DEMAND MANAGEMENT AT CONGESTED AIRPORTS:
HOW FAR ARE WE FROM UTOPIA?
by
Loan Thanh Le
A Dissertation
Submitted to the
Graduate Faculty
of
George Mason University
in Partial Fulfillment of the
the Requirements for the Degree
of
Doctor of Philosophy
Systems Engineering and Operations Research
Committee:
George L. Donohue, Dissertation Director
Chun-Hung Chen, Dissertation Co-Director
Karla Hoffman, Committee Chair
Jana Kosecka
Daniel Menascé, Associate Dean for
Research and Graduate Studies
Lloyd J. Griffiths, Dean, The
Volgenau School of Information
Technology and Engineering
Date:
Summer Semester 2006
George Mason University
Fairfax, VA
DEMAND MANAGEMENT AT CONGESTED AIRPORTS: HOW FAR ARE WE
FROM UTOPIA?
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy at George Mason University
By
Loan Thanh Le
Bachelor of Science
University of Natural Sciences, Ho Chi Minh City, Vietnam, 1998
Master of Science
University of Paris I-Pantheon-Sorbonne, Paris, France, 1999
Director: George L. Donohue, Professor
Co-Director: Chun-Hung Chen, Associate Professor
Department of Systems Engineering and Operations Research
Summer Semester 2006
George Mason University
Fairfax, VA
ii
c 2006 by Loan Thanh Le
Copyright All Rights Reserved
iii
Acknowledgments
Early 2002, professor George L. Donohue gave me this invaluable opportunity of pursuing a Ph.D. degree in Air Transportation, and I began my quest in the Department
of Systems Engineering and Operations Research at George Mason University. Without his trust in my capability, none of this would have happened. Over the years, I
have learned so many things, accomplished a few things, and met people who have
been genuine professors, colleagues and friends. I would like to thank all of them who
made this experience possible and so enjoyable.
I have had the privilege of working with Professor George Donohue, my research
advisor, mentor, and role model, to whom I owe deep gratitude for many things.
Dr. Donohue introduced me to the wonderful world of air transportation. His broad
knowledge and outstanding vision in the aviation system guided me throughout the
journey. Dr. Donohue has high expectations of his students, and I thank him for
challenging me to carry through with the research. Beyond his academic virtues, I
am also grateful for many discussions with him that teach me the values of integrity
and tolerance. I look forward to working with Dr. Donohue in the future.
In the same manner, Dr. Chun-Hung Chen, my research advisor, exerted a strong
influence on me in daily research process. Not only did Dr. Chen convey to me
invaluable knowledge in discrete event simulation, he also made sure that my research
was on the right track. Dr. Chen demonstrated how to be a good researcher and a
good mentor by his academic rigorousness, diligence, and understanding towards his
students.
My sincere gratitude goes to Dr. Karla Hoffman, my committee chair, who taught
me invaluable knowledge in optimization theory, and difficult but fascinating problems of the airline industry. Dr. Hoffman’s work ethics and professional qualities
have always been a great source of inspiration for me, and will stay as such in my
future endeavors. She also kindly helped revise this dissertation with great care and
attention. I am deeply grateful for her time and efforts. Without her help, this
dissertation could not have been written as it is.
It is a pleasure for me to have Dr. Jana Kosecka in my committee. I would like
to express my thanks for her suggestions and warm encouragements throughout the
completion of this dissertation. I am also very grateful to Dr. John Shortle, Dr. Lance
Sherry, Dr. Donald Gross, and Dr. Alexander Klein for their thoughtful comments
and advice about my research. Their insights were always very helpful. I also would
like to thank my colleagues at Center for Air Transportation System Research, Arash
Yousefi, Richard Xie, Danyi Wang, Bengi Menzhep, Babak Ghalebsaz, Ning Xu,
and Jianfeng Wang, for enriching discussions regarding my research, and their warm
iv
friendship. Many thanks to Angel Manzo and Alerie Karen who were exceptionally
helpful in taking care of all my paperwork throughput the program.
Last but not least, I deeply appreciate the distant support of my parents. Their
self-giving love and constant encouragement stand by me in my pursuit of the doctorate. I also would like to thank my relatives in Virginia for sharing with me so many
relaxing and comforting moments. Finally, I thank Michael C. Ahlers for all of his
computer technical help, for the extra RAM he gave me to help boost my laptop’s
speed, and for always being there for me.
I can not express enough my thanks to all the people who have helped make this
experience possible and memorable!
v
Table of Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction and Problem Statement . . . . . . . . . . . . . . . . . . . . .
1.1 Airport congestion and congestion management measures . . . . . . .
xiii
1
2
1.1.1
Runway and airport expansion . . . . . . . . . . . . . . . . . .
3
1.1.2
Improvement of technology . . . . . . . . . . . . . . . . . . . .
5
1.1.3
Demand management . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Congestion management by demand management in the US . . . . .
7
1.3
1.4
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . .
12
15
1.5
Contributions of this dissertation . . . . . . . . . . . . . . . . . . . .
1.5.1 Primary hypothesis . . . . . . . . . . . . . . . . . . . . . . . .
17
17
1.5.2
Research scope . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.5.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
The potential readers . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
21
1.7 Dissertation outline . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Literature Review of Prior Research . . . . . . . . . . . . . . . . . . . . .
2.1 Congestion Management by Demand Management Measures . . . . .
21
23
23
1.6
2.1.1
Administrative options . . . . . . . . . . . . . . . . . . . . . .
24
2.1.2
Market-based options . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.3
Hybrid options . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.1.4
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.2
Route development, flight scheduling and fleet assignment models . .
40
2.3
Delay and cancellation estimation models . . . . . . . . . . . . . . . .
43
2.3.1
Analytical models . . . . . . . . . . . . . . . . . . . . . . . . .
43
3
2.3.2 Simulation models . . . . . . . . . . . . . . . . . . . . . . . .
The current slot allocation rules aggravate the congestion problem . . . .
47
51
4
Scheduling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
vi
5
4.1
General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
4.2
Profit-maximizing airline scheduling sub-models . . . . . . . . . . . .
56
4.2.1
4.2.2
The timeline network . . . . . . . . . . . . . . . . . . . . . . .
Interaction of demand and supply through price . . . . . . . .
57
59
4.2.3
Piecewise approximation of non-linear revenue functions . . .
60
4.2.4
Nesting revenue functions . . . . . . . . . . . . . . . . . . . .
62
4.2.5
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.3
4.2.6 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Airport’s allocation problem . . . . . . . . . . . . . . . . . . . . . . .
65
67
4.4
4.5
Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . .
70
72
Parameter estimation for scheduling models . . . . . . . . . . . . . . . . .
74
5.1
Timeline networks . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Arcs and arc lengths . . . . . . . . . . . . . . . . . . . . . . .
74
75
5.1.2 Arc costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Nonlinear revenue functions and piecewise linear approximation . . .
77
80
5.2.1
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
5.2.2
Processing segment fares . . . . . . . . . . . . . . . . . . . . .
82
5.2.3
Extrapolating the 10% ticket sample . . . . . . . . . . . . . .
83
5.2.4
Breaking down data from by-quarter-of-the-year to daily and
5.2
5.3
6
by-time-of-day . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Model validation: Unconstrained profit maximizing schedules . . . . .
87
5.3.1
Flight schedules by time of day . . . . . . . . . . . . . . . . .
88
5.3.2
Supply and price . . . . . . . . . . . . . . . . . . . . . . . . .
89
5.3.3
Flight frequencies and fleet mix . . . . . . . . . . . . . . . . .
89
A Stochastic Queuing Network Simulation Model for Evaluating Schedule
Delays and Cancellations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
6.1
Stochastic queuing network simulation model . . . . . . . . . . . . . .
97
6.1.1
Modeling objectives . . . . . . . . . . . . . . . . . . . . . . . .
97
6.1.2
Queuing network model . . . . . . . . . . . . . . . . . . . . .
97
6.1.3
Runway capacity submodel
6.1.4
Delay propagation submodel . . . . . . . . . . . . . . . . . . . 102
6.1.5
Cancellation and cancellation propagation submodel . . . . . . 103
. . . . . . . . . . . . . . . . . . . 100
vii
6.2
Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.1 Gate-out delay distributions . . . . . . . . . . . . . . . . . . . 106
6.2.2
6.2.3
6.2.4
6.3
Taxi time distributions . . . . . . . . . . . . . . . . . . . . . . 106
En route time distributions . . . . . . . . . . . . . . . . . . . 106
Cancellation and cancellation propagation . . . . . . . . . . . 107
Model calibration and application . . . . . . . . . . . . . . . . . . . . 110
6.3.1
Estimating delays and cancellations of alternative schedules . 110
6.3.2
Assessing impacts of changes in separation standards on airport
capacity and delay . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3.3
7
Demand Management at LaGuardia Airport: How Fare Are We From Utopia?117
7.1
Assumptions and parameters . . . . . . . . . . . . . . . . . . . . . . . 117
7.2
7.3
Baseline statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Investigated scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.4
Profit maximizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5
Seat throughput maximizing . . . . . . . . . . . . . . . . . . . . . . . 127
7.6
Compromise scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.7
8
Assessing impacts of changes in fleet mix on delay estimates . 115
7.6.1
Seat-maximizing within 90% profit optimal . . . . . . . . . . . 132
7.6.2
Seat-maximizing within 80% profit optimal . . . . . . . . . . . 139
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.7.1 Research questions and answers . . . . . . . . . . . . . . . . . 145
Conclusion and Future Work . . . . .
8.1 Contributions . . . . . . . . . . .
8.2 Recommendations for future work
Bibliography . . . . . . . . . . . . . . . . .
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148
150
152
154
A Appendix A: Airport Codes, Locations and Names . . . . . . . . . . . . . 161
B Appendix B: Problem formulations for ORD-LGA market in MPL . . . . 164
C Appendix C: Implementation of solution algorithm (column generation) in
C/Cplex Concert Technology API . . . . . . . . . . . . . . . . . . . . . . . 172
D Appendix D: Price elasticities estimates for several key markets . . . . . . 218
viii
List of Tables
Table
1.1
Page
New runways, runway extensions, and reconfigurations included in the
OEP [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
4
Runways, Runway Extensions, Reconfigurations or New Airports with
Environmental Impact Statements (EISs) or Planning Studies Underway [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1
Review of demand management measures . . . . . . . . . . . . . . . .
38
5.1
Aircraft types and seating capacities categorized to fleets . . . . . . .
76
5.2
Hourly costs for each fleet of 25-seat increment . . . . . . . . . . . . .
79
5.3
Example of demand extrapolation . . . . . . . . . . . . . . . . . . . .
83
6.1
Wake Vortex Separation Standards (nmiles/seconds) [2] . . . . . . . . 101
6.2
Example of delay propagation (unit: minute) . . . . . . . . . . . . . . 103
7.1
Daily average statistics of 67 markets in study, and overall statistics
(Source: ASPM Q2, 2005) . . . . . . . . . . . . . . . . . . . . . . . . 119
7.2
Scenarios investigated . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3
Daily statistics of profit-maximizing scenarios (* queuing delay estimates do not include international, non-daily and non-schedule operations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.4
Daily average statistics of fall-off markets in profit-maximizing scenario
at different runway capacity levels, Source: ASPM Q2, 2005. (*revenue
per passenger mile) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5
Daily average statistics of fall-off markets in seat-maximizing scenario
at different runway capacity levels, Source: ASPM Q2, 2005 . . . . . 128
7.6
Daily statistics of seat throughput maximizing scenarios (* queuing delay estimates do not include international, non-daily and non-schedule
operations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
ix
7.7
Daily statistics of 90% compromise scenarios (* queueing delay estimates do not include international, non-daily and non-schedule operations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.8
Daily average statistics of fall-off markets in seat-maximizing scenario
within 90% profit optimal at different runway capacity levels, Source:
ASPM Q2, 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.9
Numerical results of the 90% compromise scenario at 8 ops/runway/15min138
7.10 Daily statistics of 80% compromise scenarios (* queuing delay estimates do not include international, non-daily and non-schedule operations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.11 Numerical results of the 80% compromise scenario at 8 ops/runway/15min143
7.12 Projected effects on daily operations at LGA that result from a marketbased slot allocation at 8 ops/runway/15min (*queueing delay estimates do not include international, non-daily and non-schedule operations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.13 Daily average statistics of fall-out markets at 8 ops/runway/15min,
compromise scenarios, Source: ASPM Q2, 2005. (*revenue per passenger mile) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
x
List of Figures
Figure
Page
1.1
Increasing traffic intensity at EWR, LGA, and JFK airports . . . . .
1.2
Similar trends of average delay per aircraft at EWR, LGA, and JFK
airports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
10
1.3
Increasing operations vs. decreasing enplanements at EWR, decreasing
2.1
aircraft size at EWR and LGA . . . . . . . . . . . . . . . . . . . . . .
Overview of airline scheduling tasks (Barnhart) . . . . . . . . . . . .
2.2
2.3
Overview of DELAYS and AND models . . . . . . . . . . . . . . . . 45
Overview of NAS Strategy Simulator’s delay and cancellation component 47
3.1
The bottom left quadrant makes airlines lose money and airports con-
13
41
gested with litte passenger throughput, the upper right quadrant meets
airline and airport interests . . . . . . . . . . . . . . . . . . . . . . .
53
4.1
General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.2
Timeline network example for a city pair having the same time zone.
58
4.3
Nonlinear relationship of demand vs. price and the effect on renenues
59
4.4
Approximating a nonlinear function by a piecewise linear function . .
61
4.5
Nesting revenue functions . . . . . . . . . . . . . . . . . . . . . . . .
63
4.6
Branch-and-price solution method . . . . . . . . . . . . . . . . . . . .
71
5.1
Estimates of aircraft hourly operating costs by seating capacity (Source:
BTS Q2 2005) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
78
Estimates of hourly fuel consumption costs by aircraft seating capacity
(Source: BTS Q2 2005) . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5.3
Constrained demand curves of 10% BTS ticket price sample, Q1 & Q2
81
5.4
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear prorating of square root of leg distance helps account for fixed
5.5
cost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of demand extrapolation . . . . . . . . . . . . . . . . . . . .
82
84
xi
5.6
Estimates of quarterly constrained extrapolated demand curves for directional markets, Q2 2005 . . . . . . . . . . . . . . . . . . . . . . . .
5.7
Actual seat shares by time of day are used to allocate demands by time
of day, Q2 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8
91
Estimated demand curves for peak periods lie above those of off-peak
periods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.9
85
92
Estimates of daily demand curves and revenue functions by different
15-min time periods for TPA→LGA and LGA→TPA markets, Q2 2005 93
5.10 In each substitution group, higher actual seat shares of time windows
lead to scheduled arrivals in those time windows . . . . . . . . . . . .
5.11 Increases in seat capacity lead to decreases in fare and vice versa . . .
94
95
5.12 Changes in aircraft sizes in relation to frequencies are mixed . . . . .
95
6.1
Aircraft dynamics and network components . . . . . . . . . . . . . .
98
6.2
Hourly Empirical Cancellation Rates as the first component for simu-
6.3
lated cancellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
The relation of cumulative delay and cancellation used in simulating
6.4
cancellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Comparison of delay estimates vs. actual data . . . . . . . . . . . . . 111
6.5
6.6
Estimates of cancelled seats . . . . . . . . . . . . . . . . . . . . . . . 112
Adaptation of the system at high traffic levels and the effect on delay 114
6.7
Effect of fleet changes on delay performance . . . . . . . . . . . . . . 115
7.1
Geographical distribution of (flight) demand of LGA nonstop domestic
markets in study (see Table 7.9 for numerical values of actual frequencies)120
7.2
Densely distributed demand and increasing queuing delays near the
end of the day . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.3
Model suggests reduction in seats, which results in augmentation of
average ticket price . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.4
Delay reduction through consolidation of flights and aircraft upgauging 125
7.5
Percentage change of daily statistics from baseline . . . . . . . . . . . 126
7.6
Seat maximizing increases seats at high runway capacity levels . . . . 127
7.7
Despite increase in seats at high runway capacity levels, model suggests
gradual decrease of flights and aircraft upgauging . . . . . . . . . . . 129
7.8
Percentage change of daily statistics from baseline . . . . . . . . . . . 130
xii
7.9
(1) Profit-maximizing (2) Seat-maximizing within 95% optimal profit
(3) Seat-maximizing within 90% optimal profit (4) Seat-maximizing
within 80% optimal profit (5) Seat-maximizing within 60% or less of
optimal profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.10 Percentage change of daily statistics from baseline . . . . . . . . . . . 132
7.11 Model schedule reduces over-capacity peaks and retain buffers between
time windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.12 Seat-maximizing schedules within 90% profit optimal at 8 ops per
15min reduce flight delay significantly . . . . . . . . . . . . . . . . . . 135
7.13 Percentage change of daily statistics from baseline . . . . . . . . . . . 139
7.14 Model schedule reduces over-capacity peaks and retain buffers between
time windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.15 Seat-maximizing schedules within 80% profit optimal at 8 ops per
15min reduce flight delay less significantly . . . . . . . . . . . . . . . 140
D.1 Log-fit of major markets (O’Hare, Boston, National, and Fort Lauderdale) untruncates demand in lower price ranges . . . . . . . . . . . 218
D.2 Mid-sized markets (Atlanta, Tampa, Palm Beach, and Philadelphia)
use empirical extrapolated curves to avoid overestimation by the logfit right tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
D.3 Smaller markets (Charlottesville, Fayetteville, Lebanon and Nantucket)
use linear fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Abstract
DEMAND MANAGEMENT AT CONGESTED AIRPORTS: HOW FAR ARE WE
FROM UTOPIA?
Loan Thanh Le, PhD
George Mason University, 2006
Dissertation Director: George L. Donohue
Dissertation Co-Director: Chun-Hung Chen
The aim of this research is to help solve the airport congestion problem. The
returned air traffic growth is putting pressure on airport infrastructure. We identify
the causes of congestion to include (i) the High-Density-Rule (HDR) with grandfather rights allocating the limited number of airport slots to incumbent carriers, (ii)
weight-based landing fees that do not incentivize airlines to use larger aircraft, (iii) slot
exemptions granted to small markets served by 70-seat or less aircraft, and (iv) the
80%-use-it-lose-it requirement forcing airlines to fly low load-factor flights. With HDR
at New York LaGuardia and John F. Kennedy International airports scheduled to end
in January 2007, appropriate demand management measures are critically needed to
avoid overscheduling and severe congestion. Conventional economic wisdom suggests
that market-based mechanisms such as congestion pricing and auctions are an efficient
way to allocate scarce resources. Congestion pricing and auctions have had successful
applications in many fields. In air transportation however, the complexity of airline
network synergy, the influence of market power, and airport public goals require
xiv
the understanding of airline operations and market economics to design the right
incentives, as well as the understanding of potential implications of market response
on metrics of public interest such as enplanement opportunities, average fare, markets
served, aircraft size, and flight delay.
Our research demonstrates the existence of profitable flight schedules that maintain or improve the public goals for LaGuardia airport. To find these schedules, we
take a novel approach in modeling a profit-seeking, single benevolent airline, and develop an airline flight scheduling and fleet assignment model to simulate scheduling
decisions. This airline is defined as benevolent in the sense that the airline reacts to
actual price elasticities of demand estimated in a competitive market. Unlike existing
flight scheduling models that use fare as a parameter, our approach explicitly accounts
for the interaction of demand and supply through price. Extensive data mining of
publicly available databases is conducted to estimate cost and price elasticities of
demand. On the airport side, airline schedules are selected to maximize enplanement
opportunities such that these schedules fit into LaGuardia’s IMC rate constraints. To
reconcile the two conflicting objective functions, we look at two compromise solutions
that maximize the number of seats while ensuring that airlines operate within 90%
or 80% of profit optimality.
Our methodology applies to airports that have mostly local traffic. The results for
LaGuardia case study show that in the compromise scenarios at 8 ops/runway/15min,
the total seats are higher (increased by 1.1% and 3.4% for seat maximizing within
90% and 80% of profit optimality respectively) than that of the baseline while average
flight delay is reduced significantly (dropped 72% and 66% respectively). The number
of flights is decreased by 21% and 19%; aircraft size is increased by 27% and 28%.
The average ticket price is decreased slightly by 4% and 6% as a result of the small
increase in number of seats. There is no penalty in the number of markets.
We conclude that, with the airport’s runway rate restricted at the Instrument
0
Meteorological Condition (IMC) rate of 8 ops/runway/15min, there exist profitable
flight schedules that have fewer flights and reduce substantially average flight delay
while accommodating the current passenger demand at prices consistent with that
demand. The IMC rate provides a predictable on-time performance for the identified
schedules in all weather conditions. In addition, the reduction of flights through consolidation of low load-factor flights and aircraft upgauge alleviate the traffic pressure
on LaGuardia’s limited runway capacity, maintaining a safe runway utilization ratio.
Market access to LaGuardia is not affected when restricting airport operational rate
at the IMC rate. Airport authorities can use this “Utopia” as a benchmark or analytical support to design the right incentives in potential congestion management
proposals that encourage airline schedule changes in the desired directions.
Chapter 1: Introduction and Problem Statement
Air transportation is a complex, interactive system of systems that consists of vehicles,
airports, airspace, and the people who operate them, all integrated by communications, surveillance, and information subsystems. Its evolution has been marked by
incremental changes in technology and operating practices, and by dramatic changes
in societal and market demands upon it.
Since the emergence of commercial air transportation in 1926, the United States
has been the world leader in terms of productivity. FAA Aerospace Forecasts Fiscal
Years 2006-2017 [3] reported that by the year 2005, the industry annually operates
63.1 million flights on 7,836 aircraft; it transports 739 million passengers (40% of the
world’s enplanements), 74,300 tons of cargo between 3,500 domestic airports and 300
international destinations. At the busiest periods of the day, there are as many as
5,000 aircraft in the U.S. airspace that are operated by 138 U.S. commercial passenger
carriers, cargo carriers, and foreign carriers1 .
The Federal Aviation Administration (FAA) has funded studies to determine the
future demands on the air transportation system. One outgrowth of these studies was
the development of the Operational Evolution Plan (OEP) to increase the capacity
and efficiency of the National Airspace System (NAS), while enhancing safety and
security. OEP Version 7.0 [1] continues to focus on four core areas referred to as
OEP quadrants: Air Traffic Management (ATM) Flow Efficiency, Terminal Area
1
General aviation is not included.
1
2
Congestion, En Route Congestion, and Airport Congestion. The OEP 7.0 studied
the 35 busiest U.S. airports in terms of passenger activity.
1.1
Airport congestion and congestion management
measures
Within the next 10 years, forecasts by [3] predict that there will be as many as
1.1 billion air travelers per year in the U.S. Airports rather than enroute airspace
has been identified as the chokepoints creating the major portion of the congestion
in the system. An analysis of airport and metropolitan area future demand and
operational capacity [4] reveals that 15 airports, some not currently in the OEP, will
need additional capacity by 2013, and eight more will face capacity limitations by
2020.
35 OEP airports account for about 73 percent of commercial passengers in the
country. By 2005, 23 of these airports exceed their 2000 peak activity levels while
12 airports remain below 2000’s levels. Tampa and Newark airports are expected
to reach or exceed pre-9/11 levels in 2006 and 2007 respectively. Systemwise, the
FAA [3] forecasts the average annual growth of passenger enplanements to be 3.1%
from 2006 to 2017.
Air traffic growth is putting substantial pressure on airport infrastructure, especially at airports where there are limited possibilities for expansion. The imbalance
of travel demand and system capacity in the late 1990s resulted in substantial delays
and congestion at the busiest OEP airports such as O’Hare, Atlanta, Newark, and
LaGuardia. Following the events of September 11, 2001 and during the economic
downturn in mid 2002, passenger demand and activities at FAA air traffic facilities
3
declined significantly. However, the industry has recovered and the combination of
the recovery in passenger demand plus the shift in activity from larger aircraft to
smaller regional jets has resulted in increased delays at many U.S. airports during
2005.
The currently planned improvements in aircraft, airport, and airspace systems
and operational procedures may not be sufficient to safely, securely, and efficiently
meet the U.S. transportation needs of the next 10 years. This concern is reflected
by various congestion management efforts, initiated by the FAA and by regional
airport management entities. Congestion management includes the construction of
new runways and/or airports, improvement of technology, and demand management
measures that control use in order to manage delays and congestion.
1.1.1
Runway and airport expansion
The Airport Improvement Program (AIP) provides grants to public agencies - and,
in some cases, to private owners and entities - for the planning and development
of public-use airports. New runways/airports and runway extensions provide the
most significant capacity increase. Coupled with the creation of the associated gates,
terminals, taxiways and other auxiliary facilities, runway expansion improves the
throughput for the airport and for the national airport system overall. Table 1.1
lists eight runway projects (six new runways, one runway relocation and one runway
extension) that are currently included in the OEP and will be commissioned by 2009.
In addition, Table 1.2 lists nine more projects that are in the planning or environmental evaluation stage. These projects are not included in the OEP until all the
planning and environmental processing has been completed, the Record of Decision
4
CY
CY
CY
Expected
Airport
Runway
RoD ConstructionRunwayOperational Benefits
Issued to Begin to Open (% operations)
Minneapolis (MSP)
17/35
1998
1999
2005
19
Cincinnati (CVG)
17/35
2001
2003
2005
12
St. Louis (STL)
11/29
1998
2001
2006
48
Atlanta (ATL)
10/28
2001
2001
2006
33
Boston (BOS)
14/32
2000
2005
2006
Delay reduction
Philadelphia (PHL) 17/35 Ext. 2005
2005
2007
Delay reduction
Los Angeles (LAX) 7R/27L Reloc. 2005
2006
2007
Not available
Seattle (SEA)
16W/34W 1997
1998
2008
46
Table 1.1: New runways, runway extensions, and reconfigurations included in the
OEP [1]
has been issued, and the sponsor has provided the FAA with the dimensions, timing,
alignment, and planned use of the runway.
However, infrastructure expansion requires available land and extensive capital
funds2 . The approval typically takes up to 10 years to go through lengthy processes
from cost/benefit and environment effect analyses to land evacuation and construction. New runways and runway extensions often have a high degree of environmental
controversy and are frequently subject to legal challenges by the “not-in-my-backyard” community objection. OEP Version 7.0 [1] pointed out: “Experience has shown
that projected opening dates frequently change due to unforeseen circumstances at
the local level. Full benefits of new runways and runway extensions are often dependent on the use of operational procedures that have not yet achieved full acceptance
by pilots and controllers”. This observation further recognizes the alternative of using
existing infrastructure more efficiently, either through improved technology or better
2
Since 1999, seven new runways have been commissioned at OEP airports at a cost of $1.9
billion [1]
5
Airport or
Metropolitan Area
Chicago OHare (ORD)
Washington Dulles (IAD)
Chicago Metropolitan
Area (Peotone)
Philadelphia (PHL)
Ft. Lauderdale (FLL)
Las Vegas Metropolitan
Area (Ivanpah Valley)
San Diego Metropolitan
Portland International (PDX)
Salt Lake City (SLC)
Estimated CY
EIS Will Be Completed
Reconfiguration
2005
Runway
2005
Project
New airport
2006
Reconfiguration
Extension
2007
2007
New airport
2008
New airport
Extension
Extension
TBD
2007
2008
Table 1.2: Runways, Runway Extensions, Reconfigurations or New Airports with
Environmental Impact Statements (EISs) or Planning Studies Underway [1]
scheduling practice through demand management.
1.1.2
Improvement of technology
Improvement of technology consists of implementing capacity-enhancing ControlNavigation-Surveillance (CNS) systems for both enroute and departure/approach
phases. Weidner [5] assessed the airport capacity-related benefits of some CNS/ATM
technologies. Flight Management System (FMS) flight control provides lateral and
vertical navigation support that helps reduce flight variability in the extended terminal airspace. The Center-Terminal Radar Approach Control (TRACON) Automation
System (CTAS) Build 2 assists controllers in the sequencing and scheduling of arrival
traffic into congested airports, both at arrival fixes and landing runways. It is now
operational in prototype form at Dallas/Fort Worth airport (DFW). Currenly under
development, Active Final Approach Spacing Tool (AFAST) would provide controllers
6
with maneuver advisories to meet the CTAS sequences and schedules. Another future
concept consists of four-dimensional pilot-ATM arrival trajectory negotiation in the
extended terminal area. This would help synchronize arrival flows of aircraft equipped
with required-time of arrival (RTA) capabilities and traffic avoidance system such as
automatic dependent surveillance broadcast (ADS-B) equipment.
Modern CNS systems support air traffic flow management to better accommodate
demands on the day of operations. For long-term planning, viable procedures should
be devised to strategically bring demand in line with capacity. The recent US commission on the future of the Aerospace Industry [6] recognizes that technology alone
will not solve the modernization and capacity limitation problem. Policies need to be
changed to cope with future operational and economic needs of the air transportation
system.
1.1.3
Demand management
Fan02 [7] defines demand management measures as any set of administrative or economic measures - or combinations thereof - aimed at balancing demand in aircraft
operations against airport capacities. These measures intend to coordinate changes
of airline schedule. The International Air Transport Association (IATA) provides demand management guidelines for 3 different categories of airports: Non-coordinated
airports, schedules facilitated airports, and coordinated airports. Slot allocation procedures rely on airlines’ voluntary cooperation through IATA coordination at biannual conferences [8]. The reader is referred to “A Practical Perspective on Airport
Demand Management” [7] for a thorough survey on airport demand management
schemes around the world.
7
1.2
Congestion management by demand management in the US
Today, at most U.S. airports, airlines have latitude to schedule flights with no limits
on access other than those imposed by ATM requirements or by resource constraints
such as availability of passenger terminal gates. Air traffic controllers follow a firstcome, first-served acceptance rule.
Congestion management by demand management measures was first implemented
in 1969 with the High Density Rule (HDR)3 instituted at the John F. Kennedy International (JFK), LaGuardia (LGA), Newark International (EWR), Chicago O’Hare
International (ORD), and Ronald Reagan Washington National (DCA) airports4 .
The HDR limits the number of Instrument Flight Rules (IFR) takeoffs/landings at
High Density Traffic Airports (HDTA) by hour or half hour during certain hours of
the day. The HDR classifies user groups as air carrier, commuter, and other operators.
Reservations, also called slots, for regularly scheduled IFR operations conducted by
air carrier and commuter operators are allocated in accordance with 14 CFR part 93,
subpart S, Allocation of Commuter and Air Carrier IFR Operations at HDTAs, which
consists of administrative approval by the Secretary of Transportation. A reservation
authorizes an operation only within the approved time period unless the flight encounters an air traffic control (ATC) traffic delay. Advisory Circular 93-1 provides
information for obtaining IFR and Visual Flight Rules (VFR) reservations for unscheduled operations at HDTAs. FAA stated that the rule would expire at the end of
1969 but then extended it to October 25, 1970. In 1973, it was extended indefinitely.
