# Parametric Optimization Study of Novel Winglets for Transonic Aircraft Wings
## 5 Optimization Studies
### Taguchi--Grey Relational Analysis Coupled with Principal Component Analysis
All the original sequences of the S/N ratio in Table 12 are then substituted in Equations (3) and (4) to obtain normalised values of \(\text{C}_{\text{L}}/\text{C}_{\text{D}}\) and \(\text{C}_{\text{M}}\), respectively. According to [PERSON] et al. [41], larger values of the normalised results correspond to better performance, and the maximum normalised results equal to 1 indicate the best performance. The estimated values of normalized responses with their corresponding grey relational coefficients are based on Equation (6), which are shown in Table 12.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model Number**} & \multicolumn{2}{c}{**Normalized Responses**} & \multicolumn{2}{c}{**Grey Relational Coefficient (GRC)**} \\ \cline{2-5}  & \(\text{C}_{\text{L}}/\text{C}_{\text{D}}\) & \(\text{C}_{\text{M}}\) & \(\text{C}_{\text{L}}/\text{C}_{\text{D}}\) & \(\text{C}_{\text{M}}\) \\ \hline
1 & 0.2101 & 0.4695 & 0.3876  
Principal component analysis (PCA) is applied to examine the weightage of each response of the winglet design to imitate its relative importance in the grey relational analysis. Equation (6) is used to produce the correlation coefficient matrix for eigenvalue and eigenvector determination based on Table 9 and the grey relational coefficients of the responses of \(\mathrm{C_{L}}/\mathrm{C_{D}}\) and \(\mathrm{C_{M}}\). \[N_{k,l}=\frac{Cov(x_{i}(k),\ x_{i}(l))}{\sqrt{Var(x_{i}(k))\cdot Var(x_{i}(l))}} \tag{6}\]  
where \(k=1,2,\ldots,n\), and \(l=1,2,\ldots,n\). \(Cov(x_{i}(k),x_{i}(l))\) is the covariance of sequence \(x_{i}(k)\) and \(x_{i}(l)\), \(Var(x_{i}(k))\) is the standard deviation of sequence \(x_{i}(k)\), and \(Var(x_{i}(l))\) is the standard deviation of sequence \(x_{i}(l)\). The eigenvalues and eigenvectors are determined from the Covariance matrix array using the following equation:  
\[\left(N_{k,l}-\lambda_{k}I_{m}\right)V_{i,k}=0 \tag{7}\]  
where \(N\) is the correlation coefficient matrix form of \(N_{k,l}\), \(\lambda_{k}\) is the \(k\)th eigenvalue, \(\sum_{k=1}^{n}\lambda_{k}=n\)\(k=1,2,\ldots n\), and \(V_{i,k}=\left[a_{k1},\ a_{k2},\ \ldots,\ a_{kn}\right]^{T}\) is the eigenvector corresponding to the eigenvalue \(\lambda_{k}\). Using Equation (7), the eigenvalues are determined from the correlation coefficient matrix of the grey relational coefficients of the transonic wing design and are shown in Table 13. The eigenvectors and contribution of winglet design responses corresponding to each eigenvalue are enumerated in Table 14.  
The weighting values of the responses are determined by principal component analysis, and the following equation gives the weighted grey relational grade of all the responses:  
\[\gamma_{i}=\frac{1}{n}\sum_{k=1}^{n}\left(\omega_{k}.\bar{\xi_{i}}(k)\right) \tag{8}\]  
where \(i=1,2,\ldots,m\), and \(k=1,2,\ldots,n\). \(\omega_{k}\) is the weighting factor for the response \(k\), which is estimated through principal component analysis. The relational degree between the ideal sequence \(x_{0}(k)\) and the given sequence \(x_{i}(k)\) is stronger the higher the grey relational grade. The first principal component has high variance contribution characteristics of 85.28% compared to other principal components. Furthermore, the squares of its corresponding eigenvectors are designated as the weighting values of \(\mathrm{C_{L}}/\mathrm{C_{D}}\) and \(\mathrm{C_{M}}\), which are 0.4999 and 0.4999, respectively. Using Equation (8), the weighted grey relational grades and their corresponding ranks are calculated as shown in Table 15. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Principal Component** & **Eigenvalues** & **Explained Variation (\%)** \\ \hline First & 1.7055 & 85.28 \\ Second & 0.2945 & 14.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Eigenvalues and explained variations for principal components of responses.