# A Collaborative Optimization Method of Flight Slots Considering Fairness Among Airports

Authors: Xianghua Tan; Shasha Wang; Weili Zeng; Zhibin Quan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

With the rapid development of civil aviation transportation, an increasing number of airport groups are formed. However, the existing literature on fairness mostly focuses on the fairness among airlines. There is no research on the realization of scheduling fairness among airports with overlapping resources in the airport group. The goal of this paper is to comprehensively consider efficiency and fairness in slot scheduling, where fairness should include both interairline and interairport fairness. Subsequently, we developed a collaborative optimization model for airport group that takes into account the above three objectives and then uses theε-constraint method to solve it. In addition to considering the basic operational constraints of the airport, the model also sets different adjustment boundaries to achieve the scheduling priorities specified by IATA. Applying the model to the Yangtze River Delta airport group, the results show that improved flight schedules can significantly reduce flight congestion, and a relatively fair scheduling result can be obtained by weighing airlines’ fairness and airports’ fairness. The model can improve the transparency of slot scheduling decisions and can be used as an auxiliary tool to help make slot scheduling decisions.


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## Body

## 1. Introduction

In recent years, with the rapid development of the civil aviation transportation industry, the contradiction between the rapidly increasing demand for flights and the limited airspace has become increasingly prominent, resulting in large-scale flight delays. According to the “Statistical Bulletin on the Development of Civil Aviation Industry,” the overall level of delays in 2016–2020 was high. The most direct cause of delays is weather, airline reasons, traffic control, etc., but the root cause is that capacity is in short supply. Expanding airport infrastructure and opening up more civil airspace to increasing capacity are the fundamental ways to mitigate delays. However, this method is costly and time-consuming. In the short term, the most effective way is to improve the operation management method so to make full use of the existing resources. The methods adopted at this stage can be roughly divided into two categories: economic means and administrative means. However, in actual operation, almost only administrative methods are used [1]. Common tactics such as ground waiting [2] and departure process control [3] have little effect in congested airports. As the simplest and most economical way to improve operation management, flight schedule optimization is more and more favored by managers [4–6].IATA has developed detailed procedures and rules for scheduling flight slots at coordinating airports. IATA’s slot coordination process begins with an airport’s published capacity, the limit on the number of takeoffs and landings per unit time, which is sometimes subdivided into runway, sector, terminal aerospace capacity, etc. The declared capacity of the coordinating airport needs to be announced about a year before the start of each flight season. About 5 months before each flight season, airlines submit their desired flight schedules to coordinators in the form of “slots series” or “ single slots.” If a flight appears at least five times in the one-week schedule of the same flight season, the flight must submit by the “ slots series” form, otherwise by the “single slots” form. Coordinator should assign the initial slots fairly and transparently based on the IATA guidelines and finally provide the airlines with the final initial slots four months before the start of each flight season. Subsequently, airline representatives, slot coordinators, airport representatives, and other interested parties will participate in the IATA slot coordination meeting to discuss possible adjustments to initial schedule allocations. Two months before the start of each season, some slots can be “returned” if the airline is no longer using them.To optimize the flight slots of an airport group, we should not only consider the operation of a single airport, but also emphasize the positioning of each airport within the airport group and the allocation of public airspace resources so as to achieve the coordinated optimization of flight slots at all airports. Different from a single airport, the optimization of flight slots in an airport group is mainly reflected in two aspects. On the one hand, it is reflected in the optimal allocation of public airspace resources. In the airport group, the public airspace resources mainly include the arrival and departure corridors and public routes. Adjacent airports in an airport group may share the same arrival and departure corridor entry points. For example, in the Beijing-Tianjin-Hebei airport group, the three airports of Daxing Airport, Capital Airport, and Tianjin Binhai Airport share two entry corridor entry points “A3” and “A4” and three departure corridor entry points “D1,” “D3,” and “D5.” Likewise, flights in a group of airports bound for similar directions may be subject to flow control on the same route or waypoint. For example, Shanghai Pudong Airport, Nanjing Lukou Airport, and Hangzhou Xiaoshan Airport belong to the Yangtze River Delta airport group, and the departure flights of these three airports are all subject to the normal flow control of the “P58” route. Therefore, how to efficiently utilize the resources of the arrival and departure corridors and public routes is of great significance to improve the flight operation efficiency of the entire airport group.On the other hand, it is reflected in the fairness of slot allocation between airports. Like the fairness between airlines, when optimizing flight slots for multiple airports in an airport group, it is necessary to pay attention to the positioning of the airports, and the issue of fairness of distribution between airports also needs to be considered. The fairness between airports is mainly reflected in the fairness of configuration deviation and operational efficiency. The fairness of configuration deviation refers to measuring the fairness according to the total deviation of each airport. That is to say, the total deviation of each airport should not be too large. This can effectively prevent the deviation of the adjusted flight volume of some airports from being too large, while the deviation of other airports is too small. Operational efficiency fairness is mainly measured from operational efficiency (such as delay, normality). The interests of one airport cannot be sacrificed to meet the high operational efficiency of another airport. Of course, for some regional hub airports, it is necessary to give priority to ensuring operational efficiency from the perspective of airport positioning to improve airport competitiveness.This paper proposes a fairness-based flight slot optimization model that weighs the equity among the airports in the airport groups in the same terminal airspace, while considering the airport efficiency. The model is targeted by the overall deviation and the fairness between airlines and the fairness between airports. Then a three-objective airport group flight slot optimization model is proposed, and the model is solved by the constraint method. The model reduces the unfairness between airports and between airlines as much as possible. Finally, this paper optimizes the flight slots of the long-Delta airports group as an example, the results indicating that the model can alleviate the excessive limit of multiple airport traffic. The deviation between airports and the deviation between airlines are slight after optimization, and the feasibility and effectiveness of the model are verified by experiments.The organizational structure of this chapter is as follows: Section1 introduces the development status of the civil aviation transportation industry, pointing out that there is a large amount of delay in civil aviation industry, and analyzes the causes of airport delays. Section 2 analyzes research status of single airport and multiairports flight slot optimization at home and abroad and proposes the main research content and technical routes in this paper according to existing research. Section 3 measured the quantitative relationship between airlines and airports, an airport groups coordination model considering fairness is established. The model will use the overall deviation, the airlines fairness, and airports fairness as the objective functions to optimize the slots, and the optimized slots are as output. The model solving algorithm based on ε-constraint law is described in Section 4. Taking the long triangular airport group as an example, Section 5 analyzes the slot configuration efficiency of the four major airports in this airport group to verify the effectiveness and practicality of the airport group coordination optimization model. Finally, the main research results of this paper are summarized in Section 6.


## 2. Literature Review

According to the number of airports considered, the flight slot optimization can be divided into single airport optimization and multiairport optimization. Single airport flight slot optimization refers to optimizing the slots of entering-outport flights only taking operating conditions and airline requirements into consideration; other airports’ impact is not directly considered. Multiairport flight slot optimization problem is based on a single airport and optimizes multiple airports at the same time. Multiairport flight slot optimization should consider multiple airport flight slots and implement overall optimization of flight slots [7].The study of single airport flight slot optimization was first started from the ground waiting policy. In 1987, Odoni established a static random model system to study the problem of surface waiting strategy [8]. Since 2012, Zografos proposes an integer linear planning model according to the slot rules developed by IATA to optimize slots [9]. Most of the later studies are based on this model, and the integer linear programming model has become a general model for flight slot optimization [10–12]. With the deepening of research, the allocation efficiency alone can no longer meet the needs of research; the model was later extended to include other objectives such as fairness [13–16], Airline’s benefit [1], delay cost [17], and more. In 2018 Zografos et al. built a biobjective flight slot optimization model, considering the different objective portfolios of the total acceptable time deviation and maximum acceptable time deviation. They found that the sacrifice part efficiency can effectively enhance airline acceptance [12]. This has also been the focus of researchers for a long time: meet airline preferences as much as possible [1, 18–20]. On this basis, Katsigiannis proposes a new model, considering the flexibility preference of the airline and its integration with constraints, thereby realizing dynamic allocation of airport resources. Comprehensive consideration of time flexibility indicators and dynamic capacity constraints can improve airport capacity utilization, significantly improving flight deviation [21]. Wang proposes a robust model that considers multicalendar day synchronous slots that consider uncertainty of airport capacity uncertainty. The model links the strategic decision-making and tactical decision-making in air traffic management (ATM) by weighing strategic difference costs and operational obstruction costs. The results show that the model effectively eliminates existing and potential scheduling conflicts and balances the airline preference and potential airport crowded risk [22].IATA proposes a hierarchy to satisfy airport slot requests, which is also a key strategy for scheduling. IATA defines moment priorities, i.e., historical, historical adjustment, new entrants, and other requests. Ribeiro proposed a priority-based multiobjective slot allocation model (PSAM), which optimizes flight slot in the case of satisfying the complex priority sets and requirements published by IATA Guide and airport declared capacity. Improve the efficiency of airline slot allocation by providing a more matching slot of airline slot needs [1]. Later, Ribeiro expanded their model to a new model, that is, PSAM-ATR, which combines a ceiling capacity. The PSAM-ATR model uses weighted dual objective functions that give priority to minimize the ratio of the maximum deviation and the total deviation [11]. Corolli viewed multiple airports are as a complete network under the premise of uncertain capacity, and the proposed model does not implement any existing IATA rules for slot assignment. However, by applying these models in order to each priority level, these models can be easily aligned with the existing IATA slot allocation program [23].Another broader focus is the improvement of airport operating efficiency. The evaluation of operational efficiency in the literature is mainly reflected in two aspects, one is the delay time [17, 24], and the other is the queue length (based on queuing theory) [25–28]. For example, Jacquillat proposed an original iterative solution algorithm that integrates the random queuing model of the airport, the dynamic planning model of the capacity utilization, and the integer planning model of the scheduled intervention, which can converge in a reasonable calculation time. The proposed airport congestion relief method’s performance is significantly better than the typical order method [10]. For studies that consider delay propagation in airport networks, the considerations of operational efficiency are more complex [29–31]. Nikola Ivanov believes that the current distribution of air traffic management (ATFM) delay does not consider whether the flight has the remaining scheduled buffer to absorb delays and reduce the spread of delayed flights. So he explores the possibility of controlling delay distribution and try to minimize the delay in propagating to subsequent flights [31].With the intensification of economic globalization, the research on the interaction between the slots of different airports has become more and more popular [20, 23, 32]. In 2012, Castelli proposed a mechanism taking the route network as a mathematical planning problem to solve the problem of European airport slots. They consider the interdependence of different airport slots and maximizing system efficiency, modeling the procedure of airport slots allocated to airlines. The cost applied by the grandfather rights was evaluated by experiment. And the cost of the monetary compensation mechanism was introduced between airlines, which proved that cancelling the grandfather right is possible [7]. Bolic also considers the entire network (i.e., airport and sector) and proposes an integer planning model for flight strategic redistribution to meet the requirements of the standard sector capacity and shorten the calculation time of large-scale instances. He uses real historical data to perform instance tests, and evaluation shows that early flight configurations can reduce delays and their cost [17]. Pellegrini et al. proposed an integer linear planning model Sosta used in Europe airport slot allocation process. The model further expanded the range of airport networks and allocated all airports’ slots at the same time; it is applicable to existing regulations and practices. The results show the benefits of synchronous assignment, as well as Sosta’s flexibility and excellent computing performance [33].However, with the gradual refinement of scheduling rules and the gradual complexity of the airspace environment, the scale of the flight slot allocation problem gradually increases. It becomes almost impossible to obtain the optimal solution through various enumeration methods and precise algorithms. Therefore, the heuristic algorithm becomes an effective method to solve the problem of simultaneous allocation of time slots [11, 34, 35]. Wang et al. proposed a new heuristic method to solve the problem of single airport slot assignment, and they introduce a constructive heuristic framework that can generate feasible solution of the model. Within this framework, many sorting heuristic algorithms have been developed to sort the slot requests to be scheduled. And an allocation algorithm was developed to assign the slot to the request for the sorting heuristic algorithm [35]. Pellegrini et al. proposed the form of simultaneous distribution of airport group flight slot (SSAP), which expands the way Europe currently distributes the airport slot in two directions. This problem is solved by using two algorithms based on metaheuristic which is iterative partial search and variable domain search and algorithms based on integer linear planning models [20]. Fairbrother introduces a model and solution framework that minimizes blocking while maintaining the ideal scheduling rules [36]. Benlic et al. almost considered all of the existing rules and guidelines for the IATA, including the priority order of flight requests, capacity restrictions on each airport, and the minimum turnaround time between the extension flights. They assign the slots to a series rather than a single flight and use heuristic search to solve this problem [29].The introduction of fairness may result in an increase in the total deviation but can avoid the problem of excessive cost of adjustment of a few airlines [19, 37]. Zografos and Jiang et al. have put forward an indicator to measure the fairness, further improving the model as a two-objective optimization model of equity and efficiency [38]. In 2020, Jamie Fairbrother et al. proposed a two-stage crowded airport slot scheduling mechanism that considers the efficiency and fair goal, which considers airlines’ preferences when distributing the total deviation associated with airline flights. The proposed slot allocation mechanism provides policymakers with useful information about fair-efficiency weighing and improves transparency and acceptability of slot allocation results [13]. Jiang and Zografos study three fair goals, namely, absolute fairness, relative equity, and Gini index, to simulate the fairness among airlines in the flight slot configuration problem. They found that, in the case of a given absolute fair objective value, it can be exported as relatively fair objective of the world [39]. Katsigiannis proposed a new three-objective slot distribution model (TOSAM) in order to fill the blank of the existing literature. The model considers the total schedule deviation, the maximum schedule deviation, and demand-based fairness and introduces a multilevel, multiobjective algorithm to solve this problem [40].From these observations, in this article, we extended Zografos [38] proposed fairness coefficients, joining the coefficients of weighing airport fairness, as biobjective functions of the model. Most of the existing literature pays more attention to efficiency or deviation, rarely mentioning fairness, even if it contains fairness, which is also a fairness between airlines and considers the benefits of airlines. However, the resources between the airports in the airport group sharing the same terminal airspace overlap each other, especially among the airports which shared the same corridor. If two flights coming from different airports arrive to the same corridor at same time, allowing A airport flights first, then B airport will have benefit loss. If at the terminal airspace, whenever the “A” and “B ” airports have a conflict, the “A” airport will always go first, so that “B” airport will bear huge losses. This slot allocation does not have acceptability. The main contribution of this study is to combine the relationship between airport and airport and between airline and airline. In addition, we will also join the classic objective function minimum deviation to ensure the practicality of the optimization configuration. The three-objective airport groups slot optimization models based on fairness are proposed, and the model is converted into a single-objective problem through the ε-constraint method. The proposed model solves the operational constraints (such as rolling capacity constraints) under the existing slot configuration rules, as well as the operational requirements of airlines and airports.


