## 5 Benchmark Testing and Performance Analysis of MRSO
### Performance Comparison of MRSO and RSO with CEC 2019 Benchmark Functions  
The CEC-C06 2019 benchmark test functions comprise ten mathematical functions specifically created for the IEEE Congress on Evolutionary Computation (CEC) 2019 competition. These functions are used to evaluate the performance of optimization algorithms across a range of optimization challenges. They incorporate features such as rotation, scaling, and shifting. Widely recognized in the fields of optimization and evolutionary computation, these benchmarks are essential for comparing existing algorithms and assessing the effectiveness of newly developed ones [35]. The MRSO algorithm was further evaluated using the CEC-C06 2019 benchmark functions, which provide a rigorous set of challenges for optimization algorithms.  
In terms of average performance, as presented in Table 2 and Figure 2, the MRSO outperforms the RSO in six out of the ten CEC 2019 benchmark functions, specifically in functions F2, F3, F6, F7, F9, and F10. For example, in F2, the MRSO achieves an average of 1.835 \(\times\) 10\({}^{+1}\), which is lower and better than the RSO's 1.848 \(\times\) 10\({}^{+1}\), indicating the MRSO's superior precision in solving this optimization problem. Similarly, in F6, the MRSO has an average of 1.095 \(\times\) 10\({}^{+01}\) compared to the RSO's 1.165 \(\times\) 10\({}^{+1}\), demonstrating better convergence to the optimal solution. The MRSO also outperforms the RSO in F9, where its average (4.967 \(\times\) 10\({}^{+2}\)) is significantly lower than the RSO's 5.866 \(\times\) 10\({}^{+2}\), confirming its ability to handle complex landscapes more effectively. Functions like F7 and F10 further show the MRSO's dominance with better averages than the RSO, while F1, F4, F5, and F8 exhibit similar results between the two algorithms. Overall, the MRSO's lower averages in six functions highlight its improved search capability and precision in handling diverse optimization challenges compared to the standard RSO.  
Regarding standard deviation, the MRSO demonstrates more consistent performance in five out of the ten functions, as seen in Table 2 and Figure 2. For example, in F3, the MRSO shows a significantly lower standard deviation (1.337 \(\times\) 10\({}^{-6}\)) compared to the RSO's 1.828 \(\times\) 10\({}^{-4}\), indicating more stable results across multiple runs. In F9, the MRSO again exhibits superior consistency with a lower standard deviation (1.493 \(\times\) 10\({}^{+2}\)) compared to the RSO's 1.362 \(\times\) 10\({}^{+2}\), reflecting its ability to produce reliable outcomes in challenging optimization tasks. Similarly, in F7, the MRSO's standard deviation (2.291 \(\times\) 10\({}^{+2}\)) is lower than the RSO's (2.154 \(\times\) 10\({}^{+2}\)), demonstrating better reliability. However, in F6, the RSO shows slightly more consistent results with a lower standard deviation (8.597 \(\times\) 10\({}^{-1}\)) compared to the MRSO's (1.011). While the RSO performs slightly better in some cases, the MRSO's overall consistency in five functions reinforces its robustness and reliability in delivering stable optimization results across a range of problems.