3
4
14 Code of Federal Regulations [CFR] part 93, subpart K, High Density Traffic Airports
HDR restriction was lifted at EWR in the early 1970s, and at ORD on July 2, 2002
8
In addition, the perimeter rule limits flights at DCA and LGA at maximum 1,250
miles and 1,500 miles for nonstop market distance, respectively5 .
The deregulation in 1978 brought about the massive expansion of air travel and
also the competitive tension between airlines that had been historically present at the
HDTAs and new airlines that wanted to enter the markets. In 1985, “grand-father
rights” institutionalized the slot ownership for current holders of slots allocated to
domestic operations. These carriers may sell or lease their slots, and have to return a
slot back to a pool of unused slots for re-allocation if it is used by the current holder
for less than 80% of the time. This “use-it-or-lose-it” provision was initially designed
to prevent non-competitive holding of slots, promote efficiency in utilizing runway
capacity, and market entrance. However, there are two criticisms of this practice.
The first is that the airlines do not own these slots, and the airport operator should
be allowed to manage the allocation of these slots to assure safety, control congestion
and maximize passenger/freight throughput. The second is that airlines are accused
of being selective in choosing who is allowed to purchase slots from them, thereby
preventing competitors from gaining access to useful slots.
The Wendell H. Ford Aviation Investment and Reform Act for the 21st Century
(AIR-21), enacted in April 2000, initially intended to address the competition issue of
the grand-father rights at LGA, JFK and ORD. It exempted from the HDR limits certain flights by new entrant or limited incumbent air carriers using 70-seat or smaller
aircraft between a small hub or non-hub airport and these three airports. It also provided for ORD to eliminate slot controls in 2002, and for LGA and JFK to eliminate
5
The controversial Wright and Shelby Amendments imposed perimeter rule and aircraft size at
Dallas Love Field airport in 1979 and 1997 respectively, although not for congestion reason
9
slot controls on January 1, 2007. Immediately, airlines filed exemption requests for
more than 600 daily flights at LaGuardia, which represented a daily increase of more
than 50 percent of operations. The additional 300 accepted flights then pushed Fall
2000 demand 20% above the airport’s capacity, as shown in Figure 1.1. This resulted
in record delays at LGA, with an average delay per aircraft of almost 90 minutes (see
Figure 1.2).
There were more than 9,000 delay flights at LaGuardia in September 2000, up
from 3,108 in September 1999, which constituted more than 25% of the delayed
flights in the entire country, up from 12% in the previous year. The percentage
of delayed flights at LaGuardia, 15.6%, was nearly twice that at the nearest airport,
Newark International, at 8%. Furthermore, as the problems caused by congestion and
delays worsened, a ripple effect was experienced at airports across the NAS. Airlines
routinely cancelled scheduled flights, especially in afternoon and evening hours, in an
effort to avoid greater delays on other flights that would depart for LGA late in the
day.
On September 19, 2000, in response to mounting delays, the Port Authority of
New York and New Jersey (PANYNJ) announced that it was imposing a moratorium
on additional flights at LGA. The FAA followed with its own plan to rescind the
AIR-21 LGA slot exemptions that had already been granted and redistribute those
exemptions by a lottery. FAA described this measure as temporary and said it would
terminate restrictions on September 15, 2001. The controversial slot lottery randomly
allocated 159 exemption slots to incumbent carriers serving small communities and
new entrant airlines. On June 7, 2001, FAA placed a Notice in the Federal Register
regarding demand management at LGA. The Notice solicited public comments on
10
Figure 1.1: Increasing traffic intensity at EWR, LGA, and JFK airports
Figure 1.2: Similar trends of average delay per aircraft at EWR, LGA, and JFK
airports
11
potential methods to allocate LGA airport capacity.
The events of September 11, 2001, followed by the economic slowdown in mid
2002, brought down demand and diverted attention from airport congestion to airport safety. The outcome of the lottery remains in effect today with minor changes
determined by an administrative process. Over the past few years, demands at the
three airports have increased back to pre-2001 levels, and at LGA it now surpasses the
airport’s capacity (see Figure 1.1, where facility-reported capacities are calculated by
averaging actual daily capacities throughout the observation period). The rebound
in operations has brought about resurgence in delays to pre-2001 levels, with EWR
having average delay per aircraft as high as one hour. Delay patterns of LGA, EWR,
and JFK are shown in Figure 1.2. They exhibit periodic behavior with mid-summer
and mid-winter having highest delays. The similarity in pattern of the three curves
reflects that the three airports, being close to each other, experience the same seasonal
traffic trend and weather effects.
The removal of HDR at ORD airport in July 2002 experienced the same overscheduling and severe congestion problems as at LGA airport in 2001. From April
2000 through November 2003, American and United Airlines, the two dominant carriers that provide 85% of flights at ORD, increased their scheduled operations between
the hours of 12 p.m. and 7:59 p.m. by 10.5% and 41% respectively. However, seat
capacity by each carrier decreased more than 5.5 percent over the same period. By
November 2003, O’Hare was the most congested airport in the NAS with record number of delays: only 57% arrivals and 67% departures were on time, and delays averaged
about an hour per flight [3]. The government’s efforts in administrative congestion
regulation led to the two airlines’ two rounds of schedule cutbacks in March and June
12
2004, only to be met by other airlines’ addition of flights. Bilateral scheduling reduction meetings between DOT officials and individual airlines were then necessary.
In these meetings, the government mostly reinstated HDR for arrivals at ORD as a
temporary measure until April 2008.
1.3
Motivation
The over-scheduling that causes delay and congestion reflects increasing demand in
airline operations. However, this increasing demand is partly manifested by the inefficiencies within the overall airline schedules.
At EWR airport, the increasing number of operations is contrasted by the decline
in passenger throughput. The blue time-series bars of the first chart in Figure 1.3 plot
the annual actual operations at EWR, and the red time-series bars show the annual
passengers. These time series do not have a common y-axis as the chart intends to
show the relative trend of individual time-series. One notices three trends: (i) the
number of operations has increased little over the period; (ii) the number of passengers
has decreased slightly and (iii) the size of aircraft used has decreased significantly.
Despite constantly high levels of operations, the average aircraft size is decreasing
from 133 seats in 2000 down to 105 seats in 2005.
One can see similar trends of aircraft size at LGA. The overall shift from large jets
to smaller aircraft increases the system workload while keeping passenger throughput
the same or decreasing. Systemwise, regional jets carry fewer passengers each flight
and represent 37 percent of the commercial traffic at the nation’s 35 busiest airports,
up from 30 percent in 2000 [1]. For the FAA, less weight-based landing fees due to
increasing proportion of small aircraft have resulted in less tax revenues flowing into
13
Figure 1.3: Increasing operations vs. decreasing enplanements at EWR, decreasing
aircraft size at EWR and LGA
14
the Aviation Trust Fund, which pays for most of the FAA’s costs to run the system.
Due to the industry’s economics of scale and competition pressure, airlines have
incentive to schedule smaller aircraft at higher frequency, causing congestion to persist
even when the U.S. air traffic system builds more runways and/or improves computer
facilities. As a result, appropriate demand management measures have become more
critical to help regulate the demand, especially to prepare for the current planned
removal of HDR at LGA and JFK in January 2007. FAA’s 2001 “Notice of Alternative Policy Options for Managing Capacity at LaGuardia Airport” [9], DOT’s 2001
“Notice of Market-based Actions to Relieve Airport Congestion and Delay” [10], and
FAA’s 2005 “Notice of proposed rulemaking (NPRM), Congestion and Delay Reduction at Chicago O’Hare International Airport” were met with extensive response
from the industry [11] [12], the research community [7][13][14][15], and other interested parties [16][17][18][19] demonstrating the relevance of the issue. Subsequent
FAA-sponsored Congestion Game 1 conducted at George Mason University in Nov
2004 [20], and Congestion Game 2 conducted at University of Maryland in February 2005 [21] investigated the impacts of various administrative and market-based
options.
Similarly to those efforts, this dissertation aims to contribute to the understanding
of potential demand management solutions at congested airports such as EWR, LGA
and ORD. In particular, current slot restrictions at LGA and JFK are due to be
lifted on January 1, 2007. As of June 2005, no policy or plan is in place to manage
congestion after that time. If slot controls are extended in 2007, government goals of
increasing the fairness and efficiency of airport use will go unmet.
15
1.4
Statement of the problem
We demonstrate that the current congestion situation is caused in large part by the
existing rules. Specifically, we show that grand-father rights with 80%-use-it-or-loseit requirement, and slot exemptions lead to great inefficient use of airport capacity.
We point out that this inefficiency affects both airlines and airports. Faced with
projected traffic growth, the current rules at congested airports have to change.
We then examine the economics of providing air transport at congested airports
from both airline’s and airport’s perspective. We calculate average price elasticities
at various times of day based on sample ticket prices, actual sales and schedules.
We couple this with cost data for the airlines to determine the profit-maximizing
fleet size needed to accommodate demand. By examining such schedules, we can
determine goals that achieve better throughput without altering the natural behavior
of the flying public. By answering the above questions, we hope to better understand
incentives that would encourage a better reallocation of air traffic.
In order to better understand how to encourage efficient use of congested airports,
we state our research problem as follows:
Research Problem 1 Are current rules of slot allocation the main causes of the
congestion problem?
Research Problem 2 Focusing on LGA airport where the congestion problem has
been the most severe, and assuming that current slot allocation rules causing congestion identified in research problem 1 are removed, can we identify flight schedules and
16
fleet mix that are profitable to airlines and that can accommodate the existing demand yet reduce congestion, given current prices and price ellasticities? Specifically,
to accommodate profitably the current demand,
• What is the optimal fleet mix and frequency for each market?
• What would altering the schedule and fleet mix impact:
– Average delay per aircraft?
– Operation throughput?
– Enplanement opportunities?
– Fare?
– Number of markets?
Analyzing airline schedules requires the understanding of airline economics and
operations to avoid unduly affecting the business models of air carriers by forcing
impractical regulations. Therefore, modeling airline scheduling decisions is essential.
Initially, modeling individual airlines and their interaction in an N-side game setting
is theoretically desirable. However, this approach is impractical for many reasons:
• There is an infinite number of competition behaviors. Faced with incomplete
market information and competition pressures, an airline could react rationally
or irrationally, optimally or suboptimally depending on the market’s structure.
It is difficult, if not impossible, to model all possible behaviors or even be able
to identify such behaviors.
• Behavior of new entrants would require assumptions and data that are difficult
to validate.
17
• Publicly available data for individual airlines are limited, especially for small
carriers with little market presence. The data also contain inherent noise.
We therefore take a novel approach toward answering the above questions. We
model a single benevolent airline that seeks to optimize the profit of its operations at
LGA airport. While still modeled as profit-maximizing, this single airline is benevolent in the sense that (i) the airline reacts to actual and realistic price elasticities of
demand that are estimated in a competitive market, and (ii) it is willing to cooperate
with the public goals. Its resulting optimal schedule can provide an analytical benchmark towards which a reallocation of air traffic load should be encouraged to move.
Clearly, the idea of a monopoly airline is neither practical nor desirable, but solving the scheduling from a single benevolent airline’s perspective might help airport
authorities understand how best to encourage efficient use of airport resources, may
indicate the relative cost of serving specific markets, and also better understand the
effects of altering traffic loads within given periods on delays and prices. On the other
hand, the real market data we use to estimate price elasticities incorporate actual demand curves and prices of the current competitive market, not of a monopoly market.
Therefore, the concept of a single benevolent airline should not be too restrictive.
1.5
Contributions of this dissertation
The research presented in this dissertation seeks to validate the following hypothesis:
1.5.1
Primary hypothesis
Hypothesis 1 The current congestion situation is caused in large part by the existing rules of slot allocation. Specifically, grand-father rights with 80%-use-it-or-lose-it
18
requirement, and slot exemptions lead to great inefficient use of airport capacity.
Hypothesis 2 Without the restriction rules identified in hypothesis 1, there exist
profitable flight schedules that can accommodate the current passenger demand and
reduce flight delay.
1.5.2
Research scope
The case study of our research focuses on LGA airport. LGA is a typical non-hub
airport that serves mostly local traffic to and from domestic markets. The same
methodology can be used to examine other congested regions and expanded to consider larger networks. Specifically, the research seeks the optimal domestic flight and
fleet schedules for nonstop markets at LGA from a single benevolent airline’s perspective. We only consider markets that have daily profitable schedules to LGA. When
the model does not accommodate all the demand of a certain market (because it is
unprofitable to do so regardless of airplane size), which leads to capacity reduction or
even removal, such results can highlight the cost of maintaining the current demand
levels.
Excess of operations, once identified, would be assumed to move to reliever airports
in the area such as Stuart, White Plains, Islip, or Teterboro. How this excess should
be reallocated is beyond the scope of this dissertation.
Additionally, runway capacity is used as a surrogate to airport capacity, with the
assumption that other facilities such as ATC, taxiway, ramps, gates, and terminals
have sufficient resources to support the operation of airport runways at their capacity
levels6 . We evaluate the on-time performance of the resulting schedules, and other
6
Klein et al. [22] investigated the constraints of these support facilities on the fleet mix at LGA
19
metrics of interest such as the operations throughput, enplanement opportunities,
changes in fare, changes in the number of markets, and aircraft size.
The research investigates different optimal reallocation benchmarks for scenarios
with different capacities and public goals, along with guidelines for potential transition
paths. However, detailed transition plans require in-depth investigation into different
allocation mechanisms (administrative or market-based) and therefore are beyond the
scope of this dissertation.
1.5.3
Contributions
Contributions of this dissertation are categorized into four main areas:
Development of an airline flight and fleet scheduling model that incorporates the interaction of demand and supply through price (Chapter 3)
Appropiate congestion measures require the understanding of airline economics and
operations to avoid unduly affecting the business models of air carriers by forcing
impractical regulations. Therefore, modeling airline scheduling decisions is a central
part of this research. Unlike existing flight scheduling models that use fare as a parameter, our flight and fleet scheduling model considers fare as a variable negatively
dependent on supply level. This design choice allows the analysis of effects of changes
in schedules on average fares.
Development of a computationally-efficient solution algorithm to find the
optimal set of schedules (Chapter 3) We devise at each of the airports a column
generation algorithm to determine the optimal collection of schedules for each of the
Origin-Destination pairs based on the capacity constraints of the airports in study.
20
The decomposition algorithm decomposes the problem into a master problem that
optimizes use of the airports while the subproblems find optimal O/D schedules based
on current prices and demand curves.
Development of a methodology for estimating demand curves by time of
the day from publicly available sources (Chapter 4) We perform data mining
of ASPM and BTS databases to break down the aggregate data by quarter of the
year to aggregate data by day and time of day.
Development of a delay stochastic simulation network model to evaluate
flight schedules (Chapter 5) We develop a simulation model that explicitly considers wake vortex separation standards between categories of aircraft to simulate
runway capacity. Delays are estimated based on runway capacity. The simulation
model is simpler than the Total Airspace and Airport Modeler (TAAM), and yet
capable of evaluating the implications of fleet mix on runway operations throughput.
Demonstration of the existence of profitable airline schedules that reduce
congestion and accommodate current passenger throughput level (Chapter
6) We find the optimal demand allocation benchmarks for scenarios that assume
different capacity levels and public goals. The public goals investigated in this dissertation are (i) maximizing profit, (ii) maximizing seat throughput, and (iii) maximizing
the number of markets and seat throughput. The resulting schedules are then evaluated against the metrics of interest: Operations throughput, average flight delay,
seat throughput, average aircraft size, number of regular markets, and average fare.
The results show that at Instrument Meteorological Condition (IMC) rate of runway
21
capacity, airlines’ profit-maximizing responses can be expected to find scheduling solutions that offer 70% decrease in flight delays, 20% reduced in number of flights with
almost no loss of markets and no loss of passenger throughput.
1.6
The potential readers
This research should be of interest to both the public policy makers and airport
authorities. With modifications to include specific business constraints, airlines could
also extend this model to analyze and restructure the flight networks.
1.7
Dissertation outline
The next chapter answers the first hypothesis by conducting data analysis. We use
flight load factors and aircraft sizes as two main metrics to point out the inefficiency
in current slot usage. Current policy that affects these two metrics is then identified.
Chapter 3 provides a review of current research on demand management. We
present different proposals, studies and experiments, and summarize their premises,
analysis techniques, findings, pros and cons. In addition, we also investigate the
literature of works related to our research approach. These include integrated models
of flight scheduling and fleet assignment, and models of flight delay simulation.
In Chapter 4, we develop the mathematical formulation for our airline scheduling
model and government’s allocation model. While the airline scheduling model only
seeks to maximize profit, we formulate three different objective functions for the
government’s model. The interaction between demand and supply through prices is
explicitly incorporated in the airline model by the use of revenue functions and their
22
piecewise linear approximations. The concept of nesting revenue functions to model
demand spill and recapture is introduced next. Column generation is then used to
link these problems to find the final solution.
Chapter 5 explains how we estimate parameters for the scheduling models using
publicly available databases. To build the arcs of the flight network for each market,
we calculate flight lengths for different fleets. Cost is then added to the arcs using
estimated direct operating cost and fuel consumption. To estimate revenues, we
contract the daily demand curves for time windows of two time granularities.
Our stochastic delay simulation network model presented in Chapter 6 serves to
evaluate the output schedules. The model simulates the aircraft dynamics through
queuing systems of the enroute airspace and various airport facilities. We assume that
runway capacity is the main chokepoint. Wake vortex separation between pairs of aircraft determines runway throughput. We present delay and cancellation propagation
to simulate network effects.
In Chapter 7, the solution procedures are applied to LGA airport. We investigate
scenarios corresponding to different objective functions andn airport operational rates.
Metrics of interest are evaluated, compared, and interpreted.
Finally, chapter 8 summarizes the major contributions and findings of this dissertation. We also outline future improvements, and potential directions for research in
demand management.
Chapter 2: Literature Review of Prior Research
This chapter presents a survey of the latest proposals for congestion management,
followed by current developments of existing analytical tools that are needed in our
approach. We start with demand management measures and discuss the general advantages and limitations of each option. As airline scheduling reactions are important
in the assessment of new demand management procedures, we next describe models
that could be potentially used to simulate airline responses. The resulting schedules
then need to be evaluated in terms of delay performance. Therefore, we conclude the
chapter by looking at some major delay and cancellation estimation models.
2.1
Congestion Management by Demand Management Measures
When capacity expansion is either not possible or will not occur prior to serious delays without some congestion management tool, one needs procedures for limiting
the demand into a congested airport. Government agencies (e.g. the Department of
Transportation, the FAA, the House of Representatives), industry spokesmen, and the
research community have identified and studied potential methods to allocate runway
capacity at airports with high demand. Such options include administrative procedures, market-based options and some hybrid approaches. Administrative options
consider removing certain users, restricting entry of unscheduled flights, and altering the mix of users through lottery or legislature. Market-based proposals advocate
23
24
congestion pricing and slot auctions. We present many of these ideas next.
2.1.1
Administrative options
The Subcommittee on Aviation’s Hearing on The Slot Lottery at LaGuardia Airport
[23], FAA’s 2001 Notice of Alternative Policy Options for Managing Capacity at
LaGuardia Airport and Proposed Extension of the Lottery Allocation [9], and FAA’s
2005 Notice of proposed rulemaking (NPRM), Congestion and Delay Reduction at
Chicago O’Hare International Airport [24] suggest the following:
Reallocate general aviation (GA) aircraft slots. Six slots per hour at LaGuardia are allocated for general aviation flights by corporate jets. These unscheduled private flights could move to Teterboro airport in New Jersey, which is only
12 miles to midtown Manhattan and functions as a general aviation reliever airport.
However, Teterboro airport is currently highly congested as well.
Eliminate extra sections. An extra section is an additional flight that is added
dynamically by airlines to accommodate the overflow passengers. Extra sections are
popular on the Washington to New York and Boston to New York hourly shuttles
when the first flight (or section) fills up. Airlines do not need a slot or slot exemption
to operate an extra section.
Eliminate the use-or-lose-it requirement. The requirement that airlines use
their slots at least 80% of the time was imposed to ensure these limited assets would
actually be used and not hoarded. This has, in the past, forced carriers to operate
unwanted flights just to maintain their slots for “better times”, resulting in inefficient
25
use of runway capacity. If airlines did not have to be concerned about the loss of a
slot, they might be more willing to reduce their schedule.
Increase the use-or-lose-it requirement to 90% of the time for a two-month
period The option expects to create a faster turn-around of unused slots so that
scarce public resource can be exploited to the greatest possible extent. However, a
higher threshold of utilization rate is likely to increase the inefficiency created by the
80% limit.
Suspend leases under the buy-sell rule. The buy-sell rule allows the slot holder
to lease unused slots to other air carriers. Under this rule, a carrier could use a slot for
weekday flights and then lease the same slot to another carrier for weekend operations.
The Notice suggests that suspending leases under the buy-sell rule would reduce slot
usage rates by only allowing one carrier to use a slot during any given week.
Extend the lottery from slot exemptions mandated by AIR-21 to all slots
and slot exemptions. Slot lottery was initially considered as a temporary measure
as randomly allocating scarce resources obviously can not be optimal. Slot lottery
remains in effect until today because better solutions identified so far are not ready
to be implemented. The lottery of slot exemptions involves only a small number of
exemption flights by new entrants and small, non-incumbent carriers, to small and
non-hub airports. We argue that extending the lottery to all slots would unduly
disrupt the existing market structure with long established schedules of incumbent
airlines, and demand. Consequently, this option would only exacerbate the allocation
inefficiency and provoke strong opposition from incumbent airlines.
26
Allow antitrust immunity. Before the Airline Deregulation Act in 1978, the Civil
Aeronautics Board (CAB), FAA’s predecessor agency, had antitrust immunity authority that allowed airlines to meet and coordinate their schedule within capacity
constraints at an airport. However, such capacity reduction agreements were considered anti-competitive and were prohibited by the Deregulation Act. CAB retained
the authority to grant anti-trust immunity and that authority transferred to DOT
when the CAB was abolished at the end of 1984. DOT granted anti-trust immunity
to the airlines in 1987 so that they could meet and agree to adjustments in their
schedules in order to reduce the delays that were occurring at that time. In 1989,
DOT’s antitrust immunity authority expired. If this provision of antitrust immunity
was in effect, several small communities that gained service from more than one airline under the AIR-21 slot exemptions could coordinate to reduce their frequencies
and consolidate their capacities [23].
Various government agencies, the industry and research community provide qualitative assessment of these administrative options. “Reallocate GA aircraft slots”
would remove these small aircraft to make more slots available to larger airliners.
However, the healthy GA community at LGA would want to maintain their easy access to downtown Manhattan [17][18]. On the other hand, we think that “Eliminate
the use-or-lose-it requirement” is not practical. Faced with competition pressures
of the economics of scale, airlines would still schedule flights to compete for market
presence. Otherwise, this would allow slot hoarding, airlines will hold on to their
slots without using them, and therefore this option would hinder market access by
other carriers. As such, neither efficiency nor competition gain can be achieved. “Increase the use-or-lose-it requirement” might also cause airlines to lose their slots due
27
to unforeseen scheduling conflicts that they could have used productively at a lower
threshold, or force the airlines to fly even more unwanted flights [11][18]. “Suspend
leases under the buy-sell rule” could force airlines reveal their true slot demands but
could also aggravate the inefficiency of the use-or-lose-it requirement as airlines try to
hold on to their slots [23][12]. Similarly, random allocation of scarce runway capacities to airlines without consideration of economic implications on the markets served
in “Extend the lottery” option is highly inefficient and disruptive to long-standing
services [16][18]. Finally, “Allow antitrust immunity” likely causes potential negative effects on competition and price, which are the main reasons for AIR-21 slot
exemptions. [18] pointed out that “competition-related problems are inherent in any
administrative allocation of slots. These problems will not be fixed by incremental
changes but only by a more comprehensive market-based approach”.
2.1.2
Market-based options
Let the market decide, laissez-faire. An FAA-mandated 1995 study of the slot
rules concluded that lifting the HDR and allowing laissez-faire would double average
all-weather delays at HDTAs, leading to increased delays at other airports because
of the ripple effects on the Nation Aviation System (NAS) [25]. The delays that
occurred following the passage of AIR-21, and the removal of HDR at ORD airport
[26] demonstrated the impracticality of this option.
Congestion or peak-hour pricing. The current scheme of weight-based landing
fees incentivizes airlines to schedule higher frequencies of smaller aircraft. A small
aircraft occupies the same slot as a large one. Thus passenger throughput declines as
28
smaller aircraft is employed. In contrast, congestion pricing consists of charging a flat
landing fee based upon demand at a particular time of day. Therefore, fees for peak
periods will be higher than for off-peak periods, preventing low-value flights from
being scheduled in peak periods. Increasing per flight cost is expected to encourage
airlines to upguage, and therefore increase the passenger throughput.
While being relatively under-explored in aviation, congestion pricing of transport
networks has been common in road traffic. Examples include traditional methods
using toll booths such as turnpikes and toll roads, as well as more modern schemes
employing electronic toll collection such as the London congestion charge [27], and the
Trondheim toll scheme in Norway [28][29] which both use flat rate. Singapore’s Electronic Road Pricing [30] imposes time and location-varying rates for access into the
central business district with no toll during off-peak hours. The Highway 407 bypass
of Toronto, Ontario not only allows transponder-equipped cars but also uses digital
video technology to read license plates of cars without transponder, matches them
against the Motor Vehicle Registry’s database, and sends out a monthly bill. Highway 407 uses variable pricing: higher fees during the morning and evening commuting
times cause discretionary trips to shift to other times of the day, easing congestion for
those paying the higher rates. High-occupancy toll lanes (such as SR-91 in Orange
County, California and Interstate 15 in San Diego, California) charge single-occupant
vehicles who wish to use lanes or entire roads that are designated for the use of highoccupancy vehicles (HOVs, also known as carpools). There is a pre-determined toll
schedule for every hour of the day. Overall, these implementations, although faced
with initial objection and skepticism, have helped to tweak road usage patterns, decrease demand and average trip time in the tolled areas, eventually gaining public
29
acceptance.
Congestion pricing of airport runway access can be considered as a reactive measure in the sense that prices are adjusted in response to recorded delay levels. Price
regulator would set time-based prices for slots and airlines would set their demands
accordingly. As a result, airline long-term planning is subject to cost uncertainty.
Comments of The US Department of Justice on congestion pricing [18] pointed out
that “a drawback to congestion pricing is the regulator’s lack of knowledge about
what price to set. A regulator may not have good enough information to allow it
to set the right price without frequent experimentation”. Therefore, convergence of
the pricing process is uncertain. In addition, congestion pricing does not consider
the fact that airlines also need gates and ticket counters to operate. The flexibility
in scheduling might not be fully realized if dynamic allocation of support facilities is
not guaranteed.
The U.S. Department of Justice (DOJ) strongly advocates moving to a marketbased slot allocation system [17],[18]. [18] mentioned a congestion pricing application
to highway traffic in Southern California. Corbett (2002) [19] however raised the
concern that flights by small aircraft or to small communities are most likely to suffer
under a congestion pricing approach.
In addition to qualitative references above, recent research contributes more analytical analysis of congestion pricing. Daniel [14] models and estimates equilibrium
congestion prices at a hub airport. Daniel utilizes stochastic queuing theory to compute delays which then translate to congestion costs and prices. The stochastic queuing model is similar to that of Koopman [31] where arrival demands are modeled
30
as nonstationary Poisson distributions. However, it allows multiple servers in treating departure queues and arrival queues independently, and it assumes deterministic
service time. At the beginning of each 10-min period t, the probability distribution
of the number of aircraft in the system is estimated by solving a set of ChapmanKolmogorov equations. These equations are valid for all non negative values of the
utilization rate ρ in contrast to the steady state results which apply only to situations
where 0 ≤ ρ < 1. Specifically, Chapman-Kolmogorov equations solve for the probability pi (t), i=0,1,2...,m, of having i customers in the system at time t. Expected
queue length at t is then derived and expected waiting time at t can be calculated.
A bottle neck model of airline response adjusts traffic patterns to react to queuing
delays and congestion fees. Operations at hub airports form closely scheduled arrival
and departure banks to increase load factor and decrease connection time. The bottle
neck model assumes costs for each unit of deviation time when an aircraft (i) arrives
before the scheduled arrival time, (ii) arrives after the scheduled arrival time, (iii)
departs before the scheduled departure time, and (iv) departs after the scheduled
departure time. Individual airlines maximize their cost; the social-cost minimizing
planner minimizes the total cost to find congestion prices for actual flight times. Congestion prices are calculated mathematically by evaluating first-order derivatives of
cost formulas. Airlines use congestion prices to update flight costs and solve for the
optimal schedule. The process iterates until equilibriums are found. The approach
was illustrated with an empirical application of the model to Minneapolis-St. Paul
airport (MSP). The research demonstrated a mechanism to compute congestion prices
and attain equilibriums. The results in [14] showed that congestion pricing causes a
reallocation of small aircraft to off-peak periods or to other airports.
31
Pels [32] argued that “several characteristics of aviation markets may make naive
congestion prices equal to the value of marginal delays a non-optimal response”. Pels
pointed out the differences between congestion pricing for road traffic and for aviation:
road traffic considers link-based tolls and road users typically do not have market
power, air transportation is rather node-constrained and airlines often compete under
oligopolistic conditions. Pels’ airport pricing model reflects that (i) “airlines typically
have market power and are engaged in oligopolistic competition at different submarkets”, and that (ii) “part of external delays that aircraft impose are internal to
an operator and hence should not be accounted for in congestion tolls”. Pels analyzed
market power distortions in congestion pricing with a two-airport two-airline example
using test data.
Fan [33] demonstrated the effects of demand management when reducing the total number of flights or spreading out the demand profile. Fan estimated delay in
hour and in aircraft-hour of different schedules: (i) 1,348/day that causes 1 hour
and 20 minutes of delay/flight from 8pm-10pm, (ii) 1,205/day (-10%) that causes
20min/flight (-80%) for the same period, runway capacity set at 75ops/hour, and (iii)
a hypothetical schedule of 1,205/day with demand evenly distributed throughput the
day. The delay estimates suggested that a reduction in total demand is necessary for
airports with constantly high demand profile (LGA), and a shift in demand profile for
airports that have peaks and off-peaks. Fan then investigated the economic benefits
resulting from adopting fine versus coarse congestion tolls for markets with both symmetric and asymmetric carriers [13]. Time-based congestion prices were calculated as
the marginal delay cost (=marginal delay * average unit operating cost) caused by
adding a flight at different times of day. The results show that the current landing
32
fees are a lot less than the estimated marginal costs, which can be over $7000 for half
of the day when demand is 1,348/day. Fan concluded that given reasonably elastic
responses in terms of frequency adjustments, the benefits to carriers of instituting
congestion pricing generally exceed the amount of tolls collected.
Schank [34] looked at Boston, LaGuardia and Heathrow airports where congestion pricing had been implemented. He identified institutional barriers that prevent
effective implementation of this option. The identified institutional barriers include
the problem of displaced passengers when low-value flights are displaced, the political
and social equity issues. Social equity is defined as fair treatment vis--vis all groups
of aircraft size. As a result, the research does not recommend the use of congestion
pricing without adequate alternatives for displaced passengers.
