A Hybrid Reverse Learning Particle Swarm Optimization Method for Aircraft Maintenance Scheduling Based on the Resource-Constrained Project Scheduling Problem Model

Abstract

Aircraft maintenance scheduling is a critical task in air transportation and national defense security, characterized by complex multi-step procedures, strict precedence dependencies, and multi-resource constraints involving personnel skills and equipment availability. Traditional scheduling methods and standard metaheuristic algorithms often suffer from insufficient model adaptability, poor population diversity, premature convergence, and complex encoding schemes that require frequent feasibility checks. To address these challenges, this paper proposes a comprehensive optimization framework based on the Resource-Constrained Project Scheduling Problem (RCPSP) model. A decimal priority-based encoding method is introduced to replace traditional integer permutation encoding, significantly reducing computational complexity and enhancing search space continuity. Furthermore, an improved hybrid Particle Swarm Optimization algorithm integrating reverse learning and partial random operations (RL-PSO) is developed. The reverse learning mechanism expands the global search space by generating reverse particles, while partial random operations maintain population diversity and prevent premature convergence. The proposed framework converts priority encoding into feasible schedules through a priority sorting and left-shift resource allocation strategy. Simulation experiments on maintenance tasks involving up to 50 aircraft demonstrate that RL-PSO achieves optimization accuracy of 332 min, convergence speed of 92.07 s, and stability of 2.8843 min in standard deviation, which are superior compared to standard PSO, Simulated Annealing, and Teaching–Learning-Based Optimization combined with the serial schedule generation scheme (SSGS). The method effectively balances global exploration and local exploitation, making it suitable for complex, large-scale aircraft maintenance scenarios. Future work will extend the framework to multi-objective optimization and dynamic scheduling environments.

1. Introduction

1.1. Background

Scheduling plays an important role in traffic timetables [1], manufacturing systems and maintenance. In the fields of air transportation and national defense security, the safe and efficient operation of aircraft directly depends on a scientifically sound maintenance scheduling system [2]. Aircraft maintenance tasks are characterized by complex procedures, stringent resource constraints, and strict process logic. Multiple maintenance steps for several aircraft must be completed with limited maintenance personnel, equipment, and time windows. Strict precedence dependencies exist between steps, and personnel and equipment must satisfy constraints such as skill matching and exclusive usage. A reasonable maintenance schedule can minimize the total maintenance period, and improve resource utilization, thereby reducing operating costs and ensuring fleet dispatch reliability. Conversely, improper scheduling can easily lead to procedure conflicts, resource idling or overload, which not only affects maintenance efficiency but may also create safety hazards.

As fleet sizes expand and maintenance processes become more complex, traditional experience-based scheduling methods are no longer suitable for complex scenarios involving multiple coupled resources and coordinated procedures [3]. There is an urgent need for intelligent scheduling through precise mathematical modeling and efficient optimization algorithms. Currently, the aircraft maintenance scheduling problem has become a research hotspot in operations research and systems engineering. Its core difficulty lies in balancing the rigidity of procedure constraints with the flexibility of resource allocation to quickly search for the optimal schedule within a complex constraint space. Therefore, conducting research on aircraft maintenance scheduling model construction and optimization algorithm design is of great significance for enhancing the intelligence level and practical engineering value of the aircraft maintenance industry.

1.2. Related Work

The aircraft maintenance scheduling problem is essentially a complex resource-constrained scheduling problem. Academic research often expands upon classical scheduling models while exploring the suitability of various optimization algorithms. This section reviews related work from the perspectives of scheduling models and optimization algorithms, based on recent research findings, to lay the theoretical foundation for this paper.

1.2.1. Job Shop Scheduling Model (JSP) and Its Improvement

The Job Shop Scheduling Problem (JSP) is one of the most fundamental scheduling models [4]. Its core characteristics are the fixed matching between operations and machines, clear operation sequence constraints, simple model structure, and low solution complexity. It was widely used in early simple maintenance scenarios with fixed processes. Šeda [5] defined the core constraints and objective of JSP through mathematical modeling, providing a basic framework for subsequent scheduling model extensions. However, traditional JSP lacks flexibility and cannot adapt to the complex requirements of aircraft maintenance, such as equipment sharing and flexible personnel allocation. Scholars have improved the model to enhance its adaptability.

In recent research, improvements to JSP have often focused on local constraint optimization and scenario adaptation. For example, some scholars have introduced task insertion mechanisms to optimize JSP’s responsiveness in multi-task parallel scenarios, but the limitation of fixed operation–machine matching remains difficult to overcome. Gao et al. [6] pointed out in their review that JSP and its variants still have efficiency advantages in simple maintenance tasks, but their suitability is significantly weaker than Flexible Job Shop Scheduling and Resource-Constrained Project Scheduling models in complex scenarios with coupled resources. Since aircraft maintenance involves dynamic allocation of multi-skilled personnel and multiple equipment types, a single JSP model cannot accurately capture its core characteristics. It is only suitable for small fleet maintenance scenarios with clear equipment division and simple procedures.

1.2.2. Flexible Job Shop Scheduling Model (FJSP) and Its Improvement

The Flexible Job Shop Scheduling Problem (FJSP) relaxes the fixed operation–machine matching constraint of JSP, allowing operations to be performed on multiple machines [7]. It possesses stronger resource allocation flexibility and is more suitable for aircraft maintenance scenarios where multiple equipment options exist. Komaki et al. [8] reviewed the application of FJSP in scenarios containing assembly operations, verifying its advantages in multi-resource selection scenarios, providing an important reference for aircraft maintenance scheduling.

Recent improvements to FJSP often focus on multi-objective optimization and refined resource constraints. Lin et al. [9], aiming to minimize maintenance completion time and balance maintenance costs, proposed a yeast reproduction propagation heuristic algorithm. Based on FJSP, they constructed a fleet maintenance decision model, improving the algorithm’s solution accuracy in multi-objective scenarios. De Bruecker et al. [10] combined simulation and optimization to build a mixed-integer linear programming FJSP model, solved using an enhanced heuristic algorithm, achieving dual-objective optimization of maintenance completion time and personnel labor cost. However, the FJSP still has inherent shortcomings: it inadequately captures project-type precedence dependencies between operations and cannot adapt well to the complex process logic of networked, coupled multi-step aircraft maintenance. Su et al. [11] found in their research on carrier-based aircraft support scheduling that the FJSP can adapt to medium-complexity maintenance tasks, but the feasibility and accuracy of the schedule decrease significantly in deeply coupled multi-step scenarios. Therefore, it is more suitable for medium-sized fleets with medium-complexity maintenance tasks.

