Strategic flight assignment approach based on multi-objective parallel evolution algorithm with dynamic migration interval

Accepted Manuscript Strategic Flight Assignment Approach based on Multi-objective Parallel Evolution Algorithm with Dynamic Migration Interval Zhang Xuejun, Guan Xiangmin, Zhu Yanbo, Lei Jiaxing PII: DOI: Reference:

S1000-9361(15)00032-1 http://dx.doi.org/10.1016/j.cja.2015.01.012 CJA 430

To appear in: Received Date: Revised Date: Accepted Date:

14 June 2014 7 August 2014 16 January 2015

Please cite this article as: Z. Xuejun, G. Xiangmin, Z. Yanbo, L. Jiaxing, Strategic Flight Assignment Approach based on Multi-objective Parallel Evolution Algorithm with Dynamic Migration Interval, (2015), doi: http:// dx.doi.org/10.1016/j.cja.2015.01.012

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Strategic Flight Assignment Approach based on Multi-objective Parallel Evolution Algorithm with Dynamic Migration Interval ZHANG Xuejun

a, b

, GUAN Xiangmina, b, ZHU Yanboa, b, c,*, LEI Jiaxinga, b,

a

School of Electronic and Information Engineering, Beihang University, Beijing 100191, China b

National Key Laboratory of CNS/ATM, Beihang University, Beijing 100191, China c

Aviation Data Communication Corporation, Beijing 100191, China

Received 14 June 2014; Revised 7 August 2014; accepted 16 January 2015

Abstract

The continuous growth of air traffic has led to acute airspace congestion and severe delays, which threatens operation safety and cause enormous economic loss. Flight assignment is an economical and effective strategic plan to reduce the flight delay and airspace congestion by reasonably regulating the air traffic flow of China. However, it is a large-scale combinatorial optimization problem which is difficult to solve. In order to improve the quality of solutions, an effective multi-objective parallel evolution algorithm (MPEA) framework with dynamic migration interval strategy is presented in this work. Firstly, multiple evolution populations are constructed to solve the problem simultaneously to enhance the optimization capability. Then a new strategy is proposed to dynamically change the migration interval among different evolution populations to improve the efficiency of the cooperation of populations. Finally, the cooperative co-evolution (CC) algorithm combined with non-dominated sorting genetic algorithm II (NSGA-II) is introduced for each population. Empirical studies using the real air traffic data of the Chinese air route network and daily flight plans show that our method outperforms the existing approaches, multi-objective genetic algorithm (MOGA), multi-objective evolutionary algorithm based on decomposition (MOEA/D), CC-based multi-objective algorithm (CCMA) as well as other two MPEAs with different migration interval strategies. Keywords: Air traffic flow management; Flight assignment; Parallel evolution algorithm; Dynamic migration interval strategy; Cooperative co-evolution

1.

Introduction

In the past decade, the air transportation of the whole world has increased rapidly. The increasing growth of the number of flights in the airspace has caused severe airspace congestion which not only threatens the safety of airspace operation, but also leads to massive economic loss. For instance, the flight delay in China has wasted billions of dollars in the past decades.1,2 How to reduce the air congestion and flight delay is always the research highlight for researchers from air traffic management.3 In general, this problem is solved by strategically adjusting the departure time and flight routes of the involved flights to balance the air traffic flow among different sectors. For instance, when a sector is predicted to undergo congestion, the involved flights are delayed at the departure airports or their routes are partly changed in order to avoid aggravating the congestion of this *

