A Market Mechanism to Assign Air Traffic Flow Management Slots

Transcription

1 A Market Mechanism to Assign Air Traffic Flow Management Slots Andrea Ranieri, Lorenzo Castelli Università degli Studi di Trieste Dipartimento di Elettrotecnica, Elettronica ed Informatica 8 th USA/Europe Air Traffic Management R&D Seminar June 29 - July 2, 2009 Napa, CA Presenter: Andrea Ranieri

2 Introduction Statement of the Problem Capacity constraints in Europe on Airports and ATC sectors; ATFM ground delays used at a pre-tactical phase to balance demand with capacity; Delays imposed on a First-Planned-First-Served (FPFS) basis; Aircraft operators are not involved in the allocation process. Objective of the study To propose a mechanism which directly involves Airlines in the ATFM delays allocation process. Total ATFM ATFM delays > 15 min Estimated cost delays (min.) (min.) (Euro Prices) En-route Airport Total En-route Airport 23.8M 11.2M 7.6M 18.9M 900M 600M Source: EUROCONTROL PRR 2008

3 Related literature Exact models to solve the ATFM problem [Odoni, 1987]: first description of the problem; [Andreatta and Romanin-Jacur, 1987]: formalization for the single airport, stochastic case; [Vranas et al., 1994]: formalization for the multi-airport case; [Hoffman and Ball, 1997]: addition of banking constraints; [Bertsimas and Stock Patterson, 1998]: inclusion of en-route capacities; [Bertsimas et al., 2008]: formalization of a more compact IP formualtion; [Lulli and Odoni, 2007]: combination of airborne and ground delays for the European case. Global objective function obtained by aggregating the direct operating costs caused to flights by ATFM regulations.

4 Related literature Slot trading [Rassenti et al., 1982]: combinatorial auction mechanism for airport slots; [Vossen and Ball, 2006a]: bartering framework for compression with singleton exchanges; [Vossen and Ball, 2006b]: formalization of more complex exchanges involving sets of slots; [Ball et al., 2005]: analysis of objectives and issues of auctions in aviation. The use of side payments associated with complex slot exchanges has not been analyzed yet.

5 Motivation for the study SESAR: Single European Sky ATM Research Programme SESAR states that airspace users will be fully involved in the process of demand and capacity balancing. Implementation of ad-hoc CDM processes: Strategically: agreements on how traffic demand or individual trajectories will be adjusted if ANSP and Airports cannot provide sufficient capacity; Tactically: in the UDPP process designed to prioritize traffic queues caused by unexpected capacity shortfalls. The airspace users will respond in a collaborative manner to the Network Management with a demand that best matches the available capacity. [SESAR D3]

6 FPFS allocation The Slot Allocation List For a given Airport or Sector s : Resource s capacity: K s (entries/hour) Capacity activation period: [st time s, end time s ] Resource s slot list: S s ={1,..., NSlot s } Slot sl = [I sl, U sl ] S s has capacity 1 ETO(F)= ETO(F1)=10.00 ETO(F2)=10.03 ETO(F3)=10.04 ETO(F4)=10.07 Original Demand CTO(F1)=10.00 CTO(F2)=10.03 CTO(F3)=10.04 CTO(F)=10.06 CTO(F4)=10.08 Regulated Demand Delay caused by the most penalizing regulation is forced on the others.

7 Problem formalization Notation Set of flights: F = {1,...,F} Set of resources (airports + sectors): R = {1,...,N} Resource s slot list: S s = {1,..., NSlot s } Resources to be used by flight f : U f = {Cr 1 f,..., Cr NC f f Request j of slots feasible for f : q j f = {sl 1,..., sl NCf } } R with sl i S Cr i f Set of feasible requests for f : Q f = MxRq j=1 q j f Valuation of a request qf i : V (f, qi f ) 0 Definition We denote by Dem(f, j, s) the slot in sector/airport s, included in request j by flight f.

8 A combinatorial allocation problem The Central allocation model Z IP = max V (f, j)x(f, j) f F j Q f x(f, j) 1 s R, sl S s f F,j Q f :Dem(f,j,s)=sl x(f, j) = j Q f x(f, j) = 1 f F x(f, j) {0, 1} f F, j Q f { 1 if demand j is assigned to flight f 0 otherwise. Maximization of the global welfare function; Each slot sl can be assigned at most once; Each flight must receive exactly 1 request j Q f.

9 Central Allocation model The LP relaxation Z LP = max V (f, j)x(f, j) f F j Q f x(f, j) 1 s R, sl S s f F,j Q f :Dem(f,j,s)=sl x(f, j) = 1 f F j Q f x(f, j) 0 f F, j Q f Where x(f,j) can either be: integer (feasible allocation) or fractional (infeasible allocation)

10 Central Allocation model The LP dual problem Z DLP = min u f + f F u f + s R,sl S s p sl s j:dem(f,j,s)=sl p sl V (f, j) f F, j Q f u f 0, p sl 0 f F, sl {S 1,.., S N } Where optimal dual variables can be interpreted as: uf is the utility for flight f ; psl is the price for slot sl.

