## 5 Benchmark Testing and Performance Analysis of MRSO
### Evaluating the MRSO Algorithm alongside Metaheuristic Methods with Classical Benchmark Functions  
By using 23 classical benchmark functions, we evaluated the performance of the MRSO algorithm in comparison with eight new metaheuristic algorithms: the Sine Cosine Algorithm (SCA) [26; 38], the Mud Ring Algorithm (MRA) [27], the Liver Cancer Algorithm (LCA) [28], the Circle Search Algorithm (CSA) [29], the Tunicate Swarm Algorithm (TSA) [30], the Dingo Optimizer Algorithm (DOA) [31] the Elk Herd Optimizer (EHO) [32], and the White Shark Optimizer (WSO) [33]. Based on the analysis of unimodal functions (F1-F7) from Table 3, the MRSO outperforms the other eight metaheuristic algorithms in two out of the seven functions. For instance, in F7, the MRSO achieves an average of \(1.98\times 10^{-4}\), which is significantly lower and better than the SCA's \(2.66\times 10^{-3}\) and the TSA's \(7.43\times 10^{-5}\). This highlights the MRSO's superior exploitation capability in certain cases. Additionally, the MRSO excels in F6 with an average of \(8.16\times 10^{-1}\), outperforming algorithms like the SCA and the DOA, which score higher averages of \(4.34\times 10^{-1}\) and \(5.58\), respectively. However, the MRSO does not perform as well in functions such as F1 and F5, where algorithms like the MRA achieve perfect averages (0) across multiple functions, indicating that the MRSO needs further refinement for simpler unimodal tasks. When considering the standard deviation metrics, which reflect the consistency of an algorithm across iterations, the MRSO outperforms the eight algorithms in three out of seven functions. For example, in F7, the MRSO has a lower standard deviation (\(1.44\times 10^{-4}\)) compared to the DOA's \(2.80\times 10^{-4}\) and the SCA's \(2.81\times 10^{-3}\), demonstrating the MRSO's more reliable performance. In F6, the MRSO's standard deviation of \(4.03\times 10^{-1}\) indicates stable results, which is comparable with the TSA but better than the DOA, whose performance fluctuates with a higher standard deviation (\(1.09\)). However, the MRA consistently outperforms the MRSO in terms of stability, achieving zero deviations across multiple functions such as F1, F2, and F3, indicating near-perfect consistency in these cases. In terms of statistical significance, represented by \(p\)-values, the MRSO shows a significant advantage in two functions. For instance, in F7, the MRSO's performance is statistically significant with a \(p\)-value of \(1.18\times 10^{-5}\) when compared to algorithms like the SCA and the DOA, highlighting a meaningful difference in performance. In F6, the MRSO's \(p\)-value of \(9.06\times 10^{-6}\) suggests that it performs significantly better than other algorithms like TSA. However, in functions like F5, the \(p\)-value of \(9.93\times 10^{-1}\) indicates that the MRSO and the competing algorithms show similar performances with no statistically significant differences. Thus, while the MRSO shows strong performance in some areas, the \(p\)-value analysis suggests that there is room for improvement in other functions to achieve consistently significant results. Table 3 summarizes these comparisons, and the results highlight the specific areas where the MRSO surpasses other metaheuristic methods, as well as the cases where it needs further optimization. In terms of average performance, as shown in Table 4, the MRSO outperforms the eight metaheuristic algorithms in five out of the nine multimodal functions (F8-F16). For example, in F9, F11, and F16, the MRSO achieves the best possible average values of 0, which are consistent with the global minima for these functions. In F8, the MRSO demonstrates competitive performance with an average of \(-8.61\times 10^{-3}\), which surpasses most algorithms except the CSA. Similarly, in F15, the MRSO delivers superior results with an average of \(8.07\times 10^{-4}\), showing its capability to avoid local optima and converge effectively.