research article an airway network flow...

Research ArticleAn Airway Network Flow Assignment Approach Based onan Efficient Multiobjective Optimization Framework

Xiangmin Guan123 Xuejun Zhang12 Yanbo Zhu12 Dengfeng Sun4 and Jiaxing Lei12

1School of Electronic and Information Engineering Beihang University Beijing 100191 China2National Key Laboratory of CNSATM Beijing 100191 China3Beijing Key Laboratory for Cooperative Vehicle Infrastructure Systems and Safety Control Beijing 100191 China4School of Aeronautics and Astronautics Purdue University West Lafayette IN 47907-2023 USA

Correspondence should be addressed to Xuejun Zhang zhxjbuaaeducn

Received 23 July 2014 Revised 7 September 2014 Accepted 18 September 2014

Academic Editor Zheng Zheng

Copyright copy 2015 Xiangmin Guan et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Considering reducing the airspace congestion and the flight delay simultaneously this paper formulates the airway network flowassignment (ANFA) problem as amultiobjective optimizationmodel and presents a newmultiobjective optimization framework tosolve it Firstly an effective multi-island parallel evolution algorithm with multiple evolution populations is employed to improvethe optimization capability Secondly the nondominated sorting genetic algorithm II is applied for each population In addition acooperative coevolution algorithm is adapted to divide the ANFA problem into several low-dimensional biobjective optimizationproblems which are easier to deal with Finally in order to maintain the diversity of solutions and to avoid prematurity a dynamicadjustment operator based on solution congestion degree is specifically designed for the ANFA problem Simulation results usingthe real traffic data from China air route network and daily flight plans demonstrate that the proposed approach can improve thesolution quality effectively showing superiority to the existing approaches such as the multiobjective genetic algorithm the well-known multiobjective evolutionary algorithm based on decomposition and a cooperative coevolution multiobjective algorithm aswell as other parallel evolution algorithms with different migration topology

1 Introduction

With the development of civil aviation the permanentlyincreasing air traffic and the limited airspace resource haveresulted inmore andmore serious congestion and flight delay[1ndash3] Meantime heavy congestion challenges the airspacesafety and the flight delay costs the airline industry billionsof dollars every year [4] Hence how to safely accommodatehigh levels of demand and to maximize the use of capacity-limited airspace and airport resources has become a majorconcern to both researchers and practitioners in air trafficmanagement (ATM)

In recent years FAA proposed the 4D-trajectory (4DT)operation concept in next generation air transportation sys-tem (NextGen) which includes three dimensions of positionand a time description of the flight Under the 4DT environ-ment flights can be accurately planned in both space and

timeThe airway network flow assignment (ANFA) approachcan provide solutions from a global point of viewwith the aimof alleviating airspace congestion and reducing time cost byoptimizing the 4D trajectories of all flights considered in theentire airspace Therefore it has become a focus of researchinterests

However the ANFA problem is a large-scale combina-torial optimization problem with complicated constraints aswell as tightly coupled variables which is difficult to solvein general For instance there are more than ten thousandflights flying over China every day on the air route networkwith more than one thousand waypoints which generatesa large number of tightly coupled decision variables andconstraints Hence in order to get optimal flight plans forall flights the ANFA problem has a very high computationalcomplex Besides with consideration of reducing congestionand minimizing the induced delay the problem usually has

Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 302615 9 pageshttpdxdoiorg1011552015302615

2 The Scientific World Journal

multiple objectives which are all nondifferentiable or evennoncontinuous

Due to the importance of the ANFA problem it hasdrawn a mass of attention of researchers [5] Earlier workhas focused on the slot-time allocation known as groundholding in a single ormultiple airport setting in which delayspropagate through the network [6 7] Besides the flight levelallocation has been considered to the flights in this problem[8 9] In addition Bertsimas et al [10 11] considered thetime slots and routes assignment simultaneously through anefficient deterministic approach From a stochastic optimiza-tion point of view Delahaye and Odoni proposed a geneticalgorithm to solve the problem [12] However it often fallsinto the local optimum because of the large-scale decisionvariables More recently a cooperative coevolution (CC)algorithm was introduced into the resolution of the ANFAproblem [13] It adopted the divide-and-conquer strategy todivide the complex problem into several low-dimensionalsubproblems which becomes easier to deal with

So far these works took the minimization of the airspacecongestion or the flight delay as the sole objective [14 15]However it might be more appropriate to consider bothairspace congestion and extra flight cost and try to seeka good trade-off between them Hence Daniel et al [16]formulated the problem into a biobjective optimization prob-lem and solved it with the multiobjective genetic algorithm(MOGA) However the application of existingmultiobjectiveevolutionary algorithms (MOEAs) to the ANFA problemmight lead to poor performance due to the scale of the prob-lem To the best of our knowledge fewworks could effectivelysolve complicatedmultiobjective optimization problemswiththousands of variables