11 Central Allocation model Assumptions Linear prices: B Q f p(b) = sl B p sl; Quasi-linear utility: u(f, B) = V (f, B) p(b); Theorem (CE prices [Bikhchandani and Mamer, 1997]) The optimal dual solutions p sl define slot prices that support competitive equilibrium, i.e. at those prices a partition of slots exists that allocates each flight a utility-maximizing request and allocates every slot with positive price exactly once. Theorem (CE existence [Bikhchandani and Ostroy, 2002]) A competitive equilibrium for the combinatorial allocation problem exists if and only if the associated primal LP problem has an integer-valued solution, in which case the dual solutions form the set of competitive prices.

12 Optimal slot exchanges Corollary Each flight will weakly increase its utility by exchanging its non optimal set of slots T assigned by FPFS, with the set S optimal for LP problem and with side payments sl T p sl sl S p sl. f 1 f 2 f 3 f 4 sl 1 sl 2 sl 1 sl 2 sl 1 sl 2 sl 1 sl 2 FPFS allocation T Market allocation S Payments 0 0 +p 1,2 0 +p 1,3 +p 2, p 1,3 p 2,4 p 1,2 p 2,5 p 1,4 p 2,7 u(f, S ) u(f, T ) = V (f, S ) p(s ) V (f, T ) + p(t ) 0 f F

13 Special Case Corollary A competitive equilibrium will always exist in the case all flights compete for the same and unique resource s, since the problem reduces to an assignment which is always integral. This is a common situation in Europe! Number of regulations per affected flight (AIRAC 311: 31 st July 2008 to 27 th August 2008)

14 Determination of Competitive Equilibrium prices Solving the central dual LP problem to determine Competitive Equilibrium prices p sl could imply: Practical complications High communication cost of sending the auction inputs over the communication network; Complete disclosure for Airlines of private information V (f, j). An iterative mechanism can discover optimal prices without requiring complete information disclosure.

15 Distributed Allocation model The lagrangian problem Z D (λ) = max V (f, j)x(f, j) + x f F j Q f + λ sl (1 s R,sl S s x(f, j) = 1 f F j Q f x(f, j) 0 f F,j Q f :Dem(f,j,s)=sl f F, j Q f x(f, j)) Lagrangian multipliers λ sl 0, sl {S 1,.., S N } represent slot prices.

16 Distributed Allocation model The Lagrangian objective function is separable into F functions: The lagrangian flight problem Z D (f, λ) = max x (V (f, j) j Q f j Q f x(f, j) = 1 i U f λ Dem(f,j,i) )x(f, j) x(f, j) 0 f F, j Q f Each Airline can solve independently this problem for each flight it operates, given prices λ Dem(f,j,i) ; The latter is an assignment problem which always gives integral solutions and can be solved in polynomial time.

17 Distributed Market Mechanism The Central problem min Z D (λ) λ 0 It can be centrally solved with: The Subgradient method SG k s λ k+1 sl = max(0, λ k sl S r k SGs k ) = 1 x(f, j) f F,j Q f :Dem(f,j,s)=sl Sr k 0, (Sr k ) 2 <, k=1 k=1 S k r =

18 Distributed Market Mechanism Iterate over 0 < k MaxK A Central Authority determines price of resources λ k+1 sl according to the current excess of demand SG k s ; Airlines respond with the utility maximizing set of slots for each flight (myopic best response). Stopping criteria A capacity-feasible global solution is achieved each flight receives sl T p sl and pays sl S p sl A specified maximum number of iterations is reached the classical FPFS solution is implemented

19 Simulations results Datasets 3 cases: Case A: 1 sector; Case B: 2 sectors constant flying time; Case C: 2 sectors variable flying time. 200 instances/case, 20 flights/instance; Cost delay U(5, 20); MaxK = 50 iterations. CE Av. cost Convergence Av. # existence saving wrt FPFS iterations Case A 100 % 28 % 26 % 21 Case B 99 % 33 % 43 % 27 Case C 95 % 33 % 39 % 27

20 Conclusions and next steps In those situations in which a Competitive Equilibrium exists, each flight increases utility with respect to the FPFS allocation, by exchanging slots at the optimal market clearing prices; Competitive Equilibrium always exists in the case of: unit-demand; gross-substitutes valuations [Kelso and Crawford 1982], which excludes complementarity in valuation functions; different price structures (non-linear and non-anonymous), which might be considered unfair by Airlines. Some special problem structures might constitute a sufficient criteria for the existence of competitive equilibria; Approximate rather than exact algorithms could provide acceptable solutions.

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