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An Analysis of Resource
Rationing Methods for
Collaborative Decision Making
Michael Ball and Thomas Vossen
R.H. Smith School of Business & Institute for Systems Research
University of Maryland
&
Robert Hoffman
Metron Aviation
1
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Ground Delay Programs
delayed
departures
delayed departures
delayed arrivals/
no airborne holding
delayed departures
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GDPs under CDM
Resource Allocation Process:
• FAA: initial “fair” slot allocation
[Ration-by-schedule]
• Airlines: flight-slot assignments/reassignments
[Cancellations and substitutions]
• FAA: periodic reallocation to maximize slot utilization
[Compression]
Note:
- reduced capacity is partitioned into sequence of arrival slots
- ground delays are derived from delays in arrival time
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Allocating Slots under CDM
Ration-By-Schedule:
Step 1: Order flights by their
original scheduled time of arrival
Step 2: Select the first flight that has not been assigned an
arrival slot.
– assign the selected flight to the earliest unassigned slot
– repeat step 2.
The resulting allocation is independent of current status of
flights and is not affected by status information given by
airlines!!
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Slot Reallocation under CDM
Need for Inter-airline slot exchange:
slots made available through flight cancellations and delays
Compression Algorithm
Initial
AAL1:1200
AAL2:1201
UAL1:1202
USA1:1203
UAL2:1204
COA1:1205
USA2:1206
AAL3:1207
S1200
S1202 CNX
S1204
S1206
S1208
S1210
S1212
S1214
Final
AAL1:1200
AAL2:1201
UAL1:1202
USA1:1203
UAL2:1204
COA1:1205
USA2:1206
AAL3:1207
S1200
S1202
S1204
S1206
S1208
S1210
S1212
S1214
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Generalizations and Extensions of CDM Procedures
•
•
•
View RBS as Process for Achieving Equitable
Allocation of Time Slots – extend to insure better
equity during dynamic GDP operation
Extension of RBS to Enroute Airspace: Priority
based on accrued delay
Interpret Compression as slot trading process:
extend from 1-for-1 trades to 2-for-2 trades
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#1. GDPs as Balanced Just-in-Time Scheduling Problem
flts
Possible
deviation measures
“ideal”
nb production rate
Xb Cumulative
production
na
time
• Airlines = products, flights = product quantities
• Minimize deviation between “ideal” rate and actual production
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GDP Situation
flts
“Release times” defined by scheduled arrivals
na
Xa
slots
Questions:
• What are appropriate “production rates” ?
• How to minimize deviations ?
• Managing program dynamics
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What is an appropriate production
rate/ideal allocation??
AVIATION OPERATIONS RESEARCH
Answer: RBS!!
It can be shown that RBS lexicographically minimizes the maximum
delay assigned to each flight.
General Principles of equity applied to a set of claimants; equity defined
relative to pair-wise comparisons:
in an equitable solution it should not be possible to improve the
allocation to a claimant at performance level p without moving
another claimant to a performance level of p or worse.
For the mini-max (RBS) solution:
if flight f has been assigned t* units of delay, it is impossible to
reduce the delay assigned to f without increasing the delay
assigned to another flight a value of t* or higher.
It can be shown that RBS satisfies some very fundamental axioms for
consistent, fair allocation process (there are other alternatives that we
are now investigating)
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Models and Algorithms for Minimizing Deviation
from Ideal Allocation
• General class of problems: minimize deviation between
actual slot allocation and ideal slot allocation
Variants based on:
– Objective function (deviation measures)
– Constraints on feasible allocations
• Minimize cumulative/maximum deviation:
– complex network flow model (based on JIT scheduling models)
can solve most variants
• Minimize sum of deviations between jth slot allocated to
airline a and ideal location for airline a’s jth slot:
– Assignment model
– Greedy algorithm for several cases
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GDPs and Flight Exemptions
• GDPs are applied to an “included set” of flights
• Two significant classes of flights destined for the
airport during the GDP time period are exempted:
– Flights in the air
– Flights originating at airports greater than a certain
distance away from the GDP airport
• Question: Do exemptions induce a systematic
bias in the relative treatment of airlines during a
GDP??
