## 6 Results
### Multimodal-Multi-Optima Problems
\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  & \multicolumn{2}{c}{**Accuracy**} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} & \multirow{2}{*}{PR} & \multirow{2}{*}{SR} \\  & \multicolumn{2}{c}{**Level \(\epsilon\)**} & & & & & & \\ \hline \multirow{5}{*}{F\({}_{1}\)} & \(10^{-1}\) & **1.000** & **1.000** & **–** & **–** & **1.000** & **1.000** \\  & \(10^{-2}\) & **1.000** & **1.000** & **–** & **–** & **1.000** & **1.000** \\  & \(10^{-3}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\  & \(10^{-4}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\  & \(10^{-5}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\ \hline \multirow{5}{*}{F\({}_{2}\)} & \(10^{-1}\) & **1.000** & **1.000** & **–** & **–** & **1.000** & **1.000** \\  & \(10^{-2}\) & **1.000** & **1.000** & **–** & **–** & **1.000** & **1.000** \\  & \(10^{-3}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\  & \(10^{-4}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\  & \(10^{-5}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\ \hline \multirow{5}{*}{F\({}_{3}\)} & \(10^{-1}\) & **1.000** & **1.000** & **–** & **–** & **1.000** & **1.000** \\  & \(10^{-2}\) & **1.000** & **1.000** & **–** & **–** & **1.000** & **1.000** \\  & \(10^{-3}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\  & \(10^{-4}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\  & \(10^{-5}\) & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** & **1.000** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results. Figure 13: F\({}_{13}\) landscape: (a) 2D plot and (b) 2D contour plot. It is worth mentioning here that, since Refs. [19; 20] already include the results of several other state-of-the-art algorithms in their validation operations, and because they demonstrate that ANDE and NMMSO perform better than these algorithms over the benchmark of [17], to maximize the validation acuteness, the present analysis voluntarily restricts the comparison to the Refs. [19; 20] results. The results in Table 4 indicate that all three algorithms successfully identify all optima for functions F\({}_{1}\) to F\({}_{5}\), and of F\({}_{10}\) and F\({}_{11}\). Summing up the best responses of each algorithm (highlighted in bold) for the three highest accuracy levels (\(\epsilon\leq 10^{-3}\)) shows that S-EPSO, ANDE and NMMSO give perfect evaluations or outperform the other two, 35, 30 and 31 times out of 51, respectively. Friedman test allows a first paired comparison of the algorithms performance across these multiple test results. The Friedman test identifies any significant difference. The null hypothesis (\(H_{0}\)) is here: _The tree algorithms give equivalent performance_; while the alternative hypothesis (\(H_{1}\)) is: _At least one of them exhibits a different performance_. Table 4 may be reformatted to present the algorithm ranking based on \(SR\) and \(PR\). The best-performing algorithm ranks first (\(R=1\)), whereas ties receive average ranks. For instance, if all three algorithms tie, each receives \(R=2\); if two algorithms tie for the first position, they both receive \(R=1.5\), and the remaining one \(R=3\), and if two tie for second, they both receive \(R=2.5\). The first step establishes the average ranking \(\overline{R}_{i}\) for each algorithm \(i\), using \(\overline{R}_{i}=\frac{1}{n_{p}}\sum_{j=1}^{n_{p}}R_{ij}\), where \(n_{p}\) represents the number of considered problems, here \(n_{p}=51\) (i.e., 17 functions at 3 precision levels). Equation (28) formulates the Friedman test, with \(k=3\) representing the number of algorithms.