# The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation
## 2 Fuzzy Best and Worst Method-Fuzzy Comprehensive Evaluation Model
### Fuzzy Comprehensive Evaluation  
The FCE method, grounded in the membership degree theory of fuzzy set theory, transforms qualitative evaluations into quantitative values, enabling a comprehensive evaluation of objects influenced by multiple factors [33]. This method effectively processes fuzzy information, thereby improving the reliability of evaluations [34]. The procedural steps of the FCE method are outlined as follows:  
**Step1. ** Constructing the factor set:  
Figure 2: Flowchart of weight calculation using FBWM. It is assumed that \(O\) is a factor set consisting of all the indicators in each layer together, which can be expressed as \(O=\{o_{1},o_{2},\ldots,o_{m}\}\), where \(o_{i}\) (\(i=1,2,\ldots,m\)) is the \(i\)-th evaluation indicator and \(m\) is the number of evaluation indicators in the factor set. **Step2. ** Determining the evaluation set:  
The proposed evaluation set \(Z\) is the set of all possible evaluation results for the evaluation object. The set can be defined as \(Z=\{z_{1},z_{2},\cdots,z_{n}\}\), where \(z_{j}\) (\(j=1\), \(2\), \(\ldots\), \(n\)) denotes the grade corresponding to the evaluation grade of the object in question. Furthermore, to accurately differentiate between the evaluation grades, the value of \(z_{j}\) can be defined in ascending order from the smallest to the largest value, represented as \(1\), \(2\), \(\ldots\), \(n\). **Step3. ** Determining the weight set of the indicators:  
The weight \(v_{i}\), corresponding to the indicator \(o_{i}\), can be determined through historical data analysis, expert evaluation methods, or fuzzy methods. In this study, the weights were calculated using the FBWM, as detailed in Section 2.2. Consequently, the weight set for the indicators is expressed as \(V=\{v_{1},v_{2},\cdots,v_{m}\}\). **Step4. ** Constructing the membership matrix  
To determine the evaluation set for a single indicator, it is essential to define the membership degree from the factor set \(O\) to the evaluation set \(Z\). Assuming that the single indicator membership matrix can be represented as \(Q_{i}=\{q_{i1},q_{i2},\ldots,q_{in}\}\). Then, the membership matrix based on the entire single-indicator matrix could be established as follows:  
\[Q=\left(q_{ij}\right)_{m\times n}= \begin{bmatrix}q_{11}&q_{12}&\cdots&q_{1n} \\ q_{21}&q_{22}&\cdots&q_{2n}\\ &\cdots&\cdots\\ q_{m1}&q_{m2}&\cdots&q_{mn}\end{bmatrix}  \tag{13}\]  
where \(q_{ij}\) is the membership degree from \(o_{i}\) to \(z_{j}\). In this study, semi-trapezoidal and trapezoidal membership functions are used to calculate the membership degree of the indicators [35; 36], as illustrated by Equations (14)-(16). \[q_{z}(t)= \begin{cases}1,&t\leq a_{i}\\ \frac{b_{i}-t}{b_{i}-a_{i}},&a_{i}<t\leq b_{i}\\ 0,&x>b_{i}\end{cases}  \tag{14}\]  
\[q_{z}(t)= \begin{cases}0,&t\leq a_{i}\\ \frac{t-a_{i}}{b_{i}-a_{i}},&a_{i}\leq t<b_{i}\\ 1,&t\geq b_{i}\end{cases}  \tag{15}\]  
\[q_{z}(t)= \begin{cases}0,&t<a_{i}\\ \frac{t-a_{i}}{b_{i}-a_{i}},&a_{i}\leq t\leq b_{i}\\ 1,&b_{i}\leq t\leq c_{i}\\ \frac{d_{i}-t}{d_{i}-c_{i}},&c_{i}\leq t<d_{i}\\ 0,&t>d_{i}\end{cases}  \tag{16}\]  
The functional characteristics of the semi-trapezoidal membership function are illustrated in Figure 3a,b, whereas the trapezoidal membership function is depicted in Figure 4.  
**Step5. ** Calculating the evaluation results  
A comprehensive evaluation vector is constructed by applying fuzzy comprehensive operations to the weight set and the membership matrix:  
\[E=V\times Q \tag{17}\]where \(E\) is the current layer's comprehensive evaluation vector, \(V\) is the set of weights of the lower layer, and \(Q\) is the membership matrix of the lower layer. To determine the comprehensive evaluation result of the bridge's condition, Equation (18) is utilized to compute the comprehensive evaluation score:  
\[\overline{E}=\frac{\sum_{i=1}^{m}\sum_{j=1}^{n}V_{i}z_{j}}{\sum_{i=1}^{m}V_{i}} \tag{18}\]  
where \(\overline{E}\) represents the comprehensive evaluation score.