1.2.3. Resource-Constrained Project Scheduling Model (RCPSP) and Its Improvement

The Resource-Constrained Project Scheduling Problem (RCPSP) centers on the project, can accurately capture precedence dependencies between operations and multi-resource constraints, and possesses greater flexibility and robustness. It has become the mainstream model for complex aircraft maintenance scheduling. This model treats the maintenance task of each aircraft as an independent project, describes operation dependencies using a network model, and considers total resource constraints and allocation rules for multiple resources like personnel and equipment, perfectly matching the core characteristics of aircraft maintenance: coupled multi-step procedures and multi-resource constraints.

Recent research on the RCPSP shows three major trends: multi-skill extension, uncertainty optimization, and multi-objective integration. Chen et al. [12] constructed a multi-objective nonlinear mixed-integer programming RCPSP model, considering skill efficiency improvement, development cycle, and cost objectives, solved using the NSGA-II algorithm, providing new ideas for multi-skill resource scheduling. Hematian et al. [13] introduced learning effects and uncertainty factors, constructed a multi-objective integrated multi-project RCPSP model, and used fuzzy mathematical programming to handle maintenance time uncertainty, fitting the actual scenario of fluctuating working hours in aircraft maintenance. Ghamginzadeh et al. [14], based on fuzzy set theory, proposed a multi-objective multi-skill RCPSP model solved by an imperialist competitive algorithm, effectively handling non-statistical uncertainty in duration and cost.

For multi-project and multi-skill coordination scenarios, Chen et al. [15] proposed an improved RCPSP model incorporating multi-skilled labor allocation and uncertainty, providing a scheduling solution for large equipment manufacturers’ maintenance. Mozhdehi et al. [16] further proposed a Multi-Skill Resource-Constrained Multi-Project Scheduling Problem (MSRCMPSP), considering worker cooperation relationships and skill proficiency, built a mixed-integer model, and designed an improved discrete biogeography-based optimization algorithm, significantly improving scheduling accuracy in complex scenarios. These studies validate the superiority of the RCPSP and its variants in complex maintenance scheduling scenarios, providing direct references for the model construction in this paper.

1.2.4. Comparison and Selection of Three Types of Models

To clearly compare the suitability of the three models for aircraft maintenance scheduling, a model feature comparison table is constructed below, systematically analyzing them from four dimensions: core features, advantages, disadvantages, and suitable scenarios, providing a basis for model selection in this paper. The comparison of three types of models is shown in

Table 1. Comparison of models.

Model Type	Core Feature	Advantage	Disadvantage	Suitable Scenario
JSP	Fixed operation–machine matching, clear sequence constraints, no resource flexibility	Simple model logic, high solution efficiency, low computational complexity, no complex resource allocation optimization	Very poor flexibility, cannot adapt to equipment sharing or cross-procedure personnel allocation, weak anti-interference ability	Simple maintenance tasks, small fleets with clear equipment division, single-type routine line maintenance
FJSP	Flexible operation–machine matching, supports multi-resource selection, basic resource allocation flexibility	Strong flexibility, adapts to equipment sharing scenarios, can account for skill differences, better solution accuracy than JSP	Inadequate description of project-type operation dependencies, high solution complexity under multi-resource coupling constraints, difficult to adapt to complex process networks	Medium-complexity maintenance tasks, medium-sized fleets with multiple equipment options, single-type deep maintenance
RCPSP	Project-centric, accurately captures operation dependency networks and multi-resource constraints, supports dynamic resource allocation	Adapts to multi-operation dependency and resource sharing scenarios, strong flexibility and robustness, accurately matches complex maintenance constraints	High model complexity, high demand on algorithm search capability, difficulty in initial schedule construction	Complex maintenance tasks, large fleets with coupled multi-step procedures and multi-resource constraints, multi-type mixed scheduled maintenance

.

Considering the multi-operation, multi-resource-constrained aircraft maintenance scheduling problem addressed in this paper, and given that aircraft maintenance tasks have clear operation precedence dependencies, dual personnel-equipment resource constraints, and skill matching requirements, the RCPSP model can accurately capture these core features. Its suitability and solution accuracy in complex scenarios are significantly better than the JSP and FJSP. Bredael et al. [17] verified through extensive benchmark experiments that the RCPSP and its variants have optimal scheduling performance in multi-resource-constrained, multi-operation coupled scenarios. Therefore, this paper constructs a mathematical model for aircraft maintenance scheduling based on the RCPSP model, providing a theoretical foundation for subsequent optimization algorithm design.

1.2.5. Research Status of Optimization Algorithms for Aircraft Maintenance Scheduling

The aircraft maintenance scheduling problem is NP-hard. As the problem scale increases, the computational complexity of exact algorithms grows exponentially, making it difficult to meet the real-time requirements of practical scheduling. Therefore, academia and industry often use heuristic, metaheuristic, and deep reinforcement learning algorithms.

Heuristic algorithms generate schedules quickly using predefined rules and have high computational efficiency, but solution quality depends on rule design, and adaptability is poor. Zan et al. [18] designed an improved max-min ant heuristic algorithm to solve wartime equipment maintenance scheduling, but it is only suitable for simple scheduling tasks in specific scenarios and difficult to adapt to the complex constraints of aircraft maintenance. Metaheuristic algorithms search for optimal solutions by simulating natural evolution, group behavior, etc. They possess stronger global search capabilities and are currently the mainstream algorithms for aircraft maintenance scheduling. Among them, Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Simulated Annealing (SA), and Teaching–Learning-Based Optimization (TLBO) are most widely applied. Jain et al. [19] reviewed the variants and progress of PSO algorithms, noting their advantage of fast convergence in scheduling problems but susceptibility to population diversity decay and premature convergence. Zeng et al. [20] combined heuristic algorithms with hybrid Petri nets, improving the dynamic response capability of traditional metaheuristics and enhancing scheduling performance under uncertainty.

Deep Reinforcement Learning (DRL) learns optimal policies through agent–environment interaction, showing potential advantages in dynamic scheduling scenarios, but its application still has limitations. On one hand, DRL environment construction is complex, requiring accurate simulation of multi-constraint coupled aircraft maintenance scenarios. On the other hand, model training cost is high, and generalization is poor, making it difficult to adapt to different scales and types of maintenance tasks. In existing research, the application of DRL in aircraft maintenance scheduling is mostly in the theoretical validation stage, not yet realized in engineering practice.

Overall, heuristic algorithms have limited effectiveness, DRL has environment construction and training difficulties, and metaheuristic algorithms remain the best choice for complex aircraft maintenance scheduling. However, traditional metaheuristic algorithms need improvement for aircraft maintenance scenarios to address issues like premature convergence and complex encoding.

1.3. Motivation and Contribution

Traditional encoding uses a full permutation of integers from 1 to m × n, requiring complex processing during evolution. We adopt decimal encoding while improving evolutionary operators.

