## 3 Multi-Objective Optimization
### Multi-Objective Optimization Design
#### 3.2.3 Analysis of Optimization Results  
The mean main effects plot of [PERSON]'s method is the average response for each combination of control factor levels. The optimization objective is to determine the level of factors that minimizes or maximizes the mean value, depending on the response. The experimental results of [PERSON]'s algorithm in the previous section were statistically analyzed, and the mean main effect plots for each variable at varying levels are shown in Figure 5, with the vertical coordinates corresponding to the level values of the factors.  
From Figure 5a, it can be concluded that when the air-gap flux density is selected as small, the parameter combination is A\({}_{1}\), B\({}_{1}\), C\({}_{1}\), and D\({}_{1}\); that is, the pole arc coefficient, permanent magnet thickness, cutting edge radius, and air-gap length are 0.6, 2.0 mm, 18 mm and 0.5 mm, respectively; From Figure 5b, when the no-load back EMF is selected as large, the parameter combination is A\({}_{2}\), B\({}_{4}\), C\({}_{4}\), and D\({}_{4}\); that is, the pole arc coefficient, permanent magnet thickness, cutting edge radius, and air-gap length are 0.7, 3.5 mm, 24 mm, and 1.5 mm, respectively. It can be concluded from Figure 5c that when the output  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Parameter**} & \multirow{2}{*}{**Flux Density**} & \multicolumn{2}{c}{**No-Load Back EMF**} & \multicolumn{2}{c}{**Output Torque**} \\ \cline{2-7}  & & **Ss** & **Specific Gravity** & **Ss** & **Specific Gravity** & **Ss** & **Specific Gravity** \\ \hline A & 0.0269 & 87.6\% & 10.145 & 67.8\% & 0.0022 & 30.3\% \\ B & 0.0010 & 3.3\% & 0.982 & 6.6\% & 0.0004 & 6.5\% \\ C & 0.0011 & 3.6\% & 1.076 & 7.2\% & 0.0017 & 23.5\% \\ D & 0.0017 & 5.5\% & 2.762 & 18.4\% & 0.0029 & 39.7\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: Proportion of influence of the parameters on target performance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Level Value/Variable**} & **A** & **B** & **C** & **D** \\ \hline
1 & 9.855 & 10.790 & 10.788 & 10.685 \\
2 & 11.925 & 11.012 & 10.965 & 10.798 \\
3 & 11.075 & 11.265 & 11.352 & 11.305 \\
4 & 11.655 & 11.443 & 11.405 & 11.723 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Average no-load back EMF for each variable at the horizontal factor.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Level Value/Variable** & **A** & **B** & **C** & **D** \\ \hline
1 & 0.4975 & 0.5075 & 0.5125 & 0.5200 \\
2 & 0.5050 & 0.4925 & 0.5100 & 0.5050 \\
3 & 0.4850 & 0.5025 & 0.4950 & 0.4825 \\
4 & 0.5175 & 0.5025 & 0.4875 & 0.4975 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Mean value of torque for each variable at the level factor.  
torque is selected as small, the parameter combination is A\({}_{3}\), B\({}_{2}\), C\({}_{4}\), and D\({}_{3}\); that is, the pole arc coefficient, permanent magnet thickness, permanent magnet arc radius, and air-gap length are 0.8, 2.5 mm, 24 mm, and 1.2 mm, respectively.  
Figure 5: Influence of the parameter influence factors on target performance. (**a**) The effect of the parameter level factor on air-gap magnetic density. (**b**) The effect of the parameter level factor on no-load back EMF. (**c**) Effect of the parameter level factor on the output torque.  
There are contradictions in the selection of the various parameters, and different parameters have different effects on the optimization objectives. The output torque and air-gap magnetic density are not the smaller the better. The no-load back EMF can meet the design requirements. Therefore, according to the proportion analysis of the influence of the four parameters on the performance of the optimization target, the final motor optimization parameters are A\({}_{2}\), B\({}_{2}\), C\({}_{4}\), and D\({}_{2}\); that is, the pole arc coefficient, thickness of the permanent magnet, permanent magnet arc radius, and air-gap length are 0.7, 2.5 mm, 24 mm, and 0.8 mm, respectively.