Planning for multiple commodities simultaneously is a challenging task arising in divers applications, including robot motion or various forms of traffic management. Separation constraints between commodities frequently have to be considered to ensure safe trajectories, i.e., paths over time. Discrete decisions to ensure at least one of often multiple possible separation conditions renders planning of best possible continuous trajectories even more complex. Hence, the resulting disjoint trajectories optimization problems are mostly solved sequentially or with restricted planning space, potentially leading to losses in the usage of sparse resources and system capacities. To tackle these drawbacks, we develop a graph-based model for disjoint trajectories optimization with general separation requirements. We present a novel technique to derive a discretization for the full available space of motion. This can depict arbitrary, potentially non-convex, restricted areas. This necessitates solving an integer linear optimization program whose size scales with the number of discretization points. Thus, even for moderately sized instances a sufficiently detailed representation of space and time leads to models too large for state of the art hard- and software. To overcome this issue, we develop an adaptive-refinement algorithm: Starting from an optimal solution to the integer program in a coarse discretization, the algorithm re-optimizes trajectories in an adaptively-refined discretized neighborhood of the current solution. This is further integrated into a rolling horizon approach. We apply our approach to the integrated trajectory optimization and runway scheduling in the surrounding of airports. Computational experiments with realistic instances demonstrate the efficiency of the method.