## 3. A Collaborative Optimization Model for Airport Group

In this section, we will introduce the collaborative optimization of airport group model. Airports flight plan schedule is the input of this optimization model. According to the provisions of the IATA, the airport published capacity, the estimated corridor capacity value, and the turnaround time, etc. are the restrictions of model. It outputs an optimized schedule to minimize the deviation of the request slot. Below we will describe the settings of the decision variables, the objective functions, and constraints required by the model and then detail the implementable functionality of the model.


### 3.1. Symbol

t: the slot assigned to flight m.tm: the requested schedule of flight m; T: A set of coordinated slot intervals.h: an airline h.H: a set of airlines.u: an airport u.U: a set of airports.m: a particular flight m.ma: an arrival particular flight ma.md: a departure particular flight md.M: a set of all flights within the airport group.Mh: a set of the flights at the airline h.Mk: a set of the flights at the specific corridor entrances k.Mu: a set of the flights at the airport u.Mud: a set of the departure flights at the airport u.Mua: a set of the arrival flights at the airport u.Mu2⊆Mua×Mud: a set of pairs of arrival and departure flights operated by the same aircraft at the airport u.KMuk: a set of flights with specific corridor entrances k∈Ku at the airport u.RMur: a set of flights using public corridors entrances r∈R at the airport u.r: a public corridor entrance r.R: a set of the public corridor entrances.Ur: a set of airports using the same public corridor entrances r∈R.k: a specific corridor entrance k.Ku: a set of the specific corridor entrances at the airport u.Cua: the entrance capacity at the airport u.Cud: the departure capacity at the airport u.Cu: the total capacity at the airport u.Vuk: the capacity with specific corridor entrances at the airport u.Wur: the capacity using public corridors entrances at the airport u.tma,mdmin: the minimum turnaround time for arrival and departure flight pairs ma,md.tma,mdmax: the maximum turnaround time for arrival and departure flight pairs ma,md.du: the deviation of the airport u.dh: the deviation of the airline h.ρu: the fairness measure of the airport u.ρh: the fairness measure of the airline h.δm: the maximum acceptable deviation slot for flight m.


### 3.2. Decision Variable

The decision variable is to assign schedule to flights. The same flight at the same airport can only be assigned one flight schedule. To coordinate and optimize the flight schedule of the airport group, it is necessary to add an indicator airport to the decision variable of the flight schedule optimization of a single airport. Therefore, the decision variable is defined as(1)Xmt=1,If the flightmis assigned to the flight schedulet,0,else,where t∈T,m∈M.


### 3.3. Objective Function

Whether it is a single airport or a group of airports, the optimized flight schedule should match the airline’s request as much as possible. Therefore, the minimum total deviation still needs to be one of the objectives of the flight schedule optimization problem for airport group. Like a single airport, the fairness between airlines involved in this paper mainly refers to the fairness of the sum of deviations of slot accepted by each airline after optimization. If the sum of deviations of one airline is concentrated to achieve the normality of other flights, then it is unfair for this airline. To be fair, such optimization results increase the difficulty of the coordinator’s work, which is meaningless and unacceptable. Fairness should be measured by the average deviations of each airline, which is the ratio of the total deviations to the total number of flights. In addition, the fairness among the airports in the airport group cannot be ignored.According to the principle of proportionality, the fairness measure of an airlineh∈H is defined as (2)ρh=dh/f1Mh/∑h∈HMh=∑m∈Mh∑t∈Tt−tmXmt/∑h∈H∑m∈Mh∑t∈Tt−tmXmtMh/∑h∈HMh,where dh/f1 represents the deviations of airline h as a proportion of the total deviations of all airlines; Mh/∑h∈HMh represents the flight numbers of airline h as a proportion of the total flight numbers of all airlines.Likewise, the fairness measure for an airportu∈U is defined as(3)ρu=du/f1Mu/∑u∈UMu=∑m∈Mu∑t∈Tt−tmXmt/∑u∈U∑m∈Mu∑t∈Tt−tmXmtMu/∑u∈UMu,where du/f1 represents the deviations of airport u as a proportion of the total deviations of all airports; Mu/∑u∈UMu represents the flight numbers of airport u as a proportion of the total flight numbers of all airports.According to the fairness measures (2) and (3), the following multiobjective functions are established considering the minimization of the total deviation, the fairness between airlines, and the fairness between airports(4)minf1,f2,f3.The first term in (4) is the total deviation of all flights in the airport group(5)f1=∑u∈U∑m∈Mu∑t∈Tt−tmXmt.The second term in (4) is the maximum deviation of each airline’s fairness index from the average fairness index(6)f2=maxρh−∑h∈HρhH,h∈H.The third term in (4) is the maximum deviation of each airport from the average fairness indicator(7)f3=maxρu−∑u∈UρuU,u∈U.


### 3.4. Constraints

In single airport flight schedule optimization, eight constraints are considered, including the uniqueness of slot allocation, acceptable deviation slot, slot capacity, runway capacity, corridor capacity, flow control of normal routes, turnaround time, and flight wave configuration. But not all constraints have to be considered. In this paper, the appropriate constraints are selected according to the characteristics of the airport. For example, if there are no airports with a flight wave configuration, then this constraint can be ignored. For the sake of simplicity, this paper only considers the following constraints: uniqueness, the acceptable deviation slot, the slot capacity (the arrival capacity, the departure capacity, and the total capacity), the turnaround time (the minimum turnaround time, the maximum turnaround time), and the capacity of the corridor (the specific corridor capacity and the public corridor capacity of each airport).(8)∑t∈TXmt=1,m∈M,u∈U,(9)∑t∈Tt−tmXmt≤δm,m∈M,(10)∑m∈Mua∑t∈ss+55Xmt≤Cua,s=0,5,…,1385,u∈U,(11)∑m∈Mud∑t∈ss+55Xmt≤Cud,s=0,5,…,1385,u∈U,(12)∑m∈Mu∑t∈ss+55Xmt≤Cu,s=0,5,…,1385,u∈U,(13)∑t∈TtXmdt−∑t∈TtXmat≥tma,mdmin,ma,md∈Mu2,u∈U,(14)∑t∈TtXmdt−∑t∈TtXmat≥tma,mdmax,ma,md∈Mu2,u∈U,(15)∑m∈KMuk∑t∈ss+15Xmt≤Vuk,s=0,5,…,1425,k∈Ku,u∈U,(16)∑u∈Ur∑m∈RMur∑t∈ss+15Xmt≤∑u∈UrWur,s=0,5,…,1385,r∈R.Equation (8) restricts each flight to be assigned only one slot at the same airport on the same day; (9) constrains the acceptable maximum deviation slot of the flight; that is, within the acceptable range of the airline the optimal solution can be found. Equations (10)–(12) are about the constraints of inbound capacity, outbound capacity, and total capacity, respectively, and these three formulas are constraints of rolling capacity for 5 minutes. Equations (13) and (14) are the constraints on the minimum turnaround time and the maximum turnaround time, respectively. The minimum turnaround time has been stipulated, but there is no corresponding provision for the maximum turnaround time. We choose the 90th percentile value of the flight history turnaround time as the maximum turnaround time. Equations (15) and (16) are restrictions on the traffic capacity, which limit the traffic capacity of each airport’s specific arrival or departure corridor and public arrival or departure corridor, respectively. It should be emphasized that the special arrival or departure corridor is only used by one airport alone, while the public corridor is used by two or more airports.In the multiairport flight schedule optimization problem, some researchers require the flight slot to be kept constant [5], while some researchers require the block time to be within a certain range [6]. It should be emphasized that, by setting the parameters in the constraints (9), it can be guaranteed that the landing slot at the destination airport will not be affected.


## 3.1. Symbol

t: the slot assigned to flight m.tm: the requested schedule of flight m; T: A set of coordinated slot intervals.h: an airline h.H: a set of airlines.u: an airport u.U: a set of airports.m: a particular flight m.ma: an arrival particular flight ma.md: a departure particular flight md.M: a set of all flights within the airport group.Mh: a set of the flights at the airline h.Mk: a set of the flights at the specific corridor entrances k.Mu: a set of the flights at the airport u.Mud: a set of the departure flights at the airport u.Mua: a set of the arrival flights at the airport u.Mu2⊆Mua×Mud: a set of pairs of arrival and departure flights operated by the same aircraft at the airport u.KMuk: a set of flights with specific corridor entrances k∈Ku at the airport u.RMur: a set of flights using public corridors entrances r∈R at the airport u.r: a public corridor entrance r.R: a set of the public corridor entrances.Ur: a set of airports using the same public corridor entrances r∈R.k: a specific corridor entrance k.Ku: a set of the specific corridor entrances at the airport u.Cua: the entrance capacity at the airport u.Cud: the departure capacity at the airport u.Cu: the total capacity at the airport u.Vuk: the capacity with specific corridor entrances at the airport u.Wur: the capacity using public corridors entrances at the airport u.tma,mdmin: the minimum turnaround time for arrival and departure flight pairs ma,md.tma,mdmax: the maximum turnaround time for arrival and departure flight pairs ma,md.du: the deviation of the airport u.dh: the deviation of the airline h.ρu: the fairness measure of the airport u.ρh: the fairness measure of the airline h.δm: the maximum acceptable deviation slot for flight m.


