MODEL FOR FLIGHT LEVEL ASSIGNMENT PROBLEM ICRAT 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

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

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.

Related works 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.

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.

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. x i,l : binary variable (0,1), takes 1 when the flight i, fly on level l and 0 otherwise. p i,l : gives the penalty associated with flight i flying on level l. a i,j : is a given constant that shows the probability of conflict for a pair of flights (flights i and j). l i,j : binary variable (0,1), takes 1 when the flight i and j fly on the same level and 0 otherwise.

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

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.

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 c ij is the cost associated with each arc (i,j ) corresponding to a level 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 (f i, f j ) involving flights f i and f j. Capacity u ij for each arc (i,j ) is u ij =1. x ij is flow variable and represents the flow on (i,j). Mathematical formulation: Minimize Subject to: (!) Kirchoff constraints; (!!) constrained conflict path constraints; (!!!) binary constraints;

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 =, d 2 and d 1 distances of aircrafts from conflict point at the moment t 0 = 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.

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. P conflict = were the mean μ and variance σ 2 are given by: μ = λ( – ρ), σ 2 = λ 2 a(τ) 2 (1 + ρ 2 ) + λ 2 b 2 ((ρcot  – cosec  ) 2 + (cot  - ρcosec  ) 2 ) and erf(x) =

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 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. Test Instances Number of Flights Used Airports Used WayPoints NET_FR NET_EU

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 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.

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.