Aircraft Routing Problem

• Aircraft Routing Problem is a family of optimization challenges where routes are optimized under constraints like maintenance, communication, and airspace limitations.
• It spans diverse applications from airline tail assignment and conflict-free terminal routing to ad hoc network information routing, all modeled via graph algorithms and MIP.
• Recent models incorporate stochastic delay propagation, energy-aware scheduling, and cooperative UAV–ground routing to enhance operational efficiency and reliability.

The Aircraft Routing Problem denotes a family of optimization problems in which routes involving aircraft are selected under operational, maintenance, resource, communication, or airspace constraints. In airline operations research, it usually refers to assigning flight legs or daily lines of flying to aircraft while ensuring regular visits to maintenance bases; in adjacent literatures, the same phrase can denote route design and allocation of aircraft trajectories, conflict-free routing in terminal airspace, routing information through aircraft in aeronautical ad hoc networks, or cooperative UAV–ground vehicle routing under communication limits (Parmentier et al., 2017, Meunier et al., 7 Aug 2025, Cui et al., 2020, Manyam et al., 2018). This suggests a family of related graph-theoretic optimization problems rather than a single canonical model.

1. Terminology and scope

In the classical airline sense, the decision object is a physical aircraft tail. Routes are sequences of flight legs operated by indistinguishable or tail-specific aircraft over a cyclic or finite horizon, with continuity, turnaround, and maintenance constraints. In this meaning, nodes are typically airports, time-space events, or maintenance states; edges are feasible leg-to-leg connections; and the problem is closely related to fleet assignment, maintenance routing, and crew pairing (Parmentier et al., 2017).

That meaning should be distinguished from several other established uses. In integrated aeronautical ad hoc networks, “aircraft routing” means routing information through aircraft, base stations, and satellites; the vertices are communication nodes and the edges are radio links satisfying signal-to-noise-ratio and visibility constraints (Cui et al., 2020). In airport and air traffic management settings, the term can refer to spatial route design and allocation of departures, or to tactical conflict-free routing and scheduling in terminal manoeuvring areas, where the decision variables are waypoint paths, speeds, holding times, and sometimes route allocations across candidate procedures (Ho-Huu et al., 2019, Zhang et al., 19 Nov 2025). In cooperative UAV settings, it denotes joint routing of an unmanned aerial vehicle and a ground vehicle under communication-radius constraints (Manyam et al., 2018).

A common source of confusion is therefore semantic rather than mathematical. The same label covers aircraft-tail rotations, route-choice over airway or terminal networks, packet routing over airborne relays, and cooperative air–ground tours. The underlying models remain recognizably combinatorial—shortest paths, cycle covers, set partitioning, mixed-integer programming, dynamic programming, and Markov decision processes—but the routed entity changes from aircraft tails to trajectories, packets, or cooperative vehicle teams.

2. Tail assignment and maintenance routing

In airline operations research, aircraft routing is often posed as a maintenance-feasible routing problem over feasible aircraft connections. At Air France, the weekly schedule is repeated cyclically, airplanes are indistinguishable within a subfleet, and each airplane must spend at least one night in a maintenance base at least every $\Delta$ days, with $\Delta = 4$ in the reported setting (Parmentier et al., 2017). The routing problem is treated operationally as a pure feasibility problem: find a partition of legs into maintenance-feasible cyclic routes using at most $n^{\mathrm a}$ airplanes.

A compact mixed-integer formulation represents maintenance through a time–maintenance-state graph. If $L$ is the set of legs and $[\Delta]=\{1,\dots,\Delta\}$, the layered vertex set is

$V = L \times [\Delta].$

A vertex $(\ell,\delta)$ means that an aircraft operates leg $\ell$ with $\delta$ days since the last night in a base just before $\ell$. Selecting arcs in this graph yields vertex-disjoint cycles, and the model enforces flow conservation, exact coverage of each leg, and a bound on the number of selected arcs crossing a weekly reference instant, which counts aircraft usage (Parmentier et al., 2017). This construction encodes maintenance feasibility in graph topology rather than in an explicit family of maintenance inequalities.

Stochastic variants extend this routing view by treating delay propagation as endogenous to the chosen routes. In the stochastic tail assignment problem at Air France, the objective is expected total cost, combining operational costs with a piecewise-linear convex delay penalty under scenarios (Baty et al., 11 Feb 2026). Delay propagation is modeled recursively through route slack: $\Delta = 4$0

$\Delta = 4$1

This turns route generation into a stochastic shortest-path pricing problem inside a column-generation framework. The reported algorithm, together with a diving heuristic, achieved an average $\Delta = 4$2 optimality gap on real-world Air France instances with up to 600 flight legs in a few hours of computing time (Baty et al., 11 Feb 2026). A plausible implication is that tail assignment can be treated as a routing problem not only over feasible connections but also over delay-propagation structures.

