## 4 Multi-Response Optimization  
The main objective of this chapter is to carry out a multi-response optimization procedure in order to identify the process parameter values that simultaneously minimize the circular kerf width, level angle, and circularity error on both the top and bottom surfaces of the workpiece sheet material. The obtained optimization results and corresponding process parameter values will be identified as those that achieve the highest possible quality of plasma jet circular cuts, thereby minimizing the need for the post-processing procedure. In order to carry this out, a hybrid grey-fuzzy approach was developed and applied. This approach combines grey relational analysis (GRA) and the artificial intelligence fuzzy logic technique. The fuzzy logic technique enhances the application of traditional GRA by incorporating human-like reasoning in the optimization procedure, making it more suitable for solving complex, real problems where uncertainty, vagueness, and subjectivity is quite often present. This novel scientific approach is already successfully utilized in multiple publications for solving optimization problems in manufacturing processes [29; 30; 31; 32; 33; 34; 35]. The GRA method is used to analyze numerical data sets from different responses in order to estimate how closely the experimental results match the ideal values. The degree of similarity between the actual and ideal responses is expressed by grey relational coefficients(GRC). In the case when the experimental and ideal response values are the same, then the GRC equals one. In order to perform multiple-response optimization, GRA converts multiple responses' GRCs into a single output defined by a grey relational grade (GRG). In this moment, a multi-objective optimization problem was transferred into a single objective optimization problem where the goal is to find an experimental trial that results in the highest value of the GRG. The highest GRG value represents the optimal solution across multiple responses, regardless of whether the objective for a given response is to maximize or minimize its value. Finally, the levels of the process parameters corresponding to the optimal solution can be easily identified. In order to conduct GRA, several steps must be completed. The first step is the normalization of the analyzed responses in the range of \([0,1]\) to decrease the variability of their actual experimental values. Depending on whether the objective of a given response is to maximize or minimize it, the appropriate normalization equation should be applied. In this case, the objective is to minimize all the circular cut quality responses: _Ckw_, \(\beta\), _Ce top_, and _Ce bottom_. Accordingly, normalization was performed following Equation (5). \[X_{i}^{*}(k)=\frac{max\ X_{i}^{0}(k)-X_{i}^{0}(k)}{max\ X_{i}^{0}(k)-min\ X_{ i}^{0}(k)} \tag{5}\]  
where \(X_{i}^{0}(k)\) is the original sequence, \(X_{i}^{*}(k)\) is the sequence after data pre-processing (normalization), \(max\ X_{i}^{0}(k)\) is the largest value in \(X_{i}^{0}(k)\), and \(min\ X_{i}^{0}(k)\) is the smallest value in \(X_{i}^{0}(k)\). The second step in the GRA optimization process is calculating the GRC. The GRC for each response can be calculated following Equation (6). \[\xi_{i}(k)=\frac{\Delta_{min}+\xi\ \Delta_{max}}{\Delta_{0i}(k)+\xi\ \Delta_{max}} \tag{6}\]  
\(\Delta_{0i}(k)\), \(\Delta_{min}\) and \(\Delta_{max}\) are calculated using Equations (7)-(9). \[\Delta_{0i}(k)=\|X_{0}^{*}(k)-X_{i}^{*}(k)\| \tag{7}\]  
\[\Delta_{max}=maxmax\|X_{0}^{*}(k)-X_{i}^{*}(k)\| \tag{8}\]  
\[\Delta_{min}=minmin\|X_{0}^{*}(k)-X_{i}^{*}(k)\| \tag{9}\]  
where \(\zeta\) is the distinguishing coefficient in the range of [0, 1] (in general, \(\zeta=0.5\)), \(\Delta_{0i}(k)\) is the deviation sequence for the reference sequence, \(X_{0}^{*}(k)\) is the reference sequence (\(X_{0}^{*}(k)=1,\ k=1,2\ldots n,\ n\) is number of responses), and \(X_{i}^{*}(k)\) is the specific comparison sequence. In the third step, the GRG is calculated following Equation (10).