# Multi-Guide Set-Based Particle Swarm Optimization for Multi-Objective Portfolio Optimization
## 6 Results
### Dax 100  
The second benchmark problem included 54 more assets than the previous benchmark problem--a notable increase. The single-objective SBPSO performed well, with the lowest average risk value and second highest average return value (refer to Table 7. MGSBPSO also performed well, with average values close to those of SPEA2. The standard deviations \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & **GD** & **IGD** & **HV** & **Overall** \\ \hline \multirow{6}{*}{SBPSO} & Wins & 4 & 1 & 0 & 5 \\  & Losses & 0 & 2 & 4 & 6 \\  & Draws & 0 & 1 & 0 & 1 \\  & Difference & 4 & \(-1\) & \(-4\) & \(-1\) \\  & Rank & 1 & 3 & 5 & 4 \\ \hline \multirow{6}{*}{MGPSO} & Wins & 2 & 1 & 2 & 5 \\  & Losses & 1 & 2 & 2 & 5 \\  & Draws & 1 & 1 & 0 & 2 \\  & Difference & 1 & \(-1\) & 0 & 0 \\  & Rank & 2 & 3 & 3 & 3 \\ \hline \multirow{6}{*}{NSGA\(-\)II} & Wins & 0 & 3 & 4 & 7 \\  & Losses & 4 & 1 & 0 & 5 \\  & Draws & 0 & 0 & 0 & 0 \\  & Difference & \(-4\) & 2 & 4 & 2 \\  & Rank & 4 & 2 & 1 & 2 \\ \hline \multirow{6}{*}{SPEA2} & Wins & 2 & 4 & 3 & 9 \\  & Losses & 1 & 0 & 1 & 2 \\  & Draws & 1 & 0 & 0 & 1 \\  & Difference & 1 & 4 & 2 & 7 \\  & Rank & 2 & 1 & 2 & 1 \\ \hline \multirow{6}{*}{MGSBPSO} & Wins & 1 & 0 & 1 & 2 \\  & Losses & 3 & 4 & 3 & 10 \\ \cline{1-1}  & Draws & 0 & 0 & 0 & 0 \\ \cline{1-1}  & Difference & \(-2\) & \(-4\) & \(-2\) & \(-8\) \\ \cline{1-1}  & Rank & 3 & 4 & 4 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Hang Seng rankings for each performance measure. Figure 3: Obtained Pareto-optimal fronts for Hang Seng. of the results for SPEA2 were smaller than those of MGSBPSO, which could explain the large difference in rankings in Table 8. SPEA2 ranked first, while MGSBPSO ranked last. NSGA-II ranked second, MGPSO ranked third, and SBPSO ranked fourth. However, Figure 4d,e show that NSGA-II and SPEA2 (respectively) were only able to approximate a part of the true POF shown in Figure 4a. The PSO algorithms were able to approximate the true POF well, particularly SBPSO and MGSBPSO. \begin{table}
\begin{tabular}{l c c c c c} \hline  & \(\mathbf{R}\) & \(\mathbf{\tilde{\sigma}}\) & \(\mathbf{GD}\) & \(\mathbf{IGD}\) & \(\mathbf{HV}\) \\ \hline \multirow{2}{*}{SBPSO} & \(\bar{x}\) & 0.007607 & 0.001897 & 0.000218 & 0.000212 & 0.781949 \\  & \(\sigma\) & 0.000003 & 0.000000 & 0.000045 & 0.000002 & 0.026784 \\ \hline \multirow{2}{*}{MGPSO} & \(\bar{x}\) & 0.007261 & 0.001967 & 0.000723 & 0.000218 & 1.192743 \\  & \(\sigma\) & 0.000705 & 0.000407 & 0.000262 & 0.000025 & 0.002658 \\ \hline \multirow{2}{*}{NSGA-II} & \(\bar{x}\) & 0.007859 & 0.002171 & 0.000824 & 0.000200 & 1.195125 \\  & \(\sigma\) & 0.000103 & 0.000063 & 0.000064 & 0.000006 & 0.000737 \\ \hline \multirow{2}{*}{SPEA2} & \(\bar{x}\) & 0.007395 & 0.001921 & 0.000659 & 0.000173 & 1.194271 \\  & \(\sigma\) & 0.000099 & 0.000046 & 0.000042 & 0.000003 & 0.000345 \\ \hline \multirow{2}{*}{MGSBPSO} & \(\bar{x}\) & 0.007382 & 0.001814 & 0.000761 & 0.000252 & 1.190891 \\  & \(\sigma\) & 0.000552 & 0.000326 & 0.000077 & 0.000029 & 0.001930 \\ \hline \end{tabular}
\end{table} 
Table 7: DAX 100 results for each performance measure.