# Payment Maintenance Decision Making Based on Optimization Models
## 3 Model Construction
### Minimum Budget-Maximum Maintenance Quality Model  
Optimal maintenance of multiple roads should seek to minimize the maintenance cost while maximizing the maintenance quality. To this end, a multiobjective optimization model named the minimum budget-maximum maintenance quality (MBMMQ) model is constructed and expressed by the following equations:  
\[maxZ_{1}=\sum_{i=1}^{n}\left(PCI_{i}^{\prime}-PCI_{i}\right)x_{i} \tag{6}\]  
\[minZ_{2}=\sum_{i=1}^{n}x_{i}Area_{i}\cdot P \tag{7}\]  
\[s.t.\ x_{i}=1\ \forall PCI_{i}<D \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Deterioration Type**} & \multicolumn{6}{c}{**Deterioration**} \\ \cline{3-8} \multicolumn{2}{c}{} & **0.01** & **0.1** & **1** & **10** & **50** & **100** \\ \hline \multirow{3}{*}{Crack} & Slippage cracks & 3 & 5 & 8 & 16 & 38 & 48 \\  & Alligator cracks & 5 & 8 & 10 & 20 & 45 & 70 \\  & Spalling & 8 & 10 & 15 & 30 & 55 & 80 \\ \cline{2-8}  & Subsidence & 3 & 5 & 12 & 25 & 47 & 63 \\  & Rut & 2 & 7 & 12 & 25 & 45 & 55 \\  & Upheaval & 3 & 10 & 15 & 30 & 52 & 65 \\ \cline{2-8}  & Pit & 10 & 15 & 25 & 40 & 65 & 72 \\  & Edge failure & 2 & 4 & 8 & 15 & 30 & 40 \\  & Stripping & 2 & 5 & 8 & 15 & 35 & 45 \\ \cline{2-8}  & \multirow{2}{*}{Others} & Poor frame & 3 & 8 & 12 & 12 & 12 & 12 \\  & Damage of repaired section & 2 & 5 & 8 & 15 & 25 & 33 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Deductions from pavement condition index (PCI) associated with different subtypes of deterioration on asphalt pavement.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Deterioration Type**} & \multicolumn{2}{c}{**Deterioration**} & \multicolumn{2}{c}{**Deterioration**} \\ \multicolumn{2}{c}{} & **Area (m2)** & **Density (\%)** & \(DP_{ij}\) & \(u_{ij}\) & \(\omega_{ij}\) & \(\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}DP_{ij}\omega_{ij}\) \\ \hline \multirow{3}{*}{Crack} & Slippage cracks & 56 & 0.21 & 5.38 & 0.080 & 0.247 & 1.33 \\ \cline{2-8}  & Alligator cracks & 0 & 0 & 0.00 & 0.000 & 0.000 & 0.00 \\ \cline{2-8}  & Spalling & 300 & 1.14 & 15.24 & 0.228 & 0.547 & 8.34 \\ \hline \multirow{3}{*}{Deformation} & Subsidence & 100 & 0.38 & 11.11 & 0.166 & 0.443 & 4.92 \\ \cline{2-8}  & Rut & 0 & 0 & 0.00 & 0.000 & 0.000 & 0.00 \\ \cline{2-8}  & Upheaval & 0 & 0 & 0.00 & 0.000 & 0.000 & 0.00 \\ \hline \multirow{3}{*}{Loosening} & Pit & 60 & 0.23 & 16.43 & 0.245 & 0.572 & 9.40 \\ \cline{2-8}  & Edge failure & 0 & 0 & 0.00 & 0.000 & 0.000 & 0.00 \\ \cline{2-8}  & Stripping & 980 & 3.74 & 10.13 & 0.151 & 0.414 & 4.19 \\ \hline \multirow{3}{*}{Others} & Poor frame & 0 & 0 & 0.00 & 0.000 & 0.000 & 0.00 \\ \cline{2-8}  & Damage of repaired section & 488 & 1.86 & 8.67 & 0.129 & 0.367 & 3.19 \\ \hline \multicolumn{2}{c}{Sum} & 1984 & 66.96 & & & 31.37 \\ \hline \hline \end{tabular}
\end{table}
Table 2: An example of calculating PCI using AnQing Rd.  
In Equations (6) and (7), the function \(Z_{1}\) represents the maintenance quality, which equals the summation of improved PCI for all involved roads, whereas \(Z_{2}\) represents the total maintenance cost. In this model, \(PCI_{i}^{\prime}\) is set as 100 to indicate that roads are assumed to be in perfect condition after maintenance. \(PCI_{i}\) denotes the original PCI of the \(i\)th road, and \(x_{i}\) is the state variable, which indicates whether the \(i\)th road requires maintenance. The value of \(x_{i}\) can be set only as 1 or 0, which signifies that the road does or does not require maintenance, respectively. Finally, \(Area_{i}\) and \(P\) have the same definitions as those in Equation (2). In Equation (8), \(D\) is the maintenance threshold; when the PCI of a specific road drops below threshold value, it is compulsory to maintain that road. Here, \(D\) is set to 70.