## 4 Mathematical Model for the CEVRPTW  
In this paper, the Capacitated Electric Vehicle Routing Problem with Time Windows is considered. The CEVRPTW is a combinatorial optimization problem. In the problem, EVs with a given payload and battery capacity leave from the depot fully charged. They leave from the depot to serve customers and visit stations to recharge their batteries throughout their routes. When an EV visits a charging station, it will be fully charged to complete its task. At the end of route, they must return to the depot. For EVs, the aim is to find routes where all customer demands will be met, and the fleet's total distance or time traveled will be minimized. The CEVRPTW is initially defined as a complete directed graph \(G=(V,E)\), as illustrated in Figure 9. \(V=\{0\cup C\cup F\}\) represents the set of nodes in the graph. The depot is denoted by \(0\), \(C\) represents the customers, and \(F\) represents the charging stations. \(E=\{(i,j)\mid i,j\in V,\,i\
eq j\}\) is the set of edges. Each edge between nodes \(i\) and \(j\) has a distance value, denoted as \(d_{ij}\). \([e_{i},l_{i}]\) represents the time window of customer \(i\). When an electric vehicle travels on an edge, it consumes an amount of energy \(e_{i,j}=h(r)\cdot d_{ij}\). \(h\) denotes the energy consumption rate. Figure 8: Summary of benchmark proposed for the CEVRPTW. This problem aims to minimize the total travel distance of all electric vehicles while serving customers within their time windows. There is a constraint that each customer can only be served one vehicle. EVs must return to the depot after fulfilling all customers' demands. It assumes that electric vehicles are always fully charged at charging stations. In the literature, [PERSON] et al. [36] propose a mathematical model for the Capacitated Electric Vehicle Routing Problem with Time Windows. The model uses the following notation. 1. Sets  
2. Parameters  
3. An example solution for the CEVRPTW problem. Figure 9: An example solution for the CEVRPTW problem. Decision Variables  
\[P_{i}\] Decision variable specifying the time of arrival at vertex i  
Decision variable specifying the remaining battery capacity on arrival at vertex \[i\]  
\[x_{ij}\] Binary decision variable indicating if arc (\(i\),\(j\)) is traveled  
The mathematical model of the CEVRPTW is formulated as Mixed-Integer Linear Programming as follows [36]:  
\[\textbf{minimize}\sum_{i\in V_{\phi}\in V_{N+1},i\
eq j}\textbf{{d}}_{ij} \textbf{{x}}_{ij} \tag{2}\]  
\[\sum_{j\in V_{N+1},\ i\
eq j}x_{ij}=1\ \ \ \forall i\,\epsilon\ V \tag{3}\]  
\[\sum_{j\in V_{N+1}^{\prime},\ i\
eq j}x_{ij}\leq 1\ \ \ \forall i\,\epsilon\ F^{\prime} \tag{4}\]  
\[\sum_{i\in V_{N+1}^{\prime},\ i\
eq j}x_{ji}-\sum_{i\in V_{0}^{\prime},\ i\
eq j}x_{ij}\ \ = \ \ \ \forall j\,\epsilon\ V^{\prime} \tag{5}\]  
Figure 8:0  
Figure 8:1  
Figure 8:2  
Figure 8:3  
Figure 8:4  
Figure 8:5  
Figure 8:6  
Figure 8:7  
The objective function (2) is to minimize the total travel distance. Constraint (3) ensures that service is provided to every customer node, constraint (4) ensures that the charging station can be visited, and constraint (5) prevents the formation of sub-tours. Constraints (6) and (7) guarantee time feasibility. Constraint (8) refers to the time window conditions. (9) and (10) are the constraints that control the vehicle's load. Here, the vehicle will leave the depot with the load as much as its capacity. Constraints (11) and (12) control the vehicle's charging status. Constraint (13) determines the number of routes. It gives the number of routes that will leave the initial warehouse and go to the nodes.