## 5 Simulation Results
### The Results and Discussion
#### 5.2.2 Comparative Analysis at \(C=40\)  
The lower the RPD, the better the performance of the algorithms. When C takes the value of 40, the average RPD of these eight algorithms is shown in Table 7. As can be seen in the last row, the average value of IHBA is 22.73%, which is significantly lower than the average value of other algorithms including the original bat algorithm. It shows that the IHBA algorithm is more efficient than the other metaheuristic algorithms, while BA performs second, with an average RPD of 23.57%. Compared with Table 7, it can be concluded that the RPD value of IHBA changes from the original 21.15 to the current 22.73, as the value of C is taken to increase, which shows that the value of RPD of IHBA increases with the increase of the number of iterations and time. In addition, PSO with a mean value of 35.76% is significantly better than GA and the other four heuristics. Meanwhile, EDA has the largest value and its performance is the worst, which is also consistent with the results in Table 4.  
Table 8 describes the results of the Friedman test calculations. As can be seen at a glance from the rankings, IHBA has a mean rank of 3.06, which is the smallest value among these algorithms. The next best performer is TDIWO, which also has a better performance, but its standard deviation is 0.89, which is larger than CMA, GA, and PSO. It can be seen from the mean and standard deviation that IHBA has values of 0.79 and 0.76, respectively, which are also the smallest values among all algorithms. The performance of EDA is still the worst, with its average rank, mean, standard deviation, and maximum value all being at a disadvantage. The performance of EDA is still the worst, with its ranking, mean, standard deviation and maximum value being at a disadvantage. In particular, the mean value is as high as 32 times that of IHBA. The \(p\)-value in the last row is 0, which is less than 0.05, indicating that the difference between the algorithms has statistical significance. This difference is not due to chance sampling, but rather, the difference between the two groups of algorithms is significant. Figure 1: Mean plot and 95% LSD interval of the bat algorithm at \(C=20\). The results obtained from the Wilcoxon paired signed test are shown in Table 9. The \(p\)-values in the last column are all equal to 0, which is less than 0.05. It shows that the differences between these algorithms are statistically significant, and that there are significant differences between IHBA and the other algorithms. This also reconfirms the conclusion that \(C=20\). More definitely, it explains that IHBA is an effective group intelligence algorithm with outstanding performance. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{6}{c}{**Parameters**} & \multicolumn{6}{c}{**Algorithms for Comparison**} \\ \hline \(m\) & \(n\) & \(s\) & \(f\) & **CMA** & **BBO** & **TDIWO** & **EDA** & **GA** & **PSO** & **BA** & **IHBA** \\ \hline  
58 & **26.59** \\ \hline \hline \multicolumn{6}{c}{average} & 36.42 & 37.64 & 38.76 & 46.72 & 39.78 & 35.76 & 23.57 & **22.73** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithms** & _Rank_ & \(n\) & _Mean_ & _Std. Deviation_ & _Min_ & _Max_ \\ \hline
**CMA** & 4.81 & 4050.00 & 1.17 & 0.92 & 0.00 & 7.05 \\
**BBO** & 4.92 & 4050.00 & 1.47 & 1.30 & 0.00 & 7.70 \\
**EDA** & 7.01 & 4050.00 & 16.89 & 12.80 & 0.00 & 50.37 \\
**TDIWO** & 3.36 & 4050.00 & 0.99 & 0.96 & 0.00 & 4.82 \\
**GA** & 3.43 & 4050.00 & 0.93 & 0.87 & 0.00 & 4.44 \\
**PSO** & 4.22 & 4050.00 & 1.11 & 0.96 & 0.00 & 5.98 \\
**BA** & 3.06 & 4050.00 & 0.79 & 0.76 & 0.00 & 4.67 \\
**IHBA** & 2.94 & 4050.00 & 0.77 & 0.75 & 0.00 & 4.71 \\ _p-value_ & 0.00 & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The Descriptive Statistics Achieved by the Friedman Test at \(C=40\) and \(\alpha=0.05\). The bold figures indicate the comparison between algorithms. Finally, in this section, EDA is excluded from the mean plot.