# Knowledge-Based Fuzzy Expert System to the Condition Assessment of Historic Road Bridges
## 2 Materials and Methods
### Fuzzy Logic and Fuzzy Weighted Geometric Mean-Theoretical Background  
A fuzzy set is a set of elements in a universal set whose boundaries are ambiguous, vagueness, and uncertain. Each element of the set is assigned a membership function \(\mu_{\widetilde{A}}(x)\), with the value within the interval \([0,1]\). The assigned value is called the degree of membership and it specifies the extent to which the element belongs to a particular fuzzy set. If the value is \(0\), then the element does not belong to the set, and if it is \(1\), then the element belongs entirely to the specified set. If the value \"lies\" within the interval \((0,1)\), the element partly belongs to the fuzzy set [33]. Therefore, each fuzzy set can be determined by a unique membership function. They can also be expressed as intervals called \(\alpha\)-cuts. **Definition 1**. : _Let \(\widetilde{A}\) be a fuzzy set within the universal set \(X\). Then the \(\alpha\)-cuts of \(\widetilde{A}\) are defined as:_  
\[A_{\alpha}=\left\{x\in X|_{\mu_{\widetilde{A}}}(x)\geq\alpha\right\}=\left[ \min\{x\in X|_{\mu_{\widetilde{A}}}(x)\geq\alpha\},\max\{x\in X|_{\mu_{ \widetilde{A}}}(x)\geq\alpha\}\right] \tag{1}\]  
According to [PERSON]'s additional principle [30], the fuzzy set \(\widetilde{A}\) can be represented as:  
\[\widetilde{A}=\cup_{\alpha}\alpha A_{\alpha},\ \ 0<\alpha\leq 1 \tag{2}\]  
Fuzzy numbers are main segments of fuzzy sets. A fuzzy number is a convex fuzzy set described by a given interval of real numbers, each with a degree of membership between \(0\) and \(1\). The membership functions of fuzzy numbers are partially continuous and satisfy the following conditions:  
1. \(\mu_{\widetilde{A}}(x)=0\) for each \(x\
otin[a,c]\);
2. \(\mu_{\widetilde{A}}(x)\) is non-decreasing (monotonically increasing) on the interval \([a,b]\) and non-increasing (monotonically decreasing) on the interval \([b,c]\);
3. \(\mu_{\widetilde{A}}(x)=1\) for each \(x\in b\),  
where \(a\leq b\leq c\) are real numbers R = \((-\infty,\ +\infty)\). Hereby, triangular fuzzy numbers are used, whose membership functions is defined as:  
\[\mu_{\widetilde{A}}(x)=\left\{ \begin{array}{ll}(x-a)/(b-a),&a\leq x\leq b \\ (c-x)/(c-b),&b\leq x\leq c\\ 0,&\text{other}\end{array} \right. \tag{3}\]  
They are often written as \((a,b,c)\). **Definition 2**. : _Let \(\widetilde{A}=(a_{1},\ a_{2},\ a_{3})\) and \(\widetilde{B}=(b_{1},\ b_{2},\ b_{3})\) be two positive triangular fuzzy numbers. Then the basic fuzzy operations are defined as [47]:__Addition:_\(\widetilde{A}+\widetilde{B}=(a_{1}+b_{1},\ a_{2}+b_{2},\ a_{3}+b_{3})\)_;  
_Subtraction:_\(\widetilde{A}-\widetilde{B}=(a_{1}-b_{3},\ a_{2}-b_{2},\ a_{3}-b_{1})\);  
_Multiplication:_\(\widetilde{A}\times\widetilde{B}\approx(a_{1}b_{1},\ a_{2}b_{2},\ a_{3}b_{3})\)_;  
_Division:_\(\widetilde{A}\div\widetilde{B}\approx\left(\frac{a_{1}}{b_{3}^{2}},\ \frac{a_{2}}{b_{2}^{2}},\ \frac{a_{3}}{b_{1}}\right)\)_._  
Fuzzy numbers often have to be transformed into a crisp value for the comparison or ranking. Such a transformation process is called defuzzification, which can be carried out in several ways. The most commonly used defuzzification process is the centroid method, also known as the center of gravity or the center of the surface of the defuzzification. The centroid method defines a centroid of fuzzy number \(\widetilde{A}\) as its defuzzified value, and according to [48] it is defined as follows:  
\[\overline{x_{0}}\ (\widetilde{A})=\ \frac{\int_{a}^{c}x\mu_{\widetilde{A}}(x) \mathrm{d}x}{\int_{a}^{c}\mu_{\widetilde{A}}(x)\mathrm{d}x} \tag{4}\]  
where \(\overline{x_{0}}(\widetilde{A})\) is a defuzzified value.