Demand Management at Congested Airports: How Far Are We from Utopia?

DEMAND MANAGEMENT AT CONGESTED AIRPORTS: HOW FAR ARE WE FROM UTOPIA?

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 Menasc´e,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

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

Acknowledgments

Early 2002, professor George L. Donohue gave me this invaluable opportunity of pur- suing 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. With- out 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 doctor- ate. 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 ...... xiii 1 Introduction and Problem Statement ...... 1 1.1 Airport congestion and congestion management measures ...... 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 Motivation ...... 12 1.4 Statement of the problem ...... 15 1.5 Contributions of this dissertation ...... 17 1.5.1 Primary hypothesis ...... 17 1.5.2 Research scope ...... 18 1.5.3 Contributions ...... 19 1.6 The potential readers ...... 21 1.7 Dissertation outline ...... 21 2 Literature Review of Prior Research ...... 23 2.1 Congestion Management by Demand Management Measures . . . . . 23 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 2.3.2 Simulation models ...... 47 3 The current slot allocation rules aggravate the congestion problem . . . . 51 4 Scheduling Models ...... 54 vi

4.1 General approach ...... 54 4.2 Profit-maximizing airline scheduling sub-models ...... 56 4.2.1 The timeline network ...... 57 4.2.2 Interaction of demand and supply through price ...... 59 4.2.3 Piecewise approximation of non-linear revenue functions . . . 60 4.2.4 Nesting revenue functions ...... 62 4.2.5 Assumptions ...... 64 4.2.6 Formulation ...... 65 4.3 Airport’s allocation problem ...... 67 4.4 Solution method ...... 70 4.5 Implementation details ...... 72 5 Parameter estimation for scheduling models ...... 74 5.1 Timeline networks ...... 74 5.1.1 Arcs and arc lengths ...... 75 5.1.2 Arc costs ...... 77 5.2 Nonlinear revenue functions and piecewise linear approximation . . . 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 by-time-of-day ...... 85 5.3 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 6 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 ...... 100 6.1.4 Delay propagation submodel ...... 102 6.1.5 Cancellation and cancellation propagation submodel ...... 103 vii

6.2 Parameter estimation ...... 105 6.2.1 Gate-out delay distributions ...... 106 6.2.2 Taxi time distributions ...... 106 6.2.3 En route time distributions ...... 106 6.2.4 Cancellation and cancellation propagation ...... 107 6.3 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 Assessing impacts of changes in fleet mix on delay estimates . 115 7 Demand Management at LaGuardia Airport: How Fare Are We From Utopia?117 7.1 Assumptions and parameters ...... 117 7.2 Baseline statistics ...... 119 7.3 Investigated scenarios ...... 121 7.4 Profit maximizing ...... 123 7.5 Seat throughput maximizing ...... 127 7.6 Compromise scenarios ...... 130 7.6.1 Seat-maximizing within 90% profit optimal ...... 132 7.6.2 Seat-maximizing within 80% profit optimal ...... 139 7.7 Discussion ...... 144 7.7.1 Research questions and answers ...... 145 8 Conclusion and Future Work ...... 148 8.1 Contributions ...... 150 8.2 Recommendations for future work ...... 152 Bibliography ...... 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 Page 1.1 New runways, runway extensions, and reconfigurations included in the OEP [1] ...... 4 1.2 Runways, Runway Extensions, Reconfigurations or New Airports with Environmental Impact Statements (EISs) or Planning Studies Under- way[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 market- based 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 . . . . . 10 1.2 Similar trends of average delay per aircraft at EWR, LGA, and JFK airports ...... 10 1.3 Increasing operations vs. decreasing enplanements at EWR, decreasing aircraft size at EWR and LGA ...... 13 2.1 Overview of airline scheduling tasks (Barnhart) ...... 41 2.2 Overview of DELAYS and AND models ...... 45 2.3 Overview of NAS Strategy Simulator’s delay and cancellation component 47 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 ...... 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) ...... 78 5.2 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 2005 ...... 81 5.4 Linear prorating of square root of leg distance helps account for fixed cost...... 82 5.5 Example of demand extrapolation ...... 84 xi

5.6 Estimates of quarterly constrained extrapolated demand curves for directional markets, Q2 2005 ...... 85 5.7 Actual seat shares by time of day are used to allocate demands by time of day, Q2 2005 ...... 91 5.8 Estimated demand curves for peak periods lie above those of off-peak periods ...... 92 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 93 5.10 In each substitution group, higher actual seat shares of time windows lead to scheduled arrivals in those time windows ...... 94 5.11 Increases in seat capacity lead to decreases in fare and vice versa . . . 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 simulated cancellations ...... 108 6.3 The relation of cumulative delay and cancellation used in simulating cancellations ...... 109 6.4 Comparison of delay estimates vs. actual data ...... 111 6.5 Estimates of cancelled seats ...... 112 6.6 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 Laud- erdale) 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 log- fit 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 grand- father 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 communica- tions, 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

1General 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-back- yard” 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

2Since 1999, seven new runways have been commissioned at OEP airports at a cost of $1.9 billion [1] 5

Airport or Estimated CY Project Metropolitan Area EIS Will Be Completed Chicago OHare (ORD) Reconﬁguration 2005 Washington Dulles (IAD) Runway 2005 Chicago Metropolitan New airport 2006 Area (Peotone) Philadelphia (PHL) Reconﬁguration 2007 Ft. Lauderdale (FLL) Extension 2007 Las Vegas Metropolitan New airport 2008 Area (Ivanpah Valley) San Diego Metropolitan New airport TBD Portland International (PDX) Extension 2007 Salt Lake City (SLC) Extension 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 Control-

Navigation-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 bian- nual 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 first- come, 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 Inter- national (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 en- counters 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.