## 3.2. Decision Variable

The decision variable is to assign schedule to flights. The same flight at the same airport can only be assigned one flight schedule. To coordinate and optimize the flight schedule of the airport group, it is necessary to add an indicator airport to the decision variable of the flight schedule optimization of a single airport. Therefore, the decision variable is defined as(1)Xmt=1,If the flightmis assigned to the flight schedulet,0,else,where t∈T,m∈M.


## 3.3. Objective Function

Whether it is a single airport or a group of airports, the optimized flight schedule should match the airline’s request as much as possible. Therefore, the minimum total deviation still needs to be one of the objectives of the flight schedule optimization problem for airport group. Like a single airport, the fairness between airlines involved in this paper mainly refers to the fairness of the sum of deviations of slot accepted by each airline after optimization. If the sum of deviations of one airline is concentrated to achieve the normality of other flights, then it is unfair for this airline. To be fair, such optimization results increase the difficulty of the coordinator’s work, which is meaningless and unacceptable. Fairness should be measured by the average deviations of each airline, which is the ratio of the total deviations to the total number of flights. In addition, the fairness among the airports in the airport group cannot be ignored.According to the principle of proportionality, the fairness measure of an airlineh∈H is defined as (2)ρh=dh/f1Mh/∑h∈HMh=∑m∈Mh∑t∈Tt−tmXmt/∑h∈H∑m∈Mh∑t∈Tt−tmXmtMh/∑h∈HMh,where dh/f1 represents the deviations of airline h as a proportion of the total deviations of all airlines; Mh/∑h∈HMh represents the flight numbers of airline h as a proportion of the total flight numbers of all airlines.Likewise, the fairness measure for an airportu∈U is defined as(3)ρu=du/f1Mu/∑u∈UMu=∑m∈Mu∑t∈Tt−tmXmt/∑u∈U∑m∈Mu∑t∈Tt−tmXmtMu/∑u∈UMu,where du/f1 represents the deviations of airport u as a proportion of the total deviations of all airports; Mu/∑u∈UMu represents the flight numbers of airport u as a proportion of the total flight numbers of all airports.According to the fairness measures (2) and (3), the following multiobjective functions are established considering the minimization of the total deviation, the fairness between airlines, and the fairness between airports(4)minf1,f2,f3.The first term in (4) is the total deviation of all flights in the airport group(5)f1=∑u∈U∑m∈Mu∑t∈Tt−tmXmt.The second term in (4) is the maximum deviation of each airline’s fairness index from the average fairness index(6)f2=maxρh−∑h∈HρhH,h∈H.The third term in (4) is the maximum deviation of each airport from the average fairness indicator(7)f3=maxρu−∑u∈UρuU,u∈U.


## 3.4. Constraints

In single airport flight schedule optimization, eight constraints are considered, including the uniqueness of slot allocation, acceptable deviation slot, slot capacity, runway capacity, corridor capacity, flow control of normal routes, turnaround time, and flight wave configuration. But not all constraints have to be considered. In this paper, the appropriate constraints are selected according to the characteristics of the airport. For example, if there are no airports with a flight wave configuration, then this constraint can be ignored. For the sake of simplicity, this paper only considers the following constraints: uniqueness, the acceptable deviation slot, the slot capacity (the arrival capacity, the departure capacity, and the total capacity), the turnaround time (the minimum turnaround time, the maximum turnaround time), and the capacity of the corridor (the specific corridor capacity and the public corridor capacity of each airport).(8)∑t∈TXmt=1,m∈M,u∈U,(9)∑t∈Tt−tmXmt≤δm,m∈M,(10)∑m∈Mua∑t∈ss+55Xmt≤Cua,s=0,5,…,1385,u∈U,(11)∑m∈Mud∑t∈ss+55Xmt≤Cud,s=0,5,…,1385,u∈U,(12)∑m∈Mu∑t∈ss+55Xmt≤Cu,s=0,5,…,1385,u∈U,(13)∑t∈TtXmdt−∑t∈TtXmat≥tma,mdmin,ma,md∈Mu2,u∈U,(14)∑t∈TtXmdt−∑t∈TtXmat≥tma,mdmax,ma,md∈Mu2,u∈U,(15)∑m∈KMuk∑t∈ss+15Xmt≤Vuk,s=0,5,…,1425,k∈Ku,u∈U,(16)∑u∈Ur∑m∈RMur∑t∈ss+15Xmt≤∑u∈UrWur,s=0,5,…,1385,r∈R.Equation (8) restricts each flight to be assigned only one slot at the same airport on the same day; (9) constrains the acceptable maximum deviation slot of the flight; that is, within the acceptable range of the airline the optimal solution can be found. Equations (10)–(12) are about the constraints of inbound capacity, outbound capacity, and total capacity, respectively, and these three formulas are constraints of rolling capacity for 5 minutes. Equations (13) and (14) are the constraints on the minimum turnaround time and the maximum turnaround time, respectively. The minimum turnaround time has been stipulated, but there is no corresponding provision for the maximum turnaround time. We choose the 90th percentile value of the flight history turnaround time as the maximum turnaround time. Equations (15) and (16) are restrictions on the traffic capacity, which limit the traffic capacity of each airport’s specific arrival or departure corridor and public arrival or departure corridor, respectively. It should be emphasized that the special arrival or departure corridor is only used by one airport alone, while the public corridor is used by two or more airports.In the multiairport flight schedule optimization problem, some researchers require the flight slot to be kept constant [5], while some researchers require the block time to be within a certain range [6]. It should be emphasized that, by setting the parameters in the constraints (9), it can be guaranteed that the landing slot at the destination airport will not be affected.


## 4. Solution Method

In order to solve the three-objective model proposed above, we adopt theε constraint method, which converts two of the objective functions into linear constraints. As an easy way to get started, the constraint method does not limit the type of objective function but requires careful selection of the objective function to convert into constraints. In this method, only one objective function is retained, and other objective functions are converted into constraints. Thus, this multiobjective programming model is transformed into a single-objective model.The objective functionf2 of airline fairness should satisfy the following constraints:(17)ρh−∑h∈HρuH≤ε1,∀h∈H,where the left side represents the deviation of slot scheduling fairness from airport u relative to the average airport fairness. But this constraint is nonlinear because the denominator of ρu contains the decision variable Xmt. By multiplying both sides of (17) by the total deviation expression f1=∑h∈H∑m∈Mh∑t∈Tt−tmXmt, it can be transformed into the following linear constraint.(18)∑m∈Mk∑t∈Tt−tmXmtMh/∑h∈HMh−∑h∈H∑m∈Mk∑t∈Tt−tmXmt/Mh/∑h∈HMhH≤ε1⋅∑h∈H∑m∈Mk∑t∈Tt−tmXmt,∀h∈H.In (18), both Mh/∑h∈HMh and H are known constants, so (18) is a linear constraint. Therefore, the airport fairness objective f3 can be transformed into the following linear constraints:(19)∑m∈Mu∑t∈Tt−tmXmtMu/∑u∈UMu−∑u∈U∑m∈Mu∑t∈Tt−tmXmt/Mu/∑u∈UMuU≤ε2⋅∑u∈U∑m∈Mu∑t∈Tt−tmXmt,∀u∈U.By transforming the two fairness objective functions into linear constraints, the original multiobjective model is simplified to the following form:(20)minZx,ε1,ε2=∑m∈M∑t∈Tt−tmXmt.The objective function (20) minimizes the total deviation at different values of parameters ε1 and ε2. The values of ε1 and ε2 represent the maximum allowable deviation between the fair value of a given airport and the average fair value of all airports. If the values of ε1 and ε2 are too large, the fairness constraint is too broad, causing its results to be indistinguishable from models that do not consider fairness. Conversely, if the values of ε1 and ε2 are too small, a feasible solution may not be found.The steps to solve theε-constrained model (objective function (20), constraints (8)–(16) and (18) and (19)) are as follows:Step 1: Solve a single-objective model (a model that does not contain two fairness objectives) that minimizes the total deviation, and substitute the optimal solution into formulas (2) and (3) to obtain the fairness coefficient of airlines and airports.Step 2: Substitute the fairness coefficients of the airline and the airport into equations (6) and (7), respectively, to obtain the fairness objective function values of the airline and the airport. Take the two fairness objective values as the initial values of ε1 and ε2.Step 3: To updateε1 and ε2, decrease the values of ε1 and ε2 iteratively to get the values of ε1n and ε2n for the n th iteration. For each ε1n and ε2n, the ε-constrained model is solved and the efficient frontier is determined according to the fairness objective value. The algorithm terminates when no feasible solutions for ε1n and ε2n are found.


## 5. Model Validation

### 5.1. Airport Group Overview

At present, the Yangtze River Delta airport group includes 16 airports. Shanghai Pudong Airport, Shanghai Hongqiao Airport, Hangzhou Xiaoshan Airport, and Nanjing Lukou Airport are the four international or regional hub airports in the Yangtze River Delta airport group, which are relatively close to each other (Figure1, the label in the figure is the distance between each airport and Pudong Airport). They restrict and influence each other on airspace resources. Shanghai Pudong Airport is positioned as an international large-scale composite hub airport. It is mainly responsible for the take-off and landing of international routes and major domestic routes. It is the core carrier of Shanghai’s aviation hub construction. It has four operating runways and an annual passenger throughput of 76.0975 million in 2019. Shanghai Hongqiao Airport plays an auxiliary role in the hub structure. Compared with Shanghai Pudong Airport, it is more inclined to point-to-point operation in domestic cities, with dual-runway operation. In 2019, the annual passenger throughput reached 45.6766 million. Hangzhou Xiaoshan Airport and Nanjing Lukou Airport are both domestic regional hub airports and important trunk airports in China. They are both dual-runway airports. In 2019, the passenger throughput of the two airports was 40.108 million and 30.581 million, respectively. The number of aircraft landings is about half of that at Pudong Airport.Figure 1
Geographical location map of the Yangtze River Delta airport group.


### 5.2. Data

The data of this experiment are the summer and autumn flight schedule in 2019 and the historical operation data of the national airports during the summer and autumn flight seasons in 2018. The flight schedule data includes flight number, schedule, aircraft type, departure airport, landing airport, planned departure time, planned landing time, etc. Historical operation data includes flight number, registration number, aircraft type, departure airport, landing airport, planned departure time, actual departure time, planned landing time, actual landing time, etc. Based on the historical operation data of the four airports in the Yangtze River Delta during the entire summer and autumn flight seasons in 2018, this paper firstly conducts a detailed statistical analysis of the delays, releases, and traffic of several airports, estimates all the parameters required for flight schedule optimization, then selects the flight schedule on Wednesday of the domestic summer and autumn flight season in 2019 for optimization, and makes a comparative analysis.