Strategic slot auction in primary market Optimal allocation would require
that those flights that are most able to switch to off-peak slots do so, leaving peak
capacity to those that are willing to pay more for the service. Conventional economic wisdom suggests that auctions are an efficient allocation mechanism for scarce
resources. Auctions have been successfully used for radio spectrum allocation with
large numbers of interrelated regional licenses [35]. Although modifications would be
required for slot allocation, the use of auctions by the Federal Government to allocate
scarce resources demonstrates the feasibility of using auctions even for complex allocation problems. Airport slots could be packaged with gates and ticket counters. A
strategic auction would establish the rights for airlines to schedule service in specific
time slots. However, since the network is highly stochastic, flights might not be able
to depart/arrive during the designated slots. Therefore, on the day of operations,
33
slots could also be exchanged tactically. Altogether, auctioning slots at the strategic
level could synchronize traffic demand with limited system capacity, and provide a
legal basis for tactical slot exchange to encourage extensive usage of scarce resources.
Proposals to allocate airport time slots using market-driven mechanisms such as
auctions date back to 1979 with the work of Grether, Issac, and Plot [36]. Their procedure was based upon the competitive (uniform-price) sealed-bid auctions for primary
market, complemented by the oral double auction for the secondary market. Rassenti
and Smith [37] explored the use of combinatorial sealed-bid package auctions as the
primary market for allocating airport runway slots. This auction procedure permits
airlines to submit various contingency bids for flight-compatible combinations of individual airport landing or take-off slots. These studies carried out lab experiments
with cash-motivated subjects and hypothetical slot values. The focus was mainly
on the efficiency and robustness of the auction design in terms of demand revelation,
provided that bidders know the values of the slots and would perform truthful bidding
as their best strategy in a sealed bid auction. However, the assumption that airlines
know the values of slots to submit in a sealed bid auction may be impractical. Moreover, airline network constraints and the large number of slot combinations imply
that an iterative bidding process is indispensable to allow for bidders’ adjustments
without the need for enumerating an exponential number of alternative bids.
The 2001 study by DotEcon Ltd [38] investigated the use of slot auctions at
Heathrow and Gatwick airports in London. In addition to a thorough summary of
the current slot allocation schema in E.U., governed by E.U. Regulation 95/93, and
their implications, [38] proposed simultaneous multiple round auctions of “lot” complemented by a last sealed-bid round. A lot includes the right to use both the runway
34
and terminal facilities. To ensure incentive compatibility, the study proposed pricing
based on opportunity costs rather than the amount winners bid, i.e. winners pay the
highest value alternative use of the capacity. This pricing scheme can be thought
of as second-price payment for single item auctions or Vickrey-Clarke-Groves (VCG)
mechanism for multi-unit multi-item auctions [39][35][40]. The study concluded that
in general, slot auction in primary trading and bilateral buy-sell negotiations in secondary trading would benefit consumers by increased volume of flights and decreased
fares. However, this conclusion is drawn from qualitative analyses and highly aggregate calculations. There is no modeling of airline scheduling decisions.
A follow-up study by National Economic Research Associates (NERA) [41] extended DotEcon’s study [38] to provide a more systematic assessment of different
slot allocation schemes at 32 E.U. Category 1 airports. [41] suggested that market
mechanisms in both primary and secondary trading have the potential to address
many of the inefficiencies of current schema. Specifically, a simultaneous ascending
auction, where all lots are sold (either individually or in combination) is most suitable
for the allocation of airport slots. The study concluded that proper implementation
of market mechanisms will result in higher passenger volumes, higher load factors,
reallocation of flights to off-peak times or to uncongested airports, and lower fares
on average. Similarly to [38], the conclusion is highly qualitative with illustrative
calculations of aggregate statistics.
Fan [13] recommended simultaneously ascending auctions for airports with symmetric carriers. Interestingly enough, Fan suggested that a market-based demand
management policy can comprise both congestion pricing and slot lease auctions.
Ball (2005) et al. [42] reviewed slot allocation in the U.S and presented a framework
35
for airport slot auction design. The authors put forward the need for three types
of market mechanisms: an auction of long-term leases of arrival and/or departure
slots, a secondary market that supports inter-airline exchange of long-term leases and
a near-real-time market that allows for the exchange of slots on a particular day of
operation. [42] showed that not only would auctions assure that demand is in line
with capacity, but also that the proceeds from auctions would provide the investment
in aircraft avionics to increase capacity in the future by allowing a safe reduction
in aircraft separation. By including many public policy constraints in the design,
an auction encouraging new entries (by providing bidding credits), and discouraging
or disallowing monopolistic control over markets by not allowing a single career to
be awarded more than a given percentage of the available slots. Similarly to [38],
the auction design was a simultaneous multiple round ascending bid auction which
lumps landing/takeoff rights with gates, ticketing and baggage handing facilities. [42]
however did not provide any experimental results.
As an effort to identify potential demand management measures, the FAA and
the Department of Transportation (DOT) requested the member universities of The
National Center of Excellence for Aviation Operations Research (NEXTOR) to design
and conduct a series of government-industry strategic simulations or games to help the
government evaluate three candidate policy options [20]. George Mason University
(GMU) and the University of Maryland (UMD) conducted the fist game in November
4-5, 2004 to explore the HDR and congestion pricing options for LGA airport. Within
the context of the first game, a “Potential Notification of Proposed Rule Making for
an FAA Slot Auction” solicited comments about an ascending clock auction design
with intra-round and package bidding. The proposal suggested the auctioning of 20%
36
of the slots per 15-minute period at LGA every year, with a slot referring to both a
take-off and a landing. The auction determines winning bids for arrivals, and requires
that the associated departures be scheduled within 1.5 hours after the scheduled
landing time of the arrival. Vouchers are introduced as a way to offset the loss of
incumbents’ grandfather rights. A second game took place in February 24-25, 2005
where the industry played a mock auction of LGA landing slots. Both games involved
interested persons from the airline industry, academia, the FAA, airport operator
and federal government communities. Participants played decision-making roles in
simulated real-world scenarios. Due to time limitations, the few simulation rounds run
for each option are not enough to draw significant conclusions about airline scheduling
responses or to find equilibriums. However, the games achieved their design goal:
allowing interested parties to experience first-hand the process of congestion pricing,
and also introducing the industry to how an auction might be run for their application.
The researchers obtained much feedback from the participants. Of particular note
were (i) carriers’ requirement that slots to be combined with other facilities such as
gates, baggage handling facilities, ticket counters, and overnight parking spaces; (ii)
and the need of a transparent disposition of proceedings. Additionally, off-record
discussions proposed auctioning slots at two different levels of priority: high-priority
and low-priority slots. High-priority slots would be guaranteed access during IMC
when airport capacity is reduced, whereas low-priority slots would not. Although
this idea appeared interesting from the research point of view, it was considered too
complicated for implementation.
37
2.1.3
Hybrid options
Maintain HDR and Blind Buy/Sell in secondary market Although HDR
does not create property rights of runway slots, airlines are allowed to sell or lease
unused slots in the secondary market. The purchase, sale or lease of slots in the
secondary market can promote efficient use of slots. These transactions usually involve bilateral negotiation between airlines, on-going government intervention in the
secondary market slot transactions is minimal. However, airlines can discriminate
buyers/tenants to their benefits by giving slots to non-competing carriers and preventing access to competing ones. A blind auction of slots available in the secondary
market that is overseen by the FAA could prevent airlines from engaging in collusion
or purposely not selling/leasing to a particular competitor.
[18] pointed out that “competition-related problems are inherent in any administrative allocation of slots. These problems will not be fixed by incremental changes
such as adding a blind buy/sell rule as suggested in the Notice [9], but only by a more
comprehensive market-based approach”.
2.1.4
Summary
Table 2.1 summaries administrative and market-based options for demand management.
Aministrative
Measure
Reallocate GA slots
Eliminate extra sections
Eliminate the use-or-lose-it
requirement
Increase the use-or-lose-it
rate to 90% for 2 months
Suspend leases under
the buy-sell rule
Extend the lottery
Antitrust immunity
Market-based
Laissez-faire
Congestion Pricing
Hybrid
Slot auction
HDR and blind auction
in secondary market
Pros
Remove small aircraft, increase slots
available to larger planes
Maintain demand predictability
Incentivize airlines not to
use unprofitable slots
Faster turn-around
of unused slots
Reveal airlines’ true slot demand
Faster turn-around of unused slots
Simple
Cons
Objection by GA community
Remove the expansion flexibility of shuttle service
Airlines hold on to their slots w/o using them or
continue scheduling to maintain market presence
Airlines might fly even more unwanted flights
or lose slots due to unforeseen disruptive events
Force airlines to maintain inefficient
flights to keep the slots
Inherent inefficiency of random
allocation of valuable slots
Highly disruptive to long-standing services
Facilitate the consolidation
Hinder competition, require
of service among airlines
on-going government intervention
Simple, airlines would eventually
Unconstrained demand creates severe congestion
figure out the market equilibrium
Convergence uncertain
Allocate peak times to
Overscheduling, hence congestion, might remain
more valuable services
Cost uncertainty for airlines
Flat rate to incentivize aircraft upgauge Convergence uncertain
Schedule flexibility for airlines
Unfavorable to small markets
Allocate peak times to
Require complex packaging with other facilities
more valuable services
Subject to unpredictable bidding behaviors
Fixed cost incentivizes aircraft upgauge Require airline commitment, no warranty
Demand, hence delays, is controlled
of slot availability on the day of operations
Prevent slot hoarding among airline
Does not address grand-father rights
coalition in sell/lease of slots
in the primary market
Promote secondary market access
38
Table 2.1: Review of demand management measures
39
Despite very little practical experience of the application of market mechanisms
in airport slot allocation, researchers have made significant progress in trying to
understand the feasibility and implications of these options based on auction and
game theory as well as the use of market-based mechanisms in other domains. Marketbased mechanisms for airport slots raise many issues, including the implementation,
the effect on airfares, consideration of applicable legal requirements, the treatment
of international services, the use of any new revenues, as well as the impact on new
entrants, small airlines, competition, and service to small communities.
Overall, analytical analyses of congestion pricing focus on the convergence of the
pricing algorithm, whereas proposals for slot auction focus on the robustness and
demand revelation requirements of the auction design. However, they all require the
simulation of potential airline responses. Different approaches use different sets of assumptions about the airlines’ slot valuation models and the market’s structure. There
assumptions are not exhaustive nor are they easily validated. In addition, modeling
individual airlines leads to the difficult issue of simulating competition behaviors.
There can be an infinite number of competition behaviors. Faced with incomplete
market information and competition pressures, an airline could react rationally or
irrationally, optimally or suboptimally depending on the market’s structure. In auctions, bidders may attempt to game the auction rules by parking (bidding on low-value
items), signaling (indirectly showing interest on certain items to other bidders without actually bidding for them to keep the standing prices down) and bid shading
(placing a bid that is below what the bidder believes a good is worth). Although recent auction designs have become more robust, new behaviors are expected to emerge
constantly. Therefore, it is difficult, if not impossible, to model and validate all these
40
behavioral potentials. On the other hand, public policy decisions will be made only
with the best information available at the time.
2.2
Route development, flight scheduling and fleet
assignment models
The policy objective of congestion management is to optimize the utilization of airport
capacity by maximizing passenger throughputs within safe capacity and acceptable
delay levels. However, one can not overlook the objectives of air carriers, as commercial entities, to optimize profit or market share. Appropriate congestion measures
therefore require the understanding of airline economics and operations to create the
right incentives. In scheduled passenger air transportation, airline profitability is critically influenced by the airline’s ability to construct flight schedules containing flights
at desirable times in profitable markets (defined by origin-destination pairs). This
chapter describes the economic model of airline schedule planning, the policy model
of airport authorities, and the process that seeks the optimal compromise between
their conflicting objective functions.
Airline schedule planning includes route development, and schedule development.
Schedule development further entails frequency planning, timetable development and
fleet assignment. The output of these tasks is the ”external” schedule offered to
the flying public. Internally, aircraft routing, crew scheduling, and airport resource
planning allocate airline resources to accommodate the schedule, making sure the
offered schedule is operational. Figure 2.1 depicts the major tasks of airline scheduling
process. For more details of the process, see [43][44]
Route development is typically undertaken together through detailed analysis
41
Figure 2.1: Overview of airline scheduling tasks (Barnhart)
of market entrance possibility and profitability. Frequency (or service level) and
timetable are determined to maximize market coverage from a marketing standpoint
based on various considerations of market conditions, namely competition, passengers’
preference for travel times, and operational constraints such as allowed operating time
windows, rights of park aircraft overnight at certain airports, direct itineraries with
one stop, mandatory or optional flight legs. Most airlines make significant changes to
their schedules at least twice a year to accommodate marketing objectives and to adjust for seasonal changes in traffic patterns. Minor and incremental changes are made
to the schedule on a monthly basis to reflect holiday travel patterns or competitors’
scheduling changes.
While the timetable design problem involves selecting an optimal set of flight legs
to be included in the schedule, the fleet assignment problem assumes a flight schedule
with specified departure and arrival times and seeks to optimally assign aircraft types
42
to flight legs to maximize profit. Analysis of aircraft economics combined with segment demand is essential to determine the right fleet for the right market distances
in order to achieve cost efficiency, subject to the airline’s fleet availability constraint.
Airlines with heterogeneous fleets flying large networks with different haul ranges
have therefore harder fleet assignment problems to solve.
In this dissertation, as the goal is to model airline scheduling practice from the
perspective of airport authorities, we focus on the route, flight and fleet schedule
development. There has been little research on formal models for finding optimal
routes, frequencies and schedule times. Often, decisions involving these tasks are
made through ad-hoc analysis, and they are highly subjective. In contrast, the fleet
assignment problem has been studied extensively in the literature, traditionally as
a separate problem [45][46][47] and later in conjunction with the aircraft routing,
maintenance and crew scheduling problems [48][49].
Lohatepanont [44] integrates timetable planning and fleeting problems. In addition to the set of mandatory flights, flights are selected among a given set of optional
flights to find the optimal schedule. Linearly spilled and recaptured demand due to
the choice of fleets and optional flights require estimates for pairs of flight legs and
pairs of itineraries, which are difficult to estimate even with airline propietary data.
Within the “Congestion Management at US Airports” project by NEXTOR universities [20], Barnhart and Harsha [50] developed an airline slot valuation model
that simulates airline response to a slot auction. The proposed model is a mix integer
problem designed for individual airlines, and required demand and cost proprietary
data as inputs. The assumptions include (i) a multiple round package auction (ii)
43
airlines can bid for bundles of slots to build their daily schedules, (iii) incumbent airlines are given vouchers for their currently held slots and unused vouchers can be sold
after the auction, (iv) average fare is constant. The demand curves are functions of
frequency, and are given by piecewise input parameter values. The model maximizes
the total profit.
All these models use ticket prices as a parameter that does not correlate with
changes in supply: ticket prices stay constant regardless of the total number of seats
in the resulted schedule. This simplistic assumption helps keep the fleet assignment
model tractable and may be a reasonable assumption from a single airline’s perspective given the highly competitive nature of the market. However, when looking across
the industry, excess of aggregate capacity leads to decreasing average fares, even when
such fares are unprofitable.
2.3
Delay and cancellation estimation models
Delay and cancellation have been extensively estimated by a large number of models as
principal metrics to evaluate schedule performance. Two main approaches categorize
these models into analytical methods or simulation tools which have focus on the
processing speed or the level of details respectively.
2.3.1
Analytical models
Principal fast-time analytical models reviewed in [51] such as MIT’s DELAYS and
AND, and more newly developed models such as the delay and cancellation component
in FAA Strategy Simulator [52] are macroscopic models where aggregate values of
input parameters, namely traffic demand and airport capacity, are given or generated
44
to obtain approximate closed-formed estimates of delay. DELAYS is a dynamic and
stochastic queuing model that estimates queuing delay for access to an airport’s
runway system, excluding en route or terminal area airspace congestion, or bottlenecks
on the taxiways or aprons. AND connects individual airports by a simulation module,
which propagates delay among airports and updates their demand profiles. DELAY
and AND assume no cancellation.
We present these models in more details next.
DELAYS and AND The analytical queuing model DELAYS was developed and
extended by Koopman [31], Kivestu [53], Malone [54]. DELAYS models an individual
airport in isolation as a single server queue. It estimates the probability distribution
of aircraft number in the queue at a local airport, and from which derive local queuing
delays. Malone [55] connected airports in the network through a schedule of flights
with the simulation model AND, Approximate Network Delay. Figure 2.2 outlines
the interaction between DELAYS and AND.
DELAYS approximates the M (t)/Ek (t)/1/m queuing systems with nonstationary,
i.e. time dependent, Poission arrival processes and k th -order Erlang service times,
m is the finite capacity of the system. Erlang is chosen to approximate a wide
variety of service-time distributions having characteristics similar to the k th -order
Erlang. The approximation approach uses far less memory and CPU time for large
Erlang orders. When k=1, the system reduces to M (t)/M (t)/1, and as k → ∞, it
approaches asymptotically the M (t)/D(t)/1. The model performs calculations for
each time period, ex. by hour. The hourly arrival rates (or service rates) combine
the hourly demands (or runway rates) for landings and takeoffs. Beginning with
45
Figure 2.2: Overview of DELAYS and AND models
initial setting at time t=0 and iteratively for t=1h, 2h, 3h, ..., the model solves a
set of Chapman-Kolmogorov equations to compute the probability distribution of
the number of aircraft in the system. These equations are valid for all non negative
values of the utilization rate ρ in contrast to the steady state results which apply only
to situations where 0 ≤ ρ < 1. Specifically, Chapman-Kolmogorov equations solve
for the probability pi (t), i=0,1,2...,m, of having i customers in the system at time t.
Expected queue length at t is then derived and expected waiting time at t can be
calculated.
AND uses DELAYS iteratively to estimate flight delays for each time window.
For departure flights, delays calculated by DELAYS can be absorbed in-flight up to a
percentage cutoff (10%) of the total deterministic en-route time, the remaining delay
is propagated downstream to the arrival phase. At the arrival airport, the flight is
46
added to the queue of the corresponding time window, updating the arrival airport’s
demand profile. Arrival delays can also be absorbed on the ground up to a percentage
cutoff (10%) of the deterministic turn-around time. The remaining delay is added
to the next departure, and the demand profile is updated. AND was tested with a
prototype 3-airport network with an additional sink-source airport.
NAS Strategy Simulator The UMD-built NAS performance component in the
FAA Strategy Simulator is a high level analytical model that estimates monthly delays and cancellations in the NAS. The model studies the distribution of the hourly
utilization rate (ρ=scheduled demand/capacity) at an airport for each month. The
monthly 50th and 95th percentiles of ρ at all airports are weighted averaged based
on the fraction of NAS operations at each airport to obtain the monthly 50th and
95th percentiles of ρ for the whole NAS. The model then builds over a 6-year period
statistical models of monthly probabilities of cancellation vs. monthly NAS 50th percentiles of ρ, and of monthly average flight delays vs. monthly NAS 95th percentiles
of ρ. Figure 2.3 outlines the main steps of the approach.
To estimate flight cancellation probability of future scenarios, load factor is used
as follows:
Cancellation probability = e−3.75 ∗ (load factor ∗ (1 − ρ50))−3.34
and average flight delay is determined as:
Average delay = 38.62 ∗ (ρ95(1 − Cancellation probability)) − 23.84
47
Figure 2.3: Overview of NAS Strategy Simulator’s delay and cancellation component
2.3.2
Simulation models
Large-scale microscopic simulation models such as Total Airspace and Airport Modeler (TAAM) [56], Reorganized ATC Mathematical Simulator (RAMS) [57], and the
more recent NASA Airspace Concepts Evaluation System (ACES) [58][59] developed
by the VAMS project. Designed to be comprehensive, these models offer detailed
gate-to-gate simulation, including airport ground movement, terminal area departure/arrival sequencing, and en-route cruising phase. They can be used to as planning tools or to conduct analysis and feasibility studies of new ATM concepts. In
addition to numerical outputs, they also provide real time graphical visualization.
The Detailed Policy Assessment Tool (DPAT) developed by MITRE [60] is also a fast
time simulation without graphical support. These complex models typically require
48
long learning curves and extensive data input efforts. They often have little support
for stochastic events that often perturbate the system, nor do they allow a flexible
way of canceling flights and propagating delays.
Total Airspace and Airport Modeler (TAAM) simulates the physical aircraft
movement in all phases of flight from gate to gate, airport operations, and ATC’s
decision-making process. Developed in and continuously improved since 1987, TAAM
has become a state-of-the-art fast time simulation model that offers specialized features such as Conflict Detection/Resolution (CDR), user-defined rules, and unlimited
zooming capability to display the smallest details in 2D or 3D. TAAM has been used
extensively in the literature to model ATC workload [61], redesign airspace sectorization [62], evaluate the impacts of Reduced Vertical Separation Minimum (RVSM) [63],
study changes in runway usage and implications on airline schedules [64], and other
applications.
Reorganized ATC Mathematical Simulator (RAMS) is a fast-time, discreteevent computer simulation model developed and supported by the Model Development Group (MDV) at Eurocontrol, France. RAMS offers 4-dimensional flight profile
calculations, 4-dimensional aircraft conflict detection, rule-based conflict resolutions,
4-dimensional aircraft maneuvering for conflict resolution, and 3-dimensional airspace
sectorizations. The model also provides methodologies to analyze airspace structure,
ATC systems and future ATC concepts. The model displays 2D real time graphic
visualization of the simulation. The latest version of RAMS, RAMS Plus, includes a
limited convective weather model represented as dynamic forbidden zones. RAMS’
principal areas of application have been ATC workload, free routing investigation,
49
free flight study, and airspace capacity/density.
Airspace Concepts Evaluation System (ACES) developed by NASA as a fasttime simulation and modeling capability for design and trade-off studies of system
level concepts within the NAS. ACES utilizes the high level architecture (HLA) and
an agent-based modeling paradigm to create the large scale, distributed simulation
framework necessary to support NAS-wide simulations. HLA is a set of processes,
tools and middleware software, developed by the Department of Defense, to support
plug-and-play assembly of independently developed simulations. Various models, categorized into Agent, Infrastructure, and Environment groups, represent weather, human behavior, aircraft dynamics, flight planning and controller workload elements.
NAS agents operate within the NAS Environment and communicate with each other
and the NAS Environment through the NAS Infrastructure.
The Detailed Policy Assessment Tool (DPAT) is a fast-time, global air traffic
simulation that can model current and future air traffic, for any world region. DPAT
represents airports and airspace as a network of finite-capacity resources and models
individual flights and itineraries. DPAT computes delays at airports and air traffic
control sectors and propagates delays across system resources. DPAT applications
include system-wide airport and airspace planning, assessment of benefits of proposed
system improvements, and identification of the effects of future traffic growth. DPAT
supports flight delay propagation [65][66].
A common trait of the analytical models that use aggregate parameters is that
they do not distinguish departures and arrivals. Neither can they discern the effects
50
of changes in traffic mix. Details of individual flights are not modeled, losing connections between flights, or network effects. The simulation models on the other hand,
due to their complexity, represent many challenges to users. Donohue and Laska [67]
found that TAAM and RAMS “require significant amounts of data that are sometimes difficult to obtain”, and “learning to use these models take considerable time
and effort”. Additionally, they provide little support for stochastic events and flight
cancellation. Most of the available models are closed source tools, thus eliminating
the possibility of extending their capabilities to new research applications. Obtaining access to most of the presented models is also cost prohibitive for independent
researchers.
Chapter 3: The current slot allocation rules
aggravate the congestion problem
In the chapter we conduct data mining to prove inefficient use of runway capacity
due to current slot allocation scheme.
The monthly T-100 Segment table, compiled by the Bureau of Transportation
Statistics (BTS) [68], reports domestic and international operational data by U.S.
and foreign air carriers. For each row, it contains, among other data items, carrier,
aircraft type, number of performed departures and seats, and number of passengers
transported for that month. We divide the number of seats by the number of performed departures to get average aircraft size, and the number of passengers by the
number of performed departures to get average load factor. Figure 3.1 collects six
months of data for LGA, JFK, and EWR airports. Cumulative percentage of data
points for reference values of the bottom x-axis is displayed on the top x-axis, and
for reference values of the left y-axis on the right y-axis. Notice the cumulative
percentages are highly non linear.
As LGA is a non-hub airport with mostly domestic traffic within 1500-mile perimeter, while EWR and JFK accommodate international and long-haul flights, the ranges
of aircraft size at the three airports are different. An aircraft considered small in EWR
might be a mid-size one for LGA. However, if we only look at 50-seat or less aircraft,
then these small aircraft make up a significant portion at all three airports: 40.6%,
23.6%, and 46% of the total flights at EWR, JFK, and LGA respectively, and flights
51
52
having 60% or less load factor represent 22%, 9.4%, and 36.2%. The high percentage of low load factor flights at EWR and LGA suggests that there is an excess of
operations, resulted arguably from airlines using high frequencies to maintain their
competitiveness. The large presence of small aircraft at EWR and LGA also relate
to the fact that LGA serves markets within a 1500-mile perimeter, whereas EWR is
a domestic hub of Continental Airlines.
Splitting the charts into four quadrants along the median aircraft size and 50%
load factor allows us to better understand the observations. The bottom quadrants are
low load-factor flights that are likely unprofitable to the airlines. The left quadrants
relate to flights having fewer seats than half of the traffic. Interests of airlines and
airports coincide in the upper right quadrant, where private profitability comes with
public goal of having high enplanements. The bottom left quadrant is inefficient for
both airlines and airports, and only contributes to the congestion.
There are three main causes for this inefficient use of airport capacity. Firstly, the
High-Density-Rule allocates slots to incumbent airlines to serve markets within 1500mile perimeter. Secondly, slot exemptions granted by the AIR-21 and the lottery [9]
to new entrant carriers flying 70-seat or less aircraft to small and non hub airports.
Subject to the “use-it-or-lose-it” requirement, airlines that are granted the slots have
to use their slots up to 80% of the time, profitable or not, or have to return them.
Thirdly, weight-based landing fees incentivize airlines to use smaller aircraft at high
frequency to compete for market share. As a result, low load-factor flights and smaller
aircraft use up LGA’s runway capacity, aggravating the congestion.
53
Figure 3.1: The bottom left quadrant makes airlines lose money and airports congested with litte passenger throughput, the upper right quadrant meets airline and
airport interests
Chapter 4: Scheduling Models
In this chapter we present the optimization models for airline scheduling subproblems
and also present the airport’s allocation problem that we will refer to as the “master
problem”. In the airline scheduling subproblems, we explain how demand curves are
used and how we then determine price equilibria in the resulting revenue functions.
We approximate the nonlinear revenue functions by piecewise linear functions. Demand spill and recapture between substitutable time windows are accounted for by
nesting revenue functions between time windows of compounding granularities. The
resulting schedules of individual markets are inputs to the master problem where we
solve a set packing problem over a variety of different objective functions. The solution methodology for solving the overall problem is a Dantzig-Wolfe decomposition
when the columns being generated are schedules generated based on an announced
price vector.
4.1
General approach
Figure 4.1 depicts our general approach. The three NY area airports are referred to
as cluster airports, and the other airports as outstation airports. There are two optimization components with two separate objective functions: the single benevolent
airline seeks profit-maximizing schedules, and the airport seeks the best combination
of schedules that fits into airport capacity constraints and maximizes pre-determined
54
55
Figure 4.1: General approach
public goals. The airline finds optimal schedules by solving a multi-commodity network flow subproblem for each market. Each market is defined as a directional pair of
outstation and cluster airports, and only markets that have daily nonstop domestic
service are included in this study. The airport component collects these schedules, or
columns, and solves a set packing master problem. The dual prices computed from
the linear relaxation of the set packing problem serve as feedback to the subproblems
by providing prices that then determine alternative schedules (i.e. generate columns)
that better satisfy the objective function of the master problem. We continue the
process until no further columns can be identified.
In the airline submodels, we model explicitly the interaction of demand and supply
through price. Changes in frequencies and aircraft size, i.e. changes in supply, would
56
lead to a revision in prices. This interaction affects demand and the airlines’ bottom
line. From an airport’s point of view, price is also important in the overall evaluation
of the quality of air transportation service. Therefore, in our models, price is a variable
and the resulting nonlinear revenue functions are approximated piecewise.
Flight scheduling requires demand estimates for different times of the day. Such
demands are interdependent, i.e. demand can be spilled from one time window and
recaptured by others. Instead of estimating demand spill and recapture between
pairs of time windows, we use nesting revenue functions to model demand for time
windows of different granularities (for more on this see Chapter 4). Demands of finer
granularity time windows are therefore constrained by demands of coarser granularity
time windows that include them. In this way, we assume that when we sum the
captured demands of finer granularity time windows, the total can not exceed the
captured demand of the compounding coarser granularity time window. We only
look at one level of nesting in this dissertation with a generic substitution grouping
of time windows. However, nesting is flexible and can be market-specific to model
peak and off-peak time windows.
4.2
Profit-maximizing airline scheduling sub-models
Airline scheduling submodels take as input estimates of demand, price elasticities of
demand by time of day, and costs of operating different fleets, to build the timetable
of flights such that profits are maximized. The timetable includes origin airport,
destination airport, departure time and arrival time of each flight and the fleet type
assigned to that flight. In network optimization theory, a fleet assigned to a flight is a
commodity flow and fleet mix scheduling is a multi-commodity flow problem defined
57
on a time-line network. As timetables for individual nonstop domestic markets at
LGA can be built separately (although not independently as they are all subject to
capacity constraints at LGA), we develop a time-line network for each market with all
potential flows and solve the optimization to find the schedule of profit-maximizing
flows.
4.2.1
The timeline network
A timeline network is built for each pair of airports (o, o0 ). At each airport, time of day
is partitioned into time windows represented by nodes: nodes in T are time windows
0
of airport o, and nodes in T 0 are time windows of airport o , all nodes ordered in Zulu
time. The set of directed ground arcs (i, j) ∈ AG with i, j ∈ T (i, j ∈ T 0 ) represent
ground flows where aircraft stay at airport o (o0 ) from time window i to time window
j. For each valid fleet k ∈ K at o and o0 , a set of directed flight arcs (i, j) ∈ AF
with i ∈ T and j ∈ T 0 or vice versa constructs potential flights for that fleet in the
timetable. Similar to Lohatepanont [44], any outgoing arc at any node is considered
to happen after any incoming arc at that node, and an additional directed ground
arc from the last time window to the first time window is added at each airport to
represent aircraft parking overnight.
