## 2 Methodology
### Multi-Objective Scheduling Optimization Model
#### 2.2.2 Constraints
A battery-swapping prompt is triggered when the remaining battery level falls below the minimum percentage of the battery capacity. The truck's initial battery level is reduced by the energy consumed during both heavy-load and no-load operations. If the remaining energy falls below the minimum requirement, a battery-swapping request is sent, and the truck proceeds to the nearest swapping station with available batteries. \[ \begin{cases}E_{1r}*x_{rij}\geq E_{r}^{now}-E_{r}*E_{\text{limit}},\ \forall r\in\{1,2,\ldots\text{k}\},\\ E_{2r}*y_{rji}\geq E_{r}^{now}-E_{r}*E_{\text{limit}},\ \forall r\in\{1,2, \ldots\text{k}\},\end{cases}  \tag{10}\]  
When the energy consumption from the loading point to the unloading point is greater than or equal to the maximum available energy of the mining truck (i.e., the current energy level of the truck minus the minimum energy requirement), a battery-swapping notification is triggered. Similarly, when the energy consumption from the unloading point to the loading point is greater than or equal to the maximum available energy of the mining truck, a battery-swapping notification is also triggered. (6) The constraint on the selection of the battery-swapping station, as follows:  
\[\sum_{\epsilon=1}^{h}E_{re}=1,\ \forall e\in\{1,2,\ldots\text{h}\}, \tag{11}\]  
Only one truck can receive swapping service at a swapping station at a time. When a battery-swapping truck arrives at the station, a mutex semaphore is set to 1, causing any other arriving trucks to wait. After the swapping is complete, the semaphore is reset to 0, allowing the station to provide swapping services to other trucks. (7) The constraint on battery capacity due to environmental temperature, as follows:  
\[E_{r}^{start}=E_{r}*[1-(t_{o}-t)*k_{o}],\ \forall r\in\{1,2,\ldots\text{k}\}, \tag{12}\]  
where t represents the mining area temperature, k0 is the fluctuation coefficient of the battery capacity with temperature, and t0 is the standard environmental temperature constant at the time of battery production. This constraint takes into account the effect of ambient temperature on the initial battery capacity, considering that battery performance significantly decreases in extremely cold climates. The nominal battery capacity is determined at 25 \({}^{\circ}\)C but fluctuates with changes in environmental temperature. (8) The constraint on battery charging time, as follows:  
\[\frac{E_{r}^{start}(1-E_{\text{limit}})}{E_{t}}\leq T_{o}\leq\frac{E_{r}(1- E_{\text{limit}})}{E_{t}},\ \forall r\in\{1,2,\ldots\text{k}\} \tag{13}\]  
\[\frac{E_{r}(1-E_{\text{limit}})}{E_{t}}\leq T_{o}\leq\frac{E_{r}^{start}(1- E_{\text{limit}})}{E_{t}},\ \forall r\in\{1,2,\ldots\text{k}\}, \tag{14}\]where \(E_{t}\) represents the standard charging speed. The constraint on the battery charging time at the swapping station ensures that, under normal conditions, the charging rate remains constant. Equation (13) applies to cold environments where the actual battery capacity is less than the rated capacity, meaning the charging time should be longer than the time required to fully charge the actual capacity but shorter than the time required to fully charge the rated capacity. Equation (14) applies to hot environments where the actual battery capacity exceeds the rated capacity, necessitating a shorter charging time than the time needed to fully charge the actual capacity but longer than the time required to fully charge the rated capacity. (9) The constraint on the production output of shovel points, as follows:  
\[\sum_{r=1}^{k}\sum_{j=1}^{m}K_{r}x_{r1j}\leq\cdots\leq\sum_{r=1}^{k}\sum_{j=1}^ {m}K_{r}x_{rij}\leq\cdots\leq\sum_{r=1}^{k}\sum_{j=1}^{m}K_{r}x_{mj},\ \forall i\in\{1,2,\ldots n\}, \tag{15}\]  
where \(K_{r}\) represents the load capacity of the \(r\)-th battery-swapping truck. This constraint applies to the production output at each shovel point, ensuring that the mining progress at different benches is consistent with actual production conditions.