# 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
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Target** & **Criteria** & **Optimal Weight** & **Comprehensive Evaluation Vector** \\ \hline \multirow{3}{*}{\(A\)} & B1 & 0.4201 & (0, 0.33, 0.59, 0.08, 0) \\  & B2 & 0.3198 & (0.17, 0.36, 0.37, 0.10, 0) \\  & B3 & 0.1570 & (0.07, 0.66, 0.27, 0, 0) \\  & B4 & 0.1032 & (0.07, 0.48, 0.45, 0, 0) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The comprehensive evaluation vectors of the target layer. A multi-level fuzzy comprehensive evaluation method was adopted to systematically evaluate bridge conditions, progressing from the bottom to the top layer. This method incorporates a hierarchical structure comprising distinct evaluation levels, each assigned to evaluate specific performance indicators. The evaluation procedure begins at the secondary indicator layer, progresses through primary indicator and criteria layers, and culminates in a comprehensive evaluation at the target layer. This tiered framework enables systematic performance analysis by simultaneously considering localized structural details and comprehensive operational performance. The comprehensive evaluation score \(\overline{E}_{A}\) was computed through weighted mean analysis, with bridge condition grades assigned numerical values from 1 (excellent) to 5 (critical). Thus, the comprehensive evaluation score is mathematically expressed as:  
\[\overline{E}_{A}=\frac{0.07\times 1+0.41\times 2+0.46\times 3+0.06\times 4+0 \times 5}{0.07+0.41+0.46+0.06+0}=2.51 \tag{31}\]  
According to Equation (31), the comprehensive score of Ding Jia Bridge is 2.51. This score falls within the third interval (2, 3), as detailed in Section 2.3; the Ding Jia Bridge was classified as a Class _III_ bridge, consistent with on-site inspections in the _2020 Bridge Inspection and Evaluation Report_ (Highway Administration of Hubei Provincial Department of Transportation) [38]. The comprehensive evaluation of Ding Jia Bridge based on BWM can be obtained from [PERSON] and [PERSON] [39]. The optimal weights of each layer's indicators based on BWM are shown in Table 8.  
Following the determination of the membership matrix and optimal weight vectors, the comprehensive evaluation vector \(E_{A}\) for target layer \(A\) is computed through the fuzzy comprehensive operation: \(E_{A}\) = (0.0315, 0.2247, 0.5148, 0.2290, 0). So, the comprehensive score \(\overline{E}_{A}\) based on BWM is expressed as:  
\[\overline{E}_{A}=\frac{0.0315\times 1+0.2247\times 2+0.5148\times 3+0.2290 \times 4+0\times 5}{0.0315+0.2247+0.5148+0.2290+0}=2.9413 \tag{32}\]  
As calculated by Equation (32), the comprehensive evaluation score for Ding Jia Bridge was determined to be 2.9413. This value falls within the third classification interval (2-3), corresponding to a Class _III_ bridge according to the evaluation set established in Section 2.3. Comparative analysis of Equations (31) and (32) reveals that both FBWM and BWM produce consistent Class _III_ bridge classifications, aligning with the standardized evaluation results stipulated in the Highway Bridge and Culvert Maintenance Code [23]. The