Dynamic Constraint Aggregation for Solving Very Large-scale Airline Crew Pairing Problems

Abstract

The monthly crew pairing problem (CPP) consists of determining a least-cost set of feasible crew pairings (sequences of flights starting and ending at a crew base) such that each flight is covered once and side constraints are satisfied. This problem has been widely studied but most works have tackled daily or weekly CPP instances with up to 3500 flights. Only a few papers have addressed monthly instances with up to 14,000 flights. In this paper, we propose an effective algorithm for solving very large-scale CPP instances. This algorithm combines, among others, column generation (CG) with dynamic constraint aggregation (DCA) that can efficiently exploit the CG master problem degeneracy. When embedded in a rolling-horizon (RH) procedure, DCA allows to consider wider time windows in RH and yields better solutions. Our computational results show, first, the potential gains that can be obtained by using wider time windows and, second, the very good performance of the proposed algorithm when compared with a standard CG/RH algorithm for solving an industrial monthly CPP instance with 46,588 flights.

Appendices

Appendix 1. Pseudo-code of DCA algorithm

In this appendix, we give the pseudo-code of the DCA algorithm that includes the multi-phase and bi-dynamic strategies described in Section 3.2.1 and that can be used for solving a linear relaxation. It uses the notation introduced in Section 3.2 and the following one.

L::

Pool of all generated columns;

ARMP(Q)::

Model (6)–(10) defined for partition Q;

(x, y)::

Current ARMP solution;

N::

Set of negative reduced cost columns generated in a column generation;

R::

Set of the pricing problem indices;

PP^r(k, h)::

Pricing problem r ∈ R subject to the restrictions imposed in phase k ∈ K of the multi-phase strategy and stage h ∈ H of the bi-dynamic strategy;

$\bar z^{C}(Q,N)$ :

(resp. $\bar z^{I}(Q,N)$): Least reduced cost of a column in N that is compatible (resp. incompatible) with Q (0 if no such column exists).

Appendix 2. Pseudo-code of IDCA algorithm

In this appendix, we give the pseudo-code of the IDCA algorithm that includes the multi-phase and bi-dynamic strategies described in Section 3.2.1 and that can be used for solving a linear relaxation. It uses the notation introduced in Section 3.2 and Appendix 1, as well as the following one.

CP(Q, k)::

Complementary problem defined for partition Q and subject to the restrictions imposed in phase k ∈ K of the multi-phase strategy.