Addressing the issues of insufficient model suitability and performance bottlenecks of optimization algorithms in existing aircraft maintenance scheduling research, this paper focuses on encoding method optimization and algorithm structure improvement. By constructing a complete model-encoding-algorithm optimization framework, it aims to achieve efficient scheduling of complex aircraft maintenance tasks. Traditional scheduling algorithms often use a full permutation integer encoding of 1 to m × n (where m is the number of aircraft and n is the number of operations per aircraft). This encoding directly corresponds to the operation execution sequence. However, its evolution process requires complex feasibility checks and repair operations, which not only increases the computational complexity of the algorithm but may also destroy the structure of good solutions. To this end, this paper adopts a decimal priority encoding method combined with improved evolutionary operator design to balance encoding complexity and search efficiency while enhancing the algorithm’s global search capability.

1.3.1. Research Motivation

Although existing research has proposed various models and algorithms for the aircraft maintenance scheduling problem, the following shortcomings still exist in practical applications, constituting the research gap addressed by this paper:

First, encoding methods have defects. Traditional scheduling algorithms often use a full permutation integer encoding of 1 to m × n, directly corresponding to the operation execution sequence. However, the evolution process requires complex feasibility checks and repair operations (e.g., handling precedence conflicts, resource allocation conflicts), which not only increases computational complexity but may also destroy the structure of good solutions, affecting search efficiency [21]. Existing encoding improvements are mostly designed for single models, lacking adaptability optimization for multi-constraint RCPSP scenarios.

Second, optimization algorithm performance is insufficient. Traditional metaheuristic algorithms (e.g., standard PSO, GA) are prone to population diversity decay and premature convergence in complex multi-resource constraint scenarios, making it difficult to explore the global optimal schedule [19]. Benchmark analyses by Bredael et al. [17] indicate that the performance of single metaheuristic algorithms in multi-objective, multi-constraint scenarios is often insufficient, requiring mechanism fusion to enhance search capability.

Third, the synergy between algorithm and model needs improvement. Existing improved algorithms are often designed for single constraints or scenarios, lacking targeted optimization for the multi-operation dependency and multi-resource coupling constraints of aircraft maintenance, leading to insufficient feasibility and practicality of the generated schedules [16]. For example, some algorithms do not fully consider personnel skill matching constraints, making the generated schedules difficult to apply directly in practical maintenance scenarios.

Based on the above research gaps, this paper aims to solve the complexity of traditional encoding methods and the premature convergence problem of traditional algorithms, construct an efficient optimization algorithm suitable for aircraft maintenance scenarios, improve the quality and feasibility of schedules, and provide technical support for practical aircraft maintenance scheduling.

1.3.2. Research Contribution

The main research contributions of this paper are as follows:

(1)

This paper proposes a decimal priority-based encoding method to replace traditional integer encoding. It avoids infeasible solutions and complex repair operations, reduces computational complexity, and improves search space continuity and population diversity.

(2)

A hybrid Particle Swarm Optimization algorithm (RL-PSO) integrating reverse learning and partial random operations is designed to overcome premature convergence and insufficient diversity in traditional PSO, balancing global exploration and local exploitation.

(3)

Constructing an RCPSP model and algorithm closed-loop adapted to aircraft maintenance scenarios. Based on the RCPSP model, fully considering the core characteristics of aircraft maintenance such as operation precedence dependencies, dual personnel-equipment resource constraints, and skill matching, combined with the proposed decimal encoding method and RL-PSO algorithm, a complete model-encoding-algorithm optimization framework is constructed.

(4)

Designing a decoding method that ensures feasibility.

Through a priority sorting and left-shift resource allocation strategy during decoding, the decimal priority encoding is converted into a feasible maintenance schedule, ensuring the feasibility and practicality of the schedule.

1.4. Organization

The section structure of this paper is arranged as follows:

Section 1: Introduction. It introduces the research background and difficulties of aircraft maintenance scheduling, reviews existing models and algorithms, selects the RCPSP model and metaheuristic approach, and states the research motivation, contributions, and structure.

Section 2: Mathematical Modeling. It defines the mathematical model for aircraft maintenance operation scheduling, including problem description, assumptions, objective function, and constraints (precedence, personnel, equipment, resource allocation).

Section 3: Optimization Algorithm Design. It presents the RL-PSO algorithm, covering the encoding method, decoding process (priority sorting and left-shift resource allocation), key improvements (reverse learning, partial random strategy), algorithm flow, and parameter settings.

Section 4: Simulation Validation. Using MATLAB, it constructs test cases with varying scales and parameters, verifies the model and algorithm via performance comparison and scalability tests, and evaluates optimization accuracy, efficiency, and robustness using data, curves, and Gantt charts.

Section 5: Conclusion and Outlook. It summarizes the main work and innovations, discusses theoretical and engineering value, identifies limitations, and suggests future research directions (e.g., dynamic resources, multi-objective optimization).

2. Mathematical Modeling

In this section, we describe in detail the mathematical model for the aircraft maintenance operation scheduling problem, including the problem description, establishment of the objective function, and setting of various constraints, providing a theoretical foundation for the subsequent optimization algorithm.

2.1. Problem Description and Assumptions

2.1.1. Problem Description

In a multi-resource-constrained aircraft maintenance scenario, the maintenance department must schedule limited multi-skilled technical personnel and specialized maintenance equipment to complete all maintenance tasks for n aircraft awaiting service. This must be done while strictly adhering to operation precedence constraints, personnel skill matching constraints, and resource exclusivity constraints, with the ultimate goal of minimizing the total maintenance period. Define the set of aircraft awaiting maintenance as $A=\left\{A_1,A_2,\cdots ,A_n\right\}$, where $A_i$ represents the i-th aircraft. Each aircraft $A_i$ must complete m maintenance operations sequentially according to a defined process network. The j-th maintenance operation of the i-th aircraft is denoted as $O_{ij}$. The set of maintenance operations for aircraft A_i is $J_i=\left\{O_{i1},O_{i2},\cdots ,O_{iM}\right\}$, and the total set of all maintenance operations for all aircraft is $J={\bigcup }_{i=1}^N{J}_i$, with a total number of operations $N=n\times m$. Each operation $O_{ij}$ has a fixed processing time $d_{ij}$ and requires a specific number of correspondingly skilled personnel and specialized maintenance equipment. The core objective of scheduling is to determine the start time $S_{ij}$ and completion time $F_{ij}$ for each operation, along with the allocation plan for personnel and equipment resources, to minimize the total maintenance period for all aircraft while satisfying all constraints.

2.1.2. Symbol Definition

This section standardizes all sets, parameters, and decision variables involved in the model to ensure consistency and uniqueness of the symbol system throughout the model:

(1)

Set Definitions:

$K$: Total set of maintenance personnel, $K=\left\{1,2,\dots ,K_{total}\right\}$, where $K_{total}$ is the total number of maintenance personnel.

