# A Sim-Learnheuristic for the Team Orienteering Problem: Applications to Unmanned Aerial Vehicles
## 1 Introduction  
The team orienteering problem (TOP) is a classic optimization problem in which a set of routes is constructed for a team of vehicles to traverse, aimed at collecting the highest possible reward from customers within the routes before a defined time limit [1]. In sectors such as search and rescue, surveillance, and logistics, addressing the TOP is crucial. One example of this could be a search and rescue mission, where a team of unmanned aerial vehicles (UAVs) must navigate through a disaster-stricken area to locate survivors and deliver supplies. While UAVs have emerged as essential assets in such missions due to their capacity to perform hazardous duties and their skill in sensing, monitoring, and navigating, the difficulties they face are difficult to overstate [2]. These challenges are multifaceted and include unpredictable environmental factors like weather or traffic conditions. Such unpredictability makes reliable behavior prediction impossible, resulting in suboptimal solutions and a reduced mission efficiency. Additionally, the natural limitations of UAVs, such as the limited battery life and cargo capacity, demand precise optimization for mission accomplishment. Despite these challenges, UAVs remain a critical technology for solving complex problems in various domains, emphasizing the importance of the TOP and its applicability to UAVs [3]. Furthermore, the integration of Internet of Things technologies stands as one of the most promising approaches for enabling UAVs to sense their surroundings and collect data. This integration facilitates UAVs to gather and exchange real-time environmental data, thereby optimizing their routes accordingly [4]. In recent years, the majority of studies focused on either deterministic or stochastic variants of the TOP. In the case of the deterministic TOP, many of the studies in the literature rely significantly on exact methods to solve this problem [5]. However, when presented withchallenges given by large-scale instances, these exact methods show limited effectiveness. On the other hand, in the case of the stochastic TOP, various studies have explored different strategies, such as stochastic programming [6], robust optimization [7], and, recently, the utilization of simheuristic approaches [8]. Furthermore, several versions of the TOP have been developed to address specific real-world scenarios, including heterogeneous vehicle capacities, customer time constraints, and unpredictable travel times and profits. For example, [PERSON] et al. [9] addressed the TOP with time windows and mandatory visits, [PERSON] and [PERSON] [10] tackled the TOP with mandatory visits, [PERSON] et al. [11] addressed the TOP with variable profits, and [PERSON] et al. [12] tackled the TOP with stochasticity. Our paper focuses on a hybrid variant of the TOP, which combines deterministic, stochastic, and dynamic characteristics, addressing a gap in the scientific literature. To solve this complex problem, a new sim-learnheuristic approach is introduced, with special applications in UAVs that can be used to aid in disaster rescue. This approach combines a biased-randomized savings-based heuristic, Monte Carlo simulations, and a multiple regression model. The main objective of this novel methodology is to combine the strengths of all three components, allowing for an agile exploration of the search space to produce high-quality solutions for the presented problem. Figure 1 depicts this new variant of the TOP, where the travel times display a wide range of features. These features include certain travel times (deterministic), uncertain travel times (stochastic), and travel times that can change over time due to dynamic factors (dynamic), such as varying traffic congestion and weather conditions. This blend of characteristics results in a more precise representation of the complexities evident in real-world situations.