# Payment Maintenance Decision Making Based on Optimization Models
## 3 Model Construction
### Maximum Maintenance Quality-Limited Budget Model  
To achieve optimal maintenance quality under a limited maintenance budget, a maximum maintenance quality-limited budget (MMQLB) model is built. The MMQLB model is a single-objective linear optimization model and is expressed as follows. \[maxZ=\sum_{i=1}^{n}\{PCI_{i}^{\prime}-PCI_{i}\}x_{i} \tag{1}\]  
\[s.t.\left\{ \begin{array}{l}\sum_{i=1}^{n}x_{i}Area_{i}{\cdot}P\leq Budget\\ x_{i}=1\qquad\forall PCI_{i}<D\end{array} \right. \tag{2}\]  
In these equations, \(x_{i}\) denotes the decision-making variable of the model, which represents whether the \(i\)th road requires maintenance. For each road, there exist two states in terms of whether maintenance is required. Setting \(x_{i}\) to 1 or 0 indicates that the road does or does not require maintenance at present, respectively. Equation (1) represents the objective function of this model, where \(n\) is the total number of roads and \(PCI_{i}^{\prime}-PCI_{i}\) represents the level of improvement in PCI (i.e., the maintenance quality) for a specific road. \(PCI_{i}^{\prime}\) indicates the PCI of the road after maintenance and is set to 100 in this model; in other words, the road is assumed to be in perfect condition after maintenance. In contrast, \(PCI_{i}\) represents the original PCI of the \(i\)th road before maintenance. Equation (2) represents the constraint condition of the model, where \(Area_{i}\) represents the area (m\({}^{2}\)) of the \(i\)th road that requires maintenance. \(P\) is a constant that represents the maintenance cost per square meter of the road and is set as 200 CNT/m\({}^{2}\). _Budget_ represents the total budget available for pavement maintenance, and \(D\) is a constant that represents the maintenance threshold of the road. When the PCI of a specific road drops below this threshold value, the road is judged to be of poor quality and require maintenance. Here, the default value of \(D\) is set to 70. \(PCI\) is selected as the parameter in the objective function because it is an important index for evaluating whether the road is in a satisfactory condition according to the Chinese \"technical code of maintenance for urban road\" [24]. In addition, \(PCI\) plays an important role in finalizing the maintenance scheme [25]. In this study, asphalt pavement is selected as the maintenance object. The equation for calculating the PCI of asphalt pavement is as follows:  
\[PCI=100-\sum_{i=1}^{n}\sum_{j=1}^{m}DP_{ij}\;\omega_{ij} \tag{3}\]  
In this equation, \(PCI\) ranges from 0 to 100; \(n\) is the total number of main deterioration types (here set as 4 for asphalt pavement exhibiting cracks, deformation, loosening, and other forms of deterioration); \(m\) is the total number of deterioration subtypes included in each main deterioration type; and \(DP_{ij}\) is the deduction caused by the \(j\)th subtype of deterioration in the \(i\)th main type. \(\omega_{ij}\) is the weight of the \(j\)th subtype of deterioration in the \(i\)th main type, which can be calculated by Equations (4) and (5) as follows:  
\[\omega_{ij}=3.0u_{ij}^{3}-5.5u_{ij}^{2}+3.5u_{ij} \tag{4}\]  
\[u_{ij}=\frac{DP_{ij}}{\sum_{i=1}^{m}DP_{ij}} \tag{5}\]  
When computing the \(DP_{ij}\) value, first, the actual deterioration density of a pavement, the ratio of the total area of the subtype j deterioration to that of the pavement, is calculated. The actual deterioration density is compared with the deterioration density of each deterioration subtype in Table 1, and \(DP_{ij}\) is found to be proportional to two numbers in two adjacent columns in Table 1. The \(u_{ij}\) is calculated by Equation (5) and w\({}_{ij}\) by Equation (4) based on the \(DP_{ij}\) in Table 2. For example, the PCI of AnQing Rd. in the research area is calculated as shown in Table 2. The paved area of AnQing road is about 26,204.1 m\({}^{2}\), with 1746.94 m of length and 4 lanes. The deterioration density is the ratio of the deteriorated area of each subtype and the paved area of AnQing road.