## 5 Benchmark Testing and Performance Analysis of MRSO
### Statistical Analysis  
A comparison between the standard RSO algorithm and our modified version, the MRSO, using the classical benchmark test functions reveals notable differences in performance, as depicted in Table 1. For instance, functions like F3 and F6 exhibit extremely low \(p\)-values (0 and 1.099 \(\times\) 10\({}^{-30}\), respectively), indicating highly significant improvements with the MRSO over the RSO. Similarly, functions F7, F8, and F12 show \(p\)-values of 8.792 \(\times\) 10\({}^{-03}\), 4.498 \(\times\) 10\({}^{-10}\), and 2.127 \(\times\) 10\({}^{-19}\), respectively, suggesting that the MRSO outperforms the RSO with high statistical significance. These results underscore the MRSO's enhanced capability to solve these functions more effectively than the standard RSO. On the other hand, several functions show \(p\)-values above 0.05, indicating no significant difference between the MRSO and the RSO. Functions F1, F2, F4, F5, and F10 have \(p\)-values of 2.030 \(\times\) 10\({}^{-1}\), 2.716 \(\times\) 10\({}^{-1}\), 3.215 \(\times\) 10\({}^{-1}\), 5.945 \(\times\) 10\({}^{-1}\), and 1.651 \(\times\) 10\({}^{-1}\), respectively. These results suggest that for these functions, the MRSO's modifications did not yield statistically significant improvements over the RSO. This indicates that while the MRSO shows promising results in many cases, its enhancements might not always provide a significant advantage, particularly for certain benchmark 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 & 1.588 \(\times\) 10\({}^{+5}\) & 3.199 \(\times\) 10\({}^{+5}\) & 6.263 \(\times\) 10\({}^{+4}\) & 1.392 \(\times\) 10\({}^{+4}\) & 1.053 \(\times\) 10\({}^{-1}\) \\ F2 & 1.835 \(\times\) 10\({}^{+1}\) & 7.231 \(\times\) 10\({}^{-3}\) & 1.848 \(\times\) 10\({}^{+1}\) & 1.981 \(\times\) 10\({}^{-1}\) & 3.797 \(\times\) 10\({}^{-4}\) \\ F3 & 1.370 \(\times\) 10\({}^{+1}\) & 1.337 \(\times\) 10\({}^{-6}\) & 1.370 \(\times\) 10\({}^{+1}\) & 1.828 \(\times\) 10\({}^{-4}\) & 2.256 \(\times\) 10\({}^{-2}\) \\ F4 & 9.205 \(\times\) 10\({}^{+3}\) & 3.203 \(\times\) 10\({}^{+3}\) & 8.861 \(\times\) 10\({}^{+3}\) & 2.152 \(\times\) 10\({}^{+3}\) & 6.275 \(\times\) 10\({}^{-1}\) \\ F5 & 4.574 & 4.133 \(\times\) 10\({}^{-1}\) & 4.631 & 4.290 \(\times\) 10\({}^{-1}\) & 6.076 \(\times\) 10\({}^{-1}\) \\ F6 & 1.095 \(\times\) 10\({}^{+1}\) & 1.011 & 1.165 \(\times\) 10\({}^{+1}\) & 8.597 \(\times\) 10\({}^{-1}\) & 4.905 \(\times\) 10\({}^{-3}\) \\ F7 & 6.113 \(\times\) 10\({}^{+2}\) & 2.291 \(\times\) 10\({}^{+2}\) & 7.898 \(\times\) 10\({}^{+2}\) & 2.154 \(\times\) 10\({}^{+2}\) & 2.915 \(\times\) 10\({}^{-3}\) \\ F8 & 6.311 & 4.138 \(\times\) 10\({}^{-1}\) & 6.321 & 4.334 \(\times\) 10\({}^{-1}\) & 9.215 \(\times\) 10\({}^{-1}\) \\ F9 & 4.967 \(\times\) 10\({}^{+2}\) & 1.493 \(\times\) 10\({}^{+2}\) & 5.866 \(\times\) 10\({}^{+2}\) & 1.362 \(\times\) 10\({}^{+2}\) & 1.791 \(\times\) 10\({}^{-2}\) \\ F10 & 2.130 \(\times\) 10\({}^{+1}\) & 1.490 \(\times\) 10\({}^{-1}\) & 2.147 \(\times\) 10\({}^{+1}\) & 1.116 \(\times\) 10\({}^{-1}\) & 5.316 \(\times\) 10\({}^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The MRSO vs. standard RSO performance on the CEC 2019 benchmark functions: average, standard Deviation, and statistical Significance. When examining the CEC 2019 benchmark test functions, several statistically significant improvements were also observed with the MRSO. Functions F2, F3, F6, F7, F9, and F10 exhibit \(p\)-values of 3.797 \(\times\) 10\({}^{-4}\), 2.256 \(\times\) 10\({}^{-2}\), 4.905 \(\times\) 10\({}^{-3}\), 2.915 \(\times\) 10\({}^{-3}\), 1.791 \(\times\) 10\({}^{-2}\), and 5.316 \(\times\) 10\({}^{-6}\), respectively, indicating significant improvements by the MRSO over the RSO.