# An Improved Multi-Objective Grey Wolf Optimizer for Aerodynamic Optimization of Axial Cooling Fans
## 3 Results and Discussion  
### Evaluation of Benchmark Functions  
#### 3.1.1 Experimental Setup  
Table 4 lists the details of five benchmark functions, comprising three two-objective [30] and two three-objective problems [31]. For the two-objective problems, each of them has 30 dimensions. For the three-objective problems, each of them has two dimensions. The true Pareto-optimal front includes four categories: convex, concave, continuous, and discontinuous. Figure 4: A flow chart of the IMOGWO algorithm. The Pareto-optimal front of the ZDT1 benchmark problem is known to be convex. The specific form is  
\[ZDT1=\left\{ \begin{array}{c}f_{1}(x)=x_{1}\\ f_{2}(x)=g(x)\Big{|}1-\sqrt{x_{1}/g(x)}\Big{|}\\ where\ g(x)=1+9\Big{(}\sum_{i=2}^{30}x_{i}\Big{)}/(30-1)\\ s.t.\ 0\leq x_{i}\leq 1,\ i=1,2,\ldots,30\end{array} \right. \tag{20}\]  
The Pareto-optimal front of the ZDT2 function exhibits a concave shape. The specific form is  
\[ZDT2=\left\{ \begin{array}{c}f_{1}(x)=x_{1}\\ f_{2}(x)=g(x)\Big{|}1-\Big{(}\frac{x_{1}}{g(x)}\Big{)}^{2}\Big{|}\\ where\ g(x)=1+9\Big{(}\sum_{i=2}^{30}x_{i}\Big{)}/(30-1)\\ s.t.\ 0\leq x_{i}\leq 1,\ i=1,2,\ldots,30\end{array} \right. \tag{21}\]  
Due to the presence of a sine component in its objective function, ZDT3 exhibits a Pareto front with multiple discontinuities. The specific form is  
\[ZDT3=\left\{ \begin{array}{c}f_{1}(x)=x_{1}\\ f_{2}(x)=g(x)\Big{|}1-\sqrt{\frac{x_{1}}{g(x)}}-\frac{x_{1}}{g(x)}\text{sin}( 10\pi x_{1})\Big{|}\\ where\ g(x)=1+9\Big{(}\sum_{i=2}^{30}x_{i}\Big{)}/(30-1)\\ s.t.\ 0\leq x_{i}\leq 1,\ i=1,2,\ldots,30\end{array} \right. \tag{22}\]  
The Viennet2 and Viennet3 have a discontinuous and continuous Pareto-optimal front, respectively. The objective functions are described as  
\[\textit{Viennet2}=\left\{ \begin{array}{c}F=(f_{1}(x,y),f_{2}(x,y),f_{3}(x,y) )\\ f_{1}(x,y)=0.5\times(x-2)^{2}+\frac{(y+1)^{2}}{13}+2\\ f_{2}(x,y)=\frac{(x+y-3)^{2}}{36}+\frac{(-x+y+2)^{2}}{8}-17\\ f_{3}(x,y)=\frac{(x+2y-1)^{2}}{175}+\frac{(2y-1)^{2}}{17^{2}}-13\\ s.t.-4\leq x,y\leq 4\end{array} \right. \tag{23}\]  
\[\textit{Viennet3}=\left\{ \begin{array}{c}F=(f_{1}(x,y),f_{2}(x,y),f_{3}(x,y) )\\ f_{1}(x,y)=0.5\times(x^{2}+y^{2})+\text{sin}\big{(}x^{2}+y^{2}\big{)}\\ f_{2}(x,y)=\frac{(3x-2y+4)^{2}}{8}+\frac{(x-y+1)^{2}}{27}+15\\ f_{3}(x,y)=\frac{1}{(x^{2}+y^{2}+1)}-1.1\exp(-x^{2}-y^{2}\big{)}\\ s.t.-30\leq x,y\leq 30\end{array} \right. \tag{24}\]  
A well-solved multi-objective optimization problem is characterized by a Pareto-optimal solution set that both approximates the true front closely and exhibits a high degree of uniformity. In order to quantitatively evaluate the performance of the proposed \begin{table}
\begin{tabular}{c c c c} \hline
**Problem** & **Pareto-Optimal Front** & **Number of Objectives** & **Constraint Condition** \\ \hline ZDT1 & convex & 2 & \(m=30;0\leq x_{i}\leq 1\) \\ \hline ZDT2 & concave & 2 & \(m=30;0\leq x_{i}\leq 1\) \\ \hline ZDT3 & discontinuous & 2 & \(m=30;0\leq x_{i}\leq 1\) \\ \hline Viennet2 & discontinuous & 3 & \(m=2;-4\leq x,y\leq 4\) \\ \hline Viennet3 & continuous & 3 & \(m=2;-30\leq x,y\leq 30\) \\ \hline \end{tabular}
\end{table} 
Table 4: The information on the benchmark functions. optimization algorithm in convergence and coverage, four performance indicators are selected, which are defined as follows [32]:  
Generalal distance (GD): The distance between the calculated Pareto front (\(PF_{know}\)) and the optimal Pareto front (\(PF_{true}\)), which is an index of the convergence evaluation. \[GD=\frac{\left(\sum_{i=1}^{n}d_{i}^{2}\right)^{\frac{1}{2}}}{n} \tag{25}\]  
where \(n\) is the number of solutions in \(PF_{know}\). \(d_{i}\) is the Euclidean distance between the i-th solution in the target space and the nearest solution in \(PF_{true}\). The smaller the \(GD\) value, the closer the found non-dominated solution set is to the real non-dominated solution set. Spacing (\(SP\)): This measure reflects the extent to which the obtained solutions are evenly distributed in proximity to the actual Pareto front.