## 5 Case Study and Analysis
### The Detailed DM Steps
#### 5.2.3 Ranking by the Improved PROMETHEE II Method  
According to the above analysis, the aggregated comprehensive evaluation matrix by the three DMs is listed in Table 5 (\(q=3\)). Then, the detailed processes of the improved PROMETHEE II method are as follows:  
**Step 1:** Normalize the original assessment decision matrix \(X=[\chi_{\varsigma\tau}]_{m\times n}\) to \(\widetilde{X}=[\widetilde{\chi}_{\varsigma\tau}]_{m\times n}\). Due to the fact that \(C_{1}\) belongs to the cost-type input, other criteria are all benefit-type inputs, then the transformation form is depicted in Table 6.  
**Step 2:** Calculate the deviations \(d_{\tau}(\tau=1,2,\ldots,n)\) based on Equation (24) (suppose \(p=2\), \(q=3\) and \(C=100\)), which are listed as:  
\[d_{1}(a_{1},a_{2})=0.1617,d_{2}(a_{1},a_{2})=-0.0341,d_{3}(a_{1},a_{2})=0.3804,d_{4}(a_{1},a_{2})=-0.1983,d_{5}(a_{1},a_{2})=-0.0183,d_{6}(a_{1 },a_{2})=-0.3280,\] \[d_{1}(a_{1},a_{3})=-0.0373,d_{2}(a_{1},a_{3})=-0.0385,d_{3}(a_{1 },a_{3})=0.3949,d_{4}(a_{1},a_{3})=-0.2608,d_{3}(a_{1},a_{3})=-0.0564,d_{6}(a_{1 },a_{3})=-0.1410,\] \[d_{1}(a_{1},a_{4})=0.2160,d_{2}(a_{1},a_{4})=0.0113,d_{3}(a_{1},a_{4})=1.0291,d_{4}(a_{1},a_{4})=-0.0735,d_{5}(a_{1},a_{4})=0.0033,d_{6}(a_{1 },a_{4})=-0.1889,\] \[d_{1}(a_{2},a_{1})=-0.1617,d_{2}(a_{2},a_{1})=0.0341,d_{3}(a_{2},a_{1})=-0.3804,d_{4}(a_{2},a_{1})=0.1983,d_{5}(a_{2},a_{1})=0.0183,d_{6}(a_{2 },a_{1})=0.3280,\] \[d_{1}(a_{2},a_{3})=-0.1991,d_{2}(a_{2},a_{3})=-0.0044,d_{3}(a_{2},a_{3})=0.0145,d_{4}(a_{2},a_{3})=-0.0625,d_{3}(a_{2},a_{3})=-0.0380,d_{6}(a_{ 2},a_{3})=0.1870,\] \[d_{1}(a_{2},a_{4})=0.0543,d_{2}(a_{2},a_{4})=0.1444,d_{3}(a_{2},a_{4})=0.6487,d_{4}(a_{2},a_{4})=0.1248,d_{5}(a_{2},a_{4})=0.0217,d_{6}(a_{ 2},a_{4})=0.1391,\] \[d_{1}(a_{3},a_{1})=0.0373,d_{2}(a_{3},a_{1})=0.0385,d_{3}(a_{3},a_{1})=-0.3949,d_{4}(a_{3},a_{1})=0.2608,d_{5}(a_{3},a_{1})=0.0564,d_{6}(a_{3 },a_{1})=0.1410,\] \[d_{1}(a_{3},a_{2})=0.1991,d_{2}(a_{3},a_{2})=0.0044,d_{3}(a_{3},a_{2})=-0.0145,d_{4}(a_{3},a_{2})=0.0625,d_{5}(a_{3},a_{2})=0.0380,d_{6}(a_{3 },a_{2})=-0.1870,\] \[d_{1}(a_{3},a_{4})=0.2533,d_{2}(a_{3},a_{4})=0.1488,d_{3}(a_{3},a_{4})=0.6342,d_{4}(a_{3},a_{4})=0.1874,d_{5}(a_{3},a_{4})=0.0597,d_{6}(a_{3 },a_{4})=-0.0479,\] \[d_{1}(a_{1},a_{2})=0.1617,d_{2}(a_{1},a_{2})=-0.0341,d_{3}(a_{1},a_{2})=0.3804,d_{4}(a_{1},a_{2})=-0.1983,d_{5}(a_{1},a_{2})=-0.0183,d_{6}(a_{1 },a_{2})=-0.3280,\]0 \[d_{1}(a_{1},a_{2})=0.1617,d_{2}(a_{1},a_{2})=-0.0341,d_{3}(a_{1},a_{2})=0.3804,d_{4}(a_{1},a_{2})=-0.1983,d_{5}(a_{1},a_{2})=-0.0183,d_{6}(a_{1 },a_{2})=-0.3280,\]1  
**Step 3:** Define the preference function \(p_{\tau}(a_{a},a_{b})\). As mentioned above, \(p_{\tau}(a_{a},a_{b})=d_{\tau}(a_{a},a_{b})\) is actively defined, so the detailed process is omitted here. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(C_{4}\) & \(C_{5}\) & \(C_{6}\) \\ \hline \(\alpha_{1}\) & \((0.702,0.470)\) & \((0.806,0.552)\) & \((0.847,0.520)\) & \((0.754,0.299)\) & \((0.795,0.364)\) & \((0.681,0.636)\) \\ \(\alpha_{2}\) & \((0.809,0.296)\) & \((0.840,0.362)\) & \((0.620,0.473)\) & \((0.825,0.651)\) & \((0.805,0.356)\) & \((0.882,0.313)\) \\ \(\alpha_{3}\) & \((0.682,0.414)\) & \((0.834,0.477)\) & \((0.625,0.314)\) & \((0.883,0.302)\) & \((0.821,0.414)\) & \((0.795,0.259)\) \\ \(\alpha_{4}\) & \((0.834,0.340)\) & \((0.769,0.298)\) & \((0.623,0.689)\) & \((0.775,0.541)\) & \((0.789,0.428)\) & \((0.796,0.568)\) \\ \hline \end{tabular}
\end{table} 
Table 5: The aggregated comprehensive evaluation matrix. **Step 4:** Determine the global preference index \(\Pi(\alpha_{a},\alpha_{b})\) by Equation (26), and the associated weights \(\omega_{\tau}=(0.2601,0.2059,0.1115,0.0325,0.1950,0.1950)\) are determined by the GBWM, shown in Table 7.  
**Step 5:** Based on Equations (27) and (28), aggregate \(\Pi(\alpha_{a},\alpha_{b})\) into the positive \(\varphi^{+}(\alpha_{a})\) and negative \(\varphi^{-}(\alpha_{a})\) outranking flows, the results are as follows:  
\[