Linear lexicographic optimization and preferential bidding system

TL;DR

This paper introduces an exact column generation method for solving linear lexicographic optimization problems, specifically applied to airline pilot scheduling, demonstrating efficiency on real-world instances from Air France.

Contribution

It develops a novel exact solution approach using column generation for lexicographic programs and extends resource-constrained longest path algorithms for this context.

Findings

Successfully solves industrial-sized instances with up to 150 pilots.

Achieves computational times compatible with operational scheduling.

Extends standard algorithms to handle lexicographic resource constraints.

Abstract

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.