# Knowledge-Based Fuzzy Expert System to the Condition Assessment of Historic Road Bridges
## 3 Results
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Element**} & \multirow{2}{*}{**Crisp Value of Rating**} & \multirow{2}{*}{**Fuzzy Value of Rating**} & \multicolumn{2}{c}{**Fuzzy Structural Importance by Each Expert (Average Value)**} \\ \cline{3-4}  & & & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} \\ \hline \multirow{2}{*}{Pavement} & \multirow{2}{*}{75} & 60, 70, 80 & 0.30; 0.55; 0.80 \\ \cline{3-4}  & & 70, 80, 90 & 0.10; 0.35; 0.60 \\ \hline \multirow{2}{*}{Fence} & \multirow{2}{*}{75} & 60, 70, 80 & 0.30; 0.55; 0.80 \\ \cline{3-4}  & & 70, 80, 90 & 0.15; 0.40; 0.65 \\ \hline \multirow{2}{*}{Drainage} & \multirow{2}{*}{75} & 60, 70, 80 & 0.30; 0.55; 0.80 \\ \cline{3-4}  & & 70, 80, 90 & 0.15; 0.40; 0.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Crisp and fuzzy values of ratings of the equipment component elements. Figure 10: Graphical representation of the equipment component rating values for all combinations. The defuzzification of rating values of equipment is obtained using the centroid method, and the final rating of the component is:  
\[\overline{x}_{0}(\widetilde{A})=\frac{1}{3}.(84.43+74.15+63.18)=73.92\]  
After the defuzzfied values of the ratings of the components are calculated, the relative importance (weights) of components are determined using the AHP method. [PERSON] et al. [52] used simplified AHP to evaluate the major rehabilitation strategies for the concrete bridge to maintain bridge assets within acceptable limits of safety, serviceability, and sustainability. Additionally, [PERSON] et al. [53] applied simplified AHP to the three-level hierarchy structure, identifying the main goal, criteria, subcriteria, and alternatives, for the remediation of the steel bridges. Hereby, the AHP method is used to the define relative importance of the bridge components according to their calculated ratings. This way, the impact of each component rating on the overall bridge condition is expressed. If the components, for example, all have the same rating each of them may not have the same impact, also if one component is rated higher than the other it still, according to an expert opinion, may have a negative impact on the overall bridge condition. First, a comparison matrix of components is established by an expert according to the Saaty's scale, as shown in Table 14. In Table 15, normalization and calculated final weights of components are presented. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\mathbf{\alpha}\)**-cut** & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1.0 \\ \hline \((R_{\text{K}\text{K}\text{O}MP})^{L}_{\text{a}}\) & 63.18 & 64.38 & 65.53 & 66.66 & 67.76 & 68.85 & 69.93 & 70.99 & 72.05 & 73.10 & 74.15 \\ \hline \(\mathbf{\alpha}\)**-cut** & 1.0 & 0.9 & 0.8 & 0.7 & 0.6 & 0.5 & 0.4 & 0.3 & 0.2 & 0.1 & 0.0 \\ \hline \((R_{\text{K}\text{O}MP})^{U}_{\text{a}}\) & 74.15 & 75.19 & 76.23 & 77.26 & 78.29 & 79.32 & 80.34 & 81.37 & 82.39 & 83.41 & 84.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: The average of all combinations—the final fuzzy value of the equipment component rating. Figure 11: Graphical representation of the membership function of equipment component final rating for all combinations. The last column in Table 15 represents the final weights of the components. After the assigned weights, the consistency of the matrix is checked as follows:  
\[\lambda_{max}=2.5\times 0.4+2.5\times 0.4+5\times 0.2=3\] \[\text{CI}=\frac{\lambda_{max}-n}{n-1}= \frac{3-3}{3-1}=0\] \[\text{RI}=0.52\text{ for }n=3\] \[\text{CR}=\frac{\text{CI}}{\text{RI}}=\frac{0}{0,52}=0\]  
The value of the consistency ratio is 0, and the consistency of the matrix is achieved, i.e., the weights are correctly assigned.