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