## 5 Sensitivity Analysis  
A variance-based technique is used to test the sensitivity of the parameters [38]. The model can be represented by:  
\[Y=F_{t}(x_{1},x_{2},...,x_{6}) \tag{35}\]  
The optimized variables \(x_{1},\)\(x_{2},\)\(...,\)\(x_{6}\) are analyzed. \(F_{t}\) stands for the objective function \(F_{1},\)\(F_{2},\)\(F_{3},\)\(F_{4}\) and \(F_{5}\) in Section 4.2. The parameter \(Y\) is the model output.  
The total variance of \(\Psi(Y)\) is expressed as follows [39]:  
\[\Psi(Y)=\sum\limits_{p=1}^{6}\,\Psi_{p}+\sum\limits_{1\leq p<q\leq 6}\Psi_{pq }+...+\Psi_{1,2,...,6} \tag{36}\]  
The parameter \(\Psi_{p}\) in Equation (36) is an index measuring the main effect of the parameter \(x_{p},\) which is expressed in Equation (37):  
\[\Psi_{p}=\Psi[E(Y|x_{p})] \tag{37}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Constraints** \\ \hline \(L_{\rm f}\) & 0.15–0.5 m \\ \(L_{\rm e}\) & 0.15–0.5 m \\ \(\theta\) & \(\pi/10\)–\(\pi/10\) rad \\ \(H\) & 0.002–0.01 m \\ \(\delta_{\rm plate}\) & 0.001–0.002 m \\ \(L_{\rm h}\) & 0.1–0.7 m \\ \hline \hline \end{tabular}
\end{table}
Table 3: Range of variables.  
Equation (37) determines the total sensitivity index for the \(p\)th parameter and is the sum of all effects involving the parameter \(x_{p}\). The parameter \(S_{Tp}\) considers the interactions between the \(p\)th parameter and the other parameters. The total sensitivity index is denoted by the symbol \(S_{Tp}\), which is defined in Equation (38).  
\[S_{Tp}=1-\frac{\Psi_{,p}}{\Psi(Y)} \tag{38}\]  
The parameter \(\Psi_{,p}\) is the sum of all of the variance terms that do not include the index \(p\).  
The sensitivity index is computed using a Monte Carlo method [40]. The principle is to randomly generate samples of parameters within their permissible ranges and to estimate \(\Psi(Y)\), \(\Psi_{p}\) and \(\Psi_{-p}\) as follows:  
(1) Choose a base sample dimension \(N\).  
(2) Generate two random input sample matrices \(M_{1}\) and \(M_{2}\) with the dimensions of \(N\times 6\).  
\[M_{1}=\left[\begin{array}{cccccccc}x_{11}&x_{12}&\cdots&x_{1i}&\cdots&x_{16} \\ x_{21}&x_{22}&\cdots&x_{2i}&\cdots&x_{26}\\ \vdots&\vdots&&\vdots&&\vdots\\ x_{31}&x_{32}&\cdots&x_{31i}&\cdots&x_{38}\\ \end{array}\right]\]  
(3) Define a matrix \(N_{p}\) formed from all of the \(M_{2}\) columns, except the \(p_{th}\) column, which is taken from matrix \(M_{1}\). The matrix \(N_{Tp}\) is complementary to \(N_{p}\).  
\[N_{p}=\left[\begin{array}{cccccccc}x_{11}^{\prime}&x_{12}^{\prime}&\cdots&x_ {1p}&\cdots&x_{16}^{\prime}\\ x_{21}&x_{22}&\cdots&x_{2p}&\cdots&x_{26}\\ \vdots&\vdots&&\vdots&&\vdots\\ x_{N1}^{\prime}&x_{N2}^{\prime}&\cdots&x_{N_{p}}&\cdots&x_{N6}^{\prime}\\ \end{array}\right]N_{Tp}=\left[\begin{array}{cccccccc}x_{11}&x_{12}& \cdots&x_{1p}^{\prime}&\cdots&x_{16}\\ x_{21}&x_{22}&\cdots&x_{2p}^{\prime}&\cdots&x_{26}\\ \vdots&\vdots&&\vdots&\vdots\\ x_{N1}&x_{N2}&\cdots&x_{N_{p}}^{\prime}&\cdots&x_{N6}^{\prime}\\ \end{array}\right]\]  
(4) Calculate the model output, such as the objective function value for all of the input values in the sample matrices \(M_{1}\), by obtaining three column vectors of the model outputs with dimensions \(N\times 1\), expressed as:  
\[Y=\Theta(M_{1})\ldots Y_{\mathrm{T}}^{\prime}=\Theta(N_{Tp}) \tag{39}\]  
(5) Compute the sensitivity indices based on the scalar products of the above defined vectors of the model outputs:  
\[\Theta_{0}=\frac{1}{N}\sum_{q=1}^{\mathrm{N}}Y^{(q)} \tag{40}\]  
The parameter \(\Theta_{0}\) is the mean value of \(Y\).  
The total variance is calculated using Equation (41) to Equation (43).  
\[\Psi=\frac{1}{N}\sum_{q=1}^{N}\left(Y^{(q)}\right)^{2}-\Theta_{0}^{2} \tag{41}\]  
\[\Psi_{p}=\frac{1}{N}\sum_{q=1}^{N}Y^{(q)}Y^{\prime(q)}-\Theta_{0}^{2} \tag{42}\]  
\[\Psi_{-p}=\frac{1}{N}\sum_{q=1}^{N}Y^{(q)}Y^{\prime(q)}-\Theta_{0}^{2} \tag{43}\]