$I$: Set of maintenance skill types, $I=\left\{1,2,\dots ,I_{total}\right\}$, where $I_{total}$ is the total number of skill types.

$K_i$: Subset of personnel possessing skill type i, $K_i\subseteq K$.

$E$: Total set of maintenance equipment, $E=\left\{1,2,\dots ,E_{total}\right\}$, where $E_{total}$ is the total number of maintenance equipment units.

$M$: Set of maintenance equipment types, $M=\left\{1,2,\dots ,M_{total}\right\}$, where $M_{total}$ is the total number of equipment types.

$E_m$: Subset of equipment of type m, $E_m\subseteq E$.

$P_{ij}$: Set of predecessor operations for operation $O_{ij}$, i.e., all operations that must be completed before $O_{ij}$ can start.

$J_t$: Set of operations being executed at time t, $J_t=\left\{O_{ij}\in J\left|S_{ij}\le t\le F_{ij}\right.\right\}$.

(2)

Parameter Definitions:

$d_{ij}$: Processing time of operation $O_{ij}$.

$r_{ij}^k$: Demand for personnel with skill type k for operation $O_{ij}$.

$q_{ij}^e$: Demand for equipment of type e for operation $O_{ij}$.

$s_{pk}$: Personnel skill matching matrix, $s_{pk}=1$ if person $p$ possesses skill type $k$, $s_{pk}=0$ otherwise.

$N_k$: Total number of personnel with skill type $k$, $N_k=\left|N_k\right|$.

$M_e$: Total number of equipment units of type $e$, $M_e=\left|M_e\right|$.

$T$: A sufficiently large positive real number, serving as the upper bound of the total maintenance period, used to formulate non-overlap constraints.

(3)

Decision Variable Definitions:

$S_{ij}$: Start time of operation $O_{ij}$ (non-negative real number).

$F_{ij}$: Completion time of operation $O_{ij}$ (non-negative real number), $F_{ij}=S_{ij}+d_{ij}$.

$x_{p,ij}$: Binary decision variable, $x_{p,ij}=1$ if person $p$ is assigned to operation $O_{ij}$, 0 otherwise.

$y_{p,ij}^k$: Binary decision variable, $y_{p,ij}^k=1$ if person $p$ uses skill type k in operation $O_{ij}$, 0 otherwise.

$z_{q,ij}$: Binary decision variable, $z_{q,ij}=1$ if equipment unit $q$ is assigned to operation $O_{ij}$, 0 otherwise.

$a_{ij,kl}$: Binary decision variable, $a_{ij,kl}=1$ if operation $O_{ij}$ is executed before operation $O_{kl}$ when sharing the same person, 0 otherwise.

$b_{ij,kl}$: Binary decision variable, $b_{ij,kl}=1$ if operation $O_{ij}$ uses a piece of equipment before operation $O_{kl}$ when sharing that equipment, 0 otherwise.

2.1.3. Assumptions

To ensure the rationality and solvability of the model, and considering the actual operational specifications of aircraft maintenance, the following assumptions are supplemented and refined based on the original ones:

(1)

All maintenance personnel and equipment are in good working condition during the scheduling period, with no sudden failures or unscheduled downtime.

(2)

Once started, a maintenance operation cannot be interrupted, and there is no parallel machine substitution or cross-equipment transfer operation.

(3)

The transfer time of personnel and equipment between different operations is negligible; i.e., the completion time of a resource’s previous operation is the earliest possible start time for its next operation.

(4)

Each maintenance person may possess one or more skills but can only use one skill per operation, and the skill level meets the operational requirements of the corresponding operation.

(5)

Each type of maintenance equipment has a unique functional attribute, can only serve operations requiring that equipment type, and can only serve one operation of one aircraft at a time.

(6)

The processing time for each operation is a fixed value determined according to the maintenance manual; the basic model does not consider fluctuations in working hours due to differences in personnel proficiency.

2.2. Objective Function

The optimization objective of this paper is to minimize the total maintenance time for all aircraft, which is referred to as the makespan $C_{max}$ in scheduling theory. It is defined as the time interval from the start moment of the earliest started operation to the end moment of the latest completed operation. This objective directly reflects the overall efficiency of maintenance scheduling. The objective function is defined as follows:

$${\mathit{minC}}_{\mathit{max}}=\mathit{max}\left\{F_{ij}\left|i=1,\dots ,n,j=1,\dots ,m\right.\right\}$$

(1)

where $F_{ij}=S_{ij}+d_{ij}$ is the completion time of operation $O_{ij}$. By optimizing the operation execution sequence and the allocation of personnel and equipment resources, minimizing the maximum completion time of all operations can effectively shorten the total maintenance period, improve maintenance resource utilization, and ensure fleet dispatch reliability.

2.3. Constraints

2.3.1. Operation Precedence Constraints

The aircraft maintenance operation process has a sequence, i.e., node-type precedence relationships. A subsequent operation can only start after all its predecessor operations are completed.

$$F_{pq}\le S_{ij},\forall O_{pq}\in P_{ij},\forall i=1,\dots ,n;j=1,\dots ,m$$

(2)

where $P_{ij}$ represents the set of predecessor operations for operation $O_{ij}$. This constraint ensures that maintenance activities strictly follow the defined process network, avoiding maintenance quality issues or safety hazards caused by operation sequence conflicts.

2.3.2. Personnel Skill Matching and Quantity Constraints

This type of constraint is the core of personnel resource scheduling, encompassing four levels: single-operation personnel quantity matching, personnel skill matching, single-person single-skill usage, and global personnel quantity constraints.

(1)

Single-Operation Personnel Quantity Matching Constraint

For any operation $O_{ij}$, the number of personnel with skill type $k$ assigned to it must be exactly equal to the operation’s demand. The mathematical expression is:

$${\sum }_{p\in K_k}{x}_{p.ij}=r_{ij}^k,\forall i=1,\dots ,n;j=1,\dots ,m;\forall k\in I$$

(3)

This constraint ensures that each operation has a sufficient number of correspondingly skilled personnel to complete the maintenance work, avoiding understaffing or redundant allocation.

(2)

Personnel Skill Matching Constraint

Only personnel possessing the corresponding skill can be assigned to an operation requiring that skill. This is a key constraint for ensuring the feasibility of the scheduling scheme. The mathematical expression is:

$$x_{p,ij}\le s_{pk},\forall p\in K;\forall k\in I;\forall O_{ij}\in J_k$$

(4)

where $J_k$ is the set of operations requiring skill type $k$. This constraint strictly defines the scope of personnel assignment, preventing the generation of infeasible schemes where personnel are assigned to tasks for which they lack the required skills.

