## 3 Methods
### CVRP: The Three-Index Formulation  
Throughout the description of the mathematical model, we denote \(Dep\) as the depot for the Capacitated Vehicle Routing Problem or CVRP approach. This latter implies that a fleet of vehicles with a corresponding cargo capacity depart from a depot, visit several customers to meet their goods demand, and return to the depot. The demand of the customers must be fully met, seeking routes at a minimum traveling cost [46]. The CVRP is represented by Equations (1)-(13), considering the relation with the candidate points of charging stations and a fixed number of EVs. Notice that \"\(\backslash\)\" is the except symbol and refers to the \"individual\" that is not included in a set. For example: \(\forall\ i\ \in\ N\ Dep\), this means: For all \(i\) that belongs the set \(N\), except the depot; in other words, the index \(i\) can take any value stored in \(N\), except the value assigned for the depot \(Dep\). \[\sum_{k\in K}\gamma_{i,k}=1,\ \forall i\varepsilon N\setminus\{Dep\} \tag{1}\]  
\[\sum_{i\varepsilon N}\sum_{k\varepsilon K}X_{i,j,k}=1,\ \forall j\varepsilon N \setminus\{Dep\} \tag{2}\]  
\[\sum_{j\varepsilon N}\sum_{k\varepsilon K}X_{i,j,k}=1,\ \forall i\varepsilon N \setminus\{Dep\} \tag{3}\]  
\[\sum_{k\varepsilon N}\sum_{j\varepsilon N}X_{Dep,j,k}=|K| \tag{4}\]  
\[\sum_{k\varepsilon K}\sum_{j\varepsilon N}X_{j,Dep,k}=|K| \tag{5}\]\[\sum_{ \begin{subarray}{c}j\
u\\ i\
eq j\end{subarray} }X_{i,j,k}-\sum_{ \begin{subarray}{c}j\
u\\ i\
eq j\end{subarray} }X_{j,i,k}=0\ \forall ieV\ \forall keK \tag{6}\]  
\[\sum_{j\in N}X_{j,i,k}=Y_{i,k}\ \forall ieN\ \forall keK \tag{7}\]  
\[\sum_{ \begin{subarray}{c}j\
u\\ i\
eq k\end{subarray} }t_{h,j,k}\leq\left[\sum_{ \begin{subarray}{c}\omega\\ i\
eq i\end{subarray} }t_{i,j,k}-D_{h}\cdot x_{i,b,k}\right]+Cap\cdot\left[1- \sum_{ \begin{subarray}{c}i\
u\\ i\
eq i\end{subarray} }x_{i,h,k}\right]\forall heN\ \backslash\{Dep\}\forall keK \tag{8}\]  
\[\sum_{ \begin{subarray}{c}i\
u\\ i\
eq k\end{subarray} }t_{i,h,k}=\sum_{ \begin{subarray}{c}j\
u\\ i\
eq j\end{subarray} }t_{h,j,k}\ \forall heC\setminus\{Dep\}\forall keK \tag{9}\]  
\[t_{i,h,k}\geq 0\ \forall ieV\ \forall heV\ \forall keK\ \ i\
eq h \tag{10}\]  
\[t_{i,h,k}\leq Cap\cdot x_{i,h,k}\ \forall ieV\ \forall heV\ \forall keK\ \ i\
eq h \tag{11}\]  
\[\sum_{ \begin{subarray}{c}\omega\
u\\ i\
eq D\
u\end{subarray} }\sum_{ \begin{subarray}{c}\omega\\ i\
eq D\
u\end{subarray} }t_{Dep,i,k}=\sum_{ \begin{subarray}{c}i\
u\\ i\
eq D\
u\end{subarray} }D_{i} \tag{13}\]  
Constraint (1) imposes that each customer is visited exactly once. Constraints (2) and (3) are indegree and outdegree constraints, which impose that exactly one arc enters and leaves each vertex associated with each customer, respectively. Likewise, constraints (4) and (5) show that, at the depot, the number of vehicles leaving and arriving after routing completion must be the same. The \(``|K|\"\) is the cardinality symbol used to represent the size of set \(K\), that is, the number of vehicles in the CVRP approach. The flow conservation constraint in (6) enforces that the number of ongoing arcs equals the number of incoming arcs at each vertex, which covers both customers and charging station candidate points. Constraint (7) prevents a customer to be visited by several vehicles. For a customer node, the change in the remaining freight load of an EV is tracked based on node sequence by constraint (8) [47]. Since a customer can be visited only once, if the customer \(h\) is visited by vehicle \(k\), the remaining load of the vehicle will be reduced by customer demand \(D_{h}\); otherwise, the constraint is relaxed. When vehicle visits a charging station, considered as a node with the absence of freight demand, constraint (9) shows that the sum of the remaining load of an EV entering the charging station is equal to the sum of the remaining load leaving the charging station. This guarantees multiple visits to a charging station and load capacity balance in the vehicles.