## 2 Materials and Methods
### Neural Network Models and SHAP Analysis Methodology  
The aim was to analyze the impact of cutting speed \(v_{c}\), feed rate \(f\), and depth of cut \(a_{p}\) on surface roughness \(R_{a}\), main cutting force \(F_{c}\), passive force \(F_{p}\), and feed force \(F_{f}\) for both types of inserts. In addition, the impact of the insert radius \(r_{i}\) on \(R_{a}\), \(F_{c}\), \(F_{p}\), and \(F_{f}\) was analyzed in this research. Therefore, three cases were created in this research. In the first case, data from an insert with a radius of 0.4 mm were used, while in the second case, data from an insert with a radius of 0.8 mm were used. The first and second cases used cutting speed \(v_{c}\), feed rate \(f\), and depth of cut \(a_{p}\) as the input variables, while average surface roughness \(R_{a}\), main cutting force \(F_{c}\), passive force \(F_{p}\), and feed force \(F_{f}\) were the output variables. In the third case, both types of inserts data were combined, and therefore, a new input variable insert radius \(r_{i}\) was added that represented an insert radius of 0.4 mm and 0.8 mm. Feed-forward artificial neural network models were created for all three cases. In the first and second cases, the number of input variables was three (\(v_{c}\), \(f\), and \(a_{p}\)), while in the third case, the number of inputs was four (\(r_{i}\), \(v_{c}\), \(f\), and \(a_{p}\)). All the developed neural network models had two hidden layers with 64 neurons each and an output layer with four output variables (\(R_{a}\), \(F_{c}\), \(F_{p}\), and \(F_{f}\)). The two hidden layers with 64 hidden neurons in each layer were created in order for the neural network to better understand the complex relations between input and output variables. In each case, the data were normalized to an interval between 0 and 1, as well as shuffled, and divided into training and test sets at a ratio of 80:20. Since the data were normalized to between 0 and 1, sigmoid activation function was used in both the hidden layers and in the output layer:  
\[s(x)=\frac{1}{1+e^{-x}} \tag{1}\]  
The total number of epochs was 300 with the prevention of overfitting, so that if the validation error did not decrease with the training error after 10 epochs, the learning process was stopped. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Input Parameter** & **Unit** & **Level 1** & **Level 2** & **Level 3** & **Level 4** & **Level 5** & **Level 6** \\ \hline \(v_{c}\) & mm/min & 100 & 130 & 160 & 200 & 230 & 260 \\ \(f\) & mm & 0.08 & 0.11 & 0.15 & 0.20 & - & - \\ \(a_{p}\) & mm & 0.5 & 0.8 & 1.1 & 1.4 & 1.7 & - \\ \(r_{i}\) & mm & 0.4 & 0.8 & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input parameters and their levels. The evaluation of performance of the developed models was conducted by calculating the metrics of coefficient of determination (\(R^{2}\)), mean squared error (\(MSE\)), and mean average error (\(MAE\)) for each output variable, respectively:  
\[R^{2}=1-\frac{\sum(y_{i}-\hat{y}_{i})^{2}}{\sum(y_{i}-\overline{y})^{2}} \tag{2}\]  
\[MSE=\frac{1}{n}{\sum(y_{i}-\hat{y}_{i})^{2}} \tag{3}\]  
\[MAE=\frac{1}{n}{\sum}|y_{i}-\hat{y}_{i}| \tag{4}\]  
where \(y_{i}\)--actual output value; \(\hat{y}_{i}\)--predicted output value; \(\overline{y}\)--mean of actual output values; and \(n\)--number of samples. SHAP values were calculated to identify how much each feature in the developed feed-forward artificial neural network prediction models influenced the prediction of \(R_{a}\), \(F_{e}\), \(F_{p}\), and \(F_{f}\). Neural network models can be considered as a 'black box', where users cannot see the relations between input and output variables, and this is why the SHAP analysis was used in this research. Neural network models are great in finding these complex relations, and SHAP analysis can interpret these findings. SHAP values measure how much each feature contributes to the model's prediction, as well as showing which features are most important for the model's prediction.