## 4 Multi-Objective Optimization
### Significance Analysis of Influencing Factors
The positive coefficients for outlet angle \(\beta_{2}\) and outlet width \(b_{2}\) indicate a positive effect on the pump head, with outlet width \(b_{2}\) having a greater effect on the pump head than outlet angle \(\beta_{2}\). The negative coefficients for inlet angle \(\beta_{1}\), blade wrap angle \(\varphi\), and impeller timing position indicate a negative effect on the pump \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Serial Number**} & **Imported Placement** & **Exit Placement** & **Outlet** & **Blade Wrap** & **Impeller Timing** & **Lift** & **Efficiency** \\  & **Angle \(\beta_{1}\)** & **Angle \(\beta_{2}\)** & **Width \(b_{2}\)** & **Angle \(\varphi\)** & **Position** & **Position** & **Lift** & **Efficiency** \\ \hline
1 & 26 & 21 & 12 & 116 & 30 & 24.6248 & 77.9006 \\
2 & 17 & 32 & 12 & 77 & 30 & 26.8351 & 68.1091 \\
3 & 26 & 21 & 8 & 77 & 30 & 22.3482 & 75.5079 \\
4 & 17 & 21 & 8 & 77 & 0 & 22.7505 & 75.1389 \\
5 & 17 & 21 & 12 & 77 & 30 & 25.7977 & 69.4718 \\
6 & 17 & 21 & 8 & 116 & 0 & 20.1328 & 78.2832 \\
7 & 26 & 32 & 8 & 116 & 30 & 20.9667 & 77.3769 \\
8 & 26 & 21 & 12 & 116 & 0 & 24.4402 & 77.9071 \\
9 & 17 & 32 & 8 & 116 & 30 & 21.5795 & 76.9954 \\
10 & 17 & 32 & 12 & 116 & 0 & 25.5931 & 75.923 \\
11 & 26 & 32 & 12 & 77 & 0 & 27.0253 & 68.0776 \\
12 & 26 & 32 & 8 & 77 & 0 & 23.176 & 74.292 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Plackett-Burman experimental design and numerical simulation results. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Serial Number**} & **Imported Placement** & **Exit Placement** & **Outlet** & **Blade Wrap** & **Impeller Timing** \\  & **Angle \(\beta_{1}\)** & **Angle \(\beta_{2}\)** & **Width \(b_{2}\)** & **Angle \(\varphi\)** & **Position** \\ \hline
2 & \(-1\) & 1 & 1 & \(-1\) & 1 \\
3 & \(1\) & \(-1\) & \(-1\) & \(-1\) & 1 \\
4 & \(-1\) & \(-1\) & \(-1\) & \(-1\) & \(-1\) \\
5 & \(-1\) & \(-1\) & 1 & \(-1\) & 1 \\
6 & \(-1\) & \(-1\) & \(-1\) & 1 & \(-1\) \\
7 & 1 & 1 & \(-1\) & 1 & 1 \\
8 & 1 & \(-1\) & 1 & 1 & \(-1\) \\
9 & \(-1\) & 1 & \(-1\) & 1 & 1 \\
10 & \(-1\) & 1 & 1 & 1 & \(-1\) \\
11 & 1 & 1 & 1 & \(-1\) & \(-1\) \\
12 & 1 & 1 & \(-1\) & \(-1\) & \(-1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Plackett-Burman experimental design permutation distribution. head, with blade wrap angle \(\varphi\) having the greatest effect on the pump head, followed by impeller timing position and inlet angle \(\beta_{1}\) having the least effect. The standard error of each influencing factor is 0.1078. The \(p\)-value represents the possibility that each factor has no influence on the head, so the smaller the \(p\)-value, the more significant the influence. When \(p<0.05\), it was a significant influence factor. From the \(p\)-value of each factor, it can be seen that the outlet placement angle \(\beta_{2}\), outlet width \(b_{2}\), and blade wrap angle \(\varphi\) are the significant influencing factors of the head. Table 6 demonstrates the degree of influence of the main geometrical parameters on efficiency. As can be seen from the table, the percentages of the sum of squares of the exit resting angle \(\beta_{2}\), exit width \(b_{2}\), and blade wrap angle \(\varphi\) are 15.04%, 34.02%, and 95.14%, respectively, much greater than the percentages of the sum of squares of the inlet resting angle \(\beta_{1}\) and impeller timing position of 4.25% and 1.51%, respectively. The coefficients of inlet angle \(\beta_{1}\) and blade cladding angle \(\varphi\) are positive, which means that they have a positive effect on efficiency, i.e., as the inlet angle \(\beta_{1}\) and blade cladding angle \(\varphi\) increase or decrease, the change in efficiency tends to rise or fall, and the degree of influence of blade cladding angle \(\varphi\) on efficiency is greater than that of inlet angle \(\beta_{1}\).