Aircraft Maintenance Scheduling Problem

• Aircraft Maintenance Scheduling Problem is defined as a set of optimization challenges that reconcile airworthiness constraints, regulatory intervals, and operational demands to minimize maintenance costs.
• It leverages diverse methods including predictive maintenance, genetic algorithms, and continuous-time MILP models to optimize scheduling in both static and dynamic environments.
• Practical applications include balancing preventive and corrective actions, reducing downtime with real-time rescheduling, and enhancing overall fleet readiness through intelligent resource allocation.

The Aircraft Maintenance Scheduling Problem (AMSP) encompasses a diverse family of optimization models and algorithms concerned with allocating maintenance actions and resources to fleets of aircraft and their components. Its primary goal is to reconcile airworthiness constraints, regulatory maintenance intervals, operational readiness, and resource limitations, while minimizing costs and disruptions. This problem landscape includes models at the aircraft, component, resource, and fleet assignment levels, spanning static, dynamic, deterministic, and stochastic formulations.

1. Mathematical Formulations Across Scheduling Dimensions

Aircraft maintenance scheduling problems are highly heterogeneous but share a focus on optimizing schedules under complex sets of operational and engineering constraints. Representative formulations include:

• Threshold-based Predictive Maintenance (Ziyad et al., 2022): Each engine $i$, observed at discrete time $t$, has sensor features $x_i(t)$. A learned prognostic model computes a log-partial-hazard score $z_i(t)$. A maintenance decision threshold $\lambda$ triggers removal if $\max_{s \leq t} z_i(s) \geq \lambda$. The objective is to minimize total cost

$$\mathrm{TotalCost}(\lambda) = \sum_{i=1}^n \left[ C_1 \cdot \mathbf{1}\{\tau_i(\lambda) < T_i\} + C_2 \cdot \mathbf{1}\{\tau_i(\lambda) \geq T_i\} \right]$$

where $C_1$ is preventive (scheduled) cost, $C_2$ is higher corrective (unscheduled) maintenance cost.

• Resource-Constrained Project Scheduling (Heavy Maintenance Checks) (Pimapunsri et al., 2022): Let $J$ be activities, $K$ resources, and $P$ precedence. Each activity $j$ uses $r_{j,k}$ of resource $k$ for duration $d_j$; project makespan $C_{\max}$ is minimized subject to

$$\sum_{j \in J, t' \leq t < t' + d_j} r_{j,k} \cdot x_{j, t'} \leq R_k, \quad \forall k, t$$

• Continuous-Time MILP for Hangar Layout and Scheduling (Pazhooh et al., 4 Aug 2025): Defines decision variables $X_a, Y_a$ (position), $\mathrm{Roll\_in}_a, \mathrm{Roll\_out}_a$ (timing), and binary spatial/temporal disjunctive variables to enforce non-overlapping, spatial, and blocking constraints.
• Flight and Maintenance Planning (Peschiera et al., 2020): For military fleets, the model schedules missions and maintenance checks, with constraints on maximum time/flight hours between checks, overlapping maintenance tasks, and fleet/cluster serviceability via a large-scale mixed-integer program.
• Dynamic Repair Shop Scheduling (Bajestani et al., 2014): Successive static sub-problems for repair shop scheduling are solved using logic-based Benders decomposition, with the objective of maximizing long-term aircraft availability under fluctuating failure rates and limited repair resources.

2. Prognostic and Optimization Methodologies

Modern AMSP research spans both data-driven prognostics and combinatorial optimization paradigms:

• Prognostics-driven Scheduling: The integration of survival analysis (e.g., Cox proportional hazards) enables maintenance timing decisions based on individualized risk predictions (Ziyad et al., 2022). Monte Carlo simulation and bootstrapping are used to optimize cost-vs-safety tradeoffs over engine populations.
• Heuristic/Metaheuristic Optimization: Permutation-based genetic algorithms (GAs) with domain-specific dispatching rules, various crossover/mutation operators, and resource assignment heuristics (EST, WEST) have demonstrated substantial improvements in maintenance makespan and crew utilization (Pimapunsri et al., 2022, Urquhart et al., 19 Dec 2025).
• Mathematical Programming: Continuous-time MILPs (Pazhooh et al., 4 Aug 2025) encode multi-aircraft, multi-resource, spatio-temporal layouts with non-overlap and blocking via pairwise disjunctions. Qualifying constraints enable integrated planning of hangar logistics with maintenance event scheduling.
• Logic-based Benders Decomposition (LBBD): For dynamic repair environments, LBBD master problems assign due dates, while subproblems handle resource-cumulative constraints; infeasibility is handled via combinatorial cuts that refine the master (Bajestani et al., 2014). This approach yields near-optimal dynamic schedules with high long-term fleet availability.
• Rolling Horizon and Rescheduling Policies: Dynamic environments benefit from overlapping planning windows and policies such as rescheduling after each wave of operations (e.g., $P_{3,1}$, i.e., plan 3 waves, reschedule after each), striking a balance between myopia and anticipatory scheduling (Bajestani et al., 2014).

