## 3 Automatic Intelligent Optimization
### Orthogonal Design Optimization
#### 3.2.1 Range and Sensitivity Analysis  
The range analysis was performed to evaluate the influence of different factors on the evaluation index. The factor with the largest range was considered as the most sensitive factor, which had the greatest impact on the evaluation index. First of all, the comprehensive average value of \(t\) at the same level of each factor was obtained by Equation (15) On this basis, the range of each influencing factor was obtained by Equation (16), subtracting the minimum value from the maximum value of \(t\) at different levels of each factors. The results are shown in Table 6.  
\[t_{i}=\frac{T_{i}}{r}, \tag{15}\]  
\[R=\max(t_{1},t_{2},t_{3})-\min(t_{1},t_{2},t_{3}), \tag{16}\]  
where \(T_{i}\) represents the sum of all test target values at the \(i\) level; \(r\) is the number of different factor levels; \(t_{i}\) denotes the average of the test target values and \(R\) is the range. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Objective** & \(\boldsymbol{\eta_{1}}\) & \(\boldsymbol{\eta_{2}}\) & \(\boldsymbol{\eta_{3}}\) & \(\boldsymbol{\omega_{i}}\) \\ \hline \(\eta_{1}\) & 1 & 2/3 & 10/11 & 0.2786 \\ \(\eta_{2}\) & 3/2 & 1 & 5/4 & 0.4059 \\ \(\eta_{3}\) & 11/10 & 4/5 & 1 & 0.3155 \\ _CR_ & & 0.000725 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Evaluation results and weight factors. _CR_, consistency ratio. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Number**} & \multicolumn{4}{c}{**Levels**} & \multicolumn{4}{c}{**Corresponding Parameters**} & **Results** \\ \cline{2-10}  & **A** & **B** & **C** & **D** & \(\boldsymbol{D_{2}}\) & \(\boldsymbol{b_{2}}\) & \(\boldsymbol{Z}\) & \(\boldsymbol{\beta_{2}}\) & \(\boldsymbol{f(X)}\) \\ \hline
1 & \(A_{1}\) & \(B_{1}\) & \(C_{1}\) & \(D_  
In order to visually reflect the impact size and trend of each factor on the evaluation target, the average value and factor level of the test target are drawn as a factors and indicators trend graph, as is shown in Figure 10. As seen from the range value analysis, the order of importance on efficiency is C, A, B, and D. The primary and secondary order of influence of each factor on the target is: \(Z>D_{2}>b_{2}\) \(>\beta_{2}\). The blade number \(Z\) has the greatest impact on pump efficiency, followed by the factor A. As the blade number \(Z\) and impeller outlet diameter \(D_{2}\) increase, the hydraulic performance improves significantly. Generally, the effect of impeller outlet width \(b_{2}\) on efficiency is relatively small. Analysis of variance (ANOVA) is an effective method to determine the significance of controllable factors on the research results [33]. It tests the variables by mean squaring and estimates experimental errors at specific levels. \(F\)-test is always used in ANOVA to analyze whether a particular design has any significant change in quality standards. In analyzing the parameters, the \(F\)-test tool based on ratio of mean square and residual error is used to find the significance of a factor. It is evaluated by using the equations below. The sum of squared total deviation \(SS_{T}\) and total degrees of freedom \(f_{T}\) are defined as:  
\[ \begin{split} SS_{T}=\sum\limits_{j=1}^{m}f_{j}^{2}(x)& -\frac{\tau^{2}}{m}(j=1,2,\ldots,m)\\ f_{T}&=m-1\end{split}  \tag{17}\]  
where \(m\) is the number of simples, \(m=9\) in this study. \(T\) is the sum of the nine scheme results. The sum of square \(SS_{k}\) and degrees of freedom of each factor \(f_{k}\) are defined as follows:  
\[ \begin{split} SS_{k}=\frac{1}{n}\sum\limits_{k=1}^{n}T_{k}^{2}- \frac{\tau^{2}}{m}(k=1,2,\ldots,n)\\ f_{k}=n-1\end{split}  \tag{18}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Target**} & \multicolumn{4}{c}{_f(X) \%_} \\ \cline{2-5}  & **A** & **B** & **C** & **D** \\ \hline \(t_{1}\) & 49.28 & 49.51 & 48.06 & 49.65 \\ \(t_{2}\) & 50.15 & 50.62 & 51.15 & 50.27 \\ \(t_{3}\) & 50.97 & 50.27 & 51.18 & 50.48 \\ \(R\) & 1.69 & 1.11 & 3.12 & 0.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Range analysis.