## 5 Benchmark Testing and Performance Analysis of MRSO
### The MRSO Algorithm and Metaheuristic Approaches: Insights from the CEC-C06 2019 Benchmarking  
As seen in Table 7, the MRSO outperformed eight metaheuristic algorithms in five out of the ten CEC-C06 2019 benchmark functions, particularly in functions F2, F3, F6, F7, and F10. For instance, the MRSO achieved an average of 1.83 \(\times\) 10\({}^{+1}\) in F2, which was better than most other algorithms. In F6, the MRSO's average of 1.09 \(\times\) 10\({}^{+1}\) also surpassed all other algorithms, demonstrating its ability to handle complex optimization problems effectively. However, the MRSO did not perform as well in F1 and F5, where its average was higher than some of the competing algorithms, indicating that there is room for improvement in certain benchmarks. In terms of standard deviation, the MRSO demonstrated consistency in its performance across multiple functions. For example, in F2 and F3, the MRSO had very low standard deviations of 7.23 \(\times\) 10\({}^{-3}\) and 1.34 \(\times\) 10\({}^{-6}\), respectively, indicating reliable and stable performance. The MRSO also performed consistently in F6 with a standard deviation of 1.01, which was lower than most other algorithms. However, in functions like F1, the MRSO had a higher standard deviation of 3.20 \(\times\) 10\({}^{+5}\), showing greater variability in its results compared to other algorithms, which suggests that further refinement may be necessary to improve its stability across all benchmarks. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**Algorithm** & **MRSO** & **SCA** & **MRA** & **LCA** & **CSA** & **TSA** & **DOA** & **EHO** & **WSO** \\ \hline Ranking & 10.2 & 12.8 & 7.5 & 15.1 & 8.1 & 11.3 & 10.6 & 12.4 & 13.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Scores comparison of MRSO, SCA, MRA, LCA, CSA, TSA, DOA, EHO, and WSO on 23 classical benchmark functions. The \(p\)-values, as shown in Table 7, further highlight the MRSO's statistical significance in several functions. In F2, the MRSO's \(p\)-value of \(1.95\times 10^{-10}\) indicates a statistically significant improvement over competing algorithms, confirming the robustness of its performance. Similarly, in F6, the \(p\)-value of \(3.06\times 10^{-6}\) demonstrates that the MRSO's results are not due to random chance and are significantly better than other algorithms. On the other hand, in functions like F5, the \(p\)-value of \(5.34\times 10^{-25}\) shows that other algorithms, like the WSO, performed better than the MRSO. Despite this, the MRSO consistently showed significant results in key functions, reinforcing its competitiveness. Table 7 highlights the MRSO's strengths in handling diverse optimization problems, showing strong performance in several key benchmarks against eight other metaheuristic algorithms, with statistically significant improvements in many areas. To systematically rank the algorithms, we again used the [PERSON] mean ranking score to evaluate the overall performance of each algorithm across all ten CEC-C06 2019 benchmark functions. As shown in Table 8, the MRSO achieved a ranking score of 3, placing it among the top performers. The SCA followed closely with a score of 3.5, and the TSA and the DOA both ranked similarly with a score of 4.4. The WSO had a slightly better score of 3.1, indicating strong performance. On the other hand, the MRA, the LCA, and the CSA ranked lower, with scores of 7.2, 7.5, and 7.9, respectively, showing that these algorithms were less effective overall compared to the MRSO and the other top-ranked algorithms. These rankings highlight the MRSO's strong capability in handling the CEC-C06 2019 benchmark functions, outperforming most of the competing algorithms. At the conclusion of Sections 5.5 and 5.6, our proposed MRSO algorithm performs better on the 10 CEC-C06 2019 benchmark functions than on the classical benchmark functions.