Corresponding author. Tel.: +86 10 82338626 E-mail address :[email protected]

sector. However, the flight assignment problem includes thousands of flights with tightly coupled decision variables and constraints, and it has been proved to be NP hard which is very difficult to deal with. In the early research, the problem is simplified as a mono-objective instance. The researchers assigned ground delays for flights in single or multiple airports, which is known as ground holding program.4,5 With the aim to ensure safety and save fuel energy, this strategy transfers the flight time in the air to the delay on the ground by optimizing the departure time for flights. Abad and Clarke proposed a routes assignment method for flights based on mixed integer linear programming to reduce airspace congestion.6 However, when the number of flights increases, it will provide more flexible solutions for controllers with the consideration of both the time and the space adjustment. Delahaye and Odoni introduced stochastic optimization techniques to optimize the routes and time slots simultaneously.7 They used genetic algorithm to solve the problem.8,9 Bertsimas and Patterson presented an efficient deterministic approach with consideration of both the time and route assignment.10 Sun et al. proposed a large-capacity cell transmission model for air traffic flow management11-13 and applied integer program to solve it. Recently, the cooperative co-evolution multi-objective algorithm (CCMA) was introduced to resolve the flight assignment problem in a simplified network.14 These works only formulate the minimization of the airspace congestion or the flight delay as a mono-objective problem. However, in the real operation, controllers are more likely to seek a good trade-off between the airspace congestion and the flight delay. Hence, Daniel et al. used a Multi-objective genetic algorithm (MOGA) to optimize the airspace congestion and flight delays at the same time.15 Real data in French airspace is used to verify their method. Similar work has been considered using the empirical data of China.16 However, MOGA is apt to fall into the local optima because of the huge search space. The parallel evolution algorithms (PEAs) show great superiority when dealing with large-scale combinatorial optimization problems because there are several populations evolving simultaneously. During the evolving process, different populations will exchange individuals, which is called migration.17-20 The migration interval is a critical problem in PEAs which can affect the solution quality dramatically. 21-23 Currently, the migration interval is considered as a constant. 24-25 In order to avoid premature and obtain better solutions, we propose a multi-objective parallel evolution algorithm to solve this problem. Firstly, multiple evolution populations are constructed to solve the problem simultaneously to improve the optimization capability. Then, a dynamic migration interval is proposed to improve the efficiency of the cooperation of populations. Finally, the cooperative co-evolution (CC) algorithm combined with non-dominated sorting genetic algorithm II （NSGA-II） is introduced for each population. Experiments with the real air traffic data from the China air route network and daily flight plans show that the proposed approach can improve the solution quality effectively and efficiently, and it is superior to the existing approaches such as the multi-objective genetic algorithm (MOGA), the multi-objective evolutionary algorithm based on decomposition (MOEA/D), CC-based multi-objective algorithm as well as other two MPEAs with different migration interval strategies. The paper is organized as follows. The problem formulation is described in Section 2. Section 3 presents the framework of the multi-objective parallel evolution algorithm and the dynamic migration interval strategy. Experimental results on the real data of the national route of China are given in Section 4. Finally, we conclude the work in Section 5. 2.

Problem formulation

A flight path will be re-arranged with the extra cost as less as possible when congestion happens.26 Fig.1 demonstrates how a flight will choose another path. The rectangular airspace includes several sectors and there are several waypoints (circle points) in the sectors. It is supposed that a flight from airport A in the left bottom will fly to airport B in the top right corner. In real operation, the aircraft does not fly along a straight line from A to B. It will fly along the waypoints between airport A and airport B, because a waypoint is a navigation marker and flights need the information, such as the desired track and heading direction, which can be provided by the ground navaids. It is assumed that there are three paths passing through different sectors from A to B in Fig. 1. For example, if congestion in sector 6 becomes severe, then the flight can choose path 3 which does not pass through sector 6. A flight plan L is previously determined by the air traffic management department, airlines and airports which can be described as follows:

L = {( S1 ,Tin1 , Tout1 ) , ( S 2 , Tin 2 , Tout 2 ) ,..., ( S k , Tin k , Tout k ) ,...}

(1)

where S k is the kth sector the aircraft will pass, Tink is the time slot it enters into the sector and Tout k is the time slot it leaves the sector.

Fig. 1 Description of airspace structure.

