Network Models, Lagrangean Relaxation and Subgradients Bundle Approach in Crew Scheduling Problems

Abstract

Mathematical models and algorithms embedded in the MTRAM package (Mass Transit Management) are described; MTRAM was installed at several mass transit companies in Italy to solve bus drivers scheduling problems, which consist in the selection of a set of duties covering the company service at the minimum cost. A duty is feasible when union constraints are satisfied; restrictions usually include the duty starting and ending at the same crew domicile. A more general class of crew scheduling problems is also addressed, where a crew domicile is given and daily duties can originate / terminate at places other than the domicile; daily duties are then grouped into pairings which last for a few days and must, in turn, originate and terminate at a single crew domicile and satisfy a variety of restrictions. Crew scheduling problems consist in the selection of a set of feasible pairings covering the service over a certain time horizon at the minimum cost. They arise in different settings such as railways and coach companies running long trips, and especially in airline companies. Mathematical models developed for MTRAM can be extended to handle the second class of crew scheduling problems and to manage the growth of the model size. Tools developed to improve the efficiency of the approach include: design and implementation of a method for the generation of feasible duties and pairings, which strongly exploits the network structure of the problem; use of a new algorithm of the bundle &trust class to solve the model’s lagrangean dual; development of a theoretical tool allowing duties deletion without affecting the optimal solution. Numerical results for both classes of problems (bus drivers and airline crew scheduling) are presented and related difficulties are discussed.