3. Periodicity, maintenance circuits, and computational hardness

Periodic aircraft routing models assume that the same set of lines of flying is repeated every day. In the formulation studied in “Aircraft routing: periodicity and complexity,” the input is an Eulerian directed graph $\Delta = 4$3, a set of bases $\Delta = 4$4, and an integer maintenance parameter $\Delta = 4$5; the solution is a collection of semi-infinite walks whose $\Delta = 4$6-th arcs are pairwise distinct and form together $\Delta = 4$7 for every day index $\Delta = 4$8, while each walk visits bases with at most $\Delta = 4$9 arcs between two visits (Meunier et al., 7 Aug 2025).

This framework distinguishes three notions of route structure. A periodic solution makes each aircraft walk periodic as a sequence of arcs. A strongly periodic solution makes each walk a repeated closed directed trail. An absolutely periodic solution further requires that any two walks are either identical or disjoint; these are exactly the Maintenance Circuit Decompositions of Talluri. The article proves that periodic solutions exist whenever a periodic instance is feasible, and that for $n^{\mathrm a}$0 every feasible periodic instance admits an absolutely periodic solution (Meunier et al., 7 Aug 2025). For practitioners, this matters because much of the earlier literature implicitly assumed that periodic instances justify this stronger cyclic decomposition.

The same work also clarifies complexity. It establishes NP-hardness for a non-periodic, finite-horizon version and proves that the finite-horizon aircraft routing problem is NP-complete even for a fixed $n^{\mathrm a}$1 (Meunier et al., 7 Aug 2025). At the same time, it identifies a polynomial special case: a “quiet night” version is polynomially solvable when the number of airplanes is fixed. This places maintenance routing in the familiar pattern of operations research: strong structural regularity can collapse complexity, whereas pathwise maintenance coupling makes the general finite-horizon problem hard.

4. Trajectory design, terminal routing, and logic-constrained flight planning

A different branch of the literature treats aircraft routing as trajectory or path design in physical airspace rather than as tail assignment. In departure-route design, the decision variables can include route geometry, vertical profile, and allocation of flights among route alternatives. At Amsterdam Airport Schiphol, a two-step framework first generates Pareto-optimal routes by multi-objective trajectory optimization and then jointly selects routes and allocates flights to minimize cumulative noise annoyance and fuel burn (Ho-Huu et al., 2019). In the 3D case, the reported solutions achieved a reduction in the number of people annoyed of up to $n^{\mathrm a}$2 and a reduction in fuel consumption of $n^{\mathrm a}$3 relative to the reference case solution (Ho-Huu et al., 2019). Here, “routing” means designing and assigning spatial departure trajectories, not rotating tails through a schedule.

At the tactical air-traffic-control level, routing can be posed over a structured terminal network. In a Changi TMA model, route choices, speed selections, and holding times are optimized over an extended 50-nautical-mile network with safety separation, speed adjustments, and holding constraints, while a rolling-horizon Model Predictive Control strategy updates decisions from real-time system states and predictions (Zhang et al., 19 Nov 2025). The objective is to minimize average landing time, equivalently increasing runway throughput, and the closed-loop framework reports a 7-fold reduction in computation time during peak congestion compared to onetime optimization (Zhang et al., 19 Nov 2025). This is a tactical, conflict-free routing-and-scheduling problem over individual aircraft rather than a strategic tail assignment problem.

Flight planning under air traffic flow restrictions introduces a third variant. The logic-constrained shortest path problem combines a one-to-one shortest-path problem with satisfiability constraints on the routing graph (Euler et al., 2024). A path is feasible if its arc-incidence assignment can be extended to satisfy a CNF formula encoding traffic flow restrictions. The resulting branch-and-bound algorithm combines dynamic shortest-path computations with SAT-style propagation, conflict extraction, and branching. On a global flight graph and a set of around 20,000 real TFRs, carefully selecting node rules, branching rules, and conflicts yields an improvement of an order of magnitude compared to an uninformed choice (Euler et al., 2024). This places route admissibility itself—not just travel cost—inside the optimization core.