314 Code of Federal Regulations [CFR] part 93, subpart K, High Density Traﬃc Airports 4HDR 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

5The 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 inef-

ﬁciencies 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 Alterna- tive 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 Re- duction 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 Febru- ary 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-lose- it 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

6Klein 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 incor- porates 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 ad- vantages 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 Manage- ment 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 La-

Guardia 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. “In- crease 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 Elec- tronic 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. High- way 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 high- occupancy 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 market- based 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 treat- ing 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 Chapman-

Kolmogorov 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. Con- gestion 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 sub- markets”, 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 sym- metric 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. More- over, 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 sym- metric 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 in- volve 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. 38 Cons Objection by GA community Remove the expansion flexibilityAirlines of hold shuttle service 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 Hinder competition, require on-going government intervention Unconstrained demand creates severe congestion Convergence uncertain Overscheduling, hence congestion, might remain Cost uncertainty for airlines Convergence uncertain Unfavorable to small markets Require complex packaging with other facilities Subject to unpredictable bidding behaviors Require airline commitment, no warranty of slot availability on the day of operations Does not address grand-father rights in the primary market Pros Remove small aircraft, increase slots Maintain demand predictability Incentivize airlines not to available to larger planes use unprofitable slots Faster turn-around of unused slots Reveal airlines’ true slot demand Faster turn-around of unused slots Simple Facilitate the consolidation of service among airlines Simple, airlines would eventually figure out the market equilibrium Allocate peak times to more valuable services Flat rate to incentivize aircraft upgauge Schedule flexibility for airlines Allocate peak times to more valuable services Fixed cost incentivizes aircraft upgauge Demand, hence delays, is controlled Prevent slot hoarding among airline coalition in sell/lease of slots Promote secondary market access Table 2.1: Review of demand management measures

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 Laissez-faire Congestion Pricing Slot auction HDR and blind auction in secondary market

Market-based Aministrative Hybrid 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. Market- based 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 kth-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 kth-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 Mod- eler (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 sectoriza- tion [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, discrete- event computer simulation model developed and supported by the Model Develop- ment 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 fast- time 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, hu- man 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 some- times 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. Obtain- ing 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 1500- mile 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. De- mand 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 of airport o, and nodes in T 0 are time windows of airport o0 , 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:

0 fk,o,o0 block time by ﬂeet k from airport o to airport o , in time windows gk minimum turnaround time of ﬂeet type k, in time windows t(i) order of time window i in Zulu time

then the directed arcs emanating from nodes in T are created as follows: 58

ground arcs

© ------?- - - airport 1 H H ¨* H ¨* H ¨* H ¨* H ¨* H ¨* H ¨* ¨* H ¨H¨ ¨H¨ ¨H¨ ¨H¨ ¨H¨ ¨H¨ ¨H¨ ¨¨ uH¨H H¨H H¨H H¨H H¨ uuH H¨ uuH H¨ uuH H¨ uuH u ¨ ¨H ¨H ¨H ¨H ¨H ¨H ¨H H flight arcs airport 2 ¨ - ¨ -Hj ¨ -Hj ¨ -Hj ¨ -Hj ¨ -Hj ¨ -Hj ¨ -Hj -Hj u uu uu uu uu u (a) subnetwork for fleet 1 that requires 2 time windows for a flight arc

------airport 1 PP PP PP1 PP1 PP1 PP1 PP1 1 1 PP PP PP PP PP PP PP uPPPPPPPP uuPP uuPP uuPP uu u P P P P P P P airport 2 - - P-Pq P-Pq -PPq P-Pq P-Pq P-Pq P-Pq u uu uu uu uu u (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.

0 F i ∈ T , j ∈ T , (i, j) ∈ A if t(i) + fk,o,o0 + gk = t(j) i, j ∈ T , (i, j) ∈ AG if t(i) + 1 = t(j) i, j ∈ T , (j, i) ∈ AG if t(i) ≤ t(k) ≤ t(j) ∀k ∈ T

Similarly, the directed arcs emanating from nodes in T 0 are created as follows:

0 F i ∈ T , j ∈ T , (i, j) ∈ A if t(i) + fk,o,o0 + gk = t(j) i, j ∈ T 0, (i, j) ∈ AG if t(i) + 1 = t(j) i, j ∈ T 0, (j, i) ∈ AG 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

Pq a function of the form y = f(x1, ..., xq) = i=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

q r X X q x = aiλi, λi = 1, λ ∈ R+. i=1 i=1

The λi are not unique, but if ai ≤ x ≤ ai+1 and λ is chosen so that x = λiai +

λi+1ai+1 and λi + λi+1 = 1, then we obtain f(x) = λif(ai) + λi+1f(ai+1). In other 61

Figure 4.4: Approximating a nonlinear function by a piecewise linear function words,

q r X X q f(x) = f(ai)λi, λi = 1, λ ∈ R+ i=1 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 (4.1)

q−1 X yi = 1 i=1

y ∈ Bq−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 non- negative 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 q λi = 1 λi ∈ B . 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)

X xi = aiqλiq q∈Q(i)

X fi(xi) = 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)

X xp = aprλpr r∈Q(p)

X fp(xp) = 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 time windows AG ground arcs AF ﬂight arcs K ﬂeet types operable at the 2 airports of the market Q(i) linear segment indexes for the revenue function of i ∈ T

Parameters:

Sk seating capacity of fleet type k ∈ K k F Cij direct operating cost for one flight of fleet type k ∈ K for (i, j) ∈ A Aiq linear segment quantities for the revenue function of i ∈ T , q ∈ Q(i) Riq linear segment revenues for the revenue function of i ∈ T , q ∈ Q(i) l average load factor

