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
8th USA/Europe Air Traffic Management R&D Seminar
June 29 - July 2, 2009
Napa, CA
Presenter: Andrea Ranieri
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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
delays (min.)
23.8M
ATFM delays > 15 min
(min.)
Estimated cost
(Euro - 2007 Prices)
En-route
Airport
Total
En-route
Airport
11.2M
7.6M
18.9M
900M
600M
Source: EUROCONTROL PRR 2008
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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]
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
FPFS allocation
The Slot Allocation List
For a given Airport or Sector s :
Resource s capacity: Ks (entries/hour)
Capacity activation period: [st times , end times ]
Resource s slot list: Ss ={1, ..., NSlots }
Slot sl = [Isl , Usl ] ∈ Ss has capacity 1
10.00
ETO(F1)=10.00
CTO(F1)=10.00
ETO(F2)=10.03
CTO(F2)=10.03
ETO(F3)=10.04
CTO(F3)=10.04
10.02
10.04
10.06
ETO(F)=10.06
ETO(F4)=10.07
CTO(F)=10.06
10.08
CTO(F4)=10.08
10.10
Original Demand
Regulated Demand
Delay caused by the most penalizing regulation is forced on
the others.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Problem formalization
Notation
Set of flights:
Set of resources (airports + sectors):
Resource s slot list:
Resources to be used by flight f :
Request j of slots feasible for f :
Set of feasible requests for f :
Valuation of a request qfi :
F
R
Ss
Uf
qfj
Qf
=
=
=
=
=
{1,...,F}
{1,...,N}
{1, ..., NSlots }
{Cr1f , ..., CrfNCf } ⊆ R
{sl1 , ..., slNCf }
SMxRq
with
qfj
sli ∈ SCrfi
= j=1
V (f , qfi ) ≤ 0
Definition
We denote by Dem(f , j, s) the slot in sector/airport s, included in
request j by flight f .
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
A combinatorial allocation problem
The Central allocation model
ZIP = max
XX
V (f , j)x(f , j)
f ∈F j∈Qf
X
x(f , j) ≤ 1
∀s ∈ R, sl ∈ Ss
x(f , j) = 1
∀f ∈ F
f ∈F ,j∈Qf :Dem(f ,j,s)=sl
X
j∈Qf
x(f , j) ∈ {0, 1}
x(f , j) =
∀f ∈ F , j ∈ Qf
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 ∈ Qf .
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Central Allocation model
The LP relaxation
ZLP = max
XX
V (f , j)x(f , j)
f ∈F j∈Qf
X
x(f , j) ≤ 1
∀s ∈ R, sl ∈ Ss
x(f , j) = 1
∀f ∈ F
x(f , j) ≥ 0
∀f ∈ F , j ∈ Qf
f ∈F ,j∈Qf :Dem(f ,j,s)=sl
X
j∈Qf
Where x(f,j) can either be:
integer (feasible allocation) or
fractional (infeasible allocation)
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Central Allocation model
The LP dual problem
ZDLP = min
X
f ∈F
uf +
uf +
X
psl
s∈R,sl∈Ss
X
psl
≥
V (f , j)
∀f ∈ F , j ∈ Qf
s∈j:Dem(f ,j,s)=sl
uf R 0, psl ≥ 0
∀f ∈ F , sl ∈ {S1 , .., SN }
Where optimal dual variables can be interpreted as:
uf∗ is the utility for flight f ;
psl∗ is the price for slot sl.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Central Allocation model
Assumptions
Linear prices: ∀B ∈ Qf
p(B) =
P
sl∈B
psl ;
Quasi-linear utility: u(f , B) = V (f , B) − p(B);
Theorem (CE prices [Bikhchandani and Mamer, 1997])
The optimal dual solutions psl∗ 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.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Optimal slot exchanges
Corollary
Each flight will weakly increase its utility by exchanging its non
optimal set of slots T assigned by FPFS,Pwith the setPS ∗ optimal
for LP problem and with side payments sl∈T psl∗ − sl∈S ∗ psl∗ .
f1
sl1
FPFS
f2
sl2
sl1
f3
sl2
sl1
f4
sl2
sl1
sl2
1
2
2
3
3
7
5
8
1
2
3
4
2
5
4
7
0
0
∗
+p1,2
0
∗
+p1,3
∗
+p2,7
0
0
0
∗
−p1,3
∗
−p2,4
∗
−p1,2
∗
−p2,5
∗
−p1,4
∗
−p2,7
allocation T
Market
allocation
S∗
Payments
0
u(f , S ∗ ) − u(f , T ) = V (f , S ∗ ) − p(S ∗ ) − V (f , T ) + p(T ) ≥ 0
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∀f ∈ F
A Market Mechanism to Assign ATFM slots
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: 31st July 2008 to 27th August 2008)
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Determination of Competitive Equilibrium prices
Solving the central dual LP problem to determine Competitive
Equilibrium prices psl∗ 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.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Distributed Allocation model
The lagrangian problem
ZD (λ) = max
x
+
XX
V (f , j)x(f , j) +
f ∈F j∈Qf
X
x(f , j))
f ∈F ,j∈Qf :Dem(f ,j,s)=sl
s∈R,sl∈Ss
X
X
λsl (1 −
x(f , j) = 1
∀f ∈ F
j∈Qf
x(f , j) ≥ 0
∀f ∈ F , j ∈ Qf
Lagrangian multipliers λsl ≥ 0, ∀sl ∈ {S1 , .., SN } represent slot
prices.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Distributed Allocation model
The Lagrangian objective function is separable into F functions:
The lagrangian flight problem
ZD (f , λ) = max
x
X
(V (f , j) −
λDem(f ,j,i) )x(f , j)
i∈Uf
j∈Qf
X
X
x(f , j) = 1
j∈Qf
x(f , j) ≥ 0
∀f ∈ F , j ∈ Qf
Each Airline can solve independently this problem for each
flight it operates, given prices λDem(f ,j,i) ;
The latter is an assignment problem which always gives
integral solutions and can be solved in polynomial time.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
Distributed Market Mechanism
The Central problem
min ZD (λ)
λ≥0
It can be centrally solved with:
The Subgradient method
λk+1
= max(0, λksl − Srk · SGsk )
sl
X
SGsk = 1 −
x(f , j)
f ∈F ,j∈Qf :Dem(f ,j,s)=sl
Srk
≥ 0,
∞
X
(Srk )2 < ∞,
∞
X
k=1
k=1
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Srk = ∞
A Market Mechanism to Assign ATFM slots
Distributed Market Mechanism
Iterate over 0 < k ≤ MaxK
A Central Authority determines price of resources λk+1
sl
according to the current excess of demand SGsk ;
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
P
sl∈T
psl and pays
P
sl∈S ∗
psl
A specified maximum number of iterations is reached
the classical FPFS solution is implemented
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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.
Case A
Case B
Case C
CE
existence
100 %
99 %
95 %
Av. cost
saving wrt FPFS
28 %
33 %
33 %
Andrea Ranieri - [email protected]
Convergence
26 %
43 %
39 %
Av. #
iterations
21
27
27
A Market Mechanism to Assign ATFM slots
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.
Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots
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Springer-Verlag, Berlin, Germany.
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Andrea Ranieri - [email protected]
A Market Mechanism to Assign ATFM slots