## 2 Materials and Methods
### Data Modeling
#### 2.2.1 Modeling with Artificial Neural Network
This technique employs an objective function to figure out how closely the outputs of the network match the target outputs in the training set of data. The result of objective function is improved by adjusting the weights of each individual processing element (neuron). The performance of ANN models developed to predict the thrust force and drilling torque was measured using the most applied indicators in literature [20; 21; 22], namely, coefficient of correlation (R), coefficient of determination (R\({}^{2}\)), root mean square error (RMSE), and the mean absolute percentage error (MAPE). The corresponding equations are:  
\[\text{R}=\frac{\sum_{i=1}^{n}(p_{i}-\overline{p})(a_{i}-\overline{a})}{\sqrt{ \sum_{i=1}^{n}(p_{i}-\overline{p})^{2}}\ \sqrt{\sum_{i=1}^{n}(a_{i}-\overline{a})^{2}}} \tag{6}\]  
\[\text{R}^{2}=1-\frac{\sum_{i=1}^{n}(a_{i}-p_{i})^{2}}{\sum_{i=1}^{n}(a_{i}- \overline{p}_{i})^{2}} \tag{7}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}{\sum_{i=1}^{n}(a_{i}-p_{i})^{2}}} \tag{8}\]  
\[\text{MAPE}=\frac{1}{n}(\sum_{i=1}^{n}\big{|}\frac{\big{|}a_{i}-p_{i}\big{|}}{ a_{i}}\big{|}\big{|})\times 100 \tag{9}\]  
where: \(a_{i}\) is the experimental value, \(p_{i}\) is the predicted value of ANN model, \(\overline{a}\) is the mean of experimental values; \(\overline{p}\) is the mean of predicted values, n is the total number of values. Coefficient of correlation and coefficient of determination measure the strength of the relationship between experimental and predicted values. When the value of R or R\({}^{2}\) is close to one, there could be concluded that there is a strong relationship between experimental and predicted values. On the other hand, a value of R or R\({}^{2}\) close to zero indicates that between the experimental and predicted values is a lower relationship. In addition, the value of R\({}^{2}\) explains what percentage of variation in the dependent variable could be explained by the analyzed independent variables. The root mean square error measures the average difference between the experimental and predicted values by the developed models. A value of RMSE close to zero tells us that the predicted values are similar with those that were obtained during the experiments. The mean absolute percentage error, which represents most performance indicator of ANN models, describes how well the developed models predict the desired outputs. According to [PERSON] (1982) cited by [PERSON] and [PERSON] [21], a value of MAPE lower than 10% indicates that the developed models have a high precision. On the other hand, when the MAPE is higher than 50%, the models are inaccurate. When the MAPE is between 10 and 20%, the models have a good accuracy. A reasonable model is obtained when the MAPE is greater than 20% and lower than 50%. 2.2.2. Modeling with Response Surface Methodology  
The first step to applying the Response Surface Methodology is to generate a central composite design. In this design, various combinations among the values of factors are generated. The corresponding response of these combinations could be revealed by means of experiments or simulations. Once the experimental design is completed, the next step consists of figuring out the regression equation that best describes the relationship between inputs and output. Usually, the ANOVA is applied to reveal which factors and coefficients are significant from a statistical point of view. The equation in coded form could be used to analyze the influence of factors (inputs) on the analyzed response (output). The equation in real form could be used to make predictions for various scenarios. Last but not least is the step that consists of using the regression equation to reveal the optimum combination of factors that corresponds to the desired output. These steps were performed by means of the Design Expert software (Version 9, Stat-Ease Inc., Minneapolis, MN, US).