Variables:

k F G xij number of ﬂights of ﬂeet type k ∈ K for (i, j) ∈ A ∪ A λiq linear segment variables for the revenue function of i ∈ T , q ∈ Q(i)

Subproblem formulation:

X X X X k k max z = Riqλiq − Cjixji (4.2) i∈T q∈Q(i) (j,i)∈AF k∈K 66

subject to:

X k X k xj,i − xij = 0 ∀ i ∈ T , k ∈ K (4.3) (j,i)∈A (i,j)∈A

X X k k X l S xji − Aiqλiq = 0 ∀ i ∈ T (4.4) k∈K (j,i)∈AF q∈Q(i)

X X X Aiqλiq − Aprβpr = 0 ∀ p ∈ P (4.5) i∈E(p) q∈Q(i) r∈Q(p)

X X X Riqλiq − Rprβpr ≤ 0 ∀ p ∈ P (4.6) i∈E(p) q∈Q(i) r∈Q(p)

X λiq = 1 ∀ i ∈ T (4.7) q∈Q(i)

X βpr = 1 ∀ p ∈ P (4.8) r∈Q(p)

|AF |x|K| |Q(i)| |Q(p)| x ∈ Z+ , λi ∈ R+ , βp ∈ R+

P P k k For any time window i, (j,i)∈AF k∈K Cjixji in the objective function (4.2) is the

P P k k total operating cost of arrivals at i. The resulting total capacity k∈K (j,i)∈AF S xji 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) Riqλiq 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. P As explained earlier, q∈Q(i) Aiqλiq 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) Riqλiq, is P constrained by the revenue of the compounding time window p, r∈Q(p) Rprβpr. (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 P k k∈K (j,i)∈AF xji, 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 schedule vector indexes T time window indexes M market indexes S(m) column indexes of market m’s schedule vectors, m ∈ M

Parameters:

|T |x|S| a matrix of arrivals by time window: aij is the number of arrival ﬂights at

time window i in schedule j |T |x|S| d matrix of departures by time window: dij is the number of departure ﬂights

at time window i in schedule j Zj coeﬃcient of the schedule vector j ∈ S, determined by the public goal to

optimize Ci arrival/departure rates of time window i ∈ T Gi ground capacities in time window i ∈ T

Variables:

yj binary variable equal to 1 if schedule vector yj is in the optimal solution

Formulation of the master problem:

X max Zjyj (4.9) j∈S

subject to: 69

X aijyj ≤ Ci ∀i ∈ T (4.10) j∈S

X dijyj ≤ Ci ∀i ∈ T (4.11) 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 k k Riqλiq − Cjixji i∈T q∈Q(i) (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 k k S xji 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 (αiaij + πidij) + µj, or when zj − i∈T (αiaij + πidij) − µj > 0. In other words, 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:

X X k X X k z − αi xj,i − πi xij − µ ≥ 1 (4.13) i∈T C k∈K,(j,i)∈AF i∈T C 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 coef-

ﬁcient 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

performed departures aircraft type by 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 Met- rics (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 Aircraft Average Size Fleet Aircraft Average Size BE-1900 19 B737-8 166 EMB-145 22 B737-9 167 DO-328 J 32 A320-1/2 168 1 SF-340/B 34 B727-200 172 7 DHC8-100 37 A321 174 EMB-135 37 B767-2/R 174 EMB-140 44 B757-200 179 200/440 47 A340-500 181 2 DHC8-300 50 A310-300 194 EMB-145 50 B757-200 194 RJ100/ER 50 A321 196 8 AV RJ85 69 B767-2/R 204 200/440 70 A330-200 206 RJ-700 70 B767-3/R 207 3 EMB-170 72 B757-200 215 MD-80 74 A321 216 BAE146-2 77 A330-200 221 B717-200 88 B757-300 222 9 B737-1/2 100 B777 222 DC-9-30 100 B767-3/R 223 4 B737-5 108 A340-200 230 MD-80 109 B767-400 235 DC-9-40 110 B757-300 245 B717-200 117 B767-400 246 A319 121 A340-200 251 B737-300 122 B767-3/R 251 5 DC-9-50 125 10 A310-300 253 B737-700 126 A340-300 255 MD-80 132 B747-400 258 A320-1/2 133 B777 258 A319 138 A330-200 261 B737-400 144 B747-400 266 MD-80 145 A300-600 267 B737-300 147 11 A340-200 272 6 A320-1/2 148 B777 283 MD-90 150 B767-400 285 B737-8 151 B757-300 154 B767-2/R 158

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: air fuels issued hourly air fuel consumption = total air hours

total air direct expenses - fuel cost hourly aircraft direct expense excluding fuel = total air hours

total air total expenses - fuel cost hourly aircraft total expense excluding fuel = 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)

Fuel consumption Direct cost Direct, maintenance and Aircraft seats (gallons/h) ($/h) depreciation cost ($/h) 25 306 703 795 50 418 840 1106 75 530 978 1417 100 759 1115 1729 125 987 1253 2040 150 1216 1390 2351 175 1445 1528 2662 200 1674 1665 2973 225 1902 1803 3284 250 2131 1940 3595 275 2360 2078 3906 350 3046 2490 4840 375 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 has the one-way ticket price of $100, then leg one is allocated $40 (=100 ∗ 10+15 ) and

15 leg two $60 (=100 ∗ 10+15 ).

Figure 5.4: Linear prorating of square root of leg distance helps account for ﬁxed cost.

1Thanks 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 Extrapolated $210 1 23.8 $200 2 47.6 $190 3 71.4 $180 4 95.2 $170 5 119 $160 6 142.9 $150 7 $140 8 $130 9 mean=$156.7 sum=45 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.

