## 2 Methodology
### Multi-Objective Scheduling Optimization Model  
#### 2.2.1 Objective Functions  
Considering the efficiency and cost requirements of the new transportation mode for battery-swapping trucks in open-pit mines, this study constructs a multi-objective scheduling optimization model with the total waiting time and total transportation costs of battery-swapping trucks as the optimization objectives. The objective functions of the constructed multi-objective scheduling model are as follows:  
\[F(x)=Min[f_{1}(x),f_{2}(x)], \tag{1}\]  
where the objective function\(f_{1}(x)\) represents the transportation costs of the mining trucks and \(f_{2}(x)\) represents the queuing waiting time of the mining trucks. To balance the two objectives in the multi-objective function, we employ a weighted linear combination of the two objectives to adjust the weights. Five different weight distributions were ultimately set, ensuring that the constructed objective model achieves both stability and rationality. The objective function for the transportation costs of mining trucks is as follows. It includes the heavy-load cost from the loading point to the unloading point, the no-load cost from the unloading point to the loading point, the battery-swapping cost, and the maintenance cost. The comprehensive transportation cost of the mining truck within a single work shift is as follows:  
\[f_{1}(x)=\sum\limits_{r=1}^{k}\left( \begin{array}{c}\sum\limits_{i=1}^{n} \sum\limits_{j=1}^{m}x_{rij}d_{ij}\text{C}_{1r}+\sum\limits_{i=1}^{n}\sum \limits_{j=1}^{m}y_{rij}d_{ij}\text{C}_{2r}+\\ \sum\limits_{j=1}^{m}z_{rji}d_{ji}\text{C}_{2r}+\sum\limits_{i=1}^{n}z_{rei}d_ {ei}\text{C}_{2r}+T_{\text{limit}}\text{C}_{3r}\end{array} \right), \tag{2}\]  
where \(x_{rij}\) denotes the number of trips made by the \(r\)-th battery-swapping truck from loading point i to unloading point j, \(y_{rji}\) denotes the number of trips made by the r-th battery-swapping truck from unloading point \(j\) to loading point \(i\), \(d_{ij}\) is the distance from loading point \(i\) to unloading point j, \(d_{je}\) is the distance from unloading point \(j\) to the battery-swapping station, \(d_{ei}\) is the distance from the battery-swapping station to loading point \(i\), \(z_{rje}\) is the number of trips made by the \(r\)-th battery-swapping truck from unloading point \(j\) to the battery-swapping station, and \(z_{rei}\) is the number of trips made by the r-th battery-swapping truck from the battery-swapping station to loading point i. The objective function for the waiting time of battery-swapping trucks is defined as follows. The waiting time for each battery-swapping truck is obtained by subtracting the transportation time from the total shift time. The transportation time includes the heavy-load running time from the loading point to the unloading point, from the loading point to the battery-swapping station, and from the battery-swapping station to the unloading point, as well as the no-load running time from the unloading point to the loading point, from the unloading point to the battery-swapping station, and from the battery-swapping station to the loading point, along with the battery-swapping time:  
\[f_{2}(x)=\sum\limits_{r=1}^{k}\left( \begin{array}{c}\text{T}_{\text{limit} }-\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}T_{rij}x_{rij}-\sum\limits_{i=1}^ {n}\sum\limits_{j=1}^{m}T_{rji}y_{rji}-\\ \sum\limits_{j=1}^{m}T_{je}z_{rje}-\sum\limits_{i=1}^{n}T_{ei}z_{rei}-\sum \limits_{i=1}^{n}T_{E}z_{rei}\end{array} \right), \tag{3}\]  
\[ \begin{cases}T_{rij}=\frac{d_{ij}}{V_{1r}}+T_{1}\\ T_{rji}=\frac{d_{ij}}{V_{2r}}+T_{2}\\ T_{je}=\frac{d_{je}}{V_{2r}}\\ T_{ei}=\frac{d_{ij}}{V_{2r}}\end{cases} , \tag{4}\]Considering the efficiency and cost requirements of the new transportation mode for battery-swapping trucks in open-pit mines, this study constructs a multi-objective scheduling optimization model with the total waiting time and total transportation costs of battery-swapping trucks as the optimization objectives.