## Materials and Methods  
### ANFIS Modeling  
ANFIS is a neural-fuzzy predictive computing system based on an adaptive neural network [24] (. With the help of a hybrid learning sequence, ANFIS is used to generate input-output relations considering fuzzy if-then rules to provide different membership functions. The parameters of each function are determined by the ANFIS technique to follow already given empirical input-output data. ANFIS uses five network layers to achieve the fuzzy interpretation steps shown below (Figure 2), where layer one is the parameters entering the system, layer two is the set database layer of the fuzzy, layer three is the fuzzy rule base structure layer, layer four is the decision making layer, and layer five is the output defuzzification layer; more information is available in the literature [25; 26; 27]. This system can be explained in terms of two suggested laws and literal values in each input, considering the following five layers:  
**Layer1:** where the output is the step to make a specified input satisfy the verbal label corresponding to the current node. In this layer, Gaussian membership functions are used to represent verbal values as a connection of aggregate production limits (see Figure 3). First parameter function  
\[A_{i}(u)=exp\left[-\frac{1}{2}\left(\frac{u-a_{i1}}{b_{i1}}\right)^{2}\right] \tag{1}\]  
Second parameter function:  
\[B_{i}(v)=exp\left[-\frac{1}{2}\left(\frac{v-a_{i2}}{b_{i2}}\right)^{2}\right] \tag{2}\]  
where \(\{a_{i},\,\ldots,\,b_{i2}\}4\) denote set of parameters. Figure 2: Adaptive Neuro-Fuzzy Interference System(ANFIS) network for a two-input ‘Sugeno’ fuzzy model. As the limit values of modification, the shapes of the model vary considerably as seen below, displaying several forms of the functions on linguistic tags \(A_{i}\), \(B_{i}\). The parameters in this layer are described as attitude limits. In **Layer 2**, each node calculates the firing strength of the related law. Here, the nodes are called the 'rule nodes'. The outputs of the top and the bottommost neurons are as follow:  
\[\text{Upper neuron}\qquad\alpha_{1}=\text{A}_{1}(\text{x})\times\text{B}_{1}( \text{y}) \tag{3}\]  
\[\text{Bottom neuron}\qquad\alpha_{2}=\text{A}_{2}(\text{x})\times\text{B}_{2}( \text{y}) \tag{4}\]  
In **Layer 3**, every node in this layer is considered by N, which indicates the regulation of the firing powers. The output of the upper and bottom neuron is made normal as follow:  
\[\text{Top neuron}\qquad\beta_{1}=\frac{\alpha_{1}}{\alpha_{1}+\alpha_{2}} \tag{5}\]  
\[\text{Bottom neuron}\qquad\beta_{2}=\frac{\alpha_{2}}{\alpha_{1}+\alpha_{2}} \tag{6}\]  
**Layer 4** provides the upper and lower neuron outputs as the result of normal firing intensity and inbuilt particle energy for the process. \[\text{Upper neuron}\qquad\beta_{1}z_{1}=\beta_{1}\left(a_{1}x+b_{1}y\right) \tag{7}\]  
\[\text{Bottom neuron}\qquad\beta_{2}z_{2}=\beta_{2}\left(a_{2}x+b_{2}y\right) \tag{8}\]  
Figure 3: Initial and final membership functions of stone strength (**s**); (**a**) initial, (**b**) final. **Layer 5** where the system overall output is determined by each node summing all the incoming signals. i.e.,  
\[B_{i}(v)=exp\left[-\frac{1}{2}\left(\frac{v-a_{i2}}{b_{i2}}\right)^{2}\right] \tag{2}\]0  
Here, the combination set of neutral parameters are learned after providing the system training set \(\{(x_{k},\ y_{k}),\ k=1,\ldots,K\}\). The corresponding sensed function for iterated k is determined by  
\[B_{i}(v)=exp\left[-\frac{1}{2}\left(\frac{v-a_{i2}}{b_{i2}}\right)^{2}\right] \tag{2}\]1  
where \(y_{k}\) is the expected output and \(o_{k}\) is the calculated output by the improved neural net. The ANFIS model was built in MATLAB using a set of 32 readings (provided in Table 1). Various related functions were used to learn ANFIS; among them, two were closed side sets ((CSS), Gape G), and the reduction ratio (RR) and four functions of the rock strength (S) were selected to generate the ANFIS model.