## 2 Materials and Methods
### Grey Fuzzy Approach  
In multi-criteria decision making, there are many methodologies that have been applied in various research [35; 36]. However, the application of certain methods requires \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{Exp. **No.**} & \multirow{2}{*}{**Machine Tool**} & \multirow{2}{*}{**Cutting Speed** [mm/min]**} & \multirow{2}{*}{**Workpiece** [**s]**} & \multirow{2}{*}{**Machine Time** [**s]**} & \multirow{2}{*}{**Dimensional** [**Accuracy** [mm]**} & \multirow{2}{*}{**Kerf Width** [**mm]**} & \multirow{2}{*}{**Surface** [**koughness** [**\(\mu\)m**]**} \\ \hline
1. & PAM & 1510 & 10 & 64.3 & 0.486 & 2.890 & 3.522 \\ \hline
2. & PAM & 2000 & 8 & 56.3 & 0.607 & 3.420 & 2.512 \\ \hline
3. & PAM & 3620 & 5 & 42.2 & 0.740 & 3.910 & 4.223 \\ \hline
4. & LBM & 900 & 10 & 58.8 & 0.064 & 0.550 & 4.185 \\ \hline
5. & LBM & 1100 & 8 & 53.2 & 0.107 & 0.770 & 5.119 \\ \hline
6. & LBM & 2270 & 5 & 39.8 & 0.179 & 1.01 & 8.075 \\ \hline
7. & AWJM & 75 & 10 & 830.8 & 0.050 & 1.14 & 6.265 \\ \hline
8. & AWJM & 120 & 8 & 646.1 & 0.079 & 1.480 & 6.511 \\ \hline
9. & AWJM & 270 & 5 & 395.2 & 0.100 & 1.890 & 6.980 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Systematized results of measurements. Figure 4: Methodology for measurement of various machining performances. Figure 3: Example of three positions for measurement surface roughness and kerf width for thickness of 10 mm (PAM). the analysis of data involved in the creation of the MCDM. Since the amount of data in this study is relatively small, the use of the Grey Fuzzy Approach is justified. The combination of grey relational analysis (GRA) and fuzzy logic (FL) is applied to select the best machine tool that has the best quality characteristics considering multiple answers. GRA has gained great popularity in the field of optimization of multi-objective problems. This method essentially analyzes the degree of relationship of multiple attributes using certain equations [37], which are presented below. To start the process, it is necessary to normalize the raw data into values from 0 to 1. Depending on the objective functions, whether maximum, nominal, or minimum, different normalization formulas apply. In this work, the objective function for all responses is minimum, so normalization is performed according to Equation (1). \[x_{i}^{*}(k)=\frac{\max_{i}^{0}(k)-x_{i}^{0}(k)}{\max_{i}^{0}(k)-\min_{i}^{0}( k)} \tag{1}\]  
In the above equations \(x_{i}^{0}(k)\) is original sequence, \(x_{i}^{*}(k)\) sequence after preprocessing, \(\max_{i}^{0}(k)\) is the largest value \(x_{i}^{0}(k)\) and smallest value of \(x_{i}^{0}(k)\). Based on the maximum normalized value, the deviation sequence is determined independently of the output variables [38]. This value is denoted by D and is given as a reference value in Equation (2). After preprocessing the data and determining the absolute difference between each normalized value and the reference value D, Equation (3), the grey relativity coefficient is calculated. The formula for the grey relational coefficient (GRC) is given in Equation (4). \[D=\max\Bigl{(}X_{ijk}\Bigr{)} \tag{2}\]  
\[\Delta_{ijk}=X_{ijk}-D \tag{3}\]  
\[\gamma_{i}(k)=\frac{\Delta_{\min}+\gamma\cdot\Delta_{\max}}{\Delta_{oi}(k)+ \gamma\cdot\Delta_{\max}},\ \gamma\in[0,1] \tag{4}\]  
In Equation (4), \(\Delta_{oi}(k)\) is \(\Delta_{oi}(k)\) of reference sequence, \(\gamma\) distinguishing coefficient (\(\gamma=0.5\)), when the \(k\)th feature is numeric. The \(\Delta_{oi}(k)\) of reference sequence is calculated by Equation (5). \[\Delta_{oi}(k)=\left\|x_{0}^{*}(k)-x_{0}^{*0}(k)\right\| \tag{5}\]  
A grey relational grade is a weighted sum of the GRC and is defined in Equation (6). \[\zeta_{i}=\frac{1}{n}\sum_{k=1}^{n}i\gamma_{i}(k) \tag{6}\]  
The second part of the GFA represents the logical system consisting of fuzzification, fuzzy inference system (rule base and membership functions) and finally a defuzzification. Fuzzification means that the crisp input values of are converted into appropriate linguistic values.