2.1. Decision variables Supposed that there are n flights (F1, F2,…, Fn) with different flight plans. For each flight i (1 £ i £ n) , there is a pair of decision variable (d i , ri ) associated with it, where d i represents the delay from the original departure time slot and

ri

is a feasible route. If

di

is negative, it means the

flight will take off ahead of the original time. Besides, for real operation consideration,

di

should not

be delayed or advanced too much and ri should not much longer than the original path. So Θ and R denote the slot set and the route set can be described as follows:

Θ = {-d q , -d q + 1,...，- 1,0,1, 2， ..., d p - 1, d p }

(2)

R = {r0 , r1, r2 ,..., rmax } where

dp

is the maximum delay,

dq

(3)

the maximum advance for a flight,

r0

the best route and

rmax

the worst one. Our goal is to find the best departure time and routes for all the flights from the two finite sets. Hence, we can see that the flight assignment problem is a large-scale combinatorial optimization problem. 2.2. Objective functions With the consideration of ensuring safety and efficiency, the airspace congestion and the total flight delay are considered as objectives which will be optimized at the same time. The two objective functions can be descripted as follows. 2.2.1. Airspace congestion

The workload of controllers of a sector can reflect the airspace congestion. The workload sector

Sk

workload

at time t includes two parts, the monitoring workload

Wco,t S

k

.8 It can be roughly expressed as

t Wmo, Sk

WStk

in a

and the coordination

t t WStk = Wmo, Sk + Wco, Sk

where

t Wmo, Sk

(4)

is equal to the number of aircraft in the sector and the coordination workload

indicates the number of aircraft passing through the boundaries of this sector at time t.

t Wmo, Sk

Wco,t Sk can be

defined by t mo,Sk

W where

M St k

t t ìï1 + M St k - Cm, if M St k > Cm, Sk Sk =í else ïî0

means the number of aircraft in sector

Sk

t,

at time

and

(5)

t Cm, Sk

the monitoring

critical capacity of the sector at time t. Similarly,

Wco,t Sk in Eq.(4) can be defined by t co,Sk

W where

Cc,t Sk

ìï1 + CSt k - Cc,t Sk if CSt k > Cc,t Sk =í else ïî0

(6)

is the critical coordination capacity of sector k at time t, and CSt the number of aircraft k

passing the boundaries of sector Sk at time t. So the first objective function can be defined as follow:15 f ææ t ö y1 = min å ç ç åWSk ÷ ´ max WStk tÎT ç k =1 è è tÎT ø

(

k=P

)

j

ö ÷ ÷ ø

(7)

where P is the number of sectors, T the considered time period, and f and j weight factors. We can find that the objective is to minimize the total workload of all sectors. Besides, the more congested sector k is, the higher probability sector k will have to reduce workload. If there are 1000 flights, 10 paths and 10 time slots for each flight, it can be concluded that the solution space will be extremely large and up to1001000. 2.2.2 Total delay The total delay consists of the delay on the ground and the delay in the air. For flight i, the ground delay can be expressed as

d s ( i ) = tn - t k

, where tk is the planned departure time slot and tn is the

actual departure time slot,. In general, the cost of the air delay is three times of the ground delay.

d r ( i ) = 3 (Tr - T0 ) , where Tr is the actual flight time and

Hence, the air delay can be presented as

T0 the shortest flight time. With the consideration of the equity between aircraft, the second objective function for all the flights is formulated by the quadratic summation of delays instead of a regular linear one:15 N

y2 = å ( d s ( i ) + d r ( i ) )

2

(8)

i =1

The search space of the flight assignment problem will be huge, and its computation complexity can be obtained by N

space = Õ ( Ri × Qi ) i =1

(9)

where

|g|

denotes the cardinality of a set, N is the number of flights, and

Θi and Ri denote the slot

set and the route set of flight i. 3.

Optimization framework

From the mathematical model mentioned above, it can be concluded that the flight assignment problem is a large-scale combinational optimization problem with tightly coupled decision variables. In addition, its computational complexity is NP-hard which is difficult to solve.15 In order to improve the quality of solutions and avoid the local optimal, an effective multi-objective parallel evolution algorithm with a dynamic migration interval strategy is proposed. The algorithm is described in Fig. 2. The main steps are given as follows. Firstly, M populations are generated to evolve simultaneously which can improve the search ability. Secondly, in each population, a cooperative co-evolution algorithm is used via dividing the complex problem into several low-dimensional sub-problems.27-28 Then, each sub-problem employs NSGA-II with the differential evolution (DE) algorithm, which is a simple yet effective algorithm for global optimization.29 At last, the optimal solutions are obtained through cooperation of different sub-problems. For each population, there is a container called archive to store the best solutions.

