# An Improved Optimization Algorithm for Aeronautical Maintenance and Repair Task Scheduling Problem

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## Abstract

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## 1. Introduction

- i.
- Slack of distributed constraints. Commercial airlines must manage a complex network of routes and the complex coupling between distributed workshops and routes, whereas the majority of military aeronautical maintenance tasks are concentrated in ship-based hanging bays on large sea platforms;
- ii.
- Differences in the maintenance cycles, civil aeronautical maintenance optimization models, and methods applied to solve problems in fleet decision optimization. A commercial fleet is highly stable and has longer maintenance cycle intervals than a military naval aircraft fleet. The carrier-based aircraft fleet aeronautical maintenance tasks investigated in this study involve military tasks with urgent task requirements [10];
- iii.
- Differences in maintenance goals. The literature on civil aeronautical maintenance mainly focuses on profitability, and the optimizations mainly consider economic benefits [11], such as balancing the maintenance cost of the fleet with the amount of hangar resources [12] or the labor costs of maintenance personnel [13]. By contrast, military aviation maintenance tasks are optimized to avoid delaying military response and to ensure appropriate conduct in both combat and training tasks. In other words, the goal is to positively impact operational effectiveness and subsequent warfare.

- i.
- Mission maintenance aspects: Han et al. [15] simulated mission maintenance for deck crews, with the number of aircraft ranging from five to nine. However, they considered only a single maintenance mode and not multimode/hybrid situations, such as preventative maintenance and failure repairs, and realistic constraints, such as maintenance coverage, parallel maintenance capacity, and maintenance workstation space. Thus, the simulation differs substantially from an actual task;
- ii.
- Optimization/scheduling of fleet maintenance tasks: Most studies on optimizing fleet maintenance tasks have focused on minimizing the maintenance completion time [16]. However, Raju et al. [17] defined a military aircraft availability index for fleet wave sortie availability; the index comprised the ratio of the number of aircraft in mission-capable states to the total number of aircraft in the fleet at a given time point. The military maintenance and operational characteristics of naval aircraft were used for closer integration by the index;
- iii.
- Optimization/Scheduling of resources: The main considerations in terms of resources have involved personnel and personnel scheduling strategies [18], resource constraints for maintenance personnel [19], and maintenance personnel time balancing [20]. No studies have been conducted to integrate limited maintenance resources, such as maintenance equipment, workshops, and space, in the models.

## 2. Problem Statement

#### 2.1. Maintenance Process

#### 2.2. Maintenance Personnel and Skills

#### 2.3. Maintenance Equipment, Workshop, and Workspace

## 3. Mathematical Model for Aeronautical Maintenance and Repair Task Scheduling Problem (AMRSP)

#### 3.1. Problem Assumptions

- i.
- The MR tasks are known with certainty and do not consider the interference of dynamic factors.
- ii.
- The MR process cannot be preempted or interrupted once started.
- iii.
- The maintenance skills are adapted to each aircraft’s MR task mode.
- iv.
- The transit time in the hangar bay is ignored.
- v.
- An adequate reserve of fixed-resource station resources is available.

#### 3.2. Constraints

#### 3.3. Objective Function

- (1)
- Maximizing fleet-wave availability (WA)

- (2)
- Minimizing the personnel load variance (PLV) results in

## 4. Algorithm for AMRSP

#### 4.1. Encoding and Serial Scheduling Generation Scheme (SSGS)

#### 4.2. Improved Teaching-Learning-Based Optimization (ITLBO) Main Loop

#### 4.2.1. Teaching Phase

#### 4.2.2. Learning Phase

#### 4.2.3. Assistant Teaching Phase

- (1)
- Fitness-distance ranking ratio

- (2)
- Assistant Teacher Teaching

#### 4.3. Complexity Analysis

## 5. Simulation Case Analysis

#### 5.1. Maintenance and Repair (MR) Task Case Generation

_{1}indicates the type of power supply station. As for the configuration of the maintenance workshops, owing to the space limitations of the compartment around the hangar bay, each workshop is equipped with one maintenance workshop covering the entire hangar bay. Ke

_{2–5}denote the aeronautical machine repair, oil and fluid inspection, ordnance maintenance, and electronic equipment maintenance workshops, respectively, and the number of parallel operations are $\left[N{e}_{1},N{e}_{2},N{e}_{3},N{e}_{4},N{e}_{5}\right]=\left[\infty ,3,2,1,4\right]$ for each workshop. The number of resources in this category usable for the parking space is indicated by [.]. The operation duration, resources, and skills required for each maintenance operation are shown in Table 4. In Table 4, Kc

_{1–4}denote special equipment, avionics, ordnance, and machinery specialties, respectively. In addition, the number of personnel is configured as $\left[5,6,4,10\right]$. Special equipment is set to be compatible with avionics, ordnance, and machinery. The first four personnel in each profession have corresponding and compatible skills. Bold numbers indicate that the operation needs two personnel. The workstation space constraint Ks considers the cockpit space; “1” indicates that the number of personnel able to work in parallel is one, and “-” indicates that there is no demand for such resources.

