Cynthia Barnhart

Branch-And-Price: Column Generation for Solving Huge Integer Programs

We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

The fleet assignment problem: solving a large-scale integer program

Given a flight schedule and set of aircraft, the fleet assignment problem is to determine which type of aircraft should fly each flight segment. This paper describes a basic daily, domestic fleet assignment problem and then presents chronologically the steps taken to solve it efficiently. Our model of the fleet assignment problem is a large multi-commodity flow problem with side constraints defined on a time-expanded network. These problems are often severely degenerate, which leads to poor performance of standard linear programming techniques. Also, the large number of integer variables can make finding optimal integer solutions difficult and time-consuming. The methods used to attack this problem include an interior-point algorithm, dual steepest edge simplex, cost perturbation, model aggregation, branching on set-partitioning constraints and prioritizing the order of branching. The computational results show that the algorithm finds solutions with a maximum optimality gap of 0.02% and is more than two orders of magnitude faster than using default options of a standard LP-based branch-and-bound code.

Flight String Models for Aircraft Fleeting and Routing

Given a schedule of flight legs to be flown by an airline, the fleet assignment problem is to determine the minimum cost assignment of flights to aircraft types, called fleets, such that each scheduled flight is assigned to exactly one fleet, and the resulting assignment is feasible to fly given a limited number of aircraft in each fleet. Then the airline must determine a sequence of flights, or routes, to be flown by individual aircraft such that assigned flights are included in exactly one route, and all aircraft can be maintained as necessary. This is referred to as the aircraft routing problem. In this paper, we present a single model and solution approach to solve simultaneously the fleet assignment and aircraft routing problems. Our approach is robust in that it can capture costs associated with aircraft connections and complicating constraints such as maintenance requirements. By setting the number of fleets to one, our approach can be used to solve the aircraft routing problem alone. We show how to extend our model and solution approach to solve aircraft routing problems with additional constraints requiring equal aircraft utilization. With data provided by airlines, we provide computational results for the combined fleet assignment and aircraft routing problems without equal utilization requirements and for aircraft routing problems requiring equal aircraft utilization.

Using Branch-and-Price-and-Cut to Solve Origin-Destination Integer Multicommodity Flow Problems

We present a column-generation model and branch-and-price-and-cut algorithm for origin-destination integer multicommodity flow problems. The origin-destination integer multicommodity flow problem is a constrained version of the linear multicommodity flow problem in which flow of a commodity (defined in this case by an origin-destination pair) may use only one path from origin to destination. Branch-and-price-and-cut is a variant of branch-and-bound, with bounds provided by solving linear programs using column-and-cut generation at nodes of the branch-and-bound tree. Because our model contains one variable for each origin destination path, for every commodity, the linear programming relaxations at nodes of the branch-and-bound tree are solved using column generation, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality. We devise a new branching rule that allows columns to be generated efficiently at each node of the branch-and-bound tree. Then, we describe cuts (cover inequalities) that can be generated at each node of the branch-and-bound tree. These cuts help to strengthen the linear programming relaxation and to mitigate the effects of problem symmetry. We detail the implementation of our combined column and- cut generation method and present computational results for a set of test problems arising from telecommunications applications. We illustrate the value of our branching rule when used to find a heuristic solution and compare branch-and-price and branch-and-price-and-cut methods to find optimal solutions for highly capacitated problems.

