# A New Chaotic-Based Approach for Multi-Objective Optimization
## 4 Computational Experiments
### Comparison with Some State-Of The-Art Evolutionary Algorithms
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Nsq2**} & \multicolumn{2}{c}{**Pes2**} & \multicolumn{2}{c}{**Moead**} & \multicolumn{2}{c}{**Xtornado-TM**} \\ \hline
**Fct** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** \\ \hline ine \(F_{1}\) & \(1.20\times 10^{-2}\) & \(4.19\times 10^{-3}\) & \(3.70\times 10^{-1}\) & \(3.51\times 10^{-2}\) & \(3.74\times 10^{-2}\) & \(1.79\times 10^{-2}\) & \(\mathbf{1.14\times 10^{-2}}\) & \(\mathbf{2.05\times 10^{-7}}\) \\ \(F_{2}\) & \(7.75\times 10^{-3}\) &archiving mechanism (which is often the case) the corresponding cost is usually not considered since it has no sensitive effect in comparison. However, in our case, dislike the methods in comparisons, X-Tornado don't use an archiving mechanism, whereas on the other side doubling the archive has considerably increased the execution time as can be observed in Table 8.  
Moreover, the distribution of the non dominated points reflects automatically the regularity of the Tchebychev decomposition and unlike the three other algorithms in comparison, a filtering mechanism such as \"crowding\" is no longer required in our X-Tornado approach. The consequence of this important advantage is that unlike the other methods in comparison, the capture of the front by the X-Tornado method is more continuous and therefore more precise, as illustrated by the Figures 13-15. In addition, the X-Tornado method is much faster (at least 5 times less expensive in CPU time) compared to other methods as we can observe in Table 9. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Nsga2**} & \multicolumn{2}{c}{**Pesa2**} & \multicolumn{2}{c}{**Moead**} & \multicolumn{2}{c}{**X-Tornado**} \\ \hline
**Problem** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** \\ \hline ine \(F_{1}\) & \(3.79\times 10^{-3}\) & \(7.68\times 10^{-4}\) & \(6.44\times 10^{-2}\) & \(1.31\times 10^{-2}\) & \(4.46\times 10^{-3}\) & \(7.26\times 10^{-3}\) & \(2.38\times 10^{-3}\) & \(7.10\times 10^{-7}\) \\ \(F_{2}\) & \(3.97\times 10^{-3}\) & \(8.13\times 10^{-4}\) & \(8.72\times 10^{-2}\) & \(6.64\times 10^{-3}\) & \(1.05\times 10^{-1}\) & \(5.58\times 10^{-2}\) & \(6.61\times 10^{-4}\) & \(4.42\times 10^{-4}\) \\ \(F_{3}\) & \(3.86\times 10^{-3}\) & \(1.07\times 10^{-3}\) & \(4.37\times 10^{-2}\) & \(2.78\times 10^{-3}\) & \(2.39\times 10^{-2}\) & \(7.76\times 10^{-3}\) & \(2.45\times 10^{-3}\) & \(1.68\times 10^{-4}\) \\ \(F_{4}\) & \(8.82\times 10^{-2}\) & \(9.24\times 10^{-2}\) & \(1.11\times 10^{+1}\) & \(6.62\times 10^{-1}\) & \(3.31\times 10^{+0}\) & \(1.05\times 10^{+0}\) & \(1.72\times 10^{-3}\) & \(3.61\times 10^{-4}\) \\ \(F_{5}\) & \(9.99\times 10^{-4}\) & \(2.52\times 10^{-5}\) & \(6.70\times 10^{-2}\) & \(0\) & \(8.52\times 10^{-4}\) & \(5.15\times 10^{-5}\) & \(3.88\times 10^{-3}\) & \(1.97\times 10^{-3}\) \\ \(F_{6}\) & \(4.44\times 10^{-4}\) & \(3.45\times 10^{-5}\) & \(4.08\times 10^{-1}\) & \(0\) & \(2.72\times 10^{-1}\) & \(1.97\times 10^{-1}\) & \(1.39\times 10^{-3}\) & \(3.41\times 10^{-5}\) \\ \(F_{7}\) & \(6.57\times 10^{-4}\) & \(2.38\times 10^{-5}\) & \(1.18\times 10^{-1}\) & \(0\) & \(6.86\times 10^{-3}\) & \(5.65\times 10^{-3}\) & \(2.38\times 10^{-3}\) & \(5.41\times 10^{-7}\) \\ \(F_{8}\) & \(7.65\times 10^{-3}\) & \(1.38\times 10^{-2}\) & \(5.27\times 10^{-2}\) & \(0\) & \(1.00\times 10^{-2}\) & \(1.45\times 10^{-3}\) & \(2.38\times 10^{-3}\) & \(1.23\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The Generational distance metric \((GD)\) comparison results for the for algorithms on the 8 selected problems.