# A New Chaotic-Based Approach for Multi-Objective Optimization
## 4 Computational Experiments
### Performances Measures  
Due to the fact that the convergence to the Pareto optimal front and the maintenance of a diverse set of solutions are two different goals of the multi-objective optimization, two performance measures were adopted in this study: the generational distance \((GD)\) to evaluate the convergence,and the Spacing \((S)\) to evaluate the diversity and cardinality.  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Problem** & **Name** & \(n\) & **Bounds** & **Objective Functions** & **Comments** \\ \hline ine \(F_{1}\) & \(ZDT1\) & \(30\) & \([0,1]^{n}\) & \(f_{1}(x)=x_{1};f_{2}(x)=g(x)(1-\sqrt{x_{1}/g(x)})\) & Convex \\  & & & & \(g(x)=1+\frac{9}{8}\sum\limits_{i=2}^{n}x_{i}\) & \\ \(F_{2}\) & \(ZDT2\) & \(30\) & \([0,1]^{n}\) & \(f_{1}(x)=x_{1};f_{2}(x)=g(x)(1-(x_{1}/g(x))^{2})\) & Non Convex \\  & & & & \(g(x)=1+\frac{9}{8}\sum\limits_{i=2}^{n}x_{i}\) & \\ \(F_{3}\) & \(ZDT3\) & \(30\) & \([0,1]^{n}\) & \(f_{1}(x)=x_{1};f_{2}(x)=g(x)(1-\sqrt{x_{1}/g(x)}-(x_{1}/g(x))\sin(10\pi x_{1})\) & Convex \\  & & & & \(g(x)=1+\frac{9}{8}\sum\limits_{i=2}^{n}x_{i}\) & \\ \(F_{4}\) & \(ZDT4\) & \(30\) & \([0,1]^{n}\) & \(f_{1}(x)=x_{1};f_{2}(x)=g(x)(1-\sqrt{x_{1}/g(x)})\) & Convex \\  & & & & \(g(x)=1+0(n-1)\sum\limits_{i=2}^{n}x_{i}^{2}-10\cos(4\pi x_{i})\) & \\ \(F_{5}\) & \(ZDT6\) & \(30\) & \([0,1]^{n}\) & \(f_{1}(x)=1-\exp(4\pi i_{1})\sin^{6}(6\pi x_{1}));f_{2}(x)=g(x)(1-\sqrt{x_{1}/g( x)})\) & Convex \\  & & & & \(g(x)=1+g\left[\frac{1}{8}\sum\limits_{i=2}^{n}x_{i}\right]^{0.025}\) & \\ \(F_{6}\) & POL & \(2\) & \([-\pi,\pi]^{2}\) & \(f_{1}(x)=1+(A_{1}-B_{1})^{2}+(A_{2}-B_{2})^{2};f_{2}(x)=(x_{1}+3)^{2}+(x_{2}+1 )^{2}\) & Convex \\  & & & & \(A_{1}=0.5\sin(1)-2\cos(1)+\sin(2)-1.5\cos(2)\) & \\  & & & & \(A_{2}=1.5\sin(1)-\cos(1)+2\sin(2)-0.5\cos(2)\) & \\  & & & & \(B_{1}=0.5\sin(x_{1})-2\cos(x_{1})+\sin(x_{2})-1.5\cos(x_{2})\) & \\  & & & & \(B_{2}=1.5\sin(x_{1})-\cos(x_{1})+2\sin(x_{2})-0.5\cos(x_{2})\) & \\ \(F_{7}\) & \(MOP1\) & \(1\) & \([-2,2]\) & \(f_{1}(x)=-x_{1,2}[-1]+(x-2)[1,3](x)+(4-x)[1,3](x)\) & Non Convex \\  & & & & \(+(x-4)[1,4](x);\)\(\ f_{2}(x)=(x-5)^{2}\) & \\ \(F_{8}\) & _No-Hole_ & \(2\) & \([-1,1]^{2}\) & \(f_{1}(x)=(t^{1})^{2}+a;f_{2}(x)=(t-1)^{2}+a\) & Convex \\ \(F_{9}\) & _Hole_ & \(2\) & \([-1,1]^{2}\) & \(f_{1}(x)=(t+1)^{2}+a+b\exp[-c(t-d)^{2}]\) & Non Convex \\  & & & & \(f_{2}(x)=(t-1)^{2}+a+b\exp[-c(t+d)^{2}]\) & \\ \(F_{10}\) & \(KUR\) & \(3\) & \([-5,5]^{3}\) & \(f_{1}(x)=\frac{x}{i=2}(-10\exp(-0.2\sqrt{x}+x_{i}^{2})\ ));f_{2}(x)=\sum\limits_{i=1}^{n}|x_{i}|^{0 8}+5\sin(x_{i}^{2})\) & Convex \\ \(F_{11}\) & \(SCH\) & \(1\) & \([-10^{+3},10^{+3}]\) & \(f_{1}(x)=x^{2};f_{2}(x)=(x-2)^{2}\) & Convex \\ \(F_{12}\) & \(FON\) & \(3\) & \([-4,4]^{3}\) & \(f_{1}(x)=1-\exp\big{(}-\sum\limits_{i=1}^{3}(x_{i}-\frac{1}{\sqrt{3}})\big{)};f_{ 2}(x)=1-\exp\big{(}-\sum\limits_{i=1}^{3}(x_{i}+\frac{1}{\sqrt{3}})\big{)}\) & Convex \\ \(F_{13}\) & \(MUR\) & \(2\) & \([0,10]\times\) & \(f_{1}(x)=2\sqrt{x_{1}};f_{2}(x)=\pi_{1}(1-x_{2})+5\) & Non Convex \\  & & & \([-10,10]\) & \(f_{1}(x)=\exp(-x)+1.4\exp(-x^{2});f_{2}(x)=\exp(x)+1.4\exp(-x^{2})\) & Non Convex \\ \(F_{14}\) & \(MSC\) & \(1\) & \([-2,2]\) & \(f_{1}(x)=\exp(-x)+1.4\exp(-x^{2});f_{2}(x)=\exp(x)+1.4\exp(-x^{2})\) & Non Convex \\ \hline \hline \end{tabular}
\end{table}
Table 1: Benchmark Problems used in our experiments.  
* The convergence metric \((GD)\) measure the extent of convergence to the true Pareto front. It is defined as: \[GD=\frac{1}{N}\sum_{i=1}^{N}d_{i},\] (32) where \(N\) is the number of solutions found and \(d_{i}\) is the Euclidean distance between each solution and its nearest point in the true Pareto front. The lower value of GD, the better convergence of the Pareto front to the real one.
* The Spacing metric \(S\) indicates how the solutions of an obtained Pareto front are spaced with respect to each other. It is defined as: \[S=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(d_{i}-\overline{d})^{2}}\] (33)