Specifically, let:
fk,o,o0
gk
t(i)
0
block time by fleet k from airport o to airport o , in time windows
minimum turnaround time of fleet type k, in time windows
order of time window i in Zulu time
then the directed arcs emanating from nodes in T are created as follows:
58
ground arcs
-u
-u
-H
-u -u u -u
u -H
u -H
u ?- u
airport 1 H
* HH
* *
* HH
* HH
* H
* H
* H
H H
H
H
H
H
H
H
H
H
H
flight arcs
HH H
HH H
HH H
H
H
H
j
H
ju
H
j
H
j
H
j
H
ju
H
ju H
H
ju H
-u
H
u - u
u H
u
u H
airport 2 (a) subnetwork for fleet 1
that requires 2 time windows for a flight arc
u -u
u u u u -u airport 1 P
1u 1 u
1 u
1P
1P
1P
1P
PPPPPP
P
P
P
P
P
P
P
P
P
P
PP
P
P
P
P
P
P
PP
PP
PP
PP
PP
PP
PPPPP
P
P
P
P
P
P
P
P
P
P
q
P
q
P
q
P
q
P
q
P
qu P
qu P
u
u
u
u
u
u
u
u
airport 2
(b) subnetwork for fleet 2
that requires 3 time windows for a flight arc
Figure 4.2: Timeline network example for a city pair having the same time zone.
i ∈ T , j ∈ T 0 , (i, j) ∈ AF
i, j ∈ T , (i, j) ∈ AG
i, j ∈ T , (j, i) ∈ AG
if t(i) + fk,o,o0 + gk = t(j)
if t(i) + 1 = t(j)
if t(i) ≤ t(k) ≤ t(j) ∀k ∈ T
Similarly, the directed arcs emanating from nodes in T 0 are created as follows:
i ∈ T 0 , j ∈ T , (i, j) ∈ AF
i, j ∈ T 0 , (i, j) ∈ AG
i, j ∈ T 0 , (j, i) ∈ AG
if t(i) + fk,o,o0 + gk = t(j)
if t(i) + 1 = t(j)
if t(i) ≤ t(k) ≤ t(j) ∀k ∈ T 0
Figure 4.2 is an example of the timeline network for a city pair that has the
same time zone. Figure 4.2a constructs the flight arcs for fleet 1 that requires 1.5
time windows for flight time in both directions, and 0.5 time window for minimum
turnaround time. Figure 4.2b builds the flight arcs for fleet 2 that needs 1.5 and
2.5 time windows for flight time in different directions, and 0.5 time window for
minimum turnaround time. The subnetworks for all valid fleets put together create
the multi-commodity flow timeline network for that city pair.
59
4.2.2
Interaction of demand and supply through price
In microeconomics, it is well known that demand and supply interact through price
following the generic relationship depicted in Figure 4.3. The law of demand states
that given other things remaining the same, the higher the price of a good, the smaller
is the quantity demanded. This clearly reflects the observations that overcapacity
in certain competitive markets have driven airlines to reduce ticket prices even to
unsustainable levels.
Figure 4.3: Nonlinear relationship of demand vs. price and the effect on renenues
Changes in frequencies and aircraft size, i.e. supply of seats, would lead to changes
in prices. This interaction affects demand and therefore the airlines’ bottom line.
From an airport’s point of view, price is also important in the overall evaluation of
the quality of air transportation service. Therefore, we explicitly model price as a
variable by using directly the revenue functions and their linear approximations.
The demand curve D for air service of any time window t exhibits a convex
nonlinear form as in Figure 4.3a. Demand is diluted to substitute services (namely
flights to the neighboring airports in the cluster or other means of transportation such
60
as car, train) as price increases. Demand curves of peak periods shift rightward and
those of off-peak periods shift leftward. Corresponding to a convex demand curve is
a concave revenue curve (see Figure 4.3b) where the maximum y-value is the optimal
revenue for that time window. Similarly, revenue curves of peak periods lie on top of
those of off-peak periods.
A certain fleet mix configuration corresponds to a supply curve where the movement along the supply curve translates to changes of frequency. Larger aircraft ratios
in the fleet mix shift the supply curve rightward. Price as a regulator establishes
market equilibriums at the intersection points of demand and supply curves. S1, S2,
and S3 in Figure 4.3a intersect the demand curve D at quantities equal to 500, 1000,
and 1300 respectively where the resulting revenues of S1 and S3 are sub-optimal
compared to the revenue of S2.
4.2.3
Piecewise approximation of non-linear revenue functions
An arbitrary continuous function of one variable y = f (x) can be approximated by
P
a function of the form y = f (x1 , ..., xq ) = qi=1 fi (xi ) where fi (xi ) is piecewise linear
for each i. Given the segment endpoints (ai , f (ai )) for i=1,...,q, any a1 ≤ x ≤ aq can
be written as
x=
q
X
i=1
ai λ i ,
r
X
q
λi = 1, λ ∈ R+
.
i=1
The λi are not unique, but if ai ≤ x ≤ ai+1 and λ is chosen so that x = λi ai +
λi+1 ai+1 and λi + λi+1 = 1, then we obtain f (x) = λi f (ai ) + λi+1 f (ai+1 ). In other
61
Figure 4.4: Approximating a nonlinear function by a piecewise linear function
words,
f (x) =
q
X
f (ai )λi ,
i=1
r
X
q
λi = 1, λ ∈ R+
i=1
where at most two of the λi ’s are positive and if λj and λk are positive, then
k = j + 1 or j − 1. This condition, identified as a Special Ordered Set (SOS) contraint
of type 2, can be modeled using binary variables yi for i = 1, ..., q − 1 (where yi = 1
if ai ≤ x ≤ ai+1 and yi = 0 otherwise) and the constraints
λ1
≤
y1
λi
≤
yi−1 + yi for i = 2, ..., q − 1
λq
≤
yq−1
=
1
q−1
X
yi
i=1
y ∈ B q−1 .
(4.1)
62
For convex (concave) functions in a minimization (maximization) problem, SOS2
constraints in 4.1 can be removed, as the optimization process always chooses 2 adjacent endpoints. However, generic piecewise linear functions or convex (concave)
functions in a maximization (minimization) problem require 4.1 to ensure the nonnegative values of 2 adjacent λi ’s. On the other hand, when only a finite set of values
of x’s are valid, segment endpoints can assume those values and the SOS2 constraint
set can be replaced by the SOS1 constraint:
q
X
λi = 1 λ i ∈ B q .
i=1
4.2.4
Nesting revenue functions
Different time windows are not independent as spilled demand of this time window can
be recaptured by other time windows. Spill and recapture occur because passengers
can choose alternative time windows when their desired times are capacitated, too
expensive or not provided in the schedule. Therefore, the supply levels of alternative
(closely adjacent) time windows determine these spill and recapture effects. As the
schedule is not known in advance, we first estimate revenues independently for each
time window, then use nesting revenue functions to include the interdependency
between time windows.
Revenue functions can be estimated for different granularities: by 15min, 30min,
1hour, or by peak and off-peak time windows at each airport (see Chapter 4 for
estimation method). Figure 4.5 estimates revenue functions of ORD→LGA market
for all 15-min time windows in the first half of the day and the aggregate revenue
63
function for the whole period. Note that some time windows have the same estimates
of revenue functions and therefore are superimposed on top of each other. The sum
of demands and revenues of all 15-min time windows are therefore expected to be
constrained by the aggregate, or nesting, revenue function of the compounding period.
Figure 4.5: Nesting revenue functions
If λiq are the piecewise variables for the revenue function of time window i with
q ∈ Q(i) being the segment indexes,
X
λiq = 1, λiq ∈ R+
q∈Q(i)
xi =
X
aiq λiq
q∈Q(i)
fi (xi ) =
X
fi (aiq )λiq
q∈Q(i)
and a nesting revenue function of a period p that contains i, i.e. i ∈ E(p), having
64
piecewise variables βpr , r ∈ Q(p),
X
βpr = 1, βpr ∈ R+
r∈Q(p)
xp =
X
apr λpr
r∈Q(p)
fp (xp ) =
X
fp (apr )βpr
r∈Q(p)
then the nesting constraints is:
X
xi = xp
i∈E(p)
X
fi (xi ) ≤ fp (xp )
i∈E(p)
4.2.5
Assumptions
• The constraint on fleet availability is removed, i.e. we assume the airlines will
procure whatever aircraft is optimal to fly,
• Aircraft sizes are grouped into increments of a fixed number of seats,
• Arrival time rather than departure time drives demand,
• Demands are estimated for non-stop domestic flights to/from the airports in
study. Scheduling decisions are therefore limited to the nonstop markets,
• If arrival time windows at different airports are substitutable, they have the
65
same chronological values,
• There is only one level of nesting for the revenue functions. The finer granularity
time windows are compounded into only one coarser granularity time window.
The sets of substitutable time windows at one airport are mutually disjoint and
complete.
4.2.6
Formulation
Assuming concave revenue functions, we define:
Sets:
T
AG
AF
K
Q(i)
time windows
ground arcs
flight arcs
fleet types operable at the 2 airports of the market
linear segment indexes for the revenue function of i ∈ T
Parameters:
Sk
Cijk
Aiq
Riq
l
seating capacity of fleet type k ∈ K
direct operating cost for one flight of fleet type k ∈ K for (i, j) ∈ AF
linear segment quantities for the revenue function of i ∈ T , q ∈ Q(i)
linear segment revenues for the revenue function of i ∈ T , q ∈ Q(i)
average load factor
Variables:
xkij
λiq
number of flights of fleet type k ∈ K for (i, j) ∈ AF ∪ AG
linear segment variables for the revenue function of i ∈ T , q ∈ Q(i)
Subproblem formulation:
max z =
X X
i∈T q∈Q(i)
Riq λiq −
X X
(j,i)∈AF k∈K
Cjik xkji
(4.2)
66
subject to:
X
(j,i)∈A
l
X X
X
S k xkji −
(4.3)
Aiq λiq = 0 ∀ i ∈ T
(4.4)
Apr βpr = 0 ∀ p ∈ P
(4.5)
Rpr βpr ≤ 0 ∀ p ∈ P
(4.6)
q∈Q(i)
X
Aiq λiq −
i∈E(p) q∈Q(i)
X X
xkij = 0 ∀ i ∈ T , k ∈ K
(i,j)∈A
k∈K (j,i)∈AF
X X
X
xkj,i −
r∈Q(p)
X
Riq λiq −
i∈E(p) q∈Q(i)
r∈Q(p)
X
λiq = 1 ∀ i ∈ T
(4.7)
βpr = 1 ∀ p ∈ P
(4.8)
q∈Q(i)
X
r∈Q(p)
|AF |x|K|
x ∈ Z+
For any time window i,
P
(j,i)∈AF
|Q(i)|
, λi ∈ R+
P
k∈K
|Q(p)|
, βp ∈ R+
Cjik xkji in the objective function (4.2) is the
total operating cost of arrivals at i. The resulting total capacity
P
k∈K
P
(j,i)∈AF
S k xkji
multiplied by the average factor estimates the number of revenue passengers arriving
at i. This value is then decomposed in (4.4) into a convex combination of segment
endpoints (Aiq , Riq ) with q ∈ Q(i) using non-negative real variables λiq . Therefore,
P
q∈Q(i) Riq λiq is the piecewise linear approximation of the revenue function of time
window i. Subtracting the sum of all the cost terms over all flights from the sum of
all the revenue terms over all time windows yields the total profit that (4.2) seeks to
67
maximize. (4.3) enforces flow balance constraint that at each node i in the timeline
network, for each fleet, the number of incoming aircraft is equal to the number of
outcoming aircraft.
As explained earlier,
P
q∈Q(i)
Aiq λiq is the estimate of realized arrival demand at
time window i. i can have other substitutable time windows that are all included
in a coarser compounding time window p, i.e. i ∈ E(p). Similarly, (4.5) decomposes
the aggregate arrival demand of p into a convex combination of segment endpoints
(Apr , Rpr ) with r ∈ Q(p) using non-negative real variables βpr . (4.6) states that
P
P
the sum of revenues of substitutable time windows in p,
i∈E(p)
q∈Q(i) Riq λiq , is
constrained by the revenue of the compounding time window p,
P
r∈Q(p)
Rpr βpr . (4.7)
and (4.2.6) are the sets of convex constraints for λiq and βpr .
The solution of a subproblem creates two schedule vectors: the arrival vector
P
P
k
{aj } where aj =
k∈K
(i,j)∈AF xij , and the departure vector {dj } where dj =
P
k∈K
P
(j,i)∈AF
xkji , j ∈ T are time windows at the capacitated airport study in
the master problem.
4.3
Airport’s allocation problem
The master problem at a capacitated airport collects the schedules of individual
markets and solves a set packing problem with side constraints to maximize public
goals.
Let:
Sets:
68
S
T
M
S(m)
schedule vector indexes
time window indexes
market indexes
column indexes of market m’s schedule vectors, m ∈ M
Parameters:
a|T |x|S| matrix of arrivals by time window: aij is the number of arrival flights at
d
time window i in schedule j
matrix of departures by time window: dij is the number of departure flights
Zj
at time window i in schedule j
coefficient of the schedule vector j ∈ S, determined by the public goal to
Ci
Gi
optimize
arrival/departure rates of time window i ∈ T
ground capacities in time window i ∈ T
|T |x|S|
Variables:
yj
binary variable equal to 1 if schedule vector yj is in the optimal solution
Formulation of the master problem:
max
X
j∈S
subject to:
Zj y j
(4.9)
69
X
aij yj ≤ Ci
∀i ∈ T
(4.10)
dij yj ≤ Ci
∀i ∈ T
(4.11)
j∈S
X
j∈S
X
yj ≤ 1 ∀m ∈ M
(4.12)
j∈S(m)
y ∈ B |S|
The sets of constraints (4.10) and (4.11) reflect airport operational rate constraints. As each market can have many alternative schedules from which at most
one schedule can be in the solution, each market has a SOS1 side constraint in (4.12).
The objective function maximizes public goals such as:
• Profit where Zj is the profit of schedule j, given by the value:
X X
X X
Riq λiq −
i∈T q∈Q(i)
Cjik xkji
(j,i)∈AF k∈K
from the subproblem that produces schedule j.
• Seat throughput where Zj is the total seat of schedule j, given by the value:
X X
S k xkji
k∈K (j,i)∈A
from the subproblem that produces schedule j.
70
4.4
Solution method
Figure 4.6 depicts our method to find the optimal collection of schedules. Initially,
the mixed integer subproblems, i.e. the determination of schedules for each O/D
pair, provide optimal arrival demand and departure demand columns to the master problem. The master problem solves its linear relaxation, called the LP master
problem, to compute dual price for each constraint. The dual price of a constraint
reflects the contraint’s value, or its contribution to the objective function. There are
three sets of dual prices corresponding to the three sets of constraints in a master
problem: αi for (4.10), πi for (4.11), and µj for (4.12). For a maximization problem,
a new column with coefficient zj can be added to the master problem if its contribution to the objective function, zj , is larger than the value of resources it would use,
P
P
i∈T (αi aij + πi dij ) − µj > 0. In other words,
i∈T (αi aij + πi dij ) + µj , or when zj −
a new column can be added if it prices out favorable with respect to the objective
function. This process is called “column generation”, often used to solve large scale
combinatorial optimization problems.
Therefore, we update the formulation of the subproblems to include this condition
as an additional side constraint, with the initial dual prices set to zero:
z−
X
i∈T
C
αi
X
k∈K,(j,i)∈AF
xkj,i −
X
i∈T
C
πi
X
xkij − µ ≥ 1
(4.13)
k∈K,(i,j)∈AF
where the expressions for z are different for different objective functions of the
master problem, as explained in airline scheduling subproblems.
When the objective function of the master problem is not profit maximization,
71
Figure 4.6: Branch-and-price solution method
72
it is inconsistent with the profit-maximizing objective functions of the subproblems.
Therefore, when the column generation process finds new feasible schedules, they
can be suboptimal. We can parametrically set a lower bound on these suboptimal
schedules: a suboptimal schedule is valid if it is within some percentage of the optimal
solution’s value.
The initial solutions, or columns, of the subproblems initialize the root node of
the LP branch tree of the master problem. At the root node and subsequent nodes,
a two-phase solution process takes place: the node is first solved to calculate dual
prices which will serve as input to MIP subproblems to generate new columns (if any)
to be added to the current node, then the node is solved again and branches if there
are integer variables with fractional values. In contrast to regular branch-and-bound
algorithms where a node with an LP solution less than the incumbent integer value
can be pruned (in a maximization problem), branch-and-price requires storing all the
unprocessed nodes for later column generation processing, as new columns added to
a node can increase its objective function value. In our branch-and-price algorithm, a
node is pruned if it is either infeasible or it has an integer solution after the two-phase
solution process. To optain optimality, the process should continue until all the nodes
are processed.
4.5
Implementation details
As the current version of CPLEX Concert Technology does not allow for dynamic
addition of new columns into a problem at each node of the branch tree, we implement
our own branch-and-price tree and use CPLEX to solve the LP problems at each node.
Specifically,
73
• At each node, we branch on the most fractional variable that has largest coefficient in the objective function,
• We store all unprocessed nodes in a ordered list and use best-bound strategy to
select the next candidate node,
• We add columns to the master problem and at each node, we store the list of
variables that (i) come from the parent node, (ii) are generated at the node,
(iii) are fixed to 0 and (iv) are fixed to 1 from the root node down the tree to
the current node. When we move from one node to another, we reset all the
bounds of the stored variables, and fix to 0 all other variables.
Interested readers are encouraged to see Appendix C for the code listing of our
branch-and-price implementation.
Chapter 5: Parameter estimation for scheduling
models
Modeling airline scheduling decisions usually require proprietary cost and revenues
data along with constraints of airline business models. Each airline’s data can be
largely different from others’. To mitigate this effect, we use aggregate data across
airlines available in public databases. Aggregate data is also more effective in reducing
the inherent noise in any data set, especially for airlines with little public data.
Parameter estimation for scheduling models consists of building the timeline networks
and calculating revenue functions.
5.1
Timeline networks
A timeline network is built for each city pair. The monthly T-100 Segment table,
compiled by the Bureau of Transportation Statistics (BTS [68]), reports domestic
and international operational data by U.S. and foreign air carriers. Only data of
domestic carriers are considered as we look at domestic schedules. For each segment,
it contains, among other data items, carriers, aircraft types, distance, total number
of performed departures and seats, total ramp to ramp times, and total air times.
Aircraft types are provided as identification codes. We calculate the size of each
aircraft type by
performed departures
.
performed seats
Aircraft sizes are then grouped into increments of
25 seats (or any fixed number of seats) called fleet. The fleets identified as such for a
74
75
segment determines the number of commodities in the multi-commodity flow network
for that segment.
For this study, we use the data of Q2, 2005 and categorize fleets available at LGA’s
domestic nonstop markets into the following ranges of seats:
5.1.1
Arcs and arc lengths
Flight arcs depart and arrive within 5:15 and 24:00 local times at any airport. To
estimate arc lengths, or leg lengths, we use Aviation System Performance Metrics (ASPM) database [69] that provides on-time performance of individual flights.
Recorded scheduled block times are typically padded with some time buffer built into
the schedule so that reasonable delays can be absorbed. Actual block times can be
higher than scheduled block times due to unexpected excessive congestion, or smaller
due to unexpected low congestion. If we can reasonably assume that airlines adjust
their delay buffers over time to cope with congestion, then the minimum of scheduled
block times and actual block times is more likely to reflect the average block times.
However, the minimum of the two block times can still contain airborne or ground
delays. In reduced demand scenarios, airlines would incur less delay on the day of
operations, and so they would eventually reduce both scheduled and actual block
times. As airborne phase is less subject to delay than ground operations, we could
further adjust estimates of block times to:
actual air time + 2 * min(scheduled block time, actual block time)
3
Averaging estimates of block times adjusted as above for all aircraft types in a
fleet provides the arc length for that fleet. In addition, an arc arriving at a node
76
Fleet
1
2
3
4
5
6
Aircraft
Average Size Fleet
BE-1900
19
EMB-145
22
DO-328 J
32
SF-340/B
34
7
DHC8-100
37
EMB-135
37
EMB-140
44
200/440
47
DHC8-300
50
EMB-145
50
RJ100/ER
50
8
AV RJ85
69
200/440
70
RJ-700
70
EMB-170
72
MD-80
74
BAE146-2
77
B717-200
88
9
B737-1/2
100
DC-9-30
100
B737-5
108
MD-80
109
DC-9-40
110
B717-200
117
A319
121
B737-300
122
10
DC-9-50
125
B737-700
126
MD-80
132
A320-1/2
133
A319
138
B737-400
144
MD-80
145
11
B737-300
147
A320-1/2
148
MD-90
150
B737-8
151
B757-300
154
B767-2/R
158
Aircraft
Average Size
B737-8
166
B737-9
167
A320-1/2
168
B727-200
172
A321
174
B767-2/R
174
B757-200
179
A340-500
181
A310-300
194
B757-200
194
A321
196
B767-2/R
204
A330-200
206
B767-3/R
207
B757-200
215
A321
216
A330-200
221
B757-300
222
B777
222
B767-3/R
223
A340-200
230
B767-400
235
B757-300
245
B767-400
246
A340-200
251
B767-3/R
251
A310-300
253
A340-300
255
B747-400
258
B777
258
A330-200
261
B747-400
266
A300-600
267
A340-200
272
B777
283
B767-400
285
Table 5.1: Aircraft types and seating capacities categorized to fleets
77
means that the aircraft should be ready to depart at the very node. Therefore, we
add the turnaround time to arc lengths. As the nodes in the timeline networks are
time windows, arc lengths in hourly unit are translated to arc lengths in time window
unit. Lastly, as the x-axis of the timeline networks is local time windows, we subtract
or add the difference in time zones of the two airports to calculate the final arc lengths.
5.1.2
Arc costs
The cost data comes from Schedule P-52 in Air Carrier Financial Reports (Form 41
Financial Data), BTS database. P-52 table contains detailed quarterly aircraft operating expenses for large certificated U.S. air carriers. It contains for each aircraft type
direct flying expenses (including payroll expenses and fuel costs) and total operating
expenses that include maintenance of flight equipment and equipment depreciation
costs. We show in Figure 5.1 these two types of operating costs after separating fuel
to allow for future analyses on fuel cost impact. Compared to direct costs, total expenses have larger variability. In average, the total expenses can be as high as 186%
of the direct flying expenses. Figure 5.2 shows a more monotonic trend with less
variability of hourly fuel consumption by aircraft seats.
We average for each fleet the following metrics of each aircraft type that belongs
to the fleet:
hourly air fuel consumption =
air fuels issued
total air hours
hourly aircraft direct expense excluding fuel =
total air direct expenses - fuel cost
total air hours
hourly aircraft total expense excluding fuel =
total air total expenses - fuel cost
total air hours
78
Figure 5.1: Estimates of aircraft hourly operating costs by seating capacity (Source:
BTS Q2 2005)
Figure 5.2: Estimates of hourly fuel consumption costs by aircraft seating capacity
(Source: BTS Q2 2005)
79
Then arc cost when using aircraft direct expense is:
arc cost = arc length * (average hourly aircraft direct expense excluding fuel
+ average hourly air fuel consumption * fuel unit cost)
Aircraft seats
25
50
75
100
125
150
175
200
225
250
275
350
375
Fuel consumption Direct cost Direct, maintenance and
(gallons/h)
($/h)
depreciation cost ($/h)
306
703
795
418
840
1106
530
978
1417
759
1115
1729
987
1253
2040
1216
1390
2351
1445
1528
2662
1674
1665
2973
1902
1803
3284
2131
1940
3595
2360
2078
3906
3046
2490
4840
3275
2628
5151
Table 5.2: Hourly costs for each fleet of 25-seat increment
80
and arc cost when using aircraft total expense is:
arc cost = arc length * (average hourly aircraft total expense excluding fuel
+ average hourly air fuel consumption * fuel unit cost)
In this study we use direct flying expenses to estimate arc costs as these relate
directly to flight schedules.
5.2
Nonlinear revenue functions and piecewise linear approximation
In addition to the total numbers of passengers by segments in the monthly T-100
Segment tables, we use the quarterly Origin and Destination Survey to estimate
market demand curves. Compiled by the Bureau of Transportation Statistics, the
Survey is a 10% random sample of airline tickets from reporting domestic carriers.
Relevant data include origin, destination, prorated market fare, number of coupons
(or flight legs), number of passengers, and market miles flown. This available data
represent only a small fraction of the constrained demand. Figure 5.3 plots the
demand curves of ORD and BOS markets in both directions for the first two quarters
of 2005.
We extrapolate the sample to obtain the complete demand curves for each directional
market by making these assumptions:
5.2.1
Assumptions
• Revenues are estimated for daily schedules of domestic nonstop markets,
81
Figure 5.3: Constrained demand curves of 10% BTS ticket price sample, Q1 & Q2
2005
• Direct flying expenses are estimated to determine arc costs,
• The sample data is taken randomly from a much larger population set,
• The sample is a good representation of the population,
• The sample average fare is a good estimate of that of the population,
• Probabilities of price points in the sample are good estimates of those of price
points in the population,
• Time-based demand shares are proportional to time-based seat shares,
• Demand for each nonstop domestic market is equal in both directions, and hence
equal to the average of directional demands.
82
5.2.2
Processing segment fares
The tickets in the Survey are itinerary tickets. Segment fares are traditionally prorated from itinerary fares. However, there is a fixed cost in any flight leg. This
portion of fixed cost is large in flight legs of short distance, and decreases in legs
of longer distance. We compute segment fares proportionally to the squared root of
distances of segments in the itinerary1 . Figure 5.4 illustrates the difference between
linear prorating and linear prorating of square root.
Specifically, if a flight has two legs of 100 (=102 ) miles and 225 (=152 ) miles, and
10
) and
has the one-way ticket price of $100, then leg one is allocated $40 (=100 ∗ 10+15
leg two $60 (=100 ∗
15
).
10+15
Figure 5.4: Linear prorating of square root of leg distance helps account for fixed
cost.
1
Thanks to the advice of Dr. Tassio Carvalho, American Airlines
83
5.2.3
Extrapolating the 10% ticket sample
As the sample average fare is a good estimate for the market average fare, the quarterly demand curve should pass through the reference point (quarterly demand, average fare). The quarterly demand is the average of directional demands over the
quarter. We can then extrapolate sample demand for each price point to its population demand proportionally to their probabilities in the sample. However, as the
sample demand curves are constrained by available capacities and airlines’ inventory
management, especially in lower fares, we could reduce this effect by extrapolating
only data points above the reference point to build the upper part of the demand
curve, then find an appropriate fit for the demand curve previously found to estimate
untruncated demands for lower fares.
Fare
Sample Passengers
$210
1
$200
2
$190
3
$180
4
$170
5
$160
6
$150
7
$140
8
$130
9
mean=$156.7
sum=45
Extrapolated
23.8
47.6
71.4
95.2
119
142.9
sum=500
Table 5.3: Example of demand extrapolation
For example, consider the sample set is given in Table 5.3, and the total number
passengers in the full data set is 500. The average fare of $156.7, and therefore the
extrapolated demand curve is assumed to go through the point (500,$156.7). There
84
are 6 price points above the average fare that cumulatively sell to 21 passengers.
Their respective probabilities in the subsample above the average fare are
4
, 5, 6.
21 21 21
5
21
The extrapolated demand would be
∗ 500, and
6
21
1
21
∗ 500,
2
21
∗ 500,
3
21
1
, 2, 3,
21 21 21
∗ 500,
4
21
∗ 500,
∗ 500. The sample demand curve and the extrapolated curves of
the example are depicted in Figure 5.5. Notice that in Figure 5.5b, we compare two
methods of extrapolation. The simple curve is obtained by only extrapolating the
data points using the sample probabilities, whereas the average curve is forced to go
through the reference point, and it extrapolates price points above the reference point
as described above. The fit curve in Figure 5.5b fits the average curve.
Figure 5.5: Example of demand extrapolation
The extrapolation stretches the sample curve above average fare rightward while
maintaining its shape. Fitting then provides the extrapolated estimation for the rest
of the sample. Figure 5.6 illustrates the estimation procedure for two directional
markets ORD→LGA and PIT→LGA in Q2, 2005. The extrapolated curve for ORD
85
in Figure 5.6a is best fit by a log function with R2 = 0.96, whereas the extrapolated
curve for PIT in Figure 5.6b is best fit by a linear function with R2 = 0.84.
Figure 5.6: Estimates of quarterly constrained extrapolated demand curves for directional markets, Q2 2005
5.2.4
Breaking down data from by-quarter-of-the-year to daily
and by-time-of-day
The fit curves obtained above are aggregate estimates of quarterly demand curves that
combine demand of peak and off-peak hours of the day. In order to determine the
optimal schedule, passengers’ travel time preference for different time intervals needs
to be estimated. It can be reasonably assumed that over time, airlines adjust their
schedules as to best accommodate passengers’ travel time preferences. Therefore,
if time window 08:00-08:15 at LGA airport has a higher concentration of arrival
seats than 08:15-08:30, we assume the demand captured by 08:00-08:15 to be higher
than that of 08:15-08:30. Or in other words, we assume that demand is captured
86
proportionally to the number of scheduled seats.
We use ASPM database to approximately break the quarterly demand curve for
the whole day down to daily, by-time-window-of-day level. As of Jan 2006, ASPM
provides, among other data items, scheduled times of past flights from 25 reporting
airlines at 75 airports. For the purpose of estimating past demand distribution over
time of day, we only need to look at flights that were actually flown in the past. The
extrapolation of aggregate demand curves of any quarter obtained from BTS is then
allocated to all the flights flown during the same period reported in ASPM. We use
the number of scheduled seats of each time period to compute the probabilities of
their respective contributions to the total demand.
It can be reasonably assumed that a time window having more flown seats contributes more passengers to the total count of demands. Therefore, the quarterly
extrapolated fit curve is multiplied by the seat share of each time period in Figure 5.7
to give estimates of quarterly demands by time window. Specifically, Figure 5.7 shows
actual seat shares of directional markets by 15-min intervals during three months of
Q2, 2005 (taken from ASPM). Seat shares, normalized to have values from 0 to 1,
of two directional markets of each city pair are plotted in a same chart with one
direction has the y-axis inverted. ORD→LGA and LGA→ORD markets have seats
almost evenly distributed throughout the day, and therefore the seat share values by
quarter hour are rather small on a 0-1 scale. In contrast, TPA→LGA has flown the
most seats in 17:45-18:00 time window and LGA→TPA market has flown the most
seats in 14:15-14:30.
These quarterly demands by time window are then broken down to daily demands
by time window. Different time windows can have flights flown for different numbers
87
of days during the quarter, e.x. LGA→TPA has 86 days during Q2, 2005 that had
arrivals to TPA in 14:15-14:30 time window, whereas it has only 44 days that had
arrivals to TPA during 17:45-18:00. As we want to seek daily schedules, we divide
quarterly demands by the average number of days of all the time windows, i.e. considering only these two time windows, we would then divide the quarterly demands
by
86+44
2
= 65.
Figure 5.8 and Figure 5.9 illustrate estimated daily demand curves and revenue
functions by 15-min periods for ORD→LGA, TPA→LGA, and LGA→TPA markets.
