## 4 Size Optimization of the PM
### Optimization Results  
The length, width and thickness of the PM in the PM-EDS structure are optimized using MOPSO, and the size range of the PM is limited to: 0.01 m \(\leqslant l\leqslant\) 0.8 m, 0.01 m \(\leqslant w\leqslant\) 0.3 m, 0.01 m \(\leqslant h\leqslant\) 0.3 m. According to Figure 12, set the deflection angle \(\theta=7^{\circ}\) in order to maximize the lift-to-drag ratio. According to Figure 13, we set the speed in the \(x\)-direction \(v_{\text{x}}\) = 30 m/s to increase the levitation force. After 100 iterations, the Pareto-optimal solutions of the lift-to-drag ratio and lift-to-weight ratio are obtained, as shown in Figure 16. Each scheme in the solution set is the optimal solution, but each scheme has a different emphasis on the two objective functions. In Figure 16, comparing the Pareto-optimal solution set with the initial design, the lift-to-drag ratio and lift-to-weight ratio at point A are higher than the initial design; the lift-to-drag ratio at point B is higher than the initial design, while the lift-to-weight ratio is lower than the initial design. At the same time, it can be seen that the lift-to-drag ratio and the lift-to-weight ratio are two conflicting objective functions, which cannot achieve the maximum value at the same time, and the optimization of one objective will lead to the degradation of the other objective. For example, point A has the highest lift-to-weight ratio but the lowest lift-to-drag ratio with the coordinate \((4.71,10.07)\), while point B has the highest lift-to-drag ratio and lowest lift-to-weight ratio with the coordinate \((5.95,5.01)\). The selections of the length, width and thickness of the PM for each solution in the solution set are shown in Figure 17, which are the decisive factor that leads to a particular intermediate solution between points A and B. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Number of particles \(N\) & 100 \\ Maximum iterative number \(n_{\text{max}}\) & 100 \\ Inertia factor \(\omega\) & \(0.9-0.4\left(\frac{2(l+1)}{n_{\text{max}}}-\left(\frac{l+1}{n_{\text{max}}} \right)^{2}\right)\) \\ Learning factor \(c_{1}\) & \(0.5\omega^{2}+3\omega\) \\ Learning factor \(c_{2}\) & \(4-c_{1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The parameters of the MOPSO. Figure 16: Pareto-optimal solution set. In order to avoid the personal subjective factors in the traditional manual selection of Pareto-optimal solutions, the fuzzy set theory is used to screen the Pareto-optimal solutions, which can effectively increase the scientificity of the selection of the optimal solution. We define the membership function \(S_{i}^{j}\):  
\[S_{i}^{j}= \begin{cases}1&,f_{i}^{j}\leqslant f_{i}^{\min}\\ \frac{f_{i}^{\max}-f_{i}^{j}}{f_{i}^{\min}-f_{i}^{\min}}&,f_{i}^{\min}\leqslant f _{i}^{j}\leqslant f_{i}^{\max}\\ 0&,f_{i}^{j}\geqslant f_{i}^{\max}\end{cases}  \tag{40}\]  
where \(f_{i}^{\max}\) and \(f_{i}^{\min}\) are the maximum and minimum values of the \(i\)-th objective function in the solution set, and \(f_{i}^{j}\) and \(S_{i}^{j}\) are the current value and the membership function value of the \(i\)th objective function of the \(j\)-th solution. Next, we define the dominating function \(\varphi_{k}\)[20], the dominant value of the \(k\)-th solution is:  
\[\varphi_{k}= \begin{cases}\sum\limits_{i=1}^{n}S_{i}^{k}\bigg{/}\bigg{(}\sum \limits_{j=1}^{M}\sum\limits_{i=1}^{n}S_{i}^{j}\bigg{)}&,F_{x}^{k}\geqslant F _{d}\\ 0&,F_{x}^{k}<F_{d}\end{cases}  \tag{41}\]  
where \(M\) is the number of nondominated solutions. According to the optimization results, \(M\) is 63, and \(n\) is the number of objective functions, \(n=2\). \(F_{x}^{k}\) is the levitation force of the \(k\)-th solution. Considering the restriction of vehicle gravity, the levitation force generated by a single PM should be greater than the expected levitation force \(F_{d}\), and its dominant value is 0 when \(F_{x}^{k}<F_{d}\).