## 2 Methodology
### Multi-Objective Scheduling Optimization Model
#### 2.2.2 Constraints  
In the multi-objective transportation scheduling model for open-pit mining electric swap trucks, several factors must be considered based on actual operating conditions, including the production demands, operational characteristics, and continuous operations. Particularly, the introduction of swap stations in the new transportation mode brings new constraints. Therefore, the model includes not only the crushing station consumption constraints and loading point production constraints but also additional constraints related to the new swap mode, such as battery selection constraints at the swap stations, the environmental impact on battery capacity constraints, swap station queue constraints, and swap speed constraints. Ensuring that these constraints are met is crucial for deriving the optimal scheduling solution in the following model optimization process:  
1. The constraint between the load of battery-swapping trucks and the consumption capacity of the crushing station, as follows: \[M_{j}^{\text{min}}\leq\sum_{r=1}^{k}\sum_{i=1}^{n}K_{r}\chi_{rij}\leq M_{j}^{ \text{max}},\ \forall j\in\{1,2,\ldots\text{m}\},\] (5) where \(M_{jmin}\) represents the production task requirement at the \(j\)-th crushing station and \(M_{jmax}\) represents the maximum consumption capacity of the \(j\)-th crushing station. The total amount of coal transported by all battery-swapping trucks to the \(j\)-th crushing station should be greater than or equal to the production task requirement and less than or equal to the maximum consumption capacity of the station. 2. The constraint between the load of battery-swapping trucks and the production capacity at the shovel point, as follows: \[N_{i}^{\text{min}}\leq\sum_{r=1}^{k}\sum_{j=1}^{m}K_{r}x_{rij}\leq N_{i}^{ \text{max}},\ \forall i\in\{1,2,\ldots\text{n}\},\] (6) where \(N_{min}\) represents the production task requirement at the \(i\)-th loading point and \(N_{\text{max}}\) represents the maximum production capacity at the \(i\)-th loading point. The total amount of coal transported by all battery-swapping trucks from the \(i\)-th loading point should be greater than or equal to the production task requirement and less than or equal to the maximum production capacity of the point. 3. The constraint on the number of available batteries at the battery-swapping station, as follows: \[Q_{e}\in\{0,1,2,\ldots\},\ \forall e\in\{1,2,\ldots\text{h}\},\] (7) The number of available batteries at each battery-swapping station is fixed. Each time the swapping station is used, the number of available batteries decreases by one. When no batteries are available, the truck must select the nearest alternative swapping station for the swapping service. 4. The constraint on the number of trips per transportation route in the open-pit mine, as follows: \[M_{j}^{\text{min}}\leq\sum_{r=1}^{k}\sum_{i=1}^{n}K_{r}x_{rij}\leq M_{j}^{ \text{max}},\ \forall j\in\{1,2,\ldots\text{m}\},\] (8) where \(M_{j}\) represents the production task requirement at the \(j\)-  
\[x_{rij},y_{rji},z_{rje},z_{rel}\in\{0,1,2,\ldots\},\ \forall r\in\{1,2,\ldots\}, \tag{8}\]  
In the equation, the constraint refers to the number of trips made by the battery-swapping trucks along the following four transportation routes: from loading points to crushing stations, from crushing stations to loading points, from crushing stations to swap stations, and from swap stations to loading points. The number of trips on each route must be a non-negative integer. (5) The constraint on the remaining battery level when a battery-swapping prompt is triggered, as follows:  
\[E_{r}^{start}-E_{1r}\sum_{i=1}^{n}\sum_{j=1}^{m}x_{rij}-E_{2r}\sum_{i=1}^{n} \sum_{j=1}^{m}y_{rji}\geq E_{r}*E_{\text{limit}},\ \forall r\in\{1,2,\ldots\text{k}\}, \tag{9}\]  
where \(E_{\text{{\it restart}}}\) is the initial battery level of the \(r\)-th battery-swapping truck, \(E_{r}\) is the rated battery capacity, and \(E_{\text{limit}}\) is the minimum battery-swap limit coefficient.