# The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation
## 2 Fuzzy Best and Worst Method-Fuzzy Comprehensive Evaluation Model
### Fuzzy Best and Worst Method
\begin{array}{l}\min\ \ \max_{j}\ \ \left\{\left|\frac{\hat{v}_{B}}{\hat{v}_ {j}}-\hat{v}_{Bj}\right|_{\prime}\left|\frac{\hat{v}_{j}}{\hat{v}_{W}}-\hat{v} _{BW}\right|\right\}\\ s.t.\left\{\begin{array}{l}\sum\limits_{j=1}^{n}Q(\hat{v}_{j})=1\\ I_{j}^{v}\leq m_{j}^{v}\leq u_{j}^{v}\\ I_{j}^{v}\geq 0\\ j=1,2,\cdots,n\end{array}\right.\end{array}  \tag{10}\]  
Equation (10) can be further transformed into Equation (11), which represents a nonlinear constrained optimization problem. By solving Equation (11), the fuzzy weight vectors \(\left(\hat{v}_{1}^{*},\hat{v}_{2}^{*},\cdots,\hat{v}_{n}^{*}\right)\) as well as the optimal target vector \(\hat{\phi}^{*}\), can be obtained. \[ \begin{array}{l}\min\hat{\phi}^{*}\\ s.t.\left\{\begin{array}{l}\sum\limits_{j=1}^{n}Q(\hat{v}_{j})=1\\ I_{j}^{v}\leq m_{j}^{v}\leq u_{j}^{v}\\ I_{j}^{v}\geq 0\\ j=1,2,\cdots,n\end{array}\right.\end{array} \right.\end{array} \tag{11}\]  
The consistency ratio (\(CR\)) serves as the core metric for evaluating the logical consistency of the FBWM and expresses the degree of consistency in fuzzy reference comparisons. The \(CR\) is calculated using Equation (12). If the \(CR\) is less than or equal to 0.1, it is considered acceptable; otherwise, it is recommended to repeat the pairwise comparisons until the \(CR\) is less than or equal to 0.1 [31]. The closer the CR value is to 0, the fewer logical contradictions exist in expert judgments, and the stronger the methodological consistency becomes, thereby ensuring the scientific validity and reliability of weight calculation results. \[CR=\frac{\hat{\phi}^{*}}{CI} \tag{12}\]  
where \(CI\) represents the Consistency index and Table 2 shows the \(CI\) corresponding to different fuzzy numbers in the FBWM. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Five Levels of** & **Identically** & **Slightly Important** & **Relatively** & **Highly Important** & **Extremely Important (EI)** \\
**Importance** & **Important (II)** & **(SI)** & **Important (RI)** & **(HI)** & **Important (EI)** \\ \hline \(\hat{\beta}_{BW}\) & \((1,1,1)\) & \(\left(\frac{2}{3},1,\frac{3}{2}\right)\) & \(\left(\frac{3}{2},2,\frac{5}{2}\right)\) & \(\left(\frac{5}{2},3,\frac{7}{2}\right)\) & \(\left(\frac{7}{2},4,\frac{9}{2}\right)\) \\ CI & 3.00 & 3.8 & 5.29 & 6.69 & 8.04 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Consistency index (CI) for FBWM. TFN [32]. The optimal weights of evaluation indicators are calculated based on the FBWM. Firstly, a comprehensive evaluation indicator system is constructed. Subsequently, through consultations with experts in design and management, the best and worst indicators are determined based on the consensus of expert discussions. Fuzzy preference comparison vectors are then established using a five-level importance scale. Finally, the fuzzy weights of each indicator are computed through a nonlinear programming model and subjected to defuzzification. After consistency verification, a weight set for the comprehensive condition indicators is established. The detailed procedural workflow is illustrated in Figure 2.