### 5.3. Parameters

Table1 align the ICAO airport codes with the airport names. For the sake of convenience, the subsequent content of the article uses the ICAO airport codes to refer to the airports. At the same time, Table 1 gives hourly capacity parameters for the four airports. Table 2 shows the arrival and departure points of each airport. AND is not only the public departure point in the southwest and northeast of Hongqiao and Hangzhou airports, but also the public arrival point of the two airports. The arrival point and the departure point have no influence on each other due to the height difference. SASAN is the westbound public arrival and departure point for Pudong and Hongqiao airports. PIKAS and LAMEN are the public departure points for Pudong and Hongqiao airports in the northeast and northwest directions, respectively. CJ is the public arrival and departure points of Nanjing and Hangzhou airports in the southeast and northbound directions, respectively. According to historical operation data, the percentile method is used to determine the capacity of each arrival and departure point. However, due to the uneven distribution of flow at each corridor, there is very little flow at some corridors. Therefore, for a nonpublic corridor with a flow rate of less than 2 in 5 minutes, 24 is used as its total hourly capacity. The specific corridor capacity settings of each airport are shown in Table 3.Table 1
Airport hourly capacity.
ICAO airport codesAirportAdvertised capacityDeparture capacityArrival capacityZSPDShanghai Pudong airport795050ZSSSShanghai Hongqiao Airport473535ZSHCHangzhou Xiaoshan airport443535ZSNJNanjing Lukou airport382525Table 2
Airport arrival and departure points.
AirportDeparture pointArrival pointZSPDBOLEX/HSN/LAMEN/NXD/ODULO/PIKAS/SASANBK/DUMET/MATNU/SASANZSSSAND/SASAN/IBEGI/LAMEN/PIKAS/NXDAND/SASAN/SUPARZSHCTOL/SUPAR/NIVIKTOL/SUPAR/IGRATZSNJTESIG/ESBAG/HFE/OFKAKIS/ESBAG/HFE/OFTable 3
Capacity parameters of each arrival and departure point.
AirportDeparture pointThe 90th percentile value of the flowCapacityArrival pointThe 90th percentile value of the flowCapacityZSPDBOLEX224BK1824HSN724MATNU724NXD911DUMET1024ODULO424SASAN11LAMEN2121PIKAS1818SASAN1212ZSSSIBEGI124SASAN1515LAMEN44AND36PIKAS66SUPAR610SASAN1010AND918NXD1012ZSHCTOL1414TOL1313SUPAR47SUPAR58NIVIK824IGRAT824ZSNJESBAG1412ESBAG1413HFE1111HFE1314OF612OF613TESIG77KAKIS33Since different aircraft types receive services for different durations, the Civil Aviation Administration has stipulated the minimum turnaround time for different aircraft types according to the support capabilities of different airports. Table4 shows the minimum turnaround time for different models. The minimum turnaround time is classified according to the number of seats. The more the seats, the longer the minimum turnaround time. The actual turnaround time at the four airports is about an hour longer than the minimum turnaround time. For the maximum turnaround time, there is no relevant document clearly stipulated. We select the 90th percentile value of the historical actual turnaround time as the maximum turnaround time (see Table 5).Table 4
Minimum turnaround time for different models.
Number of seatsCommon modelsMinimum turnaround timeAirports in Pudong, Hongqiao, and HangzhouAirports in Nanjing≤60AT72, E145, CRJ, etc.40 minutes30 minutes61∼150CRJ7, E190, A319, B737 (Type 700 and below), etc.55 minutes40 minutes151∼250B737(Type 700 and above), B752, B762, B787, A310, A320, A321, etc.65 minutes45 minutes250∼500B747, B763, B777, A300, A330, A340, A350, MD1175 minutes65 minutes>500A380120 minutes120 minutesTable 5
Maximum turnaround time.
AirportMaximum turnaround time (minutes)ZSPD210ZSSS160ZSNJ155ZSHC185


### 5.4. Experimental Results and Analysis

#### 5.4.1. Deviation Analysis

Tables6 and 7, respectively, show the airport deviations and various types of time deviations after the optimization of the flight schedule for the summer and autumn seasons in 2019. According to the adjustment priority of other flight schedule, historical adjustment schedule, and historical schedule, this paper sets the maximum deviations allowed for these three types of slots as the following times: 65 minutes, 40 minutes, and 15 minutes. This can reflect the model’s priority for different flight schedule allocation types. The higher the priority, the more the slot periods that can be shifted, the greater the possibility of being shifted, and the easier it is to be adjusted. Table 8 shows the deviation statistics of various flight slot types. As the adjustment priority decreases, the total deviation of the flight decreases in turn, and the adjusted flight amount and the average deviation also gradually decrease.Table 6
The number of flights adjusted by each airport and the average deviation.
AirportAdjusted number of flightsThe average deviation (minutes)AirportZSHC8016.13ZSHCZSNJ5222.02ZSNJZSPD11627.46ZSPDZSSS15517.77ZSSSTable 7
Adjusted number of flights and average deviation of various types of flight slots.
Type of flight slotsThe total deviation (minutes)The adjusted number of flightsThe average deviation (minutes)Historical schedule14556232.34Historical adjustment schedule303012382.45Other schedule389015482.51Table 8
The total deviation of each airline.
AirlineThe total deviation (minutes)The proportion of flights (%)AirlineThe total deviation (minutes)The proportion of flights (%)CES153023.94CHH2303.73CSN79010.38CSZ4153.64CCA4208.01CDG1801.97CSH4507.89CSC1551.94DKH6057.89JAS1601.61CQH4354.81CDC1751.58Figure2 shows the distribution of flight deviations at each airport. On the whole, as the deviation increases, the number of flights exceeding the deviation increases. But when the deviation exceeds 20 minutes, the number of flights exceeding the deviation decreases as the deviation increases. Among them, the three deviations of 5, 10, and 15 minutes are the most. Figure 3 shows the time distribution of the deviation of each airport. It can be concluded that the flight deviation has a large peak period from 12 : 00 to 17 : 00 with the change of time. This is because more flights exceed the capacity limit during the morning peak period of 9 : 00–11 : 00 in the morning and the evening inbound peak period of 16 : 00–20 : 00. As a result, the flight in this time period has to search for the optimal solution in a position with a larger front and rear span, so that the flight deviation during this period is much larger than that in other time periods. The deviations of Lukou Airport and Pudong Airport have more obvious peaks and valleys, which also shows that the slots of these two airports have obvious differences between peaks and valleys at the same time. In addition to the problem that the slot of the previous flight is occupied, the subsequent optimization interval of the peak arrival period from 16 : 00 to 20 : 00 is much more flexible. Therefore, the deviation is reduced accordingly. For example, flights after 20 : 00 are easier to optimize and have fewer deviations.Figure 2
Distribution of flight deviations at each airport.Figure 3
Time distribution of deviations for each airport.
(a)(b)(c)(d)


#### 5.4.2. Analysis of Capacity-Flow Matching

Figure4 shows the distribution of flights before and after optimization at the four airports. This paper selects the flight slot during the busy period in the schedule plan for optimization. During the peak period of 10 : 00–13 : 00, the number of optimized flights is lower than the amount before optimization, and the same result also occurs during the peak inbound period of 16 : 00–19 : 00 and the peak freight period after 22 : 00; however, the conclusion is reversed in the morning period of 5 : 00–9:00.Figure 4
Flight distribution before and after airport optimization.
(a)(b)(c)(d)It can be concluded that flight slots that exceed capacity limits during peak hours are moved to the preceding or subsequent periods of lower flow by the optimized model. As a result, the peak and trough periods of flights in a day are more “smooth,” and the number of flights is distributed as evenly as possible in each period, fluctuating around capacity. The optimization results of Xiaoshan Airport more clearly show the characteristics of “smooth fluctuation”; that is, the number of flights fluctuates evenly around the corridor and the airport capacity limit and is distributed as evenly as possible in each time period. It can be concluded that the above model is feasible and reasonable. Similarly, the optimization results of Lukou Airport are smoother and rounder. The entire planned flights are optimized to meet the airport’s capacity as much as possible, and the number of flights is evenly distributed in each time period with almost no fluctuations. The optimization performance of this model is very high. This is because the arrival and departure points of Lukou Airport are less related to the other three airports, and there are fewer flights restricted by other airports at the corridor entrance.According to the principle of whether it is shared by two or more airports, corridors can be divided into nonpublic corridors and public corridors. Due to the large number of nonpublic corridors in the studied airport group, this paper only presents the capacity-flow matching analysis of some nonpublic corridors. The flow of each period before and after the optimization of corridor TOL, HFE, and BK is shown in Figure5. Before the optimization, the congestion period of 1-hour inbound and outbound TOL at Xiaoshan Airport is 9 : 45–10 : 45, 13 : 30–15 : 00, 16 : 00–17 : 30, and 18 : 30–19 : 30. After optimization, the above time periods meet the capacity limit. Similarly, before optimization, the congestion periods of HFE at Lukou Airport are 10 : 00–13 : 00 and 16 : 30–17 : 45, and the congestion periods of BK at Pudong Airport are 12 : 45–13 : 45, 16 : 15–18 : 00, and 21 : 30–23 : 15. After optimization, the flow restrictions of these nonpublic corridor entrances have been met to a considerable extent. Compared with the hourly distribution of flights before optimization, the hourly flow is more uniform, which greatly reduces the peak load at the corridor entrance and makes full use of the time resource of the corridor entrance. From this point of view, this optimization model is very convincing, and the optimization of the corridor entrance has reached the requirements of the model.Figure 5
Statistical results of hourly capacity and flow before and after optimization of nonpublic corridors.
(a)(b)(c)(d)(e)(f)This paper makes a statistical analysis of the historical flow of public arrival and departure points. The 90th percentile of the flow of the public arrival and departure point is used as the capacity parameter of this point. However, in some cases where the flow is too small, if the capacity parameter is used directly as the capacity, it may cause a waste of corridor resources. Therefore, for the corridor entrance whose hourly capacity parameter is less than 24 sorties, 24 sorties are specified as the capacity of this point; and the capacity parameter is used as the capacity for other corridor entrances. The capacity is then allocated in proportion to the flow of each airport passing through this point, and the corresponding capacity of different airports at this point is determined.The flow of each period before and after the optimization of corridor SUPAR, SASAN, LAMEN, and NXD is shown in Figure6. Before optimization, in the public arrival and departure points of Hongqiao and Xiaoshan airports, the congestion periods of 1-hour arrivals are 11 : 00–12 : 00, 14 : 00–16 : 00, 19 : 00–20 : 00, 21 : 00–22 : 00, and 23 : 00–24 : 00, and the departure congestion periods are 7 : 00–8:00, 10 : 00–11 : 00, 13 : 00–14 : 00, 17 : 00–18:00, and 20 : 00–21 : 00. After optimization, the capacity is within the constraints. SASAN is the westbound public arrival and departure point for Pudong and Hongqiao. Similarly, before the optimization, the 1-hour arrival congestion period was 22 : 00–23 : 00, and the departure congestion period was 8 : 00–9:00. After optimization, the capacity is also within the constraints. PIKAS is the public departure point for Pudong and Hongqiao airports in the northeast. Its capacity before and after optimization is within the constraints, so it will not be described in detail. Another public departure point of Pudong and Hongqiao Airport is LAMEN. The 1-hour departure congestion period before optimization is 16 : 00–17 : 00, and it is within the capacity limit after optimization. NXD is the public departure point of Pudong and Hongqiao airports in the southeast direction, respectively. The 1-hour departure congestion period before optimization is 6 : 00–7:00, and after optimization, it is also within the constraints of capacity constraints. After the optimization, the flight distribution of the abovementioned public arrival and departure points has been significantly improved, which is mainly reflected in the fact that the traffic during peak hours is limited within the capacity requirements. Thereby, the peak load at the corridor entrance is greatly reduced. In summary, the feasibility and reliability of the model are verified.Figure 6
Statistical results of hourly capacity and flow before and after optimization of public corridor entrances.
(a)(b)(c)(d)(e)(f)(g)(h)


