## 4 Experimental and Discussion  
### Test Set and Performance Measures  
In the experiment, we use two benchmark sets ZDT [32] and SCH [33] test the performance of MGSO-BFO. For the ZDT test sets, it consists of six test instances, and we use five of them in the experiment. For more details about the test problems, please refer to Table 1.  
As we all know, convergence and diversity are two important indices for multi-objective optimization algorithms. These two indices cannot be measured adequately with one performance metric as in single objective optimization. There are many performance metrics have been proposed. In this section, we employed GD [34], SP [34] and IGD [35] to evaluate the MGSO-BFO.  
The definition of GD, SP and IGD as follow:  
\[GD=\frac{1}{n}\sqrt{\sum_{i=1}^{n}dist_{i}^{2}} \tag{20}\]  
\[SP=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\left(\overline{d}-d_{i}\right)^{2}} \tag{21}\]  
\[IGD(x,p^{*})=\frac{\sum_{x\in P^{*}}dist(x,p)}{|P^{*}|} \tag{22}\]  
where \(n\) is the number of Pareto optimal solution, \(d_{i}\) is the minimum distance from \(i\) solution to Pareto front solution. \(\overline{d}\) is the mean of \(d_{i}\). \(d(v,Q)\) is the minimum Euclidean distance between \(v\) and all the points in \(Q\). \(p\) is the true Pareto front. \(Q\) is the optimal solution set by algorithm.  
In the experiments, 30 independent runs are carried out on the machine with Intel Core i5-2400 3.10 GHz CPU, 6 GB memory, and windows 7 operating system with Matlab7.9. The stopping criterion is a fixed number of o iteration (setting to 100), while population size \(n=50\) for all algorithms. The external population of size is set as 100.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function** & **Dimension** & **Range** & **Objective Function** &
\begin{tabular}{c} **Optimal** \\ **Solution** \\ \end{tabular} & **Feature** \\ \hline SCH & 1 & \([-10^{3},10^{3}]\) & \(f_{1}(x)=x^{2}\) & \(x\in[0,2]\) & convex \\  & & & \(f_{2}(x)=(x-2)^{2}\) & & \\ \hline  & & & \(f_{1}(x)=x_{1}\) & & \\ ZDT1 & 30 & \([0,1]\) & \(f_{2}(x)=g(x)[1-\sqrt{\frac{3}{x(1)}}]\) & \(x_{1}\in[0,1]\) & convex \\  & & & \(g(x)=1+9\frac{\sum_{x}x}{9-1}\) & & \\ \hline  & & & \(f_{1}(x)=x_{1}\) & & \\ ZDT2 & 30 & \([0,1]\) & \(f_{2}(x)=g(x)[1-(\frac{x_{1}}{x(1)})^{2}]\) & \(x_{1}\in[0,1]\) & Non-convex \\  & & & \(g(x)=1+9\frac{\sum_{x}x}{9-1}\) & \(i=2\cdots n\) & \\ \hline  & & & \(f_{1}(x)=x_{1}\) & & \\ ZDT3 & 30 & \([0,1]\) & \(f_{2}(x)=g(x)[1-\sqrt{\frac{3}{x(1)}},\frac{3}{x(1)}\sin(10\pi x_{1}]\) & \(x_{1}\in[0,1]\) & convex \\  & & & \(g(x)=1+9\frac{\sum_{x}x}{9-1}\) & \(i=2\cdots n\) & \\ \hline  & & & \(f_{1}(x)=x_{1}\) & & \\ ZDT4 & 10 & \(x_{1}\in[0,1]\) & \(f_{2}(x)=g(x)[1-\sqrt{\frac{3}{x(1)}}]\) & \(x_{1}\in[0,1]\) & Non-convex \\  & & \(i=2\cdots n\) & \(g(x)=1+10(n-1)+\frac{\sum_{x}x}{2}[x_{1}^{2}-10\cos(4\pi x_{1})]\) & \(i=2\cdots n\) & \\ \hline  & & & \(f_{1}(x)=1-\exp(-4x_{1})\sin^{6}(6\pi x_{1})\) & & \\ ZDT6 & 10 & \([0,1]\) & \(f_{2}(x)=g(x)[1-(\frac{f_{1}(x)}{60\pi^{2}})^{2}]\) & \(x_{1}\in[0,1]\) & Non-convex \\  & & & \(g(x)=1+9\frac{\sum_{x}x}{9-1}\) & & \(i=2\cdots n\) & non-uniformly \\ \hline \hline \end{tabular}
\end{table}
Table 1: six test function for multi-objective optimization algorithm.