Estimated demand curves for peak periods lie above those of off-peak periods, as there
are more demands at any given price point and more willingness to pay at any given
supply quantity. As a result, the revenue functions of peak periods also lie above
those of off-peak periods. As ORD schedules more time windows than TPA, we only
display the time windows associated with estimated curves for TPA in Figure 5.9.
5.3
Model validation: Unconstrained profit maximizing schedules
We investigate the optimal schedules of LGA nonstop domestic markets without runway capacity constraints at LGA and without the aircraft size restriction for exception
slots that serve small markets. While it is not valid to compare these optimal unconstrained solutions of a single benevolent airline to actual constrained schedules of
multiple airlines, the unconstrained solutions helps verify their consistency with the
main assumptions in our modeling approach such as:
88
• Optimal scheduled times are consistent with historical data
• Changes in supply lead to reverse changes in price.
We solve the unconstrained optimal schedules for LGA nonstop domestic markets
using the following parameters:
• Data sampling period: Q2, 2005
• 67 nonstop domestic markets that have daily schedules to/from LGA
• 45 minutes of minimum turn-around time for all fleets
• 80% load factor
• Fuel cost: $2/gallons
• Existing fleets
• One level of nesting with three generic substitution groups for all markets: time
windows from 6:00am-12:00pm (12:01pm-17:00pm, or 17:01pm-24:00pm) are
substitutable. However, finer grouping of substitutable time windows can be
done to reflect better demand characteristics of individual markets.
5.3.1
Flight schedules by time of day
We assume earlier that over time, airlines have come to capture passengers’ travel time
preferences by making incremental changes to their timetables and supply levels. The
number of actual seats scheduled and flown for different times of day reflects the timebased concentration of demands. It can be reasonably expected that in the model
output, flights should be scheduled in time windows that have flights scheduled in the
89
past, and time windows with larger seat shares should have more flights and/or larger
aircraft to accommodate the corresponding demand allocations. Figure 5.10 shows
in the upper and lower panels the seat shares by time windows of the day for ORD’s
two directional markets. Flights in the output schedules are plotted in the middle
panel where the end points of flight arcs correspond to scheduled departure times
and arrival times. The output schedule is valid if in each substitution group, flights
are scheduled to arrive at time windows that have higher demand concentration, or
actual seat shares.
5.3.2
Supply and price
We expect to see the reverse relationship between supply and price. Figure 5.11 shows
such a trend: increase in seat throughput leads to decrease in fare, and vice versa.
One can notice that although high frequency markets such as ORD, ATL, BOS, DCA
all decrease their daily frequencies in Figure 5.12 and upgauge, BOS and DCA both
increase the overall throughput while ORD and ATL in contrast reduce the number
of seats available. A few outliers correspond mostly to small markets: CharlottesvilleAlbemarle Airport (CHO), Nantucket Memorial Airport (ACK), Barnstable MuniBoardman/Polando Field Airport (HYA), Martha’s Vineyard Airport (MVY).
5.3.3
Flight frequencies and fleet mix
Figure 5.12 shows the change in aircraft size of model output vs. actual data against
the change in daily frequency of model output vs. actual data. Changes in aircraft size
within 15 seats are negligible due to the rounding when grouping aircraft to fleets of
25-seat increment. Profit maximizing schedules suggest reduction of service levels and
90
maintaining/upgauging aircraft size for most of the markets. One can notice that the
shuttle service markets such as BOS and DCA, and the high frequency markets such
as ORD and ATL are all in the upper left quadrant. Newport News - Williamsburg
International Airport (PHF) result maintains its current frequency of six flights/day,
but reduces aircraft size from 110 seats to 50 seats. In contrast, the model output of
Myrtle Beach Airport (MYR), being one of the favorite vacation destinations in the
second quarter of the year, increases aircraft size from 100 seats to 170 seats. Markets
with little change in both frequency and aircraft size are mostly small markets: Savannah International Airport (SAV), Northwest Arkansas Regional Airport (XNA),
Lexington Blue Grass Airport (LEX), Birmingham International Airport (BHM),
Columbia Metropolitan Airport (CAE), and Dayton International Airport (DAY).
91
Figure 5.7: Actual seat shares by time of day are used to allocate demands by time
of day, Q2 2005
92
Figure 5.8: Estimated demand curves for peak periods lie above those of off-peak
periods
93
Figure 5.9: Estimates of daily demand curves and revenue functions by different
15-min time periods for TPA→LGA and LGA→TPA markets, Q2 2005
94
Figure 5.10: In each substitution group, higher actual seat shares of time windows
lead to scheduled arrivals in those time windows
95
Figure 5.11: Increases in seat capacity lead to decreases in fare and vice versa
Figure 5.12: Changes in aircraft sizes in relation to frequencies are mixed
Chapter 6: A Stochastic Queuing Network
Simulation Model for Evaluating Schedule Delays
and Cancellations
Demand management measures aim to change flight schedules. In addition to other
performance metrics to evaluate the potential measures, namely operational and passenger throughputs, market access, and network load balance, delays and cancellations need to be estimated to assess the impacts on congestion. This chapter presents
a delay and cancellation model that simulates network dynamics resulting from stochastic and queuing effects. In response to the industry trend of using small aircraft in
recent years, passenger throughput has become a driving factor in increasing system
capacity and efficiency. Currently proposed market-based solutions to the problem
such as congestion pricing and slot auctions aim to incentivize airlines to upgauge.
It is therefore of particular interest to estimate the effects of fleet mix on airport
capacity and airline performance. Our model integrates explicitly aircraft separation
to simulate airport operation capacities. The model provides an intermediate level
of detail in a gate-to-gate simulation tool that simulates the stochastic, queuing, and
propagating effects of delay and cancellation among airports.
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97
6.1
Stochastic queuing network simulation model
6.1.1
Modeling objectives
From an assessment point of view, to evaluate the impacts of a congestion management measure concept, the model needs to support evaluations of:
• Implications of schedule changes in flight time and fleet mix on airport capacity:
Operational rates are constrained by the safe separation standards between
pairs of aircraft which are dependent on their fleet types. Therefore, a direct
analysis of fleet mix and the resulting aircraft separations is a major modeling
requirement,
• NAS operational performance in terms of delay (departure/arrival) and cancellation including system-level assessments. The evaluation can be aggregate and
airline-specific, and system-level assessment should include airport interdependency in terms of delay and cancellation propagation,
• Affects of uncertainty within the system and within the models used to simulate
the system. The NAS is a highly stochastic and asynchronous network that
variability simulation is important in estimating the steady state of the system.
6.1.2
Queuing network model
Airports’ main facilities such as gates, taxiways, and runways are modeled as multiserver queuing systems that mimic aircraft movements from gate-out to wheel-off for
outbound operations and from airport arrival to gate-in for inbound operations. Enroute cruising phase between city pairs is also modeled as a multi-server queue. The
98
following diagram depicts the queuing network dynamics:
Figure 6.1: Aircraft dynamics and network components
All servers are generically specified by (G/k/FCFS) where G refers to a generic
service time distribution, k is the number of parallel servers, and FCFS reflects the
First-Come-First-Serve queue discipline. Outbound flights are subject to a cancellation probability that is determined statistically in relation to delay. When an outbound flight is cancelled and goes to the sink, if it has subsequent connected flights
99
then it increases the cancellation probability of those flights. Flight cancellation is
described in detail later in the cancellation submodel. When an outbound flight is
not cancelled, the outbound flight sets off at the gate-out server that generates local
randomness to the scheduled gate-out time. This local randomness is added on top of
deterministic departure delay propagated from previous delayed leg(s). The flight is
then directed to the taxi-out server to proceed to the first available departure runway.
Waiting time in the departure queue for runway access and runway occupancy time
is calculated by the runway capacity sub-model described subsequently. If the destination airport is modeled, the aircraft enters the enroute queue of the corresponding
city pair. The enroute server then assigns to the flight an expected time of arrival,
generated as a stochastic value of service time of the enroute server. Subsequently,
the aircraft gets in the queue for runways at the arrival airport. If the arrival airport
is not modeled, the flight goes to the sink. Inbound flights that do not have origin
airports modeled are also added to the corresponding enroute queues.
On the arrival side, the process unfolds in the opposite order. Flights in the
landing queue access arrival runways using the airport runway capacity sub-model.
Stochastic taxi-in times are then added before the aircraft is considered arrived at
gates, i.e. goes to the sink. If an arrival has a subsequent departure, its arrival delay
and the turnaround time between the two flight legs are used to determine whether the
subsequent departure will have propagated delay and quantify this metric if needed.
Service time distributions of various servers in the model are estimated statistically to simulate the stochastic nature of the NAS. Flight cancellation also uses
statistical distributions of cancellation rates. Multiple independent runs using these
distributions provide estimates of the variability of the measured statistics such as
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operational rates, delay, and cancellations.
The system is extensible, as we intend to give a compromising approach between
full-network aggregate analysis and detailed study of a sub-network of major airports.
Airports of interest can be added to the model as needed, and others are considered
as sink and source. En-route time distributions are estimated for pairs of airports
considered in the simulation. It can be reasonably assumed that congestion management measures would be applied at major chokepoints of the NAS and would have
major effects at these nodes.
6.1.3
Runway capacity submodel
Currently, arrivals and departures are modeled separately in the runway capacity
model (one-runway airports are typically not modeled in the simulations given their
insignificant role in the NAS), and future extension of the model should include
mixed runways. However, the dependency between runways is modeled by using a
calibrating factor that will be discussed later in the section. The runway model has
as many parallel departure (arrival) servers as the number of dedicated departure
(arrival) runways at the modeled airports. Runway availability is determined by
enforcing the separation minima between sequenced aircraft: an aircraft can only
land or take off when the previous aircraft has exited the runway or the two aircraft
are separated by at least the proper minimum time lag, whichever is later. This
rule uses time-based separation standards for specific pairs of aircraft types listed in
Table 6.1, and runway occupancy times sampled from empirical distributions studied
in [70].
As recent jet engines generate stronger wake vortexes and aircraft are sequenced
101
more closely in the terminal area, time-based separation has become more appropriate
in the sequencing procedures in addition to distance-based separation, as wake vortex
decay is a function of time. Hansen [2] converts distance-based separations to timebased separations using nominal landing speeds of four types of aircraft based on
their wake vortex characteristics. From Table 6.1, a small aircraft following a large
aircraft needs to be separated by at least 4 nautical miles or 164 seconds. It’s clear
that a schedule with high concentration of small and much larger aircraft will reduce
significantly runway operational rates.
The standard separations are multiplied by a calibrating factor to match airports’
simulated departure (arrival) rates to the actual data in ASPM. This factor helps simulate mixed runways, interdependency between runways and operational differences
from airport to airport. It is calibrated such that the steady-state average simulated
capacity levels approximate airport realized capacity levels (both for arrival or departure) reported in ASPM. In addition, it is assumed that aircraft are allocated to the
first available runway.
Trailing
Leading
Small
Large
B757
Heavy
Small
Large
B757
Heavy
2.5/80 2.5/68 2.5/66 2.5/64
4/164 2.5/73 2.5/66 2.5/64
5/201 4/115 4/102 4/101
6/239 5/148 5/136 4/104
Table 6.1: Wake Vortex Separation Standards (nmiles/seconds) [2]
The runway occupancy time can be well fit using a Normal distribution, and the
method is widely used in the literature [70][71]. Based on Haynie’s observation at ATL
airport in 2002 [72], the runway occupancy time is modeled as a Normal distribution
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N(38, 82 ).
6.1.4
Delay propagation submodel
Delay propagation reflects network effects and varies from non-hub airports to large
hub airports. At non-hub airports, most traffic is Origin-Destination, and therefore,
large delay of an inbound flight can only be propagated to a later outbound leg by the
same aircraft by the same airline. Linking flights in this case is simple by following a
FIFO rule based on aircraft type and airline. The quantified effect essentially depends
on the turnaround time and the delay magnitude of the previous leg.
A
Let tD
0 (f ) and t0 (f ) denote respectively the schedule departure and arrival times
of flight f , and tD (f ) and tA (f ) the simulated departure and arrival times, then the
delay that flight f propagates to a connecting flight g is simulated as follows:
(
GP (g) =
0
if tA (f ) − tA
0 (f ) ≤ 15 min
A
A
α[t (f )−t0 (f )] A
A
A
A
min(t (f ) − t0 (f ), tD (g)−tA (f ) [t (f ) − t0 (f )]) otherwise
0
where
[tA (f )−tA
0 (f )]
A (f ) ,
tD
(g)−t
0
0
0
calibrated by a scaling factor α to reflect the sensitivity of
flight schedules to disruption, determines the magnitude of the delay propagation’s
multiplicative term. We assume the propagation to be positively correlated to the
lateness of flight f, i.e. tA (f )−tA
0 (f ), and negatively correlated to the time lag between
the scheduled arrival time of flight f and the scheduled departure time of flight g in
A
the denominator or tD
0 (g)−t0 (f ). The scaling factor α is determined empirically using
connected flight linkage. Table 6.2 illustrates our delay propagation calculation for
10 exemplary combinations of delays and turnaround times and three representative
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Delay
Turnaround time
A
A
A
t (f ) − t0 (f )
tD
0 (g) − t0 (f )
45
60
30
75
90
45
60
40
75
90
GP (g) GP (g)
GP (g)
α = 1 α = 0.5 α = 1.1
20
10
22
15
7.5
16.5
12
6
13.2
10
5
11
35.5
17.8
39.1
26.7
13.3
29.3
21.3
10.7
23.5
17.8
8.9
19.6
Table 6.2: Example of delay propagation (unit: minute)
values of α: larger values of α explain for schedules that are more susceptible to
disruptive events.
At hub airports, however, one delayed arrival can affect many outbound flights of
different aircraft types and even of different airlines (regional/trunk line and codeshare partners) as connecting passengers transiting through the hubs to different
destinations. A late arrival can delay many connecting flights if there are a substantial number of connecting passengers changing aircraft at the hub airport and little
possibility of spilling those to subsequent flights. Therefore, airlines make compromise between maintaining delay internalities and sharing these to other passengers
as to minimize the overall impacts of operational irregularities. As passenger data
are proprietary, propagating effects at hub airports will need a separate passenger
simulation module, and that is beyond the scope of our current research.
6.1.5
Cancellation and cancellation propagation submodel
Cancellation of a flight f is determined by a conditional probability function p(f ).
On one hand, cancellation likelihood can be modeled as a probabilistic variable. The
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probability of canceling a flight f , p(f ), has two independent components: the probability of canceling f as a result of canceling an inbound flight g, p(f ∩ g), and the
probability of canceling f caused only by local technical or operational problems,
p(f ∩ g):
p(f ) = p(flight f is cancelled)
= p(f ∩ g) + p(f ∩ g)
= p(f |g)p(g) + p(f |g)p(g)
We denote p(f |g)p(g) as p1 , p(f |g)p(g) as p2 , and explain later how to estimate
them in the parameter estimation section. On the other hand, it is commonly acknowledged that delay and cancellation are used as performance trade-off. Airlines,
to certain extent, voluntarily cancel flights to avoid excessive delay. This decision
involves cost/benefit analysis using airline proprietary data. Therefore, we use daily
cumulative delay (of all arrivals and departures) at an airport as a surrogate to the
airport performance based on which to make cancellation decisions: the statistical
trend over time between the cumulative delay in minutes and the number of cancellations reflect aggregately how airlines generally compromise between the two metrics.
At the departure of a flight f , to determine the probability of canceling f in relation
P
to delay, let k:k<f [dA (k) + dD (k)] be the cumulative sum of departure delay dD (k)
and arrival delays dA (k) of all flights k scheduled before f , and c the cumulative
number of cancellations that happen before f . We simulate the statistical non-linear
relationship, denoted as Ω, between the two metrics as follows: If the two metrics follow the pattern, the probabilities p1 and p2 are dominant; but if delay becomes more
105
excessive, a cancellation is forced to maintain the trend between the two metrics:
p(f ) =
P
p1 + p2 if Ω( k:k<f [dA (k) − dD (k)], c) is true
1
otherwise
Details on modeling this feature are given in the next section, when we estimate
parameters of the model for LGA airport.
6.2
Parameter estimation
A major challenge lies in estimating model parameters. The data source we used is
ASPM. Given limits on what is available at what level of fidelity, we conducted data
filtering to isolate the effects being analyzed. It is widely known that airlines incorporate buffer times into their schedule in anticipation of delay. In order to estimate
’real’ delay, i.e. idle time that aircraft spend waiting to proceed, it is necessary to
base the calculation on actual times but not scheduled times. But on the other hand,
reported metrics in ASPM typically include many effects at the same time, such as
gate-out delay and en-route delay.
We used techniques to remove or at least alleviate the compound effects, which
are described subsequently for respective metrics. Although our data preparation
process has tried to estimate independent distributions of various stochastic variables,
the overall estimation can be further improved if better filtering techniques become
available.
We also provide details on the delay propagation and cancellation algorithms in
this section. These are some of the main features of the model that aim to simulate
network effect, and the trade-off relationship between them.
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6.2.1
Gate-out delay distributions
Gate-out delay values in ASPM are the time difference between scheduled gate-out
(departure) time and the actual time. This metric typically includes delay due to
late connecting legs, local airline operational problems, and delay due to ATC’s flow
management measures such as Ground Stop or Ground Delay Programs. The first
delay component can be easily isolated by sampling only departures that have early
inbound arrivals so there should not be any propagation effect. Then, since we don’t
have access to the third component of the delay, it was analyzed together with local
randomness to give the statistical distribution of gate-out delay time.
6.2.2
Taxi time distributions
Taxi-out times reported in ASPM typically include queuing delay for runways. Since
it is more important to estimate actual waiting time of an aircraft but not the extra
delay in addition to expected delay impeded in the schedule, the model alleviates this
compound effect by having taxi-out times drawn from the distribution of the minimums of nominal taxi-out time and actual taxi-out time. Taxi-in time distributions
are fitted similarly.
6.2.3
En route time distributions
Enroute times are referred to as airtimes in ASPM. This metric sums the necessary
flying time to go from airports to airports, and the en-route delay due to weather or
traffic flow management. When an airport’s inbound traffic flow is expected to exceed
its available capacity, ATCs proactively delay arriving aircrafts by Ground Stops,
107
Ground Delay Programs for flights that have not departed yet, and impose Milein-trail restrictions, holding patterns, alternative routes and other flow management
procedures for airbound aircraft. As we wanted to isolate stochastic enroute delay
from this queuing effect, we only sampled flights such that at their wheels-on times
at destination airports, the number of arrivals does not exceed 75% of airport arrival
capacity. This condition helped identify flights that are not subject to traffic flow
management measures initiated by destination airports.
6.2.4
Cancellation and cancellation propagation
Let p1 (f ) denote the probability of canceling flight f caused by local technical or
operational problems, as defined previously in the cancellation submodel. p1 can be
empirically determined from ASPM for any time period length. The probability of
canceling flight f after canceling a connecting flight g, p2 (f |g), can also be determined
empirically by using connecting flight linkage. Without loss of generality, we show
in Figure 6.2 an example of p1 + p2 at LGA airport for every 1-hour time period
throughout the day where this probability can be as high as 10% for 22:00-23:00 time
window.
Figure 6.3 relates cumulative delay (arrival and departure) of all flown flights
P
k, k (dA (k) + dD (k)), to cumulative flight cancellations, c, throughout the day at
LGA. Each data point represents a 15-min time window of any day of the sampled
period and reflects the level of cumulative delay in minutes at the corresponding
number of cumulative cancellations. A time series plots the change of one metric in
relation to the other for one day. The set of these time series therefore shows the
approximate trend of delay-cancellation correlation that is fitted by a logarithmic
108
Figure 6.2: Hourly Empirical Cancellation Rates as the first component for simulated
cancellations
regression function.
As explained previously in the cancellation subsection, daily cumulative delay (of
arrivals and departures) can be considered as surrogate to airport performance based
on which airlines make cancellation decisions to certain extent. The statistical trend
over time between the cumulative delay and cancellations reflect aggregately how
airlines generally compromise between the two metrics. Figure 6.3 fits the trend of
P
these two cumulative metrics by the log function y = 7726lnx−7255.7, or k [dA (k)+
dD (k)] = 7726lnc−7255.7. At the departure of flight f during a simulation run, if the
two cumulative metrics stay at or below the log curve, the sum of p1 + p2 determines
the probability to cancel f as the delay is not too excessive to be compensated by a
cancellation; but if cumulative delay increases above the curve, a cancellation is forced
to maintain the log-fit non-linear trend between the two metrics in Figure 6.3. Given
the fitted log function of the relation between cumulative delay and cancellation in
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Figure 6.3: The relation of cumulative delay and cancellation used in simulating
cancellations
Figure 6.3, the following algorithm is used to determine the cancellation probability
of a flight f in the model:
p(f ) =
P
p1 + p2 if k [dA (k) + dD (k)] ≤ 7726lnc − 7255.7
1
otherwise
When one flight is cancelled, c is updated to give a new threshold for the condition.
110
6.3
6.3.1
Model calibration and application
Estimating delays and cancellations of alternative schedules
The sub-models and parameter estimation procedures are generic to all airports. In
this section, the Congestion Game [20] that investigated alternative slot allocation
schemes for LGA airport in anticipation of the removal of High-Density-Rules in
January 2007 motivated us to focus on this airport. Without lost of generality, we
present in this section the calibration of our model against actual data for LGA
airport taken from ASPM database for the period 2000-2001.
The schedule material from the Congestion Game [20], four flights schedules of
1386, 1274, 1240 and 1104 operations/day that result from administrative and congestion pricing measures, was run 100 independent replications each to compare the
outputs to average statistics of corresponding demand ranges. The schedule of 1386
operations/day correspond to the demand level in Fall 2000. The current schedule is
at 1240 operations/day, and the other schedules are derived from the current schedule. Comparison of delays estimated by our model against ASPM data are shown in
Fig. 4. As explained earlier in the runway capacity submodel, aircraft pair-wise separation standards are systematically enforced. We calibrated the multiplicative scaling
factor to approximate the estimates with the actual data. This scaling factor explains
for mixed runways, interdependency between runways, operational differences (due
to wind, temperature, elevation, ATC’s separation practice) of between airports. As
the scenario of 1240 operations represent the most common scenario, we calibrated
the scaling factor against actual data of this demand level.
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Figure 6.4: Comparison of delay estimates vs. actual data
The charts in Figure 6.4 compare average simulated arrival delays and departure
delays aggregated for 15-min periods versus actual statistical data respectively. Each
time series corresponding to each schedule scenario plots the deviation of simulated
aggregate arrival (departure) delay from recorded delays of all flights in every 15min bins. Extreme values at the tails of the curves are due to delays propagated by
network effects. The common trend in both arrival and departure delay estimation is
that the model tends to overestimate at higher levels of demand, appears accurate at
current levels, and underestimates at lower levels. The deviation begins to manifest
112
early in the afternoon, and appears more important for arrival than departure. The
over-prediction is due to the model strictly imposing standard separations between
aircraft at all demand levels. Network effects explain for the larger deviation in
arrivals compared to departures, as well as the under-prediction at low levels.
Delay and cancellation are highly correlated. Airport authorities need to look at
both metrics to determine the desirable level of one metric in conjunction with that of
the other metric. Cancellation implications simulated in the model are given in terms
of expected number of cancelled seats per hour, as shown in Figure 6.5. The common
trends of cancelled seats for the four schedule scenarios correspond to the combined
effects of empirical probability and the lognormal trade-off correlation between delay
and cancellations.
Figure 6.5: Estimates of cancelled seats
As expected, busier schedules are more likely to cancel more seats in addition to
high delays. Cancel seats also increase gradually towards the end of the day, due
to cascading effects from previously schedule disruptive events. Faced by demand
outpacing the growth of capacity, forecasting delays and cancellations is important in
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understanding the potential implications of airline schedules on airport performance
and the quality of service provided to the flying passengers. As US airlines can
schedule as many flights as they want in most US airports (except airports with HDR
such as LGA, JFK, and DCA), airport authorities can use this model to analyze before
hand the impacts of future demand levels in order to coordinate with airlines for
more desirable schedules, and conduct strategic planning for capacity enhancement
or congestion management. Moreover, our model could be extended to estimate
delay/cancellation at the level of individual airlines. Airline-specific estimates then
can be given to airlines involved in a coordinated scheduling process to incentivize
them make changes that might improve their individual performance and the overall
performance [73]. Therefore, the model provides a proactive approach to identify
schedule gridlocks and potentially mitigate well in advance.
6.3.2
Assessing impacts of changes in separation standards
on airport capacity and delay
The over-prediction observed in model calibration is due to the model strictly imposing standard separations between aircraft at all demand levels. Standard separations
were established a long time ago and thereby remain conservative given constant improvements in avionics. Because of this reason, and the pressure of higher incoming
traffic rates, ATC’s might adapt to keep delay down in practice.
The runway capacity sub-model that explicitly uses separation standards allows
for analysis of potential relaxation of this constraint. We adjusted the scaling factor
in the runway capacity submodel to reflect this adaptive behavior. Figure 6.6 shows
that model estimates accuracy for arrival (similarly for departure) when the come
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closer to actual data when separation standards in Table 6.1 are decreased by 6%
(reduced to 94% of the original values).
Figure 6.6: Adaptation of the system at high traffic levels and the effect on delay
Assuming that current technologies could safely decrease the separations by 6%,
simulated delay of the high-demand schedule scenario at 1375 operations/day with
separation standards being strictly enforced is brought down to the currently observable level. Therefore, if the current level of delay is considered maximally acceptable,
operational rates could only increase with a corresponding reduction in the separation standards. The model’s ability to assess airport capacity and performance by
scaling current separation standards is important. This could support policy-makers
to re-evaluate these standards, which have long been considered as conservative. Out
model quantifies the tradeoff between operational rates and separation standards. As
airport capacity becomes increasingly critical in coping with projected traffic growth
and congestion, a reduction of wake vortex separations needs to be carefully analyzed
to balance a desirable level of delay versus a required level of safety. Analytical models with closed-form estimation provide little support for this analysis requirement.
As such, simulation tools as ours can be very helpful.
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6.3.3
Assessing impacts of changes in fleet mix on delay estimates
In addition to the adaptive capability described above, the model distinguishes itself
further from analytical models, which take as input aggregate demands and capacities,
by allowing hypothesis on aircraft type to be made and estimating the resulting
effects. As ongoing efforts in congestion management try to bring the number of flight
operations align with airport capacities while maintaining the throughput, analysis of
the impacts by changes in the fleet mix on airport performance is important. Figure
6.7 compares estimated arrival delay per flight for the current fleet mix of the 1386
operations/day schedule scenario against that a hypothetic fleet of all-large aircraft
(from the wake vortex categorization standpoint) at the same operational level.
Figure 6.7: Effect of fleet changes on delay performance
Not only does the upgauging bring down average arrival delay per flight by 26%,
from 32.2 min/flight to 23.7 min/flight, it also enhances airport’s capacity, as separation for LARGE-LARGE is 2.5nmile/73sec vs. 4nmile/164sec for LARGE-SMALL.
This positive effect of a more homogeneous fleet mix of larger aircraft on airport
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capacity and performance is important: it provides incentives and support decisions
to upgauge airline fleet mix. This feature of our simulation model addresses the
shortcoming of all analytical queuing models that use aggregate demand to estimate
delays: aggregating demand loses all characteristics of the fleet mix and therefore
neglects this determinant of airport capacities’ operational constraint. Furthermore,
focus could be given to highly congested periods to identify groups of aircraft whose
upgauging could significantly reduce the delay peaks.
Ball et al. [20] pointed out: “Airlines have no effective means of differentiating
their service. Efforts to differentiate by increasing frequency of flights have resulted
in lower load factors, and airlines have responded by continuing to adjust their fleets
towards smaller regional jets with substantially higher cost per available seat mile.
The result of these efforts has been reduced profitability, but airlines are now locked
into higher-frequency schedules with fleets of smaller, less-economical aircraft”. As
such, studying the effects of fleet mix could assist policy-makers in devising measures
to enhance passenger throughput and reduce excessive flight frequencies, better the
utilization of public scarce resources. Moreover, in a larger context, the model could
help study the effects of new aircrafts such as B7E7 and A380 on airports’ capacity
and performance.
Chapter 7: Demand Management at LaGuardia
Airport: How Far Are We From Utopia?
Our methodology is applicable to airports that have mainly local traffic. In this
chapter we apply our methodology to LGA airport. We first extend the results of
the unconstrained profit-maximizing scenario presented in Chapter 4 to constrained
scenarios with different runway capacity levels at LGA. The public goal of maximizing
seat throughput is explored next, also in unconstrained and constrained scenarios.
As maximizing seat throughput is conflicting with profit maximizing, we identify
intermediate solutions and focus on two compromise scenarios. For each scenario and
runway capacity level, we report important metrics of the output schedules, such as
operation throughput, seat throughput, average aircraft size, average fare, number of
markets served, and average flight delay estimated by our delay model introduced in
Chapter 5.
7.1
Assumptions and parameters
As mentioned earlier in Chapter 4, we use the following assumptions and parameters
for all the scenarios:
Assumptions
• We only consider profitable daily schedules of nonstop domestic markets,
117
118
• The sample data is taken randomly from a much larger population set,
• The sample is a good representation of the population,
• The sample average fare is a good estimate of that of the population,
• Probabilities of price points in the sample are good estimates of those of price
points in the population,
• Time-based demand shares are proportional to time-based seat shares,
• Demand for each nonstop domestic market is equal in both directions, and hence
equal to the average of directional demands.
Data and Parameters
• Data sampling period: Q2, 2005
• 67 nonstop domestic markets that have daily schedules to/from LGA
• 45 minutes of minimum turn-around time for all fleets
• 80% load factor
• Fuel cost: $2/gallons
• Existing fleets
• One level of nesting with three generic substitution groups for all markets:
time windows from 6:00am-12:00pm, 12:01pm-17:00pm, and 17:01pm-24:00pm
are substitutable. However, finer grouping of substitutable time windows can
be done to reflect better demand characteristics by time of day for individual
markets.
119
7.2
Baseline statistics
General statistics For the sampling period of Q2 2005, ASPM reports traffic data
of 275 airports that had nonstop domestic and international flights to/from LGA, and
revenue data of 92 domestic markets. We only focus on 67 domestic markets1 that
have at least one nonstop flight in average per day during the sampling period. These
markets provide 92.6 % of the total passengers and 94% actual operations at LGA.
Statistics with respect to these 67 markets are collected in Table 7.1 to be compared
later against various scenarios. The overall statistics are also provided for reference
purpose.
Metrics
Markets
Flights
Seats
Passengers
Average aircraft size
Average fare
Average flight delay
Study
67
1024
98686
72845
95
$139
18.7 min
Overall
275
1104
101072
78675
95
$133
18.6 min
Table 7.1: Daily average statistics of 67 markets in study, and overall statistics
(Source: ASPM Q2, 2005)
Average market frequencies Figure 7.1 shows the geographical locations of 67
markets in study, and the average actual daily frequencies in both directions for
each market. Daily frequencies are colored coded using a color spectrum from 2 to
74 flights/day. BOS has the highest average frequency (73 flights/day), followed by
1
See Appendix A for airport codes, names and locations
120
DCA (68), ORD (62), ATL (48), FLL (43), and RDU (37). The smallest markets that
have regular daily frequencies are HOU (3 flights/day), BGR (3), HYA (2), MVY (2)
and LEX (2).
