Efficiency and Equity Tradeoffs
in Rationing Airport Arrival Slots
Preliminary Results
Taryn Butler
butler@metronaviation.com
Robert Hoffman, Ph.D.
hoffman@metronaviation.com
Metron Aviation, Inc.
Herndon,Virginia
Single Airport GDP
• A Ground Delay Program (GDP) is a traffic
management initiative used to control the arrival
flow into a single airport
– The arrival flow is controlled by reducing the airport
acceptance rate (AAR), therefore reducing the number
of flights the airport can handle
– Arrival slots are allocated using the Ration-by-Schedule
(RBS) algorithm + compression
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RBS Algorithm in a Nutshell
•
•
RBS is a greedy algorithm
Algorithm:
1. AAR is established by traffic flow management
(TFM) for specific hours
2. Arrival slots are determined by dividing each hour
into the number of slots determined by the AAR
•
E.g. If AAR=30 flights/hour, then the hour is divided into 30
arrival slots: 1 slot every 2 minutes
3. Flights are assigned to slots based on their scheduled
and earliest arrival times, and such that the AAR is not
exceeded (essentially, first-scheduled first-served)
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Multi-fix GDP
• A Multi-fix GDP expands the control of arrivals
out to the arrival fixes for a single airport
– The AAR may be reduced at the airport and at any of
the arrival fixes
– Multiple flow constraints instead of one
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Why a Multi-fix GDP?
• More precise airport flow is needed for
– Fix load balancing (juggle flights between fixes)
– Lowered capacity may occur at some (but not all) of the
fixes
– Demand surges can occur at some fixes but not others
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Multi-fix GDP Complications
• A flight’s arrival fix is not always predictable
• Fix capacities are difficult to estimate because
they are mutually dependent
– Wx not very predictable hours in advance
• TFM might over-control the airport
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How would Multi-fix RBS work?
1. AAR and fix arrival rates (FARs) are established
2. Arrival slots are determined for the airport
3. Establish arrival bins for each fix
•
•
4.
5.
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Divided the FAR equally among the bins
E.g. If FAR=40 and 15-min bins are established, then no more
than 10 flights may arrive every 15 minutes
Assign flights to arrival slots based on scheduled and
earliest arrival times such that the AAR and the FAR are
not exceeded
If the flight can not be assigned to a slot without
exceeding the FAR, skip that flight and move to the next
flight
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Comparison
Multi-fix GDP
Single Airport GDP
Fix arrival flows
NE
NW
Airport
Airport
SW
SE
Airport arrival flow
Airport arrival flow
Airport and fix arrival flows are controlled
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Only airport arrival flow is controlled
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Counter-example
Flights
Fixes
f
A
Airport
1
Period 1
g
B
Period 2
1
1
B
1
1
A
1
B
0
Airport
1
Period 2
1
0
A
1
B
A
1 delay
unit
A
Period 3
f
g
1
B
Fixes
Period 1
1
A
2 delay
units
Flights
1
Period 3
1
B
1
1
1
Suboptimal solution from greedy algorithm. One of two flights must be delayed to a later time period, due
to airport capacity constraint in period 1. If flight g is delayed, then it must be delayed two time periods
due to constraints at fix B (left). However, if flight f is delayed, then only one time period of delay will
result (right).
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Purpose
• The purpose of this study is to examine efficiency
versus equity tradeoffs in allocating NAS
resources
– The resources are the arrival slots at an airport or at an
arrival fix
– The optimization model used in this study seeks to
allocate resources efficiently (disregards equity)
– The prototype software used allocates resources
“equitably” (in a manner similar to what is done now)
• A comparison is also made between the two
solutions
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Optimization Model
• Integer program model, similar to an assignment problem
• For this analysis, delay is defined as the difference between an
assigned arrival time/slot and the earliest scheduled arrival time/slot
that the flight could use
– The delay coefficient in the objective function is the difference between
the earliest available slot for a flight and all possible slots for the same
flight
Delay
f
 slott – earliest_eta
f
t  slott  earliest _ eta f
• Variables:
acid _ afixi _ slot t
afix _ t1
i  Arrival Fixes, t  Airport Slots
t1 is the quarter hour slot for every quarter hour withi n the GDP
• The objective is to assign flights as early as possible, therefore
minimizing delay
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Optimization Model
• The following is a mathematical description of the model
objective and constraints:
Minimize
 delay * (acid _ afix _ slot )
i
i
t
t
subject to :
c1 :
 acid _ afix _ slot
tslots
c2 : (
i
1
t
  acid _ afix _ slot
i
i flights tslots
 afixi  afix
c3 :
 acid _ afix _ slot
i flights
i
Bounds :
acid _ afixi _ slot t  1
FAR
4
Integer :
acid _ afixi _ slot t
afix f 
t
1
i  flights
t
)  afix _ t1  0
 t  (t1 , t 2 ) where (t1 , t 2 ) is the fix slot bin, f  arrival fixes
t  slots
i  flights, t  slots
f  arrival fixes
afix _ t1
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Prototype Software
• A prototype resource allocation tool was used to
execute the greedy algorithm
– RBS++ algorithm adapted to multiple fix constraints
• The tool was developed by Metron Aviation, Inc.