The global airline industry

List of Contributors. Series Preface. Notes on Contributors. Acknowledgements. 1 Introduction and Overview (P eter P. Belobaba and Amedeo Odoni). 1.1 Introduction: The Global Airline Industry. 1.2 Overview of Chapters. References. 2 The International Institutional and Regulatory Environment (Amedeo Odoni). 2.1 Introduction. 2.2 Background on the International Regulatory Environment. 2.3 Airline Privatization and International Economic Regulation. 2.4 Airports. 2.5 Air Traffic Management. 2.6 Key Organizations and Their Roles. 2.7 Summary and Conclusions. References. 3 Overview of Airline Economics, Markets and Demand (Peter P. Belobaba). 3.1 Airline Terminology and Definitions. 3.2 Air Transportation Markets. 3.3 Origin-Destination Market Demand. 3.4 Air Travel Demand Models. 3.5 Airline Competition and Market Share. 3.6 Chapter Summary. References. 4 Fundamentals of Pricing and Revenue Management ( Peter P. Belobaba). 4.1 Airline Prices and O-D Markets. 4.2 Airline Differential Pricing. 4.3 Airline Revenue Management. References. 5 Airline Operating Costs and Measures of Productivity ( Peter P. Belobaba). 5.1 Airline Cost Categorization. 5.2 Operating Expense Comparisons. 5.3 Comparisons of Airline Unit Costs. 5.4 Measures of Airline Productivity. References. 6 The Airline Planning Process (Peter P. Belobaba). 6.1 Fleet Planning. 6.2 Route Planning. 6.3 Airline Schedule Development. 6.4 The Future: Integrated Airline Planning. References. 7 Airline Schedule Optimization (Cynthia Barnhart). 7.1 Schedule Optimization Problems. 7.2 Fleet Assignment. 7.3 Schedule Design Optimization. 7.4 Crew Scheduling. 7.5 Aircraft Maintenance Routing and Crew Pairing Optimization. 7.6 Future Directions for Schedule Optimization. References. 8 Airline Flight Operations (Alan H. Midkiff, R. John Hansman and Tom G. Reynolds). 8.1 Introduction. 8.2 Regulation and Scheduling. 8.3 Flight Crew Activities During a Typical Flight. 8.4 Summary. 8.5 Appendix: List of Acronyms. References. 9 Irregular Operations: Schedule Recovery and Robustness (Cynthia Barnhart). 9.1 Introduction. 9.2 Irregular Operations. 9.3 Robust Airline Scheduling. 9.4 Directions for Ongoing and Future Work on Schedule Recovery from Irregular Operations. References. 10 Labor Relations and Human Resource Management in the Airline Industry (Jody Hoffer Gittell, Andrew von Nordenflycht, Thomas A. Kochan, Robert McKersie and Greg J. Bamber). 10.1 Alternative Strategies for the Employment Relationship. 10.2 Labor Relations in the US Airline Industry. 10.3 Labor Relations in the Airline Industry in Other Countries. 10.4 Human Resource Management at Airlines. 10.5 Conclusions. References. 11 Aviation Safety and Security (Arnold Barnett). 11.1 Safety. 11.2 Security. References. 12 Airports (Amedeo Odoni). 12.1 Introduction. 12.2 General Background. 12.3 Physical Characteristics. 12.4 Capacity, Delays and Demand Management. 12.5 Institutional, Organizational and Economic Characteristics. References. 13 Air Traffic Control (R. John Hansman and Amedeo Odoni). 13.1 Introduction. 13.2 The Generic Elements of an ATC System. 13.3 Airspace and ATC Structure. 13.4 ATC Operations. 13.5 Standard Procedures. 13.6 Capacity Constraints. 13.7 Congestion and Air Traffic Management. 13.8 Future ATC Systems. References. 14 Air Transport and the Environment (Karen Marais and Ian A. Waitz). 14.1 Introduction. 14.2 Limiting Aviations Environmental Impact: The Role of Regulatory Bodies. 14.3 Airport Water Quality Control. 14.4 Noise. 14.5 Surface Air Quality. 14.6 Impact of Aviation on Climate. 14.7 Summary and Looking Forward. References. 15 Information Technology in Airline Operations, Distribution and Passenger Processing (Peter P. Belobaba, William Swelbar and Cynthia Barnhart). 15.1 Information Technology in Airline Planning and Operations. 15.2 Airline Distribution Systems. 15.3 Distribution Costs and E-commerce Developments. 15.4 Innovations in Passenger Processing. References. 16 Critical Issues and Prospects for the Global Airline Industry (William Swelbar and Peter P. Belobaba). 16.1 Evolution of US and Global Airline Markets. 16.2 Looking Ahead: Critical Challenges for the Global Airline Industry. References. Index.

Airline Schedule Planning: Integrated Models and Algorithms for Schedule Design and Fleet Assignment

Constructing a profitable schedule is of utmost importance to an airline because its profitability is critically influenced by its flight offerings. We focus our attention on the steps of the airline schedule planning process involving schedule design and fleet assignment. Airline schedule design involves determining when and where to offer flights such that profits are maximized, and fleet assignment involves assigning aircraft types to flight legs to maximize revenue and minimize operating cost. We present integrated models and solution algorithms that simultaneously optimize the selection of flight legs for and the assignment of aircraft types to the selected flight legs. Preliminary results, based on data from a major U.S. airline, suggest that significant benefits can be achieved.

Solving binary cutting stock problems by column generation and branch-and-bound

We present an algorithm for the binary cutting stock problem that employs both column generation and branch-and-bound to obtain optimal integer solutions. We formulate a branching rule that can be incorporated into the subproblem to allow column generation at any node in the branch-and-bound tree. Implementation details and computational experience are discussed.

Applications of Operations Research in the Air Transport Industry

This paper presents an overview of several important areas of operations research applications in the air transport industry. Specific areas covered are: the various stages of aircraft and crew schedule planning; revenue management, including overbooking and leg-based and network-based seat inventory management; and the planning and operations of aviation infrastructure (airports and air traffic management). For each of these areas, the paper provides a historical perspective on OR contributions, as well as a brief summary of the state of the art. It also identifies some of the main challenges for future research.

Airline Crew Scheduling

An airline must cover each flight leg with a full complement of cabin crew in a manner consistent with safety regulations and award requirements. Methods are investigated for solving the set partitioning and covering problem. A test example illustrates the problem and the use of heuristics. The Study Group achieved an understanding of the problem and a plan for further work.

Planning for Robust Airline Operations: Optimizing Aircraft Routings and Flight Departure Times to Minimize Passenger Disruptions

Airlines typically construct their schedules assuming that every flight leg will depart and arrive as planned. Because this optimistic scenario rarely occurs, these plans are frequently disrupted and airlines often incur significant costs in addition to those originally planned. Flight delays and schedule disruptions also cause passenger delays and disruptions. A more robust plan can reduce the occurrence and impact of these delays, thereby reducing costs. In this paper, we present two new approaches to minimize passenger disruptions and achieve robust airline schedule plans. The first approach involves routing aircraft, and the second involves retiming flight departure times. Because each airplane usually flies a sequence of flight legs, delay of one flight leg might propagate along the aircraft route to downstream flight legs and cause further delays and disruptions. We propose a new approach to reduce delay propagation by intelligently routing aircraft. We formulate this problem as a mixed-integer programming problem with stochastically generated inputs. An algorithmic solution approach is presented. Computational results obtained using data from a major U.S. airline show that our approach can reduce delay propagation significantly, thus improving on-time performance and reducing the numbers of passengers disrupted. Our second area of research considers passengers who miss their flight legs due to insufficient connection time. We develop a new approach to minimize the number of passenger misconnections by retiming the departure times of flight legs within a small time window. We formulate the problem and an algorithmic solution approach is presented. Computational results obtained using data from a major U.S. airline show that this approach can substantially reduce the number of passenger misconnections without significantly increasing operational costs.