## 5 Benchmark Testing and Performance Analysis of MRSO
### Comparative Study of MRSO and RSO Using Standard Benchmarks
#### 5.2.2 Multimodal Functions (F8-F16)  
Multimodal functions, with their multiple local optima, are essential for testing an algorithm's exploration capability. Based on the results in Table 1 (and further detailed in Table A2 in Appendix A), the MRSO outperforms the RSO in six out of nine multimodal functions, indicating its superior ability to explore complex search spaces. For example, in F8, the MRSO achieves a better average (\(-8.613\times 10^{+3}\)) compared to the RSO's \(-5.709\times 10^{+3}\), demonstrating its effectiveness in avoiding local optima. Similarly, the MRSO outperforms the RSO in F12 with an average of \(5.646\times 10^{-2}\) versus the RSO's \(3.366\times 10^{-1}\), and in F14 with an average of \(1.395\) compared to the RSO's \(2.646\). The MRSO also shows improved performance in F10, F13, and F15, further reinforcing its strength in exploration. However, in F9 and F11, both algorithms achieve perfect averages of 0, showing equal performance. Overall, The MRSO demonstrates superior exploration across most of the multimodal functions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fun**} & \multicolumn{2}{c}{**MRSO**} & \multicolumn{2}{c}{**RSO**} & \multicolumn{2}{c}{**Statistical Significance**} \\ \cline{2-5}  & **Avg.** & **Std.** & **Avg.** & **Std.** & **\(p\)-Value** \\ \hline F1 & \(1.12\times 10^{-6}\) & \(4.765\times 10^{-6}\) & \(1.537\times 10^{-256}\) & 0 & \(2.03\times 10^{-1}\) \\ F2 & \(4.515\times 10^{-34}\) & \(2.228\times 10^{-33}\) & \(2.942\times 10^{-134}\) & \(1.612\times 10^{-133}\) & \(2.716\times 10^{-1}\) \\ F3 & \(5.092\times 10^{-3}\) & \(2.754\times 10^{4}\) & \(2.205\times 10^{-264}\) & \(2.205\times 10^{-264}\) & 0 \\ F4 & \(3.486\times 10^{-16}\) & \(1.91\times 10^{-15}\) & \(2.868\times 10^{-86}\) & \(1.571\times 10^{-85}\) & \(3.215\times 10^{-1}\) \\ F5 & \(2.885\times 10^{-1}\) & \(1.323\times 10^{-1}\) & \(2.882\times 10^{+1}\) & \(2.635\times 10^{-1}\) & \(5.945\times 10^{-1}\) \\ F6 & \(8.16\times 10^{-1}\) & \(4.033\times 10^{-1}\) & \(3.415\) & \(4.748\times 10^{-1}\) & \(1.099\times 10^{-30}\) \\ F7 & \(1.981\times 10^{-4}\) & \(1.438\times 10^{-4}\) & \(4.861\times 10^{-4}\) & \(5.636\times 10^{-4}\) & \(8.792\times 10^{-3}\) \\ F8 & \(-8.613\times 10^{+3}\) & \(1.835\times 10^{+3}\) & \(-5.709\times 10^{+3}\) & \(1.073\times 10^{+3}\) & \(4.498\times 10^{-10}\) \\ F9 & 0 & 0 & 0 & 0 & 0 \\ F10 & \(2.189\times 10^{-5}\) & \(8.528\times 10^{-5}\) & \(1.007\times 10^{-15}\) & \(6.486\times 10^{-16}\) & \(1.651\times 10^{-1}\) \\ F11 & 0 & 0 & 0 & 0 & 0 \\ F12 & \(5.646\times 10^{-2}\) & \(4.465\times 10^{-2}\) & \(3.366\times 10^{-1}\) & \(1.055\times 10^{-1}\) & \(2.127\times 10^{-19}\) \\ F13 & 2.824 & \(9.216\times 10^{-2}\) & \(2.862\) & \(4.32\times 10^{-2}\) & \(4.514\times 10^{-2}\) \\ F14 & 1.395 & \(8.072\times 10^{-1}\) & \(2.646\) & \(1.787\) & \(9.131\times 10^{-4}\) \\ F15 & \(8.071\times 10^{-4}\) & \(4.068\times 10^{-4}\) & \(1.178\times 10^{-3}\) & \(6.153\times 10^{-4}\) & \(7.839\times 10^{-3}\) \\ F16 & \(-1.032\) & \(1.463\times 10^{-5}\) & \(-1.031\) & \(2.045\times 10^{-4}\) & \(2.719\times 10^{-4}\) \\ F17 & \(3.99\times 10^{-1}\) & \(3.303\times 10^{-3}\) & \(4.076\times 10^{-1}\) & \(9.812\times 10^{-3}\) & \(2.831\times 10^{-5}\) \\ F18 & 3.000 & \(4.028\times 10^{-6}\) & \(3.000\) & \(5.206\times 10^{-5}\) & \(1.119\times 10^{-2}\) \\ F19 & \(-3.856\) & \(1.807\times 10^{-3}\) & \(-3.426\) & \(2.942\times 10^{-1}\) & \(6.275\times 10^{-11}\) \\ F20 & \(-2.775\) & \(3.885\times 10^{-1}\) & \(-1.723\) & \(3.772\times 10^{-1}\) & \(2.919\times 10^{-15}\) \\ F21 & \(-2.634\) & \(2.536\) & \(-7.08\times 10^{-1}\) & \(3.728\times 10^{-1}\) & \(1.236\times 10^{-4}\) \\ F22 & \(-3.719\) & \(2.774\) & \(-1.035\) & \(6.255\times 10^{-1}\) & \(3.033\times 10^{-6}\) \\ F23 & \(-2.360\) & \(2.440\) & \(-1.353\) & \(7.529\times 10^{-1}\) & \(3.504\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table}