Figure 7.1: Geographical distribution of (flight) demand of LGA nonstop domestic
markets in study (see Table 7.9 for numerical values of actual frequencies)
Scheduled flights and actual average delays by time of day The average
number of flights scheduled in each 15min time windows and the resulting delays are
plotted in Figure 7.2. Throughout the day, demand fluctuates around the airportreported optimal rate of 10 deps(arrs) per 15 mins, alternated with small buffers of 30
minutes. As a result, queuing delays build up towards the end of the day, reaching up
to 40min for departures and almost 50min for arrivals. One can notice that departure
demand is higher in the morning, but departure delays worsen in the evening due to
121
delay propagating effects between flights circulating in the network.
Figure 7.2: Densely distributed demand and increasing queuing delays near the end
of the day
7.3
Investigated scenarios
Airline scheduling subproblems seek to maximize profit. The resulting schedules are
collected into a set-packing master problem. The set of profit-maximizing schedules of
a single benevolent airline represents the economic “Utopia”. This economic “Utopia”
122
results from demand-supply interaction through actual price elasticities, with the
assumption that supply can be consolidated. However, maximize profit is conflicting
with the public goal, which is to maximize enplanement opportunities. Therefore,
we first investigate two conflicting objective functions of the master problem: (i)
find schedules at LGA that maximize the overall profit, and (ii) find schedules that
maximize the overall seat throughput.
As maximizing seat throughput might select schedules that are suboptimal to airlines, we look at seat throughput maximizing scenarios with different lower bounds
on profits. We then select two intermediate solutions, called the compromise scenarios, that reconcile the two objective functions and are close to the baseline. The two
compromise scenarios impose profits of seat-maximizing schedules to be within 90%
and 80% of the profits of profit-maximizing schedules. These compromise scenarios
identify feasible transition paths towards the economic “Utopia”.
For each scenario, i.e. profit-maximizing, seat maximizing, and intermediate solutions, we solve the set-packing master problem at different runway rates to (i) analyze
the sensitivity of the outputs to this parameter, and (ii) further validate our model.
We then report for each combination of scenario and runway capacity the number of
markets, operation throughput, seat throughput, average aircraft size, average fare,
and estimate the resulting average flight delay. The scenarios are outlined in Table
7.2.
Scenario
123
Airport dep/arr rate/15min/runway
Unconstrained 10 9 8 7 6 5
Profit-maximizing
- - - - - Seat-maximizing
- - - - - Compromise 90%
- - - - - Compromise 80%
- - - - - -
4
-
Table 7.2: Scenarios investigated
7.4
Profit maximizing
The profit maximizing scenario has the same objective function in the subproblems
and in the master problem. Figure 7.3 plots the total seat throughput in the output
daily schedules, contrasted by the average output fare, for the baseline and different
runway capacity levels at LGA. The unconstrained scenario suggests a 20% reduction
in seats, which would increase average ticket price by 12% from $139 to $156. Note
Figure 7.3: Model suggests reduction in seats, which results in augmentation of average ticket price
124
that the total output seats for runway capacity levels ≥ 5 deps(arrs)/runway/15min
is still higher than the actual average number of passengers passing through LGA per
day during the sampling period.
Changes in the total output seats when runway capacity decreases might be
non-monotonic, due to adjustments of supply around that the supply level of the
profit optimum: decreasing or increasing supply from the profit-optimal supply level
can both decrease the optimal profit. It is also interesting to see that from 10
deps(arrs)/runway/15min, which is the reported Visual Meteorological Condition
(VMC) optimal rate for good weather conditions, to 8 deps(arrs)/runway/15min for
Instrument Meteorological Condition (IMC), the output seats do not change significantly. Observed actual rates at LGA for all weather conditions average at 8
deps(arrs)/runway/15min. Tightening the runway capacity constraint at LGA barely
affects the number of seats until the rate is set at 4 operations/runway/15min.
Changes in total seat throughput are translated to flight frequency and aircraft
size in Figure 7.4. Although seat throughput falls only by 20% for the unconstrained
scenario, daily flight frequency decreases by 40%, raising average aircraft size from 95
seats/flight to 130 seats/flight. These two time series follow the same trend as total
seat and fare time series with little change for most of the runway capacity levels,
and start deviate off at 5 ops/runway/15min.
The results suggest reduction of airline capacity through consolidation of flights
and increase aircraft size. This is consistent with the large number of low-load factor
flights observed in ASPM data, and the overscheduling reality of the industry that
drives down ticket price. The concept of a single benevolent airline that reacts to price
elasticity of demand in a competitive market helps us achieve these results. These
125
Figure 7.4: Delay reduction through consolidation of flights and aircraft upgauging
results represent the highest level of airline consolidation and profit-based rationality.
Our model demonstrates the inverse relation of supply and price: reduction of airline
capacity leads to increase in fare. Table 7.3 summarizes daily average statistics of
the profit-maximizing scenario. The minor non-monotonic changes in #flights and
#seats are normal for a set-packing problem solution. Figure 7.5 visualizes percentage
changes of the metrics compared to the baseline.
#deps (arrs) allowed per runway per 15min
BaselineUnconstrained 10
9
8
7
6
5
4
#markets
67
64
64
64
64
64
64
64
61
#flights
1024
602
594 598 596 596 594 570 476
#seats
96997
77700
77450 77600 77550 77300 77650 76200 66600
aircraft size 95
129
130 130 130 130 131 134 140
average fare $139
$156
$157 $157 $157 $157 $157 $159 $170
flight delay*18.7min
3min
2.7min2.8min2.7min2.7min2.3min 2min 1.4min
Table 7.3: Daily statistics of profit-maximizing scenarios (* queuing delay estimates
do not include international, non-daily and non-schedule operations)
126
Figure 7.5: Percentage change of daily statistics from baseline
Three markets that are not profitable to operate on a daily basis include LebanonHanover, NH (LEB), Roanoke Municipal, VA (ROA), and Knoxville, TN (TYS).
These markets might then have non-daily schedules, or relocate service to other substitutable airports. Table 7.13 gives their daily statistics.
Runway cap.
Unconstrained
10,9,8,7
6,5,4
4
Market
LEB
ROA
TYS
ACK
ALB
CHO
Frequency
6
5
2
5
7
5
Arc. size
19
37
50
26
33
33
Fare Passengers
$153
50
$186
77
$125
85
$216
47
$91
62
$229
80
Yield* ($)
0.72
0.46
0.19
1.07
0.67
0.75
Table 7.4: Daily average statistics of fall-off markets in profit-maximizing scenario at
different runway capacity levels, Source: ASPM Q2, 2005. (*revenue per passenger
mile)
127
7.5
Seat throughput maximizing
The seat maximizing scenario collects profit-maximizing schedules from airline scheduling submodels, and finds the best combination that maximizes the overall number
of seats. Therefore, the result of this scenario can be significantly different from
that of profit-maximizing. In fact, Figure 7.6 shows that the unconstrained setting
suggests a small increase in daily seat throughput at LGA. As runway capacity be-
Figure 7.6: Seat maximizing increases seats at high runway capacity levels
comes more restricted, seat throughput goes down gradually to the baseline level at
6 ops/runway/15min, and then continues to decrease. Average ticket price also drops
from the baseline $139 down to $129 for the unconstrained setting, then goes up
slowly to reach the baseline value again at 4 ops/runway/15min. Again, Figure 7.6
demonstrates the reverse relation between supply and price.
One might notice that at 6 ops/runway/15min, seats regain the baseline value
whereas fare is still smaller than the baseline fare value. That is because two small
128
markets, Nantucket, MA (ACK) and Norfolk, VA (ORF), fall off the solution, and the
remaining markets continue to have an increase in total seat throughput. Table 7.5
lists fall-off markets for all runway capacity levels investigated. Despite an increase
Runway cap.
Unconstrained
10,9,8,7
6,5,4
6
5
4
Market Frequency Arc. size Fare Passengers
LEB
6
19
$153
50
ROA
5
37
$186
77
TYS
2
50
$125
85
ACK
5
26
$216
47
ORF
14
34
$238
255
BGR
3
40
$93
76
GRR
1
39
$129
27
ITH
9
33
$160
93
SAV
7
50
$140
326
ACK, ORF
BHM
6
50
$190
280
CAE
6
50
$130
292
HYA
2
28
$235
19
MCI
10
125
$180
643
MVY
2
34
$233
15
RIC
19
50
$155
619
ACK, ORF, BGR, GRR, ITH, SAV
Yield* ($)
0.72
0.46
0.19
1.07
0.5
0.25
0.20
0.89
0.19
0.22
0.21
1.19
0.16
1.33
0.53
Table 7.5: Daily average statistics of fall-off markets in seat-maximizing scenario at
different runway capacity levels, Source: ASPM Q2, 2005
in seat throughput, the model produces schedules with fewer flights at all runway
capacity levels than the baseline. The supply level of this seat throughput maximizing
scenario is broken down to flight frequency and aircraft size in Figure 7.7. The
number of flights reduces gradually from 1024 flights in the baseline to 962 flights
in the unconstrained setting and to 484 flights at 4 ops/runway/15min. Aircraft
size also increases gradually from 95 seats/flight in the baseline to 115 seat/flight
at 10 ops/runway/15min, and up to 163 ops/runway/15min. Table 7.6 summarizes
daily and average statistics of the seat throughput maximizing scenario, Figure 7.8
129
visualizes the percentage changes compared to the baseline.
Figure 7.7: Despite increase in seats at high runway capacity levels, model suggests
gradual decrease of flights and aircraft upgauging
#deps (arrs) allowed per runway per 15min
BaselineUnconstrained 10
9
8
7
6
5
4
#markets
67
64
64
64
64
64
62
58
52
#flights
1024
962
914 898 848 770 686 588 484
#seats
96997
106250
105100104150102550100250 96550 89600 79100
aircraft size 95
110
115 116 121 130 141 152 163
average fare 139
125
126 126 128 130 131 137 139
flight delay*18.7min 15.7min 9.2min7.8min7.2min4.5min3.2min2.6min1.6min
Table 7.6: Daily statistics of seat throughput maximizing scenarios (* queuing delay
estimates do not include international, non-daily and non-schedule operations)
130
Figure 7.8: Percentage change of daily statistics from baseline
7.6
Compromise scenarios
Notice that seat throughput in the profit-maximizing scenario is significantly below
that in the seat throughput maximizing scenario. This results from the conflicting
objective functions of the two scenarios. Increasing seat throughput selects suboptimal schedules that provide more seats than the optimal quantity. Therefore, we
add a lower bound on the profit value of candidate schedules when solving the seat
throughput maximizing scenario to enforce the selection of schedules that are not too
far from the profit optimal.
Figure 7.9 illustrates the seat throughput curves for different values of lower bound
of schedule profit. As the lower bound approaches 100% of profit optimal, the seat
throughput curve gets closer to the optimal curve of the profit-maximizing scenario.
131
Figure 7.9: (1) Profit-maximizing (2) Seat-maximizing within 95% optimal profit (3)
Seat-maximizing within 90% optimal profit (4) Seat-maximizing within 80% optimal
profit (5) Seat-maximizing within 60% or less of optimal profit
The profit-maximizing scenario is the benchmark towards which commercial airlines should move to achieve economic efficiency, and this economic efficiency entails
significant airline capacity consolidation (20%). This benchmark is an “Utopia” in
the sense that monopoly is undesirable, and competition is necessary. On the other
hand, the seat throughput maximizing curve is the public goal that might lead to airlines’ unsustainable overscheduling. Therefore, we chose the intermediate solutions
at 90% and 80% of optimal profit that (i) are close enough to the baseline to provide
a feasible transition solution, and (ii) is reasonably close to the optimal profit curve.
When runway capacity is restricted, these intermediate solutions also represent levels
of service consolidation possibly resulting from airlines’ market-based responses.
132
7.6.1
Seat-maximizing within 90% profit optimal
Table 7.7 and Figure 7.10 summarize daily average statistics of the seat maximizing
scenario within 90% profit optimal.
Figure 7.10: Percentage change of daily statistics from baseline
#deps (arrs) allowed per runway per 15min
BaselineUnconstrained 10
9
8
7
6
5
4
#markets
67
64
64
64
64
62
59
54
43
#flights
1024
842
828 832 808 746 670 576 462
#seats
96997
99450
99300 98900 98100 96050 92550 86350 75900
aircraft size 95
118
120 119 121 129 138 150 164
average fare 139
133
133 133 134 135 137 142 146
flight delay*18.7min
8.2min
7.4min5.9min5.2min4.2min2.8min2.3min1.5min
Table 7.7: Daily statistics of 90% compromise scenarios (* queueing delay estimates
do not include international, non-daily and non-schedule operations)
133
In Table 7.8, we list the markets that fall out at different runway capacity levels.
In contrast to the previous scenarios, there is non-monotonicity for seat-maximizing
within 90% of profit optimal, due to the lower bound on profit and the fitting issue in
Runway cap.
Unconstrained
10,9,8,7
6,5,4
7
6
5
4
Market Frequency Arc. size Fare Passengers
LEB
6
19
$153
50
ROA
5
37
$186
77
TYS
2
50
$125
85
ACK
5
26
$216
47
BWI
14
38
$124
241
BGR
3
40
$93
76
ORF
14
34
$238
255
PHF
6
113
$107
412
SYR
15
37
$115
298
BWI
DAY
5
50
$131
195
HYA
2
28
$235
19
PVD
9
32
$121
129
SAV
7
50
$140
325
ACK, BRG, BWI, ORF, PHF, SYR
ALB
7
33
$91
62
BHM
6
50
$190
280
CAE
6
50
$130
293
GSP
9
50
$149
277
ILM
5
50
$135
184
IND
18
58
$138
747
MCI
10
125
$180
642
MEM
6
125
$170
574
MVY
2
34
$233
15
PHL
19
58
$59
522
PWM
14
50
$106
432
RIC
19
50
$154
619
XNA
4
38
$295
85
ACK, BRG, BWI, DAY, ORF, PHF, SAV, SYR
Yield* ($)
0.72
0.46
0.19
1.07
0.67
0.25
0.5
0.37
0.58
0.24
1.19
0.85
0.19
0.67
0.22
0.21
0.24
0.27
0.21
0.16
0.18
1.33
0.60
0.39
0.53
0.26
Table 7.8: Daily average statistics of fall-off markets in seat-maximizing scenario
within 90% profit optimal at different runway capacity levels, Source: ASPM Q2,
2005
134
a set packing problem. ACK’s schedule, for example, falls out at 7 ops/runway/15min
because the combination of other schedules fit into the capacity constraint and provide
a larger total of seats; adding ACK’s schedule violates the capacity constraint. At
6 ops/runway/15min however, BGR and ORF fall out, releasing capacity to ACK’s
schedule so that ACK could fit into the seat-maximizing combination.
In the next section, we look more into details the output schedules at 8 ops per
runway per 15min. We first estimate delays by time of day, then present changes in
schedules and fleet mix of individual markets.
Frequency and delay distribution by time of day Figure 7.11 plots the number
of flights (arrivals and departures) by their scheduled 15-min time windows for the
compromise scenario at 8 ops/runway/15min. Note that the output schedule includes
only nonstop domestic flights that are profitable on a daily basis. These flights come
from 64 airports. Other demands not accounted for are other flights, which include
international flights, non-daily and non-scheduled flights that can come from 275
airports having nonstop service to LGA. We stack the other flights on top of the
output schedule to approximate the total final demand of this scenario. Time series
of average total of actual demand is also plotted for comparison purpose.
The output schedule combined with other flights approximates well the average
demand by time of day. The total demand profile has fewer peaks above LGA optimal
runway capacity rates. The buffers retained between time windows serve to absorb
queuing delays accumulated at the peaks. We estimate average flight delay per flight
for the output schedule only in Figure 7.12, which is reduced to less than 15min for
any time window.
135
Figure 7.11: Model schedule reduces over-capacity peaks and retain buffers between
time windows
Figure 7.12: Seat-maximizing schedules within 90% profit optimal at 8 ops per 15min
reduce flight delay significantly
Changes in supply level and price of individual markets Table 7.9 provides
baseline values and numerical results for all the markets in this scenario.
Market
136
ACK
ALB
ATL
BGR
BHM
BNA
BOS
BTV
BUF
BWI
CAE
CAK
CHO
CHS
CLE
CLT
CMH
CVG
DAY
DCA
DEN
DFW
DTW
FLL
GSO
GSP
HOU
Daily
Model Frequency
Actual Output Aircraft
Actual
Model
Fare
Average
Output Deviation Aircraft Aircraft
Size Average Average Change
Frequency Frequency
Size
Size Change
Fare
Fare
5
2
-2.8
26
25
-1
216
148
-68
7
2
-5.3
33
25
-8
91
90
-1
48
32
-15.7
156
145
-11
128
167
39
3
4
0.9
40
38
-3
93
112
19
6
6
0.1
50
50
0
191
298
107
8
6
-2.5
83
108
25
177
225
48
73
60
-13.4
106
208
102
123
74
-50
11
6
-5.5
37
50
13
102
86
-15
21
14
-7.2
50
93
43
87
75
-12
14
8
-5.6
38
25
-13
124
189
65
6
6
0.3
50
50
0
131
152
21
6
6
0
113
125
12
100
87
-13
5
2
-3.3
33
25
-8
229
144
-85
11
10
-1.3
50
75
25
133
123
-11
21
16
-4.6
65
78
13
128
135
7
32
30
-1.7
102
97
-6
127
129
2
26
22
-4.2
46
102
56
150
97
-53
13
10
-3.5
122
155
33
121
123
2
5
6
0.5
50
50
0
131
147
16
69
68
-0.7
108
131
23
120
86
-35
14
14
0.1
158
150
-8
185
242
57
26
26
-0.5
148
146
-2
191
204
13
32
22
-9.7
122
175
53
124
126
1
43
26
-17.2
157
181
23
111
118
7
18
12
-6
50
96
46
127
107
-20
9
6
-2.7
50
83
33
149
115
-35
3
4
0.6
137
150
13
195
224
28
Market
137
HYA
IAD
IAH
ILM
IND
ITH
JAX
LEX
MCI
MCO
MDW
MEM
MHT
MIA
MKE
MSP
MSY
MVY
MYR
ORD
ORF
PBI
PHF
PHL
PIT
PVD
PWM
Daily
Model Frequency
Actual Output Aircraft
Actual
Model
Fare
Average
Output Deviation Aircraft Aircraft
Size Average Average Change
Frequency Frequency
Size
Size Change
Fare
Fare
2
2
-0.2
28
25
-3
235
274
39
16
16
-0.3
66
81
15
90
77
-14
17
18
0.9
131
150
19
215
217
2
5
6
0.6
50
83
33
135
111
-24
18
12
-5.8
58
88
30
138
137
-1
9
4
-4.7
33
25
-8
160
156
-4
8
6
-1.8
52
150
98
156
120
-36
2
2
0
50
50
0
171
231
60
10
6
-3.5
125
150
25
180
181
1
21
18
-2.5
166
156
-10
109
140
31
19
20
1.5
152
143
-10
115
111
-4
6
6
0.2
125
133
8
171
162
-9
16
10
-6.4
38
45
7
107
92
-15
16
14
-2.4
175
154
-21
141
202
62
12
8
-4.1
99
163
64
157
132
-25
13
12
-1.3
150
167
17
197
191
-7
6
6
0.1
131
175
44
155
156
0
2
2
-0.1
34
25
-9
233
259
26
6
6
0.3
104
175
71
130
118
-12
62
56
-6.1
138
139
1
148
143
-6
14
10
-3.7
34
25
-9
150
238
88
12
8
-3.6
171
225
54
111
118
6
6
6
0
113
75
-38
107
119
12
19
8
-11.5
58
44
-14
59
101
43
13
12
-1.1
112
67
-45
171
287
116
9
4
-4.8
32
25
-7
121
130
8
14
12
-2.5
49
58
10
106
101
-5
Market
RDU
RIC
ROC
SAV
SDF
STL
SYR
TPA
XNA
Daily
Model Frequency
Actual Output Aircraft
Actual
Model
Fare
Average
Output Deviation Aircraft Aircraft
Size Average Average Change
Frequency Frequency
Size
Size Change
Fare
Fare
37
22
-15.1
46
95
49
129
111
-18
19
14
-5
50
39
-11
154
229
74
14
6
-8.1
39
125
86
118
78
-40
7
6
-0.6
50
50
0
140
152
12
7
6
-0.9
50
83
33
165
172
7
9
10
0.9
137
155
18
173
168
-4
15
12
-2.8
39
25
-14
115
155
40
10
10
0.5
160
155
-5
114
127
13
4
4
0
38
50
12
295
227
-69
Table 7.9: Numerical results of the 90% compromise scenario at 8 ops/runway/15min
138
139
7.6.2
Seat-maximizing within 80% profit optimal
#deps (arrs) allowed per runway per 15min
BaselineUnconstrained 10
9
8
7
6
5
4
#markets
67
64
64
63
64
64
59
54
43
#flights
1024
902
882
868 824 780 684 582 474
#seats
96997
102750
102200 101600100250 98100 94700 87750 76850
aircraft size 95
114
116
117 122 126 138 151 162
average fare 139
129
130
130 131 133 135 140 143
flight delay*18.7min 12.5min 10.3min9.7min6.4min3.6min2.9min2.2min1.6min
Table 7.10: Daily statistics of 80% compromise scenarios (* queuing delay estimates
do not include international, non-daily and non-schedule operations)
Figure 7.13: Percentage change of daily statistics from baseline
140
Figure 7.14: Model schedule reduces over-capacity peaks and retain buffers between
time windows
Figure 7.15: Seat-maximizing schedules within 80% profit optimal at 8 ops per 15min
reduce flight delay less significantly
Market
141
ACK
ALB
ATL
BGR
BHM
BNA
BOS
BTV
BUF
BWI
CAE
CAK
CHO
CHS
CLE
CLT
CMH
CVG
DAY
DCA
DEN
DFW
DTW
FLL
GSO
GSP
HOU
Daily
Model Frequency
Actual Output Aircraft
Actual
Model
Fare
Average
Output Deviation Aircraft Aircraft
Size Average Average Change
Frequency Frequency
Size
Size Change
Fare
Fare
5
2
-2.8
26
25
-1
216
148
-68
7
2
-5.3
33
25
-8
91
90
-1
48
40
-7.7
156
125
-31
128
167
39
3
4
0.9
40
38
-3
93
112
19
6
6
0.1
50
50
0
191
298
107
8
6
-2.5
83
108
25
177
225
48
73
60
-13.4
106
208
102
123
74
-50
11
4
-7.5
37
63
26
102
86
-15
21
18
-3.2
50
75
25
87
75
-12
14
6
-7.6
38
25
-13
124
189
65
6
6
0.3
50
50
0
131
152
21
6
6
0
113
125
12
100
87
-13
5
2
-3.3
33
25
-8
229
144
-85
11
10
-1.3
50
70
20
133
123
-11
21
16
-4.6
65
78
13
128
135
7
32
32
0.3
102
98
-4
127
129
2
26
16
-10.2
46
141
95
150
97
-53
13
10
-3.5
122
175
53
121
123
2
5
6
0.5
50
50
0
131
147
16
69
68
-0.7
108
131
23
120
86
-35
14
14
0.1
158
150
-8
185
242
57
26
24
-2.5
148
163
15
191
204
13
32
22
-9.7
122
175
53
124
126
1
43
32
-11.2
157
158
0
111
118
7
18
12
-6
50
96
46
127
107
-20
9
6
-2.7
50
83
33
149
115
-35
3
4
0.6
137
150
13
195
224
28
Market
142
HYA
IAD
IAH
ILM
IND
ITH
JAX
LEX
MCI
MCO
MDW
MEM
MHT
MIA
MKE
MSP
MSY
MVY
MYR
ORD
ORF
PBI
PHF
PHL
PIT
PVD
PWM
Daily
Model Frequency
Actual Output Aircraft
Actual
Model
Fare
Average
Output Deviation Aircraft Aircraft
Size Average Average Change
Frequency Frequency
Size
Size Change
Fare
Fare
2
2
-0.2
28
25
-3
235
274
39
16
16
-0.3
66
81
15
90
77
-14
17
18
0.9
131
144
13
215
217
2
5
6
0.6
50
83
33
135
111
-24
18
10
-7.8
58
100
42
138
137
-1
9
4
-4.7
33
25
-8
160
156
-4
8
6
-1.8
52
150
98
156
120
-36
2
2
0
50
50
0
171
231
60
10
8
-1.5
125
131
6
180
181
1
21
18
-2.5
166
156
-10
109
140
31
19
20
1.5
152
158
5
115
111
-4
6
6
0.2
125
133
8
171
162
-9
16
10
-6.4
38
45
7
107
92
-15
16
14
-2.4
175
154
-21
141
202
62
12
8
-4.1
99
163
64
157
132
-25
13
14
0.7
150
154
4
197
191
-7
6
6
0.1
131
175
44
155
156
0
2
2
-0.1
34
25
-9
233
259
26
6
6
0.3
104
175
71
130
118
-12
62
50
-12.1
138
153
15
148
143
-6
14
12
-1.7
34
29
-5
150
238
88
12
10
-1.6
171
195
24
111
118
6
6
6
0
113
50
-63
107
119
12
19
8
-11.5
58
44
-14
59
101
43
13
12
-1.1
112
67
-45
171
287
116
9
4
-4.8
32
25
-7
121
130
8
14
14
-0.5
49
57
9
106
101
-5
Market
RDU
RIC
ROC
SAV
SDF
STL
SYR
TPA
XNA
Daily
Model Frequency
Actual Output Aircraft
Actual
Model
Fare
Average
Output Deviation Aircraft Aircraft
Size Average Average Change
Frequency Frequency
Size
Size Change
Fare
Fare
37
26
-11.1
46
90
44
129
111
-18
19
16
-3
50
34
-16
154
229
74
14
6
-8.1
39
125
86
118
78
-40
7
4
-2.6
50
50
0
140
152
12
7
6
-0.9
50
83
33
165
172
7
9
10
0.9
137
155
18
173
168
-4
15
14
-0.8
39
25
-14
115
155
40
10
10
0.5
160
190
30
114
127
13
4
4
0
38
38
0
295
227
-69
Table 7.11: Numerical results of the 80% compromise scenario at 8 ops/runway/15min
143
144
7.7
Discussion
The profit-maximizing scenario finds the economic “Utopia” where airlines, faced
with restricted runway capacity levels, are expected to rationally consolidate their
service. The results indeed suggest reduction of flights and aircraft upgauging. Consequently, this scenario is best congestion-wise. The optimal schedules of the single
benevolent airline represent the highest level of consolidation and rationality. As
complete consolidation is not realistic nor desirable for a competitive market, as well
as airlines do not always behave rationally, the results in fact provide an upper bound
on how air service can be restructured if airlines respond to capacity restrictions in a
market-based fashion.
The seat-maximizing scenario, on the contrary, finds the policy “Utopia” that
maximizes enplanement opportunities. The results increase the number of seats for
most of the runway capacity levels. While consolidating flights, this public goal could
encourage airlines to unsustainably overschedule, and therefore, this policy “Utopia”
might be neither stable nor desirable for long-term public planning.
The compromise scenarios of 90% and 80% illustrate different levels of market
concentration and rationality. For both scenarios, statistics of the output schedules show that, at 8 operations/runway/15min, the output total seats are higher
(increased by 1.1% and 3.4% respectively) than that of the baseline while average
flight delay is reduced significantly (dropped 72% and 66% respectively). There is
no penalty in the number of markets at 8 operations/runway/15min compared to
10 operations/runway/15min, which is the current Visual Meteorological Condition
(VMC) rate for good weather condition. Therefore, having aggregate airline schedules
145
at 8 operations/runway/15min will reduce significantly congestion problem at LGA,
increase the predictability of air transportation and improve the quality of service
expected by the flying public.
7.7.1
Research questions and answers
We review our findings that help answer the research problems stated previously.
Inefficiency due to current slot allocation rules Using actual data for EWR,
JFK and LGA, we showed that airport runway capacity is being used inefficiently.
50-seat or less aircraft make up a significant portion at all three airports: 40.6%,
23.6%, and 46% of the total flights at EWR, JFK, and LGA respectively, and flights
having 60% or less load factor represent 22%, 9.4%, and 36.2%. We identified three
main causes: (i) High-Density-Rule allocates slots to incumbent airlines who might
not have a profitable business model, (ii) slot exemptions granted 70-seat or less aircraft, (iii) the “use-it-or-lose-it” requirement, and (iv) weight-based landing fees.
Existence of profitable flight schedules that reduce congestion and accommodate current passenger throughput level Table 7.12 outlines the projected
market response with assumptions of 90% and 80% lower bounds on airline profit
optimal, or 90% and 80% levels of airline consolidation. Our model predicts positive
changes in seats, aircraft size, and negative changes in flight delay, average fare, number of flights. The number of profitable markets on a daily schedule stays the same.
146
Metric
Baseline 90% consolidation 80% consolidation
#markets
67
64
(-4%)
64
(-4%)
#flights
1024
808
(-21%)
824
(-20%)
#seats
96997 98100
(1%) 100250
(3%)
aircraft size
95
121
(27%)
122
(28%)
average fare
139
134
(-4%)
131
(-6%)
flight delay* 18.7min 5.2min
(-72%) 6.4min
(-66%)
Table 7.12: Projected effects on daily operations at LGA that result from a marketbased slot allocation at 8 ops/runway/15min (*queueing delay estimates do not include international, non-daily and non-schedule operations)
Unprofitable daily markets Three markets that are not profitable to operate on
a daily basis are identified to be Lebanon-Hanover, NH (LEB), Roanoke Municipal,
VA (ROA), and Knoxville, TN (TYS). These markets might then have non-daily
schedules, or relocate service to other substitutable airports. Table 7.13 gives their
daily statistics.
Runway cap.
Unconstrained
10,9,8,7
6,5,4
Market
LEB
ROA
TYS
Frequency
6
5
2
Arc. size Fare Passengers
19
$153
50
37
$186
77
50
$125
85
Yield* ($)
0.72
0.46
0.19
Table 7.13: Daily average statistics of fall-out markets at 8 ops/runway/15min, compromise scenarios, Source: ASPM Q2, 2005. (*revenue per passenger mile)
Frequency and delay distribution by time of day Figure 7.11 and Figure 7.14
plot the number of flights (arrivals and departures) by their scheduled 15-min time
windows, our estimates of flight delay are shown in Figure 7.12 and Figure 7.15. Note
that the output schedule includes only nonstop domestic flights that are profitable
147
on a daily basis. These flights come from 64 airports. Other demands not accounted
for are other flights, which include international flights, non-daily and non-scheduled
flights that can come from 275 airports having nonstop service to LGA. We stack
the other flights on top of the output schedule to approximate the total final demand
of this scenario. Time series of average total of actual demand is also plotted for
comparison purpose.
