# The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation
## 4 Case Study
### Case Study 1
#### 4.1.1 Weight Calculation for Dingjia Bridge  
The FBWM was employed to determine weight coefficients within the novel bridge condition evaluation model. Specifically, the weighting calculation process for the criteria layer is demonstrated using safety (\(B1\)), durability (\(B2\)), suitability (\(B3\)), and reinforcement economy (\(B4\)) as representative indicators. Within this set, safety (\(B1\)) was selected as the best indicator, while reinforcement economy (\(B4\)) served as the worst indicator. Table 3 summarizes the expert-derived fuzzy reference comparisons between the best indicator (\(B1\)) and other indicators, as well as comparisons of all indicators against the worst indicator (\(B4\)). Figure 8: (a) Defects on the bridge deck; (b) defects on the bottom of the T-beam; (c) defects on abutments. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Indicator** & \(B1\) & \(B2\) & \(B3\) & \(B4\) \\ \hline Best Indicator \(B1\) & II & SI & HI & EI \\ Worst Indicator \(B4\) & EI & HI & RI & II \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The fuzzy reference comparisons of the best and the worst indicator. Figure 7: Ding Jia bridge location schematic. The fuzzy vector of comparing the best indicator to all other indicators is shown in Equation (19):  
\[\hat{P}_{B}=\left((1,1,1),\left(\frac{2}{3},1,\frac{3}{2}\right),\left(\frac{5}{2 },3,\frac{7}{2}\right),\left(\frac{7}{2},4,\frac{9}{2}\right)\right) \tag{19}\]  
The fuzzy vector of comparing all other indicators to the worst indicator is shown in Equation (20):  
\[\hat{P}_{W}=\left(\left(\frac{7}{2},4,\frac{9}{2}\right),\left(\frac{5}{2},3, \frac{7}{2}\right),\left(\frac{3}{2},2,\frac{5}{2}\right),\left(1,1,1\right)\right) \tag{20}\]  
Following the identification of the best and worst indicators through the FBWM, Equation (11) was mathematically reformulated as Equation (21) through algebraic manipulation. This reformulated equation was subsequently solved numerically via constrained nonlinear optimization implemented in MATLAB (2022), yielding optimized fuzzy weight vectors for the four evaluation indicators (safety, durability, suitability, and reinforcement economy), along with the derived optimal target vector: \(\hat{\sigma}_{B1}^{*}=\left(0.4045,\,0.4197,\,0.4372\right)\); \(\hat{\sigma}_{B2}^{*}=\left(0.2840,\,0.3213,\,0.3494\right)\); \(\hat{\sigma}_{B3}^{*}=\left(0.1293,\,0.1540,\,0.1966\right)\); \(\hat{\sigma}_{B4}^{*}=\left(0.0956,\,0.1027,\,0.1125\right)\); \(\hat{\sigma}^{*}=\left(0.2893,\,0.2893,\,0.2893\right)\). After determining fuzzy weight vectors for one hierarchical layer via the FBWM, subsequent indicator weights were systematically calculated according to the established protocol. Criteria layer fuzzy weight vectors were defuzzified through the application of Equation (7). This procedure yielded optimal weights for the four indicators, namely \"Safety (\(B1\)), Durability (\(B2\)), Suitability (\(B3\)), and Economic Reinforcement (\(B4\))\", are, respectively, 0.4201, 0.3198, 0.1570, and 0.1032, with the corresponding optimal target value calculated as 0.2893. \[