# MILP-Based Approach for High-Altitude Region Pavement Maintenance Decision Optimization
## 3 Mathematical Model
### Constraints  
For pavement maintenance decisions, the optimization is carried out under certain constraints and decision objectives. Based on previous research and the context of pavement maintenance in high-altitude regions, the constraints considered in this paper are as follows. 1. Calculation of the pavement condition index (PCI): The value of PCI is dynamic and influenced by factors such as time, pavement loads, and maintenance work. It ranges from 0 to 100, with higher values indicating better pavement conditions [13]. Let \(u*\) represent the year in which maintenance may be performed, and its calculation formula can be expressed as a piecewise function, that is: \[s_{i,u}=\left\{ \begin{array}{l}S_{i,0}\times e^{u_{i,1}\times u +\beta_{i,1}}+\sum\limits_{m\in\mathbf{M}}\sum\limits_{u*\in\{u*\}|u*\leq u\} \sum\limits_{j\in\mathcal{I}}x_{i,j}\times E_{j}\times y_{i,u*,m}\times e^{u_{ i,1}\times(u-u*)+\beta_{i,1}},\;s_{i,u}\leq S_{\max}\\ S_{\max},\;s_{i,u}>S_{\max}\end{array} \right.,\forall i\in\mathbf{I},u\in\mathbf{U}\] (1)2. Requirement for PCI value: For pavement technical conditions, different decision-makers have similar requirements, especially desiring that the PCI values of each segment meet the requirement of the minimum value, which can be expressed as: \[s_{i,u}\geq S_{\min},\ \forall i\in\mathbf{I},u\in\mathbf{U}\] (2) Furthermore, certain standards impose requirements on pavement conditions at the network level. For instance, the \"Technical Standard of Highway Maintenance (JTG 5110-2023)\" issued by the Ministry of Transport of China stipulates that the segments meeting certain pavement quality index (PQI) requirements should account for a corresponding proportion in a road network [30]. Based on this, this paper further imposes requirements on PCI at the network level to ensure the overall performance of a network, thereby guaranteeing overall service levels and user experience. Specifically, it is required that the average PCI value of all segments in the network should not be lower than a certain threshold, which can be expressed by: \[\sum_{i\in I}s_{i,u}\times L_{i}\geq S_{ave}\times\sum_{i\in I}L_{i},\ \forall u\in\mathbf{U}\] (3) Calculation of the international roughness index (IRI): IRI is an index used to standardize the evaluation of pavement smoothness, reflecting the level of vibration and bumpiness when traveling on a road [31; 32]. Clearly, the better the smoothness, the smaller the IRI, resulting in a better driving experience and a positive impact on traffic safety. Previous studies have shown that IRI is closely related to PCI, indicating that pavement condition affects smoothness. Based on Ref. [31], the relationship between the two can be expressed as: \[r_{i,u}=\alpha_{i,2}\times e^{\beta_{i,2}\times s_{i,u}}\] (4) Requirement for working time: In high-altitude regions, maintenance conditions are generally unfavorable during cold seasons, which tend to last relatively long. Therefore, this paper considers the impact of seasons on maintenance work. Based on this, some months are set as non-maintenance periods, and the corresponding constraint can be expressed as: \[y_{i,u,m}\geq 0,\ \forall i\in\mathbf{I},u\in\mathbf{U},m\in\mathbf{M}_{s}\] (5) \[y_{i,u,m}=0,\ \forall i\in\mathbf{I},u\in\mathbf{U},m\in\mathbf{M}_{u}\] (6)
5. Restricted monthly maintenance task: When the number of maintenance workers is limited, the maintenance task that can be completed within a unit of time is also restricted. When scheduling maintenance projects, it is necessary to consider manpower and time requirements. Combining the ceiling function \(\lceil\ \rceil\), the man-hours for any segment, and the total number of maintenance workers, the monthly maintenance task can be expressed as: \[\sum_{j\in I}\sum_{i\in I}\bigg{(}\bigg{\lceil}\frac{G_{j}\times x_{ij}\times L _{i}\times B_{i}}{Q\times O}\bigg{\rceil}\times y_{i,u,m}\bigg{)}\leq T_{u,m},\ \forall u\in\mathbf{U},m\in\mathbf{M}\] (7) Based on the above constraints, we also restrict any single maintenance task to be completed within one month.