# A Hybrid Multi-Objective Optimization Method and Its Application to Electromagnetic Device Designs
## 4 Numerical Examples
### Performance Validation  
Standard test functions, including KUR, ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6 [13], were used to validate the performance of the proposed algorithm. The details of these test functions are shown in Table 2. Each algorithm was run 30 times independently for different test functions in order to obtain its statistical properties. The comparison results of PFs obtained by the aforementioned four algorithms for each test function are presented in Figures 2-7, respectively. It can be observed that the Pareto solutions obtained by the proposed hybrid algorithm are much closer to the true PFs as compared to those of its three counterparts, especially for ZDT4, which has a large number of local optima, and ZDT6, which has a non-uniformly distributed PF. \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Hybrid** & **NSGA-MOPSO** & **MOPSO** & **NSGA-II** \\ \hline \(\tau_{max}\) & 125 & 125 & 250 & 250 \\ _popsize_ & 100 & 100 & 100 & 100 \\ \(p_{c}\) & 0.9 & 0.9 & — & 0.9 \\ \(p_{m}\) & \(p_{ng}\) \& \(p_{mp}\) & \(p_{ng}\) \& \(p_{mp}\) & \(\left(1-\frac{\tau-1}{\tau_{max}}\right)^{10}\) & 0.1 \\ \(\omega\) & 0.5 & 0.5 & 0.5 & — \\ \(i_{r}\) & 0.1 & 0.1 & 0.1 & — \\ \(\mu_{u}\) & 20 & 20 & — & 20 \\ \(c_{1}\) & 1 & 1 & 1 & — \\ \(c_{2}\) & 2 & 2 & 2 & — \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter settings. \begin{table}
\begin{tabular}{c c c c c} \hline
**Problem** & **n** & **Variable Bounds** & **Objective Functions** & **Optimal Solutions** & **PF Characteristics** \\ \hline KUR & 3 & [-5, 5] & \(f_{1}(x)=\sum_{i=1}^{n-1}\left(-10\exp\left(-0.2\sqrt{\chi_{i}^{2}+x_{i+1}^{2} }\right)\right)\) & Ref [12] & nonconvex \\  & & \(f_{2}(x)=\sum_{i=1}^{n}\left(\left|x_{i}\right|^{08}+5 sinx_{i}^{0}\right)\) & & \\ ZDT1 & 30 & [0, 1] & \(f_{2}(x)=g(x)\left[1-\sqrt{\frac{x_{1}}{g(x)}}\right]\) & \(x_{1}\in[0,1]\) & convex \\  & & \(g(x)=1+g\left(\sum_{i=2}^{n}x_{i}\right)/(n-1)\) & \(i=2,\cdots,n\) & \\  & & \(f_{1}(x)=x_{1}\) & \(x_{1}\in  
Figure 4: Comparison results for ZDT2. Figure 3: Comparison results for ZDT1. Figure 2: Comparison results for KUR. To quantitatively evaluate the performances of an algorithm, its convergence metric \(\gamma\) and diversity metric \(\Delta\) are calculated and given in Tables 3 and 4. In the tables, E(\(\cdot\)) denotes the mean value and \(\sigma(\cdot)\) stands for the variance. Smaller values of both mean and variance indicate a better performance. Figure 5: Comparison results for ZDT3. Figure 6: Comparison results for ZDT4. Figure 7: Comparison results for ZDT6. From Table 3, the convergence performance of the proposed hybrid algorithm is better than that of the latter three for most of the test functions, especially for ZDT4 and ZDT6, which have a large number of local optima. From Table 4, the proposed hybrid algorithm generally outperformed NSGA-II, MOPSO, and NSGA-MOPSO in terms of diversity for most of the test functions. Only the mean of the diversity metric for KUR failed to perform as well as NSGA-II. Meanwhile, it can be observed that the simple hybrid algorithm NSGA-MOPSO obtains better results than NSGA-II and MOPSO when handling the ZDT4 problem. However, there was no improvement in other test functions, and in some problems, it obtained results slightly worse than NSGA and MOPSO individually. In comparison, the proposed algorithm possesses better statistical results in both convergence and diversity metrics. This is owed to the proposed adaptive operator; in the initial stage, two-point crossover and polynomial mutation dominated to improve the global search ability of the algorithm, which contributes to finding non-dominated individuals. As the evolution went on, the SBX and Gaussian mutation improved the local search ability of the algorithm, which prompted it to obtain more uniformly distributed PFs. This design helps to achieve a balance between global and local search. The excellent individuals can be reserved and the population diversity can be improved at the same time.