MODEL FOR FLIGHT LEVEL ASSIGNMENT PROBLEM

1. MODEL FOR FLIGHT LEVEL ASSIGNMENT PROBLEM ICRAT 2004 - International Conference on Research in Air Transportation Alfred Bashllari National & Kapodistrian University of Athens, School of Sciences, Faculty of Mathematics Athens, Greece. Dritan Nace, HEUDIASYC Laboratory, UMR CNRS 6599 University of Technology of Compiègne, France

2. Outline • Introduction • Related works • Models for the Flight Level Assignment (FLA) problem • A mixed integer linear programming model • A constrained min-cost flow model • Conflict probability estimation • Numerical results • Concluding Remarks

3. Introduction • The route and the level assignment problem, aiming at global flight plan optimization, has already become a key issue owing to the growth of air traffic. • The principal objective of most of research work is minimizing the delay. • Obviously there is no a unique solution to this problem but there are a lot of partial solutions that could bring the actual ATM to meet all requirements. • One direction to reduce congestion is to modify the Flight Plans in a way to adapt the demand to the available capacity. • Our aim in this work, is to reduce the number of potential en-route conflicts through a better assignment of flight levels, called the FLA problem.

4. Relatedworks • Bertsimas and Stock have considered the Air Traffic Flow Management Rerouting Problem (TFMRP), treating simultaneously the time and the route assignment problem through a dynamic approach. • Doan and all have presented a deterministic model intended to optimize route and flight-level assignment in a trajectory-based ATM environment. • Delahaye and Odoni represent the problem of airspace congestion by the stochastic optimization point of view and particularly, using the genetic algorithm. • Genetic Algorithms (GA) are problem solving systems based on principles of evolution and heredity. • The level assignment problem is specifically addressed in some work at CENA, INRETS. • A common idea: representing the problem as a graph coloring one.

5. Models for the Flight Level Assignment (FLA) problem • We will introduce two models for the FLA problem. Both of them use potential conflict probabilities as input data. Thus a preliminary stage was elaborating a computation method for conflict probability. • In this work, not only space but also time proximity have been taken into account. • We have considered flights instead of flows and we then associate with two crossing routes the conflict probability. • We are restricted to the case of fixed single routes and we do not consider the possibility of route’s change.

6. A mixed integer linear programming model • Notation • L denotes the set of possible flight-levels l and Li the set of eligible flight levels for flight i. • The set of flights is noted with F. • xi,l: binary variable (0,1), takes 1 when the flight i, fly on level l and 0 otherwise. • pi,l: gives the penalty associated with flight i flying on level l. • ai,j : is a given constant that shows the probability of conflict for a pair of flights (flights i and j). • li,j: binary variable (0,1), takes 1 when the flight i and j fly on the same level and 0 otherwise.

7. A mixed integer linear programming model the number of potential conflicts global cost induced by chosen level • Mathematical formulation Minimize subject to: (1) (2) (3) (4) • Mathematical formulation Minimize subject to: (1) (2) (3) (4) unique level for each flight li,j to be a binary one

8. A constrained min-cost flow model We propose in the following to model the FLA problem as a min-cost flow problem augmented with a certain number of additional constraints, called constrained min-cost flow model.

9. A constrained min-cost flow model • Let G=(N, A) be a directed network defined by a set N of nodes and a set A of directed arcs • cijis the cost associated with each arc (i,j ) corresponding to alevel choice or a conflict arc. • Let F be the set of flights, L the set of eligible flight levels and S the set of potential conflicts (fi, fj) involving flights fiand fj. • Capacity uij for each arc (i,j ) is uij =1. • xijis flow variable and represents the flow on (i,j). Mathematical formulation: Minimize Subject to: (!) Kirchoff constraints; (!!) constrained conflict path constraints; (!!!) binary constraints;

10. Conflict probability estimation • In this annex we present a method for estimating the conflict probability between two aircraft by taking into account the uncertainties in their flight trajectories. • Initially we calculate the «minimum distance» between them: where ρ = , d2and d1distances of aircrafts from conflict point at the moment t0 = 0 and the angle between their trajectories. • Then we integrate in it the along-track and cross-track errors (inspired by works of Herzberg, Pajelli, Irvine). • Under certain assumptions, it is shown that the «minimum distance» is a random variable with normal distribution. Consequently the probability of conflict can be calculated as the cumulative probability with known distribution.

11. Conflict probability estimation • The probability of conflict corresponds to the part of the distribution of the «minimum distance» that lies between -s and +s,where s is the minimum allowed horizontal separation for on-route airspace. Pconflict = werethe mean μ and variance σ2 are given by: μ = λ( – ρ), σ2 = λ2 a(τ)2(1 + ρ2 ) + λ2b2((ρcot – cosec)2 + (cot - ρcosec)2) and erf(x) =

12. Numerical Results • These numerical results are for the mixed integer linear programming model • The test data, provided by EUROCONTROL, correspond to a real instance for a French and a European air transportation network, both as of August 12th 1999. • The granularities of time used for the tests was 10 min. • We considered just one preferred route for each flight. • The number of candidate levels is limited to 3.

13. Numerical Results • The program we developed is composed of two parts: • Network modeling, finding conflicts and calculation of conflict probability for each conflict. • Computing the optimized level assignment, using the ILOG MIP solver CPLEX 8.0 • In the case of European network from 15410 conflicts detected before the optimization, we reduce to 5275 conflicts, about 67% of conflicts are removed. • In the case of French network from 756 conflicts before the optimization, remain 229 conflicts after optimization, about 69% of conflicts been eliminated thanks to optimization.

14. Concluding Remarks • We propose in this study two mathematical models for the FLA problem. • The first model is an O-1 linear programming problem and doesn’t seem to be appropriate for large instances. • The experience of implementation has shown that the first model becomes infeasible for larger instances, as European network. • We are now studying how to build appropriate cutting planes. • The second model for the FLA problem is formulated as a min-cost flow problem augmented with additional constraints. • We believe that the second model can deal with larger instances of FLA problem, at least provide an efficient heuristic.