## 2 Problem Statement and Model Formulation
### Time-Discretized Multi-Commodity Flow Model  
In this section, we build a time-discrete multi-commodity flow model (denoted as M2) for multi-task ISRPTWC. The decision variable of the M2 model is defined as \(y(d,i,j,s,e,w,w^{\prime})\) which takes the value one when a task \(d\) is transmitted using the link \((i,j,s,e,w,w^{\prime})\) and zero otherwise. The M2 model is as follows:  
\[Min\ Z=\sum_{d\in D}\frac{1}{Pri_{d}}\sum_{(i,j,s,e,w,w^{\prime})\in D_{d}}c \big{(}d,i,j,s,e,w,w^{\prime})y\big{(}d,i,j,s,e,w,w^{\prime}\big{)} \tag{13}\]  
\[\sum_{(i,j,s,e,w,w^{\prime})\in D_{d}}y\big{(}d,i,j,s,e,w,w^{\prime}\big{)}=1, \quad(i,s)=st_{i},\ w=w_{s},\forall d\in D \tag{14}\]  
\[\sum_{(i,j,s,e,w,w^{\prime})\in D_{d}}y\big{(}d,i,j,s,e,w,w^{\prime}\big{)}=1, \quad(j,e)=ar_{i},\ w=w_{e},\forall d\in D \tag{15}\]  
\[\sum_{(j,e,w^{\prime})}y\big{(}d,i,j,s,e,w,w^{\prime}\big{)}-\sum_{(i^{\prime },s^{\prime},w^{\prime\prime})}y\big{(}d,i^{\prime},i,s^{\prime},s,w^{\prime \prime},w\big{)}=0, \tag{16}\]  
\[\sum_{d\in D}\sum_{(i,j,s,e,w,w^{\prime})\in\Phi(j,e,w^{\prime})}y\big{(}d,i,j,s,e,w,w^{\prime}\big{)}\leq Cap_{(j,e,w^{\prime})} \tag{17}\]  
\[\sum_{d\in D}y\big{(}d,i,j,s,e,w,w^{\prime}\big{)}\leq Cap_{(i,j,s,e,w,w^{ \prime})} \tag{18}\]  
\[y\big{(}d,i,j,s,e,w,w^{\prime}\big{)}\in\{0,1\}\quad\forall\big{(}i,j,s,e,w,w ^{\prime}\big{)}\in D_{d},\forall d\in D \tag{19}\]  
The objective function (13) is to minimize the total transmission cost. Constraints (14), (15), and (16) represent flow balance constraints in the time-space state network. Constraint (17) represents the node capacity constraint, and (18) represents the link capacity constraint. Constraint (19) indicates that the decision variables are binary.  
Intuitively, we can find that the time-dependent constraints are dissipated in the M2 model because they can be ensured not to be violated in the time-space state network. As shown in Table 4, the time-discrete multi-commodity flow model (M2) contains all constraints of the integer linear programming model (M1).  
\begin{table}
\begin{tabular}{c c} \hline
**M1 Model** & **M2 Model** \\ \hline The decision variable is \(x_{ij}^{d}\). & The decision variable is \(y(d,i,j,s,e,w,w^{\prime})\). \\ The objective function (M1-1) is the minimum transmission cost. & The objective function (M2-13) is the minimum transmission cost. \\ (M1-2) and (M1-3) are to ensure that each task start node and end node are matched. & (M2-14) and (M2-16) are to ensure that each task start node and end node are matched. \\ (M1-4) is to ensure that each task is transmitted. & (M2-14) is to ensure that each task is transmitted. \\ (M1-4) and (M1-6) are to ensure the flow balance constraint of the network. & (M2-14) and (M2-16) are to ensure the flow balance constraint of the network \\ (M1-6) represents that each task is transmitted to the termination node. & (M2-16) represents that each task is transmitted to the termination node. \\ (M1-7) represents each task must be transmitted to the terminating node within the maximum transmission time. & Defined task transmission time using time-space state network. \\ (M1-8) represents the capacity of each node to be less than the maximum capacity. & (M2-17) represents the capacity of each node to be less than the maximum capacity. \\ Dissipating the maximum congestion time constraint according to the task state in the time-space state network (Theorem 1). & Dissipating the maximum congestion time constraint according to the task state in the time-space state network (Theorem 1). \\ (M2-18) represents the capacity of each link to be less than the maximum capacity. & (M2-18) represents the capacity of each link to be less than the maximum capacity. \\ Time-space state network structure ensures a set of & Time-space state network structure ensures a set of \\ feasible links. & (M2-19) defines the range of values of the decision variables. \\ \hline \end{tabular}
\end{table}
Table 4: Comparison of M1 model and M2 model constraints.