1. 1 EUROCONTROL UMR CNRS 6599 Heuristique et Diagnostic des Systèmes Complexes © 2002 HEUDIASYC © 2002 HEUDIASYC A LINEAR PROGRAMMING APPROACH FOR ROUTE AND LEVEL FLIGHT ASSIGNMENT Dritan Nace, Jacques Carlier, Nhat Linh Doan, University of Technology of Compiègne, France & Vu Duong Eurocontrol, Brétigny/Orge, France. 5 th EUROCONTROL / FAA ATM R&D SEMINAR, Budapest, Hungary, 23 rd - 27 th June 2003

2. © 2002 HEUDIASYC © 2002 HEUDIASYC 2 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Content  Context  Situation of Air Transportation in Europe  Route and level flight assignment  Research Motivation and Goals  Related works  Dynamic Network Modeling  Mathematical model Assumptions, Notation, Mathematical formulation  Experimental Results  Global approach  Summary and future work

3. © 2002 HEUDIASYC © 2002 HEUDIASYC 3 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Situation of Air Transportation in Europe  Increasing Pressure on the Air Traffic Systems  “The number of flights in Europe is growing by about 5% each year, and is expected to nearly double by 2015 when compared to 1998 level.” More congestion is expected SOLUTION:  Modify the structure of network New airports, new routes or route’s reorganization. Increase number of runways. Increase number of sectors (reduce their size) – reorganization (super-sector, sectorless,...)  Modify the flight plans in order to adapt the demand to the available capacity. Slot-time allocation (GHP) Routing  Optimal global conflict resolution  Scheduling

4. © 2002 HEUDIASYC © 2002 HEUDIASYC 4 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Research motivation and goals  Problem of global flight plan optimization from a routing point of view (assigning the appropriate route and level to each flight) in a given airway network.  Principal goal: reducing the number of potential en-route conflicts.  Secondary goals: equitability, cost reducing.  Proposing a model particularly suitable in the trajectory-based airspace context that easy the end-to-end control of the flight (route).  Build an exploratory tool useful for strategic and pre-tactical ATFM planning.

5. © 2002 HEUDIASYC © 2002 HEUDIASYC 5 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Related works  In most of related works one can distinguish some modeling variations as deterministic versus stochastic and static versus dynamic models.  Bertsimas and Stock (98, 00), Dell’Olmo&Lulli (02), : a deterministic approach.  Delahaye and Odoni (97): stochastic optimization.  Ground Holding problem (Vranas, Odoni, etc.), single or multi-airport, static or dynamic, etc.  (Direct) Route Network (Letrouit, Barnier & Brisset,..)  Transport area (CRT Montreal)  Soumis et al.(00)

6. © 2002 HEUDIASYC © 2002 HEUDIASYC 6 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Relative research in telecommunications Relative research in telecommunications  Several routing problems in Telecommunication networks are similar:  Modeled as flow problems in a graph.  Flight level assignment can be stated as wave-length routing in optical network : graph coloring problem.  Both need global or local rerouting to avoid conflicts or reduce congestion.  Inter-sector coordination similar to hand-off in wireless networks, macro-cell ~ super-sector.  In contrast, problems in airway network are generally harder : more complicated constraints and integer variables, huge size.  All relative models need to be adapted for ATM area: (flows versus packets, single routing, uncertainties…)

7. © 2002 HEUDIASYC © 2002 HEUDIASYC 7 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Level and route flight assignment  Dynamic Network Modeling  Mathematical formulation and resolution approach

8. © 2002 HEUDIASYC © 2002 HEUDIASYC 8 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Dynamic Network Modeling  Transform a dynamic network to a static one (general case):  Discretize the time horizon into a finite number of periods  Replicating the network for each period: Connecting all nodes v t in G t with a node w t’ in G t’ iff (t’-t) = traversing time from v to w. Connecting all nodes in v t in G t with v t+1 in G t+1  Size of the obtained graph: |T|*|G|.

9. © 2002 HEUDIASYC © 2002 HEUDIASYC 9 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Mathematical model: Assumptions  For each flight, a set of preferred routes is given.  The flight level has to be determined in the set of a given preferred flight-levels.  The departure times are considered given at this stage (variation and/or perturbation will be considered in the next stage).  Each trajectory is determined by: the route (2D), the level flight, the speed (known) and the departure time.  The granularity of time is also fixed.

10. © 2002 HEUDIASYC © 2002 HEUDIASYC 10 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Mathematical model  Input data:  Airway network - the topology consists of a number of initial nodes, corresponding to airports and beacons, and a set of fixed links corresponding to the probable used ones.  All potential conflict areas (corresponding to nodes) will be represented by links.  All candidate flight-routes are supposed known, H(f) denotes the set of preferred routes for a given flight f.  Objectives:  Finding the route (j) and the level flight (l) for  t,  f  F(t) decision variables x t j,l,f.

