# 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 l l l l l l} \hline \hline  & \multicolumn{3}{c}{**XTornado-TS**} & \multicolumn{3}{c}{**XTornado-TM**} & \multicolumn{3}{c}{**XTornado-ATS**} \\ \hline
**Problem** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** \\ \hline ine \(F_{1}\) & \(1.42\times 10^{-2}\) & \(3.06\times 10^{-5}\) & \(1.14\times 10^{-2}\) & \(2.05\times 10^{-7}\) & \(1.43\times 10^{-2}\) & \(3.71\times 10^{-6}\) \\ \(F_{2}\) & \(6.74\times 10^{-3}\) & \(8.33\times 10^{-3}\) & \(1.61\times 10^{-2}\) & \(1.08\times 10^{-2}\) & \(1.88\times 10^{-2}\) & \(2.89\times 10^{-2}\) \\ \(F_{3}\) & \(5.64\times 10^{-2}\) & \(1.72\times 10^{-3}\) & \(4.69\times 10^{-2}\) & \(2.31\times 10^{-3}\) & \(5.61\times 10^{-2}\) & \(7.01\times 10^{-3}\) \\ \(F_{4}\) & \(3.90\times 10^{-2}\) & \(4.27\times 10^{-3}\) & \(1.45\times 10^{-2}\) & \(2.25\times 10^{-3}\) & \(4.16\times 10^{-2}\) & \(3.39\times 10^{-3}\) \\ \(F_{5}\) & \(7.47\times 10^{-2}\) & \(2.20\times 10^{-2}\) & \(6.46\times 10^{-2}\) & \(1.80\times 10^{-2}\) & \(1.53\times 10^{-2}\) & \(3.43\times 10^{-3}\) \\ \(F_{6}\) & \(1.43\times 10^{-2}\) & \(2.36\times 10^{-7}\) & \(1.14\times 10^{-2}\) & \(3.23\times 10^{-6}\) & \(1.43\times 10^{-2}\) & \(9.73\times 10^{-7}\) \\ \(F_{7}\) & \(1.43\times 10^{-2}\) & \(2.01\times 10^{-6}\) & \(1.14\times 10^{-2}\) & \(2.80\times 10^{-6}\) & \(1.43\times 10^{-2}\) & \(3.45\times 10^{-6}\) \\ \(F_{8}\) & \(1.01\times 10^{-2}\) & \(4.76\times 10^{-4}\) & \(2.44\times 10^{-2}\) & \(3.36\times 10^{-3}\) & \(9.93\times 10^{-2}\) & \(2.92\times 10^{-2}\) \\ \(F_{9}\) & \(5.95\times 10^{-2}\) & \(5.04\times 10^{-3}\) & \(4.94\times 10^{-2}\) & \(5.23\times 10^{-3}\) & \(5.68\times 10^{-2}\) & \(3.55\times 10^{-3}\) \\ \(F_{10}\) & \(2.70\times 10^{-1}\) & \(1.68\times 10^{-2}\) & \(3.15\times 10^{-1}\) & \(2.61\times 10^{-2}\) & \(2.57\times 10^{-1}\) & \(1.49\times 10^{-2}\) \\ \(F_{11}\) & \(9.03\times 10^{-2}\) & \(1.28\times 10^{-8}\) & \(1.14\times 10^{-2}\) & \(1.52\times 10^{-6}\) & \(9.03\times 10^{-2}\) & \(2.75\times 10^{-8}\) \\ \(F_{12}\) & \(4.67\times 10^{-3}\) & \(8.65\times 10^{-7}\) & \(1.86\times 10^{-2}\) & \(4.59\times 10^{-7}\) & \(4.67\times 10^{-3}\) & \(2.88\times 10^{-7}\) \\ \(F_{13}\) & \(1.48\times 10^{-2}\) & \(1.15\times 10^{-12}\) & \(4.46\times 10^{-2}\) & \(1.43\times 10^{-11}\) & \(1.48\times 10^{-2}\) & \(3.34\times 10^{-12}\) \\ \(F_{14}\) & \(1.44\times 10^{-1}\) & \(1.14\times 10^{-9}\) & \(4.57\times 10^{-2}\) & \(8.71\times 10^{-11}\) & \(1.44\times 10^{-1}\) & \(2.13\times 10^{-9}\) \\ \hline \hline \end{tabular}  
* TS: Standard Tchebytcheff, TM: Tchebytcheff with multiple references points, STS: Augmented Tchebytcheff.  
\end{table} 
Table 3: Results of the Spacing metric \((S)\) for the three versions of X-Tornado on the 14 selected problems. By analysing the obtained results in Table 4, it is clear that the X-Tornado approach has the best performance in term of convergence to the front. Indeed the proposed X-Tornado obtains the lowest \(GD\) metric value for twelve out of the 18 test problems and with small standard deviation in almost all problems. A low \(GD\) metric value of an algorithm on a problem is significant for accuracy of the obtained Pareto front. That is to say, the proposed algorithm has a good accuracy and stability on the performance of these problems. In Table 5 X-Tornado outperforms all other algorithms on the mean of the spacing metric in almost all test problems except in ZDT3, KUR, MSC and DTLZ4. The next Tables 6 and 7 show the comparisons result in case of doubling the archive length used by the three meta heuristics involved in the comparison with our X-Tornado method. Indeed, as w can see, X-Tornado is still having slightly better performance among the compared methods. But, the most interesting thing that had motived this additional simulation is to demonstrate the superiority of our method over the classical approach adopting an archiving mechanism in term of Time execution.