5. Aircraft as network nodes and cooperative air–ground routing

In integrated aeronautical ad hoc networks, the aircraft routing problem becomes a shortest-delay routing problem over a ground–air–space communication graph. The node set is

$n^{\mathrm a}$4

where $n^{\mathrm a}$5 are aircraft, $n^{\mathrm a}$6 are ground base stations, and $n^{\mathrm a}$7 are satellites. Directed edges correspond to aircraft-to-aircraft, direct air-to-ground, and satellite-to-aircraft links that satisfy an SNR threshold $n^{\mathrm a}$8 and a visibility condition $n^{\mathrm a}$9 (Cui et al., 2020). The route from a source base station $L$0 to a destination aircraft $L$1 minimizes the sum of link delays

$L$2

subject to SNR, visibility, flow-conservation, and degree constraints. With edge weights combining transmission, propagation, and decode-and-forward delay, the problem reduces to a Dijkstra-type shortest-path search on each topology snapshot (Cui et al., 2020). The North Atlantic case study shows that aircraft-aided multi-hop communications can reduce the total delay of satellite communications when evaluated on real historical flight data (Cui et al., 2020).

Cooperative air–ground routing introduces another aircraft-specific network model. In the Cooperative Aerial-Ground Vehicle Routing Problem, one ground vehicle and one UAV must jointly visit a set of targets while the UAV remains within communication range $L$3 of the ground vehicle (Manyam et al., 2018). The model uses ground-vehicle edge variables, UAV arc variables, and assignment variables that indicate whether a target is visited by the ground vehicle or assigned to a UAV subtour rooted at a ground-vehicle stop. A mixed-integer linear programming formulation, a branch-and-cut algorithm, and a transformation to a generalized traveling salesman problem are developed (Manyam et al., 2018). The transformed GTSP is exact: any feasible GTSP tour on the configuration graph can be mapped to a feasible cooperative solution with the same cost, and vice versa (Manyam et al., 2018). This is aircraft routing in a genuinely coupled sense: the aircraft route is feasible only relative to the mobile position of another vehicle.

6. Energy-aware, cargo-integrated, and predictive extensions

Electrification has made routing inseparable from charging and infrastructure. In a regional electric-aircraft setting, aircraft assignment, routing, and charge scheduling are modeled jointly with airport renewable generation and stationary batteries on a time-expanded directed acyclic graph (Vehlhaber et al., 2023). Binary edge-use variables determine where each aircraft flies or waits, continuous charging variables determine when and where energy is taken on, and airport energy-balance constraints couple aircraft operations to photovoltaic production, batteries, and grid draw. The objective is

$L$4

In the Dutch Leeward Antilles case study, optimizing flights and operations in a renewable-energy-aware manner reduced grid dependency from 18 to 100% relative to current schedules, depending on weather conditions (Vehlhaber et al., 2023).

Urban air mobility routing brings similar couplings under stochastic dynamics. In a UAM network, a single electric aircraft is routed from origin to destination with minimal expected total travel time while accounting for limited battery capacity, stochastic travel times on corridors, stochastic queueing delays, and a limited number of charging stations at vertistops (Wei et al., 2023). The routing strategy is computed as an optimal policy of a Markov decision process over an augmented state space that includes vertistop, link status, elapsed link time, and battery level. The case study uses 29 vertistops and 137 flight corridors (Wei et al., 2023). This suggests that, once queueing and charging decisions become endogenous, fixed paths are often less informative than state-feedback routing policies.

Cargo operations add another layer of integration. The Air Cargo Load Planning with Routing, Pickup, and Delivery Problem combines a TSP-type aircraft tour over hubs with pickup–delivery, palletization, and weight-and-balance constraints based on standardized pallets in fixed positions (Mesquita et al., 24 Jun 2026). The routing variable is the permutation

$L$5

and the objective maximizes a benefit–cost ratio,

$L$6

where $L$7 is transported score and $L$8 includes fuel cost and a center-of-gravity penalty (Mesquita et al., 24 Jun 2026). This extends aircraft routing beyond aggregate payload capacities to detailed stowage, pallet positions, and loading-induced fuel penalties.

A further adjacent development is predictive decision support for rerouting. A reroute prediction service uses historical FAA reroute data and weather forecasts to predict whether an ARTCC-specific or advisory-specific reroute will be active in future time buckets, reaching mean accuracy values higher than 90% in the reported experiments (Oliveira et al., 2023). It does not compute an optimal route, but it supplies time-dependent disruption information that can be embedded in subsequent routing models.

Taken together, these strands show that the Aircraft Routing Problem is no longer confined to a single formulation. It includes maintenance-feasible tail assignment, periodic circuit decomposition, logic-constrained flight planning, tactical TMA routing, airborne communication shortest paths, cooperative UAV routing, renewable-aware electric routing, and cargo-integrated tour design. The unifying theme is the selection of feasible paths, cycles, or policies on graphs whose states and constraints encode the physics, infrastructure, and operational rules of aircraft systems.