## 3 A Formal Description of the Problem  
The TOP can be mathematically formalized as follows. Let us assume an undirected graph \(G=(N,A)\), where: (\(i\)) \(N=\{0,1,2,\ldots,n+1\}\) includes the set of nodes accounting for \(n\) customers, \(N^{\prime}=\{1,2,\ldots,n\}\), a source node \(0\), and a sink node \(n+1\); and (\(ii\)) \(A\) is the set of edges \((i,j)\), with \(i\
eq j\), connecting nodes \(i\in N\setminus\{n+1\}\) and \(j\in N\setminus\{0\}\). At the source, node \(0\), the team is composed of a limited number of vehicles \(m\geq 1\). Furthermore, there is a maximum time, \(t_{0}>0\), for completing each open route (from its source node to the sink node, with at least one customer node in between). Each customer node \(i\in N^{\prime}\) has a constant reward \(u_{i}>0\), while \(u_{0}=u_{n+1}=0\). Rewards can only be collected on the first visit to a node. Each edge \((i,j)\in A\) has an associated travel time, \(t_{ij}>0\). It is assumed that \(t_{ij}=t_{ji}\), i.e., for \((i,j)\in A\), the matrix of travel times, \(T=[t_{ij}]\), is a symmetric one that satisfies the triangle inequality.  
A solution to the TOP is a set of feasible routes departing from the source node, visiting a subset of customers in a specified order, and arriving at the sink node. In other words, each route starts at the source node \(0\), collects the reward from one or more customers in \(N^{\prime}\), and ends at the sink node, without exceeding the maximum travel time allowed, \(t_{0}\). Let us consider the binary decision variable \(x_{ij}\), which takes the value \(1\) if the edge \((i,j)\in A\) is used by a vehicle to collect the reward at node \(j\), and \(0\) otherwise. In addition, let us define the continuous variable \(w_{i}\), which represents the total travel time that the vehicle has spent after visiting customer \(i\). The objective function is given by the maximization of the total collected reward:  
\[\max\sum_{(i,j)\in A}u_{i}x_{ij}. \tag{1}\]  
This objective function is subject to the following constraints:  
\[\sum_{j\in N\setminus\{0\}}x_{ij}\leq 1,\quad\forall i\in N^{\prime}, \tag{2}\]  
\[\sum_{i\in N\setminus\{n+1\}}x_{ij}\leq 1,\quad\forall j\in N^{\prime}, \tag{3}\]\[\sum_{k\in N\setminus\{n+1\}}x_{ki}=\sum_{j\in N\setminus\{0\}}x_{ij},\quad\forall i \in N^{\prime}, \tag{4}\]  
\[\sum_{j\in N^{\prime}}x_{0j}=\sum_{i\in N^{\prime}}x_{i\,(n+1)}, \tag{5}\]  
\[\sum_{j\in N^{\prime}}x_{0j}\leq m, \tag{6}\]  
\[w_{j}\leq t_{0},\quad\forall j\in N\setminus\{0\}, \tag{7}\]  
\[w_{i}+t_{ij}\;x_{ij}-w_{j}\leq t_{0}-t_{0}\;x_{ij},\quad\forall i\in N \setminus\{n+1\},\forall j\in N\setminus\{0,i\}, \tag{8}\]  
\[\sum_{i\in N}x_{ii}+x_{0\,(n+1)}=0, \tag{9}\]  
\[x_{ij}\in\{0,1\},\quad\forall(i,j)\in A, \tag{10}\]  
\[w_{i}\geq 0,\quad\forall i\in N\setminus\{0\}. \tag{11}\]  
Constraints (2) and (3) impose that each customer node has at most one edge departing from it or entering it, respectively. In addition, constraint (4) imposes that, for each customer node, the number of incoming edges is equal to the number of outgoing edges (due to the previous constraints, this value will be either 0 or 1). Constraint (5) ensures that the number of routes starting at the source node is the same as the number of routes arriving at the sink node. Constraint (6) forces that the number of routes must be less or equal to the number of available vehicles \(m\). Two constraints, (7) and (8), are introduced for both the connectivity of the solution and the maximum travel time requirement. Constraint (9) avoids degenerated routes. Finally, Constraints (10) and (11) define the range of the associated variables.