  Some airlines use the preferential bidding system to construct the schedules
of their pilots. In this system, the pilots bid on the different activities and
the schedules that lexicographically maximize the scores of the pilots
according to their seniority are selected. A sequential approach to solve this
maximization problem is natural: the problem is first solved with the bids of
the most senior pilot; then it is solved with those of the second most senior
without decreasing the score of the most senior, and so on. The literature
admits that the structure of the problem somehow imposes such an approach. The
problem can be modeled as an integer linear lexicographic program. We propose a
new exact method, which relies on column generation for solving its continuous
relaxation. To design this column generation, we prove that bounded linear
lexicographic programs admit "primal-dual" feasible bases and we show how to
compute such bases efficiently. Another contribution on which our exact method
relies consists in the extension of standard tools for resource-constrained
longest path problems to their lexicographic versions. This is useful in our
context since the generation of new columns is modeled as a lexicographic
resource-constrained longest path problem. Numerical experiments show that this
new method is already able to solve industrial instances provided by Air
France, with up to 150 pilots. By adding a last ingredient in the resolution of
the longest path problems, which exploits the specificity of the preferential
bidding system, the method achieves for these instances computational times
that are compatible with operational constraints.
