# 2-Phase NSGA II: An Optimized Reward and Risk Measurements Algorithm in Portfolio Optimization
## 5 Performance Evaluation
### Experimental Results  
This sub-section compares the proposed 2-phase NSGA II with the known NSGA II algorithm. Both algorithms were implemented in MATLAB on a laptop Core 2 Duo at 2 GHz with 4 GB RAM under Windows Vista operating system.  
Each algorithm was performed ten times and all results of these ten runs were overlapped. Then, the non-dominated solutions were chosen. Figure 3 presents the non-dominated solutions of accumulated runs by NSGA II and 2-phase NSGA II for five test problems. As it can be seen, the introduced algorithm worked impressively in all problem sizes. The values obtained for all metrics for all test problems are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Problem Size**} & \multicolumn{4}{c}{**Algorithm and Index**} \\ \cline{2-5}  & & \multicolumn{2}{c}{_MID_} & \multicolumn{2}{c}{_NPS_} \\ \hline
**Port 1** & **N** & **NSGA II** & **2-p NSGA II** & **NSGA II** & **2-p NSGA II** \\ \hline
**Port 1** & 31 & 0.2025 & 0.11623 & 0.20062 & 0.01591 \\
**Port 2** & 85 & 0.06574 & 0.06453 & 0.06887 & 0.00284 \\
**Port 3** & 89 & 0.1132 & 0.11276 & 0.15946 & 0.0056 \\
**Port 4** & 98 & 0.19875 & 0.19848 & 0.34357 & 0.00777 \\
**Port 5** & 225 & 0.31602 & 0.31365 & 0.22209 & 0.00558 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Port 1**} & **N** & **NSGA II** & **2-p NSGA II** \\ \hline
**Port 2** & 85 & 0.01813 & 0.77739 \\
**Port 3** & 89 & 0.03507 & 0.8947 \\
**Port 4** & 98 & 0.01618 & 0.82006 \\
**Port 5** & 225 & 0.01861 & 0.90701 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Evaluation of non-dominated solution.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Problems** & _NoG_ & _popsize_ & \(k\) & \(M\) \\ \hline
**Port 1** & 400 & 100 & 10 & 8 \\
**Port 2** & 500 & 200 & 20 & 17 \\
**Port 3** & 300 & 200 & 20 & 18 \\
**Port 4** & 500 & 200 & 20 & 14 \\
**Port 5** & 300 & 200 & 30 & 23 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Distinguished parameters for each problem size.  
Obviously, according to \(S\), \(CS\) and \(NPS\) criteria the 2-phase NSGA II acquired better results than NSGA II, meanwhile based on \(MID\) and \(MS\) factors there are not any clear differences between the two algorithms.  
Analysis of the Results  
We used non-parametric statistical test of Mann-Whitney to analyze the outputs of the algorithms. \(P_{value}\) of each metric for all five test problems is represented in Table 5, according to which no significant differences were observed between the two algorithms based on MID and MS metrics while the significant differences between the two algorithms was clear according to \(NPS\), \(S\) and \(CS\).  
Figure 3: Pareto-front of non-dominated solutions yielded by both algorithms for five test problems.  
Also, Figure 4 shows means plot for the interaction among the metrics of algorithm and test problems. As can be seen, the proposed algorithm has generally better performance based on \(S\), \(CS\) and \(NPS\) metrics. According to \(MID\) and \(MS\) metrics, the graphs of both algorithms are very close to each other.