#### 5.4.3. Utilization Rate of Schedule Resources

According to the capacity of the airport schedule, the data before and after the optimization of the schedule plan in 2019 are used to calculate the resource utilization rate of the four airports between 6 : 00 and 24 : 00, respectively. The results are shown in Table9. Pudong Airport is 98.54% and 99.25%; Hongqiao Airport is 94.01% and 94.47%; Lukou Airport is 80.65% and 94.93%; Xiaoshan Airport is 97.70% and 97.70%. It can be concluded that, before and after the optimization, only Xiaoshan Airport’s time resource utilization remains unchanged, and it is slightly improved in other airports. The optimization model in this paper can improve the utilization rate of time resources, which verifies the reliability of the model.Table 9
Utilization of airport slots.
AirportBefore optimization (%)After optimization (%)ZSHC97.7097.70ZSNJ80.6594.93ZSPD98.5499.25ZSSS94.0194.47Table10 shows the utilization rate of corridor entrances of each airport. In general, the time resources at the airport are tighter than those at the corridor entrance. After optimization, the planned utilization rate of time resources at the airport is close to 95%, and the flexibility space is limited. The utilization rate of the entry and departure points is not much different before and after the optimization but both have improved. Except for the corridor entrance points with very few flights, the utilization rate of the public arrival and departure points has increased significantly.Table 10
Utilization rate of airport corridors.
AirportDeparture pointSchedule plan (%)Actual operation (%)Arrival pointSchedule plan (%)Actual operation (%)ZSPDBOLEX1.841.84BK65.9065.90HSN29.0329.49DUMET55.7657.14NXD46.0846.54MATNU37.7937.79ODULO32.2632.26SASAN33.3315.79LAMEN4.173.26PIKAS1.161.17SASAN68.0674.56ZSSSIBEGI0.350.35SASAN53.0667.02LAMEN8.3310.53AND27.7831.75PIKAS47.2259.65SUPAR55.5563.50SASAN49.1762.11NXD47.5749.33AND83.3395.26ZSHCTOL74.7083.46TOL79.8181.38SUPAR43.7550.00SUPAR37.5037.89NIVIK71.8879.61IGRAT71.8876.97ZSNJESBAG5.566.58ESBAG5.456.88HFE67.5082.30HFE66.6775.94OF50.6962.28OF50.0053.51TESIG54.1766.17KAKIS58.3370.18


#### 5.4.4. Analysis of Operational Efficiency

This paper uses the daily historical operation data of the same flight in the previous season to calculate the actual delay for statistical optimization. Take this data as the real delay amount, and compare and analyze it with the simulated delay data of the optimized flight plan. The results are shown in Table11. It can be concluded that the delay time of each time period after optimization is significantly reduced, and the operating efficiency has been significantly improved.Table 11
Situations of airport delays.
AirportBefore optimizationAfter optimizationAverage delay time (min)Punctuality rate (%)Average delay time (min)Punctuality rate (%)ZSPD15.5172.265.3288.89ZSSS11.1883.194.1293.41ZSHC14.9176.294.5395.73ZSNJ13.2575.044.9498.18The following table shows that the average delay time of the Yangtze River Delta airport group in 2018 is 14.96 minutes, and the flight punctuality rate is 76%. Among them, the average delay time at Shanghai Pudong Airport, Hongqiao Airport, Hangzhou Xiaoshan Airport, and Nanjing Lukou Airport are 15.51 minutes, 11.18 minutes, 14.91 minutes, and 13.25 minutes, respectively. The flight regularity rates of the above four airports are 72.26%, 83.19%, 76.29%, and 75.04%, respectively. After optimization, the simulated average delay time of the airport group is 4.88 minutes, and the flight punctuality rate is 95.67%. The average delay times of the four airports are 5.32 minutes, 4.12 minutes, 4.53 minutes, and 4.94 minutes, respectively, and the flight punctuality rates are 88.89%, 93.41%, 95.73%, and 98.18%, respectively. Through the comparison before and after optimization, the average delay time of the four airports has been reduced by nearly 10 minutes, and the normal rate has increased by nearly 20%. This leads to the conclusion that the operating efficiency has increased significantly.Delayed absorptive capacity can also characterize the effectiveness of flight scheduling. The amount of delayed absorption is defined as the difference between the departure delay and the arrival delay; that is, the amount of delayed absorption = (off-block time−planned departure time)−(block time−planned arrival time). A negative value indicates that part of the delay is absorbed; conversely, a positive value indicates that the delay is propagated and amplified. Therefore, the delayed absorption capacity can alleviate the ripple effect of inbound delays to a certain extent.The amount of delayed absorption is mainly used for arrival delays that have occurred in turnaround flights, which may affect subsequent departure flights. When passing through the terminal, the airport controller can adjust the turnaround time to absorb some of the delayed flights. In this way, the arrival of delayed flights can be tracked. According to the calculation formula of the amount of delayed absorption, it can be obtained that the average delay absorption amount of each inbound delayed flight at Pudong Airport, Hongqiao Airport, Lukou Airport, and Xiaoshan Airport is −3.57 minutes, −8.66 minutes, −6.58 minutes, and −4.24 minutes. The specific delay absorption of each airport is listed in Table12.Table 12
The airport delayed absorption and the average turnaround time.
AirportAverage amount of delayed absorption (min)Average turnaround time (min)ZSPD−3.5788ZSSS−8.6692ZSHC−6.5876ZSNJ−4.2473The delayed absorption capacity is closely related to planned turnaround time. At Xiaoshan Airport and Lukou Airport, which are both dual-runway airports and have relatively close traffic, the average turnaround time at Xiaoshan Airport is 1 : 16 : 23, which is significantly greater than 1 : 13 : 17 at Lukou Airport. The average delay at Xiaoshan Airport is 1 : 16 : 23. The average amount of delayed absorption is −18.58 minutes, which is also significantly higher than that of Lukou Airport (−4.24 minutes).


#### 5.4.5. Analysis of Fairness

In this paper, the optimized results are first substituted into formula (3) to calculate the fairness coefficient of each airport. The results are shown in Table 13. Then, the four fairness coefficients are, respectively, substituted into the airport fairness objective function. The initial value ε20  = 2.8751 is calculated from the initial solution obtained when the fairness objective is not added. After many iterations, the difference between the airport fairness coefficient and the average fairness coefficient is obtained. The maximum difference is 0.51; that is, the final value is ε2  = 0.51, which verifies the controllable relative fairness of the model. In the same way, this paper substitutes the results into formula (2) to calculate the fairness coefficient of airlines whose schedule accounts for more than 1%. The results are shown in Table 14. The fairness coefficients of the fifteen airlines that account for more than 1% all fluctuate around 1, which is not much different. Then, these fairness coefficients are substituted into the airline fairness objective, and the initial value ε10  = 3.0662 is calculated from the initial solution obtained when the fairness objective is not added. And the final value ε1 = 0.409 is obtained after many iterations. According to the indicators, the airlines are also relatively balanced. Although some airlines have “disadvantages” and some have “advantages,” these differences are small, and the model achieves the greatest degree of fairness among these airlines.Table 13
Fairness index of various airports.
AirportThe total deviation (min)Adjusted number of flightsProportion of flights (%)Fairness indexZSHC12908021.560.714ZSNJ11455219.570.699ZSPD318511636.991.028ZSSS275515521.881.503Table 14
Fairness index of various airlines.
AirlineFairness indexAirlineFairness indexAirlineFairness indexCES0.763CQH1.080CDG1.094CSN0.908CXA1.055CSC0.956CCA0.626CYZ1.163JAS1.184CSH0.681CHH0.737CDC1.319In general, by optimizing the flight slots of Shanghai Pudong Airport, Shanghai Hongqiao Airport, Nanjing Lukou Airport, and Hangzhou Xiaoshan Airport on Wednesday during the summer and autumn flight seasons in 2019, high-quality results can be obtained within 15 minutes, which proves that the algorithm has good convergence. First, the comparative flight distribution before optimization and the flight deviation after optimization are compared and analyzed. Then, the average deviation of each airline and airport is counted, and the resource utilization and operation efficiency at the time are compared and analyzed, which verifies the feasibility and reliability of the model. Finally, the fairness index of each airline and airport is calculated. On this basis, the fairness objective of the airline and the airport is calculated. By comparing the values before and after the iteration, the ability of the model to balance fairness is verified.


## 5.1. Airport Group Overview

At present, the Yangtze River Delta airport group includes 16 airports. Shanghai Pudong Airport, Shanghai Hongqiao Airport, Hangzhou Xiaoshan Airport, and Nanjing Lukou Airport are the four international or regional hub airports in the Yangtze River Delta airport group, which are relatively close to each other (Figure1, the label in the figure is the distance between each airport and Pudong Airport). They restrict and influence each other on airspace resources. Shanghai Pudong Airport is positioned as an international large-scale composite hub airport. It is mainly responsible for the take-off and landing of international routes and major domestic routes. It is the core carrier of Shanghai’s aviation hub construction. It has four operating runways and an annual passenger throughput of 76.0975 million in 2019. Shanghai Hongqiao Airport plays an auxiliary role in the hub structure. Compared with Shanghai Pudong Airport, it is more inclined to point-to-point operation in domestic cities, with dual-runway operation. In 2019, the annual passenger throughput reached 45.6766 million. Hangzhou Xiaoshan Airport and Nanjing Lukou Airport are both domestic regional hub airports and important trunk airports in China. They are both dual-runway airports. In 2019, the passenger throughput of the two airports was 40.108 million and 30.581 million, respectively. The number of aircraft landings is about half of that at Pudong Airport.Figure 1
Geographical location map of the Yangtze River Delta airport group.


## 5.2. Data

The data of this experiment are the summer and autumn flight schedule in 2019 and the historical operation data of the national airports during the summer and autumn flight seasons in 2018. The flight schedule data includes flight number, schedule, aircraft type, departure airport, landing airport, planned departure time, planned landing time, etc. Historical operation data includes flight number, registration number, aircraft type, departure airport, landing airport, planned departure time, actual departure time, planned landing time, actual landing time, etc. Based on the historical operation data of the four airports in the Yangtze River Delta during the entire summer and autumn flight seasons in 2018, this paper firstly conducts a detailed statistical analysis of the delays, releases, and traffic of several airports, estimates all the parameters required for flight schedule optimization, then selects the flight schedule on Wednesday of the domestic summer and autumn flight season in 2019 for optimization, and makes a comparative analysis.


## 5.3. Parameters

Table1 align the ICAO airport codes with the airport names. For the sake of convenience, the subsequent content of the article uses the ICAO airport codes to refer to the airports. At the same time, Table 1 gives hourly capacity parameters for the four airports. Table 2 shows the arrival and departure points of each airport. AND is not only the public departure point in the southwest and northeast of Hongqiao and Hangzhou airports, but also the public arrival point of the two airports. The arrival point and the departure point have no influence on each other due to the height difference. SASAN is the westbound public arrival and departure point for Pudong and Hongqiao airports. PIKAS and LAMEN are the public departure points for Pudong and Hongqiao airports in the northeast and northwest directions, respectively. CJ is the public arrival and departure points of Nanjing and Hangzhou airports in the southeast and northbound directions, respectively. According to historical operation data, the percentile method is used to determine the capacity of each arrival and departure point. However, due to the uneven distribution of flow at each corridor, there is very little flow at some corridors. Therefore, for a nonpublic corridor with a flow rate of less than 2 in 5 minutes, 24 is used as its total hourly capacity. The specific corridor capacity settings of each airport are shown in Table 3.Table 1
Airport hourly capacity.
ICAO airport codesAirportAdvertised capacityDeparture capacityArrival capacityZSPDShanghai Pudong airport795050ZSSSShanghai Hongqiao Airport473535ZSHCHangzhou Xiaoshan airport443535ZSNJNanjing Lukou airport382525Table 2
Airport arrival and departure points.
AirportDeparture pointArrival pointZSPDBOLEX/HSN/LAMEN/NXD/ODULO/PIKAS/SASANBK/DUMET/MATNU/SASANZSSSAND/SASAN/IBEGI/LAMEN/PIKAS/NXDAND/SASAN/SUPARZSHCTOL/SUPAR/NIVIKTOL/SUPAR/IGRATZSNJTESIG/ESBAG/HFE/OFKAKIS/ESBAG/HFE/OFTable 3
Capacity parameters of each arrival and departure point.
AirportDeparture pointThe 90th percentile value of the flowCapacityArrival pointThe 90th percentile value of the flowCapacityZSPDBOLEX224BK1824HSN724MATNU724NXD911DUMET1024ODULO424SASAN11LAMEN2121PIKAS1818SASAN1212ZSSSIBEGI124SASAN1515LAMEN44AND36PIKAS66SUPAR610SASAN1010AND918NXD1012ZSHCTOL1414TOL1313SUPAR47SUPAR58NIVIK824IGRAT824ZSNJESBAG1412ESBAG1413HFE1111HFE1314OF612OF613TESIG77KAKIS33Since different aircraft types receive services for different durations, the Civil Aviation Administration has stipulated the minimum turnaround time for different aircraft types according to the support capabilities of different airports. Table4 shows the minimum turnaround time for different models. The minimum turnaround time is classified according to the number of seats. The more the seats, the longer the minimum turnaround time. The actual turnaround time at the four airports is about an hour longer than the minimum turnaround time. For the maximum turnaround time, there is no relevant document clearly stipulated. We select the 90th percentile value of the historical actual turnaround time as the maximum turnaround time (see Table 5).Table 4
Minimum turnaround time for different models.
Number of seatsCommon modelsMinimum turnaround timeAirports in Pudong, Hongqiao, and HangzhouAirports in Nanjing≤60AT72, E145, CRJ, etc.40 minutes30 minutes61∼150CRJ7, E190, A319, B737 (Type 700 and below), etc.55 minutes40 minutes151∼250B737(Type 700 and above), B752, B762, B787, A310, A320, A321, etc.65 minutes45 minutes250∼500B747, B763, B777, A300, A330, A340, A350, MD1175 minutes65 minutes>500A380120 minutes120 minutesTable 5
Maximum turnaround time.
AirportMaximum turnaround time (minutes)ZSPD210ZSSS160ZSNJ155ZSHC185


