## 3 Results and Discussion  
### Development of the RSM Model  
For the purposes of this study, laser cutting experiments were designed using a Box-Behnken experimental design with three factors at three levels. The independent variables were cutting speed (A), laser power (B), and assist gas pressure (C). The dependent variable, or system response, was surface roughness. Initially, the significance of the model and its terms was determined through analysis of variance (ANOVA). Based on the statistical evaluation of the model's adequacy and the significance of deviations from the model, a quadratic model was recommended for surface roughness and subsequently selected for further analysis. A backward elimination method was applied to remove nonsignificant terms, resulting in a reduced model containing only influential parameters. Table 3 presents the reduced ANOVA table with the significant factors retained.  
The analysis of variance (ANOVA) indicated that the developed model is statistically significant, with a model F-value of 13.93 and only a 0.02% probability that such a high F-value could occur due to noise.  
Factors A and B, as well as their quadratic terms (A\({}^{2}\) and B\({}^{2}\)), were found to be significant (\(p\)-values less than 0.05), while factor C was excluded from the model due to its lack of statistical significance, according to the RSM analysis. Since the inclusion of nonsignificant terms can degrade model quality, a model reduction was performed to enhance predictive performance and simplify interpretation.  
The lack of fit test revealed a significant mismatch between the model and the experimental data (lack of fit F-value of 11.08, \(p=0.0171\)), suggesting that the model does not fully capture the variability in the response and could be further improved.  
The reduced model, obtained by eliminating nonsignificant factors, will be used for further process optimization and prediction of surface roughness during laser processing. Its improved simplicity and maintained statistical significance make it suitable for reliable interpretation and practical application. The final model, in terms of actual factors, is given in Equation (7):  
\[R_{a}=236.534-30.738\cdot v-5.143\cdot P_{w}+14.658\cdot v^{2}+0.030\cdot P_{w} ^{2} \tag{7}\]  
The basic statistical data of the reduced quadratic model for processing productivity are presented in Table 4. The coefficient of determination (R-squared) is 0.8227, indicating a strong relationship between the model predictions and experimental data. Both the adjusted R-squared (0.7637) and the predicted R-squared (0.5898) are reasonably close to each other, which confirms the significance and reliability of the model. The adequacy of the model is further supported by the Adeq Precision value of 11.399, which is well above the recommended minimum of 4, indicating an adequate signal-to-noise ratio. In addition, the coefficient of variation (C.V. %) of 9.98% and the predicted residual sum of squares (PRESS) value of 1.85 further confirm the model's significance and stability.  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Source** &
\begin{tabular}{c} **Sum of** \\ **Squares** \\ \end{tabular} & **df** & **Mean Square** & **F Value** & **\(p\)-Value** & **Significance** \\ \hline Model & 3.71 & 4 & 0.93 & 13.93 & 0.0002 & Significant \\ A-\(v\) & 0.73 & 1 & 0.73 & 10.96 & 0.0062 & None \\ B-\(P_{w}\) & 0.4 & 1 & 0.4 & 5.95 & 0.0313 & None \\ A\({}^{2}\) & 1.45 & 1 & 1.45 & 21.79 & 0.0005 & None \\ B\({}^{2}\) & 0.99 & 1 & 0.99 & 14.88 & 0.0023 & None \\ Residual & 0.8 & 12 & 0.067 & None & None & None \\ Lack of Fit & 0.76 & 8 & 0.096 & 11.08 & 0.0171 & Significant \\ Pure Error & 0.035 & 4 & 0.008626 & None & None & None \\ Cor Total & 4.51 & 16 & None & None & None & None \\ \hline \hline \end{tabular}
\end{table}
Table 3: ANOVA for response surface reduced quadratic model.