#### 5.2. Simulation Comparison Analysis

#### 5.2.1. Algorithm Comparison

#### 5.2.2. Adaptation Verification of Algorithms

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Example of fitness-distance ranking ratio. (

**a**) Individual position distribution; (

**b**) Fitness and distance distribution between teachers and assistant teachers.

**Figure 9.**Flowchart for improved teaching-learning-based optimization algorithm with a serial scheduling generation scheme (ITLBO-S).

Notations | Descriptions |
---|---|

$I$ | $\mathrm{The}\mathrm{set}\mathrm{of}\text{carrier-based}\mathrm{aircraft},I=\left\{1,2,\cdots ,i,\cdots ,\left|I\right|\right\}$. |

${p}_{i}$ | $\mathrm{The}\mathrm{parking}\mathrm{spot}\mathrm{of}\mathrm{the}i\mathrm{th}$ carrier-based aircraft. |

${J}_{i}$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{maintenance}\mathrm{operations}\mathrm{of}\mathrm{the}i\mathrm{th}$$\text{carrier-based}\mathrm{aircraft},{J}_{i}=\left\{1,2,\cdots ,j,\cdots ,\left|{J}_{i}\right|\right\}$. |

$J$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{maintenance}\mathrm{operations}\mathrm{of}\mathrm{the}\mathrm{fleet},J=\left\{\left(i,j\right)\left|i\in I,j\in {J}_{i}\right.\right\}$. |

${A}_{t}$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{all}\mathrm{maintenance}\mathrm{operations}\mathrm{of}\mathrm{the}\mathrm{fleet}\mathrm{in}\mathrm{the}\mathrm{execution}\mathrm{state}\mathrm{at}\mathrm{time}\mathrm{point}t$. |

${A}_{it}$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{maintenance}\mathrm{operations}\mathrm{of}\mathrm{the}i\mathrm{th}$$\text{carrier-based}\mathrm{aircraft}\mathrm{in}\mathrm{the}\mathrm{execution}\mathrm{state}\mathrm{at}\mathrm{time}\mathrm{point}t$. |

${O}_{ij}$ | $\mathrm{The}j\mathrm{th}$$\mathrm{maintenance}\mathrm{operation}\mathrm{of}\mathrm{the}i\mathrm{th}$ carrier-based aircraft. |

$P{s}_{ij}$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{immediately}\mathrm{preceding}\mathrm{operations}\mathrm{of}{O}_{ij}$. |

$E{x}_{i}$ | $\mathrm{The}\mathrm{tethering}\mathrm{completion}\mathrm{time}\mathrm{of}\mathrm{the}i\mathrm{th}$ carrier-based aircraft. |

${d}_{ij}$ | $\mathrm{The}\mathrm{operation}\mathrm{duration}\mathrm{of}{O}_{ij}$. |

$BM$ | A sufficiently large real number. |

$Lp$ | The set of maintenance personnel. |

$Le$ | The set of maintenance equipment/workshops. |

$Kc$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{skill}\mathrm{categories}\mathrm{of}\mathrm{the}\mathrm{maintenance}\mathrm{personnel},Kc=\left\{1,2,\cdots ,\left|Kc\right|\right\}$. |

$Ke$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{skill}\mathrm{categories}\mathrm{for}\mathrm{maintenance}\mathrm{equipment}/\mathrm{workshops},Ke=\left\{1,2,\cdots ,\left|Ke\right|\right\}$. |

$Ks$ | $\mathrm{The}\mathrm{set}\mathrm{of}\mathrm{skill}\mathrm{categories}\mathrm{at}\mathrm{the}\mathrm{maintenance}\mathrm{workspace},Ks=\left\{1,2,\cdots ,\left|Ks\right|\right\}$. |

$r{c}_{ijk}$ | $\mathrm{An}\mathrm{indicator}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{has}\mathrm{a}\mathrm{demand}\mathrm{for}\mathrm{the}k\mathrm{th}$ skill category, whereas 0 indicates otherwise. |

$r{e}_{ijk}$ | $\mathrm{An}\mathrm{indicator}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{has}\mathrm{a}\mathrm{demand}\mathrm{for}\mathrm{the}k\mathrm{th}$ maintenance equipment/workshop category, whereas 0 indicates otherwise. |

$r{s}_{ijk}$ | $\mathrm{An}\mathrm{indicator}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{has}\mathrm{a}\mathrm{demand}\mathrm{for}\mathrm{the}k\mathrm{th}$ maintenance workspace, whereas 0 indicates otherwise. |

