# Decision Support Model for Allocating Maintenance Budgets for Bridges
## 5 Example Project
### Discussion of the Results  
The first step is to conduct the prioritization for the four bridges in the network level to determine the overall score (rank) for each bridge. The rank will then be used in the second step as a weight for each bridge. To do so, the bridge's data shown in Table 6, along with the utility curves depicted in Figure 2, are used to convert the data into utility values from the curves. Alternatively, the trendline equation and the utility value of each criterion in each bridge is multiplied by the weight of the criterion and then the product is summed to find the overall score for each bridge. Table 9 summarizes the overall score/rank for each bridge. The priority of structural element is computed by multiplying the utility value of condition assessment for that structural element by the weight/rank of the bridge and by the weight of the structural element. The values for the structural elements Priority in Table 10 are multiplied by 1000 for a better presentation. An example calculation for Bridge Deck and Parapet priority is as follows:  
\[\text{Priority of Bridge Deck and Parapet for Bridge 1 is = 0.439 \times 0.540\times 17.3\%\times 1000=40.9}\]  
where 0.439 is the priority of Bridge 1 Deck and Parapet Bridge as presented in Table 10; 0.540 is priority score as presented in Table 10; 17.3% is the weight of Bridge Deck and Parapet as presented in Table 5. The higher value means higher priority. After that, the maintenance strategy weight was incorporated according to the condition state. Table 11 lists the recommended maintenance strategy that corresponds to the condition state of each structural element. It should be noted that these weights are assigned by the owner depending on their preferences. These recommended maintenance strategies are proposed by the bridge maintenance manual of MOMRA (MOMRA, 2014). Following that, the result, which is a list of the structural elements to be maintained with its maintenance cost and priority, is further used as an input to the optimization-based simulation module to look for the near optimum solution or the most feasible solution which considers the uncertainty in the main two variables of the model. Table 12 summarizes the list of structural elements after incorporating the maintenance strategy. The uncertainty \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Prioritization Factor** & **Weight** & **Bridge 1 (B1)** & **Bridge 2 (B2)** & **Bridge 3 (B3)** & **Bridge 4 (B4)** \\ \hline Condition assessment & 25.9\% & 0.438 & 0.067 & 0.637 & 0.297 \\ Age (Years) & 12.7\% & 0.377 & 0.193 & 0.628 & 0.723 \\ Location & 17.0\% & 0.750 & 0.750 & 0.500 & 1.000 \\ Previous of maintenance & 11.2\% & 0.225 & 0.000 & 0.391 & 0.065 \\ Scheduled maintenance & 11.9\% & 1.000 & 0.973 & 1.000 & 0.733 \\ Traffic volume (ADT) & 21.3\% & 0.500 & 0.750 & 0.250 & 1.000 \\ Overall score & & 0.540 & 0.445 & 0.546 & 0.646 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Overall score of each bridge. is examined in three different scenarios in which each case has different considerations. The results are compared to assess the effect of uncertainty on the model, along with the optimal solution and the final decision. The first scenario is a deterministic case in which no uncertainty is considered, and the estimated cost is considered as a crisp number. The second scenario considers uncertainty in the two variables of priority and estimated maintenance cost. The third scenario considers uncertainty in one variable, which is the priority as the main variable. Table 13 shows the model variables' definitions for the first scenario (deterministic). Table 14 summarizes the results of the three scenarios. It can be noticed that each scenario resulted in a different optimal solution. Owing to the characteristics of the variables, the probability distribution is assigned. The more suitable distribution requires gathering a considerable amount of data to create the proper probability distribution.