With the consideration of the reduction of the airspacecongestion and the flight delay simultaneously this paperformulates the ANFA problem as a multiobjective optimiza-tion model Then an efficient multiobjective optimizationframework is presented to solve it This framework employsthe parallel computation and population diversity adjustmenttechniques to improve the optimization capability Firstlyan effective multi-island parallel evolution algorithm (PEA)with multiple evolution populations is adopted Besidesone-way ring migration topology is applied to exchangeindividuals among populations to improve the efficiency ofthe cooperation of populations Secondly the multiobjectiveevolutionary algorithm NSGA2 is used to optimize eachpopulation In addition a cooperative coevolution algorithmis adapted to divide the ANFA problem into several low-dimensional biobjective optimization problems which areeasier to deal with Finally in order to maintain the diversityof solutions and avoid prematurity a dynamic adjustmentoperator based on solution congestion degree is specifi-cally designed It can greatly improve the distribution ofthe nondominated solutions in the archive and maintainthe population diversity by inheriting the nondominatedsolutions with low congestion degree in a high probabilitySimulation results using the real traffic data from Chinaair route network and daily flight plans demonstrate thatthe proposed approach can improve the solution qualityeffectively showing superiority to the existing approaches

such as the multiobjective genetic algorithm the well-knownmultiobjective evolutionary algorithm based on decomposi-tion (MOEAD) and a CC-based multiobjective algorithmas well as other parallel evolution algorithms with differentmigration topology

The rest of this paper is organized as follows Section 2introduces the formulation of the investigated biobjectiveANFA problem Section 3 describes the proposed multiob-jective optimization framework in detail Experimental studyis presented in Section 4 to evaluate the effectiveness ofour algorithm Finally Section 5 concludes this paper anddiscusses directions for further research

2 Problem Formulation

The airway network can be modeled as a directed graphincluding the waypoints as nodes and segments as edgesEach flight can be considered as a single commodity onthe network with a defined pair of origin-destination nodesBesides each flight has its predefined departure time slots andthe optional routes In this paper with the consideration ofsafety and efficiency the ANFA problem is formulated as abiobjective problem to reduce the airspace congestion and thetotal flight delay simultaneously

Each flight (flight 119896) is associated with a pair of decisionvariables (119903119896 120591119896) where 119903119896 is a possible route and 120591119896 is afeasible departure time slot In addition they are subject tothe following

(1) 119903119896 isin Path119896 = 1199031015840

1 1199031015840

2 119903

1015840

max where Pathk is the setof all possible paths of flight 119896

(2) 119905min119896

le 120591119896 le 119905max119896

where 119905max119896

is the latest departuretime slot and 119905

min119896

is the earliest departure time slot

We can see that constraint (1) denotes a feasible route of eachflight from a defined route set while constraint (2) enforcesthe flight to depart at a predefined departure time slot

The first objective function is the minimization of theairspace congestion Here the workload of the sectors isused to indicate the airspace congestion which is defined asfollows

It is generally known that the workload of a sector mainlydepends on the monitoring workload and the coordinationworkload Hence the total workload of a sector 119878119896 at time 119905

can be roughly expressed by the following [16]

119882119905

119878119896= 119882119905

mo119878119896 + 119882119905

co119878119896 (1)

where 119882119905mo119878119896 is the monitoring workload and can be numer-ically estimated by

119882mo119904119896 = 1 + 119872

119905

119878119896minus 119862119905

119898119878119896 if 119872119905

119878119896gt 119862119905

119898119878119896

0 else(2)

where 119872119905119878119896is related to the number of aircraft in sector 119878119896 at

time 119905 and119862119905

119898119878119896is themonitoring critical capacity of the sector

at time 119905

The Scientific World Journal 3

BEGINInitialize119872 populations of size ps each 119892 = 0 Archive = NULLWHILE 119892 lt max gen

FOR population 119895 = 0 (119872 minus 1)Divide the problem into 119899 subcomponents based on cooperative co-evolution

FOR each subcomponentUse the NSGA2 framework and differential evolution as the evolutionaryalgorithm to solve the subcomponent

END FORArchive = Obtain and update the non-dominated solutions based on the dynamicadjustment operator for solutions congestion degree

END FORIF (migration condition meet)

Exchange individuals selected among populations based on a migration topologyEND IF119892 = 119892 + 1

ENDWHILEEND

Algorithm 1 The framework of the proposed algorithm

Besides 119882119905co119878119896 is the coordination workload and can benumerically estimated by

119882co119904119896 = 1 + 119862119905

119878119896minus 119862119905

119888119878119896 if 119862119905

119878119896gt 119862119905

119888119878119896

0 else(3)

where 119862119905

119878119896is related to the flights passing through the

boundary of sector 119878119896 at time 119905 and 119862119905

119878119896is related to the

number of aircraft passing the boundaries of sector 119878119896 at time119905

The goal of the optimization is to minimize the airspacecongestion via spreading the congestion over several sectorsHence the objective is defined by the following [16]

min119860119862 =

119896=119899119878

sum

119896=1

((sum

119905isin119879

119882119905

119878119896)

1minus120593

times (max119905isin119879

119882119905

119878119896)

