## 6 Simulations
### Simulation Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Mig.** & \multirow{2}{*}{**\# of**} & \multicolumn{5}{c}{**Ring**} & \multicolumn{5}{c}{**Trigonal Pyramid/Cube/Hyper-Cube**} \\ \cline{3-14}  & & & \multirow{2}{*}{**Sub-pop.**} & \multirow{2}{*}{**Mean**} & & \multirow{2}{*}{**Min.**} & & \multirow{2}{*}{**Max.**} & & & **Std.** & **Ave.** & & \multirow{2}{*}{**Mean**} \\  & & & & & & & & & & & & & Rank \\ \hline - & - & 1 & 100.00 & 99.90 & 100.17 & 0.06 & 18.82 & - & - & - & - & - \\ \hline \multirow{4}{*}{Abest} & \multirow{4}{*}{-} & \multirow{4}{*}{1} & 2 & 99.94 & 99.82 & 100.06 & 0.05 & 17.44 & - & - & - & - & - \\ \cline{3-14}  & & 4 & 99.89 & 99.79 & 99.99 & 0.04 & 16.22 & - & - & - & - & - \\ \cline{3-14}  & & 8 & 99.80 & 99.73 & 99.89 & 0.03 & 11.66 & - & - & - & - & - \\ \cline{3-14}  & & 16 & 99.71 & 99.59 & 99.83 & 0.04 & 5.64 & - & - & - & - & - \\ \hline \multirow{4}{*}{Abest \& Mig.} & \multirow{4}{*}{W-B} & 2 & 99.85 & 99.73 & 99.92 & 0.05 & 14.28 & - & - & - & - & - \\ \cline{3-14}  & & 4 & 99.79 & 99.67 & 99.87 & 0.05 & 10.52 & 99.80 & 99.71 & 99.91 & 0.05 & 11.02 \\ \cline{3-14}  & & 8 & 99.73 & 99.62 & 99.86 & 0.05 & 6.76 & 99.75 & 99.53 & 99.86 & 0.06 & 8.3 \\ \cline{3-14}  & & 16 & 99.67 & 99.5 & 99.83 & 0.08 & 4.52 & 99.73 & 99.53 & 99.87 & 0.08 & 7.04 \\ \hline \multirow{4}{*}{Mig.} & \multirow{4}{*}{W-B} & 2 & 99.86 & 99.78 & 99.97 & 0.04 & 15.46 & - & - & - & - & - \\ \cline{3-14}  & & 4 & 99.78 & 99.69 & 99.89 & 0.05 & 9.94 & 99.79 & 99.71 & 99.87 & 0.04 & 11.18 \\ \cline{3-14}  & & 8 & 99.72 & 99.59 & 99.82 & 0.06 & 6.18 & 99.73 & 99.6 & 99.89 & 0.06 & 6.8 \\ \cline{3-14}  & & **16** & **99.63** & **99.45** & **99.82** & **0.09** & **3.14** & 99.68 & 99.52 & 99.91 & 0.08 & 5.08 \\ \hline \multicolumn{14}{c}{\(p\)-value} & \multicolumn{5}{c}{\(4.89\times 10^{-120}\)} \\ \hline \hline \end{tabular}  
* All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%.  
\end{table} 
Table 4: Comparison of the mean, the minimum, the maximum, the standard deviation, and average rank of the objective function value of the optimal objective function values among various numbers of sub-populations, and topologies using both migration with the _W-B_ policy and abest, only migration with the _W-B_ policy, and only abest with 640 individuals through 50 trials for Case 1 and a \(p\)-value by Friedman test. 1 \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Mig. Policy** & \multirow{2}{*}{**\# of sub-pop.**} & \multicolumn{5}{c}{**Ring**} \\ \cline{3-14}  & & & **Mean** & & **Min.** & **Max.** & **Std.** & **Ave. Rank** \\ \hline - & - & 1 & 100.00 & 99.90 & 100.17 & 0.06 & 4 \\ \hline Abest & - & 16 & 99.71 & 99.59 & 99.83 & 0.04 & 2.4 \\ \hline Abest \& Mig. & _W-B_ & 16 & 99.67 & 99.5 & 99.83 & 0.08 & 2.02 \\ \hline
**Mig.** & _W-B_ & **16** & **99.63** & **99.45** & **99.82** & **0.09** & **1.58** \\ \hline \multicolumn{14}{c}{\(p\)-value} & \multicolumn{5}{c}{\(1.48\times 10^{-21}\)} \\ \hline \hline \end{tabular}  
* All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%.  
\end{table} 
Table 5: The best values of the mean, the minimum, the maximum, the standard deviation, and the average rank of the objective function value of the optimal objective function values of each migration model with 640 individuals through 50 trials for Case 1 from Table 4 and a \(p\)-value by Friedman test. 1  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Policy**} & \multirow{2}{*}{**NSP**} & \multicolumn{4}{c}{**Ring**} & \multicolumn{4}{c}{**Cube/Trigonal Pyramid/Hypercube**} & \multirow{2}{*}{**Ave. Rank**} \\ \cline{2-2} \cline{4-11}  & & **Mean** & **Min. ** & **Max. ** & **Std. ** & & & **Mean** & **Min. ** & **Max. ** & **Std.