## 4 Results and Discussion
### Assignment to the Classes  
Based on the whitenization weight function \(f_{r}^{j}(\cdot)\) formulated in Equations (5)-(13) and the clustering global weights \(\eta\), the grey fixed weight cluster coefficients for each class \((\sigma_{s}^{j})\) can be found. The calculation of the grey fixed weight clustering coefficients as decision indicators is summarized in Table 12. What stands out in Table 12 is that cores \(a_{1},a_{3},\ a_{4},\ a_{5},\ a_{6},\ a_{8},\ a_{9}\) were mostly classified into class 1. Meanwhile, cores \(a_{2},\ a_{7}\) were essentially classified into class 2. Noticeably, no core was classified to class 3. These results indicate that the majority of the used hydraulic cylinders were of considerable quality. The clustering weights also could be achieved through the objective weighting method, i.e., the Shannon entropy method. To establish the clustering weights for each criterion \(r\left(\eta_{r}\right)\), the Shannon entropy weights (\(w_{ro}^{r}\)) were used. The Shannon entropy weights are summarized by the three steps given as follows [26; 27]:  
Step 1. Normalization  
At the beginning, due to the data having different scales and units, the data should first be normalized into dimensionless data as follows:  
\[p_{sr}=\frac{x_{sr}}{\sum_{s=1}^{S}x_{sr}} \tag{14}\]  
The results are presented in Table 13.  
Step 2. Calculation of the entropy (\(E_{r}\)) and the degree of diversity (\(d_{r}\))  
Afterwards, entropy within the data of the normalized decision matrix for the \(r^{th}\) criterion can be obtained using Equation (15). Meanwhile, the results are given in Table 14.  
\[E_{r}=-c\sum_{s=1}^{S}p_{sr}\ln(p_{sr}) \tag{15}\]  
where the constant \(c\) is defined as:  
\[c=\frac{1}{\ln(S)} \tag{16}\]  
Later on, in order to calculate the amount of uncertainty, it can be stated by the degree of divergence (\(d_{r}\)), which is calculated as follows:  
\[d_{r}=1-E_{r} \tag{17}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Core (\(a_{s}\))**} & \multicolumn{8}{c}{**Sub-Criticia (\(c_{ro}\))**} \\ \cline{2-11}  & **c11** & **c12** & **c13** & **c14** & **c21** & **c22** & **c23** & **c24** & **c31** & **c32** & **c33** \\ \hline \(a_{1}\) & 0.100 & 0.100 & 0.133 & 0.067 & 0.067 & 0.027 & 0.100 & 0.067 & 0.100 & 0.133 & 0.133 \\ \(a_{2}\) & 0.133 & 0.100 & 0.079 & 0.111 & 0.100 & 0.097 & 0.083 & 0.067 & 0.067 & 0.100 & 0.100 \\ \(a_{3}\) & 0.067 & 0.133 & 0.132 & 0.056 & 0.100 & 0.129 & 0.111 & 0.033 & 0.067 & 0.133 & 0.133 \\ \(a_{4}\) & 0.100 & 0.100 & 0.132 & 0.167 & 0.150 & 0.097 & 0.139 & 0.100 & 0.100 & 0.167 \\ \(a_{5}\) & 0.100 & 0.133 & 0.105 & 0.111 & 0.050 & 0.129 & 0.111 & 0.067 & 0.033 & 0.133 & 0.133 \\ \(a_{6}\) & 0.167 & 0.133 & 0.079 & 0.056 & 0.100 & 0.097 & 0.083 & 0.033 & 0.067 & 0.100 & 0.100 \\ \(a_{7}\) & 0.133 & 0.067 & 0.105 & 0.111 & 0.100 & 0.129 & 0.111 & 0.067 & 0.067 & 0.133 & 0.133 \\ \(a_{8}\) & 0.167 & 0.100 & 0.132 & 0.167 & 0.100 & 0.097 & 0.139 & 0.100 & 0.067 & 0.100 & 0.167 \\ \(a_{9}\) & 0.067 & 0.133 & 0.132 & 0.111 & 0.150 & 0.097 & 0.111 & 0.067 & 0.100 & 0.100 & 0.133 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The normalized values of each criterion (\(p_{sr}\)) for each incoming core.