## 5.4. Experimental Results and Analysis

### 5.4.1. Deviation Analysis

Tables6 and 7, respectively, show the airport deviations and various types of time deviations after the optimization of the flight schedule for the summer and autumn seasons in 2019. According to the adjustment priority of other flight schedule, historical adjustment schedule, and historical schedule, this paper sets the maximum deviations allowed for these three types of slots as the following times: 65 minutes, 40 minutes, and 15 minutes. This can reflect the model’s priority for different flight schedule allocation types. The higher the priority, the more the slot periods that can be shifted, the greater the possibility of being shifted, and the easier it is to be adjusted. Table 8 shows the deviation statistics of various flight slot types. As the adjustment priority decreases, the total deviation of the flight decreases in turn, and the adjusted flight amount and the average deviation also gradually decrease.Table 6
The number of flights adjusted by each airport and the average deviation.
AirportAdjusted number of flightsThe average deviation (minutes)AirportZSHC8016.13ZSHCZSNJ5222.02ZSNJZSPD11627.46ZSPDZSSS15517.77ZSSSTable 7
Adjusted number of flights and average deviation of various types of flight slots.
Type of flight slotsThe total deviation (minutes)The adjusted number of flightsThe average deviation (minutes)Historical schedule14556232.34Historical adjustment schedule303012382.45Other schedule389015482.51Table 8
The total deviation of each airline.
AirlineThe total deviation (minutes)The proportion of flights (%)AirlineThe total deviation (minutes)The proportion of flights (%)CES153023.94CHH2303.73CSN79010.38CSZ4153.64CCA4208.01CDG1801.97CSH4507.89CSC1551.94DKH6057.89JAS1601.61CQH4354.81CDC1751.58Figure2 shows the distribution of flight deviations at each airport. On the whole, as the deviation increases, the number of flights exceeding the deviation increases. But when the deviation exceeds 20 minutes, the number of flights exceeding the deviation decreases as the deviation increases. Among them, the three deviations of 5, 10, and 15 minutes are the most. Figure 3 shows the time distribution of the deviation of each airport. It can be concluded that the flight deviation has a large peak period from 12 : 00 to 17 : 00 with the change of time. This is because more flights exceed the capacity limit during the morning peak period of 9 : 00–11 : 00 in the morning and the evening inbound peak period of 16 : 00–20 : 00. As a result, the flight in this time period has to search for the optimal solution in a position with a larger front and rear span, so that the flight deviation during this period is much larger than that in other time periods. The deviations of Lukou Airport and Pudong Airport have more obvious peaks and valleys, which also shows that the slots of these two airports have obvious differences between peaks and valleys at the same time. In addition to the problem that the slot of the previous flight is occupied, the subsequent optimization interval of the peak arrival period from 16 : 00 to 20 : 00 is much more flexible. Therefore, the deviation is reduced accordingly. For example, flights after 20 : 00 are easier to optimize and have fewer deviations.Figure 2
Distribution of flight deviations at each airport.Figure 3
Time distribution of deviations for each airport.
(a)(b)(c)(d)


### 5.4.2. Analysis of Capacity-Flow Matching

Figure4 shows the distribution of flights before and after optimization at the four airports. This paper selects the flight slot during the busy period in the schedule plan for optimization. During the peak period of 10 : 00–13 : 00, the number of optimized flights is lower than the amount before optimization, and the same result also occurs during the peak inbound period of 16 : 00–19 : 00 and the peak freight period after 22 : 00; however, the conclusion is reversed in the morning period of 5 : 00–9:00.Figure 4
Flight distribution before and after airport optimization.
(a)(b)(c)(d)It can be concluded that flight slots that exceed capacity limits during peak hours are moved to the preceding or subsequent periods of lower flow by the optimized model. As a result, the peak and trough periods of flights in a day are more “smooth,” and the number of flights is distributed as evenly as possible in each period, fluctuating around capacity. The optimization results of Xiaoshan Airport more clearly show the characteristics of “smooth fluctuation”; that is, the number of flights fluctuates evenly around the corridor and the airport capacity limit and is distributed as evenly as possible in each time period. It can be concluded that the above model is feasible and reasonable. Similarly, the optimization results of Lukou Airport are smoother and rounder. The entire planned flights are optimized to meet the airport’s capacity as much as possible, and the number of flights is evenly distributed in each time period with almost no fluctuations. The optimization performance of this model is very high. This is because the arrival and departure points of Lukou Airport are less related to the other three airports, and there are fewer flights restricted by other airports at the corridor entrance.According to the principle of whether it is shared by two or more airports, corridors can be divided into nonpublic corridors and public corridors. Due to the large number of nonpublic corridors in the studied airport group, this paper only presents the capacity-flow matching analysis of some nonpublic corridors. The flow of each period before and after the optimization of corridor TOL, HFE, and BK is shown in Figure5. Before the optimization, the congestion period of 1-hour inbound and outbound TOL at Xiaoshan Airport is 9 : 45–10 : 45, 13 : 30–15 : 00, 16 : 00–17 : 30, and 18 : 30–19 : 30. After optimization, the above time periods meet the capacity limit. Similarly, before optimization, the congestion periods of HFE at Lukou Airport are 10 : 00–13 : 00 and 16 : 30–17 : 45, and the congestion periods of BK at Pudong Airport are 12 : 45–13 : 45, 16 : 15–18 : 00, and 21 : 30–23 : 15. After optimization, the flow restrictions of these nonpublic corridor entrances have been met to a considerable extent. Compared with the hourly distribution of flights before optimization, the hourly flow is more uniform, which greatly reduces the peak load at the corridor entrance and makes full use of the time resource of the corridor entrance. From this point of view, this optimization model is very convincing, and the optimization of the corridor entrance has reached the requirements of the model.Figure 5
Statistical results of hourly capacity and flow before and after optimization of nonpublic corridors.
(a)(b)(c)(d)(e)(f)This paper makes a statistical analysis of the historical flow of public arrival and departure points. The 90th percentile of the flow of the public arrival and departure point is used as the capacity parameter of this point. However, in some cases where the flow is too small, if the capacity parameter is used directly as the capacity, it may cause a waste of corridor resources. Therefore, for the corridor entrance whose hourly capacity parameter is less than 24 sorties, 24 sorties are specified as the capacity of this point; and the capacity parameter is used as the capacity for other corridor entrances. The capacity is then allocated in proportion to the flow of each airport passing through this point, and the corresponding capacity of different airports at this point is determined.The flow of each period before and after the optimization of corridor SUPAR, SASAN, LAMEN, and NXD is shown in Figure6. Before optimization, in the public arrival and departure points of Hongqiao and Xiaoshan airports, the congestion periods of 1-hour arrivals are 11 : 00–12 : 00, 14 : 00–16 : 00, 19 : 00–20 : 00, 21 : 00–22 : 00, and 23 : 00–24 : 00, and the departure congestion periods are 7 : 00–8:00, 10 : 00–11 : 00, 13 : 00–14 : 00, 17 : 00–18:00, and 20 : 00–21 : 00. After optimization, the capacity is within the constraints. SASAN is the westbound public arrival and departure point for Pudong and Hongqiao. Similarly, before the optimization, the 1-hour arrival congestion period was 22 : 00–23 : 00, and the departure congestion period was 8 : 00–9:00. After optimization, the capacity is also within the constraints. PIKAS is the public departure point for Pudong and Hongqiao airports in the northeast. Its capacity before and after optimization is within the constraints, so it will not be described in detail. Another public departure point of Pudong and Hongqiao Airport is LAMEN. The 1-hour departure congestion period before optimization is 16 : 00–17 : 00, and it is within the capacity limit after optimization. NXD is the public departure point of Pudong and Hongqiao airports in the southeast direction, respectively. The 1-hour departure congestion period before optimization is 6 : 00–7:00, and after optimization, it is also within the constraints of capacity constraints. After the optimization, the flight distribution of the abovementioned public arrival and departure points has been significantly improved, which is mainly reflected in the fact that the traffic during peak hours is limited within the capacity requirements. Thereby, the peak load at the corridor entrance is greatly reduced. In summary, the feasibility and reliability of the model are verified.Figure 6
Statistical results of hourly capacity and flow before and after optimization of public corridor entrances.
(a)(b)(c)(d)(e)(f)(g)(h)


### 5.4.3. Utilization Rate of Schedule Resources

According to the capacity of the airport schedule, the data before and after the optimization of the schedule plan in 2019 are used to calculate the resource utilization rate of the four airports between 6 : 00 and 24 : 00, respectively. The results are shown in Table9. Pudong Airport is 98.54% and 99.25%; Hongqiao Airport is 94.01% and 94.47%; Lukou Airport is 80.65% and 94.93%; Xiaoshan Airport is 97.70% and 97.70%. It can be concluded that, before and after the optimization, only Xiaoshan Airport’s time resource utilization remains unchanged, and it is slightly improved in other airports. The optimization model in this paper can improve the utilization rate of time resources, which verifies the reliability of the model.Table 9
Utilization of airport slots.
AirportBefore optimization (%)After optimization (%)ZSHC97.7097.70ZSNJ80.6594.93ZSPD98.5499.25ZSSS94.0194.47Table10 shows the utilization rate of corridor entrances of each airport. In general, the time resources at the airport are tighter than those at the corridor entrance. After optimization, the planned utilization rate of time resources at the airport is close to 95%, and the flexibility space is limited. The utilization rate of the entry and departure points is not much different before and after the optimization but both have improved. Except for the corridor entrance points with very few flights, the utilization rate of the public arrival and departure points has increased significantly.Table 10
Utilization rate of airport corridors.
AirportDeparture pointSchedule plan (%)Actual operation (%)Arrival pointSchedule plan (%)Actual operation (%)ZSPDBOLEX1.841.84BK65.9065.90HSN29.0329.49DUMET55.7657.14NXD46.0846.54MATNU37.7937.79ODULO32.2632.26SASAN33.3315.79LAMEN4.173.26PIKAS1.161.17SASAN68.0674.56ZSSSIBEGI0.350.35SASAN53.0667.02LAMEN8.3310.53AND27.7831.75PIKAS47.2259.65SUPAR55.5563.50SASAN49.1762.11NXD47.5749.33AND83.3395.26ZSHCTOL74.7083.46TOL79.8181.38SUPAR43.7550.00SUPAR37.5037.89NIVIK71.8879.61IGRAT71.8876.97ZSNJESBAG5.566.58ESBAG5.456.88HFE67.5082.30HFE66.6775.94OF50.6962.28OF50.0053.51TESIG54.1766.17KAKIS58.3370.18


