# The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation
## 4 Case Study
### Case Study 1
#### 4.1.2 Condition Rating of Dingjia Bridge
\right) \tag{26}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Criteria** & **Primary Indicator** & **Secondary Indicator** & \multicolumn{5}{c}{**Membership Matrix**} \\
**Layer** & **Layer** & **Excellent** & **Good** & **Fail** & **Poor** & **Critical** \\ \hline \multirow{10}{*}{\(B1\)} & & & \(D1\) & 0 & 0.4 & 0.6 & 0 & 0 \\  & & \(D2\) & 0 & 0.5 & 0.5 & 0 & 0 \\  & C1 & & \(D3\) & 0 & 0.3 & 0.7 & 0 & 0 \\  & & \(D4\) & 0 & 0 & 0.4 & 0.6 & 0 & 0 \\  & & \(D5\) & 0 & 0 & 0.7 & 0.3 & 0 \\ \hline \multirow{10}{*}{\(B2\)} & & & \(D7\) & 0 & 0.5 & 0.5 & 0 & 0 \\  & C2 & & \(D7\) & 0 & 0.4 & 0.6 & 0 & 0 \\  & & \(D8\) & 0 & 0.6 & 0.4 & 0 & 0 \\ \cline{1-1}  & C3 & & \(D9\) & 0 & 0.4 & 0.6 & 0 & 0 \\ \cline{1-1}  & C4 & & \(D10\) & 0 & 0 & 0.7 & 0.3 & 0 \\ \hline \multirow{10}{*}{\(B1\)} & & & \(D11\) & 0 & 0.6 & 0.4 & 0 & 0 \\  & & \(D12\) & 0 & 0.5 & 0.5 & 0 & 0 \\  & C5 & & \(D13\) & 0 & 0.3 & 0.7 & 0 & 0 \\  & & \(D14\) & 0.8 & 0.2 & 0 & 0 & 0 \\  & C6 & & \(D15\) & 0 & 0.7 & 0.3 & 0 & 0 \\  & & \(D16\) & 0 & 0.4 & 0.6 & 0 & 0 \\  & C6 & & \(D16\) & 0 & 0.4 & 0.6 & 0 & 0 \\  & & \(D17\) & 0 & 0 & 0.6 & 0.4 & 0 \\  & & \(D18\) & 0 & 0 & 0.5 & 0.5 & 0 \\ \cline{1-1}  & & \(D19\) & 0 & 0.7 & 0.3 & 0 & 0 \\  & C7 & & \(D20\) & 0 & 0.5 & 0.5 & 0 & 0 \\  & & \(D21\) & 0 & 0.6 & 0.4 & 0.6 & 0 \\ \hline \multirow{10}{*}{\(B3\)} & & & \(D22\) & 0.2 & 0.8 & 0 & 0 & 0 \\  & & \(D23\) & 0 & 0.4 & 0.6 & 0 & 0 \\  & C8 & & \(D24\) & 0 & 0.7 & 0.3 & 0 & 0 \\  & & \(D25\) & 0 & 0.6 & 0.4 & 0 & 0 \\  & & \(D26\) & 0 & 0.7 & 0.3 & 0 & 0 \\ \hline \multirow{10 The comprehensive evaluation vectors of the primary indicators, C1-C4, constitute the membership matrix for the criteria layer indicator \(B1\), as expressed in Table 6.  
Thus, the membership matrix for \(B1\) can be obtained as expressed in Equation (27). \[Q_{B1}= \begin{pmatrix}0&0.32&0.58&0.10&0\\ 0&0.49&0.51&0&0\\ 0&0.40&0.60&0&0\\ 0&0&0.70&0.30&0\end{pmatrix}  \tag{27}\]  
Following the fuzzy comprehensive operation of each indicator in the primary indicator layer, the fuzzy vectors for each indicator in the criteria layer can be derived. The calculation process for the criteria layer indicator \(B1\) is presented as follows:  
\[E_{B1}=V_{C1-C4}\times Q_{B1}=\left( \begin{array}{c}0.3823\\ 0.1793\\ 0.2987\\ 0.1397\end{array} \right)^{T}\times\left( \begin{array}{cccccc}0&0.32&0.58&0.1 0&0\\ 0&0.49&0.51&0&0\\ 0&0.40&0.60&0&0\\ 0&0&0.70&0.30&0\end{array} \right)=(0,0.32,0.59,0.08,0) \tag{28}\]  
Following the process outlined above, the comprehensive evaluation vectors for each layer can be obtained. Finally, by applying the weights from the criteria layer and the membership matrices, the comprehensive evaluation vector for the target layer is calculated, as shown in Equation (28), with the specific fuzzy computation results for the target layer indicator \"Bridge Comprehensive Condition (\(A\))\" presented in Table 7.  
Thus, the membership matrix and the comprehensive evaluation vector of the target layer \(A\) can be obtained as expressed in Equations (29) and (30). \[Q_{A}= \begin{pmatrix}0&0.33&0.59&0.08&0\\ 0.17&0.36&0.37&0.10&0\\ 0.07&0.66&0.27&0&0\\ 0.07&0.48&0.45&0&0\end{pmatrix}  \tag{29}\]  
\[E_{A}=V_{B1-B4}\times Q_{A}=\left( \begin{array}{c}0.4201\\ 0.3198\\ 0.1570\\ 0.1032\end{array} \right)^{T}\times\left( \begin{array}{cccccc}0&0.33&0.59&0. 08&0\\ 0.17&0.36&0.37&0.10&0\\ 0.07&0.66&0.27&0&0\\ 0.07&0.48&0.45&0&0\end{array} \right)=(0.007,0.41,0.46,0.06,0) \tag{30}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Criteria** & **Primary Indicator Layer** & **Optimal Weight** & **Comprehensive Evaluation Vector** \\ \hline \multirow{3}{*}{\(B1\)} & C1 & 0.3823 & (0, 0.32, 0.58, 0.10, 0) \\  & C2 & 0.1793 & (0, 0.49, 0.51, 0, 0) \\  & C3 & 0.2987 & (0, 0.40, 0.60, 0, 0) \\  & C4 & 0.1397 & (0, 0, 0.70, 0.30, 0) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The comprehensive evaluation vectors of the primary indicators, C1–C4.