# Knowledge-Based Fuzzy Expert System to the Condition Assessment of Historic Road Bridges
## 3 Results
The input data for the substructure component are given in Table 10.  
The number of obtained combinations based on the total number of elements and fuzzy sets to which the crisp value of the element rating belongs for the substructure component is the matrix \(N_{k}\) = [12 \(\times\) 3]. Fuzzy values of the element ratings of the substructure for all combinations are shown in Figure 8.  
The final fuzzy value of the component rating is shown in Table 11, whit the lower and upper boundaries of the \(\alpha\)-cuts are given, and in Figure 9, a graphical representation of the membership function with all its \(\alpha\)-cuts is given. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\boldsymbol{\alpha}\)-cut & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\ \hline \((R_{KOMP})_{\boldsymbol{a}}^{L}\) & 61.28 & 62.47 & 63.63 & 64.77 & 65.89 & 66.99 & 68.08 & 69.17 & 70.24 & 71.31 & 72.37 \\ \hline \(\boldsymbol{\alpha}\)-cut & 1.0 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 & 0.4 & 0.3 & 0.2 & 0.1 & 0.0 \\ \hline \((R_{KOMP})_{\boldsymbol{a}}^{\boldsymbol{U}}\) & 72.37 & 73.46 & 74.54 & 75.62 & 76.70 & 77.78 & 78.85 & 79.92 & 80.98 & 82.04 & 83.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The average of all combinations—the final fuzzy value of the superstructure component rating. Figure 7: Graphical representation of the membership function of superstructure component final rating for all combinations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element** & **Crisp Value of Rating** & **Fuzzy Value of Rating** & **Fuzzy Structural Importance by Each Expert (Average Value)** \\ \hline \multirow{3}{*}{Pier} & \multirow{3}{*}{70} & 50, 60, 70 & 0.65; 0.90; 1.00 \\ \cline{2-3}  & & 60, 70, 80 & 0.30; 0.55; 0.80 \\ \cline{2-3}  & & 70, 80, 90 & 0.10; 0.35; 0.60 \\ \hline \multirow{3}{*}{Abutment 1} & \multirow{3}{*}{75} & 60, 70, 80 & 0.30; 0.55; 0.80 \\ \cline{2-3}  & & 70, 80, 90 & 0.10; 0.35; 0.60 \\ \hline \multirow{3}{*}{Abutment 2} & \multirow{3}{*}{75} & 60, 70, 80 & 0.30; 0.55; 0.80 \\ \cline{2-3}  & & 70, 80, 90 & 0.10; 0.35; 0.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Crisp and fuzzy values of ratings of the substructure component elements. Figure 8: Graphical representation of the substructure component rating values for all combinations. Figure 9: Graphical representation of the membership function of substructure component final rating for all combinations. The defuzzification of rating values of substructure is obtained using the centroid method, and the final rating of the component is:  
\[\overline{x}_{0}(\widetilde{A})=\frac{1}{3}(82.30+71.63+60.24)=71.39\]  
Elements of the equipment component are curtain, fence, drainage and lighting. As there is no lighting for the observed bridge, it could not be assessed. The input data for the equipment component are given in Table 12.  
The number of obtained combinations based on the total number of elements and fuzzy sets to which the crisp value of the rating of an element belongs for the equipment component is a matrix \(N_{k}=[8\times 3]\). Fuzzy values of ratings of equipment elements for all combinations are shown in Figure 10.  
The final fuzzy value of the equipment component rating is shown in Table 13 with the lower and upper boundaries of the \(\alpha\)-cuts, and in Figure 11, a graphical representation of the membership function with all \(\alpha\)-cuts is given.