## 3 Problem Definition and Formulations
### Mathematical Formulation
#### 3.3.2 Decision Variables  
\(x_{ijk}\): Binary variable denoting if truck \(k\) traverses on arc \((i,j)\)  
\(y_{ij}\): The time a truck leaves depot \(i\) to proceed to a non-depot vertex \(j\)  
\(\gamma_{ij}\): The time a truck starts moving back to depot \(j\) from a non-depot vertex \(i\)  
\(\tau_{i}\): The start time of the activities at the request vertex \(i\)  
\(s_{k}\): The start time of truck \(k\)'s operation  
\(e_{k}\): The end time of truck \(k\)'s operation \(w_{k}\): The duration by which truck \(k\) surpasses the maximum truck working hours  
3.3.3. Formulation  
\[\min Z=C1\times\sum_{i\in N_{r}}\sum_{j\in N_{r}}\sum_{k\in K}x_{ijk}\times t_{ ij}+C_{2}\times\sum_{k\in K}w_{k}+C_{3}\times\sum_{k\in K}\left(e_{k}-s_{k}-\sum_{i\in N _{r}}\sum_{j\in N_{r},i\
eq j}x_{ijk}\times t_{ij}\right) \tag{1}\]  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]  
\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]  
\[\sum_{j\in N_{r}}\sum_{k\in K}x_{ijk}\leq n_{i}\quad\forall i\in N_{d} \tag{4}\]  
\[\sum_  
\[\sum_{j\in N_{DFC}\cup N_{DFC}\cup N_{Jd},i\
eq j}\sum_{k\in K}x_{ijk}= 1\quad\forall i\in N_{DFC} \tag{19}\] \[\sum_{i\in N_{DFC}\cup N_{DFC}\cup N_{DFC}\cup N_{Jd},i\
eq j}\sum_{k \in K}x_{ijk}= 1\quad\forall i\in N_{DFC}\] (20) \[\sum_{j\in N_{DFC}\cup N_{DFC}\cup N_{DFC}\cup N_{Jd},i\
eq j}\sum_{k \in K}x_{ijk}= 1\quad\forall i\in N_{PFC}\] (21) \[\sum_{j\in N_{DFC}\cup N_{DFC}\cup N_{DFC}\cup N_{Jd},i\
eq j}\sum_{k \in K}x_{ijk}= 1\quad\forall i\in N_{PFC}\] (22) \[y_{ij}=\sum_{k\in K}s_{k}x_{ijk}\quad\forall i\in N_{d},\;\forall j \in N_{DEC}\] (23) \[\gamma_{ji}=\sum_{k\in K}e_{k}x_{ijk}\quad\forall i\in N_{d}, \forall j\in N_{PEC} \tag{24}\]  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]0  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]1  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]2  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]3  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]4  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]5  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]6  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]7  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]8  
\[\text{s.t.}\sum_{j\in N,i\
eq j}x_{ijk}=\sum_{j\in N,i\
eq j}x_{ijk}\quad\forall i \in N_{r},\forall k\in K \tag{2}\]9  
\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]0  
\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]1  
\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]2\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]3  
\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]4  
\[\sum_{j\in N}\sum_{k\in K}x_{ijk}=1\quad\forall i\in N_{r} \tag{3}\]5  
The goal of function (1) is to minimize the total cost associated with trucks. Three components, respectively, indicate truck travel cost, truck penalty cost, and truck dwelling cost. Constraint (2) ensures the uninterrupted route of each truck. Constraint (3) enforces each request vertex is served only once. Constraints (4) and (5) determine the maximum number of departures and returns allowed for trucks from and to each depot. Constraints (6)-(8) govern the processes for truck departures and returns.