# Elite Multi-Criteria Decision Making--Pareto Front Optimization in Multi-Objective Optimization
## Appendix A  
This section contains all the test MOO functions used for this paper, including the constraints and search domains of each MOO function. Figure 5: Pareto front for the MHHM functions. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Function** & **Test Function** & **Constraints** & **Search Domain** \\ \hline Binh and Korn Function & \(f_{1}(x)=4x_{1}^{2}+4x_{2}^{2}\) & \((x_{1}-5)^{2}+x_{2}^{2}\leq 25\) & \(0\leq x_{1}\leq 5\) \\  & \(f_{2}(x)=(x_{1}-5)^{2}+(x_{2}-5)^{2}\) & \((x_{1}-8)^{2}+(x_{2}+3)^{2}\geq\) & \(0\leq x_{2}\leq 3\) \\  & & 7.7 & \\ \hline Chankong and Haimes Function & \(f_{1}(x_{1},x_{2})=2+(x_{1}-2)^{2}+(x_{2}-1)^{2}\) & \(x_{1}^{2}+x_{2}^{2}\leq 225\) & \(-20\leq x_{1}\) \\  & \(f_{2}(x_{1},x_{2})=9x_{1}-(x_{2}-1)^{2}\) & \(x_{1}-3x_{2}+10\leq 0\) & \(x_{2}\leq 20\) \\ \hline Fonseca–Fleming Function & \(f_{1}(\mathbf{x})=1-\exp\left(-\sum_{i=1}^{n}\left(x_{i}-\frac{1}{\sqrt{n}} \right)^{2}\right)\) & & \(-4\leq x_{i}\leq 4\) \\  & \(f_{2}(\mathbf{x})=1-\exp\left(-\sum_{i=1}^{n}\left(x_{i}+\frac{1}{\sqrt{n}} \right)^{2}\right)\) & & \(1\leq i\leq n\) \\ \hline Test Function 4 & \(f_{1}(x_{1},x_{2})=x_{1}^{2}-x_{2}\) & \(6.5-\frac{x_{1}}{6}-x_{2}\geq 0\) & \(-7\leq x_{1}\) \\  & \(f_{2}(x_{1},x_{2})=-0.5x_{1}-x_{2}-1\) & \(7.5-0.5x_{1}-x_{2}\geq 0\) & \(x_{2}\leq 4\) \\  & \(30-5x_{1}-x_{2}\geq 0\) & & \\ \hline Kursawe Function & \(f_{1}(\mathbf{x})=\sum_{i=1}^{n-1}\left(-10\exp\left(-0.2\sqrt{x_{i}^{2}+x_{i+ 1}^{2}}\right)\right)\) & & \(-5\leq x_{i}\leq 5\) \\  & \(f_{2}(\mathbf{x})=\sum_{i=1}^{n}\left(|x_{i}|^{0.8}+5\sin(x_{i}^{3})\right)\) & & \(1\leq i\leq 3\) \\ \hline Schaffer Function N1 & \(f_{1}(x)=x^{2}\) & & \(-A\leq x\leq A\) \\  & \(f_{2}(x)=(x-2)^{2}\) & & Values of A from \\  & & & 10 to \(10^{5}\) \\ \hline Schaffer Function N2 & \(f_{1}(x)= \begin{cases}-x&x\leq 1\\ x-2&1<x\leq 3\\ 4-x&3<x\leq 4\\ x-4&x>4\end{cases} \) & & \(-5\leq x\leq 10\) \\  & \(f_{2}(x)=(x-5)^{2}\) & & \\ \hline [PERSON]’s Two-Objective Function & \(f_{1}(x_{1},x_{2})=[1+(A_{1}-B_{1}(x_{1},x_{2}))^{2}+(A_{2}-\pi\leq x_{1};x_{2} \leq\pi\) \\  & \(B_{2}(x_{1},x_{2}))^{2}]\) & & \\  & \(f_{2}(x_{1},x_{2})=(x_{1}+3)^{2}+(x_{2}+1)^{2}\) & & \\  & where & \(A_{1}=0.5\sin(1)-2\cos(1)+\sin(2)-1.5\cos(2)\), & \\  & \(A_{2}=1.5\sin(1)-\cos(1)+2\sin(2)-0.5\cos(2)\), & \\  & \(B_{1}(x_{ \begin{table}
\begin{tabular}{l l c c c c} \hline \hline
**Test Function** & **Method** & \multicolumn{3}{c}{**Minimum**} \\ \hline  & & **Function 1** & **Function 2** & **Function 3** \\ \hline Viennet & NSGA-II & \(0.000051\) & \(15.000000\) & \(-0.099997\) \\  & NSGA-III & \(131,609.806452\) & \(443,378.967742\) & \(0.000004\) \\  & M-PF & \(4292.110524\) & \(10,870.875716\) & \(-0.002940\) \\  & eMPF & \(0.000028\) & \(15.000026\) & \(-0.099998\) \\ \hline MHHM1 & NSGA-II & \(9.6666\times 10^{-11}\) & \(7.3333\times 10^{-11}\) & \(6.33333\times 10^{-11}\) \\  & NSGA-III & \(6.193743\times 10^{-5}\) & \(4.7659963\times 10^{-5}\) & \(3.433997\times 10^{-5}\) \\  & M-PF & \(2.4781883\times 10^{-5}\) & \(7.064937\times 10^{-5}\) & \(7.134097\times 10^{-5}\) \\  & eMPF & \(1.0666\times 10^{-10}\) & \(9.6666\times 10^{-11}\) & \(5.3333\times 10^{-11}\) \\ \hline MHHM2 & NSGA-II & \(6.18220\times 10^{-6}\) & \(1.7605\times 10^{-6}\) & \(2.837620\times 10^{-6}\) \\  & NSGA-III & \(0.001283\) & \(0.001229\) & \(0.00249529\) \\  & M-PF & \(0.00029338\) & \(0.0004046536\) & \(0.0006900754\) \\  & eMPF & \(6.868939\times 10^{-6}\) & \(1.63898\times 10^{-6}\) & \(4.5422\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the minimum values for three objective functions using each optimization technique.