# 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  
The systematic construction of evaluation factor sets constitutes a critical foundation for ensuring the validity and precision of the FCE method. By systematically analyzing the bridge condition determinants, as schematically represented in Figure 1, the evaluation factor set \(O\) is formally defined as: \(O_{A}=\{o_{B1},o_{B2},o_{B3},o_{B4}\}\), \(O_{B1}=\{o_{C1},o_{C2},o_{C3},o_{C4}\}\), \(O_{C1}=\{o_{D1},o_{D2},o_{D3},o_{D4},o_{D5}\}\), and so forth. In accordance with the _Specifications for Maintenance of Highway Bridges and Culverts_ (JTG 5120-2004) [23] and the _Standards for Technical Condition Evaluation of Highway Bridges_ (JTG/T H21-2011) [37], the comprehensive condition of bridge's is classified into five distinct grades. Accordingly, the evaluation set is formally defined as:  
\(Z=\{z_{1},z_{2},z_{3},z_{4},z_{5}\}=\{\text{Class }I\text{ bridge, Class }II\text{ bridge, Class }IV\text{ bridge, Class }V\text{ bridge}\}=\{\text{Excellent, Good, Fail, Poor, Critical}\}\). Following the membership function determination method detailed in Section 2.3 and leveraging data from the _2020 Bridge Inspection and Evaluation Report_ issued by the Highway Administration of the Hubei Provincial Department of Transportation [38], the membership matrices were systematically constructed, as shown in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline
**Criteria** & **Final Weights** & **Primary** & **Final** & **Set of Weights for** **Secondary Indicators Layer** \\ \hline \multirow{4}{*}{_B1_} & & C1 & 0.3823 & \{0.3152, 0.2304, 0.2478, 0.1186, 0.0787\} \\  & & C2 & 0.1793 & \{0.4287, 0.3244, 0.2469\} \\  & & C3 & 0.2987 & \{1\} \\  & & C4 & 0.1397 & \{1\} \\ \hline \multirow{4}{*}{_B2_} & & C5 & 0.5510 & \{0.0860, 0.0657, 0.1337, 0.2660, 0.2047, \\  & & C6 & 0.2551 & \{0.0878, 0.2686, 0.2171, 0.1388, 0.0707, \\  & & C7 & 0.1939 & 0.2171\} \\ \hline \multirow{4}{*}{_B3_} & 0.1570 & C8 & 1 & \{0.3452, 0.2140, 0.1125, 0.0892, 0.2392\} \\  & & C9 & 0.5265 & \{1\} \\ \cline{1-1}  & & C10 & 0.1265 & \{0.3359, 0.6642\} \\ \cline{1-1}  & & C11 & 0.3471 & \{0.2509, 0.7491\} \\ \hline \end{tabular}
\end{table} 
Table 4: The optimal weights of each layer’s indicators based on FBWM. The fuzzy comprehensive operation is performed through the matrix multiplication by using Equation (17), where the weight vector \(V\) and membership matrix \(Q\) of the secondary indicator layer generate the comprehensive evaluation vector \(E\) for primary indicators. Specifically, the computational procedures for the \(E_{\text{C1}}\)-\(E_{\text{C4}}\) are mathematically expressed as in Equations (23)-(26):  
\[E_{\text{C1}}=V_{\text{D1}-\text{D5}}\times Q_{\text{C1}}=\left( \begin{array}[] {c}0.3152\\ 0.2304\\ 0.2478\\ 0.1186\\ 0.0787\end{array} \right)^{T}\times\left( \begin{array}{cccc}0&0.4&0.6&0&0\\ 0&0.5&0.5&0&0\\ 0&0.3&0.7&0&0\\ 0&0&0.4&0.6&0\\ 0&0.7&0.3&0\end{array} \right)=\left( \begin{array}{cccc}0&0.32&0.58&0.10&0 \end{array} \right) \tag{23}\]  
\[E_{\text{C2}}=V_{\text{D6}-\text{D8}}\times Q_{\text{C2}}=\left( \begin{array}[] {c}0.4287\\ 0.3244\\ 0.2468\end{array} \right)^{T}\times\left( \begin{array}{cccc}0&0.5&0.5&0&0\\ 0&0.4&0.6&0&0\\ 0&0.6&0.4&0&0\end{array} \right)=\left( \begin{array}{cccc}0&0.49&0.51&0&0 \end{array} \right) \tag{24}\]  
\[E_{\text{C3}}=V_{\text{D9}}\times Q_{\text{C3}}=\left(1\right)^{T}\times \left( \begin{array}{cccc}0&0.4&0.6&0&0\\ 0&0.4&0.6&0&0\\ 0&0.6&0.4&0&0\end{array} \right)=\left( \begin{array}{cccc}0&0.40&0.60&0&0 \end{array} \right) \tag{25}\]  
\[E_{\text{C4}}=V_{\text{D10}}\times Q_{\text{C4}}=\left(1\right)^{T}\times \left( \begin{array}{cccc}0&0&0.7&0.3&0\end{array} \right)=\left( \begin{array}[ ]{cccc}0&0&0.70&0.30&0\end{array}