Fig. 2 Framework of the algorithm. 3.1. Multi-island parallel evolution algorithm The multi-island parallel evolution algorithm is a popular heuristic algorithm, which consists of several populations. These populations can fully search the solution space and obtain better solutions. Supposed that there are M islands (populations), then each population Pi can be denoted as

{

Pi = id i _1 , id i _ 2 , K , id i _ ps

}

(1 £ i £ M )

(10)

where ps is the number of individuals in a population, and id is a chromosome of the population. If there are N flights involving in this problem, the individual idij in each population indicating the chromosome coding can be defined by

id ij = {rij1 , d ij1 , rij 2 , d ij 2 ,K , rijN , d ijN }

(1 £ j £ ps )

(11)

The fitness functions are the two objectives in Eqs. (7) and (8). In each island, the selection operator, crossover operator and mutation operator are the same as the ones proposed by Daniel et al.15 3.2. Dynamic migration interval (DMI) The migration interval, determining how often migration occurs, plays an important role in the effect of the MPEA framework24. In general, the migration interval is set as a constant. However, with the proceeding of evolution, the best migration interval may change dynamically. In this paper, we present a strategy to change the migration interval dynamically based on the diversity of the non-dominated solutions obtained by parent populations. Next, the population diversity and the dynamic migration interval strategy will be described in detail. 3.2.1. Population diversity Population diversity reflecting the spread of solutions is an important indicator. As the process of evolution, population diversity degrades gradually,23, 24 which could result in premature. Although the entropy of each population is used to measure the diversity of the population for the mono-objective function,25 it cannot be suitable for the multi-objective problem. Hence, a method to calculate the population diversity for a multi-objective problem is presented, which is similar to the classical method used to calculate the diversity of the nondominated solutions in NSGA-II.30 Suppose that the non-dominated solution set in Fig. 3 is

{ x1 , x2 ,K, xn }

and di represents the

distance between xi and xi+1 which is defined by

di = f ( xi ) - f ( xi +1 ) where f is the vector of objective functions, and

(0
2

(12)

f = ( f1 , f 2 ) . Hence, the diversity indicator hgth of

non-dominated solutions at the gth generation is defined by n -1

h gth = -

where

-

å di - d i =1

-

( n - 1) d

(13)

n -1

d = å di / ( n - 1) . We can conclude that hgth indicates the spread of the solutions. i =1

-

Especially, when

d = di

population diversity is.

for all the solutions,

h gth = 0 .

The smaller h gth is, the larger the

Fig. 3 Non-dominated solution set.

3.2.2 Dynamic migration interval strategy The migration interval determines the frequency of individual migration which plays an important role in the quality of solutions under the MPEA framework. A constant migration interval is used in most experiments. With the proceeding of evolution, the diversity of solutions may deteriorate sharply, so constant migration interval is not the best choice for all the generations. In this paper a dynamic migration interval strategy based on the diversity of the parent solutions is presented to solve this problem. After each generation, the diversity of solutions obtained from current population will be calculated. When it deteriorates obviously, which may cause the local optima, the migration will be applied. The individuals from other population can help to jump out from the local optima. A flag variable F(g) is introduced to control the migration which is defined by

ì1 if g £ k or h gth >p ´h( g -k )th F (g) = í î0 otherwise

(14)

where h gth is the diversity of solutions in the gth generation and k is a constant integer which represents the generation gap. In order to avoid frequent migration, a scale factor p is multiplied with h( g -k )th . F(g) controls the migration frequency following the rules: when F(g) is 1, migration will be applied; when F(g) is 0, migration is not used. 4.