1 2 3 Their respective probabilities in the subsample above the average fare are 21 , 21 , 21 ,

4 5 6 1 2 3 4 21 , 21 , 21 . The extrapolated demand would be 21 ∗ 500, 21 ∗ 500, 21 ∗ 500, 21 ∗ 500,

5 6 21 ∗ 500, and 21 ∗ 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. con- sidering only these two time windows, we would then divide the quarterly demands

86+44 by 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 time- based 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: Charlottesville-

Albemarle Airport (CHO), Nantucket Memorial Airport (ACK), Barnstable Muni-

Boardman/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: Sa- vannah 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.

96 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 cancel-

lation including system-level assessments. The evaluation can be aggregate and

airline-speciﬁc, and system-level assessment should include airport interdepen-

dency 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 multi- server 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. En- route 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 100

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 time- based 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 Small Large B757 Heavy Leading Small 2.5/80 2.5/68 2.5/66 2.5/64 Large 4/164 2.5/73 2.5/66 2.5/64 B757 5/201 4/115 4/102 4/101 Heavy 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 102

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.

D A Let t0 (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:

( A A 0 if t (f) − t0 (f) ≤ 15 min P A A G (g) = A A α[t (f)−t0 (f)] A A min(t (f) − t0 (f), D A [t (f) − t0 (f)]) otherwise t0 (g)−t0 (f)

A A [t (f)−t0 (f)] where D A , calibrated by a scaling factor α to reﬂect the sensitivity of t0 (g)−t0 (f) ﬂight schedules to disruption, determines the magnitude of the delay propagation’s multiplicative term. We assume the propagation to be positively correlated to the

A A lateness of flight f, i.e. t (f)−t0 (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

D A the denominator or t0 (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 103

Delay Turnaround time GP (g) GP (g) GP (g) A A D A t (f) − t0 (f) t0 (g) − t0 (f) α = 1 α = 0.5 α = 1.1 45 20 10 22 60 15 7.5 16.5 30 75 12 6 13.2 90 10 5 11 45 35.5 17.8 39.1 60 26.7 13.3 29.3 40 75 21.3 10.7 23.5 90 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 code- share 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 104

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 ac- knowledged 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 A D D to delay, let k:k

excessive, a cancellation is forced to maintain the trend between the two metrics:

p + p if Ω(P [dA(k) − dD(k)], c) is true p(f) = 1 2 k:k

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. 106

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 mini- mums 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 Mile- in-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 A D k, k(d (k) + d (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 A these two cumulative metrics by the log function y = 7726lnx−7255.7, or k[d (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 109

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 + p if P [dA(k) + dD(k)] ≤ 7726lnc − 7255.7 p(f) = 1 2 k 1 otherwise

When one ﬂight is cancelled, c is updated to give a new threshold for the condition. 110

6.3 Model calibration and application

6.3.1 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. 111

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 15- min 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 113

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 114

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 observ- able 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. 115

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 116

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 Study Overall Markets 67 275 Flights 1024 1104 Seats 98686 101072 Passengers 72845 78675 Average aircraft size 95 95 Average fare $139 $133 Average ﬂight delay 18.7 min 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

1See 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 airport- reported 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. 123

Airport dep/arr rate/15min/runway Unconstrained 10 9 8 7 6 5 4 Proﬁt-maximizing ------Seat-maximizing ------Compromise 90% ------Scenario Compromise 80% ------

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 sig- niﬁcantly. 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 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. Market Frequency Arc. size Fare Passengers Yield* ($) Unconstrained LEB 6 19 $153 50 0.72 10,9,8,7 ROA 5 37 $186 77 0.46 6,5,4 TYS 2 50 $125 85 0.19 ACK 5 26 $216 47 1.07 4 ALB 7 33 $91 62 0.67 CHO 5 33 $229 80 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. Market Frequency Arc. size Fare Passengers Yield* ($) Unconstrained LEB 6 19 $153 50 0.72 10,9,8,7 ROA 5 37 $186 77 0.46 6,5,4 TYS 2 50 $125 85 0.19 ACK 5 26 $216 47 1.07 6 ORF 14 34 $238 255 0.5 BGR 3 40 $93 76 0.25 GRR 1 39 $129 27 0.20 5 ITH 9 33 $160 93 0.89 SAV 7 50 $140 326 0.19 ACK, ORF BHM 6 50 $190 280 0.22 CAE 6 50 $130 292 0.21 HYA 2 28 $235 19 1.19 4 MCI 10 125 $180 643 0.16 MVY 2 34 $233 15 1.33 RIC 19 50 $155 619 0.53 ACK, ORF, BGR, GRR, ITH, SAV

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. Market Frequency Arc. size Fare Passengers Yield* ($) Unconstrained LEB 6 19 $153 50 0.72 10,9,8,7 ROA 5 37 $186 77 0.46 6,5,4 TYS 2 50 $125 85 0.19 ACK 5 26 $216 47 1.07 7 BWI 14 38 $124 241 0.67 BGR 3 40 $93 76 0.25 ORF 14 34 $238 255 0.5 6 PHF 6 113 $107 412 0.37 SYR 15 37 $115 298 0.58 BWI DAY 5 50 $131 195 0.24 HYA 2 28 $235 19 1.19 5 PVD 9 32 $121 129 0.85 SAV 7 50 $140 325 0.19 ACK, BRG, BWI, ORF, PHF, SYR ALB 7 33 $91 62 0.67 BHM 6 50 $190 280 0.22 CAE 6 50 $130 293 0.21 GSP 9 50 $149 277 0.24 ILM 5 50 $135 184 0.27 IND 18 58 $138 747 0.21 MCI 10 125 $180 642 0.16 4 MEM 6 125 $170 574 0.18 MVY 2 34 $233 15 1.33 PHL 19 58 $59 522 0.60 PWM 14 50 $106 432 0.39 RIC 19 50 $154 619 0.53 XNA 4 38 $295 85 0.26 ACK, BRG, BWI, DAY, ORF, PHF, SAV, SYR

