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Managing delay in tail assignment: from minimum turn time to stochastic routing at Air France

Léo Baty, Axel Parmentier

Abstract

On-time performance is a critical challenge in the airline industry, leading to large operational and customer dissatisfaction costs. The tail assignment problem builds the sequences of flights or routes followed by individual airplanes. While airlines cannot avoid some sources of delay, choosing routes wisely limits propagation along these. This paper addresses the stochastic tail assignment problem at Air France. We propose a column generation approach for this problem. The key ingredient is the pricing algorithm, which is a stochastic shortest path problem. We use dedicated bounds to discard paths in an enumeration algorithm, and introduce new bounds based on a lattice ordering of the set of piecewise linear convex functions to strike a balance between bounds quality and computational cost. A diving heuristic enables us to retrieve integer solutions. Numerical experiments on real-world Air France instances demonstrate that our algorithms lead to an average 0.28% optimality gap on instances with up to 600 flight legs in a few hours of computing time. The resulting solutions effectively balance operational costs and delay resilience, outperforming previous approaches based on minimum turn time.

Managing delay in tail assignment: from minimum turn time to stochastic routing at Air France

Abstract

On-time performance is a critical challenge in the airline industry, leading to large operational and customer dissatisfaction costs. The tail assignment problem builds the sequences of flights or routes followed by individual airplanes. While airlines cannot avoid some sources of delay, choosing routes wisely limits propagation along these. This paper addresses the stochastic tail assignment problem at Air France. We propose a column generation approach for this problem. The key ingredient is the pricing algorithm, which is a stochastic shortest path problem. We use dedicated bounds to discard paths in an enumeration algorithm, and introduce new bounds based on a lattice ordering of the set of piecewise linear convex functions to strike a balance between bounds quality and computational cost. A diving heuristic enables us to retrieve integer solutions. Numerical experiments on real-world Air France instances demonstrate that our algorithms lead to an average 0.28% optimality gap on instances with up to 600 flight legs in a few hours of computing time. The resulting solutions effectively balance operational costs and delay resilience, outperforming previous approaches based on minimum turn time.
Paper Structure (45 sections, 2 theorems, 38 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 45 sections, 2 theorems, 38 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions
  4. Outline
  5. Problem statement
  6. Activity schedule and fleet
  7. Connection graph
  8. Feasible decisions
  9. Preprocessing the graph
  10. Operational costs
  11. Delay costs
  12. Stochastic tail assignment
  13. Delay propagation model and sample average approximation
  14. Measuring delays
  15. Delay propagation equations
  16. ...and 30 more sections

Key Result

Proposition 1

Let $r = p_1 + p_2$ an optimal solution of eq:pricing-sub-problem, with $p_1$ an $s-v$ path and $p_2$ a $v-t$ path. If $p'_1$ is an $s-v$ path such that $\vec{q}_{p'_1}\preceq\vec{q}_{p_1}$, then $r' = p'_1 + p_2$ is an optimal solution of eq:pricing-sub-problem.

Figures (3)

  • Figure 1: Arrival-departure propagation between two flight legs.
  • Figure 2: Example of a tail assignment instance and a feasible solution. This instance has 115 legs, 17 maintenances, and 7 aircraft. Each row correspond to an aircraft route, each box corresponding to an activity. Grey boxes are maintenances, colored boxes are flight legs. Green legs are legs that depart from the Charles de Gaulle airport, red legs are legs that arrive at the Charles de Gaulle airport, and yellow legs are legs that are not connected to the Charles de Gaulle airport.
  • Figure 3: MIP gap values with respect to the number of legs in the instances.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 1
  • proof
  • Proposition 2
  • Remark 5
  • Remark 6
  • Remark 7