## 5 Benchmark Testing and Performance Analysis of MRSO
### Evaluating the MRSO Algorithm alongside Metaheuristic Methods with Classical Benchmark Functions
However, the MRSO does not perform as well in functions like F14, where it scores an average of \(1.39\), slightly underperforming compared to the CSA and other algorithms. Overall, the MRSO proves to be highly competitive in most multimodal functions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithm** & **Metric/Function** & **F1** & **F2** & **F3** & **F4** & **FS** & **F6** & **F7** \\ \hline \multirow{2}{*}{MRSO} & Average & \(1.12\times 10^{-4}\) & \(4.52\times 10^{-34}\) & \(5.09\times 10^{-3}\) & \(3.69\times 10^{-48}\) & \(2.89\times 10^{-4}\) & \(8.16\times 10^{-1}\) & \(1.88\times 10^{-4}\) \\  & Std. & \(4.76\times 10^{-4}\) & \(2.23\times 10^{-30}\) & \(2.27\times 10^{-4}\) & \(1.91\times 10^{-3}\) & \(1.32\times 10^{-1}\) & \(4.40\times 10^{-1}\) & \(1.44\times 10^{-1}\) \\  & Ranking & 6 & 5 & 9 & 5 & 5 & 6 & 3 \\ \hline \multirow{4}{*}{SCA} & Average & \(3.31\times 10^{-2}\) & \(5.82\times 10^{-10}\) & \(6.55\times 10^{-3}\) & \(3.21\times 10^{-3}\) & \(2.90\times 10^{-4}\) & \(4.34\times 10^{-1}\) & \(2.66\times 10^{-3}\) \\  & Std. & \(8.99\times 10^{-10}\) & \(7.90\times 10^{-3}\) & \(6.55\times 10^{-3}\) & \(1.42\times 10^{-1}\) & \(1.19\times 10^{-3}\) & \(1.49\times 10^{-1}\) & \(2.81\times 10^{-3}\) \\  & \(p\),value & \(2.63\times 10^{-1}\) & \(1.83\times 10^{-4}\) & \(1.77\times 10^{-2}\) & \(2.2\times 10^{-1}\) & \(9.93\times 10^{-4}\) & \(9.80\times 10^{-4}\) & \(1.18\times 10^{-1}\) \\ \hline \multirow{4}{*}{MRA} & Average & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0.64\times 10^{-7}\) & \(1.01\times 10^{-4}\) \\  & Std. & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(1.46\times 10^{-4}\) & \(1.07\times 10^{-4}\) \\  & \(p\),value & \(2.03\times 10^{-1}\) & \(2.72\times 10^{-1}\) & \(0\) & \(3.21\times 10^{-4}\) & \(4.69\times 10^{-10}\) & \(5.87\times 10^{-4}\) & \(4.57\times 10^{-3}\) \\  & Ranking & 1 & 1 & 1 & 1 & 1 & 2 & 2 \\ \hline \multirow{4}{*}{ICA} & Average & \(2.42\times 10^{-1}\) & \(1.87\times 10^{-1}\) & \(4.1\times 10^{-1}\) & \(7.3\times 10^{-2}\) & \(1.25\times 10^{-1}\) & \(2.35\times 10^{-1}\) & \(6.39\times 10^{-4}\) \\  & Std. & \(2.99\times 10^{-1}\) & \(1.11\times 10^{-4}\) & \(4.43  
Regarding standard deviation, the MRSO exhibits stable and consistent performance across several functions, outperforming other algorithms in four of the nine multimodal functions (F8-F16). For instance, in F9, F11, and F16, the MRSO achieves a standard deviation of 0, indicating perfect consistency across runs, outperforming algorithms such as the DOA, the SCA, and the TSA. In F12, the MRSO shows better stability with a lower standard deviation of 4.46 \(\times\) 10\({}^{-2}\) compared to the SCA's 3.83 \(\times\) 10\({}^{-2}\) and the TSA's 3.81 \(\times\) 10\({}^{-1}\), highlighting its reliability in delivering consistent results. However, in F14, the MRSO shows slightly higher variation with a standard deviation of 8.07 \(\times\) 10\({}^{-1}\), indicating that it may struggle with consistency in this function compared to some other algorithms like the CSA and the DOA. Despite these minor fluctuations, the MRSO remains one of the more stable performers in this evaluation. In terms of statistical significance, based on the \(p\)-values in Table 4, the MRSO demonstrates statistically significant performance in four out of nine functions. For example, in F9 and F11, the MRSO's \(p\)-values indicate that its results are statistically indistinguishable from the global optimum (\(p\)-values of 7.76 \(\times\) 10\({}^{-2}\) and 4.55 \(\times\) 10\({}^{-3}\), respectively). In F16, the MRSO's \(p\)-value of 9.85 \(\times\) 10\({}^{-5}\) demonstrates its significant superiority over many of the compared algorithms. However, in functions like F13 and F14, the MRSO does not show as much statistical significance, with higher \(p\)-values, such as 2.80 \(\times\) 10\({}^{-68}\) for F13 and 3.77 \(\times\) 10\({}^{-1}\) for F14, indicating that its performance is less conclusive when compared to the other metaheuristic methods.