## 2 Materials and Methods
### Data Modeling
#### 2.2.1 Modeling with Artificial Neural Network
The experimental design generated by the software is a two-level face centered design (FCCD), which is pictured in Figure 4.0 The FCCD generates four combinations for the flat drill (#1, #16, #23, and #24) and four combinations for helical drill (#6, #8, #13, and #25) among the low (\(-1\)) and high (\(+1\)) levels of analyzed factors, namely, drill tip angle (X\({}_{1}\)) and tooth bite (X\({}_{2}\)) (Tables 2 and 3). The two-level factorial points are needed to estimate the linear effect of factors on the analyzed responses. On the other hand, to reveal whether there is a non-linear effect of factors, the FCD design requires taking into account the axial points (\(\alpha\)). In Table 4, the axial points are combinations #3, #10, #17, and #18 for the flat drill and combination #4, #14, #19, and #22 \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Independent Variable**} & \multicolumn{2}{c}{**Number of Neurons**} & \multirow{2}{*}{**Testing Performance Indicator, R**} \\ \cline{2-2} \cline{4-5}  & **ANN Model for** & **ANN Model for** & **ANN Model for** & **ANN Model for** \\  & **Thrust Force** & **Drilling Torque** & **Thrust Force** & **Drilling Torque** \\ \hline Drill tip angle (\(X_{1}\)) & 1 & 3 & 0.074 & 0.389 \\ \hline \hline Tooth bite (\(X_{2}\)) & 0 & 3 & 0.377 & 0.768 \\ \hline Drill type (\(X_{3}\)) & 0 & 0 & 0.866 & 0.436 \\ \hline Drill tip angle (\(X_{1}\)) \(\times\) Tooth bite (\(X_{2}\)) & 4 & 2 & 0.372 & 0.839 \\ \hline Drill tip angle (\(X_{1}\)) \(\times\) Drill type (\(X_{3}\)) & 2 & 6 & 0.872 & 0.627 \\ \hline Tooth bite (\(X_{2}\)) \(\times\) Drill type (\(X_{3}\)) & 2 & 7 & 0.977 & 0.890 \\ \hline Drill tip angle (\(X_{1}\)) \(\times\) Tooth bite (\(X_{2}\)) & 14 & 4 & 0.986 & 0.988 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results obtained during the influence of independent variable on the performance of ANN models. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Combination \#**} & \multicolumn{2}{c}{**Factors (Inputs)**} & \multicolumn{2}{c}{**Responses (Outputs)**} \\ \cline{2-5}  & **Drill Tip Angle (\(X_{1}\)),**\({}^{\circ}\) & **Tooth Bite (\(X_{2}\)), mm** & **Drill Type (\(X_{3}\))** & **Thrust Force (\(Y_{1}\)), N** & **Drilling Torque (\(Y_{2}\)), Nm** \\ \hline
1 & 30 (\(-\)1) & 0.1 (\(-\)1) & Flat (\(-\)1) & 173.87 & 1.14 \\
2 & 75 (0) & 0.4 (0) & Flat (\(-\)1) & 313.98  
Once the values of analyzed responses were figured out, a quadratic model could be obtained. The general form is presented in Equation (10) [23]:  
\[Y=\beta_{0}+\sum_{i=1}^{k}\beta_{i}X_{i}+\sum_{i=1}^{k}\beta_{ii}X_{i}^{2}+\sum _{i=1}^{k}\sum_{j=1}^{k}\beta_{ij}X_{i}X_{j}+\varepsilon \tag{10}\]  
where \(Y\) is the dependent variable or the response (thrust force or drilling torque), \(X_{i}\) and \(X_{j}\) are the analyzed factors, namely, drill tip angle (\(X_{1}\)) and tooth bite (\(X_{2}\)), \(\beta_{0}\) is the constant, \(\beta_{i}\), \(\beta_{ii}\), and \(\beta_{ij}\) are the coefficients of the equation, \(\varepsilon\) is the error term, and \(k\) represents the number of analyzed factors. The coefficient of mathematical models were developed through regression analysis. In order to figure out if the model parameters of each factor are significant, the analysis of variance (ANOVA) was involved. ANOVA was performed at a 1% and 5% level of significance. The coefficient of determination (R\({}^{2}\)) was used to establish the model accuracy. For the optimization study, a various combinations of analyzed factors are analyzed to fulfill the optimization criteria, namely, to minimize or maximize the investigated response or responses. In this study, the optimization criteria was to minimize both the thrust force and drilling torque. To determine the best combination among factors, the Design Expert software uses the desirability function approach, which is one of the most widely used methods in the industry for the optimization of multiple response processes. The method finds operating conditions that provide the most desirable response values.