## 4 Multi-Response Optimization
The defuzzified exact values of the GFRG are presented in Figure 15. The dark green columns represent the experimental trials (fourth, fifth, and ninth) with the three highest GFRG values. These solutions lie on the Pareto front, representing a set of compromise solutions for the simultaneous optimization of multiple objective functions: (_Ckw_) min, (_\(\beta\)_) min, (_Ce top_) min, and (_Ce bottom_) min. The highest GFRG value was achieved in the fifth experiment. The corresponding input process parameter levels are v = 2000 mm/min, I = 65 A, and H = 1.5 mm. This result corresponds to the GRG rank presented in Table 6's last column. In order to validate THE prediction accuracy of the developed fuzzy logic model, a comparison between the calculated GRG and predicted GFRG values was performed. The coefficient of determination (R\({}^{2}\)) and mean absolute percentage error (MAPE) were applied as validation measures. The comparison results are presented in Figure 16. The obtained R\({}^{2}\) value of 0.9089 and MAPE of 5.68% indicate a strong agreement between the calculated GRG and predicted GFRG values, further confirming the accuracy of the developed fuzzy logic model. In the next step, after the prediction accuracy of the developed model was verified, it was additionally applied to generate 3D surface plots. These plots offer a significant advantage by simplifying the evaluation and interpretation of the results and by enabling a clearer analysis of how the input variables affect the predicted output. Figure 17 shows the 3D surface plots of the GFRG, illustrating its variation based on the interaction of two circular cut quality response GRCs, while the remaining two GRCs are held constant at  
Figure 13: Gaussian membership functions for (**a**) input: GRC circular kerf width and (**b**) output GFRG. specific levels. The input settings were chosen to highlight the regions (red color) on the plots corresponding to the highest predicted GFRG values. Figure 16: Comparison results between calculated GRG and predicted GFRG values with validation measures R\({}^{2}\) and MAPE. Figure 14: Graphical representation of fuzzy IF-THEN rules between inputs: GRCs of circular quality responses and GFRG. Figure 15: GFRG values for all experimental trials. Finally, in order to discuss the influence of the plasma jet cutting process parameters, the cutting speed, arc current, and cutting height on the GFRG, and to determine the optimal process regions, the same Mamdani fuzzy inference system was applied. The inputs and output of this system were also described by Gaussian membership functions. These functions were already presented in Figures 5 and 13b. The graphical representation of a developed set of 27 fuzzy IF-THEN rules is shown in Figure 18. The settings of the process parameters levels were chosen to define the highest value of the GFRG. Figure 19 reveals the comparison results between the calculated GRG and fuzzy-logic-predicted GFRG values. These results and calculated validation measures (R\({}^{2}\): 0.838, MAPE: 5.97) confirm the prediction accuracy of this GFRG fuzzy logic model. In order to analyze more precisely the influences of individual process parameters on the GFRG output, the main effects plots were created, Figure 20. These plots were generated based on the average fuzzy-logic-predicted GFRG values and serve to enhance the determination of the optimal process conditions and the identification of the most significant process parameters. The steep slope of the cutting speed's blue line from Figure 20 reveals that this parameter has the highest influence on the GFRG while the other two parameters are less significant. The blue line points from this plot, that has the highest mean value, define the optimal process parameter levels. This corresponds to the next process parameter settings: _v_: 2000 mm/min, _I_: 65 A, _H_: 1.5 mm. Table 7 confirms these findings. The bold numbers in this table define the highest mean GFRG values for each parameter and, consequently, their optimal operating levels.