Truck drone arc covering problem with an application and case study in disaster management

River exploration during, before, or after floods enables operators in civil protection and disaster control to better prepare for or even prevent disasters. While typically, this river exploration is done by boat, truck, helicopter, or even not at all, autonomous flying drones equipped with a camera can enhance this process. Moreover, interaction between a truck and a drone can enable the drone to be used flexibly and extend its short range. Thus, the Bavarian Red Cross equipped a truck with a drone for river coverage. Based on this real case, we introduce a truck drone arc covering problem (TD-ACP) for the application of river coverage. We formulate the TD-ACP as a mixed-integer linear program and introduce valid inequalities that strengthen the formulation and allow us to solve realistic-sized instances to optimality. In a real-world case study involving an actual river, we demonstrate that using drones for river coverage can reduce coverage time by 56.3% compared to boats and by 28.1% compared to trucks. Additionally, we propose a manual planning heuristic that is straightforward for practitioners to apply and achieves an optimality gap of 4.0% on this specific river.