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. 136 7 2 2 1 7 -1 39 19 48 65 21 16 57 13 28 -68 -50 -15 -12 -13 -85 -11 -53 -35 -20 -35 107 Fare Change 90 74 86 75 87 97 86 148 167 112 298 225 189 152 144 123 135 129 123 147 242 204 126 118 107 115 224 Fare Model Average 91 93 87 216 128 191 177 123 102 124 131 100 229 133 128 127 150 121 131 120 185 191 124 111 127 149 195 Fare Actual Average 0 0 0 -1 -8 -3 -8 -6 -8 -2 25 13 43 12 25 13 56 33 23 53 23 46 33 13 -11 -13 102 Size Change Aircraft 25 25 38 50 50 93 25 50 25 75 78 97 50 96 83 145 108 208 125 102 155 131 150 146 175 181 150 Size Output Aircraft 26 33 40 50 83 37 50 38 50 33 50 65 46 50 50 50 156 106 113 102 122 108 158 148 122 157 137 Size Actual Aircraft 0 -6 0.9 0.1 0.3 0.5 0.1 0.6 -2.8 -5.3 -2.5 -5.5 -7.2 -5.6 -3.3 -1.3 -4.6 -1.7 -4.2 -3.5 -0.7 -0.5 -9.7 -2.7 -15.7 -13.4 -17.2 Deviation Frequency 2 2 4 6 6 6 8 6 6 2 6 6 4 32 60 14 10 16 30 22 10 68 14 26 22 26 12 Model Output Frequency 5 7 3 6 8 6 6 5 5 9 3 48 73 11 21 14 11 21 32 26 13 69 14 26 32 43 18 Daily Average Frequency FLL CLT ATL GSP CHS CLE BOS ALB BWI BUF GSO DAY CAE BTV DEN BNA ACK BGR CVG DCA CAK CHO HOU BHM CMH DFW DTW Market 137 2 1 0 6 8 -1 -4 -4 -9 -7 -6 -5 39 60 31 62 26 88 12 43 -14 -24 -36 -15 -25 -12 116 Fare Change 77 92 274 217 111 137 156 120 231 181 140 111 162 202 132 191 156 259 118 143 238 118 119 101 287 130 101 Fare Model Average 90 59 235 215 135 138 160 156 171 180 109 115 171 107 141 157 197 155 233 130 148 150 111 107 171 121 106 Fare Actual Average 0 8 7 1 -3 -8 -9 -9 -7 15 19 33 30 98 25 64 17 44 71 54 10 -10 -10 -21 -38 -14 -45 Size Change Aircraft 25 81 83 88 25 50 45 25 25 75 44 67 25 58 150 150 150 156 143 133 154 163 167 175 175 139 225 Size Output Aircraft 28 66 50 58 33 52 50 38 99 34 34 58 32 49 131 125 166 152 125 175 150 131 104 138 171 113 112 Size Actual Aircraft 0 0 0.9 0.6 1.5 0.2 0.1 0.3 -0.2 -0.3 -5.8 -4.7 -1.8 -3.5 -2.5 -6.4 -2.4 -4.1 -1.3 -0.1 -6.1 -3.7 -3.6 -1.1 -4.8 -2.5 -11.5 Deviation Frequency 2 6 4 6 2 6 6 8 6 2 6 8 6 8 4 16 18 12 18 20 10 14 12 56 10 12 12 Model Output Frequency 2 5 9 8 2 6 6 2 6 6 9 16 17 18 10 21 19 16 16 12 13 62 14 12 19 13 14 Daily Average Frequency PBI PIT ITH IAH IAD IND ILM MCI JAX MIA LEX PHL PHF MSP HYA ORF PVD MSY ORD MKE MHT MYR MVY MCO MEM PWM MDW Market 138 7 -4 74 12 40 13 -18 -40 -69 Fare Change 78 111 229 152 172 168 155 127 227 Fare Model Average 129 154 118 140 165 173 115 114 295 Fare Actual Average 0 -5 49 86 33 18 12 -11 -14 Size Change Aircraft 95 39 50 83 25 50 125 155 155 Size Output Aircraft 46 50 39 50 50 39 38 137 160 Size Actual Aircraft 0 -5 0.9 0.5 -8.1 -0.6 -0.9 -2.8 -15.1 Deviation Frequency 6 6 6 4 22 14 10 12 10 Model Output Frequency 7 7 9 4 37 19 14 15 10 Daily Average Table 7.9: Numerical results of the 90% compromise scenario at 8 ops/runway/15min Frequency RIC STL SAV SDF SYR TPA ROC XNA RDU Market 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 141 7 2 2 1 7 -1 39 19 48 65 21 16 57 13 28 -68 -50 -15 -12 -13 -85 -11 -53 -35 -20 -35 107 Fare Change 90 74 86 75 87 97 86 148 167 112 298 225 189 152 144 123 135 129 123 147 242 204 126 118 107 115 224 Fare Model Average 91 93 87 216 128 191 177 123 102 124 131 100 229 133 128 127 150 121 131 120 185 191 124 111 127 149 195 Fare Actual Average 0 0 0 0 -1 -8 -3 -8 -4 -8 25 26 25 12 20 13 95 53 23 15 53 46 33 13 -31 -13 102 Size Change Aircraft 25 25 38 50 63 75 25 50 25 70 78 98 50 96 83 125 108 208 125 141 175 131 150 163 175 158 150 Size Output Aircraft 26 33 40 50 83 37 50 38 50 33 50 65 46 50 50 50 156 106 113 102 122 108 158 148 122 157 137 Size Actual Aircraft 0 -6 0.9 0.1 0.3 0.3 0.5 0.1 0.6 -2.8 -5.3 -7.7 -2.5 -7.5 -3.2 -7.6 -3.3 -1.3 -4.6 -3.5 -0.7 -2.5 -9.7 -2.7 -13.4 -10.2 -11.2 Deviation Frequency 2 2 4 6 6 4 6 6 6 2 6 6 4 40 60 18 10 16 32 16 10 68 14 24 22 32 12 Model Output Frequency 5 7 3 6 8 6 6 5 5 9 3 48 73 11 21 14 11 21 32 26 13 69 14 26 32 43 18 Daily Average Frequency FLL CLT ATL GSP CHS CLE BOS ALB BWI BUF GSO DAY CAE BTV DEN BNA ACK BGR CVG DCA CAK CHO HOU BHM CMH DFW DTW Market 142 2 1 0 6 8 -1 -4 -4 -9 -7 -6 -5 39 60 31 62 26 88 12 43 -14 -24 -36 -15 -25 -12 116 Fare Change 77 92 274 217 111 137 156 120 231 181 140 111 162 202 132 191 156 259 118 143 238 118 119 101 287 130 101 Fare Model Average 90 59 235 215 135 138 160 156 171 180 109 115 171 107 141 157 197 155 233 130 148 150 111 107 171 121 106 Fare Actual Average 0 6 5 8 7 4 9 -3 -8 -9 -5 -7 15 13 33 42 98 64 44 71 15 24 -10 -21 -63 -14 -45 Size Change Aircraft 25 81 83 25 50 45 25 29 50 44 67 25 57 144 100 150 131 156 158 133 154 163 154 175 175 153 195 Size Output Aircraft 28 66 50 58 33 52 50 38 99 34 34 58 32 49 131 125 166 152 125 175 150 131 104 138 171 113 112 Size Actual Aircraft 0 0 0.9 0.6 1.5 0.2 0.7 0.1 0.3 -0.2 -0.3 -7.8 -4.7 -1.8 -1.5 -2.5 -6.4 -2.4 -4.1 -0.1 -1.7 -1.6 -1.1 -4.8 -0.5 -12.1 -11.5 Deviation Frequency 2 6 4 6 2 8 6 8 6 2 6 6 8 4 16 18 10 18 20 10 14 14 50 12 10 12 14 Model Output Frequency 2 5 9 8 2 6 6 2 6 6 9 16 17 18 10 21 19 16 16 12 13 62 14 12 19 13 14 Daily Average Frequency PBI PIT ITH IAH IAD IND ILM MCI JAX MIA LEX PHL PHF MSP HYA ORF PVD MSY ORD MKE MHT MYR MVY MCO MEM PWM MDW Market 143 7 -4 74 12 40 13 -18 -40 -69 Fare Change 78 111 229 152 172 168 155 127 227 Fare Model Average 129 154 118 140 165 173 115 114 295 Fare Actual Average 0 0 44 86 33 18 30 -16 -14 Size Change Aircraft 90 34 50 83 25 38 125 155 190 Size Output Aircraft 46 50 39 50 50 39 38 137 160 Size Actual Aircraft 0 -3 0.9 0.5 -8.1 -2.6 -0.9 -0.8 -11.1 Deviation Frequency 6 4 6 4 26 16 10 14 10 Model Output Frequency 7 7 9 4 37 19 14 15 10 Daily Average Table 7.11: Numerical results of the 80% compromise scenario at 8 ops/runway/15min Frequency RIC STL SAV SDF SYR TPA ROC XNA RDU Market 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. Con- sequently, 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 market- based 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. Market Frequency Arc. size Fare Passengers Yield* ($) Unconstrained LEB 6 19 $153 50 0.72 10,9,8,7 ROA 5 37 $186 77 0.46 6,5,4 TYS 2 50 $125 85 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.