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Analysis of Flight Exemptions (Logan Airport)
4/
21
/0
1
AAL
4/
14
/0
1
4/
7/
01
USA
3/
31
/0
1
DAL
3/
24
/0
1
3/
17
/0
1
UCA
3/
10
/0
1
3/
3/
01
UAL
2/
24
/0
1
2/
17
/0
1
COA
2/
10
/0
1
CJC
2/
3/
01
1/
27
/0
1
1/
13
/0
1
1/
20
/0
1
TWA
40
30
20
10
0
-10
-20
-30
-40
1/
6/
01
Minutes/Flight
Deviation RBS (standard) vs RBS (+exemptions), Boston
GDPs
Flight exemptions introduce systematic biases:
• USA (11m/flt), UCA (18m/flt) “lose” under exemptions
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Reducing Exemption Bias
Objective :
• Use deviation model to mitigate exemption bias
– i.e. “inverse” compression
Approach:
• RBS applied to all flights whose arrival times fall within
GDP time window Î ideal allocation
• Set of exempted flights are defined as before (there are good
reasons they are exempted)
• Time slots given to exempted flights “count against”
allocation
• Delays allocated to non-exempted flights so as to minimize
overall deviation from ideal allocation
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Flight Exemptions
USA
AAL
4/
21
/0
1
DAL
4/
14
/0
1
UCA
4/
7/
01
UAL
3/
31
/0
1
COA
3/
24
/0
1
-40
1/
6/
01
-40
4/
21
/0
1
-30
4/
14
/0
1
-30
4/
7/
01
-20
3/
31
/0
1
-20
3/
24
/0
1
-10
3/
17
/0
1
-10
3/
10
/0
1
0
3/
3/
01
0
2/
24
/0
1
10
2/
17
/0
1
10
2/
10
/0
1
20
2/
3/
01
20
1/
27
/0
1
30
1/
20
/0
1
30
1/
13
/0
1
40
1/
6/
01
40
CJC
3/
17
/0
1
TWA
3/
10
/0
1
AAL
3/
3/
01
USA
2/
24
/0
1
DAL
2/
17
/0
1
UCA
2/
10
/0
1
UAL
1/
27
/0
1
COA
1/
20
/0
1
CJC
1/
13
/0
1
TWA
Deviation RBS ideal-Opt. model
2/
3/
01
Deviation RBS ideal-RBS actual
• Minimize deviations using optimization model that incorporates
exemptions
• reduces systematic biases, e.g. USA from 11m/flt to 2m/flt,
UCA from 18m/flt to 5m/flt
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Discussion
Approach yields system where:
• airlines are assigned priority lists
– based on sched. arr. times, constant during GDP
• dynamic changes (capacity, airline data) initiate (re)rationing
– ration according to airline priorities (compression)
• priority scheme cannot (completely) be maintained with
flight exemptions
– deviation model shows potential to reduce exemption bias
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#2. Extension of RBS to Enroute Airspace:
Priority Based on Accrued Delay
Priority given to
flight with largest
accrued delay;
accrued delay
(downstream) defined
as difference between
ETA and scheduled
arrival time.
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Scenario
Scenario: flights in air and on the ground; need to reroute flights around
flow constrained area:
Which flights are rerouted?? Which flights are delayed on ground??
Flow Constrained
Area (FCA)
Targeted Region (NY area)
???
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Impact on Allocated Delay
Motivation for Priority
Based on Accrued
Delay:
• Generalization of RBS
– equivalent relative to
equity principles
• Implicitly coordinates
multiple initiatives
since previously
delayed flights are
given priority.
1st come/
1st served
Priority
Based on
Accrued
Delay
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Implementation Alternatives
F-Based
S-Based
Step 1: For each unassigned flight,
calculate its total accrued delay
assuming it is assigned to the earliest
available slot for which it is eligible.
