# Decision Support Model for Allocating Maintenance Budgets for Bridges
## 5 Example Project
### Discussion of the Results
The model after the simulation of uncertainty searches for the optimal solution considering the objective function which maximizes the priority sum, while total maintenance cost needs to be below or equal to the budget. The results show that considering the uncertainties results in yielding different solutions, with each solution constituting a maintenance plan. This means that the structural elements involved in each maintenance plan are different and thus the owner decision will involve an inherent risk in each solution. Therefore, the owner may wish to look for a more feasible solution than the optimal solution since the optimal solution has the most uncertainty and thus there is a risk at maximum compared with all solutions that satisfy the objective function and the constraint. The owner may look for a solution in which she/he can anticipate the final list of structural elements to be maintained with a certain level of confidence. In other words, the owner wants to select a maintenance plan by which she/he can maintain a maximum set of structural elements with the available budget without requiring an additional budget. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Bridge** & \multicolumn{2}{c}{**Bridge 1 (B1)**} & \multicolumn{2}{c}{**Bridge 2 (B2)**} & \multicolumn{2}{c}{**Bridge 3 (B3)**} & \multicolumn{2}{c}{**Bridge 4 (B4)**} \\ \hline Components & Weight & 0.540 & 0.445 & 0.546 & 0.646 \\ Bridge Deck and Parapet & 17.3\% & 0.439 & 0.118 & 0.618 & 0.272 \\ Bearing Pads & 20.7\% & 0.618 & 0.272 & 0.439 & 0.272 \\ Expansion Joint & 16.9\% & 0.272 & 0.439 & 0.618 & 0.439 \\ Abutment Walls & 13.6\% & 0.439 & 0.045 & 0.809 & 0.618 \\ Pier Columns & 16.4\% & 0.439 & 0.045 & 0.809 & 0.118 \\ Foundations & 15.1\% & 0.439 & 0.000 & 0.439 & 0.118 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Structural element priority of each bridge. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Recommended Maintenanceances** & **Preventive** & **Remedial/Curative** & **Rehabilitation** & **Replacement** \\ \hline Score & 0.5 & 0.75 & 1 & 0.25 \\ Condition State & More than 5 & 5 to 4 & 4 to 3 & Less than 3 \\ Utility Value & 0.272 or less & 0.272–0.439 & 0.439–0.618 & 0.618 or more \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Recommended maintenance strategies (plan). Efficient Frontier Analysis is used to assist the owner in judging the risk inherent in each solution (maintenance plan). The risk here is measured by the standard deviation of the sum of the priority for each solution in the maintenance plan. Efficient Frontier depicts the relation between the mean of the sum priority and the standard deviation for a set of optimal solutions (i.e., maintenance plans). Thus, each optimal maintenance plan has a sum of priority and a standard deviation which represents the level of risk inherent in that maintenance plan. Plans with higher standard deviation values means they are accompanied with higher potential of variation compared to the selected plan. In other words, the bridge management authority may end up maintaining different structural elements other than the selected structural elements in the originally selected plan. Figure 5 shows the outputs of the Efficient Frontier Analysis for the example project, while Figure 6 shows the model steady state after 630 simulations. Table 15 shows the selected bridges and structural elements in each solution presented in Figure 5. The optimal solution, highlighted in the table, consists of selecting 19 different bridge structural elements from five different bridges with a total maintenance cost of 7,987,228.42 SAR, which is very close to the total available budget. The most feasible solution, which is ranked (291),