### 5.4.4. Analysis of Operational Efficiency

This paper uses the daily historical operation data of the same flight in the previous season to calculate the actual delay for statistical optimization. Take this data as the real delay amount, and compare and analyze it with the simulated delay data of the optimized flight plan. The results are shown in Table11. It can be concluded that the delay time of each time period after optimization is significantly reduced, and the operating efficiency has been significantly improved.Table 11
Situations of airport delays.
AirportBefore optimizationAfter optimizationAverage delay time (min)Punctuality rate (%)Average delay time (min)Punctuality rate (%)ZSPD15.5172.265.3288.89ZSSS11.1883.194.1293.41ZSHC14.9176.294.5395.73ZSNJ13.2575.044.9498.18The following table shows that the average delay time of the Yangtze River Delta airport group in 2018 is 14.96 minutes, and the flight punctuality rate is 76%. Among them, the average delay time at Shanghai Pudong Airport, Hongqiao Airport, Hangzhou Xiaoshan Airport, and Nanjing Lukou Airport are 15.51 minutes, 11.18 minutes, 14.91 minutes, and 13.25 minutes, respectively. The flight regularity rates of the above four airports are 72.26%, 83.19%, 76.29%, and 75.04%, respectively. After optimization, the simulated average delay time of the airport group is 4.88 minutes, and the flight punctuality rate is 95.67%. The average delay times of the four airports are 5.32 minutes, 4.12 minutes, 4.53 minutes, and 4.94 minutes, respectively, and the flight punctuality rates are 88.89%, 93.41%, 95.73%, and 98.18%, respectively. Through the comparison before and after optimization, the average delay time of the four airports has been reduced by nearly 10 minutes, and the normal rate has increased by nearly 20%. This leads to the conclusion that the operating efficiency has increased significantly.Delayed absorptive capacity can also characterize the effectiveness of flight scheduling. The amount of delayed absorption is defined as the difference between the departure delay and the arrival delay; that is, the amount of delayed absorption = (off-block time−planned departure time)−(block time−planned arrival time). A negative value indicates that part of the delay is absorbed; conversely, a positive value indicates that the delay is propagated and amplified. Therefore, the delayed absorption capacity can alleviate the ripple effect of inbound delays to a certain extent.The amount of delayed absorption is mainly used for arrival delays that have occurred in turnaround flights, which may affect subsequent departure flights. When passing through the terminal, the airport controller can adjust the turnaround time to absorb some of the delayed flights. In this way, the arrival of delayed flights can be tracked. According to the calculation formula of the amount of delayed absorption, it can be obtained that the average delay absorption amount of each inbound delayed flight at Pudong Airport, Hongqiao Airport, Lukou Airport, and Xiaoshan Airport is −3.57 minutes, −8.66 minutes, −6.58 minutes, and −4.24 minutes. The specific delay absorption of each airport is listed in Table12.Table 12
The airport delayed absorption and the average turnaround time.
AirportAverage amount of delayed absorption (min)Average turnaround time (min)ZSPD−3.5788ZSSS−8.6692ZSHC−6.5876ZSNJ−4.2473The delayed absorption capacity is closely related to planned turnaround time. At Xiaoshan Airport and Lukou Airport, which are both dual-runway airports and have relatively close traffic, the average turnaround time at Xiaoshan Airport is 1 : 16 : 23, which is significantly greater than 1 : 13 : 17 at Lukou Airport. The average delay at Xiaoshan Airport is 1 : 16 : 23. The average amount of delayed absorption is −18.58 minutes, which is also significantly higher than that of Lukou Airport (−4.24 minutes).


### 5.4.5. Analysis of Fairness

In this paper, the optimized results are first substituted into formula (3) to calculate the fairness coefficient of each airport. The results are shown in Table 13. Then, the four fairness coefficients are, respectively, substituted into the airport fairness objective function. The initial value ε20  = 2.8751 is calculated from the initial solution obtained when the fairness objective is not added. After many iterations, the difference between the airport fairness coefficient and the average fairness coefficient is obtained. The maximum difference is 0.51; that is, the final value is ε2  = 0.51, which verifies the controllable relative fairness of the model. In the same way, this paper substitutes the results into formula (2) to calculate the fairness coefficient of airlines whose schedule accounts for more than 1%. The results are shown in Table 14. The fairness coefficients of the fifteen airlines that account for more than 1% all fluctuate around 1, which is not much different. Then, these fairness coefficients are substituted into the airline fairness objective, and the initial value ε10  = 3.0662 is calculated from the initial solution obtained when the fairness objective is not added. And the final value ε1 = 0.409 is obtained after many iterations. According to the indicators, the airlines are also relatively balanced. Although some airlines have “disadvantages” and some have “advantages,” these differences are small, and the model achieves the greatest degree of fairness among these airlines.Table 13
Fairness index of various airports.
AirportThe total deviation (min)Adjusted number of flightsProportion of flights (%)Fairness indexZSHC12908021.560.714ZSNJ11455219.570.699ZSPD318511636.991.028ZSSS275515521.881.503Table 14
Fairness index of various airlines.
AirlineFairness indexAirlineFairness indexAirlineFairness indexCES0.763CQH1.080CDG1.094CSN0.908CXA1.055CSC0.956CCA0.626CYZ1.163JAS1.184CSH0.681CHH0.737CDC1.319In general, by optimizing the flight slots of Shanghai Pudong Airport, Shanghai Hongqiao Airport, Nanjing Lukou Airport, and Hangzhou Xiaoshan Airport on Wednesday during the summer and autumn flight seasons in 2019, high-quality results can be obtained within 15 minutes, which proves that the algorithm has good convergence. First, the comparative flight distribution before optimization and the flight deviation after optimization are compared and analyzed. Then, the average deviation of each airline and airport is counted, and the resource utilization and operation efficiency at the time are compared and analyzed, which verifies the feasibility and reliability of the model. Finally, the fairness index of each airline and airport is calculated. On this basis, the fairness objective of the airline and the airport is calculated. By comparing the values before and after the iteration, the ability of the model to balance fairness is verified.


## 5.4.1. Deviation Analysis

Tables6 and 7, respectively, show the airport deviations and various types of time deviations after the optimization of the flight schedule for the summer and autumn seasons in 2019. According to the adjustment priority of other flight schedule, historical adjustment schedule, and historical schedule, this paper sets the maximum deviations allowed for these three types of slots as the following times: 65 minutes, 40 minutes, and 15 minutes. This can reflect the model’s priority for different flight schedule allocation types. The higher the priority, the more the slot periods that can be shifted, the greater the possibility of being shifted, and the easier it is to be adjusted. Table 8 shows the deviation statistics of various flight slot types. As the adjustment priority decreases, the total deviation of the flight decreases in turn, and the adjusted flight amount and the average deviation also gradually decrease.Table 6
The number of flights adjusted by each airport and the average deviation.
AirportAdjusted number of flightsThe average deviation (minutes)AirportZSHC8016.13ZSHCZSNJ5222.02ZSNJZSPD11627.46ZSPDZSSS15517.77ZSSSTable 7
Adjusted number of flights and average deviation of various types of flight slots.
Type of flight slotsThe total deviation (minutes)The adjusted number of flightsThe average deviation (minutes)Historical schedule14556232.34Historical adjustment schedule303012382.45Other schedule389015482.51Table 8
The total deviation of each airline.
AirlineThe total deviation (minutes)The proportion of flights (%)AirlineThe total deviation (minutes)The proportion of flights (%)CES153023.94CHH2303.73CSN79010.38CSZ4153.64CCA4208.01CDG1801.97CSH4507.89CSC1551.94DKH6057.89JAS1601.61CQH4354.81CDC1751.58Figure2 shows the distribution of flight deviations at each airport. On the whole, as the deviation increases, the number of flights exceeding the deviation increases. But when the deviation exceeds 20 minutes, the number of flights exceeding the deviation decreases as the deviation increases. Among them, the three deviations of 5, 10, and 15 minutes are the most. Figure 3 shows the time distribution of the deviation of each airport. It can be concluded that the flight deviation has a large peak period from 12 : 00 to 17 : 00 with the change of time. This is because more flights exceed the capacity limit during the morning peak period of 9 : 00–11 : 00 in the morning and the evening inbound peak period of 16 : 00–20 : 00. As a result, the flight in this time period has to search for the optimal solution in a position with a larger front and rear span, so that the flight deviation during this period is much larger than that in other time periods. The deviations of Lukou Airport and Pudong Airport have more obvious peaks and valleys, which also shows that the slots of these two airports have obvious differences between peaks and valleys at the same time. In addition to the problem that the slot of the previous flight is occupied, the subsequent optimization interval of the peak arrival period from 16 : 00 to 20 : 00 is much more flexible. Therefore, the deviation is reduced accordingly. For example, flights after 20 : 00 are easier to optimize and have fewer deviations.Figure 2
Distribution of flight deviations at each airport.Figure 3
Time distribution of deviations for each airport.
(a)(b)(c)(d)


## 5.4.2. Analysis of Capacity-Flow Matching

Figure4 shows the distribution of flights before and after optimization at the four airports. This paper selects the flight slot during the busy period in the schedule plan for optimization. During the peak period of 10 : 00–13 : 00, the number of optimized flights is lower than the amount before optimization, and the same result also occurs during the peak inbound period of 16 : 00–19 : 00 and the peak freight period after 22 : 00; however, the conclusion is reversed in the morning period of 5 : 00–9:00.Figure 4
Flight distribution before and after airport optimization.
(a)(b)(c)(d)It can be concluded that flight slots that exceed capacity limits during peak hours are moved to the preceding or subsequent periods of lower flow by the optimized model. As a result, the peak and trough periods of flights in a day are more “smooth,” and the number of flights is distributed as evenly as possible in each period, fluctuating around capacity. The optimization results of Xiaoshan Airport more clearly show the characteristics of “smooth fluctuation”; that is, the number of flights fluctuates evenly around the corridor and the airport capacity limit and is distributed as evenly as possible in each time period. It can be concluded that the above model is feasible and reasonable. Similarly, the optimization results of Lukou Airport are smoother and rounder. The entire planned flights are optimized to meet the airport’s capacity as much as possible, and the number of flights is evenly distributed in each time period with almost no fluctuations. The optimization performance of this model is very high. This is because the arrival and departure points of Lukou Airport are less related to the other three airports, and there are fewer flights restricted by other airports at the corridor entrance.According to the principle of whether it is shared by two or more airports, corridors can be divided into nonpublic corridors and public corridors. Due to the large number of nonpublic corridors in the studied airport group, this paper only presents the capacity-flow matching analysis of some nonpublic corridors. The flow of each period before and after the optimization of corridor TOL, HFE, and BK is shown in Figure5. Before the optimization, the congestion period of 1-hour inbound and outbound TOL at Xiaoshan Airport is 9 : 45–10 : 45, 13 : 30–15 : 00, 16 : 00–17 : 30, and 18 : 30–19 : 30. After optimization, the above time periods meet the capacity limit. Similarly, before optimization, the congestion periods of HFE at Lukou Airport are 10 : 00–13 : 00 and 16 : 30–17 : 45, and the congestion periods of BK at Pudong Airport are 12 : 45–13 : 45, 16 : 15–18 : 00, and 21 : 30–23 : 15. After optimization, the flow restrictions of these nonpublic corridor entrances have been met to a considerable extent. Compared with the hourly distribution of flights before optimization, the hourly flow is more uniform, which greatly reduces the peak load at the corridor entrance and makes full use of the time resource of the corridor entrance. From this point of view, this optimization model is very convincing, and the optimization of the corridor entrance has reached the requirements of the model.Figure 5
Statistical results of hourly capacity and flow before and after optimization of nonpublic corridors.
(a)(b)(c)(d)(e)(f)This paper makes a statistical analysis of the historical flow of public arrival and departure points. The 90th percentile of the flow of the public arrival and departure point is used as the capacity parameter of this point. However, in some cases where the flow is too small, if the capacity parameter is used directly as the capacity, it may cause a waste of corridor resources. Therefore, for the corridor entrance whose hourly capacity parameter is less than 24 sorties, 24 sorties are specified as the capacity of this point; and the capacity parameter is used as the capacity for other corridor entrances. The capacity is then allocated in proportion to the flow of each airport passing through this point, and the corresponding capacity of different airports at this point is determined.The flow of each period before and after the optimization of corridor SUPAR, SASAN, LAMEN, and NXD is shown in Figure6. Before optimization, in the public arrival and departure points of Hongqiao and Xiaoshan airports, the congestion periods of 1-hour arrivals are 11 : 00–12 : 00, 14 : 00–16 : 00, 19 : 00–20 : 00, 21 : 00–22 : 00, and 23 : 00–24 : 00, and the departure congestion periods are 7 : 00–8:00, 10 : 00–11 : 00, 13 : 00–14 : 00, 17 : 00–18:00, and 20 : 00–21 : 00. After optimization, the capacity is within the constraints. SASAN is the westbound public arrival and departure point for Pudong and Hongqiao. Similarly, before the optimization, the 1-hour arrival congestion period was 22 : 00–23 : 00, and the departure congestion period was 8 : 00–9:00. After optimization, the capacity is also within the constraints. PIKAS is the public departure point for Pudong and Hongqiao airports in the northeast. Its capacity before and after optimization is within the constraints, so it will not be described in detail. Another public departure point of Pudong and Hongqiao Airport is LAMEN. The 1-hour departure congestion period before optimization is 16 : 00–17 : 00, and it is within the capacity limit after optimization. NXD is the public departure point of Pudong and Hongqiao airports in the southeast direction, respectively. The 1-hour departure congestion period before optimization is 6 : 00–7:00, and after optimization, it is also within the constraints of capacity constraints. After the optimization, the flight distribution of the abovementioned public arrival and departure points has been significantly improved, which is mainly reflected in the fact that the traffic during peak hours is limited within the capacity requirements. Thereby, the peak load at the corridor entrance is greatly reduced. In summary, the feasibility and reliability of the model are verified.Figure 6
Statistical results of hourly capacity and flow before and after optimization of public corridor entrances.
(a)(b)(c)(d)(e)(f)(g)(h)


