MILP formulation and polynomial time algorithm for an aircraft scheduling problem

This paper presents a polynomial time algorithm used for solving a mixed integer linear program (MILP) formulation of a scheduling problem applicable to air traffic control. We first relate the general MILP (which we believe to be NP-hard) to the air traffic control problem, which consists of performing maneuver assignments to achieve scheduling constraints for airport arrival traffic. This MILP can be solved with CPLEX, yet there is no guarantee on the running time. We show that a specific case of this air traffic control problem, which is of interest in its own right, may be solved using an exact polynomial-time algorithm. The case of interest consists of finding the largest achievable time separation between aircraft upon arrival, compatible with airspace restrictions and aircraft performance. Our algorithm transforms the problem to a single machine scheduling problem, and then embeds its solution into a bisection algorithm. We establish the polynomial complexity of the resulting algorithm by proving an algebraic property of its optimal solution. We compare the running times of CPLEX and our algorithm for 1800 cases with up to 20 aircraft. The results show numerical evidence of the guaranteed running time of our algorithm, by contrast with CPLEX whose average performance is good, but also shows a significant number of instances with unpredictably large computational time. We perform 8100 additional runs of our algorithm with up to 100 aircraft, to numerically confirm the predicted worst case running time of our algorithm.