Experiments

In this paper, we use the real air traffic data of China including flight plans, the national route network and the sectors to test and verify the efficiency of the algorithm. The national route network of China consists of 1706 legs, 940 waypoints and 150 airports. The flights are classified as light, medium and heavy with different speeds of 700 km/h, 800 km/h and 900 km/h. The algorithms, such as our proposed method, MOGA, MOEA/D31 and cooperative co-evolution-based algorithm, in this work were implemented in C++, and the simulations were performed on a server with an E5620 2.4GHz CPU with 12 GB RAM. For each algorithm, the results were collected and analyzed on the basis of 15 independent runs. Besides, the proposed approach was realized by multithreaded programming. Then, the optimization of all the islands and sub-problems of each population can proceed separately and simultaneously which can reduce the computation time. The parameter M is 5, the maximum delayed time is 45 mins, and the maximum number of routes of each flight is 10. Two weight factors in Eq.(7) are set as f =0.9, j =0.1. In Eq.(14), k=5, p=1.05. Other parameters used in all the experiments are listed in Table 1. Table 1 Parameters of

experiments.

Parameters

Description

MOGA

MOEA/D

CCMA

MPEA

popsize

Population size

100

100

100

100

maxgen

Max generation

150

150

150

150

pc

Crossover probability

0.9

0.9

0.9

0.9

pm

Mutate probability

0.1

0.1

0.1

0.1

4.1. Comparison with other algorithms In order to test the effectiveness of MPEA, in this part, we will compare the proposed MPEA with some existing algorithms, including the classical MOGA, MOEA/D, and a CC-based multi-objective algorithm (CCMA). In the experiment, we consider two scenarios, 960 flights and 1664 flights. The results calculated based on 15 independent runs were analyzed statistically. For evaluating the performance of the solutions obtained by each of the algorithm, three typical metrics are adopted: the convergence metric γ,30 the spread metric Δ,31 and the Hypervolume metric IH.32 γ suggests the average Euclidean distance from the obtained non-dominated solution set to the actual Pareto front. Δ indicates the diversity of solutions along the Pareto front. IH can evaluate the convergence and the extent of spread of the solutions without the real Pareto front. Table 2 and Table 3 show the average value of IH, ID and Δ over 15 independent runs of the algorithms for the two scenarios respectively. In each row of the table, the best value is highlighted in boldface. It can be seen from the tables that MPEA outperforms the other three algorithms in terms of IH, ID and Δ. Moreover, when the number of flights increases, MPEA performs much better. It can be concluded that MPMA has superiority to solve this large-scale problem. Table 2 Comparison of different existing algorithms for 960 flights. Algorithm

IH

g

Δ

MOGA

1.2970 ´ 1013

6.6865 ´ 10 6

1.0077

MOEA/D

2.1090 ´ 1013

1.9350 ´ 10 6

1.2706

CCMA

3.4922 ´ 1013

3.4252 ´ 105

1.0524

MPEA

3.6274 ´ 1013

9.4610 ´ 103

0.9980

Table 3 Comparison of different existing algorithms for 1664 flights. Algorithm

IH

g

Δ

MOGA

3.6509 ´ 1012

2.1931 ´ 10 7

1.0113

MOEA/D

7.7791 ´ 1013

1.1373 ´ 10 7

1.0409

CCMA

2.0523 ´ 1014

3.8542 ´ 10 6

1.0243

MPEA

2.2950 ´ 1014

6.1393 ´ 10 4

0.5597

Fig. 4 shows the non-dominated solutions obtained by the four algorithms. The horizontal axis represents airspace congestion which is related to the congestion in each sector as the Eq. (7) describes. The vertical axis indicates the cost of delay which includes the cost of delay on the ground and the cost of extra flying distance in the air. Safety and cost is a pair of contradiction. In practical operation, with consideration of the workload, the air traffic controllers in general choose the suitable solutions to balance the cost and congestion. From Fig. 4 it can be concluded that MPEA performs the best because its solutions can dominate those obtained by other algorithms. Besides, it can be seen that MOGA has the worst performance, and CCMA performs better than MOEA/D, but MOEA/D has good performance in terms of diversity.

