## 3 Problem Formulation  
The mathematical model of our problem is based on the one proposed by [PERSON] et al. [1] for the TOP. In this study, we manage a fleet \(D=\{1,2,\ldots,d\}\) consisting of \(d\) homogeneous UAVs. Each UAV aims to visit as many nodes as possible within a specified time limit, starting at an initial node and ending at a final node. Upon visiting a node, a UAV receives a corresponding reward, and no other UAV can collect the reward from the same node. Thus, each team member must select a subset of nodes to visit, minimizing overlapamong UAVs while complying with time constraints and maximizing the team's total score. The TOP is formalized within a directed graph \(G=(V,A)\), where (i) the set of nodes is \(V=\{0,1,2,\ldots,n\}\), with node 0 representing the initial depot and node \(n\) representing the final depot; (ii) the set of arcs is defined as \(A=\{(i,j)\mid i,j\in V,i\
eq j\}\); and (iii) the set of rewards is \(R=\{r_{0},r_{1},r_{2},\ldots,r_{n}\}\), where \(r_{i}>0\) is the reward a UAV receives when visiting node \(i\).  
Each UAV \(d\in D\) starts its route at the initial depot, visits several intermediate nodes, and ends at the final depot. This implies that the maximum number of routes in any solution equals the fleet size \(D\). The travel time between nodes \(i\) and \(j\) is denoted by \(T_{ij}>0\) (if \(i\
eq j\)), and the parameter \(t_{\max}>0\) represents the maximum allowed duration for a route. It is important to note that rewards are accumulated only the first time a node is visited, i.e., revisiting a node does not grant additional benefits. The objective is to maximize the total sum of rewards obtained by all UAVs.  
To model the problem, we define the following binary variables: \(y_{id}\) takes the value 1 if node \(i\in V\) is visited by UAV \(d\in D\), and 0 otherwise; \(x_{ijd}\) takes the value 1 if arc \((i,j)\) is traversed by UAV \(d\in D\), and 0 otherwise. Under these conditions, the goal is to maximize the total collected reward:  
\[\max\sum_{d\in D}\sum_{i\in V}r_{i}y_{id} \tag{1}\]  
In addition, the following constraints apply:  
If a node \(i\) is visited, there must exist an arc connecting it, as follows:  
\[\sum_{j\in V,j\
eq i}x_{ijd}=y_{id},\quad\forall i\in V,d\in D \tag{2}\]  
The initial and final depots should not be visited by more UAVs than the fleet size:  
\[\sum_{d\in D}y_{0d}\leq|D|,\quad\sum_{d\in D}y_{nd}\leq|D| \tag{3}\]  
Each intermediate node \(i\) must be visited by only one UAV:  
\[\sum_{d\in D}y_{id}\leq 1,\quad\forall i\in V\setminus\{0,n\} \tag{4}\]  
The total travel time of any UAV must not exceed \(t_{\max}\):  
\[\sum_{(i,j)\in A}T_{ij}x_{ijd}\leq t_{\max},\quad\forall d\in D \tag{5}\]  
The variables \(y_{id}\) and \(x_{ijd}\) must be binary:  
\[y_{id}\in\{0,1\},\quad x_{ijd}\in\{0,1\},\quad\forall i,j\in V,d\in D \tag{6}\]  
It is important to note that in real-world UAV operations, travel times between nodes are often subject to uncertainty due to environmental factors. In our model, we consider this uncertainty by classifying edges into deterministic and stochastic categories. Specifically, edges whose end node has an even ID (\(i\in V\) where \(imod2=0\)) or is divisible by three (\(imod3=0\)) are considered to have stochastic travel times, while the remaining edges have deterministic travel times. This classification allows us to model realistic scenarios where travel time variability may affect route feasibility and reward collection.