## 6 Example Analysis
### Single-Level Fuzzy Comprehensive Evaluation  
(1) Determine the subjective weights  
Firstly, the subjective weights of different levels of indicators are determined based on AHP. By consulting experts' opinions, the pairwise comparison judgment matrix is constructed. Then, the maximum eigenvalues and corresponding eigenvectors are solved according to Equation (4). The weights of the evaluation indicators are obtained by normalizing the maximum eigenvectors, as displayed in Table 5.  
It can be seen from Table 5 that weights of the criterion hierarchy indicators that are determined by AHP have passed the consistency check. In the same way, the consistency check of weights of the program hierarchy indicators are conducted. There are \(\lambda_{\text{max}}\) = 5.3839, \(CI\) = 0.096, \(CR\) = 0.0857 \(<\) 1, as a result, the consistency check is also passed. (2) Determine the objective weights  
The normalized judgment matrices are constructed from the normalized data, and the standardized judgment matrices are processed according to the Equations (7)-(10) by the entropy weight method. The objective weights of evaluation indicators are shown in Table 6. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Three-Level** & \multicolumn{8}{c}{**Electricity Retailers**} \\ \cline{2-10}
**Indicators** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** & **I** & **J** \\ \hline \(U_{31}\) & 0.8521 & 0.6549 & 0.2993 & 0.9085 & 1.0000 & 0.5880 & 0.4366 & 0.7852 & 0.6127 & 0.8627 \\ \(U_{32}\) & 0.3064 & 0.4641 & 0.2368 & 0.6855 & 1.0000 & 0.5685 & 0.3371 & 0.3832 & 0.6552 & 0.8838 \\ \(U_{33}\) & 0.9595 & 0.8639 & 0.8155 & 0.9269 & 0.7402 & 0.6862 & 0.8493 & 0.7750 & 0.9291 & 1.0000 \\ \(U_{34}\) & 0.8904 & 0.8800 & 0.7774 & 1.0000 & 0.9814 & 0.8403 & 0.9709 & 0.8834 & 0.9592 & 0.9825 \\ \(U_{41}\) & 0.9016 & 0.9147 & 0.9357 & 0.9921 & 0.9501 & 0.8635 & 1.0000 & 0.8898 & 0.9777 & 0.9606 \\ \(U_{42}\) & 0.9265 & 0.9027 & 0.9870 & 0.8519 & 0.9146 & 0.8249 & 1.0000 & 0.8303 & 0.8908 & 0.9243 \\ \(U_{51}\) & 0.7123 & 0.8630 & 0.7511 & 0.9372 & 1.0000 & 0.6644 & 0.7477 & 0.8196 & 0.7123 & 0.8345 \\ \(U_{52}\) & 0.9615 & 0.9707 & 0.9791 & 1.0000 & 0.9791 & 0.8678 & 0.9637 & 0.9749 & 0.9791 & 0.9791 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Program** & **Weights of** & **Criterion** & **Weights of** & **Consistency Check** \\
**Hierarchy** & **Indicators** & **Hierarchy** & **Indicators** & **Consistency Check** \\ \hline \multirow{4}{*}{\(U_{1}\)} & \multirow{4}{*}{0.0939} & \(U_{11}\) & 0.1443 & \multirow{4}{*}{\(\lambda_{\text{max}}\) = 5.2461} \\  & & \(U_{12}\) & 0.4309 & & \(\lambda_{\text{max}}\) = 5.2461 \\  & & \(U_{13}\) & 0.3155 & \(CI\) = 0.0615 \\  & & \(U_{14}\) & 0.0665 & \(CR\) = 0.0549 \\ \hline \multirow{4}{*}{\(U_{2}\)} & \multirow{4}{*}{0.2188} & \(U_{21}\) & 0.5280 & \multirow{4}{*}{\(\lambda_{\text{max}}\) = 4.2496} \\  & & \(U_{22}\) & 0.3221 & & \(\lambda_{\text{max}}\) = 4.2496 \\  & & \(U_{23}\) & 0.0604 & & \(CI\) = 0.0832 \\  & & \(U_{24}\) & 0.0895 & & \(CR\) = 0.0936 \\ \hline \multirow{4}{*}{\(U_{3}\)} & \multirow{4}{*}{0.5800} & \(U_{31}\) & 0.3326 & \multirow{4}{*}{\(\lambda_{\text{max}}\) = 4.1179} \\  & & \(U_{32}\) & 0.1136 & & \(\lambda_{\text{max}}\) = 4.1179 \\  & & \(U_{33}\) & 0.2572 & \(CI\) = 0.0393 \\  & & \(U_{34}\) & 0.2975 & & \(CR\) = 0.0442 \\ \hline \multirow{4}{*}{\(U_{4}\)} & \multirow{4}{*}{0.0412} & \(U_{41}\) & 0.2500 & \multirow{4}{*}{\(\lambda_{\text{max}}\) = 2} \\  & & \(U_{42}\) & 0.7500 & Complete consistency \\ \hline \multirow{4}{*}{\(U_{5}\)} & \multirow{4}{*}{0.0661} & \(U_{51}\) & 0.1429 & \multirow{4}{*}{\(\lambda_{\text{max}}\) = 2} \\  & & \(U_{52}\) & 0.8571 & Complete consistency \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Subjective weights of energy saving and emission reduction evaluation indicators of electricity retailers.