## 6 Results
### Multimodal-Multi-Optima Problems
\[\chi_{F}^{2}=\frac{12n_{p}}{k(k+1)}\left[\sum_{l=1}^{k}\left(\overline{R}_{i} \right)^{2}\right]-3n_{p}(k+1) \tag{28}\]  
The critical chi-square value \(\chi_{critical}^{2}\) is 5.991 at a significance level \(\alpha=0.05\) and 2 degrees of freedom (\(k-1\)). The computed statistics \(\chi_{F}^{2}\) are 0.775 and 0.824 for \(PR\) and \(SR\), respectively. Since both values are below the critical threshold, \(H_{0}\) cannot be rejected, meaning there is no overall statistically significant difference among the three algorithms. The Friedman test includes 21 evaluations in which all three algorithms achieved perfect performance. Thus, excluding these data results in 30 remaining evaluations. S-EPSO, ANDE and NMMSO achieve the best performance on 14, 9 and 10 of them, respectively. These numbers include three cases of equal responses by S-EPSO and ANDE on F\({}_{16}\). Although a Friedman test applied to this reduced dataset would still not reject \(H_{0}\), the following analysis provides targeted descriptions and comparisons. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**S-EPSO**} & \multicolumn{3}{c}{**ANDE**} & \multicolumn{3}{c}{**NMMSO**} \\ \hline  & **Accuracy** & **PR** & **SR** & **PR** & **SR** & **PR** & **SR** \\  & **Level \(\epsilon\)** & & & & & & \\ \hline \multirow{4}{*}{F\({}_{16}\)} & \(10^{-1}\) & 0.667 & 0.000 & **–** & **–** & **1.000** & 0.000 \\  & \(10^{-2}\) & 0.667 & 0.000 & **–** & **–** & **0.703** & 0.000 \\  & \(10^{-3}\) & **0.667** & 0.000 & **0.667** & 0.000 & 0.653 & 0.000 \\  & \(10^{-4}\) & **0.667** & 0.000 & **0.667** & 0.000 & 0.653 & 0.000 \\  & \(10^{-5}\) & **0.667** & 0.000 & **0.667** & 0.000 & 0.633 & 0.000 \\ \hline \multirow{4}{*}{F\({}_{17}\)} & \(10^{-1}\) & **0.728** & 0.000 & **–** & **–** & 0.553 & 0.000 \\  & \(10^{-2}\) & **0.625** & 0.000 & **–** & **–** & 0.548 & 0.000 \\ \cline{1-1}  & \(10^{-3}\) & **0.615** & 0.000 & 0.397 & 0.000 & 0.543 & 0.000 \\ \cline{1-1}  & \(10^{-4}\) & **0.588** & 0.000 & 0.397 & 0.000 & 0.538 & 0.000 \\ \cline{1-1}  & \(10^{-5}\) & **0.515** & 0.000 & 0.397 & 0.000 & 0.238 & 0.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._To complete the description, Table 5 presents pairwise comparisons of the S-EPSO performance. These results amalgamate 136 comparisons conducted across the 17 functions evaluated at 3 and 5 accuracy levels for the S-EPSO/ANDE and the S-EPSO/NMMSO pairs, respectively. The presented amounts are the number of times each side outperforms the other, as well as the number of tied results. Table 5 also reports the corresponding percentages. The best result is again indicated in boldface. This evaluation confirms that, across the considered benchmark functions, the overall S-EPSO performance exceeds that of the two other algorithms, demonstrating its competitive capacities. On the other hand, when considering the 2D version of the F\({}_{6}\) Shubert and the F\({}_{7}\) Vincent functions, the ability of S-EPSO to locate the optima with very high precision decreases slightly as compared to the two other algorithms. In reality, the S-EPSO performance remains very good but is below that of the two reference state-of-the-art algorithms. The difference also increases when considering F\({}_{8}\) and F\({}_{9}\), the 3D versions of the same functions. In these cases, S-EPSO does not offer competitive responses. This accuracy loss may be attributed to the nature of the algorithm, where the particles do not belong to any restrictive niche. These two specific landscapes concentrate many optima in small areas of their domains. To illustrate this description, Figure 14 shows the optimal positions of F\({}_{8}\), the 3D version of the Shubert function. First, to locate optima, S-EPSO necessitates a sufficient number of particles able to move and reach their positions. Later on, the medium haziness \(\gamma^{*}\) controls the ability of the algorithm to preserve the found locations.