Aircraft routing and crew pairing problems aims at building the sequences of flight legs operated respectively by airplanes and by crews of an airline. Given their impact on airlines operating costs, both have been extensively studied for decades. Our goal is to provide reliable and easy to maintain frameworks for both problems at Air France. We propose simple approaches to deal with Air France current setting. For routing, we introduce a compact MIP formulation that can be solved by current MIP solvers in at most a few minutes even on Air France largest instances. Regarding crew pairing, we provide a standard methodology to model the column generation pricing subproblem within a new resource constrained shortest path framework recently introduced by the first author. This new framework, which can be used as a black-box, leverages on bounds on paths' resource to discard partial solutions and speed-up the resolution. The resulting approach enables to solve to optimality Air France largest instances. Recent literature has focused on integrating aircraft routing and crew pairing problems. As a side result, we have been able to solve to near optimality large industrial instances of the integrated problem by combining the aforementioned algorithms within a simple cut generating method.
Resource constrained shortest path problems are usually solved thanks to a smart enumeration of all the non-dominated paths. Recent improvements of these enumeration algorithms rely on the use of bounds on path resources to discard partial solutions. The quality of the bounds determines the performance of the algorithm. The main contribution of this paper is to introduce a standard procedure to generate bounds on paths resources in a general setting which covers most resource constrained shortest path problems, among which stochastic versions.In that purpose, we introduce a generalization of the resource constrained shortest path problem where the resources are taken in a monoid. The resource of a path is the monoid sum of the resources of its arcs. The problem consists in finding a path whose resource minimizes a nondecreasing cost function of the path resource among the paths that respect a given constraint. Enumeration algorithms are generalized to this framework. We use lattice theory to provide polynomial procedures to find good quality bounds. These procedures solve a generalization of the algebraic path problem, where arc resources belong to a lattice ordered monoid. The practical efficiency of the approach is proved through an extensive numerical study on some deterministic and stochastic resource constrained shortest path problems.
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