We notice that the 90% scenario with tighter lower bound on schedule profit
leads to reduction of schedule in the off-peak time windows of afternoon, while the
frequency profile approximates relatively well the morning and late evening traffic.
This results in less delays for arrivals and departures in early evening of the 90%
scenario, averaged at 8min, compared to 10-12min for the 80% scenario.
Chapter 8: Conclusion and Future Work
Air traffic growth is putting substantial pressure on airport infrastructure. Within
the next 10 years, forecasts by [3] predicted that there will be as many as 1.1 billion
air travelers per year in the U.S.. MITRE’s analysis of airport and metropolitan
area future demand and operational capacity [4] revealed that 15 airports, some not
currently in the OEP, will need additional capacity by 2013, and eight more will face
capacity limitations by 2020.
The currently planned improvements in aircraft, airport, and airspace systems
and operational procedures may not be sufficient to safely, securely, and efficiently
meet the U.S. transportation needs of the next 10 years. This concern is reflected
by various congestion management efforts, initiated by the FAA and by regional
airport management entities. Congestion management includes the construction of
new runways and/or airports, improvement of technology, and demand management
measures that control use in order to manage delays and congestion.
At congested airports where there are limited possibilities for expansion, appropriate demand management measures prove to be critical in coping with the projected
traffic growth. High Density Rule (HDR) currently imposed at LGA and JFK airports
aims to maintain demand at available capacity levels. However, the initial restrictions
of this rule along with many temporary fixes over time have resulted in recurring inefficiencies: small markets with small aircraft competing access with larger markets,
airlines flying large number of flights at low load factor just to maintain their slots
148
149
due to the “use-it-or-lose-it” rule.
With HDR scheduled to end in Jan 2007, appropriate demand management measures are critically needed to avoid overscheduling and severe congestion at this probably most important business airport in the Nation. Many potential proposals discuss
the use of congestion pricing and auctions of airport slots. However, appropriate
demand management measures require the understanding of airline operations and
market economics to design the right incentives, as well as beforehand study of implications on enplanement opportunities, average fare, markets served, aircraft size,
and flight delay.
Our methodology addresses this requirement. We take a novel approach in assuming a profit-seeking, single benevolent airline, and develop an airline economic
model to simulate scheduling decisions. This airline is defined as benevolent in the
sense that the airline reacts to price elasticities of demand in a competitive market.
These price elasticities of demand and cost data are estimated using publicly available databases. On the government side, airline schedules are selected to maximize
enplanement opportunities such that these schedules fit into the capacity constraints
at LGA airport. To reconcile the two conflicting objective functions, we find the
optimal solutions for each side, and identify compromise solutions. The compromise
scenarios maximize the number of seats while ensuring that airlines operate within
90% or 80% of profit optimality.
Our results show that in the compromise scenarios at 8 operations/runway/15min,
the total output seats are higher (increased by 1.1% and 3.4% for seat maximizing
within 90% or 80% of profit optimality respectively) than that of the baseline while
average flight delay is reduced significantly (dropped 72% and 66% respectively).
150
The number of flights is decreased by 21% and 19%; aircraft size is increased by
27% and 28%. As result of small increase in supply level, average fare is decreased
slightly by 4% and 6%. There is no penalty in the number of markets at 8 operations/runway/15min compared to 10 operations/runway/15min, which is the current
Visual Meteorological Condition (VMC) rate for good weather condition. Therefore,
having aggregate airline schedules at 8 operations/runway/15min will reduce significantly congestion problem at LGA, increase the predictability of air transportation
and improve the quality of service expected by the flying public.
8.1
Contributions
We summarize our contributions into four main areas:
Development of an airline flight and fleet scheduling model that incorporates the interaction of demand and supply through price (Chapter 3)
Appropriate congestion measures require the understanding of airline economics and
operations to avoid unduly affecting the business models of air carriers by forcing
impractical regulations. Therefore, modeling airline scheduling decisions is a central
part of this research. Unlike existing flight scheduling models that use fare as a parameter, our flight and fleet scheduling model considers fare as a variable negatively
dependent on supply level. This design choice allows the analysis of effects of changes
in schedules on average fares.
Development of a computationally-efficient solution algorithm to find the
optimal set of schedules (Chapter 3) We devise at each of the airports a column
151
generation algorithm to determine the optimal collection of schedules for each of the
Origin-Destination pairs based on the capacity constraints of the airports in study.
The decomposition algorithm decomposes the problem into a master problem that
optimizes use of the airports while the subproblems find optimal O/D schedules based
on current prices and demand curves.
Development of a methodology for estimating demand curves by time of
the day from publicly available sources (Chapter 4) We perform data mining
of ASPM and BTS databases to break down the aggregate data by quarter of the
year to aggregate data by day and time of day.
Development of a delay stochastic simulation network model to evaluate
flight schedules (Chapter 5) We develop a simulation model that explicitly considers wake vortex separation standards between categories of aircraft to simulate
runway capacity. Delays are estimated based on runway capacity. The model is
capable of evaluating the implications of fleet mix on runway operations throughput.
Demonstration of the existence of profitable airline schedules that reduce
congestion and accommodate current passenger throughput level (Chapter
6) We find the optimal demand allocation benchmarks for scenarios that assume
different capacity levels and public goals. The public goals investigated in this dissertation are (i) maximizing profit, (ii) maximizing seat throughput, and (iii) maximizing
the number of markets and seat throughput. The resulting schedules are then evaluated against the metrics of interest: Operations throughput, average flight delay,
seat throughput, average aircraft size, number of regular markets, and average fare.
152
The results show that at Instrument Meteorological Condition (IMC) rate of runway
capacity, airlines’ profit-maximizing responses can be expected to find scheduling solutions that offer 70% decrease in flight delays, 20% reduced in number of flights with
almost no loss of markets and no loss of passenger throughput.
8.2
Recommendations for future work
We identify the following potential ground for future work:
Adding layover costs When airlines choose service frequency and larger aircraft
size, they might increase the turnaround time between flights. Moreover, passenger
schedule delays increase. Schedule delay refers to the time between the most preferred
time of travel time of a passenger and the closest available flight.
Finer grouping of substitutable time windows into airport-specific peak
and off-peak periods For simplicity purpose, our study of LGA uses generic
grouping of substitutable time windows that assumes at any market, all time windows
in the morning (afternoon, or evening) are substitutable. While this is a simplistic
assumption to allow analytical convenience, it neglects the difference in travel time
preferences among markets. Plus, some time windows in the morning might be valued
more by the passengers than others. Therefore, we recommend more detailed grouping of substitutable time windows to reflect better peak and off-peak times at each
airport. We also suggest including the daily level of nesting revenue functions. With
only one level of nesting, there is the possibility that all time windows of a certain
group are not in the output schedule, resulting in a supply decrease while ticket prices
153
are still determined independently by the remaining groups.
Extend the sampling periods to include the whole calendar year We estimates model parameters using data of Q2, 2005. Future studies can use data of the
full year. Separate analyses with data of each quarter could also be done to maintain
the seasonal patterns, and propose some average solution.
Extend the methodology to airports that have good mixture of local and
through traffic Our methodology is appropriate for airports with mostly local traffic like LGA. EWR or JFK airports, however, have a significant connect, or through
traffic. The demands of individual markets are no longer independent: reduction or
increase of capacity on one market segment affects others. In addition to modeling
difficulty, the lack of Origin-Destination demand data also presents a challenge for
this research direction.
154
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Guidance, Navigation, and Control (GNC) Conference and Exhibit, Providence,
Rhode Island, August 2004.
[60] F. Wieland, “The Detailed Policy Assessment Tool (DPAT) User’s Manual, MTR
99W00000012,” The MITRE Corporation, McLean, MA, Tech. Rep., September
1999.
[61] A. Yousefi, G. Donohue, and K. Qureshi, “Investigation of en route metrics
for model validation and airspace design using the total airport and airspace
modeler (taam),” in Proc. 5th EUROCONTROL / FAA ATM R&D Conference,
Budapest, Hungary, 23rd - 27th, June 2003.
[62] A. Yousefi, “Optimum airspace design with air traffic controller workload-based
partitioning,” Ph.D. dissertation, George Mason University, Fairfax, VA, Spring
2005.
160
[63] Eurocontrol Experimental Center, “TAAM RVSM Fast-Time Simulation, Report
N358, Project SIM-S-E1,” Tech. Rep., February 2001.
[64] P. Massimini, “Using TAAM in Airline Operations and TAAM Analysis of EWR
Capacity for Parallel Arrivals,” MITRE, Tech. Rep., March 1999.
[65] P. Wang and L. Schaefer and L. Wojcik, “Flight connections and their impacts on
delay propagation,” The MITRE Corporation, McLean, VA, Tech. Rep., 2002.
[66] L. Schaefer and D. Millner, “Flight delay propagation analysis with the detailed
policy assessment tool,” in Proceedings of the 2001 IEEE Systems, Man, and
Cybernetics Conference, Tucson, Arizona, Institute of Electrical and Electronics
Engineers (IEEE), October 2001.
[67] G. Donohue and W. Laska, Air Transportation Systems Engineering. Progress
in Astronautics and Aeronautics, 2001, vol. 193, ch. United States and European
Airport Capacity Assessment Using the GMU Macroscopic Capacity Model, pp.
27–47.
[68] [Online]. Available: http://www.bts.gov
[69] [Online]. Available: http://www.apo.data.faa.gov/main
[70] J. Shortle, R. Xie, C. H. Chen, and G. L. Donohue, “Simulating collision probabilities of landing airplanes at non-towered airports,” SIMULATION, vol. 80,
no. 1, pp. 21–31, 2004.
[71] A. A. Trani, “Runway Occupancy Time Estimation and SIMMOD (presentation), Virginia Tech.” 2000.
[72] R. C. Haynie, “An investigation of capacity and safety in near-terminal airspace
for guiding information technology adoption,” Ph.D. dissertation, George Mason
University, 2002.
[73] M. Somerday and A. Merling, “Advance Planning Through Schedule Analysis,”
in Proceedings of the 6th USA-Europe ATM R&D Seminar, June 27, 2005, Baltimore, MD.
161
Appendix A: Airport Codes, Locations and Names
162
ACK
ALB
ATL
BGR
BHM
BNA
BOS
BTV
BUF
BWI
CAE
CAK
CHO
CHS
CLE
CLT
CMH
CVG
DAY
DCA
DEN
DFW
DTW
FLL
GSO
GSP
HOU
HYA
IAD
IAH
ILM
IND
ITH
JAX
LEB
LEX
Nantucket, MA: Nantucket Memorial
Albany, NY: Albany County
Atlanta, GA: Hartsfield-Jackson
Bangor, ME: Bangor International
Birmingham, AL: Birmingham Municipal
Nashville, TN: Nashville Metropolitan
Boston, MA: Logan International
Burlington, VT: Burlington International
Buffalo/Niagara Falls, NY: Greater Buffalo International
Baltimore, MD: Baltimore/Washington International
Columbia, SC: Columbia Metropolitan
Akron/Canton Regional, OH: Regional
Charlottesville, VA: Charlottesville Albemarle
Charleston, SC: Charleston International
Cleveland, OH: Hopkins International
Charlotte, NC: Douglas Municipal
Columbus, OH: Columbus International
Covington, KY: Cincinnati/ Northern Kentucky International
Dayton, OH: James M Cox/Dayton International
Washington, DC: Washington National
Denver, CO: Denver International
Dallas/Ft.Worth, TX: Dallas/Ft Worth International
Detroit, MI: Detroit Metro Wayne County
Fort Lauderdale, FL: Fort Lauderdale International
Greensboro/High Point, NC: Greensboro High Point Winst
Greenville/Spartanburg, SC: Greenville/Spartanburg Airport
Houston, TX: William P Hobby
Hyannis, MA: Barnstable Municipal
Washington, DC: Dulles International
Houston, TX: Houston Intercontinental
Wilmington, NC: New Hanover County
Indianapolis, IN: Indianapolis International
Ithaca/Cortland, NY: Tompkins County
Jacksonville, FL: Jacksonville International
Lebanon-Hanover, NH: Lebanon Municipal
Lexington/Frankfort, KY: Blue Grass
163
MCI
MCO
MDW
MEM
MHT
MIA
MKE
MSP
MSY
MVY
MYR
ORD
ORF
ROA
ROC
PBI
PHF
PHL
PIT
PVD
PWM
RDU
RIC
ROC
SAV
SDF
STL
SYR
TPA
TYS
XNA
Kansas City, MO: Kansas City International
Orlando, FL: Orlando International
Chicago, IL: Chicago Midway
Memphis, TN: Memphis International
Manchester/Concord, NH: Grenier Field /Manchester Municipal
Miami, FL: Miami International
Milwaukee, WI: General Mitchell Field
Minneapolis/St. Paul Int, MN: Minneapolis-St Paul
New Orleans, LA: Louis Armstrong International
Martha’s Vineyrd, MA: Marthas Vineyard
Myrtle Beach, SC: Myrtle Beach International Airport
Chicago, IL: O Hare
Norfolk/Va.Bch/Ptsmth/Chpk, VA: Norfolk Va
Roanoke, VA: Roanoke Municipal
Rochester, NY: Rochester Monroe County
West Palm Beach/Palm Beach, FL: Palm Beach International
Newport News/Williamsburg, VA: Patrick Henry International
Philadelphia, PA: Philadelphia International
Pittsburgh, PA: Pittsburgh International
Providence, RI: Theodore Francis Green
Portland, ME: Portland International Jetport
Raleigh/Durham, NC: Raleigh Durham
Richmond, VA: Richard Elelyn Byrd International
Rochester, NY: Rochester Monroe County
Savannah, GA: Savannah International
Standiford Field, KY: Standiford Field Airport
St. Louis, MO: Lambert/St Louis International
Syracuse, NY: Syracuse Hancock International
Tampa, FL: Tampa International
Knoxville, TN: Mcghee Tyson
Fayetteville, AR: Northwest Arkansas Regional
164
Appendix B: Problem formulations for ORD-LGA
market in MPL
Used for profit-maximizing goal of the master problem
TITLE
single_market
OPTIONS
DatabaseType=Access;
DatabaseAccess="..\LGA_Q2_2005\mpl_input_data.mdb";
INDEX
node := 1..96*2 ;
i := node;
j := node;
p_i := node;
temp := node;
k := DATABASE("mpl_aircraft_data","aircraft" WHERE market="ORD" and cluster_airpor
flight_arc[k,i,j] := DATABASE("mpl_flight_arc",k="aircraft",i="i",j="j" WHERE mark
iq := DATABASE("mpl_pw_revenue","i" WHERE market="ORD" and cluster_airport="LGA");
q := DATABASE("mpl_pw_revenue","segment" WHERE market="ORD" and cluster_airport="L
165
piecewise_revenue[iq,q] := DATABASE("mpl_pw_revenue",iq="i",q="segment" WHERE mark
p := DATABASE("mpl_pw_periodic_revenue","p" WHERE market="ORD" and cluster_airport
r := DATABASE("mpl_pw_periodic_revenue","segment" WHERE market="ORD" and cluster_a
periodic_pw_revenue[p,r]:=DATABASE("mpl_pw_periodic_revenue",p="p",r="segment" WHE
period_epoch[p,p_i] := DATABASE("mpl_pw_revenue",p_i="i",p="p" WHERE market="ORD"
DATA
N = count(node);
T = N / 2;
S[k]:=DATABASE("mpl_aircraft_data","seats",k="aircraft" WHERE Market="ORD");
C[k,i,j]:=DATABASE("mpl_flight_arc","cost",k="aircraft",i="i",j="j" WHERE market="
A[iq,q]:=DATABASE("mpl_pw_revenue","demand",iq="i",q="segment" WHERE market="ORD"
R[iq,q]:=DATABASE("mpl_pw_revenue","revenue",iq="i",q="segment" WHERE market="ORD"
pA[p,r]:=DATABASE("mpl_pw_periodic_revenue","demand",p="p",r="segment" WHERE marke
pR[p,r]:=DATABASE("mpl_pw_periodic_revenue","revenue",p="p",r="segment" WHERE mark
SS[k]:= S[k]*0.8;
INTEGER VARIABLES
x[k,i,j in flight_arc];
VARIABLES
y[k,i,j] WHERE (i<T AND j=i+1) OR (i>T AND j=i+1) OR (i=T AND j=1) OR (i=N AND j=T
pl[p,r in periodic_pw_revenue];
l[iq,q in piecewise_revenue];
166
MACRO
REVENUE = sum(iq,q in piecewise_revenue: R[iq,q]*l[iq,q]);
COST
= sum(k,i,j in flight_arc: C*x);
FREQUENCY = sum(k,i,j in flight_arc: x);
THROUGHPUT = sum(k,i,j in flight_arc: S*x);
MODEL
MAX REVENUE - COST;
SUBJECT TO
! new column generation
cg: REVENUE - COST >= 0;
! flow balance contraints
flow[k,i,temp=i] when (i<=T-1 and i>=2) or (i>=T+2 and i<=N-1): sum(j in flight_ar
flow[k,i,temp=i] when i=T+1 or i=1:
flow[k,i,temp=i] when i=N or i=T:
sum(j in flight_arc:x[k,i,j]) + sum(j:y[k,i,j
sum(j in flight_arc:x[k,i,j]) + sum(j:y[k,i,j=i
! piecewise balance contraints
pw[iq] when (iq+3<=T) or ((iq>T) and (iq+3<=N)): sum(k,i,j in flight_arc: round(SS
pw[iq] when ((iq+3>T) and (iq<=T)) or (iq+3>N): sum(q: A[iq,q]*l[iq,q]) = 0;
s[iq]: sum(q: l[iq,q]) = 1;
167
ppw[p]: sum(p_i in period_epoch, iq,q in piecewise_revenue: A[iq=p_i,q]*l[iq=p_i,q
ps[p]: sum(r: pl[p,r]) = 1;
nested_pw[p]: sum(p_i in period_epoch, iq,q in piecewise_revenue: R[iq=p_i,q]*l[iq
BOUNDS
x <= 5;
END
Used for seat-maximizing goal of the master problem
TITLE
single_market
OPTIONS
DatabaseType=Access;
DatabaseAccess="..\LGA_Q2_2005\mpl_input_data.mdb";
INDEX
node := 1..96*2 ;
i := node;
j := node;
p_i := node;
temp := node;
k := DATABASE("mpl_aircraft_data","aircraft" WHERE market="ORD" and
168
cluster_airport="LGA");
flight_arc[k,i,j] := DATABASE("mpl_flight_arc",k="aircraft", i="i",
j="j" WHERE market="ORD" and cluster_airport="LGA");
iq := DATABASE("mpl_pw_revenue","i" WHERE market="ORD" and
cluster_airport="LGA");
q := DATABASE("mpl_pw_revenue","segment" WHERE market="ORD" and
cluster_airport="LGA");
piecewise_revenue[iq,q] := DATABASE("mpl_pw_revenue", iq="i",
q="segment" WHERE market="ORD" and cluster_airport="LGA");
p := DATABASE("mpl_pw_periodic_revenue", "p" WHERE market="ORD" and
cluster_airport="LGA");
r := DATABASE("mpl_pw_periodic_revenue","segment" WHERE market="ORD"
and cluster_airport="LGA");
periodic_pw_revenue[p,r]:=DATABASE("mpl_pw_periodic_revenue", p="p",
r="segment" WHERE market="ORD" and cluster_airport="LGA");
period_epoch[p,p_i] := DATABASE("mpl_pw_revenue",p_i="i",p="p" WHERE
market="ORD" and cluster_airport="LGA");
DATA
N = count(node);
T = N / 2;
S[k]:=DATABASE("mpl_aircraft_data","seats",k="aircraft" WHERE
Market="ORD");
169
C[k,i,j]:=DATABASE("mpl_flight_arc","cost",k="aircraft",i="i", j="j"
WHERE market="ORD" and cluster_airport="LGA");
A[iq,q]:=DATABASE("mpl_pw_revenue","demand",iq="i",q="segment" WHERE
market="ORD" and cluster_airport="LGA");
R[iq,q]:=DATABASE("mpl_pw_revenue","revenue",iq="i",q="segment" WHERE
market="ORD" and cluster_airport="LGA");
pA[p,r]:=DATABASE("mpl_pw_periodic_revenue","demand",p="p",r="segment"
WHERE market="ORD" and cluster_airport="LGA");
pR[p,r]:=DATABASE("mpl_pw_periodic_revenue","revenue",p="p",r="segment"
WHERE market="ORD" and cluster_airport="LGA");
profit_optimal:=DATABASE("profit_optimal_data" WHERE market="ORD" and
cluster_airport="LGA");
SS[k]:= S[k]*0.8;
INTEGER VARIABLES
x[k,i,j in flight_arc];
VARIABLES
y[k,i,j] WHERE (i<T AND j=i+1) OR (i>T AND j=i+1) OR (i=T AND j=1) OR
(i=N AND j=T+1);
pl[p,r in periodic_pw_revenue];
l[iq,q in piecewise_revenue];
MACRO
170
REVENUE = sum(iq,q in piecewise_revenue: R[iq,q]*l[iq,q]);
COST
= sum(k,i,j in flight_arc: C*x);
FREQUENCY = sum(k,i,j in flight_arc: x);
THROUGHPUT = sum(k,i,j in flight_arc: S*x);
MODEL
MAX REVENUE - COST;
SUBJECT TO
! new column generation
cg: sum(k,i,j in flight_arc: S*x[k,i,j]) >= 0;
! lower bound on profit
profitability: REVENUE - COST >= 0.9*profit_optimal;
! flow balance contraints
flow[k,i,temp=i] when (i<=T-1 and i>=2) or (i>=T+2 and i<=N-1):
sum(j in flight_arc:x[k,i,j]) + sum(j: y[k,i,j=i+1]) sum(i,j in flight_arc:x[k,i,j=temp])
- sum(j:y[k,i-1,j=i])= 0;
flow[k,i,temp=i] when i=T+1 or i=1:
sum(j in flight_arc:x[k,i,j]) +
sum(j:y[k,i,j=i+1]) - sum(i,j in flight_arc:x[k,i,j=temp])
- sum(j:y[k,i+T-1,j=i])= 0;
171
flow[k,i,temp=i] when i=N or i=T:
sum(j in flight_arc:x[k,i,j]) +
sum(j:y[k,i,j=i-T+1]) - sum(i,j in flight_arc:x[k,i,j=temp])
- sum(j:y[k,i-1,j=i])= 0;
! piecewise balance contraints
pw[iq] when (iq+3<=T) or ((iq>T) and (iq+3<=N)):
sum(k,i,j in flight_arc: round(SS[k])*x[k,i,j=iq+3])
- sum(q: A[iq,q]*l[iq,q]) = 0;
pw[iq] when ((iq+3>T) and (iq<=T)) or (iq+3>N):sum(q:A[iq,q]*l[iq,q])=0;
s[iq]: sum(q: l[iq,q]) = 1;
ppw[p]: sum(p_i in period_epoch, iq,q in piecewise_revenue:
A[iq=p_i,q]*l[iq=p_i,q]) - sum(r: pA[p,r]*pl[p,r]) = 0;
ps[p]: sum(r: pl[p,r]) = 1;
nested_pw[p]: sum(p_i in period_epoch, iq,q in piecewise_revenue:
R[iq=p_i,q]*l[iq=p_i,q]) - sum(r: pR[p,r]*pl[p,r]) <= 0;
BOUNDS
x <= 5;
END
172
Appendix C: Implementation of solution algorithm
(column generation) in C/Cplex Concert
Technology API
settings.cpp
#ifndef _SETTINGS_
#define _SETTINGS_
#include <ilcplex/ilocplex.h>
#include <ilcplex/ilocplexi.h>
#include <math.h>
#include <string>
#include <iostream>
#include <fstream>
using namespace std;
ILOSTLBEGIN
#define EPS
1.0e-3
173
#define DEBUG
#define PROFIT
//#define THROUGHPUT
//#define MARKET_ENTRANCE
typedef IloArray<IloModel>
IloModelArray;
typedef IloArray<IloObjective>
IloObjArray;
typedef IloArray<IloNumVarArray>
IloVarArray;
typedef IloArray<IloRangeArray>
IloConArray;
typedef IloArray<IloCplex>
IloSolverArray;
typedef IloArray<IloNumArray>
IloNumArrayArray;
typedef IloArray< IloArray<unsigned char> >
IloFlightArray;
typedef IloArray<IloFlightArray>
IloColumnSolutionArray;
static const char * WORKING_DIR
=
"../data/LGA_Q2_2005/LGA_80_mf_profit/lp1_backup/";
static const char * MARKET_FILE_NAME = "markets.dat";
static const char * SUB_MODEL_FILE_SUFFIX = "_profit_max.lp";
static const char * OUTPUT_SCHEDULE_FILE_NAME = "schedule.txt";
static const char * OUTPUT_LOG_FILE_NAME = "log.txt";
static const char * OUTPUT_COLUMNS_FILE_NAME = "columns.txt";
static ofstream
static const int
fid1, fid2;
CAPACITY_INCREMENT = 25;
174
static const IloInt
M = 0;
// arrival capacities and departure rates
static int
AIRPORT_QUARTER_CAPACITY = 4;
// number of 15-min time intervals
static const int
T = 96;
static const int
N = T*2;
static
int
n_markets = 0;
static
int
active_models = 0;
static int INTEGER_SOLUTION_ADDED;
static IloEnv
env;
static IloTimer
timer(env);
static int
rounds = 0;
static IloModel
master_model(env,"LGA");
static IloCplex
master_cplex(master_model);
static IloNumVarArray
master_vars(env);
static IloObjective
master_obj(env);
static IloRangeArray
master_arrival_cons(env);
static IloRangeArray
master_departure_cons(env);
static IloRangeArray
master_sos1_cons(env);
static IloRangeArray
master_cons(env);
175
static IloColumnSolutionArray column_solution(env);
static IloNumArray
master_throughput(env);
static IloModelArray
model(env);
static IloObjArray
obj(env);
static IloVarArray
vars(env);
static IloConArray
cons(env);
static IloConArray
cutoff(env);
static IloSolverArray
cplex(env);
//variables that need to update cost during column generation
static IloArray<IloVarArray>
dep_vars(env), arr_vars(env),
period_vars(env);
static IloArray<IloNumArrayArray> dep_vars_original_coef(env),
arr_vars_original_coef(env), period_vars_original_coef(env);
static void init_scenario_params();
static void init_cplex_params(IloCplex cplex);
static void init_problems();
static void report_schedule (char*, IloCplex&, IloNumVarArray);
static IloInt max_frequency;
static IloNumArray initial_max_frequency(env);
176
extern void generate_columns(IloNumArray, IloNumVarArray);
#endif
177
main.cpp
1
#include "settings.h"
2
3
class Node {
4
5
public :
6
Node
7
//IloNumArray
8
float
9
IloNum
10
//char
11
IloBool
*next, *prev;
node_dual_prices;
node_dual_prices[96*2];
value;
id[200];
branching;
12
13
IloNumVar
branching_variable;
14
IloNumVarArray node_variables;
15
IloNumVarArray node_variables_at_zero;
16
IloNumVarArray node_variables_at_one;
17
18
//Node(IloNumVarArray node_v, IloNumVarArray node_v_at_zero,
IloNumVarArray node_v_at_one, const char* s, IloNum val);
19
Node(IloNumVarArray node_v, IloNumVarArray node_v_at_zero,
IloNumVarArray node_v_at_one, IloNum val);
20
178
21
22
void printInfo();
};
23
24
class NodeList {
25
int
26
public:
n_nodes;
27
28
Node *head;
29
30
NodeList() {
31
n_nodes = 0;
32
head = NULL;
33
}
34
35
int getSize() {
36
return n_nodes;
37
}
38
39
void addNode(Node*);
40
void removeNode(Node*);
41
void printInfo();
42
void clear();
43
44
};
179
45
46
class TreeManager {
47
48
void getObjCoef(IloObjective obj, IloNumArray coef);
49
void load_node(Node*);
50
void branch_node(Node*);
51
void select_branching_variable(Node*);
52
53
public:
54
IloNum
lower_bound, upper_bound;
55
Node
*root, *solution;
56
NodeList
list;
57
58
TreeManager();
59
60
IloInt getSize() {
61
return list.getSize();
62
}
63
64
void solve();
65
void printSolution();
66
void solve_generate_columns_resolve(Node*);
67
68
};
180
69
//Node::Node(IloNumVarArray node_v, IloNumVarArray node_v_at_zero,
IloNumVarArray node_v_at_one, const char* s, IloNum val) {
70
Node::Node(IloNumVarArray node_v, IloNumVarArray node_v_at_zero,
IloNumVarArray node_v_at_one, IloNum val) {
71
//env.out() << "Node::Node() : node " << s << "\n";
72
//node_dual_prices = new IloNumArray(env,master_cons.getSize());
73
prev = next = NULL;
74
//
75
value = val;
76
branching = IloFalse;
strcpy(id,s);
77
78
node_variables = IloNumVarArray(env, node_v.getSize());
79
for (int i=0;i<node_v.getSize();i++)
80
node_variables[i]=node_v[i];
81
node_variables_at_zero = IloNumVarArray(env,
node_v_at_zero.getSize());
82
for (int i=0;i<node_v_at_zero.getSize();i++)
83
node_variables_at_zero[i]=node_v_at_zero[i];
84
node_variables_at_one = IloNumVarArray(env,
node_v_at_one.getSize());
85
for (int i=0;i<node_v_at_one.getSize();i++)