3. Performance Results and Computational Analysis

Detailed empirical studies reveal the operational impacts across scheduling paradigms:

Model / Reference	Key Results / Performance Metrics
Predictive Maintenance (Ziyad et al., 2022)	Up to 50% cost reduction vs naive/threshold extremes; 13% savings with clustered, mode-tailored strategies
GA for RCPSP (Pimapunsri et al., 2022)	Makespan reduction of ~30% vs planner; peak resource demand cut by ~15%; 5–35 min runtime
Continuous-Time MILP (Pazhooh et al., 4 Aug 2025)	Solved up to 40-aircraft instances to 5% optimality in <3 hours; heuristic gap 9-59%
Dynamic Shop + LBBD (Bajestani et al., 2014)	10% higher long-term aircraft availability, ~4x speed vs monolithic MIP, near-zero coverage variance
Military FMP (Peschiera et al., 2020)	Fleets up to 200 aircraft (180 months) solved with <5% optimality gap; warm-started heuristics reduce solve time by 10-30%

Performance is sensitive to problem size, mission concurrency, constraint tightness, and resource slack. Partitioned or decomposed models with tailored heuristics enable tractability at scale.

4. Model Assumptions, Limitations, and Extensions

• Assumptions Vary by Model: Many models assume deterministic durations (Pimapunsri et al., 2022), fixed crew/resource homogeneity, and ignore unscheduled/emergent tasks unless explicitly handled (e.g., via online insertion heuristics (Aboussalim et al., 2010)).
• Multi-Component and Multi-Objective Extensions: Advanced models add modules for joint component/aircraft scheduling, time-dependent costs/availabilities, and per-resource or per-component reliability constraints, often via integer/MIP generalizations or metaheuristics (Ziyad et al., 2022).
• Dynamic Real-Time Updates: For in-service corrective insertions or urgent repairs, fast heuristics or local shift algorithms guarantee real-time response without global re-optimization (Aboussalim et al., 2010).
• Uncertainty: Current industrial models often neglect stochastic processing times and unpredicted failures; robust/stochastic variants are research frontiers (Pimapunsri et al., 2022).
• Integration of Prognostics and Discrete Optimization: There is increasing movement toward architectures that wrap AI/ML-based health estimation with simulation-based or mathematical programming scheduling layers, especially for highly customized and context-sensitive maintenance regimes (Ziyad et al., 2022).

5. Complexity and Structural Results

Theoretical complexity results inform both solvability and model design:

• Periodic Aircraft Routing with Maintenance (Meunier et al., 7 Aug 2025): The periodic fleet assignment problem with maintenance at most every γ=4 days admits a polynomial-time solution as a disjoint-cycle decomposition. The finite-horizon (non-periodic) variant is NP-complete for γ≥4, implying the need for decomposition or heuristics in operational settings.
• Military Aircraft FMP (Peschiera et al., 2020): The scheduling variant with minimum-duration missions, maintenance windows, and resource capacities is NP-hard (shown via reduction from fixed-interval personnel scheduling).
• Dynamic Repair Shop Scheduling (Bajestani et al., 2014): The use of logic-based Benders yields optimal solutions for static subproblems and, when embedded in rolling planning, results in provable bounds for dynamic long-term policies.

These results clarify under which operational regimes cycle-based models suffice, and when combinatorial explosion necessitates decomposition, heuristics, or rolling-horizon strategies.

6. Industrial Integration and Practical Guidelines

Implemented AMSP systems often combine exact and heuristic methods within decision support pipelines:

• GA-based Scheduling: Deployed as modules in airline Maintenance Planning Systems, providing schedules, call-up lists, and real-time resource assignments.
• MILP/CP+Heuristic Compositions: Exact models supplemented by fast greedy algorithms (for large N or urgent scenarios), with visualization dashboards to improve buy-in from maintenance managers (Pazhooh et al., 4 Aug 2025).
• LBBD with Rolling Horizon: Applied in military shop scheduling, combining MIP-based due date assignment with CP for resource-cumulative shaping (Bajestani et al., 2014).
• Real-Time Task Insertion: Online algorithms efficiently insert dynamic (unscheduled) tasks without global timing disruptions (Aboussalim et al., 2010).

Tuning of metaheuristic parameters, proper heuristic initializations (warm-starts), and careful modeling of slack/availability variables are critical for real-world scalability. Visualization and interpretability facilitate stakeholder adoption, especially in resource-constrained or availability-critical environments.

7. Open Problems and Future Directions

• Stochastic and Robust Maintenance Scheduling: Incorporating uncertainty in durations, equipment failures, and resource fluctuations.
• Integrated Prognostics-Optimization Architectures: Broadening the use of machine learning and survival models in scheduling layers, moving from static thresholds to adaptive, context-sensitive policies.
• Multi-Aircraft and Multi-Resource Coordination: Extension to scenarios with interactions, coupling between aircraft/components, and tiered spare-part constraints.
• Multi-Objective Optimization: Simultaneous consideration of availability, maintenance cost, risk, and crew overtime, possibly via Pareto front computation.
• Hybrid Matheuristics and Real-Time Rescheduling: Dynamic integration of metaheuristics, rolling-horizon optimization, and local search for irregular/emergent disruptions.

Ongoing research continues to drive advances in both mathematical and AI-driven approaches to the Aircraft Maintenance Scheduling Problem, establishing new standards for cost-effectiveness, operational continuity, and safety compliance (Ziyad et al., 2022, Pimapunsri et al., 2022, Pazhooh et al., 4 Aug 2025, Urquhart et al., 19 Dec 2025, Bajestani et al., 2014, Meunier et al., 7 Aug 2025, Peschiera et al., 2020, Aboussalim et al., 2010, Kalosi et al., 2016).