# Complexity Modeling of Steel-Laser-Hardened Surface Microstructures
## 3 Complexity Prediction Model
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Groups of** & **Specimen** & \multicolumn{2}{c}{**Temperature**} & **Velocity** & **Impact Angle** & **Complexity** \\ \cline{3-8}
**Specimens** & **(X1) [K]** & **(X2) [mm/s]** & **(up) [\({}^{\circ}\)]** & **(X3) [\({}^{\circ}\)]** & **(CC) [-]** \\ \hline \multirow{4}{*}{I} & P1\({}^{\prime}\) & 1373.0 & 2.0 & 30 & 30 & 1.8681 \\  & P2\({}^{\prime}\) & 1473.0 & 3.0 & 60 & 60 & 1.8524 \\  & P3\({}^{\prime}\) & 1573.0 & 4.0 & 75 & 75 & 1.8275 \\  & P4\({}^{\prime}\) & 1673.0 & 5.0 & 90 & 90 & 1.8735 \\ \hline \multirow{4}{*}{II} & P5\({}^{\prime}\) & 1373.0 & \(-2.0\) & 150 & \(-30\) & 1.8916 \\  & P6\({}^{\prime}\) & 1473.0 & \(-3.0\) & 120 & \(-60\) & 1.8365 \\  & P7\({}^{\prime}\) & 1573.0 & \(-4.0\) & 105 & \(-75\) & 1.8921 \\  & P8\({}^{\prime}\) & 1673.0 & \(-5.0\) & 90 & \(-90\) & 1.8863 \\ \hline \multirow{4}{*}{III} & P9\({}^{\prime}\) & 1373.0 & 2.0 & 30 & 30 & 1.8809 \\  & P10\({}^{\prime}\) & 1473.0 & 3.0 & 60 & 60 & 1.8923 \\  & P11\({}^{\prime}\) & 1573.0 & 4.0 & 75 & 75 & 1.7253 \\  & P12\({}^{\prime}\) & 1673.0 & 5.0 & 90 & 90 & 1.8771 \\ \hline \multirow{4}{*}{IV} & P13\({}^{\prime}\) & 1373.0 & \(-2.0\) & 150 & \(-30\) & 1.8924 \\  & P14\({}^{\prime}\) & 1473.0 & \(-3.0\) & 120 & \(-60\) & 1.7952 \\  & P15\({}^{\prime}\) & 1573.0 & \(-4.0\) & 105 & \(-75\) & 1.893 \\  & P16\({}^{\prime}\) & 1673.0 & \(-5.0\) & 90 & \(-90\) & 1.8846 \\ \hline \multirow{4}{*}{V} & P12\({}^{\prime}\) & 1373.0 & 2.0 & 30 & 30 & 1.8793 \\  & P18\({}^{\prime}\) & 1473.0 & 3.0 & 60 & 60 & 1.8552 \\  & P19\({}^{\prime}\) & 1573.0 & 4.0 & 75 & 75 & 1.8614 \\  & P20\({}^{\prime}\) & 1673.0 & 5.0 & 90 & 90 & 1.7721 \\ \hline \multirow{4}{*}{VI} & P21\({}^{\prime}\) & 1373.0 & \(-2.0\) & 150 & \(-30\) & 1.8832 \\  & P22\({}^{\prime}\) & 1473.0 & \(-3.0\) & 120 & \(-60\) & 1.8382 \\ \cline{1-1}  & P23\({}^{\prime}\) & 1573.0 & \(-4.0\) & 105 & \(-75\) & 1.8718 \\ \cline{1-1}  & P24\({}^{\prime}\) & 1673.0 & \(-5.0\) & 90 & \(-90\) & 1.8857 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Process parameters and complexity index for the first set of RLH specimens. From the 24 \(\times\) 6 _CC_ specimens' evaluations, the following genetic programming model (Equation (5)), able to predict the complexity of robot laser-hardened steel specimens, was obtained. \[Y=A\bigg{(}B+\frac{1}{X_{1}}(X_{2}+X_{3}+C)\bigg{)}(-9.7938+D+E) \tag{5}\]  
where:  
\[A=\frac{1}{X_{2}}\bigg{(}2X_{2}+\frac{1}{X_{3}}X_{3}\bigg{)}\]  
\[B=-9.7983+\frac{X_{2}+\frac{X_{2}+\frac{X_{1}+X_{3}}{X_{2}}+\frac{X_{1}+X_{3}}{ X_{2}}}{X_{3}}}{X_{3}}\]  
\[C=\frac{X_{3}\Bigg{(}-9.7938+X_{2}+\frac{X_{3}}{2X_{2}-0.10210X_{3}+\frac{X_{3} }{X_{1}}}\Bigg{)}}{X_{1}}\]  
\[D=\frac{X_{3}+\frac{\big{(}-9.7938+\frac{1}{X_{1}}+X_{2}\big{)}X_{3}}{X_{1}X_{3 }}}{X_{1}X_{3}}\]  
\[E=\frac{X_{2}-0.10210X_{3}-\frac{0.32354X_{3}\big{(}X_{3}+\frac{X_{3}}{X_{2}} \big{)}}{X_{1}\big{(}2X_{2}-0.10210X_{3}+\frac{X_{3}}{X_{1}}\big{)}}}{X_{2}}\]  
To verify the effectiveness of the GP prediction model, 24 \(\times\) 6 evaluations of complexity were made using the same process parameters (\(X_{1}\), \(X_{2}\), and \(X_{3}\)) used in the experimental tests. The results were summarized in Table 3 which shows the values of minimum, maximum, and medium values obtained with the complexity GP prediction model. Method result accuracy was studied in detail in the next section, where the calculation of the statistical error (mean values and standard deviation) of the method was carried out through an ANOVA. It is worth noting that although the process of the laser heat treatment was simple, the investigations revealed that the evolution of the microstructure during the laser-hardening process was rather complex. The main objective of this research work is to show to what extent different angles of laser beam have an impact on the complexity factor of the microstructure of laser-hardened specimens. In this regard, the new method proposed made it possible to determine the same complex structure. What emerges is that the