## 4 Experimental Studies
### Simulation of Benchmark Functions
#### 4.1.2 Performance Metrics  
To comprehensively evaluate the performance of multiobjective optimization algorithms, it is necessary to analyze them from three aspects: convergence performance, diversity and width of solution set distribution [37]. Therefore, the following three measurement metrics are used to compare the multiobjective optimization methods:  
Closeness to \(PF^{*}\)  
Generational distance (\(GD\)) [38] metric is used to measure the distance between the non-dominated solution set (\(PF_{know}\)) searched by the optimization algorithm and the true Pareto front (\(PF^{*}\)), which is described as follows:  
\[GD=\frac{\sqrt{\sum_{i=1}^{n}D_{i}^{2}}}{\eta} \tag{17}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Objective Functions** & **Dimension** & **Variable Bounds** & **Characteristics** \\ \hline SCH & \(\min f_{1}(x)=x^{2}\) & 1 & \(x\in\left[-10^{3},10^{3}\right]\) & convex \\  & \(\min f_{2}(x)=(x-2)^{2}\) & & & \\ \hline FON & \(\min f_{1}(x)=1-\exp\Bigl{(}-\sum\limits_{i=1}^{3}\left(x_{i}-\frac{1}{\sqrt{3 }}\right)^{2}\Bigr{)}\) & 3 & \(\left[-4,4\right]\) & nonconvex \\  & \(\min f_{2}(x)=1-\exp\Bigl{(}-\sum\limits_{i=1}^{3}\left(x_{i}+\frac{1}{\sqrt{3 }}\right)^{2}\Bigr{)}\) & & & \\ \hline KUR & \(\min f_{1}(x)=\sum\limits_{i=1}^{n-1}\Bigl{(}-10\exp\Bigl{(}-0.2\sqrt{x_{i}^{2 }+x_{i+1}^{2}}\Bigr{)}\Bigr{)}\) & 3 & \(\left[-5,5\right]\) & discontinuous \\  & \(\min f_{2}(x)=\sum\limits_{i=1}^{n}\left(\left|x_{i}\right|^{0.8}+5\sin\left( x_{i}^{3}\right)\right)\) & & & \\ \hline ZDT1 & \(\min f_{1}(x)=x_{1}\) & 30 & \(\left[0,1\right]\) & convex \\  & \(g(x)=1+9\sum\limits_{i=2}^{n}x_{i}/(n-1)\) & & & \\ \hline ZDT2 & \(\min f_{1}(x)=x_{1}\) & 30 & \(\left[0,1\right]\) & concave \\  & \(\min f_{2}(x)=g\Bigl{(}1-\left(f_{1}/g\right)^{2}\Bigr{)}\) & & & \\  & \(g(x)=1+9\sum\limits_{i=2}^{n}x_{i}/(n-1)\) & & & \\ \hline ZDT3 & \(\min f_{1}(x)=x_{1}\) & 30 & \(\left[0,1\right]\) & convex, \\  & \(\min f_{2}(x)=g\Bigl{(}1-\sqrt{\frac{\hbar}{8}}-\Bigl{(}\frac{\hbar}{8}\Bigr{)} \sin(10\pi f_{1})\Bigr{)}\) & & & discontinuous \\  & \(g(x)=1+9\sum\limits_{i=2}^{n}x_{i}/(n-1)\) & & & \\ \hline ZDT4 & \(\min f_{1}(x)=x_{1}\) & 10 & \(x_{1}\in\left[0,\,1\right]\) & nonconvex \\  & \(\min f_{2}(x)=g\bigl{(}1-\sqrt{f_{1}/g}\bigr{)}\) & & \(x_{i}\in\left[-5,\,5\right]\) & \\  & \(g(x)=1+10(n-1)+\sum\limits_{i=2}^{n}\left(x_{i}^{2}-10\cos(4\pi x_{i})\right)\) & & \(i=2,\cdots n\) & \\ \hline ZDT6 & \(\min f_{1}(x)=1-\exp(-4x_{1})sin^{6}(6\pi x_{1})\) & 10 & \(\left[0,1\right]\) & nonconvex \\  & \(\min f_{2}(x)=g\Bigl{(}1-\left(f_{1}/g\right)^{2}\Bigr{)}\) & & & \\  & \(g(x)=1+9\left(\frac{\sum_{i=2}^{n}x_{i}}{n-1}\right)^{0.25}\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Descriptions of benchmark functions. where \(n\) represents the number of individuals in \(PF_{know}\), and \(D_{i}\) refers to the Euclidean distance between the \(i\)-th individual in \(PF_{know}\) and the nearest individual in \(PF^{*}\). The smaller \(GD\) value indicates the better convergence performance. Diversity  
In 2002, [PERSON] [39] introduced an index to evaluate the diversity of distribution of non-inferior solution sets, and its formula is defined as follows:  
\[\Delta=\frac{D_{f}+D_{l}+\sum_{i}^{n-1}\lvert D_{i}-\overline{D}\rvert}{D_{f} +D_{l}+(n-1)\overline{D}} \tag{18}\]  
Parameters \(D_{f}\) and \(D_{l}\) in the formula are Euclidean distances of extreme solutions in true Pareto front set \(PF^{*}\) and boundary solutions of non-dominated solution set \(PF_{know}\) obtained by multiobjective optimization algorithm. \(D_{i}\) is the distance between two adjacent solutions in the \(PF_{know}\), and \(\overline{D}\) is the average of all the \(D_{i}\). A smaller value of \(\Delta\) means that the better the performance of the algorithm. [PERSON] et al.