Step 2: Choose the flight with the
maximum accrued delay that has
not been assigned to a slot.
– assign the selected flight to the
earliest unassigned slot for which
it is eligible
– if there are remaining unassigned
flights, go to step 1; otherwise
stop
Step 1: Choose the earliest available slot,
s.
Step 2:
Find the set of flights, F, that can be assigned
to s.
For each flight in F, calculate the total
accrued delay assuming the flight is
assigned to s.
Choose the flight in F with the maximum
calculated accrued delay and assign it to
s.
If there are remaining unassigned flights go to
step 1; otherwise stop.
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F-Based vs S-Based: The Leap Frog Principle
F-Based:
• Utilization = 100%
• Avg. delay = 30 min
• Avg. of top 20% = 103
min
• Std. Dev. = 43 min.
S-Based:
• Utilization = 100%
• Avg. total delay = 30
min
• Avg. of top 20% = 50
min
• Std. Dev = 14 min.
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#3. Compression vs Trading
RBS allocates slots to airlines during a GDP
Compression provides a mechanism whereby
an airline gives up a slot it cannot use and
get one (from another airline) that it can use
…
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An Alternate View of Compression:
Inter-Airline Bartering
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UAL
AAL
Mediator:
FAA
DAL
NWA
SWA
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Mediated Slot Exchange
• Offer:
– slot_O: slot willing to give up
– slot_A1,…, slot_An: slots willing to accept in
return
• Each airline submits a set of offers
• Mediator determines set of offers to accept
and for each accepted offer, the returned
slot
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Default Offers
earliest time
of arrival
slot_An
slot_A1
slot_O
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Offer Associated with Canceled or Delayed Flights
time slot from
canceled flight
earliest time of
arrival for earliest
available flight
occupied
time slot
occupied
time slot
slot_O
slot_A1
slot_An
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1-for-1 trades to 2-for-2 trades
• Compression ⇔ 1-for-1 trading system, i.e. offers involve
giving up one slot and getting one in return (many offers
processed simultaneously)
• What about k-for-k or k-for-n offers, e.g. 2-for-2:
Trade??
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Initial Results
• Compression Benefits
– performance improvement if compression executed
after flts with excessive delay (>2hrs) are canceled
Compression Improvement
40
Global Max.
Compression
% improvement
35
30
25
20
15
10
5
bo
s0
106
-0
bo
1
s0
115
-0
bo
1
s0
119
-0
bo
1
s0
130
-0
bo
1
s0
208
-0
bo
1
s0
214
-0
bo
1
s0
221
-0
bo
1
s0
226
-0
bo
1
s0
310
-0
bo
1
s0
313
-0
bo
1
s0
321
-0
bo
1
s0
323
-0
bo
1
s0
330
-0
bo
1
s0
408
-0
bo
1
s0
418
-0
1
0
GDP
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Initial Results
• 2-for-2 Trading Model:
– proposed offers: all at-least, at-most pairs that improve
on-time performance
Trading Improvement
40
Global Max
Trading Model
30
25
20
15
10
5
06
-0
bo
1
s0
115
-0
bo
1
s0
119
-0
bo
1
s0
130
-0
bo
1
s0
208
-0
bo
1
s0
214
-0
bo
1
s0
221
-0
bo
1
s0
226
-0
bo
1
s0
310
-0
bo
1
s0
313
-0
bo
1
s0
321
-0
bo
1
s0
323
-0
bo
1
s0
330
-0
bo
1
s0
408
-0
bo
1
s0
418
-0
1
0
bo
s0
1-
%Improvement
35
GDP
Computational
Efficiency:
• 13sec avg.
• 67% solved by
LP relaxation
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Generic View of CDM GDP Procedures
• Equitable allocation among airlines
• Intra-airline maximization of economic
efficiency
• Inter-airline resource exchange