Fig. 4 Comparison of different algorithms for 960 flights and1664 flights

From the experimental results, we find that MPEA performs better than the other three methods for the two scenarios. While the flight assignment problem has a large searching space which also increases exponentially over the number of flights, MOGA has difficulty in finding feasible solutions in this problem. Though MOEA/D can get the solutions with better spread, it can easily fall into local optimum and evolves slowly in the later generation. CCMA divides the complex problem into several low-dimensional sub-problems which is easier to solve, and it performs better than MOEA/D. However, the variables and constraints are so tightly coupled to find better solutions. The proposed MPEA adopts an effective multi-island parallel evolution framework which can improve the optimization capability. Besides, the dynamic migration interval strategy can further avoid premature and improve the solution quality. 4.2. Comparison of dynamic migration interval strategy with others The first experiment has justified the superiority of MPEA over the existing methods. It is still unclear how the dynamic migration interval strategy works. To investigate this issue, we have compared MPEA with other two migration interval strategies. These algorithms share the same settings except for the migration interval strategy. They are briefly described as follows: (1) Constant migration interval (CMI): The migration interval is set as a constant. It means that the subpopulations will exchange individuals after a constant interval. In the experiment, the constant interval is set as 5. (2) Random migration interval (RMI): The migration interval is controlled by a parameter pr between 0 and 1. In each generation, if pr is larger than a random number between 0 and 1, the migration will be applied. Here, pr is set as 0.2. Table 4 and Table 5 show the results by the three compared methods in terms of the values of the metrics over 15 independent runs of the algorithms when the number of flights is 960 and 1664 respectively. From the tables we can see that in the two scenarios, DMI has the least Δ which means the solutions distribute uniformly. Furthermore, it can get the least g , indicating that the solutions are closer to Pareto Front. With respect to IH, DMI shows superior performance than other two strategies. It can be concluded from the tables that the dynamic migration interval strategy outperforms other two strategies. The superiority of DMI can be analyzed from Table 4 and Table 5. During the process of evolving, after each generation, the diversity of solutions will be calculated. When the solutions will converge to a local optima, the individual migration operation will be applied which can improve the search capability. So DMI can get better solutions in convergence and diversity. Table 4 Comparison of different migration interval strategies for 960 flights.

Algorithm

IH

g

Δ

DMI

1.9275 ´ 1011

1.3418 ´ 10 4

1.0219

CMI

1.6156 ´ 1011

3.1351 ´ 10 4

1.1707

RMI

1.7743 ´ 1011

2.1757 ´ 10 4

1.2152

Table 5 Comparison of different migration interval strategies for 1664 flights. Algorithm

IH

g

Δ

DMI

2.4138 ´ 1012

6.1393 ´ 10 4

0.5597

CMI

1.7418 ´ 1012

3.3502 ´ 10 5

0.9049

RMI

1.7138 ´ 1012

3.6342 ´ 105

0.9358

Fig. 5 Comparison of different migration interval strategies for 960 flights and 1664 flights

Furthermore, like the first experiment, the non-dominated solutions of the compared strategies are depicted in Fig. 5. The vertical axis indicates the cost of delay which includes the cost of delay on the ground and the cost of extra flying distance in the air. Fig. 5 shows that the dynamic migration interval strategy performs better than others and almost all non-dominated solutions obtained by DMI can dominate the ones obtained by other strategies especially for scenario 2. When the number of flight is 960, DMI could not achieve obvious advantages compared with other strategies. When the number comes to 1664, DMI performs much better than the other two strategies especially with respect to the total delay. With the same airspace congestion, DMI can reduce total delay sharply. It can be concluded that when the scale of the problem increases, the diversity of solutions may deteriorate sharply, which can cause local optima. However, the CMI strategy and the RMI strategy are not so efficient by adjusting the frequence of individual migration in a blind and passive way. On the contrary, the DMI strategy based on the diversity of the parent solutions can effectively avoid local optima. 5.

