# The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation
## 4 Case Study
### Case Study 2
#### 4.2.1 Weight Calculation for Jigongling Bridge
To validate the scientific validity and accuracy of the FBWM-derived results, this case study also employed the AHP and BWM methods to calculate the optimal weights and the CRs. This comparative verification process, whose quantitative findings are systematically summarized in Table 10.  
The FBWM employs TFNs to quantify linguistic uncertainties in bridge decision-making contexts. A comparative analysis among the AHP, BWM, and FBWM models revealed smaller discrepancies in optimal weight allocations; however, significant differences emerged in their consistency ratios (_CRs_): FBWM achieved a \(CR\) value of 0.0448 \(<\) 0.05, which was 19.7% lower than that of the BWM (0.0608) and 22% lower than that of the AHP (0.0625). This result demonstrates that the fuzzy extension strategy effectively alleviates subjective cognitive biases arising from the limitations of rigid scales in traditional decision models. Furthermore, the comparative analysis reveals that FBWM achieves a 20% reduction in required pairwise comparisons relative to the AHP while maintaining decision consistency. The FBWM significantly reduces decision-making complexity by \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Secondary Indicators Layer** & **Optimal Weights** & **CR** & **Number of Comparisons** \\ \hline \multirow{8}{*}{AHP} & \(D22\) & 0.3811 & \multirow{8}{*}{0.0625} & \multirow{8}{*}{\(\frac{n(n-1)}{2}\)} \\  & \(D23\) & 0.2869 & & \\  & \(D24\) & 0.1203 & & \\  & \(D25\) & 0.0820 & & \\  & \(D26\) & 0.1297 & & \\ \hline \multirow{8}{*}{BWM} & \(D22\) & 0.4664 & \multirow{8}{*}{0.0608} & \multirow{8}{*}{\(2(n-1)\)} \\  & \(D23\) & 0.2291 & & \\ \cline{1-1}  & \(D24\) & 0.1161 & & \\ \cline{1-1}  & \(D25\) & 0.0570 & & \\ \cline{1-1}  & \(D26\) & 0.1314 & & \\ \hline \multirow{8}{*}{FBWM} & \(D22\) & 0.3317 & \multirow{8}{*}{0.0448} & \multirow{8}{*}{\(2(n-1)\)} \\  & \(D23\) & 0.2865 & & \\ \cline{1-1}  & \(D24\) & 0.1437 & & \\ \cline{1-1}  & \(D25\) & 0.0944 & & \\ \cline{1-1}  & \(D26\) & 0.1437 & & \\ \hline \hline \end{tabular} Where \(n\) represents the number of indicators, with \(n\) = 5 in Table 3 of the case study.  
\end{table} 
Table 10: Comparative analysis of the optimal weights for \(D22\)–\(D26\) indicators and the \(CR\) derived from AHP, BWM, and FBWM. decreasing the number of pairwise comparisons required. The specific reduction percentage depends on the number of indicators, typically ranging from 20% to 60%, with the reduction percentage increasing as the number of indicators grows. These methodological advancements collectively validate the robustness and practical applicability of FBWM in sustainable bridge management, providing bridge practitioners with a standardized computational framework that complies with specifications, significantly enhancing infrastructure decision-making efficiency. Therefore, the FBWM can be systematically applied to compute the optimal weights and target values for each layer's indicators, with the optimal weights of each layer's indicators are shown in Table 11.