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Test Sets
• The prototype program was used to output flight information
for the following airports, dates, times (Zulu):
AIRPORT
DATE
GDP BEGIN
ATL
11/13/2002
1800
0200
585
DFW
JFK
11/13/2002
11/13/2002
1600
2100
2300
0100
477
134
ORD
SFO
11/13/2002
11/14/2002
1900
1700
0100
0100
547
208
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GDP END # FLIGHTS
Experiments
• There were two cases explored for each
experiment:
Case 1
Reduced airport capacity
– Case 1
• The airport is constrained during the GDP and
then returns to the maximum capacity after the
GDP
• The fixes are not constrained
• Analogous to a single airport GDP
• This case is used to determine if the CPLEX
model and greedy algorithm agree on the single
airport, single constraint case
Consistent fix capacity
Case 2
Reduced airport capacity
– Case 2
• The airport is constrained during the GDP and
then returns to the maximum capacity after the
GDP
• The arrival fixes are constrained during the GDP
and then return to the maximum capacity after the
GDP
• Analogous to a multi-fix GDP
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Reduced fix capacity
ATL Results
• Case 1
– % difference = 0.270
– Run time = 1113.89 sec
• Case 2
– % difference = 0.268
– Run time = 1141.81 sec
Solutions are essentially
the same
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DFW Results
• Case 1
– % difference = 0.012%
– Run time = 409.93 sec
• Case 2
– % difference = -5.521%
– Run time = 494.14 sec
Greedy algorithm is
slightly suboptimal
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JFK Results
• Case 1
– % difference = 0.156%
– Run time = 5.93 sec
• Case 2
– % difference = -9.525%
– Run time = 6.29 sec
Greedy algorithm is
slightly suboptimal
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ORD Results
• Case 1
– % difference = 1.131%
– Run time = 1229.76 sec
• Case 2
– % difference = -13.199%
– Run time = 853.16
Greedy algorithm is
substantially suboptimal
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SFO Results
• Case 1
– % difference = 1.759%
– Run time = 34.68 sec
• Case 2
– % difference = -24.563%
– Run time = 26.86 sec
Greedy algorithm is highly
suboptimal
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Additional SFO Experiments
• Additional experiments were conducted for SFO to further
investigate the large percent difference in Case 2
• The following are the parameters used:
AIRPORT
DATE
SFO
11/14/2002
1700
0100
208
SFO
11/19/2002
1700
0100
211
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GDP BEGIN GDP END # FLIGHTS
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SFO Experiment 2
• Case 1
– % difference = -0.359%
– Run time = 16.05 sec
• Case 2
– % difference = -31.031%
– Run time = 15.40 sec
Greedy algorithm is highly
suboptimal
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SFO Experiment 3
• Case 1
– % difference = 1.561%
– Run time = 40.32 sec
• Case 2
– % difference = -19.335
– Run time = 26.42 sec
Greedy algorithm is highly
suboptimal
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All Results
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Airport
Case 1
Case 2
ATL
0.270%
0.268%
DFW
0.012%
-5.521%
JFK
0.156%
-9.525%
ORD
1.131%
-13.199%
SFO 1
1.759%
-24.563%
SFO 2
-0.359%
-31.031%
SFO 3
1.561%
-19.335
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Conclusions
• The greedy algorithm assigned slightly less delay in all but
one Case 1 experiment
– Assume greedy algorithm is optimal
– Optimization model is a good match
– Little, if any, tradeoff between equity and efficiency in the singleconstraint case
• The model performed better than the greedy algorithm in
all but one Case 2 experiment
– Greedy algorithm is suboptimal
– Sizeable tradeoff between equity and efficiency in the multiconstraint case
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Conclusions
• RBS approach greedy algorithm is not an optimization
model and is quite complicated
• There are some differences in the way the model and the
prototype software create available slots at the airport,
which may account for the large differences in Case 2
– The CPLEX model does RBS and Compression in one step but the
greedy algorithm does these in two separate steps
• RBS throws away slots that flights do not get assigned to and
therefore, when Compression looks to move flights to earlier slots,
those earlier slots are no longer there
• The CPLEX model does not throw away any slots and can therefore
move flights to slots as early as the earliest_eta for the flight
• RBS does not use the earliest_eta, but Compression does
– Cancelled flights are handled a little differently in the greedy algorithm
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Conclusions
• A flight-by-flight analysis and an in-depth analysis
of the greedy algorithm is necessary to determine
why certain flights were assigned to certain slots
• Greedy Algorithm
– Multi-queue problem may not make optimal use of the
airport slots
– Single queue problem is almost always optimal
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