## 4 Modeling Methodology  
The first step of the AHP is to structure the problem with respect to its criteria and goals. For this research, the criteria were the crash types and reasons, while the goal was the crash severity. The structure can be observed from Figures 4 and 5; the process was applied exclusively for the crash type and then for crash reason. The ranking for each criterion and goal were calculated, as shown by Equations (4) and (5), while the total weight for each crash severity was calculated as per Equation (6). As stated earlier, this research employed the PBAHP method for the analysis. The difference between this method and traditional AHP is the use of proportions for calculating the ranking of the factors (crash types and reasons, in this case), as shown in Equations (4) and (5). The traditional AHP method uses a survey or interview and asks the respondents to carry out a pairwise comparison [39]. To achieve this, the respondent hypothetically generates the ratio of importance of each factor while answering the comparison questions. The same approach can be applied to quantitative data, such as crash data, by applying a method to evaluate the relative importance of one event over the other, which can be used in place of the ratings given by respondents in the traditional AHP method. The present study used a simple and efficient method to evaluate the relative importance of each crash event (type or reason) using the proportion of its occurrence in the crash data. This method makes use of the available data without the risk of receiving biased responses and does not require complicated or lengthy preprocessing for using the data. In the present research, these rankings were used to determine the relative weights (based upon their occurrence) for the crash severity. Higher rankings/ratios indicated a higher proportion of certain crash types or reasons in comparison to the other, which shows their comparatively higher importance or significance for crash severity. For example, if the ratio is 3 between crash type A and B, then this indicates that type A has 3 times more  
Figure 5: Hierarchy structure for crash reasons. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Crash Types** & **Number of** & **Frequency of** & **Crash Reasons** & **Number of** & **Frequency of** \\  & **Crashes** & **Crashes** & **Crashes** & **Crashes** & **Crashes** \\ \hline Undefined category & 37 & 0.92\% & Drifting & 5 & 0.12\% \\ Hit fixed object & 34 & 0.85\% & Falling asleep & 2 & 0.05\% \\ Hit tree & 25 & 0.62\% & Getting out of moving vehicle & 2 & 0.05\% \\ Fell off the slope & 7 & 0.17\% & Hanging on the outside of vehicle & 2 & 0.05\% \\ Hit plate & 4 & 0.10\% & Unsafe road works & 2 & 0.05\% \\ Fell off bridge & 4 & 0.10\% & Exhaustion & 1 & 0.02\% \\ Fire on vehicle & 3 & 0.07\% & Violating pedestrian sign & 1 & 0.02\% \\ Hit signal & 2 & 0.05\% & Downhill & 1 & 0.02\% \\ Hit waste container & 0 & 0.00\% & No warning signs & 1 & 0.02\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._importance (weightage) than type B, and vice versa, in the available dataset. This is further explained in Section 5. Rij = Pi/Pj  
where  
Rij = Ranking of reason/type 'i' in comparison to \"i\";  
Pi = Proportion of crashes by reason/type \"i\";  
Pj = Proportion of crashes by reason/type \"j\". Rxyi = Pxi/Pyi  
where  
Rxyi = Ranking of crash severity \"x\" over \"y\" based on crash type/reason \"i\";  
Pxi = Proportion of crashes belonging to severity \"x\" out of those caused by crash type/reason \"i\";  
Pyi = Proportion of crashes belonging to severity \"y\" out of those caused by crash type/reason \"i\". The total weight for each crash severity was the weighted sum of probability of each crash severity based on the contributing factor, as shown in Equation (6). This was used to develop the utility function for the crash severities.