# 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  
The BWM is widely employed for determining indicator weights. However, it often struggles to accurately capture relative preferences due to the inherent vagueness and intangibility of human qualitative judgments, as well as the uncertainties present in real-world data. To address these limitations, the FBWM replaces crisp numerical values with TFNs, thereby offering a more robust and flexible comparative framework [27]. This method effectively accounts for uncertainties in the comparison process, namely fuzzy reference comparison, reduces the influence of subjective factors on evaluation indicators, and ensures that the resulting evaluation more accurately reflects the complexities of real-world conditions. Unlike the traditional 1-9 scale used in BWM [28], the FBWM employs a five-level importance scale to perform fuzzy reference comparisons, as shown in Table 1. The procedural steps of the FBWM, along with their detailed descriptions, are based on the method developed by [PERSON] et al. [29]. The steps of the FBWM are as follows:  
**Step1. ** Identifying the best indicator \(K_{B}\) and the worst indicator \(K_{W}\). In this step, the best indicator, \(K_{B}\) and the worst indicator, \(K_{W}\) should be determined in general, without making any comparisons. **Step2. ** Performing the fuzzy reference comparisons for the best indicator. The best indicator \(K_{B}\) is compared with all other indicators using a five-level importance scale provided in Table 2. The resulting fuzzy comparison vector is derived as Equation (8):  
\[\hat{P}_{B}=\left(\hat{\beta}_{B1},\hat{\beta}_{B2},\ldots,\hat{\beta}_{Bn}\right) \tag{8}\]  
where \(\hat{\beta}_{Bj}\) represents the fuzzy reference comparisons of the best indicator over all other indicators, \(\hat{\beta}_{Bj}=\left(I_{Bj},\ m_{Bj},\ u_{Bj}\right)\), \(j=1,2,\ldots,n\). **Step3. ** Performing the fuzzy reference comparisons s for the worst indicator. All indicators are compared with the worst indicator \(K_{W}\) using a five-level importance scale provided in Table 2. The resulting fuzzy comparison vector is derived as Equation (9):  
\[\hat{P}_{W}=\left(\hat{\beta}_{W1},\hat{\beta}_{W2},\ldots,\hat{\beta}_{Wn}\right) \tag{9}\]  
where \(\hat{\beta}_{W}\) represents the fuzzy reference comparisons of all indicators over the worst indicator, \(\hat{\beta}_{BW}=\left(I_{jW},\ m_{jW},\ u_{jW}\right),j=1,2,\ldots,n\). **Step4. ** Calculating the optimal fuzzy weights vector \(\hat{\phi}^{*}=\left(\hat{\phi}^{*}_{1},\hat{\phi}^{*}_{2},\cdots,\hat{\phi} ^{*}_{n}\right)\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Five-Level Importance Scale** & **TFN** \\ \hline Identically Important (II) & \((1,1,1)\) \\ Slightly Important (SI) & \(\left(\frac{2}{3},1,\frac{3}{2}\right)\) \\ Relatively Important (RI) & \(\left(\frac{3}{2},2,\frac{5}{2}\right)\) \\ Highly Important (HI) & \(\left(\frac{5}{2},3,\frac{7}{2}\right)\) \\ Extremely Important (EI) & \(\left(\frac{7}{2},4,\frac{9}{2}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Five-level importance scale and its associated TFN [30]. The obtained weights are optimal when \(\frac{\hat{v}_{B}}{\hat{v}_{j}}=\hat{\beta}_{Bj}\) and \(\frac{\hat{v}_{j}}{\hat{v}_{B}}=\hat{\beta}_{BW}\). In addition, in order to obtain the optimal weights, it should minimize the maximum absolute differences of the \(\left|\frac{\hat{v}_{B}}{\hat{v}_{j}}-\hat{\beta}_{Bj}\right|\) and \(\left|\frac{\hat{v}_{j}}{\hat{v}_{W}}-\hat{\beta}_{BW}\right|\) (\(\hat{v}_{B}\), \(\hat{v}_{j}\) and \(\hat{v}_{W}\) are TFNs). In practical applications, it is often necessary to convert fuzzy weight vectors into crisp values for each indicator. Consequently, after obtaining the fuzzy weights represented by TFNs, the GMIR formula in Equation (7) can be applied to determine the optimal weights of the indicators. Equation (10) formulates the optimization problem designed to determine the fuzzy weight vectors of the indicators. \[