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 proba- bly 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 incor- porates 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.

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

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 traf-

fic 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

Bibliography 155

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Appendix A: Airport Codes, Locations and Names 162

ACK Nantucket, MA: Nantucket Memorial ALB Albany, NY: Albany County ATL Atlanta, GA: Hartsfield-Jackson BGR Bangor, ME: Bangor International BHM Birmingham, AL: Birmingham Municipal BNA Nashville, TN: Nashville Metropolitan BOS Boston, MA: Logan International BTV Burlington, VT: Burlington International BUF Buffalo/Niagara Falls, NY: Greater Buffalo International BWI Baltimore, MD: Baltimore/Washington International CAE Columbia, SC: Columbia Metropolitan CAK Akron/Canton Regional, OH: Regional CHO Charlottesville, VA: Charlottesville Albemarle CHS Charleston, SC: Charleston International CLE Cleveland, OH: Hopkins International CLT Charlotte, NC: Douglas Municipal CMH Columbus, OH: Columbus International CVG Covington, KY: Cincinnati/ Northern Kentucky International DAY Dayton, OH: James M Cox/Dayton International DCA Washington, DC: Washington National DEN Denver, CO: Denver International DFW Dallas/Ft.Worth, TX: Dallas/Ft Worth International DTW Detroit, MI: Detroit Metro Wayne County FLL Fort Lauderdale, FL: Fort Lauderdale International GSO Greensboro/High Point, NC: Greensboro High Point Winst GSP Greenville/Spartanburg, SC: Greenville/Spartanburg Airport HOU Houston, TX: William P Hobby HYA Hyannis, MA: Barnstable Municipal IAD Washington, DC: Dulles International IAH Houston, TX: Houston Intercontinental ILM Wilmington, NC: New Hanover County IND Indianapolis, IN: Indianapolis International ITH Ithaca/Cortland, NY: Tompkins County JAX Jacksonville, FL: Jacksonville International LEB Lebanon-Hanover, NH: Lebanon Municipal LEX Lexington/Frankfort, KY: Blue Grass 163