Conclusion With the rapid development of air transportation, flight delay and airspace congestion have become

more and more serious which lead to huge economic loss and threaten the safety of passengers, so it is very important to assign the flight economically and effectively. (1) An effective multi-objective parallel evolution algorithm (MPEA) framework is developed in this paper to deal with the flight assignment problem. In this framework, there are several populations evolving simultaneously to different directions, so it can avoid sinking into local

optima. (2) Firstly, multiple evolution populations are constructed and in each population, the cooperative co-evolution algorithm combined with NSGA-II is introduced to solve the problem. Cooperative co-evolution algorithm adopting the idea of divide-and-conquer strategy is used in each population as the global optimization method. (3) Moreover, a specially designed dynamic migration interval strategy is presented to further improve the searching capability. The strategy dynamically changes the migration interval according to the diversity of obtained solutions which will guarantee the distribution of solutions at a satisfied level. In order to test and verify the effectiveness and feasibility of the proposed method, empirical studies using the real traffic data of the Chinese air route network and daily flight plans are used which show that our approach outperforms the existing approaches including the multi-objective genetic algorithm, the well-known multi-objective evolutionary algorithm based on decomposition, a CC-based multi-objective algorithm and MPEAs with other two migration interval strategies. In the future, we will improve the model to consider the influence of severe weather. Acknowledgements This study was co-supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No.60921001). References 1. Guan XM, Zhang XJ, Han D, Zhu YB, Lv J, Su J. A strategic flight conflict avoidance approach based on memetic algorithm. Chinese Journal of Aeronautics 2014;27(1):93–101. 2. Zhang XJ, Guan XM, Hwang I, Cai KQ. A hybrid distributed-centralized conflict resolution approach for multi-aircraft based on cooperative co-evolutionary. Science China Information Sciences 2013;56(12):1-16. 3. Ye B, Hu M, Shortle JF. Collision risk-capacity tradeoff analysis of an en-route corridor model. Chinese Journal of Aeronautics 2014;27(1):124-35. 4. Vranas PB, Bertsimas DJ, Odoni AR. The multi-airport ground-holding problem in air traffic control. Operations Research 1994;42(2):249-61. 5. Vranas PBM, Bertsimas D, Odoni AR. Dynamic ground-holding policies for a network of airports. Transportation Science 1994;28(4):275-91. 6. Abad AM, Clarke B. Using tactical flight level allocation to alleviate airspace corridor congestion. AIAA 4th aviation technology,integration and operations(ATIO) forum; 2004 Sep 20-22; Chicago, USA; 2004. p.1-9. 7. Delahaye D, Odoni AR. Airspace congestion smoothing by stochastic optimization. Evolutionary programming VI; 1997 Aug 15; Indianapolis, USA; 1997. p.163-76. 8. Oussedik S, Delahaye D. Reduction of air traffic congestion by genetic algorithms. Lecture Notes in Computer Science 1998;1498:855-64. 9. Oussedik S, Delahaye D, Schoenauer M. Dynamic air traffic planning by genetic algorithms. Proceedings of the congress on evolutionary computation; 1999 July 6-9; Washing D.C., USA; 1999.p.1110-7. 10. Bertsimas D, Patterson SS. The traffic flow management rerouting problem in air traffic control: a dynamic network flow approach. Transportation Science 2000;34(3):239-55.

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ZHANG Xuejun currently is a professor in School of Electronic and Information Engineering, Beihang University where he received the B.S. and Ph.D. degrees in 1994 and 2000 respectively. His main research interests are air traffic management, data communication and air surveillance. GUAN Xiangmin is a Ph.D. student at School of Electronic and Information Engineering, Beihang University. He received his B.S. degree from Hubei University in 2007. His area of research includes air traffic control, air traffic flow management and optimization. ZHU Yanbo received the Ph.D. degree from Beihang University. He joined the Aviation Data Communication Corporation (ADCC) in China in 1996 when the company was founded, and now he is the vice president of ADCC. His major research fields include air ground broadband communication, satellite navigation, collaborative surveillance and air traffic management.