## 3 1-Principal-1-Assistant Scenario
### Problem Formalization
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline \(x_{i,t}\) & Binary decision variable equals to one if the principal agent arrives \(v_{i}\) at time \(t\); zero otherwise. \\ \(y_{i,k,j}\) & Binary decision variable equals to one if the assistant agent is released at \(v_{i}\), retrieved at \(v_{j}\), and serves \(v_{k}\); zero otherwise. \\ \(z_{i,j}\) & Binary decision variable equals to one if the principal agent moves from \(v_{i}\) to \(v_{j}\); zero otherwise. \\ \(b_{i,t}\) & Binary decision variable equals to one if \(t\) is equal or greater than the released time and less than the retrieved time in the sortie serving \(v_{i}\); zero otherwise. \\ \(s_{i}\) & Integer intermediate variables representing the principal agent leaving time from \(v_{i}\). \\ \(e_{i}\) & Integer intermediate variables representing the principal agent arrival time to \(v_{i}\). \\ \(t\) & The time step. \\ \(M\) & A very large number. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters and variables for the 1-principal-1-assistant scenario. \[\forall v_{i},v_{k},v_{j}\in V,0\leq t\leq d_{0} \tag{25}\] \[b_{k,t}\leq 1+(e_{j}-1-t)/M+(1-y_{i,k,j})M,\] \[\forall v_{i},v_{k},v_{j}\in V,0\leq t\leq d_{0}\] (26) \[\sum_{0\leq t\leq d_{0}}b_{k,t}\geq e_{j}-s_{i}+(y_{i,k,j}-1)M, \forall v_{k}\in V\] (27) \[\sum_{v_{i}\in V}\sum_{v_{j}\in V}y_{i,k,j}+\sum_{1\leq t\leq d_{0 }}x_{k,t}\leq 1,\forall v_{k}\in V \tag{28}\]  
The objective function (1) seeks to maximize the number of served tasks. Constraints (2)-(4) establish the relationship between \(s_{i}\), \(e_{i}\), and \(x_{i,t}\). Explicitly, except initial vertex \(v_{0}\), for other \(v_{i},s_{i}=e_{i}\). Constraints on the principal agent routesThe principal agent travels in the graph, holding constraints (5)-(14). Constraints (5)-(7) ensure that the principal agent departs from the initial vertex \(v_{0}\) at time \(0\) and finally returns to it before \(d_{0}\). Constraint (8) indicates that the principal agent serves at most one vertex at any time, and constraint (9) indicates that the principal agent serves a task at a vertex except initial vertex \(v_{0}\) at most one time. Constraint (10) states that the moving distance of the principal agent from \(v_{i}\) to \(v_{j}\) is not less than the minimum distance between them. Constraints (11)-(14) link \(z_{i,j}\) and \(x_{i,t},x_{j,t}\). Constraint (11) ensures \(z_{i,j}\) cannot be \(1\) if the principal agent visits \(v_{j}\) after its deadline. Constraints (12) and (13) ensure that the principal agent serves any tasks at most once. Constraint (14) ensures that if \(z_{i,j}=1\), the moving distance between any two vertices is not less than the minimum distance between them. Constraints (15)-(27) are the constraints on the assistant agent routes. Constraint (15) ensures the assistant agent serves tasks that are not served by the principal agent. Constraint (16) guarantees that the assistant agent will be retrieved by the principal agent and Constraint (17) guarantees that the assistant agent moves in the range. Constraints (18)-(20) ensure the assistant agent serves any task at a vertex at most once. Constraints (21) and (22) ensure that the assistant agent serves the task vertex \(v_{k}\) before its deadline and arrives at retrieved vertex \(v_{j}\) before the principal agent arrives. Constraints (23)-(27) link \(b_{k,t}\) and \(y_{i,k,j}\). Constraint (23) requires the assistant agent executes at most one task at any time. Constraint (24) ensures \(b_{k,t}=0,0\leq t\leq d_{0}\) for unserved vertex \(v_{k}\). Constraints (25)-(27) ensure that if the assistant agent serves \(v_{k}\), departing from the principal agent at \(v_{i}\) and returning to the principal agent at \(v_{j}\), \(b_{k,t}=1,s_{i}\leq t<e_{j}\); otherwise, \(b_{k,t}=0\). Constraint (28) indicates that \(v_{i}\) can only be served by the principal agent and the assistant agent once.