}\) \\  & \(10\times 1000\) & \(5.4835385\) & \(5.299\times 10^{-27}\) & \(3.3074\times 10^{-16}\) & \(1.224803\) & \(8.3354\times 10^{-07}\) & \(2.26476\times 10^{-27}\) \\ F16 & \(20\times 2000\) & \(83.467\) & \(1.6344\) & \(0.18037\) & \(49.16841\) & \(5.1322\) & \(7.17014\times 10^{-72}\) \\  & \(30\times 3000\) & \(265.90708\) & \(282.1864\) & \(45.0408\) & \(133.9679\) & \(67.0301\) & \(5.45179\times 10^{-251}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(Cont.\) \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Friedman Value** & **Kruskal–Wallis** \\ \hline PSO & 39.09 & 39.33 \\ SO-PSO & 37.47 & 38.39 \\ H-PSO & 38.50 & 38.91 \\ TO-PSO & 41.79 & 42.67 \\ WE-PSO & 41.88 & 42.50 \\ KN-PSO & 18.24 & 23.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean ranks obtained by Kruskal–Wallis and Friedman test for all PSO-based approaches. We can see that majority of the figures contains better convergence curve for KN-PSO on functions F1, F2, F3, F4, F5, F6, F17, F8, F9, F10, F11, F12, F13, F14, and F15 over WE-PSO, TO-PSO, H-PSO, SO-PSO, and standard PSO on all dimensions comprehensively. The other proposed approach TO-PSO provides better results over WE-PSO on functions F1, F2, F4, F6, F17, F8, F9, F10, F14, F15, and F16 and beats on all functions for PSO, SO-PSO, and TO-PSO. Effect of Using Different Initializing PSO Approaches

In this simulation, PSO is initialized with the WELL sequence (WE-PSO), Torus Sequence (TO-PSO), and Knuth Sequence (KN-PSO) instead of uniform distribution.