Column Generation and the Airline Crew Pairing Problem

The cost of ight crews is the second largest operating cost of an airline. Minimizing it is a fundamental problem in airline planning and operations, and one which has leant itself to mathematical optimization. We discuss several recent advances in the methods used to solve these problems. After describing the general approach taken, we discuss a new method which can be used to obtain approximate solutions to linear programs, dramatically improving the solution time of these problems. This is the so-called volume algorithm. We also describe several other ideas used to make it routinely possible to get very good solutions to these large mixed integer programs. 1 Airline Operations and the Crew Pairing Problem The Airline Crew Pairing problem has become well known as a prototypical applied problem which lends itself to a column generation approach. A pairing, or tour of duty, is a sequence of ights to be own by a crew, starting and ending at the crew member's home base. Typically the length of a pairing will range from a single day to four or ve days. The pairing problem is to compute a set of pairings covering all the scheduled ights of an airline, which has minimum, or near minimum, cost. This problem normally forms the third major stage in a four part planning process: 1. Schedule Creation. Flights are planned, based on market demands and competitive analysis. For example, we may schedule a ight from New York's La Guardia Airport to Chicago's O'Hare Airport every non-holiday weekday departing at 6:15 P.M. 2. Fleet Scheduling. Airlines normally have several diierent types ((eets) of aircraft. A eet is chosen for each ight scheduled in the rst stage, subject to the constraints that the type of aircraft chosen for each ight be of suitable capacity and ying characteristics and subject to \Kirchoo's law"