Using dynamic flow network modeling for global flight plan optimization CARE workshop 14th –15th March 2001, EUROCONTROL Brussels. Dritan Nace Heudiasyc Laboratory, UMR CNRS 6599, University of Technology of Compiègne, France.

Outline of the problem Global flight plan optimization and a better coordination of all existing flights are particularly interesting in conditions of air traffic growth. The aim of this work is to reduce the number of potential en-route conflicts. Global flight plan optimization is a multi-period (dynamic) problem which can be transformed into a static one by using standard technique of time-expanding the underlying network (Ahuja et al, 1993). Modeling the problem as an integer linear programming one.

Telecommunication network versus airspace network The telecommunication network: 1) High costs of infrastructure; 2) The quantity of data transmitted is such that we consider flows (no packets) routed through the network, so multi-routing is permitted; The airspace network: 1) No infrastructure: free links; 2) We consider only "packets" corresponding to aircraft so a unique route for each aircraft : mono-routing; Note the need of considering multi-period models;

Assumptions Airspace network : the topology consists of a number of initial nodes, corresponding to the ground-based stations emitting navigation signals, and a set of fixed links corresponding to the most probable used ones. All potential conflict areas (corresponding to nodes) will be represented by links. The time will be discretized in periods, the duration will be fixed according to the accuracy of the modeling and this of the disposed data. All candidate trajectories are supposed to be known.

Mathematical formulation Let G (V,A) be a directed network defined by a set V of nodes (v) and a set A of directed arcs (k). Let A be a specified set of arcs corresponding to potential en-route conflict areas. Let F be the set of flights to be routed. For each flight f, we suppose the origin/destination and the period of taking-off to be known. H(f) denotes the set of eligible routes for a given flight f. Let T be the set of periods t. x t j,f gives the traffic value (0/1) using the route j for the flight f taking-off at period t c t j,f denotes the cost associated with the route j for the flight f taking-off at period t. a t,p j,f,k takes value (0/1) according to the predictions of the trajectory of flight f, route j, link k j, taking-off at period t, during the period t+p. r t k,p corresponds to the number of aircraft beginning flying at period t and flying through arc k during the period t+p. y t k corresponds to the number of aircraft flying simultaneously through arc k during the period t. R gives the maximum number of aircraft involved in the same conflict.

Mathematical formulation Minimize subject to: (1) t T, k A, R - y t k 0 ; (2) t T, f F(t), ; (3) t T, p {1,2,..,|T|-t}, k A, ; (4) t T, k A, ; (5) t T, f F(t), j h(f), x t j,f binary; (6) t T, k A, r t k,p, y t k N*;

Criticizing the model 1) The problem is formulated as an integer linear programming one... 2) General case : taking into account the uncertainty in the delays and different flying speeds. Our model can handle these two problems : it is sufficient to fix for each trajectory the duration (or period(s)) that the aircraft would be in potential conflict area (through a t,p j,f,k values). 3) Due to cost considerations, only a limited number of routes for each pair origin-destination have to be considered (preferred routes). 4) On the other hand, only the congestion arcs are concerned by the model, reducing so drastically the number of effective variables and constraints. 5) Decomposition methods as benders one could be envisaged, but it requires relaxing the mono-routing constraints.

Proposition of solution methodology The problem has a large number of integer variables, consequently a branch & bound method could provide good results due to the limited number of routes to be considered for each pair origin-destination. The second direction of our work is to envisage some decomposition methods (Benders) in order to reduce the size of the considered problem.

Concluding remarks Using advanced linear programming methods... Getting inspired of related telecommunication problems… Considering general case… Other problems (eg. slot-time allocation) could be coupled with global flight plan optimization… Related issues to be studied... Further investigations are needed...