MCI Kansas City, MO: Kansas City International MCO Orlando, FL: Orlando International MDW Chicago, IL: Chicago Midway MEM Memphis, TN: Memphis International MHT Manchester/Concord, NH: Grenier Field /Manchester Municipal MIA Miami, FL: Miami International MKE Milwaukee, WI: General Mitchell Field MSP Minneapolis/St. Paul Int, MN: Minneapolis-St Paul MSY New Orleans, LA: Louis Armstrong International MVY Martha’s Vineyrd, MA: Marthas Vineyard MYR Myrtle Beach, SC: Myrtle Beach International Airport ORD Chicago, IL: O Hare ORF Norfolk/Va.Bch/Ptsmth/Chpk, VA: Norfolk Va ROA Roanoke, VA: Roanoke Municipal ROC Rochester, NY: Rochester Monroe County PBI West Palm Beach/Palm Beach, FL: Palm Beach International PHF Newport News/Williamsburg, VA: Patrick Henry International PHL Philadelphia, PA: Philadelphia International PIT Pittsburgh, PA: Pittsburgh International PVD Providence, RI: Theodore Francis Green PWM Portland, ME: Portland International Jetport RDU Raleigh/Durham, NC: Raleigh Durham RIC Richmond, VA: Richard Elelyn Byrd International ROC Rochester, NY: Rochester Monroe County SAV Savannah, GA: Savannah International SDF Standiford Field, KY: Standiford Field Airport STL St. Louis, MO: Lambert/St Louis International SYR Syracuse, NY: Syracuse Hancock International TPA Tampa, FL: Tampa International TYS Knoxville, TN: Mcghee Tyson XNA 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_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"); 165

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");

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");

SS[k]:= S[k]*0.8;

INTEGER VARIABLES x[k,i,j in flight_arc];

VARIABLES y[k,i,j] WHERE (iT 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]; 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_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; 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; 167

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

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 (iT 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

using namespace std;

ILOSTLBEGIN

#define EPS 1.0e-3 173

#define DEBUG

#define PROFIT

//#define THROUGHPUT

//#define MARKET_ENTRANCE

typedef IloArray IloModelArray; typedef IloArray IloObjArray; typedef IloArray IloVarArray; typedef IloArray IloConArray; typedef IloArray IloSolverArray; typedef IloArray IloNumArrayArray; typedef IloArray< IloArray > IloFlightArray; typedef IloArray 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 fid1, fid2;

static const int 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 dep_vars(env), arr_vars(env), period_vars(env); static IloArray 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"

3 class Node {

5 public :

6 Node *next, *prev;

7 //IloNumArray node_dual_prices;

8 float node_dual_prices[96*2];

9 IloNum value;

10 //char id[200];

11 IloBool branching;

13 IloNumVar branching_variable;

14 IloNumVarArray node_variables;

15 IloNumVarArray node_variables_at_zero;

16 IloNumVarArray node_variables_at_one;

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 void printInfo();

22 };

24 class NodeList {

25 int n_nodes;

26 public:

28 Node *head;

30 NodeList() {

31 n_nodes = 0;

32 head = NULL;

33 }

35 int getSize() {

36 return n_nodes;

37 }

39 void addNode(Node*);

40 void removeNode(Node*);

41 void printInfo();

42 void clear();

44 }; 179

46 class TreeManager {

48 void getObjCoef(IloObjective obj, IloNumArray coef);

49 void load_node(Node*);

50 void branch_node(Node*);

51 void select_branching_variable(Node*);

53 public:

54 IloNum lower_bound, upper_bound;

55 Node *root, *solution;

56 NodeList list;

58 TreeManager();

60 IloInt getSize() {

61 return list.getSize();

62 }

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 // strcpy(id,s);

75 value = val;

76 branching = IloFalse;

78 node_variables = IloNumVarArray(env, node_v.getSize());

79 for (int i=0;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

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

86 node_variables_at_one[i]=node_v_at_one[i];

88 } 181

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;

98 }

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

<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 = IloNumArray(env);

147 obj_coef = IloNumArray(env, master_vars.getSize());

148 master_cplex.getValues(x, master_vars);

149 getObjCoef(master_obj, obj_coef);

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 xj_inf = 1.0 - xj_inf;

159 if ( xj_inf >= maxinf && (xj_inf > maxinf ||

IloAbs (obj_coef[j]) >= maxobj) ) {

160 bestj = j;

161 maxinf = xj_inf;

162 maxobj = IloAbs (obj_coef[j]);

163 }

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

183 master_vars[i].setUB(0);

184 master_vars[i].setLB(0);

185 }

186 for (int i=0;inode_variables.getSize();i++) {

187 n->node_variables[i].setUB(1);

188 n->node_variables[i].setLB(0);

189 }

190 for (int i=0;inode_variables_at_zero.getSize();i++)

191 n->node_variables_at_zero[i].setUB(0);

192 for (int i=0;inode_variables_at_one.getSize();i++)

193 n->node_variables_at_one[i].setLB(1);

194 }

195

196 void TreeManager::branch_node(Node* n) {

197 if ((n->branching) && (IloFloor(n->value)>lower_bound)){

198 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 delete solution;