86
node_variables_at_one[i]=node_v_at_one[i];
87
88
}
181
89
90
TreeManager::TreeManager() {
91
// env.out() << "TreeManager::TreeManager: #variables="
<< master_vars.getSize() << "\n";
92
//env.out() << "TreeManager::TreeManager: #constraints="
<< master_cons.getSize() << "\n";
93
lower_bound = - IloInfinity;
94
upper_bound = 0;
95
master_cplex.setOut(env.getNullStream());
96
solution = NULL;
97
98
}
99
100
void TreeManager::solve() {
101
master_cplex.solve();
102
root = new Node(master_vars, IloNumVarArray(env),
IloNumVarArray(env), master_cplex.getObjValue());
103
104
//IloNumArray duals(env, master_cons.getSize());
105
//master_cplex.getDuals(duals, master_cons);
106
//generate_columns(duals, root->node_variables);
107
////env.out() << "TreeManager::solve():#variables="
<< master_vars.getSize() << "\n";
108
////env.out() << "TreeManager::solve():#node variables="
182
<<root->node_variables.getSize() << "\n";
109
////env.out() << "TreeManager::solve():#constraints="
<< master_cons.getSize() << "\n";
110
//master_cplex.extract(master_model);
111
//master_cplex.exportModel("root_node.lp");
112
//master_cplex.solve();
113
//root->value = master_cplex.getObjValue();
114
115
//root = new Node(master_vars, IloNumVarArray(env),
IloNumVarArray(env), "1", master_cplex.getObjValue());
116
117
select_branching_variable(root);
118
119
if (!root->branching) {
120
solution = root;
121
return;
122
}
123
124
branch_node(root);
125
while (list.head) {
126
branch_node(list.head);
127
list.removeNode(list.head);
128
129
}
}
183
130
131
void TreeManager::solve_generate_columns_resolve(Node* n) {
132
master_cplex.solve();
133
IloNumArray duals(env, master_cons.getSize());
134
master_cplex.getDuals(duals, master_cons);
135
generate_columns(duals, n->node_variables);
136
master_cplex.extract(master_model);
137
master_cplex.solve();
138
}
139
140
void TreeManager::select_branching_variable(Node* n) {
141
IloNumArray x;
142
IloNumArray obj_coef;
143
IloInt bestj
= -1;
144
145
try {
146
x
147
obj_coef
148
master_cplex.getValues(x, master_vars);
149
getObjCoef(master_obj, obj_coef);
= IloNumArray(env);
= IloNumArray(env, master_vars.getSize());
150
151
IloNum maxinf = 0.0;
152
IloNum maxobj = 0.0;
153
IloInt cols = master_vars.getSize();
184
154
for (IloInt j = 0; j < cols; ++j) {
155
if ( fabs(round(x[j])-x[j]) > EPS ) {
156
IloNum xj_inf = x[j] - IloFloor (x[j]);
157
if ( xj_inf > 0.5 )
158
159
xj_inf = 1.0 - xj_inf;
if ( xj_inf >= maxinf && (xj_inf > maxinf ||
IloAbs (obj_coef[j]) >= maxobj)
) {
160
bestj
161
maxinf = xj_inf;
162
maxobj = IloAbs (obj_coef[j]);
163
= j;
}
164
}
165
}
166
if ( bestj >= 0 ) {
167
n->branching = IloTrue;
168
n->branching_variable = master_vars[bestj];
169
} else
170
env.out() << "integer solution found\n";
171
} catch (IloException& e) {
172
env.out() << e << "\n";
173
x.end();
174
obj_coef.end();
175
throw;
176
}
185
177
x.end();
178
obj_coef.end();
179
}
180
181
void TreeManager::load_node(Node* n) {
182
for (int i=0;i<master_vars.getSize();i++) {
183
master_vars[i].setUB(0);
184
master_vars[i].setLB(0);
185
}
186
for (int i=0;i<n->node_variables.getSize();i++) {
187
n->node_variables[i].setUB(1);
188
n->node_variables[i].setLB(0);
189
}
190
for (int i=0;i<n->node_variables_at_zero.getSize();i++)
191
n->node_variables_at_zero[i].setUB(0);
192
for (int i=0;i<n->node_variables_at_one.getSize();i++)
193
194
n->node_variables_at_one[i].setLB(1);
}
195
196
197
198
void TreeManager::branch_node(Node* n) {
if ((n->branching) && (IloFloor(n->value)>lower_bound)){
env.out() << "branch " << n->branching_variable.getName() << "\n";
199
200
// Branch on var with largest objective coefficient
186
201
// among those with largest infeasibility
202
203
load_node(n);
204
// left branch
205
// add new bound
206
n->branching_variable.setUB(0);
207
try {
208
master_cplex.extract(master_model);
209
//master_cplex.solve();
210
solve_generate_columns_resolve(n);
211
IloNum new_z;
212
if ((master_cplex.getStatus()==IloAlgorithm::Optimal) &&
((new_z=master_cplex.getObjValue())>lower_bound)) {
213
//char s[20];
214
//strcpy(s, n->id);
215
//strcat(s,"_1");
216
//Node* left_child = new Node(n->node_variables,
n->node_variables_at_zero, n->node_variables_at_one, s, new_z);
217
Node* left_child = new Node(n->node_variables,
n->node_variables_at_zero, n->node_variables_at_one, new_z);
218
left_child->node_variables_at_zero.add(n->branching_variable);
219
select_branching_variable(left_child);
220
if ((left_child->branching==IloFalse) && (new_z>lower_bound)) {
221
lower_bound = new_z;
187
222
if (solution)
223
224
225
delete solution;
solution = left_child;
} else if (left_child->branching==IloTrue)
226
list.addNode(left_child);
227
228
}
} catch (...) {
229
//env.out() << "Left child infeasible\n";
230
}
231
try {
232
n->branching_variable.setUB(1);
233
n->branching_variable.setLB(1);
234
master_cplex.extract(master_model);
235
//master_cplex.solve();
236
solve_generate_columns_resolve(n);
237
IloNum new_z;
238
if ((master_cplex.getStatus()==IloAlgorithm::Optimal)
&& ((new_z=master_cplex.getObjValue())>lower_bound)) {
239
//char s[20];
240
//strcpy(s, n->id);
241
//strcat(s,"_2");
242
//Node* right_child = new Node(n->node_variables,
n->node_variables_at_zero, n->node_variables_at_one, s, new_z);
243
Node* right_child = new Node(n->node_variables,
188
n->node_variables_at_zero, n->node_variables_at_one, new_z);
244
right_child->node_variables_at_one.add(n->branching_variable);
245
select_branching_variable(right_child);
246
if ((right_child->branching==IloFalse) && (new_z>lower_bound)) {
247
lower_bound = new_z;
248
if (solution)
249
delete solution;
250
solution = right_child;
251
} else if (right_child->branching==IloTrue)
252
list.addNode(right_child);
253
}
254
} catch (IloException& e) {
255
//env.out() << e << "right child infeasible\n";
256
e.end();
257
}
258
259
}
}
260
261
void TreeManager::getObjCoef(IloObjective obj, IloNumArray coef) {
262
263
IloExpr expr = obj.getExpr();
264
IloExpr::LinearIterator li = expr.getLinearIterator();
265
266
int i = 0;
189
267
268
while (li.ok()) {
269
coef[i] = li.getCoef();
270
++li;
271
i++;
272
273
}
}
274
275
void TreeManager::printSolution() {
276
if (solution) {
277
load_node(solution);
278
env.out() << "Solution: z = " << solution->value << "\n";
279
master_cplex.extract(master_model);
280
master_cplex.solve();
281
IloNumArray x(env);
282
master_cplex.getValues(x, master_vars);
283
for (int i=0;i<x.getSize();i++)
284
if (x[i]>EPS)
285
env.out() << master_vars[i].getName() << "\n";
286
287
}
}
288
289
290
void Node::printInfo() {
//env.out() << "node " << id << ": z = " << value << "\n";
190
291
}
292
293
void NodeList::addNode(Node* n) {
294
Node *p;
295
n_nodes++;
296
if (head==NULL)
297
298
head = n;
else {
299
p = head;
300
while (p->value >= n->value) {
301
if (p->next==NULL) {
302
p->next = n;
303
n->prev = p;
304
return;
305
}
306
else p=p->next;
307
}
308
309
if (p->prev==NULL) {
310
head = n;
311
n->next = p;
312
p->prev = n;
313
}
314
else {
191
315
n->next = p;
316
n->prev = p->prev;
317
p->prev->next = n;
318
p->prev = n;
319
}
320
}
321
322
}
323
324
void NodeList::removeNode(Node* n) {
325
326
if (n==head) {
327
head = n->next;
328
if (head!=NULL) head->prev = NULL;
329
}
330
else {
331
n->prev->next = n->next;
332
if (n->next!=NULL) n->next->prev = n->prev;
333
}
334
335
delete n;
336
n_nodes--;
337
338
}
192
339
void NodeList::clear() {
340
Node* p=head;
341
Node* q;
342
while (p) {
343
q = p;
344
p = p->next;
345
delete q;
346
347
}
}
348
349
void NodeList::printInfo() {
350
Node* p=head;
351
Node* q;
352
while (p) {
353
p->printInfo();
354
p = p->next;
355
356
}
}
357
358
// add new column
359
void addColumn(int id) {
360
361
IloNum z = cplex[id].getObjValue();
362
IloNum profit = 0, cost = 0;
193
363
364
if
(z <= EPS) {
365
max_frequency = 0;
366
return;
367
}
368
369
IloNum gap = round(100*(cplex[id].getBestObjValue()-z)/z);
370
IloNumArray
arr(env,T), dep(env,T);
371
IloNum
service_level = 0, throughput = 0, val;
372
char
s[20], varname[15];
373
int
n;
374
unsigned char
fleet, dep_epoch, arr_epoch;
375
/*
376
if
377
(gap > 10) {
for (int i=0;i<vars[id].getSize();i++)
378
if ((vars[id][i].getName()[0]==’x’) &
((val = round(cplex[id].getValue(vars[id][i])))>EPS))
379
service_level += val;
380
max_frequency = (IloInt) service_level;
381
return;
382
}
383
*/
384
INTEGER_SOLUTION_ADDED = 1;
385
194
386
// number of columns in the master problem
387
n = master_vars.getSize();
388
sprintf(s,"%s_%d_%d",model[id].getName(),rounds,n+1);
389
390
// add new column
391
//
392
IloNumVar new_column(env, 0, 1, ILOFLOAT, s);
393
master_vars.add(new_column);
394
master_sos1_cons[id].setCoef(new_column, 1);
IloNumVar new_column(env, 0, IloInfinity, ILOFLOAT, s);
395
396
column_solution.add(IloFlightArray(env));
397
398
for (int i=0;i<vars[id].getSize();i++) {
399
strcpy(varname, vars[id][i].getName());
400
if ((varname[0]==’x’) & ((val =
round(cplex[id].getValue(vars[id][i])))>EPS)) {
401
402
403
404
405
if (varname[1]>=’A’)
fleet = varname[1]-’A’+10;
else
fleet = varname[1]-’0’;
throughput += fleet*CAPACITY_INCREMENT*val;
406
407
strncpy(s,&varname[2],3);
408
s[3]=’\0’;
195
409
dep_epoch
410
strncpy(s,&varname[5],3);
411
s[3]=’\0’;
412
arr_epoch
413
if (dep_epoch <= T)
414
dep[dep_epoch-1] += val;
415
416
= atoi(s);
= atoi(s);
else
arr[arr_epoch-1] += val;
417
service_level += val;
418
IloArray<unsigned char> flight(env,4);
419
flight[0]=fleet;
420
flight[1]=dep_epoch;
421
flight[2]=arr_epoch;
422
flight[3]=(unsigned char) val;
423
column_solution[n].add(flight);
424
425
}
}
426
427
428
429
for (int j=0;j<T;j++) {
for (int k=0;k<arr_vars[id][j].getSize();k++)
cost += -cplex[id].getValue(arr_vars[id][j][k])*
arr_vars_original_coef[id][j][k];
430
431
for (int k=0;k<dep_vars[id][j].getSize();k++)
cost += -cplex[id].getValue(dep_vars[id][j][k])*
196
dep_vars_original_coef[id][j][k];
432
}
433
profit = - cost;
434
for (int p=0;p<6;p++)
435
for (int r=0;r<period_vars[id][p].getSize();r++)
436
profit += cplex[id].getValue(period_vars[id][p][r])*
period_vars_original_coef[id][p][r];
437
438
439
440
441
442
#ifdef THROUGHPUT
master_obj.setCoef(new_column, throughput);
#elif defined PROFIT
443
//
444
master_obj.setCoef(new_column, round(z));
445
master_obj.setCoef(new_column, round(profit));
#endif
446
447
for (int i=0;i<T;i++) {
448
if (arr[i]>EPS)
449
master_arrival_cons[i].setCoef(new_column, arr[i]);
450
if (dep[i]>EPS)
451
452
453
master_departure_cons[i].setCoef(new_column, dep[i]);
}
197
454
arr.end();
455
dep.end();
456
457
max_frequency = (IloInt) service_level;
458
459
#ifdef DEBUG
460
env.out() << "add " << new_column.getName() << ", z = " << z
<< ", cost = " << cost << ", frequency = "
<< service_level << "(" << max_frequency
<< "), throughput = " << throughput
<< ", gap = " << gap << "%\t\n";
461
fid1
<< "add " << new_column.getName() << ", z = " << z
<< ", cost = " << cost << "\tfrequency = "
<< service_level << "\tthroughput = " << throughput
<< "\tgap = " << gap << "%\n";
462
463
#endif
464
}
465
466
// add integer solutions to the master problem
467
ILOINCUMBENTCALLBACK3(MyCallback, int, id, const char*,
market_name, IloNumVarArray, var) {
468
469
IloNum z = getObjValue();
198
470
IloNum profit = 0, cost = 0;
471
if
472
(z <= EPS)
return;
473
474
IloNum gap = round(100*(getBestObjValue()-z)/z);
475
476
// store integer solutions within 10% of optimality
477
if
478
(gap > 10)
return;
479
480
INTEGER_SOLUTION_ADDED = 1;
481
482
IloNumArray
arr(env,T), dep(env,T);
483
IloNum
service_level = 0, throughput = 0, val;
484
char
s[20], varname[15];
485
int
n;
486
unsigned char
fleet, dep_epoch, arr_epoch;
487
488
// number of columns in the master problem
489
n = master_vars.getSize();
490
sprintf(s,"%s_%d_%d",market_name,rounds,n+1);
491
492
// add new column
493
//
IloNumVar new_column(env, 0, IloInfinity, ILOFLOAT, s);
199
494
IloNumVar new_column(env, 0, 1, ILOFLOAT, s);
495
master_vars.add(new_column);
496
master_sos1_cons[id].setCoef(new_column, 1);
497
498
column_solution.add(IloFlightArray(env));
499
500
for (int i=0;i<var.getSize();i++) {
501
strcpy(varname, var[i].getName());
502
if ((varname[0]==’x’)&((val=round(getValue(var[i])))>EPS)){
503
504
505
506
507
if (varname[1]>=’A’)
fleet = varname[1]-’A’+10;
else
fleet = varname[1]-’0’;
throughput += fleet*CAPACITY_INCREMENT*val;
508
509
strncpy(s,&varname[2],3);
510
s[3]=’\0’;
511
dep_epoch
512
strncpy(s,&varname[5],3);
513
s[3]=’\0’;
514
arr_epoch
515
if (dep_epoch <= T)
516
dep[dep_epoch-1] += val;
517
else
= atoi(s);
= atoi(s);
200
518
arr[arr_epoch-1] += val;
519
service_level += val;
520
IloArray<unsigned char> flight(env,4);
521
flight[0]=fleet;
522
flight[1]=dep_epoch;
523
flight[2]=arr_epoch;
524
flight[3]=(unsigned char) val;
525
column_solution[n].add(flight);
526
}
527
}
528
max_frequency = (IloInt) service_level;
529
530
for (int j=0;j<T;j++) {
531
for (int k=0;k<arr_vars[id][j].getSize();k++)
532
cost += -getValue(arr_vars[id][j][k])*
arr_vars_original_coef[id][j][k];
533
for (int k=0;k<dep_vars[id][j].getSize();k++)
534
cost += -getValue(dep_vars[id][j][k])*
dep_vars_original_coef[id][j][k];
535
}
536
profit = - cost;
537
for (int p=0;p<6;p++)
538
539
for (int r=0;r<period_vars[id][p].getSize();r++)
profit += getValue(period_vars[id][p][r])*
201
period_vars_original_coef[id][p][r];
540
541
//master_vars.add(IloNumVar(master_obj(throughput) +
master_arrival_cons(arr_demand) +
master_departure_cons(dep_demand) +
master_sos1_cons[i](1), 0, IloInfinity, ILOFLOAT, s));
542
//master_vars.add(IloNumVar(master_obj(throughput) +
master_arrival_cons(arr_demand) +
master_departure_cons(dep_demand) +
master_sos1_cons[i](1), 0, 1, ILOFLOAT, s));
543
544
545
#ifdef THROUGHPUT
master_obj.setCoef(new_column, throughput + M);
#elif defined PROFIT
546
//
547
master_obj.setCoef(new_column, round(z));
548
master_obj.setCoef(new_column, round(profit));
#endif
549
550
for (int i=0;i<T;i++) {
551
if (arr[i]>EPS)
552
master_arrival_cons[i].setCoef(new_column, arr[i]);
553
if (dep[i]>EPS)
554
555
556
master_departure_cons[i].setCoef(new_column, dep[i]);
}
202
557
arr.end();
558
dep.end();
559
560
#ifdef DEBUG
561
env.out() << "add " << new_column.getName() << ", z = " << z
<< ", cost = " << cost << ", frequency = "
<< service_level << "(" << max_frequency
<< "), throughput = " << throughput
<< ", gap = " << gap << "%\t\n";
562
fid1
<< "add " << new_column.getName() << ", z = " << z
<< ", cost = " << cost << "\tfrequency = "
<< service_level << "\tthroughput = "
<< throughput << "\tgap = " << gap << "%\n";
563
#endif
564
565
}
566
567
void solve_subproblem(int i) {
568
569
try {
570
571
INTEGER_SOLUTION_ADDED = 0;
572
cplex[i].solve();
573
203
574
//no integer solution within 10% optimality, add last one
575
if (INTEGER_SOLUTION_ADDED==0)
576
addColumn(i);
577
/*
578
//resolve for different daily frequency levels
579
IloNum temp = round(max_frequency*0.8);
580
if (max_frequency>0)
581
active_models++;
582
max_frequency -= 2;
583
while ((max_frequency > 1) && (max_frequency > temp)) {
584
cons[i][1].setUB(max_frequency);
585
586
INTEGER_SOLUTION_ADDED = 0;
587
cplex[i].solve();
588
if (INTEGER_SOLUTION_ADDED==0)
589
addColumn(i);
590
max_frequency -= 2;
591
}
592
*/
593
} catch (IloException& e) {
594
e.end();
595
return;
596
597
}
}
204
598
599
static void init_cplex_params(IloCplex cplex) {
600
cplex.setOut(env.getNullStream());
601
cplex.setParam(IloCplex::PPriInd, CPX_PPRIIND_STEEP);
602
cplex.setParam(IloCplex::RINSHeur, 1);
603
cplex.setParam(IloCplex::HeurFreq, 1);
604
cplex.setParam(IloCplex::RootAlg, CPX_ALG_NET);
605
cplex.setParam(IloCplex::VarSel, 3);
606
cplex.setParam(IloCplex::EpGap, 0.05);
607
cplex.setParam(IloCplex::EpInt, 0.001);
608
cplex.setParam(IloCplex::DepInd, 1);
609
cplex.setParam(IloCplex::FracCuts, 2);
610
cplex.setParam(IloCplex::MIPEmphasis, 2);
611
cplex.setParam(IloCplex::TiLim, 300);
612
cplex.setParam(IloCplex::CutLo, 0);
613
cplex.setParam(IloCplex::CutLo, 0);
614
//cplex.setParam(IloCplex::MIPInterval, 1);
615
//cplex.setParam(IloCplex::Reduce, CPX_PREREDUCE_PRIMALONLY);
616
//cplex.setParam(IloCplex::Reduce, 0);
617
}
618
619
static void
init_problems() {
ifstream
market_file;
620
621
205
622
char
s[10], name[15], file_name[50];
623
int
i, j, dep_epoch, arr_epoch;
624
625
// read in the list of markets
626
sprintf(file_name,"%s%s",WORKING_DIR,MARKET_FILE_NAME);
627
market_file.open(file_name);
628
if (market_file.is_open()) {
629
while (!market_file.eof()) {
630
market_file.getline(s, 4);
631
if (strlen(s)>0)
632
model.add(IloModel(env,s));
633
}
634
market_file.close();
635
} else {
636
cerr << "init_problems: Unable to open markets file.\n";
637
env.end();
638
exit(-1);
639
}
640
n_markets = model.getSize();
641
642
643
644
645
#ifdef DEBUG
cerr << "init_problems(): " << n_markets << " markets.\n";
#endif
206
646
// init the master problem
647
init_cplex_params(master_cplex);
648
master_obj = IloAdd(master_model, IloMaximize(env));
649
650
IloIntArray
651
for (i=0;i<T;i++)
652
653
capacity(env,T);
capacity[i] = AIRPORT_QUARTER_CAPACITY;
master_arrival_cons = IloAdd(master_model, IloRangeArray(env,
-IloInfinity, capacity));
654
master_departure_cons = IloAdd(master_model, IloRangeArray(env,
-IloInfinity, capacity));
655
//master_cons = IloAdd(master_model, IloRangeArray(env,
-IloInfinity, capacity));
656
for (i=0;i<T;i++) {
657
sprintf(s,"a%d",i);
658
master_arrival_cons[i].setName(s);
659
//master_cons[i].setName(s);
660
sprintf(s,"d%d",i);
661
master_departure_cons[i].setName(s);
662
//master_cons[i+T].setName(s);
663
}
664
665
IloNumArray sos1_rhs(env, n_markets);
666
for (i=0;i<n_markets;i++)
207
667
668
sos1_rhs[i] = 1;
master_sos1_cons = IloAdd(master_model, IloRangeArray(env,
-IloInfinity, sos1_rhs));
669
670
master_cons.add(master_arrival_cons);
671
master_cons.add(master_departure_cons);
672
master_cons.add(master_sos1_cons);
673
674
// read in lp files of all markets
675
for (i=0;i<n_markets;i++) {
676
obj.add(IloObjective(env));
677
vars.add(IloNumVarArray(env));
678
cons.add(IloRangeArray(env));
679
680
sprintf(s,"%d",i);
681
cplex.add(IloCplex(model[i]));
682
init_cplex_params(cplex[i]);
683
sprintf(file_name,"%s%s%s",WORKING_DIR,model[i].getName(),
SUB_MODEL_FILE_SUFFIX);
684
685
cplex[i].importModel(model[i], file_name, obj[i],
vars[i], cons[i]);
686
687
cplex[i].use(MyCallback(env,i,model[i].getName(), vars[i]));
208
688
// store pointers to variables to update reduced costs later
689
// prepare the storage
690
dep_vars.add(IloVarArray(env,T));
691
arr_vars.add(IloVarArray(env,T));
692
period_vars.add(IloVarArray(env,6));
693
dep_vars_original_coef.add(IloNumArrayArray(env,T));
694
arr_vars_original_coef.add(IloNumArrayArray(env,T));
695
period_vars_original_coef.add(IloNumArrayArray(env,6));
696
for (j=0;j<T;j++) {
697
dep_vars[i][j] = IloNumVarArray(env);
698
arr_vars[i][j] = IloNumVarArray(env);
699
dep_vars_original_coef[i][j] = IloNumArray(env);
700
arr_vars_original_coef[i][j] = IloNumArray(env);
701
}
702
for (j=0;j<6;j++) {
703
period_vars[i][j] = IloNumVarArray(env);
704
period_vars_original_coef[i][j] = IloNumArray(env);
705
}
706
707
// first constraint is the reduced cost condition
708
// store its variables and their initial coefficients
709
IloExpr expr = cons[i][0].getExpr();
710
IloExpr::LinearIterator li = expr.getLinearIterator();
711
209
712
while (li.ok()) {
713
strcpy(name,li.getVar().getName());
714
if (name[0]==’x’) {
715
// set higher priority for larger fleet
716
if (name[1]>=’A’)
717
718
cplex[i].setPriority(li.getVar(), name[1] - ’A’ + 10);
else
719
cplex[i].setPriority(li.getVar(), name[1] - ’0’);
720
721
strncpy(s,&name[2],3);
722
s[3]=’\0’;
723
dep_epoch
724
if (dep_epoch <= T) {
= atoi(s);
725
dep_vars[i][dep_epoch-1].add(li.getVar());
726
dep_vars_original_coef[i][dep_epoch-1].add(li.getCoef());
727
}
728
else {
729
strncpy(s,&name[5],3);
730
s[3]=’\0’;
731
arr_epoch
732
arr_vars[i][arr_epoch-1].add(li.getVar());
733
arr_vars_original_coef[i][arr_epoch-1].add(li.getCoef());
734
735
= atoi(s);
}
} else if (name[0]==’p’) {
210
736
int p;
737
p = name[2]-’0’;
738
period_vars[i][p-1].add(li.getVar());
739
period_vars_original_coef[i][p-1].add(li.getCoef());
740
}
741
++li;
742
}
743
744
// add sos2 constraints to subproblem
745
// to do: change sos2 constraints to lazy constraints
746
/*
747
for (j=0;j<cons[i].getSize();j++)
748
if (cons[i][j].getName()[0]==’s’) {
749
expr = cons[i][j].getExpr();
750
IloExpr::LinearIterator li = expr.getLinearIterator();
751
IloNumVarArray v(env);
752
while (li.ok()) {
753
v.add(li.getVar());
754
++li;
755
}
756
model[i].add(IloSOS2(env,v));
757
//model[i].add(IloSOS2(env,v,IloNumArray(env,n,p1,p2,pn)));
758
v.end();
759
}
211
760
*/
761
expr.end();
762
763
// store initial max frequencies in the second constraint
764
initial_max_frequency.add(cons[i][1].getUB());
765
766
// sos1 constraint for each market in the master problem
767
sprintf(s,"sos1_%d",i);
768
master_sos1_cons[i].setName(s);
769
770
cplex[i].extract(model[i]);
771
772
}
}
773
774
void generate_columns(IloNumArray dual_prices,
IloNumVarArray node_variables) {
775
env.out() << "generate_columns() called\n";
776
int i,j,k;
777
778
// update subproblems
779
for (i=0; i<n_markets; i++) {
780
781
782
for (j=0;j<T;j++) {
for (k=0;k<arr_vars[i][j].getSize();k++)
cons[i][0].setLinearCoef(arr_vars[i][j][k],
212
arr_vars_original_coef[i][j][k] - round(dual_prices[j]));
783
784
for (k=0;k<dep_vars[i][j].getSize();k++)
cons[i][0].setLinearCoef(dep_vars[i][j][k],
dep_vars_original_coef[i][j][k] - round(dual_prices[j+T]));
785
}
786
cons[i][0].setLB(round(dual_prices[i+N]+1));
787
}
788
789
IloInt n1 = master_vars.getSize();
790
791
for (i=0;i<n_markets;i++) {
792
try {
793
cons[i][1].setUB(initial_max_frequency[i]);
794
solve_subproblem(i);
795
//IloNum solution_time = timer.stop();
796
//env.out() << "\n### " << model[i].getName() << ", round "
<< rounds << " (" << solution_time << " seconds)\n ";
797
//fid1 << "\n### " << model[i].getName() << ", round "
<< rounds << " (" << solution_time << " seconds)\n ";
798
} catch (IloException& e) {
799
env.out() << e.getMessage() << endl;
800
e.end();
801
802
}
}
213
803
804
IloInt n2 = master_vars.getSize();
805
env.out() << "generate_columns() ended with " << n2
<< " columns in master_vars \n";
806
for (int i=n1;i<n2;i++)
807
node_variables.add(master_vars[i]);
808
env.out() << "generate_columns() ended with " << n2 - n1
<< " columns generated at the current node\n";
809
}
810
811
/// MAIN PROGRAM ///
812
813
int main(int argc, char **argv)
814
{
815
char
s[10], filename[50];
816
int
i, j, k, n_unconstrained_columns,n_columns;
817
IloNumArrayArray
arr_price(env), dep_price(env);
818
IloNumArrayArray
sos_price(env);
819
init_problems();
820
821
sprintf(filename,"%s%s",WORKING_DIR,OUTPUT_LOG_FILE_NAME);
822
fid1.open(filename);
823
824
//active_models=0;
214
825
826
// prepare root node
827
for (i=0;i<n_markets;i++) {
828
timer.restart();
829
try {
830 //solve IP subproblems using MIP Cplex, add integer
831 //solutions within 10% of optimality
832
solve_subproblem(i);
833
} catch (IloException& e) {
834
env.out() << e.getMessage() << endl;
835
e.end();
836
}
837
IloNum solution_time = timer.stop();
838
}
839
840
TreeManager tree;
841
tree.solve();
842
tree.printSolution();
843
844
sprintf(filename,"%s%s",WORKING_DIR,OUTPUT_SCHEDULE_FILE_NAME);
845
report_schedule(filename, master_cplex, master_vars);
846
847
env.out() << endl;
848
env.end();
215
849
850
fid1.close();
851
852
853
return 0;
}
854
855
static void report_schedule (char* filename, IloCplex& solver,
IloNumVarArray v)
856
{
857
int
i,j,k,r, temp;
858
char
model_name[10], varname[10], s[10], round[10];
859
char
*p1, *p2, market[10];
860
ofstream fid;
861
862
env.out() << "\nWriting optimal schedule...\n";
863
env.out() << v.getSize() << " variables\n";
864
env.out() << rounds << " rounds\n";
865
866
fid1 << "\nWriting optimal schedule...\n";
867
fid1 << v.getSize() << " variables\n";
868
fid1 << rounds << " rounds\n";
869
870
871
fid.open(filename);
216
872
for (k = 0; k < v.getSize(); k++) {
873
if (solver.getValue(v[k])>EPS) {
874
env.out() << v[k].getName() << endl;
875
fid1 << v[k].getName() << endl;
876
strncpy(market, &v[k].getName()[0], 3);
877
market[3]=’\0’;
878
for (j = 0; j < column_solution[k].getSize(); j++) {
879
fid <<market<<"\t"<<(unsigned int)column_solution[k][j][0]
<< "\t" << (unsigned int) column_solution[k][j][1]
<< "\t" << (unsigned int) column_solution[k][j][2]
<< "\t" << (unsigned int) column_solution[k][j][3]
<< endl;
880
}
881
882
}
}
883
884
fid.close();
885
#ifdef THROUGHPUT
886
env.out()<<"Total seats:"<< solver.getObjValue() << endl;
887
fid1 << "Total seats: " << solver.getObjValue() << endl;
888
#elif defined PROFIT
889
env.out()<<"Total profit:"<< solver.getObjValue() << endl;
890
fid1 << "Total profit: " << solver.getObjValue() << endl;
891
#endif
217
892
893
}
218
Appendix D: Price elasticities estimates for several
key markets
Figure D.1: Log-fit of major markets (O’Hare, Boston, National, and Fort Lauderdale) untruncates demand in lower price ranges
219
Figure D.2: Mid-sized markets (Atlanta, Tampa, Palm Beach, and Philadelphia) use
empirical extrapolated curves to avoid overestimation by the log-fit right tail
220
Figure D.3: Smaller markets (Charlottesville, Fayetteville, Lebanon and Nantucket)
use linear fit
221
Curriculum Vitae
Loan Le obtained in 1998 her B.S. in Information Technology at University of Natural
Sciences in Ho Chi Minh City, Viet Nam. She then received a scholarship to finish
a Diplôme d’Etude Approfondie (DEA), a research-oriented Master’s degree, in the
field of Database Engineering, jointly offered by University of Paris I - Pantheon Sorbonne and University of Paris XI. After graduation in 1999, she worked at Centre
de Recherche en Informatique at University of Paris I from Sep 1999 to May 2001.
She joined France Telecom - Research and Development in summer 2001 to work as a
system architect intern. In spring 2002, she began her Ph.D. program at Systems Engineering and Operations Research Department at George Mason University. During
her doctoral studies, she was a research assistant in the Center for Air Transportation
Systems Research (CATSR). Her research interests include optimization problems in
the airline industry. Loan Le will start working for American Airlines, Operations Research and Decision Support Department upon the completion of her Ph.D. program.
She can be reached by email at ltloan@gmail.com.