## 5 Case Study
### Validation of Model-II: Annual Budget Allocation Model  
From the results of Model-I, the budget value required for the first year can be calculated, totaling NTD 12.6 million (USD 420 thousand), as shown in Table 11 below. Letter M stands for maintenance and E for effect period:  
Assuming that the actual budget is approved, and the amount of the actual budget is reduced to NTD 10 million, then the calculation results for Model-II will be the same as those shown in Table 12:  
It can be seen from the comparison between Tables 11 and 12, that the lack of an actual budget has caused some facilities to be unable to carry out maintenance work, as calculated using Model-I. Different maintenance plans have been chosen for a total of four facilities, two sup, seven sup, and eight sub, which should undergo maintenance. However, maintenance could not be carried out as scheduled due to an insufficient budget. Taking two sup as an example, the calculation results of Model-II show that the facility will continue to be used for 6 years before maintenance in plan three is carried out, and the effect period will be 6 years.  
It is assumed in the study that all bridges are not allowed to be in an unusable state. Thus, when the annual maintenance budget is insufficient for the facilities to be maintained as originally planned, the maintenance will have to be delayed. If the budget for the next year is again insufficient, the calculation using Model-II will be carried out again, and those that cannot be maintained will also be delayed. When the maintenance cannot be delayed any more in a certain year, the model will have no solution. The failure to obtain a solution is not an optimization failure but reflects that the annual budget must be higher than a certain value so that all facilities can be maintained. The next section will analyze such a situation.  
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Facility** & **1 sup** & **1 sub** & **2 sup** & **2 sub** & **3 sup** & **3 sub** & **4 sup** & **4 sub** \\ \hline state & E & E & M & E & E & E & M & E \\ \hline Cost \({}^{*}\) & 0.67 & 1 & 43 & 0.33 & 1.33 & 1.33 & 0.67 & 58 \\ \hline
**Facility** & **5 sup** & **5 sub** & **6 sup** & **6 sub** & **7 sup** & **7 sub** & **8 sup** & **8 sub** \\ \hline state & E & M & M & M & M & M & M & M \\ \hline Cost \({}^{*}\) & 0.33 & 3.67 & 7 & 8 & 3.33 & 3.33 & 2.67 & 3.33 \\ \hline \hline \end{tabular} \({}^{*}\) Unit: 1 thousand (USD).  
\end{table}
Table 11: Budget demand for the first year.  
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Facility** & **1 sup** & **1 sub** & **2 sup** & **2 sub** & **3 sup** & **3 sub** & **4 sup** & **4 sub** \\ \hline Current plan & 2 & 2 & 1 & 2 & 3 & 2 & 2 & 2 \\ \hline
1 st period plan & 1 & 1 & 3 & 1 & 1 & 2 & 1 & 3 \\ \hline Start time (year) & 1 & 2 & 6 & 1 & 6 & 5 & 5 & 0 \\ \hline
1 st Effect period & 15 & 15 & 6 & 15 & 15 & 15 & 11 & 8 \\ \hline
**Facility** & **5 sup** & **5 sub** & **6 sup** & **6 sub** & **7 sup** & **7 sub** & **8 sup** & **8 sub** \\ \hline Current plan & 2 & 3 & 1 & 1 & 3 & 2 & 2 & 2 \\ \hline
1 st period plan & 1 & 3 & 3 & 3 & 3 & 3 & 1 & 2 \\ \hline Start time (year) & 3 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \\ \hline
1 st Effect period & 10 & 6 & 8 & 10 & 8 & 9 & 14 & 11 \\ \hline \hline \end{tabular}
\end{table}
Table 12: Calculation results of Model-II.