## 2 Optimization Design Process and Modeling Method
### Pump Model
#### 2.2.2 Optimization Objectives  
The objective of the optimization is to build the relationships between the geometrical parameters of the impeller and the various kinds of energy losses in the pump, with the ultimate goal of minimizing the total energy loss. Thus, the following geometric parameters of impeller were selected as optimization variables based on the results of previous practice, such as the impeller outlet diameter \(D_{2}\), blade outlet width \(b_{2}\), blade number \(Z\), blade outlet angle \(\beta_{2}\), blade inlet angle \(\beta_{1}\), leading edge tangential angle \(t_{3}\), trailing edge tangential angle \(\phi\), and blade thickness \(\delta\). Moreover, the electronic water pump needs to operate under multiple working flowrates according to the working state of the cooling system. Therefore, a multi-point optimization is necessary. To broaden the high efficiency range of the model pump, the weighted average efficiency of three flow conditions, namely 0.8Q\({}_{d}\), 1.0Q\({}_{d}\), 1.25Q\({}_{d}\), were set as the objective function, and the design head was set as a constraint. In general, the time of optimization with the intelligent method will be increased with both the number of parameters and the range of each parameter. In this study, exactly four parameters, impeller outlet diameter \(D_{2}\), blade outlet width \(b_{2}\), blade number \(Z\), blade outlet angle \(\beta_{2}\), were selected for the orthogonal experiment at the first step. These four parameters are considered to play most important roles in the comprehensive performance of centrifugal pumps [6; 13]. Local optimization can be achieved, and then the range of parameters can be reduced in this step. However, because the optimization is also a multi-objective problem, it is necessary to establish the model with the highest weighted average efficiency under three working flowrates. The optimization problem can be formulated as:  
Solve \(\mathbf{X}=[D_{2},\,b_{2},\,Z,\,\beta_{2}]^{\mathrm{T}}\) make:  
\[f(\mathbf{X})=\frac{\sum\limits_{i=1}^{3}\omega_{i}\eta_{i}(\mathbf{X})}{\sum\limits_{ i=1}^{3}\omega_{i}}\rightarrow\max, \tag{1}\]  
\[\text{subject to}\quad \begin{array}{c}\mathbf{X}\in\mathrm{R}\\ H_{\mathrm{d}}\geq 7.5\mathrm{m}\end{array} , \tag{2}\]  
where \(\mathbf{X}\) is a vector of all design variables, \(R\) is the set of variable ranges, \(\eta_{1}\), \(\eta_{2}\), \(\eta_{3}\) represent the efficiency of 0.8Q\({}_{d}\), 1.0Q\({}_{d}\), and 1.25Q\({}_{d}\), respectively. Accordingly, \(\omega_{i}\) represents the weighting factors. Analytic hierarchy process (AHP) was developed in the late 1970s by [PERSON] [27]. Due to its simplicity, ease of use, and great flexibility, this technique has been widely used as a decision model to deal with multi-criteria evaluation. However, AHP does not take into account the inherent uncertainty and imprecision. Moreover, the comparison matrices used in AHP are combined with crisp scales. To deal with the uncertainty or vagueness of data, fuzzy analytic hierarchy process (FAHP) was utilized to derive the weight of every objective [28; 29]. In FAHP, the importance of factor A to factor B is obtained and the fuzzy judgment matrix is generated after comparing different factors. This study use FAHP with the following steps. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Description** & **Parameter** & **Symbol** & **Unit** & **Value** \\ \hline \multirow{6}{*}{Impeller} & Inlet diameter & \(D_{1}\) & mm & 20 \\  & Outlet diameter & \(D_{2}\) & mm & 47 \\  & Outlet width & \(b_{2}\) & mm & 2.2 \\  & Blade inlet angle & \(\beta_{1}\) & \(\,Firstly, build and construct a fuzzy complementary judgment matrix. The pairwise comparison among factors is made to determine the importance of one factor relative to the other factors, and then the judgment matrix is generated:  
\[A=\left(a_{ij}\right)_{N\times N^{\prime}} \tag{3}\]  
where \(a_{ij}\) represents the evaluation of the relative importance of the factor \(i\) to the factor \(j\), \(a_{j\,i}=1/a_{ij}\).