(3)

Single-Person Single-Skill Usage Constraint

As stated in the assumptions, the same maintenance person can only use one skill in a given operation. The mathematical expression is:

$${\sum }_{k\in I}{y}_{p,ij}^k\le 1,\forall p\in K;\forall O_{ij}\in J$$

(5)

Furthermore, there is a logical correspondence between the skill usage variable $y_{p,ij}^k$ and the personnel assignment variable $x_{p,ij}$: if person $p$ is assigned to operation $O_{ij}$, they must use one of the skills required for that operation. The mathematical expression is:

$${\sum }_{k\in I}{y}_{p,ij}^k=x_{p,ij},\forall p\in K;\forall O_{ij}\in J$$

(6)

(4)

Global Personnel Quantity Constraint

At any time $t$, the total number of personnel with skill type $k$ engaged in maintenance work must not exceed the total available number of personnel with that skill. The mathematical expression is:

$${\sum }_{O_{ij\in J_t}}{\sum }_{p\in K_k}{x}_{p,ij}\le N_k,\forall t\ge 0;\forall k\in I$$

(7)

This constraint limits the global occupancy of personnel resources over time, avoiding personnel overload at any given moment and ensuring the rationality of personnel scheduling throughout the maintenance period.

2.3.3. Equipment Exclusivity and Quantity Constraints

This type of constraint standardizes the usage rules for equipment resources, encompassing three levels: single-operation equipment quantity matching, global equipment quantity constraints, and equipment time exclusivity constraints.

(1)

Single-Operation Equipment Quantity Matching Constraint

For any operation $O_{ij}$, the number of equipment units of type $e$ assigned to it must be exactly equal to the operation’s demand. The mathematical expression is:

$${\sum }_{q\in E_e}{z}_{q,ij}=q_{ij}^e, \forall i=1,\dots ,n;j=1,\dots ,m,\forall e\in M$$

(8)

(2)

Global Equipment Quantity Constraint

At any time $t$, the total number of equipment units of type $e$ being used must not exceed the total available number of equipment units of that type. The mathematical expression is:

$${\sum }_{O_{ij}\in J_t}{\sum }_{q\in E_e}{z}_{q,ij}\le M_e,\forall t\ge 0;\forall e\in M$$

(9)

(3)

Equipment Time Exclusivity Constraint

The same piece of maintenance equipment can only serve one operation at a time; i.e., the time intervals during which the same equipment is used for different operations cannot overlap. For any equipment unit $q$, if it is assigned to both operation $O_{ij}$ and operation $O_{kl}\left(O_{ij}\ne O_{kl}\right)$, then one operation must start after the other operation is completed. The mathematical expression is:

$$\left\{\begin{array}{c}{S}_{ij}+d_{ij}\le S_{kl}+T\left(1-b_{ij,kl}\right)\\ S_{kl}+d_{kl}\le S_{ij}+T{b}_{ij,kl}\end{array}\right.,\forall q\in E;\forall O_{ij},O_{kl}\in J,O_{ij}\ne O_{kl}$$

(10)

where $T$ is a sufficiently large positive real number, and $b_{ij,kl}$ is a binary variable indicating the usage order of the equipment. When $b_{ij,kl}=1$, operation $O_{ij}$ uses equipment $q$ before operation $O_{kl}$, $b_{ij,kl}=0$ otherwise. This constraint strictly guarantees the exclusivity of equipment usage, completely avoiding equipment usage conflicts.

2.3.4. Personnel Time Exclusivity Constraint

Similarly to equipment, the same maintenance person can only participate in one operation at a time; i.e., the time intervals during which the same person participates in different operations cannot overlap. For any person $p$, if they are assigned to both operation $O_{ij}$ and operation $O_{kl}\left(O_{ij}\ne O_{kl}\right)$, then one operation must start after the other operation is completed. The mathematical expression is:

$$\left\{\begin{array}{c}{S}_{ij}+d_{ij}\le S_{kl}+T\left(1-a_{ij,kl}\right)\\ S_{kl}+d_{kl}\le S_{ij}+T{a}_{ij,kl}\end{array}\right.,\forall p\in K;\forall O_{ij},O_{kl}\in J,O_{ij}\ne O_{kl}$$

(11)

where $a_{ij,kl}$ is a binary variable indicating the order of personnel participation. This constraint ensures that maintenance personnel are not scheduled to perform multiple tasks simultaneously, fully conforming to actual work rules.

2.3.5. Decision Variable Constraints

All binary decision variables involved in the model can only take values of 0 or 1, and time variables are non-negative real numbers. The mathematical expression is:

$$\left\{\begin{array}{c}{x}_{p,ij},y_{p,ij}^k,z_{q,ij},a_{ij,kl},b_{ij,kl}\in \left\{0,1\right\}\\ S_{kl},F_{ij}\ge 0,\forall i,p,q,j,l\end{array}\right.$$

(12)

3. Optimization Algorithm

This section will provide a detailed introduction to the optimization algorithm for solving the aircraft maintenance operation scheduling problem, including the encoding method, individual decoding process, and the design and implementation steps of the Reverse Learning Particle Swarm Optimization algorithm.

3.1. Encoding Method

Encoding quality directly affects algorithm search efficiency. This paper adopts an encoding method based on maintenance operation priority. Let the G-th generation population be $P_G=\left[X_{1,G},X_{2,G},\cdots ,X_{n,G}\right]$, where $n=1,2,\cdots ,N_P$, $N_P$ represents the number of individuals in the population. $X_{n,G}$ represents the encoding of the n-th individual in the G-th generation, defined as $X_{n,G}=\left[x_{11,n,G},x_{12,n,G},\cdots ,x_{NM,n,G},\cdots \right]$. Here, $x_{ij,n,G}$ represents the priority number of the j-th maintenance operation of the i-th aircraft in the n-th individual of the G-th generation. Values are randomly generated within the interval (0, 1). A smaller value indicates a higher execution priority for the operation. The encoding matrix is arranged by aircraft number and operation sequence. The encoding dimension for each individual is N × M (N: number of aircraft, M: number of operations per aircraft), ensuring a clear correspondence between operations and aircraft, as shown in Figure 1.

3.2. Individual Decoding of Populations

The core of the decoding process is to convert the priority encoding into a feasible maintenance schedule, achieving the goal of minimizing total maintenance time through a priority sorting and resource optimization allocation strategy. The specific steps are as follows:

1.

Initialize the completed operation set S (empty) and the schedulable operation set D (initially contains all operations without predecessor constraints).

2.

Sort the operations in D in ascending order based on their priority value. Select the operation $O_{ij}$ with the highest priority.

3.

