# A New Chaotic-Based Approach for Multi-Objective Optimization
## 4 Computational Experiments
### Comparison with Some State-Of The-Art Evolutionary Algorithms
#### 4.5.1 Application to Multi-Objective Structural Optimization Problem  
In this section, we consider two applications: The first one is the four bar truss problem proposed by [PERSON] in [28]. The goal is to find the optimal truss structure while simultaneously minimizing the total mass and static displacement at point C. These two criteria are in conflict since minimizing the mass of a structure tends to increase displacement. So the best solution is to find a trade off between the two criteria. For this, we consider two cost functions to minimize. the total volume ( \(f_{1}\) (cm\({}^{3}\))) and displacement ( \(f_{2}\) (cm)). The four bar truss problem is shown in Figure 16.  
Figure 15: Pareto Front captured by X-Tornado for problems \(F_{13}-F_{18}\)The second application is the two bar truss structure subjected to random Gaussian loading [29]. The two bar problem is a bi-objective problem where the areas of the two bars are decision variables of the optimization. The left end of the bars is fixed while the other end is subjected to a mean plus a fluctuating load. Two objectives are considered for the optimization problem: the mean and the standard deviation of the vertical displacement. Figure 17 illustrates the two bar problem and further technical details can be found in [29].  
Table 10 show the comparison results for the four methods in comparisons in term of GD and Spacing metrics for 300,000 Fe. By analysing these results we observe that X-Tornado is performing as well as Nsga2 in term of convergence and spacing metrics and outperforms Pesa2 and MOEA/D for this two problems. Moreover, X-Tornado is much faster in comparison to the three algorithms.  
\begin{table}
\begin{tabular}{l l c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{} & \multicolumn{4}{c}{**GD**} & \multicolumn{4}{c}{**Spacing**} & \multicolumn{4}{c}{**Spread**} & \multicolumn{1}{c}{**CPU(s)**} \\ \hline
**Problem** & **Method** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** & **Std** & **Mean** \\ \hline ine & Nsga2 & \(\mathbf{1.49\times 10^{-3}}\) & \(\mathbf{5.08\times 10^{-5}}\) & \(\mathbf{1.10\times 10^{+1}}\) & \(\mathbf{1.06\times 10^{+0}}\) & \(\mathbf{8.53\times 10^{-1}}\) & \(9.00\times 10^{-3}\) & 299.0 \\ TR4 & Pesa2 & \(6.75\times 10^{-3}\) & \(1.40\times 10^{-4}\) & \(2.19\times 10^{+1}\) & \(6.18\times 10^{+0}\) & \(9.72\times 10^{-1}\) & \(4.37\times 10^{-2}\) & 141.3 \\ Problem & MOEA/D & \(1.44\times 10^{-2}\) & \(8.43\times 10^{-3}\) & \(2.03\times 10^{-18}\) & \(3.52\times 10^{-18}\) & \(1.00\times 10^{+0}\) & 0 & 257.3 \\  & X-Tornado & \(2.27\times 10^{-3}\) & \(4.90\times 10^{-5}\) & \(1.63\times 10^{+1}\) & \(7.36\times 10^{-1}\) & \(3.74\times 10^{-5}\) & \(5.35\times 10^{-8}\) & **15.8** \\ ine & Nsga2 & \(\mathbf{1.09\times 10^{-3}}\) & \(\mathbf{2.26\times 10^{-5}}\) & \(\mathbf{1.45\times 10^{-4}}\) & \(\mathbf{5.96\times 10^{-6}}\) & \(9.95\times 10^{-1}\) & \(1.23\times 10^{-4}\) & 324.9 \\ TR2 & Pesa2 & \(7.50\times 10^{-0}\) & \(1.98\times 10^{+0}\) & \(3.05\times 10^{-2}\) & \(8.45\times 10^{-3}\) & \(1.09\times 10^{+0}\) & \(1.29\times 10^{-2}\) & 189.0 \\ Problem & MOEA/D & \(1.80\times 10^{-5}\) & \(9.91\times 10^{-6}\) & \(2.14\times 10^{-3}\) & \(1.76\times 10^{-6}\) & \(1.01\times 10^{+0}\) & \(7.62\times 10^{-6}\) & 294.5 \\  & X-Tornado & \(1.94\times 10^{-3}\) & \(5.09\times 10^{-10}\) & \(1.52\times 10^{-4}\) & \(2.45\times 10^{-11}\) & \(9.91\times 10^{-1}\) & \(1.02\times 10^{-9}\) & **74.6** \\ \hline \hline \end{tabular}
\end{table}
Table 10: Comparison Results obtained by the four methods for TR4 and TR2 problems.  
Figure 16: Four-bar Truss Problem (TR4)  
Figure 17: Two-bar Truss Problem (TR2)