# 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
To defuzzify triangular fuzzy number \(\widetilde{A}=(a,\ b,\ c)\) centroid expression is used as follows:  
\[\overline{x_{0}}\ (\widetilde{A})=\ \frac{a+b+c}{3} \tag{5}\]  
When fuzzy number \(\widetilde{A}\) is presented by \(\alpha\)-cuts, i.e., \(\widetilde{A}=\ \cup_{a}\alpha\cdot A_{\alpha}=\ \cup_{a}\alpha\cdot\left[(x_{L})_{ \alpha^{\prime}}(x_{U})_{a}\right]\), (\(0<\alpha\leq 1\)), its defuzzifyed centroid form can be expressed as follows [49]:  
\[\int_{a}^{c}\mu_{\widetilde{A}}(x)\mathrm{d}x=\frac{\left(x_{U}\right)_{a_{0} }-\left(x_{L}\right)_{a_{0}}}{2} \tag{6}\]  
\[\int_{a}^{c}x\mu_{\widetilde{A}}(x)\mathrm{d}x=\frac{\left(x_{U}\right)^{2}_{ a_{0}}-\left(x_{L}\right)^{2}_{a_{0}}+\left(x_{U}\right)_{a_{i}}\cdot\left(x_{U} \right)_{a_{i+1}}-\left(x_{L}\right)_{a_{i}}\cdot\left(x_{L}\right)_{a_{i+1}} }{6} \tag{7}\]  
\[\overline{x}_{0}(\widetilde{A})=\frac{\left(x_{U}\right)^{2}_{a_{0}}-\left(x _{L}\right)^{2}_{a_{0}}+\left(x_{U}\right)_{a_{i}}\cdot\left(x_{U}\right)_{a_ {n}}-\left(x_{L}\right)_{a_{i}}\cdot\left(x_{L}\right)_{a_{n}}}{3\left(\left( x_{U}\right)_{a_{0}}-\left(x_{L}\right)_{a_{0}}\right)}=\frac{\left(x_{U} \right)_{a_{0}}+\left(x_{U}\right)_{a_{n}}+\left(x_{L}\right)_{a_{0}}}{3} \tag{8}\]  
Expression (8) represents the defuzzification by the centroid method for a triangular fuzzy number. The average weight of \(n\) fuzzy numbers has so far been obtained using the fuzzy weighted average (FWA) when assessing the condition of concrete bridges [33]. To assess the condition of historic road bridges, the FWGM approach will be used for \(n\) fuzzy numbers expressed as follows:  
\[ \begin{array}{l}\widetilde{y_{G}}{=f_{G}(\widetilde{x_{1}},\ldots, \widetilde{x_{n}};\ \widetilde{w_{1}},\ldots,\widetilde{w_{n}})}\\ {=(\widetilde{x_{1}})^{\frac{\widetilde{w_{1}}}{\widetilde{w_{1}+\widetilde{w _{2}}}}}_{\widetilde{w_{1}+\widetilde{w_{2}}}}(\widetilde{x_{2}})^{\frac{ \widetilde{w_{2}}}{\widetilde{w_{1}+\widetilde{w_{2}}}-\widetilde{w_{n}}}} \ldots(\widetilde{x_{n}})^{\frac{\widetilde{w_{n}}}{\widetilde{w_{1}+ \widetilde{w_{2}}}}}_{\widetilde{w_{1}+\widetilde{w_{2}}}_{\widetilde{w_{2}}}_{ \widetilde{w_{1}}+\widetilde{w_{2}}}_{\widetilde{w_{1}}}+\widetilde{w_{n}}}}\\ {=\prod_{i=1}^{n}(\widetilde{x_{i}})^{\frac{\widetilde{w_{i}}}{\widetilde{w_{ 1}}}}_{\widetilde{w_{i}}}}\end{array}  \tag{9}\]  
where \(\widetilde{x_{1}},\ldots,\widetilde{x_{n}}\)\(n\) are positive fuzzy numbers, and \(\widetilde{w_{1}},\ldots,\widetilde{w_{n}}\) are their fuzzy weights. It is obvious that \(\widetilde{y_{G}}\) is also fuzzy number and it can be calculated using \(\alpha-\) cuts and [PERSON]'s additional principle. Let \(\left(y_{G}\right)_{a}=\left[\left(y_{L}\right)_{a^{\prime}}\ \left(y_{U}\right)_{a}\right]\), where \(\alpha-\)cut is a set of \(\widetilde{y_{G}}\), then lower and upper value of set can be determined according to following mathematical models:  
\[ \begin{array}{l}\left(y_{L}\right)_{a}=\text{Min}\ \prod\limits_{i=1}^{n}(x_{i})^{\frac{w_{i}}{ \widetilde{w_{1}}-\widetilde{w_{1}}}}\\ {\left[\left(w_{L}\right)_{i}\right]_{a}\approx w_{i}\leq\left[\left(w_{U} \right)_{i}\right]_{a^{\prime}},\ \ i=1,\ldots,n}\\ {\left[\left(x_{L}\right)_{i}\right]_{a}\leq x_{i}\leq\left[\left(x_{U} \right)_{i}\right]_{a^{\prime}},\ \ \ i=1,\ldots,n}\end{array}  \tag{10}\]\[ \begin{array}{c}\left(y_{U}\right)_{\alpha}=\text{Max}\ \prod\limits_{i=1}^{n}\left(x_{i}\right)^{\frac{w_{i}}{\sum_{i=1}^{n}w_{i}}}\\ \left[\left(w_{L}\right)_{i}\right]_{\alpha}\leq w_{i}\leq\left[\left(w_{U} \right)_{i}\right]_{\alpha},\quad i=1,\ldots,n\\ \left[\left(x_{L}\right)_{i}\right]_{\alpha}\leq x_{i}\leq\left[\left(x_{U} \right)_{i}\right]_{\alpha},\quad i=1,\ldots,n\end{array}  \tag{11}\]  
Considering that:  
\[f_{\text{G}}(x_{1},\ldots,x_{n};\ w_{1},\ldots,w_{n})=\prod\limits_{i=1}^{n}( x_{i})^{\frac{w_{i}}{\sum_{i=1}^{n}w_{i}}} \tag{12}\]  
is increasing function of variables \(x_{i}(i=1,\ldots,n)\), the above mathematical models can therefore be expressed as:  
\[