224 solution = left_child;

225 } 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

284 if (x[i]>EPS)

285 env.out() << master_vars[i].getName() << "\n";

286 }

287 }

288

289 void Node::printInfo() {

290 //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 head = n;

298 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 (gap > 10) {

377 for (int i=0;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 // IloNumVar new_column(env, 0, IloInfinity, ILOFLOAT, s);

392 IloNumVar new_column(env, 0, 1, ILOFLOAT, s);

393 master_vars.add(new_column);

394 master_sos1_cons[id].setCoef(new_column, 1);

395

396 column_solution.add(IloFlightArray(env));

397

398 for (int i=0;i

399 strcpy(varname, vars[id][i].getName());

400 if ((varname[0]==’x’) & ((val =

round(cplex[id].getValue(vars[id][i])))>EPS)) {

401 if (varname[1]>=’A’)

402 fleet = varname[1]-’A’+10;

403 else

404 fleet = varname[1]-’0’;

405 throughput += fleet*CAPACITY_INCREMENT*val;

406

407 strncpy(s,&varname[2],3);

408 s[3]=’\0’; 195

409 dep_epoch = atoi(s);

410 strncpy(s,&varname[5],3);

411 s[3]=’\0’;

412 arr_epoch = atoi(s);

413 if (dep_epoch <= T)

414 dep[dep_epoch-1] += val;

415 else

416 arr[arr_epoch-1] += val;

417 service_level += val;

418 IloArray 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 for (int j=0;j

428 for (int k=0;k

429 cost += -cplex[id].getValue(arr_vars[id][j][k])*

arr_vars_original_coef[id][j][k];

430 for (int k=0;k

431 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

436 profit += cplex[id].getValue(period_vars[id][p][r])*

period_vars_original_coef[id][p][r];

437

438

439

440 #ifdef THROUGHPUT

441 master_obj.setCoef(new_column, throughput);

442 #elif defined PROFIT

443 // master_obj.setCoef(new_column, round(profit));

444 master_obj.setCoef(new_column, round(z));

445 #endif

446

447 for (int i=0;i

448 if (arr[i]>EPS)

449 master_arrival_cons[i].setCoef(new_column, arr[i]);

450 if (dep[i]>EPS)

451 master_departure_cons[i].setCoef(new_column, dep[i]);

452 }

453 197

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 (z <= EPS)

472 return;

473

474 IloNum gap = round(100*(getBestObjValue()-z)/z);

475

476 // store integer solutions within 10% of optimality

477 if (gap > 10)

478 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

501 strcpy(varname, var[i].getName());

502 if ((varname[0]==’x’)&((val=round(getValue(var[i])))>EPS)){

503 if (varname[1]>=’A’)

504 fleet = varname[1]-’A’+10;

505 else

506 fleet = varname[1]-’0’;

507 throughput += fleet*CAPACITY_INCREMENT*val;

508

509 strncpy(s,&varname[2],3);

510 s[3]=’\0’;

511 dep_epoch = atoi(s);

512 strncpy(s,&varname[5],3);

513 s[3]=’\0’;

514 arr_epoch = atoi(s);

515 if (dep_epoch <= T)

516 dep[dep_epoch-1] += val;

517 else 200

518 arr[arr_epoch-1] += val;

519 service_level += val;

520 IloArray 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

531 for (int k=0;k

532 cost += -getValue(arr_vars[id][j][k])*

arr_vars_original_coef[id][j][k];

533 for (int k=0;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 for (int r=0;r

539 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 #ifdef THROUGHPUT

544 master_obj.setCoef(new_column, throughput + M);

545 #elif defined PROFIT

546 // master_obj.setCoef(new_column, round(profit));

547 master_obj.setCoef(new_column, round(z));

548 #endif

549

550 for (int i=0;i

551 if (arr[i]>EPS)

552 master_arrival_cons[i].setCoef(new_column, arr[i]);

553 if (dep[i]>EPS)

554 master_departure_cons[i].setCoef(new_column, dep[i]);

555 }

556 202

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() {

620

621 ifstream market_file; 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 #ifdef DEBUG

643 cerr << "init_problems(): " << n_markets << " markets.\n";

644 #endif

645 206

646 // init the master problem

647 init_cplex_params(master_cplex);

648 master_obj = IloAdd(master_model, IloMaximize(env));

649

650 IloIntArray capacity(env,T);

651 for (i=0;i

652 capacity[i] = AIRPORT_QUARTER_CAPACITY;

653 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

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

667 sos1_rhs[i] = 1;

668 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

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 cplex[i].use(MyCallback(env,i,model[i].getName(), vars[i]));

687 208

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

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 cplex[i].setPriority(li.getVar(), name[1] - ’A’ + 10);

718 else

719 cplex[i].setPriority(li.getVar(), name[1] - ’0’);

720

721 strncpy(s,&name[2],3);

722 s[3]=’\0’;

723 dep_epoch = atoi(s);

724 if (dep_epoch <= T) {

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 = atoi(s);

732 arr_vars[i][arr_epoch-1].add(li.getVar());

733 arr_vars_original_coef[i][arr_epoch-1].add(li.getCoef());

734 }

735 } 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

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

780 for (j=0;j

781 for (k=0;k

782 cons[i][0].setLinearCoef(arr_vars[i][j][k], 212

arr_vars_original_coef[i][j][k] - round(dual_prices[j]));

783 for (k=0;k

784 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

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

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

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 return 0;

853 }

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 fid.open(filename);

871 216

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 <

<< "\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 Laud- erdale) 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 Diplômed’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 En- gineering 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 Re- search and Decision Support Department upon the completion of her Ph.D. program. She can be reached by email at [email protected].