# Knowledge-Based Fuzzy Expert System to the Condition Assessment of Historic Road Bridges
## 2 Materials and Methods
### Proposed Model to Historic Road Bridge Condition Rating  
Hereby, the model to historic road bridge condition rating is explained in ten steps, and its illustrative presentation is given in Figure 1.  
Figure 1: Model to historic road bridge condition rating. Step 1. Assigning a crisp grade to elements ranging from 0 to 100. The grading of bridge condition is set by the structural expert. Given the subjectivity that occurs when assessing the condition of the bridge, in order to achieve accuracy and objectivity, division into 11 fuzzy sets is proposed. According to the defined rating, the fuzzy sets within which the rating is located are singled out as follows:  
**Definition 3**. : _Let crisp value of grade x be a part of set S = {10, 20, 30, 40, 50, 60, 70, 80, 90}, then for each x is There is: \(\left(\widetilde{R}_{EL_{i}}\right)_{k}=\left(\left(\left(R_{EL_{i}}\right)_{k} \right)_{L^{\prime}}\,\left(\left(R_{EL_{i}}\right)_{k}\right)_{M^{\prime}}\, \left(\left(R_{EL_{i}}\right)_{k}\right)_{U}\right)\), \(k=1\), \(2\) and \(3\); \(i=1\),..., \(n\), where \(\left(\widetilde{R}_{EL_{i}}\right)_{k}\), \(k\)-th fuzzy set of \(i\)-th element. For each x \(\
otin\) S, there is:_  
Demonstrated fuzzy sets present all possible conditions of elements according to the expert's grade. Step 2. Assigning structural importance to elements with respect to all possible conditions given according to determined crisp grade. Importance are assigned by experts as follows:  
\[\left(\widetilde{w}_{EL_{i}}\right)_{j}=\left(\frac{1}{n}\sum_{j=1}^{n}\left( \left(w_{EL_{i}}\right)_{j}\right)_{L^{\prime}}\frac{1}{n}\sum_{j=1}^{n}\left( \left(w_{EL_{i}}\right)_{j}\right)_{M^{\prime}}\frac{1}{n}\sum_{j=1}^{n}\left( \left(w_{EL_{i}}\right)_{j}\right)_{U}\right)\text{, }i=1,\ldots,n,\]  
where \(\left(\widetilde{w}_{EL_{i}}\right)_{j}\) is a structural importance of \(i\)-th element given by \(j\)-th expert. Step 3. Determining the total number of combinations of all elements in all graded conditions defined by crisp values:  
\[N_{komb}=\prod_{i=1}^{n}\left(\left(\widetilde{R}_{EL_{i}}\right)_{k^{\prime} }\,\left(\widetilde{w}_{EL_{i}}\right)_{j}\right) \tag{18}\]  
Step 4. Determining the fuzzy grade of component by all combinations as follows:  
\[\left(\widetilde{R}_{KOMP}\right)_{l}=\prod_{i=1}^{n}\left(\widetilde{R}_{EL_{ i}}\right)_{k}\overset{\left(\widetilde{w}_{EL_{i}}\right)_{j}}{\overset{ \left(\widetilde{w}_{EL_{i}}\right)_{j}}{\overset{\left(\widetilde{w}_{EL_{i} }\right)_{j}}{\overset{\left(\widetilde{w}_{EL_{i}}\right)_{j}}{  
importance of the elements (\(A_{ij}\)) is obtained by comparing the element \(i\) with the element \(j\). The relationship \(\alpha_{ij}\) and the comparison matrix of the order \(p\times p\) can be represented as [51]:  
\[A_{p\times p}=\left[\alpha_{ij}\right]=\left[\frac{w_{i}}{w_{j}}\right],\ i,j=1,p \tag{26}\]  
Each data of matrix \([A]\) represents an expert judgment by comparison. The consistency of the matrix \([A]\) satisfies the condition:  
\[A_{ij}=A_{ik}A_{kj}\ \ \ i,j,k=1,2,\ldots,n \tag{27}\]  
The consistency of matrix has rank 1 with an eigenvector (\(\lambda\)) = \(n\). Then there is:  
\[[A]\langle w\rangle=n\langle w\rangle \tag{28}\]  
where \(w\) is eigenvector. In most cases, element comparisons are not ideal, i.e., the comparison matrix does not always have to achieve consistency. Expression (28) does not have to be satisfied for all possible assigned weights. The maximum eigenvalue of the matrix is close to \(n\) (number of elements), while the other values are close to 0. Therefore, in inconsistent cases it is necessary to determine the eigenvector corresponding to the largest eigenvalue \(\lambda_{max}\), i.e., the eigenvector \(w\) must be defined to satisfy the following:  
\[A\langle w\rangle=\lambda_{max}\langle w\rangle \tag{29}\]  
where \(\lambda_{max}\approx n\). The consistency of the matrix is achieved if the value of the consistency ratio (CR) is less than or equal to 0.1.