## 4 Experimental Simulation and Results Analysis
### Performance Comparison Between MSEChoA and Other Algorithms
#### 4.3.1 Solution Accuracy Analysis
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Algorithm** & **Standard** & **Best Fitness** & **Average** & **Worst Fitness** & **Standard Deviation of** \\  & **Function** & **Value** & **Fitness Value** & **Value** & **Fitness Value** \\ \hline GWO & & 2.6014 \(\times\) 10\({}^{1}\) & 2.6887 \(\times\) 10\({}^{1}\) & 2.7976 \(\times\) 10\({}^{1}\) & 6.6254 \(\times\) 10\({}^{1}\) \\ EO & & 2.4195 \(\times\) 10\({}^{1}\) & 2.4447 \(\times\) 10\({}^{1}\) & 2.4845 \(\times\) 10\({}^{1}\) & 1.6682 \(\times\) 10\({}^{1}\) \\ ALO & & 2.6732 \(\times\) 10\({}^{1}\) & 2.6905 \(\times\) 10\({}^{1}\) & 1.7752 \(\times\) 10\({}^{1}\) & 4.3966 \(\times\) 10\({}^{2}\) \\ MSGWO & & 2.8824 \(\times\) 10\({}^{1}\) & 2.8955 \(\times\) 10\({}^{1}\) & 2.8997 \(\times\) 10\({}^{1}\) & 3.5780 \(\times\) 10\({}^{2}\) \\ CHOA & F5 & 2.8062 \(\times\) 10\({}^{1}\) & 2.8836 \(\times\) 10\({}^{1}\) & 2.8948 \(\times\) 10\({}^{1}\) & 1.7312 \(\times\) 10\({}^{1}\) \\ ICHOA & & 2.8743 \(\times\) 10\({}^{1}\) & 2.8794 \(\times\) 10\({}^{1}\) & 2.8868 \(\times\) 10\({}^{1}\) & 2.7903 \(\times\) 10\({}^{2}\) \\ WChOA & & 29 & 29 & 29 & 5.7969 \(\times\) 10\({}^{1}\) \\ AWChOA & & 29 & 29 & 29 & 6.5972 \(\times\) 10\({}^{1}\) \\ MSEChOA & & 2.7710 \(\times\) 10\({}^{1}\) & 2.8460 \(\times\) 10\({}^{1}\) & 2.8821 \(\times\) 10\({}^{1}\) & 4.2720 \(\times\) 10\({}^{1}\) \\ \hline GWO & & 2.9949 \(\times\) 10\({}^{1}\) & 6.6693 \(\times\) 10\({}^{1}\) & 1.5180 \(\times\) 10\({}^{1}\) & 3.4501 \(\times\) 10\({}^{1}\) \\ EO & & 4.8801 \(\times\) 10\({}^{1}\) & 2.0216 \(\times\) 10\({}^{8}\) & 1.8594 \(\times\) 10\({}^{8}\) & 1.3300 \(\times\) 10\({}^{9}\) \\ ALO & & 8.4663 \(\times\) 10\({}^{7}\) & 9.1442 \(\times\) 10\({}^{6}\) & 8.3931 \(\times\) 10\({}^{5}\) & 8.3129 \(\times\) 10\({}^{6}\) \\ MSGWO & & 5.7383 & 6.0207 & 6.5039 & 9.4713 \(\times\) 10\({}^{1}\) \\ CHOA & F6 & 3.0442 & 3.3381 & 3.6401 & 2.3  
In the multimodal functions (F6 to F9), the MSECHOA algorithm also demonstrated superior performance. On the F6 function, while the fitness value of MSECHOA was slightly lower than that of AWChOA, it had a lower standard deviation, indicating better stability. In the F7 and F8 functions, MSECHOA showed good optimization results, especially in the F8 function, where MSECHOA achieved the optimal result among all tested algorithms, with a standard deviation close to zero, indicating high stability and accuracy. On the F9 function, MSECHOA, like several other improved algorithms, achieved the optimal value of 0, and its performance was comparable to other top algorithms such as AWChOA. In the fixed-dimension multimodal functions (F10, F11), MSECHOA continued to exhibit excellent performance, particularly in complex high-dimensional search spaces. In both of these functions, MSECHOA achieved the theoretical optimal value and displayed a low standard deviation value, demonstrating its advantage in high-dimensional multimodal optimization problems. In summary, the MSECHOA algorithm has demonstrated excellent optimization capabilities across various benchmark test functions, especially in handling complex optimization problems.