${\lambda}_{kl}^{p}$ | $\mathrm{An}\mathrm{indicator}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}\mathrm{the}l\mathrm{th}$$\mathrm{of}\mathrm{the}k\mathrm{th}$$\mathrm{maintenance}\mathrm{equipment}/\mathrm{workshop}\mathrm{category}\mathrm{has}\mathrm{a}\mathrm{reachability}\mathrm{relation}\mathrm{with}p$, whereas 0 indicates otherwise. |

$n{s}_{ik}$ | $\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{personnel}\mathrm{who}\mathrm{can}\mathrm{work}\mathrm{in}\mathrm{parallel}\mathrm{with}\mathrm{the}i\mathrm{th}$$\text{carrier-based}\mathrm{aircraft}k\mathrm{th}$ workspace category. |

$N{e}_{kl}$ | $\mathrm{The}\mathrm{number}\mathrm{of}\mathrm{operations}\mathrm{that}\mathrm{can}\mathrm{be}\mathrm{accommodated}\mathrm{in}\mathrm{parallel}\mathrm{in}\mathrm{the}l\mathrm{th}$$\mathrm{workshop}\mathrm{of}\mathrm{the}k\mathrm{th}$ category. |

$S{m}_{ij}$ | $\mathrm{A}\mathrm{decision}\mathrm{variable}\mathrm{indicating}\mathrm{the}\mathrm{start}\mathrm{time}\mathrm{of}{O}_{ij}$. |

$E{m}_{ij}$ | $\mathrm{A}\mathrm{decision}\mathrm{variable}\mathrm{indicating}\mathrm{the}\mathrm{end}\mathrm{time}\mathrm{of}{O}_{ij}$. |

$X{p}_{ijkl}$ | $\mathrm{A}\mathrm{decision}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{the}l\mathrm{th}$$\mathrm{maintenance}\mathrm{personnel}\mathrm{using}\mathrm{the}k\mathrm{th}$ skill category, whereas 0 indicates otherwise. |

$X{e}_{ijkl}$ | $\mathrm{A}\mathrm{decision}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{the}l\mathrm{th}$$\mathrm{of}\mathrm{the}k\mathrm{th}$ maintenance equipment/workshop category, whereas 0 indicates otherwise. |

$Y{p}_{ijeg}$ | $\mathrm{A}\mathrm{decision}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{the}\mathrm{same}\mathrm{maintenance}\mathrm{personnel}\mathrm{as}{O}_{eg}$$,\mathrm{and}{O}_{ij}$$\mathrm{is}\mathrm{prioritized}\mathrm{over}{O}_{eg}$, whereas 0 indicates otherwise. |

$Y{e}_{ijeg}$ | $\mathrm{A}\mathrm{decision}\mathrm{variable}\mathrm{valued}0\mathrm{or}1,\mathrm{where}1\mathrm{indicates}\mathrm{that}{O}_{ij}$$\mathrm{is}\mathrm{assigned}\mathrm{to}\mathrm{the}\mathrm{same}\mathrm{maintenance}\mathrm{equipment}/\mathrm{workshop}\mathrm{as}{O}_{eg}$$,\mathrm{and}{O}_{ij}$$\mathrm{is}\mathrm{prioritized}\mathrm{over}{O}_{eg}$, whereas 0 indicates otherwise. |

P. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MR Tasks | Carrier-based aircraft no.; Ex (min) MR tasks modes | |||||||||||||

Case 1 | A; 2 2 | B; 8 3 | C; 0 5 | D; 0 4 | E; 9 1 | F; 16 1 | G; 0 6 | H; 0 5 | I; 3 1 | J; 15 2 | - | - | - | - |

Case 2 | A; 2 2 | B; 8 3 | C; 0 5 | D; 0 4 | E; 9 1 | F; 16 1 | G; 0 6 | H; 0 5 | I; 3 1 | J; 15 2 | K; 21 5 | L; 22 3 | - | - |

Case 3 | A; 2 2 | B; 8 3 | C; 0 5 | D; 0 4 | E; 9 1 | F; 16 1 | G; 0 6 | H; 0 5 | I; 3 1 | J; 15 2 | K; 21 3 | L; 22 4 | M; 27 4 | N; 29 4 |

P. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ke_{1} | [3] | [4] | [3,5] | [5] | [9] | [10] | [9] | [10] | [6] | [1] | [3] | [7] | [2] | [8] |

Ke_{2–5} | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] | [1] |

MR Task Modes | Operation No. | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | ||

Operation Duration (min) | |||||||||||||||||||||

Mechanical failure | 0 | 0 | 0 | 0 | 24 | 0 | 0 | 0 | 0 | 44 | 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 | |

Avionics system failure | 0 | 0 | 0 | 0 | 19 | 0 | 0 | 0 | 0 | 53 | 0 | 0 | 0 | 0 | 0 | 22 | 0 | 0 | 0 | 0 | |

