The duration of missions that can be accomplished by a system of unmanned aerial vehicles (UAVs) is limited by the battery or fuel capacity of its constituent UAVs. However, a system of UAVs that is supported by automated refueling stations may support long term or even indefinite duration missions. We develop a mixed integer linear program (MILP) model to formalize the problem of scheduling a system of UAVs and multiple shared bases in disparate geographic locations. There are mission trajectories that must be followed by at least one UAV. A UAV may hand off the mission to another in order to return to base for fuel. To address the computational complexity of the MILP formulation, we develop a genetic algorithm to find feasible solutions when a state-of-the-art solver such as CPLEX cannot. In practice, the approach allows for a long-term mission to receive uninterrupted UAV service by successively handing off the task to replacement UAVs served by geographically distributed shared bases.