# Parametric Optimization Study of Novel Winglets for Transonic Aircraft Wings
## 5 Optimization Studies
### Taguchi--Grey Relational Analysis Coupled with Principal Component Analysis  
The multi-input, discrete, and uncertain data problems are compactly analyzed by grey relational analysis (GRA), which gives the relationship between known and unknown information. This analysis appraises the absolute difference between data sequences while predicting the approximate grade of correlation between the responses. The grey relational grade (GRG) computes the degree of influence of similar sequences with the reference sequences [42]. GRA substantially aims to convert the multiple responses into single GRG values of all the trails. GRA synergizes with principal component analysis (PCA) in optimizing the process parameters in order to optimize the responses by finding the weightage of the responses, which determines the good results anticipated [43; 44; 45]. The designing and analysis of the transonic wing can be performed under several design parameters and with various responses of the winglet. In this study, to achieve the optimum responses of the transonic design, the Taguchi-based grey relational analysis (GRA) coupled with principal component analysis (PCA) for multi-response optimization are applied to improve the performance of the winglet design in the transonic wing. The grey relational grade (GRG) is used to identify the relationship between parameters in the sequence [46]. A higher GRG value level informs the optimum parameter levels in winglet design for maximizing the \(C_{L}/C_{D}\) ratio (the larger-the-better concept) and minimizing the \(C_{M}\) (the smaller-the-better) value. The reduction in variability in the experimental data are initially normalized, which is known as data preprocessing, and it helps to reduce the large variations in the responses between other responses. In other words, the normalizing experimental responses lie in the range of 0 to 1 [47]. Due to the above reason, two different approaches are discussed, depending on the characteristics of the data sequence, which may be either the larger-the-better or the smaller-the-better concept. For the larger-the-better concept, the sequence is normalized using the following equation:  
\[x_{i}(k)^{*}=\frac{x_{i}(k)-x_{i}(k)^{-}}{x_{i}(k)^{+}-x_{i}(k)^{-}} \tag{3}\]  
For the smaller-the-better concept, the sequence is normalized using the following equation:  
\[x_{i}(k)^{*}=\frac{x_{i}(k)^{+}-x_{i}(k)}{x_{i}(k)^{+}-x_{i}(k)^{-}} \tag{4}\]  
\(x_{i}(k)^{*}\) is the normalised response, \(x_{0}(k)\) is the desired response, \(x_{i}(k)^{+}\) is the maximum of \(x_{i}(k)\), and \(x_{i}(k)^{-}\) is the minimum of \(x_{i}(k)\). After pre-processing of the data, the corresponding grey relational coefficient (GRC) is calculated to express the relationship between the predicted and the actual experimental responses of the experiments. The grey relational coefficient \(\zeta_{i}\) (\(k\)) can be calculated using the following equation:  
\[\zeta_{i}(k)=\frac{\Delta_{min}+\psi\Delta_{max}}{\Delta_{oi}(k)+\psi\Delta_{ max}} \tag{5}\]  
where the difference in the absolute value \(x_{0}\) (\(k\)) and \(x_{i}(k)\) is \(\Delta_{oi}=\|x_{0}(k)-x_{i}(k)\|\). The smallest value of \(\Delta_{0i}\), \(\Delta_{min}=\underset{\forall j\in i}{min}\underset{\forall k}{min}\|x_{0}( k)-x_{j}(k)\|\). The largest value of \(\Delta_{0i}\), \(\Delta_{max}=\underset{\forall j\in i}{max}\underset{\forall k}{max}\|x_{0}( k)-x_{j}(k)\|\), and the distinguishing coefficient, \(\zeta=0.5\), is widely accepted for the analysis [41]. The experimental results are primarily used to obtain S/N ratios for the performance characteristics to examine the required effect with the best performance and the smallest variance. In this work, \(\text{C}_{\text{L}}/\text{C}_{\text{D}}\) is considered with the larger-the-better concept, while \(\text{C}_{\text{M}}\) is for the smaller-the-better concept for the winglet design of the transonic wing.