## 3 Results and Discussions
### RSM Hardness Modeling  
The hardness of the hardened layer was used as the response value, and the data from the BBD experiments conducted at different induction frequencies were modeled by multivariate regression fitting. The model was then analyzed using the analysis of variance (ANOVA) method, and the results of the ANOVA for the hardness of the hardened layer are presented in Tables 6 and 7. It was found that the prediction model of the hardness of the hardened layer using the response surface method was worse compared to that of the depth. Table 6 illustrates that at an induction frequency of 10 kHz, the quadratic term A\({}^{2}\), the cubic term AB\({}^{2}\), and the three interaction terms significantly influence the hardness of the hardened layer (\(p\)! 0.05), while the remaining terms are not statistically significant. The process parameters are ranked in terms of their influence on hardness as follows: B? C? A. The \(p\)-value for the lack-of-fit term is 0.2819 (\(p\)? 0.05), suggesting that the model effectively \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source** & **Sum of Squares** & **Mean Squares** & **F-Value** & \(p\)**-Value** & **Significance** \\ \hline Model & 22.96 & 2.55 & 272.85 & \textless{}0.0001 & Yes \\ A-P & 1.41 & 1.41 & 150.88 & \textless{}0.0001 & Yes \\ B-V & 2.58 & 2.58 & 275.88 & \textless{}0.0001 & Yes \\ C-IP & 15.75 & 15.75 & 1684.75 & \textless{}0.0001 & Yes \\ A\({}^{2}\) & 0.0622 & 0.0622 & 6.65 & 0.0366 & Yes \\ B\({}^{2}\) & 0.1402 & 0.1402 & 14.99 & 0.0061 & Yes \\ C\({}^{2}\) & 0.1603 & 0.1603 & 111.89 & \textless{}0.0001 & Yes \\ AB & 0.0063 & 0.0063 & 0.6772 & 0.4377 & No \\ AC & 0.1278 & 0.1278 & 13.67 & 0.0077 & Yes \\ BC & 0.0002 & 0.0002 & 0.0253 & 0.8781 & No \\ Residual & 0.0655 & 0.0094 & & & \\ Lack of fit & 0.0655 & 0.0218 & 2.86 & 0.6217 & No \\ Pure error & \textless{}0.0001 & \textless{}0.0001 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: ANOVA of hardened layer depth predicted by RSM model (induction frequency = 15 kHz). describes the relationship between the hardness of the laser-induced hybrid hardening hardened layer and the influencing factors. In addition, the coefficient of determination of the RSM model has a value of 0.850 for R\({}^{2}\) and 0.758 for R2 adjust. After calculating Equations (1)-(3), the RMSE is 10.098 HV\({}_{0.3}\), the MAE is 8.504 HV\({}_{0.3}\), and the MAPE is 1.15%. The final response regression prediction of the quadratic polynomial model of the actual parameters is established as follows:  
\(HV_{0.3}=-1465.76338+1.19704\times P+401.487\times v-26.671\times IP+0.03187 \times P\times v+\\ 0.004864\times P\times IP+4.221\times v\times IP-0.000211\times P^{2}-110.575 \times v^{2}-0.02455\times P\times v^{2}\)  
Table 7 indicates that at an induction frequency of 15 kHz, scanning speed (B), the cubic term AB\({}^{2}\), and the interaction term BC significantly affect the hardness of the hardened layer (\(p<0.05\)), while the remaining terms are non-significant. The ranking of process parameters influencing hardness is B > A > C. The \(p\)-value for the lack-of-fit term is 0.2819 (\(p>0.05\)), suggesting that the model effectively describes the relationship between the hardness of the laser-induced hybrid hardening hardened layer and the influencing factors. In addition, the coefficient of determination of the RSM model has a value of 0.9085 for R\({}^{2}\) and 0.7910 for R2 adjust. After calculating Equations (1)-(3), the RMSE is 12.691 HV\({}_{0.3}\)