## 3 Establishing a Mathematical Model
### Establishing a Minimum Cost Mathematical Model  
Under a deterministic traffic network, candidate paths \(p_{ij}\) belonging to each arc \(A^{c}\) are first processed according to the statistical analysis of the data of the expected traffic network. Next, the planning of the vehicle travel path is carried out in order to minimize the total cost.  
Firstly, the decision variables are defined as follows:  
\[x_{ij}=\begin{cases}1&if\text{ {the selected route is the best route in }arc(i,\ j)}\\ 0&\text{others}\end{cases}\]  
\[x_{ij}^{p}=\begin{cases}1&if\text{ {the vehicle is driving on }arc(i,\ j)},\\ 1&\text{selecting a path in the road set}\\ 0&\text{others}\end{cases}\]  
\[f_{ij}=\text{The load of {the vehicle via }arc(i,j)}\]  
\[f_{ij}^{p}=\text{The load of {the vehicle on the road }\rho}\]  
Secondly, the penalty function needs to be defined as follows:  
\[u_{i}(S_{i})\begin{cases}a_{i}(E_{i}-S_{i})&if\ EE_{i}<S_{i}<E_{i}\\ 0&if\ E_{i}<S_{i}<L_{i}\\ b_{i}(S_{i}-L_{i})&if\ L_{i}<S_{i}<LL_{i}\\ +\infty&if\ S_{i}<EE_{i}\ \text{or}\ S_{i}>LL_{i}\end{cases}\]  
\(E_{0}\) indicates the departure time from the distribution center. Since the early penalty cost is generally less than the late penalty cost, the time window here is asymmetrical, i.e., \(a_{i}\
eq b_{i}\).  
Then, the following mathematical model is formulated:  
\[\begin{array}{ll}MinZ&=\sum\limits_{(i,j)\in A}\sum\limits_{p\in P}\bigg{(} \lambda\bigg{(}\omega\bigg{(}d_{ij}^{p}/s\bigg{)}X_{ij}^{p}+vd_{ij}^{p}s^{2}X_{ ij}^{p}+Q\bigg{(}\mu X_{ij}^{p}+f_{ij}^{p}\bigg{)}d_{ij}^{p}\bigg{)}\bigg{)} \bigg{)}\\ &+\sum\limits_{(i,j)\in A}\sum\limits_{p\in P}\bigg{(}d_{ij}^{p}/s\bigg{)}X  
\[\sum_{i\in N}x_{ij}=1\quad\forall j\in N\backslash\{0\} \tag{15}\]  
\[\sum_{i\in N}f_{ij}-\sum_{k\in N}f_{jk}=q_{j}\quad\forall j\in N \backslash\{0\} \tag{16}\]  
\[q_{j}x_{ij}\leq f_{ij}\leq(Q-q)x_{ij}\quad\forall(i,j)\in A \tag{17}\]  
\[\sum_{p\in P}x_{ij}^{p}=x_{ij}\quad\forall(i,j)\in A \tag{18}\]  
\[\sum_{p\in P}f_{ij}^{p}=f_{ij}\quad\forall(i,j)\in A \tag{19}\]  
\[q_{j}x_{ij}^{p}\leq f_{ij}^{p}\leq(Q-q)x_{ij}^{p}\quad\forall(i,j)\in A. \forall p\in P \tag{20}\]  
\[EE_{j}\leq S_{i}+T_{i}+t_{ij}\leq LL_{j}\quad i,j=1\ldots n \tag{21}\]  
In the objective function, Equation (7) concerns the minimization of the total fuel cost, driver wage, vehicle depreciation expense related to travel distance and time-window penalty cost, respectively. Constraints (8) and (9) ensure that the number of vehicles used cannot exceed the number of available vehicles \(K\), and Constraints (10)-(13) are variable specification constraints. Constraints (14)-(17) are the standard constraints for a double-index single commodity flow model in vehicle routing problems with limited capacity, the first two of which (Constraints (14) and (15)) are conservation constraints of vehicle flow. Constraint (16) is the conservation constraint of vehicle commodity flow, and constraint (17) guarantees that the loaded cargo does not exceed the capacity of the vehicle. Constraint (18) implies that the vehicle can select exactly one of the paths under the customer connection arc \((i,j)\in A\). Constraints (19) and (20) also ensure that the goods are transported along a path, and that the loaded cargo does not exceed the capacity of the vehicle if it travels over an arc \((i,j)\in A\). Moreover, constraint (21) is a time-window constraint.