11. © 2002 HEUDIASYC © 2002 HEUDIASYC 11 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Mathematical model: Notation 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 subset 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. Let T be the set of periods t. Let L be the set of possible flight levels l. x t j,l,f gives the traffic value (0/1) using the route j for the flight f taking-off at period t. c t j,l,f denotes the cost associated with the route j for the flight f taking-off at period t. a t,p j,l,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. y t k,l corresponds to the number of aircraft flying simultaneously through arc k, level l, during the period t. R gives the maximum number of aircraft involved in the same conflict.

12. © 2002 HEUDIASYC © 2002 HEUDIASYC 12 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Mathematical formulation Minimize subject to: (1)  t  T,  k  A’,  l  L, R – (y t k,l+ +1)  0 ; (2)  t  T,  f  F(t), ; (3)  t  T,  k  A’,  l  L, ; (4)  t  T,  f  F(t),  j  h(f),  l  L, x t j,l,f binary; (5)  t  T,  k  A’,  l  L, y t k,l  N*; Maximum number of aircraft in one conflict Total number of aircraft passing through conflict nodes Routing cost Only one route is chosen Total number of aircraft passing through arc k

13. © 2002 HEUDIASYC © 2002 HEUDIASYC 13 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Properties of the model  A mixed 0-1 linear programming model, important number of variables and constraints.  Novelty:  Sector-less oriented.  A «finer» model, considering en-route conflicts instead of en-route sector capacities.  Considering simultaneously route and flight levels.  (LP) Path formulation model.

14. © 2002 HEUDIASYC © 2002 HEUDIASYC 14 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Implementation  Simple Model  There are generally 3 preferred routes for each flight,  The departure time is also given.  Data Using  Real data from EUROCONTROL.  French (resp. European) Network 769, (resp. 5010), beacons & waypoints 1722 (resp. 22427) flights on 12 August 1999  Graph created by using links existing in flight plans.  Actually, a program composed of two parts:  Network modeling and preferred route generation.  Computing the optimized route and level flight assignment, (CPLEX).

15. © 2002 HEUDIASYC © 2002 HEUDIASYC 15 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Numerical Results data and hypothesis  European data air traffic on 12 August 1999,  22427 flights in a network with 5010 nodes.  Two time periods : 5 and 10 minutes.  Route generation is done on two ways:  Three shortest paths, (yellow line)  The optimal solution with respect to a multi-commodity flow problem, (blue line)  Computational times: ~ 3 hours

16. © 2002 HEUDIASYC © 2002 HEUDIASYC 16 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Numerical results Figure 2: Comparison of Number of Conflict Points Before and After Optimization (period of 5 minutes)

17. © 2002 HEUDIASYC © 2002 HEUDIASYC 17 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Numerical results Figure 1: Comparison of Number of Conflict Points Before and After Optimization (period of 10 minutes)

18. © 2002 HEUDIASYC © 2002 HEUDIASYC 18 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Global approach  The problem becomes unfeasible for large instances.  Level of optimization depends on the “quality” of preferred routes.  Global approach : a heuristic

19. © 2002 HEUDIASYC © 2002 HEUDIASYC 19 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Sketch of a heuristic  I. Resolving a multi-commodity flow problem.  A flow represents aggregation of flights with the same source, destination.  Obtaining a set of a limited number of candidate routes  II. Route and level assignment 1) Resolving the problem for each flight-level All possible flights for each level are considered. 2) Associate some penalty to each flight 3) Remove the most penalizing flights for each level.  repeat until there is a single level assigned to each flight.

20. © 2002 HEUDIASYC © 2002 HEUDIASYC 20 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Summary and concluding remarks  Deterministic model.  The model can be slightly modified in order to handle some “equitability”.  Uncertainties relative to the time granularity.  Simple model works well with medium size data.  Encouraging performance  Level of optimization depends on the “quality” of preferred routes.

21. © 2002 HEUDIASYC © 2002 HEUDIASYC 21 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003. Future works  Studying the feasibility of decomposition methods,  (i.e. on level flights).  Looking for dynamic route generation method.  Integrating the weather factor.  Extending the model with departure. scheduling time variation.  Adapt this approach for operational needs.

22. © 2002 HEUDIASYC © 2002 HEUDIASYC 22 EUROCONTROL 24 june 2003, 5 th ATM R&D seminar, Budapest 2003.  Thank you!!  Questions ?