Allocate resources using a left-shift strategy: Analyze idle time slots of maintenance personnel, equipment, and space. Through time series analysis and intersection operations, select the earliest available time window that meets the operation’s duration requirement.

4.

Determine the start time $S{T}_{ij}$ of operation $O_{ij}$ (take the maximum of the completion time of its predecessors and the earliest available time of the resources). Calculate the completion time $E{T}_{ij}=S{T}_{ij}+d_{ij}$.

5.

Move $O_{ij}$ from D to S. Update D by adding new operations whose predecessors have all been completed.

6.

Repeat steps 2–5 until all operations are scheduled.

In step 1, the schedulable operation set D is initialized to contain all operations without predecessor constraints. Each time step 5 is conducted, schedulable operations of which predecessors are completed is added to set D. These ensure that all operations in set D are feasible.

This decoding method effectively shortens operation waiting time by utilizing resource idle time, ensuring the minimization of total maintenance time.

3.3. Hybridized Particle Swarm Optimization

The basic Particle Swarm Optimization (PSO) algorithm is prone to issues like insufficient population diversity and premature convergence when applied to aircraft maintenance scheduling optimization, making it difficult to explore better schedules. Therefore, this paper introduces a reverse learning mechanism and partial random operations into the basic PSO framework, designing an improved RL-PSO algorithm. Its core improvement logic revolves around enhancing exploration capability and ensuring convergence efficiency.

The introduced reverse learning mechanism expands the population search space by constructing reverse particles. Dynamically generating reverse particles corresponding to the global best particle during iteration effectively improves the overall quality of the initial population and helps the population escape local optima. Simultaneously, partial random operations periodically perform local random resets on some non-optimal particles, preventing the population from over-converging to a single search area. By controlling the random intensity appropriately, the structure of good solutions already found is protected.

The two mechanisms work synergistically: reverse learning is responsible for global diversity expansion, and partial random operations are responsible for local area fine exploration. This ultimately achieves a balance between fast convergence and global optimum, adapting to the optimization requirement of minimizing time under multi-operation, multi-resource constraints in aircraft maintenance scheduling. The algorithm flowchart is shown in Figure 2.

Step 1: Population Initialization. First, according to the aircraft maintenance task, randomly initialize the positions and velocities of a particle population of size Np. Calculate the fitness function values corresponding to the population using the designed decoding method. Select the individual best ($Pbest$) and the global best particle ($Gbest$). Set the iteration counter G = 1.

Step 2: Following the Optimum. The particle swarm updates individuals based on the current $Pbest$ and $Gbest$ according to the particle velocity update rules.

Step 3: Reverse Learning Phase. The particle swarm generates reverse particles through the reverse learning mechanism, performs optimization judgment, and retains the better individuals.

Step 4: Random Particle Perturbation Phase. Based on a random perturbation probability, some position vectors of some particles are re-randomized. This increases the diversity of particles in the population, enhancing the global search capability.

Step 5: Termination Judgment Phase. Check whether the termination condition for the optimization iteration is met. If yes, end the iteration and output the best particle found and its corresponding maintenance schedule. Otherwise, set G = G + 1 and go to Step 2 to continue the optimization iteration.

3.3.1. Following the Optimum Phase

This phase operates on the principle of following the current optimal particles. It uses individual best and global best particles to guide the iteration of individual particles to improve the overall swarm level. Let the position of the i-th particle be $x_i={\left(x_{i1},x_{i2},\cdots x_{iM}\right)}^T$ and its velocity be $v_i={\left(v_{i1},v_{i2},\cdots ,v_{iM}\right)}^T$. Its individual extreme value ($Pbest$) is defined as $p_i={\left(p_{i1},p_{i2},\cdots ,p_{iM}\right)}^T$, and the global extreme value ($Gbest$) is defined as $p_g={\left(p_{g1},p_{g2},\cdots ,p_{gM}\right)}^T$. Particle $x_i$ updates its velocity and position according to the following formulas:

$$v_i\left(t+1\right)=w{v}_i\left(t\right)+c_1{r}_1\left(p_i\left(t\right)-x_i\left(t\right)\right)+c_2{r}_2\left(p_g\left(t\right)-x_i\left(t\right)\right)$$

(13)

$$x_i\left(t+1\right)=x_i\left(t\right)+a{v}_i\left(t+1\right)$$

(14)

where $x_i$, $v_i$, $p_i$ are the position, velocity, and individual best of the i-th particle, $p_g$ is the global best particle. $r_1$ and $r_2$ are random numbers uniformly distributed in [0, 1], $c_1$ and $c_2$ are acceleration coefficients, typically set to 2, $a$ is a constriction factor to control the weight of the velocity. To prevent the velocity from becoming too large, a maximum velocity $V_{max}$ is set. If $\left|v_i\right|>V_{max}$, then $\left|v_i\right|=V_{max}$. $w$ is the inertia weight, whose variation determines the global/local search capability during the following phase. Therefore, a random inertia weight strategy is adopted to change the single search pattern.

3.3.2. Reverse Learning Phase

The reverse learning mechanism is implemented through two steps: reverse particle generation and optimal solution selection. Considering the characteristics of aircraft maintenance scheduling encoding (particle positions are operation priority values in the (0, 1) interval), the specific design is as follows:

(1)

Definition and Calculation of Reverse Particles

For the i-th particle with position $x_i={\left(x_{i1},x_{i2},\cdots x_{iM}\right)}^T$ (M is the total number of operations, $x_{im}\in \left(0,1\right)$ represents the priority of the m-th operation, smaller value means higher priority), its reverse particle $x_i^{opp}={\left(x_{i1}^{opp},x_{i2}^{opp},\cdots x_{iM}^{opp}\right)}^T$ is calculated as:

$$x_{im}^{opp}=L+U-x_{im}$$

(15)

where $L=0$ and $U=1$ are the lower and upper bounds of the particle position. This formula ensures that the reverse particle is symmetrically distributed in the search space relative to the original particle, potentially covering areas not explored by the original particle.

(2)

Reverse Learning during Iteration Phase

To prevent the population from converging to a local optimum during iteration, the RL-PSO algorithm generates global best reverse particles every 10 generations. It extracts the current global best particle $Gbest$, calculates its reverse particle $Gbes{t}^{opp}$, compares the fitness of the two particles, and updates and retains the global best. By periodically generating the reverse of the global best, it explores potentially better solutions in the symmetric search area, enhancing global search capability.

3.3.3. Partial Random Strategy

The core of the partial random operation is controllable random perturbation. It only resets local dimensions of some non-optimal particles in the population, avoiding breaking good solutions while increasing flexibility in local exploration. By applying a random strategy judgment to particles, some of their position vectors are reset.

