# A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Our Contributions

- (1)
- The three-layer networks, including air route network, city-pair airline network and flight operation network, are built to model the air transportation system from physical facilities to application, where the air route network is regarded as the physical layer, and the city-pair airline network and flight operation network are the application layers. In addition, the topology characteristics of the three-layer networks are analysed using the complex network theory.
- (2)
- A simulated annealing-based method is proposed to optimize the routing paths of each city-pair airline based on the three-layer air transport networks, such that the congestion issue of air route segments in the air route network can be reduced.

## 2. Multi-Layer Air Transport Networks

#### 2.1. Construction of Air Transport Networks

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- (1)
- t < ${t}_{f}^{dep}$ and ${t}_{f}^{arr}$ < $t+T$
- (2)
- ${t}_{f}^{dep}$ < t and ${t}_{f}^{arr}$ < $t+T$
- (3)
- t < ${t}_{f}^{dep}$ and $t+T$ < ${t}_{f}^{arr}$
- (4)
- ${t}_{f}^{dep}$ < t and $t+T$ < ${t}_{f}^{arr}$

#### 2.2. Mapping of Air Transport Networks

#### 2.2.1. Coupling Relationship between ${G}_{AN}$ and ${G}_{ARN}$

#### 2.2.2. Coupling Relationship between ${G}_{AN}$ and ${G}_{FN}$

#### 2.2.3. Coupling Relationship between ${G}_{ARN}$ and ${G}_{FN}$

#### 2.3. Problem Description

- (1)
- JiNan → P200 → P64 → P223 → P181 → ATV → P42 → Hefei
- (2)
- JiNan → P86 → P182 → ATV → P42 → Hefei
- (3)
- JiNan → P197 → P182 → ATV → P42 → Hefei

- (1)
- JiNan → P197 → NJHLD → SZPOI → ShangHai
- (2)
- JiNan → P58 → WZPYI → SZPOI → ShangHai
- (3)
- JiNan → P60 → YC → SZPOI → ShangHai

- -
- the air route network, ${G}_{ARN}$, which is the basis of national airspace and will affect the flight distance and operational capacity of the air transportation system;
- -
- the city-pair airline network, ${G}_{AN}$, which represents the existence of flights between a pair of airports or not;
- -
- the flight operation network, ${G}_{FN}^{t}$, at different time intervals $[t,t+1)$, $1\le t\le 24$, which represents the air traffic flows based on the timetable of flights;
- -
- the coupling relationship between the three-layer air transport networks, as discussed in Section 2.2.
- -
- the candidate routing paths ${S}_{crp}^{i,j}$ for the each city-pair airline ${l}_{i,j}$ in ${G}_{AN}$.

## 3. Routing Paths Optimization of City-Pair Airlines

- (a)
- A city-pair airline ${l}_{rev}$ of ${G}_{AN}$ is selected, randomly. Furthermore, the traffic flows in the corresponding routing path (air route segments) on air route network ${G}_{ARN}$ should be updated.
- (b)
- Each candidate routing path in the set ${S}_{crp}^{i,j}$ is evaluated for the removed airline ${l}_{rev}$, and the best one is selected.
- (c)
- According to the selected best routing path, the traffic flows on the corresponding air route segments are updated actually.

#### Objective Function

## 4. Experiments

#### 4.1. Analysis of Aviation Networks

#### 4.1.1. Air Route Network

#### 4.1.2. City-Pair Airline Network

#### 4.2. Optimization Results

- (1)
- “$Default$”, which takes the routing path used by air traffic managers in practice;
- (2)
- “$Shortest$”, which takes the shortest length routing path from the candidate set ${S}_{crp}^{i,j}$ for airline ${l}_{i,j}$;
- (3)
- “$Random$”, which takes a routing path for airline ${l}_{i,j}$ from its candidate set ${S}_{crp}^{i,j}$;
- (4)
- “$Proposed$”, which is the proposed simulated annealing-based method to select the optimal routing paths.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${G}_{ARN}$ | Graph of the air route network |

${G}_{FN}^{t}$ | Graph of the flight operation network at time t |

${G}_{AN}$ | Graph of the city-pair airline network |

${r}_{i,j}$ | air route segment connecting city i and city j |

${l}_{i,j}$ | city-pair airline connecting city i and city j |

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**Figure 2.**The flight operation network ${G}_{FN}^{t}$ in different time intervals based on the flight schedule of one day in 2019.

