## 2 Problem Definition and Formulation  
The VRPTW is defined by a set of identical vehicles denoted by K and by a direct network \(G\) (\(N\), \(A\)), where N is the set of nodes and \(A\ \ =\ (i,\ j):i\
eq j,\ i,\ j\ \in N\) is the set of arcs. Node \(0\) represents the central depot, while \(N^{*}\ =\ \{N/0\}\) represents customers. Every arc \((i,j)\), which is a path from node \(i\) to node \(j\), is characterized by a distance, indicated as \(d_{ij}\). \(t_{ij}\) stands for the travel time from node \(i\) to node \(j\) and has the same value with \(d_{ij}\), as the assumption that each distance unit corresponds to one specific time unit is made. Each customer \(i\) is characterized by the demand \(q_{i}\), the service time needed \(s_{i}\) and the time window \((e_{i},l_{i})\), where \(\mathrm{e_{i}}\) is the opening time and \(l_{i}\) the closing time. Customers must be served by exactly one vehicle that may arrive any time within the time window. The arrival time is given by \(t_{i}\). Additionally, if the vehicle arrives before the beginning of the time window \((e_{i})\), it must wait for \(w_{i}^{k}\) time, until service is possible, while no vehicle may arrive after the end of the time interval \((l_{i})\).  
The decision variable \(x_{ij}^{k}\) is equal to 1 if vehicle \(k\) drives from node \(i\) to node \(j\), and \(0\) otherwise. The objective of the VRPTW is to service all customers, minimizing the number of vehicles and the total traveled distance while simultaneously ensuring that all the constraints are satisfied. The mathematical model of the VRPTW is presented below:  
\[Objective\ 1:\ \sum_{k\in K}\sum_{j\in N^{k}}x_{0j}^{k} \tag{1}\]  
\[Objective\ 2:\ \sum_{k\in K}\sum_{(i,j)\in A}d_{ij}x_{ij}^{k} \tag{2}\]  
\[\sum_{i\in N}q_{i}\sum_{j\in N}x_{ij}^{k}\ \leq\ q_{max}^{k},\ \forall k\ \in\ K \tag{3}\]  
\[\sum_{k\in K}\sum_{j\in N}x_{ij}^{k}\ =\ 1,\ \forall i\in N \tag{4}\]  
\[\sum_{j\in N^{*}}x_{0j}^{k}\ =\ 1,\ \forall k\ \in\ K \tag{5}\]  
\[\sum_{i\in N}x_{ij}^{k}\ -\ \sum_{j\in N}x_{ij}^{k}\ =\ 0,\ \forall l\ \in\ N,\ \forall k\ \in\ K \tag{6}\]  
\[\sum_{j\in N^{*}}x_{j0}^{k}\ =\ 1,\ \forall k\ \in\ K \tag{7}\]  
\[t_{0}\ =w_{0}^{k}\ =s_{0}\ =0 \tag{8}\]  
\[\sum_{j\in N}x_{ij}^{k}\left(t_{i}\ +\ w_{i}^{k}\ +\ s_{i}\ +\ t_{ij}\right)\leq\ t_{j},\ \forall i\in N,\ \forall k\ \in\ K \tag{9}\]  
\[e_{i}\leq\ t_{i}\ +\ w_{i}^{k}\ \leq\ l_{i},\ \forall i\in N,\ \forall k\ \in\ K \tag{10}\]  
\[x_{ij}^{k}\ \in\ \{0,1\},\ \forall\ (i,j)\ \in\ A,\ \forall k\ \in\ K \tag{11}\]  
The objective function in Equation (1) states that the total number of vehicles and the total traveled distance should both be minimized based on the objective function (2). The constraint (3) states that a vehicle's capacity cannot be exceeded and (4) that each customer must be served exactly once and by one vehicle. Constraints (5)-(7) make sure that each vehicle starts from the depot _[0]_, visits and serves a certain number of customers and finally returns to the depot {0}. Constraints (8)-(10) indicate that, for a trip from node \(i\) to node \(j\), no vehicle may arrive at customer \(j\) after the end of the time window (\(\rm{I_{i}}\)). Simultaneously, the time of arrival to a customer depends on the time of arrival, the waiting time and the service time of the previous served customer.