# The Condition Evaluation of Bridges Based on Fuzzy BWM and Fuzzy Comprehensive Evaluation
## 2 Fuzzy Best and Worst Method-Fuzzy Comprehensive Evaluation Model
### Fuzzy Set and Membership Function  
Fuzzy theory was pioneered by professor [PERSON], a distinguished American cybernetics expert, in 1965 [24]. Its primary objective is to provide an effective method for analyzing and addressing complex problems that are challenging to describe using traditional precise mathematical models. Fuzzy set theory, which forms the foundation and core of fuzzy theory, represents an extension of classical set theory. Specifically, fuzzy sets generalize the membership degree of elements to sets from a binary value (either 0 or 1) to a continuous interval (between 0 and 1). This generalization enables fuzzy sets to provide a more accurate and flexible representation of objects with ambiguous or uncertain characteristics. The mathematical formulation of a fuzzy set is expressed as Equation (1):  
\[\hat{P}=\big{\{}\big{(}\sigma,\mu_{\hat{P}}(\sigma)\big{)}\big{|}\sigma\in U \big{\}} \tag{1}\]  
where \(U\) represents the universe of discourse; \(\hat{P}\) represents the fuzzy set; \(\sigma\) signifies an element within the universe; and \(\mu_{\hat{P}}(\sigma)\) defines the membership function of \(\sigma\) with respect to \(\hat{P}\). Triangular and trapezoidal functions are the most widely utilized membership functions in fuzzy theory, with their mathematical expressions presented in Equation (2) and Equation (3), respectively. Triangular Function:  
\[\mu_{f}(t) \begin{cases}0_{i}&t\leq a_{i}\\ \frac{t-a_{i}}{b_{i}-a_{i}},&a_{i}\leq t\leq b_{i}\\ \frac{c_{i}-t}{c_{i}-b_{i}},&b_{i}\leq t\leq c_{i}\\ 0,&t\geq c_{i}\end{cases}  \tag{2}\]  
A triangular membership function \(\mu_{f}(t)\) is defined by three line segments forming either a symmetric or asymmetric triangle. The vertex of the triangle is located at coordinates (\(b_{i}\), 1), while the base endpoints are positioned at (\(a_{i}\), 0) and (\(c_{i}\), 0). Trapezoidal Function:  
\[\mu_{f}(t)= \begin{cases}0,&t\leq a_{i}\\ \frac{t-a_{i}}{b_{i}-a_{i}},&a_{i}\leq t\leq b_{i}\\ 1,&b_{i}\leq t\leq c_{i}\\ \frac{d_{i}-t}{d_{i}-c_{i}},&c_{i}\leq t\leq d_{i}\\ 0,&x\geq d_{i}\end{cases}  \tag{3}\]  
A trapezoidal membership function \(\mu_{f}(t)\) is characterized by four line segments forming a trapezoid. The top plateau spans the interval [\(b_{i}\), \(c_{i}\)], while the base endpoints are located at coordinates (\(a_{i}\), 0) and (\(d_{i}\), 0). A triangular fuzzy number (TFN) is a specialized type of fuzzy number derived from fuzzy set theory, which employs a triangular membership function to characterize the variation in an element's degree of membership within a set. The triangular structure of TFNs provides a precise and quantitative extension of the fuzzy set concept, making it particularly suitable for modeling fuzzy phenomena that exhibit a distinct central tendency and ambiguous boundaries. This characteristic renders TFNs a valuable and practical tool for addressing uncertainty in a wide range of applications. The TFN \(\hat{T}_{F}\) can be mathematically represented as \(\hat{T}_{F}=\left(\hat{t}_{F},\hat{m}_{F},\hat{a}_{F}\right)\), and its corresponding triangular membership function \(\mu_{T_{F}}(t)\) is illustrated in Figure 1. Where \(\hat{t}_{F}\leq\hat{m}_{F}\leq\hat{a}_{F}\), the parameters \(\hat{t}_{F}\), \(\hat{m}_{F}\) and \(\hat{a}_{F}\) denote the lower bound, the most likely value, and the upper bound of the \(\hat{T}_{F}\), respectively. Given two triangular fuzzy numbers \(\hat{T}_{F1}=\left(\hat{t}_{F1},\hat{m}_{F1},\hat{a}_{F1}\right)\) and \(\hat{T}_{F2}=\left(\hat{t}_{F2},\hat{m}_{F2},\hat{a}_{F2}\right)\), the fundamental principles of arithmetic operations on TFNs are demonstrated in Equations (4)-(6) [25].