## 5.4.3. Utilization Rate of Schedule Resources

According to the capacity of the airport schedule, the data before and after the optimization of the schedule plan in 2019 are used to calculate the resource utilization rate of the four airports between 6 : 00 and 24 : 00, respectively. The results are shown in Table9. Pudong Airport is 98.54% and 99.25%; Hongqiao Airport is 94.01% and 94.47%; Lukou Airport is 80.65% and 94.93%; Xiaoshan Airport is 97.70% and 97.70%. It can be concluded that, before and after the optimization, only Xiaoshan Airport’s time resource utilization remains unchanged, and it is slightly improved in other airports. The optimization model in this paper can improve the utilization rate of time resources, which verifies the reliability of the model.Table 9
Utilization of airport slots.
AirportBefore optimization (%)After optimization (%)ZSHC97.7097.70ZSNJ80.6594.93ZSPD98.5499.25ZSSS94.0194.47Table10 shows the utilization rate of corridor entrances of each airport. In general, the time resources at the airport are tighter than those at the corridor entrance. After optimization, the planned utilization rate of time resources at the airport is close to 95%, and the flexibility space is limited. The utilization rate of the entry and departure points is not much different before and after the optimization but both have improved. Except for the corridor entrance points with very few flights, the utilization rate of the public arrival and departure points has increased significantly.Table 10
Utilization rate of airport corridors.
AirportDeparture pointSchedule plan (%)Actual operation (%)Arrival pointSchedule plan (%)Actual operation (%)ZSPDBOLEX1.841.84BK65.9065.90HSN29.0329.49DUMET55.7657.14NXD46.0846.54MATNU37.7937.79ODULO32.2632.26SASAN33.3315.79LAMEN4.173.26PIKAS1.161.17SASAN68.0674.56ZSSSIBEGI0.350.35SASAN53.0667.02LAMEN8.3310.53AND27.7831.75PIKAS47.2259.65SUPAR55.5563.50SASAN49.1762.11NXD47.5749.33AND83.3395.26ZSHCTOL74.7083.46TOL79.8181.38SUPAR43.7550.00SUPAR37.5037.89NIVIK71.8879.61IGRAT71.8876.97ZSNJESBAG5.566.58ESBAG5.456.88HFE67.5082.30HFE66.6775.94OF50.6962.28OF50.0053.51TESIG54.1766.17KAKIS58.3370.18


## 5.4.4. Analysis of Operational Efficiency

This paper uses the daily historical operation data of the same flight in the previous season to calculate the actual delay for statistical optimization. Take this data as the real delay amount, and compare and analyze it with the simulated delay data of the optimized flight plan. The results are shown in Table11. It can be concluded that the delay time of each time period after optimization is significantly reduced, and the operating efficiency has been significantly improved.Table 11
Situations of airport delays.
AirportBefore optimizationAfter optimizationAverage delay time (min)Punctuality rate (%)Average delay time (min)Punctuality rate (%)ZSPD15.5172.265.3288.89ZSSS11.1883.194.1293.41ZSHC14.9176.294.5395.73ZSNJ13.2575.044.9498.18The following table shows that the average delay time of the Yangtze River Delta airport group in 2018 is 14.96 minutes, and the flight punctuality rate is 76%. Among them, the average delay time at Shanghai Pudong Airport, Hongqiao Airport, Hangzhou Xiaoshan Airport, and Nanjing Lukou Airport are 15.51 minutes, 11.18 minutes, 14.91 minutes, and 13.25 minutes, respectively. The flight regularity rates of the above four airports are 72.26%, 83.19%, 76.29%, and 75.04%, respectively. After optimization, the simulated average delay time of the airport group is 4.88 minutes, and the flight punctuality rate is 95.67%. The average delay times of the four airports are 5.32 minutes, 4.12 minutes, 4.53 minutes, and 4.94 minutes, respectively, and the flight punctuality rates are 88.89%, 93.41%, 95.73%, and 98.18%, respectively. Through the comparison before and after optimization, the average delay time of the four airports has been reduced by nearly 10 minutes, and the normal rate has increased by nearly 20%. This leads to the conclusion that the operating efficiency has increased significantly.Delayed absorptive capacity can also characterize the effectiveness of flight scheduling. The amount of delayed absorption is defined as the difference between the departure delay and the arrival delay; that is, the amount of delayed absorption = (off-block time−planned departure time)−(block time−planned arrival time). A negative value indicates that part of the delay is absorbed; conversely, a positive value indicates that the delay is propagated and amplified. Therefore, the delayed absorption capacity can alleviate the ripple effect of inbound delays to a certain extent.The amount of delayed absorption is mainly used for arrival delays that have occurred in turnaround flights, which may affect subsequent departure flights. When passing through the terminal, the airport controller can adjust the turnaround time to absorb some of the delayed flights. In this way, the arrival of delayed flights can be tracked. According to the calculation formula of the amount of delayed absorption, it can be obtained that the average delay absorption amount of each inbound delayed flight at Pudong Airport, Hongqiao Airport, Lukou Airport, and Xiaoshan Airport is −3.57 minutes, −8.66 minutes, −6.58 minutes, and −4.24 minutes. The specific delay absorption of each airport is listed in Table12.Table 12
The airport delayed absorption and the average turnaround time.
AirportAverage amount of delayed absorption (min)Average turnaround time (min)ZSPD−3.5788ZSSS−8.6692ZSHC−6.5876ZSNJ−4.2473The delayed absorption capacity is closely related to planned turnaround time. At Xiaoshan Airport and Lukou Airport, which are both dual-runway airports and have relatively close traffic, the average turnaround time at Xiaoshan Airport is 1 : 16 : 23, which is significantly greater than 1 : 13 : 17 at Lukou Airport. The average delay at Xiaoshan Airport is 1 : 16 : 23. The average amount of delayed absorption is −18.58 minutes, which is also significantly higher than that of Lukou Airport (−4.24 minutes).


## 5.4.5. Analysis of Fairness

In this paper, the optimized results are first substituted into formula (3) to calculate the fairness coefficient of each airport. The results are shown in Table 13. Then, the four fairness coefficients are, respectively, substituted into the airport fairness objective function. The initial value ε20  = 2.8751 is calculated from the initial solution obtained when the fairness objective is not added. After many iterations, the difference between the airport fairness coefficient and the average fairness coefficient is obtained. The maximum difference is 0.51; that is, the final value is ε2  = 0.51, which verifies the controllable relative fairness of the model. In the same way, this paper substitutes the results into formula (2) to calculate the fairness coefficient of airlines whose schedule accounts for more than 1%. The results are shown in Table 14. The fairness coefficients of the fifteen airlines that account for more than 1% all fluctuate around 1, which is not much different. Then, these fairness coefficients are substituted into the airline fairness objective, and the initial value ε10  = 3.0662 is calculated from the initial solution obtained when the fairness objective is not added. And the final value ε1 = 0.409 is obtained after many iterations. According to the indicators, the airlines are also relatively balanced. Although some airlines have “disadvantages” and some have “advantages,” these differences are small, and the model achieves the greatest degree of fairness among these airlines.Table 13
Fairness index of various airports.
AirportThe total deviation (min)Adjusted number of flightsProportion of flights (%)Fairness indexZSHC12908021.560.714ZSNJ11455219.570.699ZSPD318511636.991.028ZSSS275515521.881.503Table 14
Fairness index of various airlines.
AirlineFairness indexAirlineFairness indexAirlineFairness indexCES0.763CQH1.080CDG1.094CSN0.908CXA1.055CSC0.956CCA0.626CYZ1.163JAS1.184CSH0.681CHH0.737CDC1.319In general, by optimizing the flight slots of Shanghai Pudong Airport, Shanghai Hongqiao Airport, Nanjing Lukou Airport, and Hangzhou Xiaoshan Airport on Wednesday during the summer and autumn flight seasons in 2019, high-quality results can be obtained within 15 minutes, which proves that the algorithm has good convergence. First, the comparative flight distribution before optimization and the flight deviation after optimization are compared and analyzed. Then, the average deviation of each airline and airport is counted, and the resource utilization and operation efficiency at the time are compared and analyzed, which verifies the feasibility and reliability of the model. Finally, the fairness index of each airline and airport is calculated. On this basis, the fairness objective of the airline and the airport is calculated. By comparing the values before and after the iteration, the ability of the model to balance fairness is verified.


## 6. Conclusions

This paper first briefly introduces the airport group and then introduces a collaborative optimization model for the flight schedule of the airport group considering the allocation efficiency and fairness and designs a model solving algorithm based on the constraint method. Finally, the optimization model is verified by taking the world-class airport cluster in the Yangtze River Delta as an example.First of all, this paper designs a feasible solution to the problem of overlapping public resources between airports and the realization of fairness, discusses the fairness between airports and between airlines and airlines, and proposes fairness indicators. This indicator not only achieves fairness among airlines, but also balances fairness among airports. Then a three-objective model is established with the goal of minimizing the regional overall runtime deviation and fairness. In addition to the basic restrictions on airport operations, the restrictions on the capacity of corridors in the terminal area of the airport are also considered. In order to solve the model, the constraint method is used to transform the multiobjective optimization problem into a single-objective optimization problem and solve it.Based on the historical operation data of the airport group in the Yangtze River Delta, the capacity of each corridor is estimated, and the capacity share of the public corridor in each involved airport is allocated, and the maximum turnaround time of aircraft types in each airport is calculated. Based on the above parameters, the model was flexibly used to optimize the flight schedule for the 2019 summer and autumn flight season, and a relatively reliable feasible solution was obtained. Finally, the results are statistically analyzed in terms of deviation distribution, capacity-flow matching, resource utilization, operating efficiency, and fairness. The feasibility and reliability of the model are verified by comparing the flight schedule statistics before optimization.



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*Source: 1418911-2022-09-10.xml*