## 5 Benchmark Testing and Performance Analysis of MRSO
### Evaluating the MRSO Algorithm alongside Metaheuristic Methods with Classical Benchmark Functions
Nevertheless, the MRSO consistently shows significant results in the majority of the tested functions, establishing its competitiveness. Table 4 illustrates that the MRSO is a strong performer across multimodal functions in terms of average values, stability (standard deviation), and statistical significance (\(p\)-values), outperforming the eight metaheuristic algorithms in several key areas. In terms of average performance, as shown in Table 5, the MRSO outperforms the eight metaheuristic algorithms in three out of the seven fixed-dimension multimodal functions (F17-F23). For example, in F17, the MRSO achieves the exact global minimum with an average value of 3.99 \(\times\) 10\({}^{-1}\), which ties with the TSA in terms of best performance. Similarly, the MRSO reaches the global minimum in F18 with an average of 3.00, outperforming most other algorithms, except for the SCA and the EHO, which show comparable results. However, the MRSO lags behind in functions like F21, where it achieves an average of \(-\)2.63, significantly worse than the CSA and the MRA, which achieve much better results. Overall, the MRSO demonstrates solid performance in three functions but leaves room for improvement in the remaining four. When analyzing the standard deviation, which indicates the consistency of the algorithm, the MRSO performs best in four out of seven functions. For instance, in F18, the MRSO achieves a standard deviation of 4.03 \(\times\) 10\({}^{-6}\), highlighting its excellent stability and consistency in this function. In F17 and F19, the MRSO also shows relatively low standard deviations (3.30 \(\times\) 10\({}^{-3}\) and 1.81 \(\times\) 10\({}^{-3}\), respectively), demonstrating that the MRSO consistently performs well across different runs. However, the MRSO exhibits higher variability in functions such as F21 and F22, with standard deviations of 2.54 and 2.77, respectively, showing that its performance fluctuates more significantly in these cases compared to algorithms like the CSA and the MRA, which demonstrate greater stability. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Algorithm** & **Metric/Function** & **P17** & **F18** & **F19** & **F20** & **F21** & **F22** & **P23** \\ \hline \multirow{2}{*}{MRSO} & Average & \(3.99\times 10^{-1}\) & 3 & –3.86 & –2.78 & –2.63 & –3.72 & –2.36 \\  & Std. & \(3.33\times 10^{-3}\) & \(4.03\times 10^{-4}\) & \(18.11\times 10^{-9}\) & \(3.89\times 10^{-1}\) & 2.54 & 2.77 & 2.44 \\  & Ranking & 1 & 1 & 4 & 6 & 9 & 8 & 9 \\ \hline \multirow{4}{*}{SCA} & Average & \(4\times 10^{-1}\) & 3 & –3.85 & –2.92 & –2.9 & –3.1 & –4.03 \\  & Std. & \(2.86\times 10^{-1}\) & \(9.16\times 10^{-5}\) & \(3\times 10^{-1}\) & \(2.27\times 10^{-1}\) & 1.89 & 1.77 & 1.40 \\  & \(p\),value & \(2.97\times 10^{-1}\) & \(3.36\times 10^{-4}\) & \(8.21\times 10^{-3}\) & \(1.05\times 10^{-1}\) & \(6.44\times 10^{-1}\) & \(311\times 10^{-1}\) & \(1.89\times 10^{-3}\) \\  & Ranking & 3 & 2 & 5 & 5 & 8 & 9 & 8 \\ \hline \multirow{4}{*}{MRA} & Average & \(42\times 10^{-1}\) & 7.62 & –3.72 & –2.73 & –\(1.02\times 10^{-1}\) & \(-1.04\times 10^{-1}\) & \(-1.05\times 10^{-1}\) \\  & Std. & \(2.8\times 10^{-2}\) & 4.46 & \(7.33\times 10^{-2}\) & \(1.86\times 10^{-1}\) & \(2.88\times 10^{-2}\) & \(1.27\times 10^{-3}\) & \(49\times 10^{-3}\) \\  & p,value & \(2.77\times 10^{-1}\) & \(4.68\times 10^{-7}\) & \(3.53\times 10^{-7}\) & \(5.38\times 10^{-7}\) & \(3.08\times 10^{-3}\) & \(41\times 10^{-7}\) & \(7.97\times 10^{-3}\) \\  & Ranking & 7 & 6 & 7 & 7 & 2 & 3 & 3 \\ \hline \multirow{4}{*}{ICA} & Average & \(2.77\times 10^{-1}\) & \(2.35\times 10^{-1}\) & –3.21 & –1.78 & -5.02 & –4.97 & –4.45 \\  & Std.