## 3 Integrated Disaster Resource Scheduling Modeling
### Constraints of Mobile Water Storage Vehicles  
Compared to the charge-discharge process of mobile energy storage vehicles, the loading and unloading of water can be completed relatively quickly. Without loss of generality, it is assumed that the total time required for water loading and unloading is \(1\) h.  
Figure 6 illustrates two typical scheduling modes for mobile water storage vehicles in a post-disaster emergency scenario. The horizontal axis represents discrete time steps \((t,t+1,t+2,\cdots,t+h,t+h+1,t+h+2)\), with the rows labeled \"Mode 1\" and \"Mode 2\" corresponding to different routing strategies. In the diagram, rectangular blocks filled in gray represent the transiting state (vehicles traveling between nodes), while those filled in orange represent the parking state (vehicles stopping for water loading/unloading). Mode 1 depicts continuous transit followed by a single parking interval, whereas Mode 2 features segmented transit interspersed with multiple parking events. The flexible scheduling framework enables mobile water storage vehicles to meet diverse water supply demands under dynamic emergency conditions. The complete mode of transportation (represented by the gray sections in Figure 6) reflects the long-duration operation of mobile water storage vehicles, primarily designed for rapid response to water resource demands in the target areas. In contrast, the segmented transportation and operation mode depicts scenarios where mobile water storage vehicles need to make multiple stops for water loading or unloading during transit. This mode enhances scheduling flexibility and improves the capability to meet multi-site water supply demands. To better characterize the scheduling process, the specific spatiotemporal transition constraints are as provided as follows:  
\[T_{m,n}(t)+T_{c}^{\text{load/unload}}\leq T_{\text{max}},\quad \forall m,n,c,t, \tag{40}\] \[T_{c}^{\text{load}}\leq T_{c}^{\text{transat}}(t+1)\leq T_{c}^{ \text{unload}}(t+2),\quad\forall t,c,\] (41) \[\sum_{m}p_{i,t}^{\text{priority}}\cdot T_{i,t}^{\text{priority}} \geq p_{j,t}^{\text{priority}},\quad\forall i,j,t,\text{where}p_{i,t}^{\text{ priority}}>p_{j,t}^{\text{priority}},\] (42) \[T_{c}^{\text{load/unload}}\leq h,\quad\forall t,c,\] (43) \[T_{m,n}(t)=\frac{D_{m,n}}{v_{ij}(t)},\quad\forall t,m,n, \tag{44}\]  
where Equation (40) limits the total operation time of the water storage vehicle \(c\) within a given time period \(t\); Equation (41) ensures that task execution follows a logical sequence, guaranteeing that loading, transportation, and unloading are carried out in the correct order; Equation (42) enforces a scheduling strategy that prioritizes high-priority tasks, i.e., or a given set of tasks \(i\) and \(j\) within the same time period \(t\), if task \(i\) has a higher priority than task \(j\), then the system must execute task \(i\) first; Equation (43) limits the total duration of loading and unloading tasks to not exceed the threshold \(h\); finally, Equation (44) ensures that the transportation time is dynamically adjusted based on the distance \(D_{m,n}\) and road conditions along the selected path. At the water storage stations, mobile water storage vehicles can transport water between reservoirs.