## 4 Multi-Response Optimization
The last column in Table 7 determines the significance rank of all parameters based on the range between the highest and lowest mean GFRG value. According to that, it can be concluded that the cutting speed has the highest significance for the GFRG, followed by the cutting height and, finally, the arc current. This is affirmed also by the ANOVA results, Figure 14.0 ANOVA was performed  
Figure 14.1 Three-dimensional surface plots of GRC circular quality response interactions: (**a**) circular ker width and circular bevel angle, (**b**) circular ker width and circularity error—top, (**c**) circular ker width and circularity error—bottom, (**d**) circular bevel angle and circularity error—top, (**e**) circular bevel angle and circularity error—bottom, and (**f**) circularity error—top and circularity error—bottom effects on GFRG values. with a confidence level of 95%. From the last column, it is visible that the \(p\)-value of the cutting speed is lower than 0.05, which confirms the previous conclusion that it has the greatest influence on the GFRG. Figure 14.2 Comparison results between calculated GRG and predicted GFRG values with validation measures R\({}^{2}\) and MAPE. Figure 14.3 Main effects plot for grey–fuzzy reasoning grade (GFRG). Figure 14.4 Graphical representation of fuzzy IF-THEN rules between process parameters: cutting speed, arc current, cutting height, and predicted GFRG values. In order to determine the Pareto front of optimal solutions and to discuss the process parameter interactions' effects on the GFRG values, 3D surface plots were constructed, Figure 14.5 In these plots, two parameters were varied while the third one was kept constant on a certain level. In Figure 21, the dark red color marks the areas with the highest GFRG values. These areas correspond to the most desirable solutions for the simultaneous optimization of all objectives, the circular quality response functions: (_Ckw_) min, (\(\beta\)) min, (_Ce top_) min, and (_Ce bottom_) min. These multi-objective optimization solutions correspond to the next plasma jet cutting process conditions: \(v\) = 2000 mm/min, \(I\) = 65 A, \(H\) = 1-1.5 mm/\(v\) = 2000 mm/min, \(I\) = 85 A, \(H\) = 2 mm. Finally, the validation experimental test was performed to verify the prediction capabilities of the developed fuzzy logic models and to check their generalization ability. The process conditions for this additional test were chosen according to the parameters levels that correspond to the multi-objective optimization solutions. Accordingly, three additional experimental trials were conducted and the analyzed cut quality characteristics were measured. First, the prediction capability of the fuzzy logic model for the cut quality responses was tested, followed by the evaluation of the fuzzy logic model used to estimate the GFRG values. As a validation measure, MAPE was applied. The validation test results are presented in Figure 14.6 The calculated MAPE values indicate that the previously described development process of the fuzzy logic models was successfully carried out and that they possess a very good prediction capability. \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Process Parameters**} & \multicolumn{4}{c}{**Operating Level**} & \multirow{2}{*}{**Max-Min**} & \multirow{2}{*}{**Rank**} \\ \cline{2-2} \cline{4-6}  & & **1** & & & **2** & **3** \\ \hline Cutting speed, \(v\) (mm/min) & **0.6093** & 0.5131 & 0.5000 & 0.1093 & 1 \\ Arc current, \(I\) (A) & 0.5262 & **0.5584** & 0.5378 & 0.0322 & 3 \\ Cutting height, \(H\) (mm) & 0.5378 & **0.5599** & 0.5247 & 0.0352 & 2 \\ \hline \end{tabular}
\end{table} 
Table 7: Response table for grey–fuzzy reasoning grade (GFRG). Figure 21: Three-dimensional surface plots of process parameters interactions: (**a**) cutting speed and arc current, (**b**) cutting speed and cutting height, and (**c**) arc current and cutting height effects on GFRG values.