## 4 Multi-Objective Optimization
### Multi-Objective Particle Swarm Optimization Algorithm Optimization
It can be seen from Table 9 that the head of the optimal individual is 5.1344 m higher than the initial model, but the efficiency is reduced by 14.2099 percentage points. The efficiency of the optimal individual is 4.5793 m lower than the initial model, but the efficiency is increased by 5.5803 percentage points. If the head optimum and the efficiency optimum are chosen as the final optimization results, although there is an increase in the pump head and efficiency, there is also a significant decrease in efficiency and head. Therefore, two solutions are to be found on the Pareto front, denoted as head-optimal and efficiency-optimal, such that they each satisfy: (1) Under the condition that the efficiency is not reduced, the head is improved to the greatest extent; (2) The efficiency is improved to the greatest extent without reducing the head. It is easy to see that the two points are the horizontal and vertical coordinates of the initial model as indicators, and the intersection of these two lines with the Pareto front curve is the intersection of the vertical and horizontal lines, respectively. This is shown in Figure 10 below. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Models** & **Lift (m)** & **Efficiency (\%)** \\ \hline Initial model & 24.2396 & 75.3835 \\ Individuals with optimum head & 29.3740 & 61.1736 \\ Optimal efficiency individual & 19.6603 & 80.9638 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparison of external characteristics before and after optimization. Figure 9: Pareto frontier distribution. The Pareto solution at the boundary point within the right-angle intersection is defined as the better point. The higher-head individual is modeled by the structural parameters at the optimal head point and calculated by the CFD method of simulation. This individual is the model with the best head point predicted by the MOPSO algorithm. The higher-efficiency individual is modeled by the structural parameters at the optimal efficiency point and simulated by the CFD method. This individual is the model with the best efficiency point predicted by the MOPSO algorithm. The structural parameters corresponding to the higher head and higher efficiency are modeled and simulated by CFD calculations according to the obtained higher head and higher efficiency, and the structural parameters are shown in Table 10. From the table, it can be seen that whether it is to increase the head or improve the efficiency, it is necessary to reduce the outlet angle and increase the outlet width and blade wrap angle. The individuals with the best head or best efficiency were compared with the head and efficiency of the initial model, as shown in Table 11. From the table, it can be seen that the selected individuals with the best head or best efficiency satisfy the maximum increase in head without a decrease in efficiency and the maximum increase in efficiency without a decrease in head, respectively. The individual with the best head has an increase of 2.653 m compared to the initial model head and a decrease of about 0.6 percentage points in efficiency, which is a small and acceptable decrease in efficiency. The individual with the best efficiency has an increase of about 0.6 m compared to the initial model head and an increase of about 2.3 percentage points in efficiency. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Structural Parameters**} & **Exit Placement** & \multirow{2}{*}{**Outlet Width \(b_{2}\)**} & **Blade Wrap Angle \(\varphi\)** \\  & **Angle \(\beta_{2}\)** & & \\ \hline Initial model & 27 & 10 & 97 \\ Individuals with better head & 17 & 15 & 122 \\ Individuals with better efficiency & 17 & 13 & 130 \\ \hline \end{tabular}
\end{table} 
Table 10: Comparison of structural parameters before and after optimization. Figure 10: Pareto frontier distribution truncation.