3.4. Overall Framework

The overall framework of the RL-PSO algorithm follows the core logic of encoding–decoding–iterative optimization, forming a complete optimization closed-loop adapted to the RCPSP model for aircraft maintenance scheduling. First, an initial population is generated using decimal priority encoding, avoiding the complex feasibility checks of traditional integer encoding. Next, through the decoding strategy of priority sorting and left-shift resource allocation, particle encodings are transformed into actual schedules that satisfy multiple constraints. Then, during the iteration process, the local exploitation of the following optimum phase, the global exploration of the reverse learning phase, and the local perturbation of the partial random operation are integrated to dynamically balance optimization accuracy and efficiency. Finally, upon meeting the termination condition, the schedule with the optimal total maintenance time is output. The entire framework closely connects the constraint requirements of the mathematical model with the algorithm improvement mechanisms, ensuring that each step serves the core objective of minimizing the total maintenance period under multi-resource constraints, achieving deep synergy between the model, encoding, and algorithm.

4. Simulation Validation

This section evaluates the effectiveness of the proposed optimization model and algorithm through simulation validation. Different simulation parameters and scenarios are set, the performance of different algorithms is compared, and the results are analyzed.

4.1. Data Source

The data of operation times and resource requirements used in the simulation is based on the research on carrier-based aircraft maintenance scheduling [22].

4.2. Parameter Setting Configuration

This experiment is based on the MATLAB R2021a software platform, with a hardware environment of Intel^® Core™ i7-14650HX processor (2.20 GHz) and 32 GB RAM. The simulation maintenance case involves a total of 20 aircraft labeled J1–J20 of three types. There are four maintenance types in total, with the number of aircraft requiring maintenance for each type being [5, 5, 5, 5]. The process includes 13 maintenance operations (including dummy start and end operations). The maintenance process for each aircraft type follows a node-type precedence network. Operation times refer to actual maintenance standards, as detailed in Figure 3.

The operation times for each maintenance task type are shown in

Table 2. Operation times and resource requirements by maintenance task type.

Maintenance Task Type	Operation Time (min)
	1	2	3	4	5	6	7	8	9	10	11	12	13
1	0	19	16	2	10	18	4	15	13	5	15	9	0
2	0	6	12	16	16	4	5	10	11	6	6	5	0
3	0	17	12	15	23	12	18	2	6	15	4	1	0
4	0	14	4	8	9	17	23	1	12	15	2	2	0

. The required support equipment/workshop and workspace requirements for each operation are shown in

Table 3. Resource requirements for maintenance support operations.

Operations	Skilled Personnel (Number)	Maintenance Equipment
1	--	--
2	Type 3 skilled personnel (2)	--
3	Type 1 skilled personnel (2)	--
4	Type 4 skilled personnel (2)	Type 1 equipment
5	Type 2 skilled personnel (2)	--
6	Type 3 skilled personnel (2)	--
7	Type 3 skilled personnel (1)	--
8	Type 1 skilled personnel (2)	Type 3 equipment
9	Type 1 skilled personnel (2)	Type 3 equipment
10	Type 4 skilled personnel (1)	--
11	Type 2 skilled personnel (1)	Type 2 equipment
12	Type 4 skilled personnel (2)	Type 1 equipment
13	--	--

. In this case, the number of technical personnel for the four types is [6, 4, 6, 5], and the quantities for the three types of maintenance equipment are Ne = [3, 2, 4].

4.3. Performance Comparison

To comprehensively verify the solution performance of the proposed RL-PSO algorithm for the aircraft maintenance scheduling problem, three mainstream metaheuristic algorithms were selected for comparison: Standard Particle Swarm Optimization (PSO), Simulated Annealing (SA), and Teaching–Learning-Based Optimization (TLBO). All of these mainstream metaheuristic algorithms are based on the serial schedule generation scheme (SSGS), which is the baseline method for the RCPSP, for decoding.

All algorithms were tested on the 20-aircraft maintenance case using the same simulation environment and parameter settings. The superiority of RL-PSO was evaluated from three dimensions: optimization accuracy, convergence speed, and stability.

To ensure fairness, core parameters such as population size, number of iterations, and number of evaluations were kept consistent across all algorithms. The specific parameter settings are as follows: Population size $N=30$, learning factors $c_1=c_2=2$, inertia weight $\omega =0.5$, partial re-randomization probability $F=0.3$, particle velocity limits $V_{max}=-V_{min}=0.2$, the class size in TLBO $Np=30$, elite teacher group size $Nt=1$. The initial temperature in SA $T_{max}=100$, final temperature $T_{min}=0$, cooling rate $Cr=1$. All algorithms use the number of evaluations $Q=100$ as the iteration termination criterion. Each algorithm was run independently 30 times. After simulation optimization, the experimental results are summarized in

Table 4. Algorithm comparison.

Algorithm	Optimal (min)	Average (min)	Std (min)	Time Consumption (s)
PSO	346	349.7	2.1932	63.33
SA	348	351.8	3.2496	55.78
TLBO	338	346.5	4.3643	199.82
RL-PSO	332	344.4	2.8843	92.07

, and the iterative optimization curves are shown in Figure 4.

The experimental results show that the RL-PSO algorithm performs best in the aircraft maintenance scheduling scenario. Its solution quality is better than the standard traditional algorithms, demonstrating superior optimization accuracy and stability, except for a slightly higher std than the PSO algorithm. Furthermore, the algorithm converges faster, balancing optimization effectiveness and computational cost effectively, with average runtime within a reasonable range.

Analyzed from its mechanism, standard PSO suffers from premature convergence and loss of population diversity due to its sole reliance on following the global and personal best particles, while SA and TLBO, though capable of escaping local optima to some extent, lack explicit mechanisms to maintain search space coverage under complex multi-resource constraints. With the synergistic effect of the reverse learning mechanism and partial random operations, the RL-PSO method successfully addresses the issues of insufficient population diversity and premature convergence in traditional algorithms. These advantages are further reflected in the actual schedules generated. Gantt charts for the resulting maintenance personnel and equipment baseline schedules for the 20 aircraft are shown in Figure 5 and Figure 6.

4.4. Scalability Test

To verify the scalability and applicability of the RL-PSO algorithm for different scales of aircraft maintenance tasks, five test cases with varying numbers of aircraft (10, 20, 30, 40, 50) were designed. Each aircraft had 13 maintenance operations (consistent with the case in Section 4.1). Resource allocations were scaled proportionally: technical personnel numbers increased by a factor of 1:1.5, and equipment numbers increased by a factor of 1:1.2. For each case, 30 independent runs were conducted.

The optimal solution, average solution, standard deviation, and average runtime were recorded to evaluate the algorithm’s scalability. The average and the standard deviation (for better display, the standard deviation is twice the actual value) of the experimental results are shown in Figure 7. The test results are summarized in

Table 5. Algorithm comparison under different aircraft scales.

