## 5 Simulation Results
### The Results and Discussion
#### 5.2.1 Comparative Analysis at \(C\) = 20  
When \(C\) = 20, the values of RPD for various heuristic algorithms are shown in Table 4. It can be seen from the bold figures that the value of PRD of the proposed algorithm IHBA in this paper is smaller than that of other algorithms in the vast majority of combinatorial solutions, including the recognized TDIWO algorithm and the original bat algorithm. Only in these 5 sets of data (1) \(m=5,n=100,s=30\) and \(f=4\), (2) \(m=5,n=200,s=40\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Comparison** & **Parameters** & **Range** \\ \hline \multirow{8}{*}{PSO} & Number of populations & \(2n\) \\  & Decreasing coefficient & 0.975 \\  & Crossover rate & [0,1] \\  & Variation rate & [0,1] \\  & Uniform random number & [0,1] \\  & Inertia weights & [0,4,0.9] \\ \hline \multirow{2}{*}{BA} & Frequency & [0,1] \\  & Loudness & [1,2] \\  & pulse emission rate & [\(-\)1,1] \\ \hline \multirow{4}{*}{IHBA} & Frequency & [0,1] \\  & Loudness & [1,2] \\ \cline{1-1}  & pulse emission rate & [0,1] \\ \cline{1-1}  & Step size & [0,1] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._and \(f=4\), (3) \(m=10,n=100,s=50\) and \(f=8\), (4) \(m=15,n=200,s=50\) and \(f=4\), (5) \(m=15,n=200,s=40\) and \(f=8\), IHBA is 5-10% smaller than the other algorithms. Meanwhile, IHBA not only obtains the lowest relative percentage deviation value, but also the smallest average RPD value, which is 21.15%, with an error of about 0.83% from the optimal value. This is good proof that the IHBA algorithm has a strong advantage and superiority. It can also be seen that the GA and PSO performance is also very good, with average RPD values of 32.09% and 31.52%. Although TDIWO is not satisfactory in this comparison, it is significantly better than BBO and EDA. The most surprising is the original bat algorithm, with a result of 27.69, which is second only to the algorithm proposed in this paper. This also shows that the bat algorithm itself has better performance and has stronger search capability. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{10}{c}{**Parameters**} \\ \hline \(m\) & \(n\) & \(s\) & \(f\) & **CMA** & **BBO** & **TDIWO** & **EDA** & **GA** & **PSO** & **BA** & **IHBA** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: The Average RPD Values at \(C=20\). It can be seen from the bold figures that the value of PRD of the proposed algorithm IHBA in this paper is smaller than that of other algorithms in the vast majority of combinatorial solutions, including the recognized TDIWO algorithm and the original bat algorithm. Then, the results of the descriptive statistics of the Friedman test were calculated for the case of \(C=20\), as shown in Table 5, where \(N\) is the number of test cases and takes the value of 4050. The minimum value of each item derived in the algorithm is marked in bold. The \(p\)-value calculated for the test statistic is 0 when the significance level \(\alpha=0.05\), which is less than or equal to 0.05. This indicates that there are significant differences between the algorithms used for comparison. The following conclusions can be clearly seen through Table 5. The rank of IHBA is only 2.60, which is close to one-half of CMA, BBO and TDIWO, while it is smaller than the ranks of GA, PSO and the original bat algorithm by 1.39, 1.09 and 0.78, respectively. Although IHBA does not take the best value in terms of maximum value, it means that standard deviation and minimum value are the smallest among all algorithms. The comparison between the groups also shows that the ranking of EDA is even more than three times that of IHBA, indicating the worst performance of the algorithm, which echoes the results and analysis in Table 4. In addition, it can also be seen by the mean and standard deviation that the three algorithms of DIWO are very close to the proposed algorithm. Next, the non-parametric test for paired samples is the Wilcoxon paired signed test method used.