**Figure 3.**An example of the mapping relations of a three-layer air transport network with six cities, JiNan, HeFei, YanCheng, NanTong, NanJing, and ShangHai. (

**a**–

**c**) show the flight operation networks in the time intervals of 9:00–10:00, 18:00–19:00, and 21:00–22:00, respectively. (

**d**) shows the city-pair airline network among these six cities. (

**e**) shows the air route network of China, and (

**f**) shows the routing path on the air route network for the city-pair airline “Hefei → Shanghai ”.

**Figure 6.**The first ten traffic flows of an air route network in the time interval 9:00–10:00 under different routing-path strategies.

**Table 1.**Three different factors of air transportation network and most researches hardly cover flight operation network.

Factors Discussed in Research | Air Route | City-Pair Airline | Flight Operation |
---|---|---|---|

[8,10,12,13,15,16,17,18,19,20,21,22,23] | √ | - | - |

[9,25,27] | - | √ | - |

[2,24,26,28,29] | √ | √ | - |

This paper | √ | √ | √ |

Parameters | 2012 [24] | 2015 | 2018 | 2019 |
---|---|---|---|---|

Nodes | 147 | 198 | 227 | 233 |

Edges | 1055 | 2657 | 3919 | 4498 |

Average degree | 14.35 | 16.83 | 22.52 | 24.98 |

Average shortest path length | 2.20 | 2.29 | 2.29 | 2.25 |

Diameter | 4 | 3 | 3 | 3 |

Mixing coefficient | −0.15 | −0.45 | −0.38 | −0.36 |

Clustering coefficient | 0.79 | 0.62 | 0.55 | 0.58 |

**Table 3.**The maximum traffic flows of air route segments in different time intervals using different routing path methods.

Time Interval | Maximum Traffic Flows | |||
---|---|---|---|---|

$\mathit{Default}$ | $\mathit{Shortest}$ | $\mathit{Random}$ | $\mathit{Proposed}$ | |

0:00–1:00 | 5 | 5 | 5 | 4 |

1:00–2:00 | 9 | 9 | 9 | 8 |

2:00–3:00 | 8 | 8 | 8 | 9 |

3:00–4:00 | 8 | 9 | 8 | 8 |

4:00–5:00 | 5 | 5 | 5 | 6 |

5:00–6:00 | 5 | 5 | 5 | 4 |

6:00–7:00 | 24 | 22 | 24 | 23 |

7:00–8:00 | 43 | 41 | 43 | 42 |

8:00–9:00 | 52 | 56 | 56 | 51 |

9:00–10:00 | 54 | 56 | 56 | 52 |

10:00–11:00 | 45 | 49 | 49 | 41 |

11:00–12:00 | 38 | 38 | 36 | 37 |

12:00–13:00 | 40 | 40 | 40 | 39 |

13:00–14:00 | 44 | 47 | 47 | 43 |

14:00–15:00 | 41 | 45 | 45 | 40 |

15:00–16:00 | 45 | 42 | 42 | 42 |

16:00–17:00 | 45 | 45 | 45 | 48 |

17:00–18:00 | 46 | 48 | 48 | 47 |

18:00–19:00 | 43 | 44 | 44 | 43 |

19:00–20:00 | 43 | 45 | 45 | 42 |

20:00–21:00 | 40 | 41 | 41 | 39 |

21:00–22:00 | 38 | 40 | 40 | 37 |

22:00–23:00 | 42 | 41 | 42 | 41 |

23:00–00:00 | 31 | 32 | 32 | 30 |

$Average$ | 33.08 | 33.88 | 33.96 | 32.33 |

1.0 | - | - | −2.4% | |

- | 1.0 | - | −4.6% | |

- | - | 1.0 | −4.8% |

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## Share and Cite

**MDPI and ACS Style**

Ding, H.; Hu, M.; Xu, Q.; Tian, Y.; Yin, J.
A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks. *Appl. Sci.* **2023**, *13*, 866.
https://doi.org/10.3390/app13020866

**AMA Style**

Ding H, Hu M, Xu Q, Tian Y, Yin J.
A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks. *Applied Sciences*. 2023; 13(2):866.
https://doi.org/10.3390/app13020866

**Chicago/Turabian Style**

Ding, Hui, Minghua Hu, Qiucheng Xu, Yungang Tian, and Jianan Yin.
2023. "A Method to Optimize Routing Paths for City-Pair Airlines on Three-Layer Air Transport Networks" *Applied Sciences* 13, no. 2: 866.
https://doi.org/10.3390/app13020866