## 3 Method
### Analytical Hierarchy Process (AHP)
After this step, the values of Table 6 were entered in Expert Choice software, and the relative weights of all eight indexes were calculated (Table 7). As indicated in Table 7, the relative weight of indices was calculated. This table contains additional information called compatibility rating. It is the mechanism that determines the adaptability of comparisons, indicating the extent to which the priorities selected by the group or the priorities of the mixed table are reliable. According to the experience, if the compatibility rate is less than 0.1, the adaptability of comparisons is acceptable; otherwise, the comparisons need be repeated [46]. The next step is to examine all indices in detail. In Figure 3, each index is divided into some sub-indices. In this step, the relative importance/weight of all sub-indices must be determined. Figure 3 shows that the structural index consists of four sub-indices. These four sub-indices have various relative importance. Therefore, Table A2 in the questionnaire \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} & \multicolumn{1}{c}{**Index**} \\ \hline Structural & 1 & 5.271 & 3.152 & 4.581 & 1.877 & 3.13 & 6.075 & 3.578 \\ Hydrology and Climate & & 1 & 1.037 & 1.382 & 0.788 & 1.377 & 1.871 & 0.941 \\ Safety & & & 1 & 2.613 & 1.633 & 1.489 & 3.318 & 2.074 \\ Bridge Performance (load impact) & & & & 1 & 0.761 & 1.164 & 2 & 1.154 \\ Geotechnical and Seismic & & & & & 1 & 2.859 & 3.133 & 2.216 \\ Strategic Importance & & & & & & 1 & 2.766 & 1.75 \\ Facilities & & & & & & & 1 & 0.975 \\ Traffic and Pavement & & & & & & & & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Relative weights of indices and compatibility rating. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline Relative weight & 0.331 & 0.097 & 0.146 & 0.080 & 0.143 & 0.088 & 0.046 & 0.068 \\ compatibility rating & & & & & 0.03 & & \\ \hline \hline Relative weight & 0.331 & 0.097 & 0.146 & 0.080 & 0.143 & 0.088 & 0.046 & 0.068 \\ compatibility rating & & & & & 0.03 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Relative weight of indices based on a survey of experts. was assigned to the relative importance of these four sub-indices. In other indices of Figure 3, all sub-indices have the same importance/weight. By determining the relative importance/weight of all indices and sub-indices, it is time to determine the score of each sub-index in various conditions. Tables A3-A10 were designed for this purpose. In these tables, experts rated sub-indices in different states. Considering the given explanations, they filled the blank cells of the tables with a number in the range of 0 to 100. Details of this step are presented in the next sections of the paper. Figure 4 depicts the flow chart of AHP in this study. Now, the BCI can be calculated. For each bridge, first, the score of each sub-index is determined based on the bridge inspection. After that, the score of each index is computed as the sum of weighted scores of its sub-indices. Finally, BCI can be determined based on Equation (13):  
\[\text{BCI}=\sum\left(\text{X}_{\text{i}}\times\text{W}_{\text{i}}\right)\text{, i = 1, \ldots,8} \tag{13}\]  
where BCI is the bridge condition index, X\({}_{\text{i}}\) is the score of ith index, and W\({}_{\text{i}}\) is the weight of ith index.