# MILP-Based Approach for High-Altitude Region Pavement Maintenance Decision Optimization
## 3 Mathematical Model
### Objective Function  
Based on previous research, it can be found that most studies adopt multi-objective optimization methods [3, 13], which means there are multiple objective functions. Therefore, we first construct different objective functions from various perspectives.  
1. Maximize maintenance effectiveness: The primary goal of pavement maintenance is to provide better driving conditions for as many users as possible. This means that the decision should consider not only the immediate effects of maintenance (i.e., how is the pavement after maintenance?) but also the overall effectiveness (i.e., how many users can use roads in better condition?). Therefore, this paper uses the product of traffic flow and pavement condition as a decision objective to make maintenance decisions more scientific. Multiplying pavement condition by traffic flow considersthat segments with higher traffic flow contribute more to overall effectiveness and, thus, are worth focusing on. This can be expressed as: \[\text{max}f_{1}=\sum_{m\in\mathbf{M}}\sum_{u\in\mathbf{U}}\sum_{i\in\mathbf{I}}s_{i,u}\times V _{i,u,m}\times T_{u,m}\] (15) 2. Minimize carbon emissions: For pavement maintenance, while improving pavement conditions, there may be environmental impacts, including greenhouse gases, noise, dust, etc. Compared with low-altitude regions, the ecological environment in high-altitude regions is more fragile. Therefore, when formulating pavement maintenance plans for high-altitude regions, these adverse impacts should be given greater consideration. Hence, this paper sets minimizing carbon emissions as an optimization objective, which can be expressed as: \[\text{min}f_{2}=\sum_{j\in I}\sum_{i\in\mathbf{I}}P_{j}\times L_{i}\times B_{i} \times x_{i,j}\] (16) 3. Minimize the affected traffic volume: During pavement maintenance, construction activities inevitably have adverse effects on traffic, especially when traffic flow is high. To reduce the interference of pavement maintenance on traffic, it is necessary to consider traffic management during construction and to schedule maintenance time reasonably. Based on this, this paper constructs an objective function to minimize the impact on traffic, combining the predicted monthly traffic volume, the working hours required for different maintenance methods, and the protection time required after maintenance, specifically expressed as: \[\text{min}f_{3}=\sum_{m\in\mathbf{M}}\sum_{u\in\mathbf{U}}\sum_{j\in I}\sum_{i\in\mathbf{ I}}\left[\left(\left\lceil\frac{G_{j}\times x_{i,j}\times L_{i}\times B_{i}}{Q \times O}\right\rceil+D_{j}\times x_{i,j}\right)\times V_{i,u,m}\times y_{i,u, m}\right]\] (17) 4. Minimizing the roughness index: The international roughness index (IRI) reflects the level of vibration and bumpiness experienced by vehicles traveling on the road [30; 32], indicating that roughness affects the comfort of the driving experience. Additionally, roughness has a significant impact on energy consumption, with higher IRI values corresponding to higher energy consumption [32]. Therefore, to meet comfort needs and environmental protection requirements, minimizing IRI value as much as possible can be set as an optimization objective: \[\text{min}f_{4}=\sum_{u\in\mathbf{U}}\sum_{i\in\mathbf{I}}r_{i,u}\] (18) 5. Minimize maintenance costs: Minimizing maintenance costs as an optimization objective is a common approach in pavement maintenance decision [3]. The calculation of maintenance costs is related to the maintenance approach and maintenance area, and the corresponding objective function can be expressed as: \[\text{min}f_{5}=\sum_{j\in I}\sum_{i\in I}C_{j}\times L_{i}\times B_{i}\times x _{i,j}\] (19)