## 6 Simulations
### Simulation Results  
Firstly, the comparison of the mean, the minimum, the maximum, and the standard deviation of the objective function values for Case 1 among DEEPSO, BSO, MBSO, GBSO, and GMBSO based methods are shown in Table 2. Eighty individuals are utilized in Table 2. The number of individuals is set to 80 because the condition is the most severe for all methods. The results in the table are percentages of the objective function value when the mean of the objective function value by DEEPSO is set to 100%. It is verified that the mean, the minimum, the maximum, and the standard deviation of GMBSO based method can be the most reduced among all methods. Table 3 shows results of the mean ranks for Friedman test and \(p\)-value among the conventional DEEPSO, BSO, GBSO, MBSO, and GMBSO based methods. It is verified that there are significant differences at 0.05 significance level among all methods and the mean ranks by the GMBSO based method for each case are the best in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & & **Mean** & **Min.** & **Max.** & **Std.** \\ \hline \multirow{8}{*}{\(1\)} & DEEPSO & 100.00 & 98.75 & 101.63 & 0.57 \\ \cline{2-6}  & BSO & 97.13 & 96.46 & 97.96 & 0.30 \\ \cline{2-6}  & GBSO & 95.94 & 95.55 & 97.03 & 0.26 \\ \cline{2-6}  & MBSO & 97.20 & 96.75 & 97.66 & 0.20 \\ \cline{2-6}  & **GMBSO** & **95.06** & **94.90** & **95.29** & **0.09** \\ \hline \multirow{8}{*}{\(2\)} & DEEPSO & 100.00 & 99.53 & 100.58 & 0.20 \\ \cline{2-6}  & BSO & 99.28 & 98.98 & 99.60 & 0.14 \\ \cline{1-1} \cline{2-6}  & GBSO & 98.29 & 98.22 & 98.42 & 0.04 \\ \cline{1-1} \cline{2-6}  & MBSO & 99.38 & 99.15 & 99.50 & 0.06 \\ \cline{1-1} \cline{2-6}  & **GMBSO** & **98.26** & **98.17** & **98.36** & **0.04** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the mean, the minimum, the maximum, and the standard deviation of the objective function value among conventional DEEPSO, BSO, MBSO, GBSO, and the proposed GMBSO based methods with 80 individuals. \({}^{1}\)Secondly, the best multi-population model is investigated among three models, namely, only migration model, only abest model, or a model using both migration and abest. Table 4 shows comparison of the mean, the minimum, the maximum, the standard deviation, and the mean rank of the objective function value 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 for Case 1 through 50 trials and a _p_-value by Friedman test. Six hundred and forty individuals are utilized in the table because there is a possibility to obtain better results by the MP-GMBSO based method through previous studies. The results of Table 4 are calculated when the mean of the objective function value using one sub-population is set to 100%. It is verified that the mean, the minimum, the maximum, and the standard deviation can be the most reduced by the only migration model with 16 sub-populations (**bold numbers**). It can be considered that the GMBSO based method can focus on intensification more than the other methods including the multi-swarm DEEPSO based method with a model using both migration and abest [26]. Therefore, the only migration model is the best model especially for the proposed MP-GMBSO based model in order to balance diversification and intensification. In addition, the mean rank by the only migration model with 16 sub-populations is the best among all parameters in the table. Table 5 shows 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 and a _p_-value by Friedman test. It is verified that the mean, the minimum, the maximum value, and the average rank using only migration model are the best among all multi-population models.