Convex Formulations of Air Traffic Flow Optimization Problems

The problem of regulating air traffic in the en route airspace of the National Airspace System is studied using a Eulerian network model to describe air traffic flow. The evolution of traffic on each edge of the network is modeled by a modified Lighthill-Whitham-Richards partial differential equation. The equation is transformed with a variable change, which makes it linear and enables us to use linear finite difference schemes to discretize the problem. We pose the problem of optimal traffic flow regulation as a continuous optimization program in which the partial differential equation appears in the constraints. We propose a discrete formulation of this problem, which makes all constraints (the discretized partial differential equations, boundary, and initial conditions) linear. Corresponding linear programming and quadratic programming based solutions to this convex optimization program yield globally optimal solutions to various air traffic management objectives. The proposed method is applied to the maximization of aircraft arrivals and minimization of delays in the arrival airspace due to exogenous capacity reductions. The corresponding linear and quadratic programs are solved numerically using CPLEX for a benchmark scenario in the Oakland Air Route Traffic Control Center. Several computational aspects of the method are assessed—in particular, accuracy of the numerical discretization, computational time, and storage space required by the method.