Special equipment failure | 0 | 0 | 0 | 0 | 26 | 0 | 0 | 0 | 0 | 47 | 0 | 0 | 0 | 0 | 0 | 17 | 0 | 0 | 0 | 0 | |

Maintenance after 25 flight hours | 0 | 18 | 30 | 8 | 6 | 8 | 10 | 6 | 8 | 15 | 20 | 0 | 16 | 18 | 0 | 3 | 10 | 8 | 6 | 0 | |

Maintenance after 50 flight hours | 0 | 25 | 45 | 8 | 8 | 8 | 12 | 6 | 8 | 30 | 30 | 26 | 26 | 28 | 16 | 8 | 18 | 10 | 10 | 0 | |

Maintenance after 100 flight hours | 0 | 34 | 66 | 10 | 12 | 10 | 15 | 10 | 12 | 48 | 40 | 45 | 33 | 44 | 46 | 16 | 26 | 18 | 14 | 0 | |

Required resource type | Kc | - | 4 | 4 | 3 | 1,2,3,4 | 2 | 2 | 1 | 1 | 1,2,4 | 4 | 4 | 2 | 1 | 3 | 1,2,4 | 2 | 1 | 3 | - |

Ke | - | - | - | - | 1 | - | 1 | - | 1 | 2,5 | 3 | 3 | 5 | 5 | 4 | 1 | 1 | 1 | 1 | - | |

Ks | - | - | - | - | 1 | - | 1 | - | 1 | - | - | - | - | - | - | 1 | 1 | 1 | 1 | - |

Cases | Objective Functions | Evaluating Indicators | Algorithms | |||
---|---|---|---|---|---|---|

ITLBO-S | TLBO | DE | PSO | |||

Case 1 | WA | Best. | 0.750 | 0.720 | 0.720 | 0.720 |

Avg. | 0.750 | 0.709 | 0.714 | 0.713 | ||

Std. | 0 | 0.010 | 0.008 | 0.009 | ||

PLV | Best. | 52.382 | 59.380 | 66.107 | 65.866 | |

Avg. | 53.720 | 64.663 | 69.068 | 69.424 | ||

Std. | 1.264 | 3.383 | 1.999 | 1.802 | ||

Case 2 | WA | Best. | 0.710 | 0.690 | 0.600 | 0.620 |

Avg. | 0.702 | 0.648 | 0.592 | 0.598 | ||

Std. | 0.012 | 0.021 | 0.019 | 0.028 | ||

PLV | Best. | 5.626 | 16.186 | 25.946 | 20.026 | |

Avg. | 10.612 | 33.983 | 51.695 | 55.754 | ||

Std. | 5.113 | 13.326 | 21.837 | 26.416 | ||

Case 3 | WA | Best. | 0.650 | 0.620 | 0.600 | 0.600 |

Avg. | 0.627 | 0.620 | 0.582 | 0.584 | ||

Std. | 0.015 | 0 | 0.031 | 0.018 | ||

PLV | Best. | 19.280 | 24.480 | 66.582 | 60.720 | |

Avg. | 45.088 | 54.560 | 83.964 | 100.624 | ||

Std. | 12.503 | 21.056 | 16.966 | 31.059 |

ObjectiveFunctions | Evaluating Indicators | MR Tasks | |||

Case 1 | Case 2 | Case 3 | |||

WA | Best. | 0.520 | 0.492 | 0.450 | |

Avg. | 0.520 | 0.461 | 0.430 | ||

Std. | 0 | 0.023 | 0.008 | ||

PLV | Best. | 57.862 | 16.580 | 6.400 | |

Avg. | 65.7 | 21.337 | 11.447 | ||

Std. | 3.276 | 2.789 | 2.686 | ||

Time/s | Avg. | 173.2 | 240.8 | 321.8 |

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## Share and Cite

**MDPI and ACS Style**

Li, C.; Zhang, Y.; Su, X.; Wang, X.
An Improved Optimization Algorithm for Aeronautical Maintenance and Repair Task Scheduling Problem. *Mathematics* **2022**, *10*, 3777.
https://doi.org/10.3390/math10203777

**AMA Style**

Li C, Zhang Y, Su X, Wang X.
An Improved Optimization Algorithm for Aeronautical Maintenance and Repair Task Scheduling Problem. *Mathematics*. 2022; 10(20):3777.
https://doi.org/10.3390/math10203777

**Chicago/Turabian Style**

Li, Changjiu, Yong Zhang, Xichao Su, and Xinwei Wang.
2022. "An Improved Optimization Algorithm for Aeronautical Maintenance and Repair Task Scheduling Problem" *Mathematics* 10, no. 20: 3777.
https://doi.org/10.3390/math10203777