## 4 Results and Discussion
### Assignment to the Classes
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(a_{s}\)} & \multicolumn{2}{c}{\(c_{s}^{j}=\sum\limits_{r=1}^{R}\mathbf{f_{r}(X_{r})\cdot\eta_{r}}\)} & \multicolumn{2}{c}{**Maximum**} \\  & \(j=1\) & \(j=2\) & \(j=3\) & \(\sigma_{s}^{j^{*}}=max_{1:j\leq j\leq 3}[\sigma_{s}^{j}]\) \\ \hline \(a_{1}\) & **0.7092** & 0.5945 & 0.6804 & 0.7092 & \(j^{*}=1\) \\ \(a_{2}\) & 0.6778 & **0.7249** & 0.6399 & 0.7249 & \(j^{*}=2\) \\ \(a_{3}\) & **0.7888** & 0.4601 & 0.4443 & 0.7888 & \(j^{*}=1\) \\ \(a_{4}\) & **0.789** & 0.4862 & 0.4078 & h0.789 & \(j^{*}=1\) \\ \(a_{5}\) & **0.7362** & 0.5405 & 0.5071 & 0.7362 & \(j^{*}=1\) \\ \(a_{6}\) & **0.6798** & 0.6757 & 0.65815 & 0.6798 & \(j^{*}=1\) \\ \(a_{7}\) & 0.598 & **0.671** & 0.57804 & 0.671 & \(j^{*}=2\) \\ \(a_{8}\) & **0.833** & 0.4037 & 0.57412 & 0.833 & \(j^{*}=1\) \\ \(a_{9}\) & **0.7478** & 0.42325 & 0.43905 & 0.7478 & \(j^{*}=1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The value of the grey fixed weight cluster coefficient for each class \(\left(\sigma_{s}^{j}\right)\) with subjective weighting. Step 3. Calculation of objective weights (\(w_{r0}^{e}\))  
The last step is the linear normalization of \(d_{r}\) to find the relative weight of each criterion is calculated using Equation (18). The results are presented in Table 15.  
\[w_{r0}^{e}=\frac{d_{r}}{\sum_{r=1}^{R}d_{r}} \tag{18}\]  
Finally, according to the AHP global weights \(\left(w_{r0}^{h}\right)\) which are equal to \(\eta_{r0}\) in Table 11 and the entropy weights (\(w_{r0}^{e}\)) in Equation (18), the weight of each criterion can be acquired on the principle of combined weighting methods as established by Equation (15). The weights of each criterion were derived by the combined weighting methods (\(w_{r0}^{e}\)) given in Table 16.  
\[w_{r0}^{e}=\frac{w_{r0}^{h}\times w_{r0}^{e}}{\sum_{r=1}^{R}w_{r0}^{h}\times w _{r0}^{e}} \tag{19}\]  
Figure 7 shows a clustered bar chart that was used to compare the weighting values between the pairwise comparison judgment, Shannon entropy, and combined weighting methods. As can be seen, the weighting methods gave different results for all three methods. What is interesting in the chart is the differences between weighted results between methods. It is clear from the graph that the sub-criterion of multiple life cycles is the most important factor in evaluating the quality of incoming cores based on subjective weighting. It is respectively followed by disassembablability as the second highest, and damage level as third highest. In contrast, according to objective weighting, it is clear from the graph that the sub-criterion of geometric and dimensional tolerance is more important than the multiple life cycle sub-criterion. Subsequently, sub-criterion of the geometric \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{8}{c}{**Sub-Criteria (\(\epsilon_{r0}\))**} \\ \cline{2-13}  & \(\epsilon_{11}\) & \(\epsilon_{12}\) & \(\epsilon_{13}\) & \(\epsilon_{14}\) & \(\epsilon_{21}\) & \(\epsilon_{22}\) & \(\epsilon_{23}\) & \(\epsilon_{24}\) & \(\epsilon_{31}\) & \(\epsilon_{32}\) & \(\epsilon_{33}\) \\ \hline AHP weights & 0.062 & 0.080 & **0.384** & **0.099** & **0.082** & 0.077 & 0.070 & 0.012 & 0.055 & 0.072 & 0.014 \\ Entropy weights & 0.073 & 0.077 & 0.077 & 0.098 & 0.098 & 0.093 & 0.081 & 0.151 & 0.134 & 0.064 & 0.054 \\ Combined weighting & 0.06 & 0.08 & **0.39** & **0.13** & 0.11 & 0.09 & 0.07 & 0.02 & 0.10 & 0.06 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 16: The combined weighting methods (\(w_{r0}^{e}\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Core (\(a_{a}\))**} & \multicolumn{8}{c}{**Sub-Criteria (\(\epsilon_{r0}\))**} \\ \cline{2-13}  & \(\epsilon_{11}\) & \(\epsilon_{12}\) & \(\epsilon_{13}\) & \(\epsilon_{14}\) & \(\epsilon_{21}\) & \(\epsilon_{22}\) & \(\epsilon_{23}\) & \  
and dimensional tolerance is slightly more important than sub-criterion of the frequency of use.