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

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.

💡 Research Summary

This paper tackles the stochastic tail‑assignment problem faced by Air France, extending traditional deterministic models that rely on fixed minimum turn times. Tail assignment is a tactical planning task performed about a week before operations, where each aircraft must be assigned a sequence of flight legs and mandatory maintenances (collectively called activities) that together cover the entire schedule. Once the routes are fixed, they cannot be altered during the day, so any flight delay that occurs propagates along the pre‑determined aircraft routes, generating additional costs. The authors therefore propose a scenario‑based stochastic optimization framework that explicitly accounts for delay propagation and balances operational (fuel, connection) costs against delay‑related costs.

Problem formulation
The activity set V consists of flight legs L and maintenances M. Each activity v has a scheduled departure/arrival time ( (\bar T_d^v,\bar T_a^v) ) and airports ( (k_d^v,k_a^v) ). A directed acyclic connection graph D = ((\bar V), A) is built, where (\bar V = V \cup {s,t}) adds a source and a sink. An arc ( (v^-,v) ) exists only if the airports match and the scheduled times respect a minimum turn time (\Delta_{\min}^v). For each aircraft i, a sub‑graph (D_i) is derived by removing arcs that would violate aircraft‑specific constraints such as skipping its mandatory maintenances or violating its first‑activity location. A feasible route for aircraft i is any s‑t path in (D_i); the set of all feasible routes is denoted (R_i). A feasible tail‑assignment solution selects exactly one route from each (R_i) such that the routes partition V (every activity is covered once) and all mandatory connections are respected.

Cost structure
Two cost components are considered. Operational cost (c_{\text{op}}) is the sum of leg‑specific fuel/engine costs (c_{i\ell}) and connection costs (w_{ia}). Delay cost (c_{\text{delay}}) is modeled as a piecewise‑linear, non‑decreasing function of the arrival delay of a flight leg. The function is defined by breakpoints (\alpha_j) and slopes (\beta_j), reflecting the airline’s practice that short delays incur modest penalties while longer delays trigger steep cost increases (missed connections, passenger compensation, crew overtime, etc.). Delay variables (\xi) (departure and arrival delays for each activity) are random; their joint distribution is conditioned on the chosen routes because delay propagation depends on the aircraft sequence. The authors adopt a Sample Average Approximation (SAA) approach: a finite set of scenarios (\mathcal S) (derived from historical data) is used to approximate the expected delay cost.

Optimization approach
The stochastic tail‑assignment problem is tackled with a column‑generation algorithm. The master problem is a set‑partitioning formulation where each column corresponds to a feasible route r for some aircraft i, with a binary variable (x_{ir}). The objective combines the deterministic operational cost of the selected routes and the average delay cost over the scenarios. The master problem is solved as a linear program (LP) to obtain dual values for the activity‑covering constraints. These dual values feed a pricing sub‑problem: for each aircraft i, find a route r that minimizes reduced cost, i.e., operational cost plus the sum of dual prices of covered activities plus the expected incremental delay cost. This sub‑problem is a stochastic constrained shortest‑path problem.

Bounding techniques
Enumerating all possible routes is infeasible, so the authors introduce two bounding mechanisms to prune the search space during pricing:

1. A classic lower bound based on the sum of the smallest possible operational costs plus dual prices, which discards any partial path whose lower bound already exceeds the best reduced cost found.
2. A novel bound exploiting the lattice ordering of piecewise‑linear convex functions. If a partial path’s delay‑cost function f is dominated (≤) by another already‑explored function g (meaning f’s value is never larger than g’s for any delay), then any extension of f cannot improve upon the best reduced cost and can be safely eliminated. This lattice‑based pruning is computationally cheap because the functions are stored in a dedicated piecewise‑linear library and compared via their breakpoint‑slope representation.

Diving heuristic
Column generation yields a fractional LP solution. To obtain an integer feasible solution, the authors apply a diving heuristic. Starting from the LP solution, they iteratively fix the most fractional route variable to 1 (or 0) and resolve the reduced master problem, respecting the set‑partitioning constraints. This “branch‑and‑price‑like” dive continues until all variables are integral. The heuristic dramatically improves solution quality: without diving the average optimality gap is 2–3 %, whereas with diving it drops to ≤0.3 %.

Computational study
Experiments use real Air France data: instances with up to 600 flight legs, 30–40 aircraft, and 3–5 delay scenarios. The proposed algorithm is benchmarked against (i) a compact mixed‑integer linear programming (MILP) model that enforces a fixed minimum turn time, and (ii) a column‑generation version without the lattice bounds. Results show:

• The stochastic column‑generation approach solves all instances within a few hours, achieving an average optimality gap of 0.28 %.
• The MILP model often fails to finish within a 24‑hour limit or runs out of memory on the larger instances.
• The lattice‑based bounds reduce the number of explored paths by an order of magnitude, making the pricing step tractable.
• The diving heuristic is essential for obtaining high‑quality integer solutions; it consistently outperforms a naïve rounding of the LP solution.

Software contribution
Two open‑source Julia packages are released:

• ConstrainedShortestPaths.jl – a generic implementation of constrained shortest‑path algorithms, used for the pricing sub‑problem.
• PiecewiseLinearFunctions.jl – utilities for constructing, manipulating, and comparing piecewise‑linear convex functions, enabling the lattice‑ordering bounds.

Conclusions and future work
The paper demonstrates that a stochastic tail‑assignment model, combined with sophisticated column generation, lattice‑based pruning, and a diving heuristic, can efficiently handle realistic airline instances while explicitly accounting for delay propagation. This provides airlines with a decision‑support tool that balances fuel/operational savings against the risk of costly delays, something deterministic minimum‑turn‑time models cannot achieve. Future research directions include: integrating real‑time delay forecasts, expanding the scenario set via importance sampling, coupling tail assignment with crew‑pairing or maintenance‑scheduling, and exploring decomposition techniques for even larger planning horizons.