# 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  
Comprehensive field investigations conducted on the Jigongling Bridge revealed significant structural anomalies within the deck system, including transverse cracking, localized spalling, and expansion joint blockages. These defects, characterized by accelerated material degradation kinetics and service life expectancy reduction risks, necessitate a systematic contributory factor analysis. This research integrates certified engineering measurements (acquired through laser scanning and sampling techniques) with FBWM-based weight de  
Figure 10: The plan view of the bridge structure. Figure 9: Jigongling bridge location schematic. termination to establish a multi-criteria decision-making framework. The resultant priority ranking of five secondary indicators under the Bridge Deck System (C8)--including bridge deck pavement (_D_22), expansion and contraction joint (_D_23), pedestrian walk and handrail (_D_24), illuminated sign (_D_25), and drainage systems (_D_26)--provides evidence-based guidance for cost-effective rehabilitation strategies that balance short-term maintenance expenditures with long-term structural sustainability. Among these, \"drainage systems (_D_22)\" and \"illuminated sign (_D_25)\" are, respectively, determined to be the best indicator and the worst indicator based on the Bridge Deck System (C8). The linguistic terms of decision-maker for fuzzy reference comparisons of the best indicator over all the indicators and all the indicators over the worst indicator are listed in Table 9.  
The fuzzy vector of comparing the best indicator against all other indicators is shown in Equation (33):  
\[\hat{P}_{B}=\left((1,1,1),\left(\frac{2}{3},1,\frac{3}{2}\right),\left(\frac{3} {2},2,\frac{5}{2}\right),\left(\frac{7}{2},4,\frac{9}{2}\right),\left(\frac{3} {2},2,\frac{5}{2}\right)\right) \tag{33}\]  
The fuzzy vector of comparing all other indicators against the worst indicator is shown in Equation (34):  
\[\hat{P}_{W}=\left(\left(\frac{7}{2},4,\frac{9}{2}\right),\left(\frac{5}{2},3, \frac{7}{2}\right),\left(\frac{2}{3},1,\frac{3}{2}\right),\left(1,1,1\right),\left(\frac{2}{3},1,\frac{3}{2}\right)\right) \tag{34}\]  
Figure 11: (**a**) Transverse crack pattern at beam bottom; (**b**) vertical crack detail at left wall of bridge abutment; (**c**) transverse crack detail on bridge deck pavement; (**d**) severe blockage detail in expansion joint. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator** & _D_**22** & _D_**23** & _D_**24** & _D_**25** & _D_**26** \\ \hline Best Indicator \(D\)22 & II & SI & RI & EI & RI \\ Worst Indicator \(D\)25 & EI & HI & SI & II & SI \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The fuzzy reference comparisons of the best and the worst indicators. After solving Equation (21) using MATLAB, the optimal fuzzy weight vectors for the five indicators \(D22\)-\(D26\) and the optimal target vector can be determined: \(\hat{o}_{B1}^{*}\) = (0.2881, 0.3384, 0.3485); \(\hat{o}_{B2}^{*}\) = (0.2393, 0.2943, 0.3022); \(\hat{o}_{B3}^{*}\) = (0.1249, 0.1453, 0.1560); \(\hat{o}_{B4}^{*}\) = (0.0845, 0.0963, 0.0967); \(\hat{o}^{*}\) = (0.1249, 0.1453, 0.1560). Following the optimal weights of the five indicators can be obtained: \(v_{D22}^{*}\) = 0.3317, \(v_{D23}^{*}\) = 0.2865, \(v_{D24}^{*}\) = 0.1437, \(v_{D25}^{*}\) = 0.0944, \(v_{D26}^{*}\) = 0.1437, and the optimal target values are denoted by \(\varphi^{*}\) = 0.3605. Given that \(\hat{\beta}_{BW}=\hat{\beta}_{14}=\left(\frac{7}{2},4,\frac{9}{2}\right)\), It can be seen from Table 2 that the \(CI\) for case study 2 is \(CI\) = 8.04. The \(CI\) and corresponding optimal target value, obtained through the FBWM, are systematically substituted into Equation (12) to calculate the CR. The detailed computational procedure is mathematically formulated in Equation (22). \[CR=\frac{\varphi^{*}}{CI}=\frac{0.3605}{8.04}=0.0448<0.1 \tag{35}\]  
The value of \(CR\) (0.0448) is below 0.1, signifying that the outcomes have successfully passed the consistency examination.