## 5 The Case Study Results and Analysis
### Constructing a Multi-Attribute Evaluation System for Alternative Schemes
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) & \(\mathbf{P_{4}}\) & \(\mathbf{P_{5}}\) & \(\mathbf{P_{6}}\) & \(\mathbf{P_{7}}\) & \(\mathbf{P_{8}}\) \\ \hline F\({}_{1}\) & 0.224288 & 0.497183 & 0.184793 & 0.321727 & 0.341332 & 0.590379 & 0.640773 & 0.473467 \\ F\({}_{2}\) & 0.356094 & 0.484521 & 0.198694 & 0.491200 & 0.463439 & 0.410996 & 0.612154 & 0.283782 \\ F\({}_{3}\) & 0.374902 & 0.409930 & 0.415529 & 0.347796 & 0.272320 & 0.413849 & 0.533784 & 0.578542 \\ F\({}_{4}\) & 0.271011 & 0.380614 & 0.387004 & 0.378025 & 0.433941 & 0.413410 & 0.365342 & 0.515332 \\ F\({}_{5}\) & 0.202519 & 0.342664 & 0.533862 & 0.316413 & 0.514885 & 0.489927 & 0.315685 & 0.377816 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Score of attributes for each alternative scheme based on expert knowledge. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{F_{1}}\) & \(\mathbf{F_{2}}\) & \(\mathbf{F_{3}}\) & \(\mathbf{F_{4}}\) & \(\mathbf{F_{5}}\) \\ \hline
0.177493 & 0.183588 & 0.201573 & 0.220597 & 0.216750 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comprehensive score of schemes based on expert knowledge. \[SIM= \begin{bmatrix}1&0.6356&0.6630&0.6680&0.6695&0.6578&0.6743&0.6464&0.6544&0.6804 \\ 0.6352&1&0.7054&0.7138&0.7396&0.7144&0.6673&0.6800&0.7194&0.7119\\ 0.6631&0.7054&1&0.7696&0.6888&0.7228&0.7251&0.6845&0.7359&0.7403\\ 0.6680&0.7138&0.7696&1&0.7466&0.7239&0.7570&0.7082&0.7155&0.7570\\ 0.6693&0.7396&0.6888&0.7466&1&0.7431&0.7167&0.7329&0.7264&0.7745\\ 0.6576&0.7144&0.7228&0.7329&0.7431&1&0.6862&0.7109&0.7040&0.7313\\ 0.6743&0.6673&0.7251&0.7570&0.7167&0.6862&1&0.7079&0.7052&0.7579\\ 0.6464&0.6800&0.6845&0.7028&0.7329&0.7190&0.7097&1&0.7092&0.7147\\ 0.6544&0.7149&0.7359&0.7155&0.7264&0.7040&0.7052&0.7092&1&0.7315\\ 0.6804&0.7119&0.7430&0.7572&0.7746&0.7313&0.7579&0.7147&0.7315&1\\ \end{bmatrix} \]  
Step 8: We obtained each user's support sup and user weight \(\gamma\) using Equations (20) and (21), as shown in Tables 6 and 7. Step 9: The score function for each attribute was calculated by applying user weights to the normalized decision matrix, as shown in Table 8.  
Step 10: We calculated the scheme score \(\tau\) based on user preferences according to Equations (22) and (23), as shown in Table 9.  
Step 11: After that, we normalized \(\sigma_{i}\) and \(\tau_{i}\). Using a deviation minimization optimization function, we determined the optimal values of \(\mathbf{\varphi}\), which resulted in expert weights of 0.52 and user weights of 0.42. We calculated the comprehensive score of the scheme according to Equation (24), as shown in Table 10. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \(\mathbf{U_{1}}\) & \(\mathbf{U_{2}}\) & \(\mathbf{U_{4}}\) & \(\mathbf{U_{5}}\) & \(\mathbf{U_{6}}\) & \(\mathbf{U_{7}}\) & \(\mathbf{U_{8}}\) & \(\mathbf{U_{9}}\) & \(\mathbf{U_{10}}\) \\ \hline
5.9487 & 6.2829 & 6.4354 & 6.5596 & 6.5382 & 6.4025 & 6.3994 & 6.3046 & 6.3970 & 6.5996 \\ \hline \end{tabular}
\end{table} 
Table 6: User support. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\mathbf{U_{1}}\) & \(\mathbf{U_{2}}\) & \(\mathbf{U_{4}}\) & \(\mathbf{U_{5}}\) & \(\mathbf{U_{6}}\) & \(\mathbf{U_{7}}\) & \(\mathbf{U_{8}}\) & \(\mathbf{U_{9}}\) & \(\mathbf{U_{10}}\) \\ \hline
5.9487 & 6.2829 & 6.4354 & 6.5596 & 6.5382 & 6.4025 & 6.3994 & 6.3046 & 6.3970 & 6.5996 \\ \hline \end{tabular}
\end{table} 
Table 7: User weights. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) & \(\mathbf{P_{4}}\) & \(\mathbf{P_{5}}\) & \(\mathbf{P_{6}}\) & \(\mathbf{P_{7}}\) & \(\mathbf{P_{8}}\) \\ \hline
0.255068 & 0.596740 & 0.175258 & 0.497878 & 0.265392 & 1.000000 & 0.715532 & 0.752909 \\ \hline \end{tabular}
\end{table} 
Table 8: Score of attributes for each alternative scheme based on user preferences.