Number of Aircrafts	Optimal (min)	Average (min)	Std (min)	Time Consumption (s)
10	261	266.4	2.4576	46.73
20	332	344.4	2.8843	92.07
30	369	371.6	2.0100	150.79
40	433	435.3	1.6763	231.97
50	431	433.3	1.3454	424.90

.

Based on the test data, the RL-PSO algorithm shows outstanding scalability. As the number of aircraft increases from 10 to 50 (total operations scaling 5-fold), the average runtime increases smoothly from 46.73 s to 424.90 s, verifying the effective control of computational complexity by the decimal encoding. The optimal and average solutions increase synchronously with scale, and the standard deviation remains within a narrow range of about 3 min, reflecting the role of reverse learning and partial random operations in ensuring optimization stability. Even in the large-scale 50-aircraft scenario, the algorithm’s runtime is only 7.1 min, well within practical scheduling requirements. In summary, the RL-PSO algorithm maintains efficient and stable performance across different scales of maintenance tasks, making it adaptable to scheduling needs ranging from small to large fleets.

4.5. Sensitivity Analysis

A sensitivity analysis is conducted to further analyze the influence of personnel and equipment on the RL-PSO algorithm.

In the personnel sensitivity analysis, two test cases with halved and doubled skilled personnel were designed and for each case, 30 independent runs were conducted. Gantt charts for the resulting maintenance personnel and equipment baseline schedules for halved and doubled personnel are shown in Figure 8, Figure 9, Figure 10 and Figure 11.

The optimal solution, average solution and standard deviation were used to evaluate the sensitivity of personnel. The test results are summarized in

Table 6. Personnel sensitivity tests.

Number of Personnel	Optimal (min)	Average (min)	Std (min)
11 (halved)	556	574.8	5.8085
21 (origin)	332	344.4	2.8843
42 (doubled)	287	291.6	2.2163

.

In the equipment sensitivity analysis, two test cases with halved and doubled equipment were designed and for each case, 30 independent runs were conducted. Gantt charts for the resulting maintenance personnel and equipment baseline schedules for halved and doubled equipment are shown in Figure 12, Figure 13, Figure 14 and Figure 15.

The optimal solution, average solution and standard deviation were used to evaluate the sensitivity of equipment. The test results are summarized in

Table 7. Equipment sensitivity tests.

Number of Equipment	Optimal (min)	Average (min)	Std (min)
5 (halved)	422	433.5	4.4850
9 (origin)	332	344.4	2.8843
18 (doubled)	335	339.0	2.7506

.

It can be seen from Table 6 and Table 7 that as the number of personnel or equipment increases, the optimal solution, average solution and standard deviation drops. This indicates that with more personnel or equipment, the RL-PSO algorithm can achieve solutions with better stability. Besides, the optimal solution, average solution and standard deviation of the doubled equipment are close to those of the origin equipment, which indicates that increasing the number of personnel or equipment has a marginal effect, as the number of maintenance operations is limited.

4.6. Discussion

The simulation results presented in Section 4.3, Section 4.4 and Section 4.5 demonstrate the effectiveness, scalability, and sensitivity of the proposed RL-PSO algorithm for carrier-based aircraft maintenance scheduling. This discussion interprets the key findings, compares them with the existing literature, and addresses the practical implications and limitations of the study.

The RL-PSO algorithm achieves the best optimal (332 min) and average (344.4 min) among all compared algorithms (PSO, SA, TLBO), demonstrating superior optimization accuracy and convergence speed. Although its standard deviation (2.8843) is slightly higher than that of PSO (2.1932), this trade-off is acceptable as the primary objective is minimization. The synergy between reverse learning and partial random operations effectively mitigates premature convergence and enhances population diversity, explaining RL-PSO’s consistent advantage over TLBO and other methods.

As the number of aircraft increases from 10 to 50, RL-PSO maintains stable performance, with average runtime rising smoothly from 46.73 s to 424.90 s (approx. 7.1 min), well within practical scheduling limits. The standard deviation remains within a narrow range (1.3–2.9 min), indicating excellent stability. The near-linear runtime growth confirms that the decimal encoding scheme effectively controls computational complexity, making RL-PSO suitable for fleet-level maintenance planning across varying scales.

Personnel and equipment sensitivity tests reveal clear diminishing returns: halving personnel increases by 67.5% (from 332 to 556 min), while doubling personnel reduces it by only 13.6% (from 332 to 287 min); similarly, halving equipment increases by 27.1% (from 332 to 422 min), whereas doubling equipment yields negligible improvement (from 335 min to 332 min). This marginal effect suggests that beyond a critical threshold, additional resources provide limited scheduling benefits, offering practical guidance for carrier fleet resource allocation decisions.

5. Conclusions

This paper focused on the aircraft maintenance scheduling problem characterized by coupled multi-step procedures and multi-resource constraints. Addressing pain points of traditional algorithms such as insufficient population diversity, premature convergence, and complex encoding, a complete scheduling optimization solution was constructed.

The study adopted the RCPSP model to build a mathematical formulation, comprehensively covering core constraints such as operation sequences, personnel–skill matching, and equipment exclusivity, accurately capturing the complex characteristics of maintenance scheduling. To overcome the defects of traditional integer full-permutation encoding, a decimal priority encoding within the (0, 1) interval was proposed, avoiding complex feasibility checks during evolution, reducing computational complexity while enriching the search space. Based on this, an RL-PSO algorithm integrating a reverse learning mechanism and partial random operations was designed. The global search space is expanded via reverse particles, while local random perturbations enhance exploration flexibility, effectively balancing optimization accuracy and efficiency. Simulation validation demonstrated that the RL-PSO algorithm performs excellently in the test scenarios. RL-PSO achieves optimization accuracy of 332 min, convergence speed of 92.07 s, and stability of 2.8843 min in standard deviation, which are superior compared to standard PSO, Simulated Annealing, and Teaching–Learning-Based Optimization combined with the serial schedule generation scheme (SSGS). Its stability and convergence speed are superior to traditional heuristic algorithms, maintaining stable optimization accuracy and adapting to scheduling needs for different fleet sizes. A sensitivity analysis proved that increasing the number of persons or equipment can reduce the optimal solution time, but there is a marginal effect which requires a trade-off.

Although certain research contributions have been made, some limitations need further refinement: The study focused solely on minimizing total maintenance time without considering multi-objective optimization needs, and validation was conducted mainly in static scenarios. Future work will expand to multi-objective models, comprehensively considering resource utilization and cost. The algorithm’s dynamic response mechanism will be improved to adapt to dynamic scenarios such as sudden failures. Furthermore, engineering validation using actual maintenance data will be pursued to promote the application of the research outcomes.