## 5 Verification Case Study and Discussions
### Pareto-Optimal Designs of Planar Miniaturized Multiband Antenna  
The fast multi-objective optimization of the planar antenna geometry given in Figure 8 is implemented using MOEA/D [36] and the improved PSO-RBFNN model. Two design goals are to be achieved: (i) the reflection coefficients are lower than \(-10\)dB within three frequency bands of 2.40\(-\)2.60 GHz, 3.30\(-\)3.80 GHz, 5.00\(-\)5.85 GHz, covering the entire WLAN2.4/5.2/5.8 GHz and WiMAX3.5 GHz bands (objective \(F_{1}\)); and (ii) the size of antenna structure is reduced to satisfy the need of antenna miniaturization in portable devices (objective \(F_{2}\)). The objective function of \(F_{1}\) is specified as  
\[F_{1}=\frac{1}{N}\sum_{i=1}^{n}Q(f_{i}) \tag{14}\]  
\[Q(f_{i})=\left\{ \begin{array}{c|c}\left|S_{11}(f_{i})\right|&\left|S_{11}(f_ {i})\right|>-10\\ -10&S_{11}(f_{i})\right|\leq-10\end{array} \right. \tag{15}\]  
where \(f_{i}\) is the \(i\)th sample within the given operation bands; \(S_{11}(f_{i})\) is the reflection coefficient at \(f_{i}\); \(N\) is the total number of sampling frequencies. The objective function of \(F_{2}\) is defined as  
\[F_{2}=w\times l \tag{16}\]  
According to the principles of setting parameters for MOEA/D in Reference [36], the number of population is chosen as 100, and the maximum number of iterations is chosen as 150. The representations of the Pareto set during the optimization process for the planar miniaturized multiband antenna is given in Figure 11, which displays the evolution behavior of the objective functions. Table 3 shows the detailed antenna parameters on the final selected designs. It can be seen from Figure 11 and Table 3 that achievements of the two design objectives are actually in conflict. To show the fitting ability of the PSO-RBFNN model, the HFSS simulated and predicted \(S_{11}\) for the Pareto-optimal designs are given in Figure 12. It can be observed that the predicted \(S_{11}\) well match the simulated \(S_{11}\). To validate the feasibility of the proposed model, the predicted results of fitness \(F_{1}\) and percentage error relative to the HFSS simulations are given in Figure 13.  
Furthermore, the proposed PSO-RBFNN model is compared with other competitive techniques in terms of the computation time. Method 1 is direct MOEA/D-based optimization combined with HFSS simulation, and Method 2 is MOEA/D with RBF model [11]. The detailed computational cost is shown in Table 4. One EM simulation takes about 51 s under the high-fidelity running environment. This work and Method 2 use only 1.37% and 1.40% of the computational time of Method 1, 2.93 h and 2.98 h, respectively. This work takes a bit longer time than Method 2 due to the PSO optimization of RBF network parameters, however, it is still more computationally efficient than Method 1 with 213.51 h. In conclusion, the proposed PSO-RBFNN method enables realization of antenna design solutions satisfying multiple requirements within affordable computational cost. This method significantly reduces the design cycles of multi-parameter multi-objective antenna optimizations. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **HFSS** & **Kriging [5]** & **RBFNN [11]** & **PSO-RBFNN [24]** &
\begin{tabular}{c} **Improved** \\ **PSO-RBFNN** \\ \end{tabular} \\ \hline Total time & 1017.820 & 0.413 & 0.165 & 0.043 & 0.039 \\ Average time & 50.891 & 0.021 & 0.008 & 0.002 & 0.002 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time cost of different surrogate models and HFSS simulations (units: s). Figure 11: The obtained representations of the Pareto set during the optimization process for the planar miniaturized multiband antenna.