120593

) (4)

where 120593 (1 minus 120593) isin [0 1] indicates the relative importance ofthe maximum congestion and the average congestion

The second objective is the minimization of the extraflight cost (EFC) which includes the departure delay and theairborne delay caused by choosing a longer path than theshortest one Then the second objective can be expressed asfollows [16]

min EFC = sum

119894isin119865

(10038161003816100381610038161003816120591119894 minus 119905

prf119896

10038161003816100381610038161003816+ (119905119903119894

minus 119905119903prf119894

))2

(5)

where 119905119903119894 and 119905119903prf119894

denote the flight time of 119903119894 and the preferred

shortest path respectively and 119905prf119896

indicates the preferreddeparture time The first part is the airborne delay and thesecond part is the ground delay

3 Optimization MethodIt can be seen that the ANFA problem is a large-scale com-binatorial optimization problem and the objective functionsare nonlinear and nondifferentiable Black-box optimizationmethods such as EAs appeared to be promising to deal withthis kind of problem [15ndash19] Given the multiobjective modelof the ANFA problem basically any existing MOEA fordiscrete multiobjective optimization can be readily utilizedHowever the ANFA problem involves thousands of tightlycoupled decision variables which are difficult to solve Inaddition few existing MOEAs have been extensively inves-tigated on problems of such a large scale while the scalabilityof EAs with respect to the number of decision variables is ingeneral deemed to be poor therefore a direct application ofexisting MOEAs might not obtain satisfactory solutions

With the aim of avoiding prematurity and improving theconvergence rate of this complex problem a newmultiobjec-tive optimization framework is proposed Firstly an effectivemulti-island parallel evolution algorithm with multiple evo-lution populations is employed and one-way ring migrationtopology for exchange individuals among populations isapplied to improve the efficiency of the cooperation of popu-lations Secondly the multiobjective evolutionary algorithmNSGA2 is used to optimize each population In addition thispaper introduces the idea of cooperative coevolutionary (CC)into the resolution of the multiobjective ANFA problemsThe main idea is to divide the high-dimensional probleminto low-dimensional subcomponents The subcomponentswork cooperatively to obtain better solutions Finally all thenondominated solutions are stored in an archive As theANFA problem is of such a large scale with thousands ofvariables the main difficulty is how to maintain the diversityof the population to get better solutions and to improve thedistribution of the solutions in the archive Hence in order tomaintain the diversity of solutions and to avoid prematuritya dynamic adjustment operator based on solution congestiondegree is specifically designed for the ANFA problem Theframework is presented in Algorithm 1


31 Multi-Island Parallel Evolution Algorithm As parallelcomputers become more commonplace in scientific com-puting it becomes more feasible to harness their power foruse with evolutionary algorithms (EAs) [20] Multi-islandPEAs consist of several populations which can optimizesimultaneously to avoid premature convergence They havebeen successfully applied to find acceptable solutions toproblems in different engineering domains [21 22]

Suppose that there are119872 islands and119873 flightsThen eachpopulation can be denoted as

pop119894= 119894119889119894V1198941 119894119889119894V1198942 119894119889119894V119894119901119904 1 le 119894 le 119872 (6)

where 119894119889119894V119894119895 is defined by

119894119889119894V119894119895 = (1199031198941198951 1205911198941198951 1199031198941198952 1205911198941198952 119903119894119895119873 120591119894119895119873) 1 le 119895 le ps (7)

where ps is the size of population

32 Migration Topology The migration topology is a keyfeature of the island model which determines the destinationof the migrants and it could greatly affect the quality of thesolutions and the efficiency of algorithms If two populationsrarely communicate with each other it is difficult for the bestsolution to spread which may prevent populations findingbetter solutions

Currently themainmigration topologies are the one-wayring topology and the random topology [22] The randomtopology delivers the migrants to a randomly selected pop-ulation However it may result in inefficient communicationAfter many times of migration under the random topologythe best solution cannot be spread effectively among popu-lations In the one-way ring topology populations are num-bered and the worst individual of a population is replacedby the best individual of the next population The one-wayring topology can provide sufficient communication amongpopulations and maintain the population diversity whichcan effectively avoid premature convergence and improve theoptimization capability

33 Cooperative Coevolution for Each Population Though themulti-island PEA uses several populations simultaneouslyin fact each population is hard to avoid falling into a localoptimum because the ANFA problem involves large-scaletightly coupled decision variables as presented in the previoussections Hence we introduce the cooperative coevolutionalgorithm for the optimization of each population to furtherimprove the solution quality The cooperative coevolution(CC) algorithm adopting the divide-and-conquer strategydivides the complex problem into several low-dimensionalsubproblems [23ndash25] There are two critical issues in thisapproach

(1) Decomposition Strategy The decomposition strategy is akey feature of the cooperative coevolution framework whichcan greatly affect the capability and the efficiency of algo-rithms In this work the random grouping strategy is usedwhich has been both theoretically and experimentally provedto be effective for the large-scale complex problem [23]

At each generation each population is randomly divided into119899119904 disjoint subpopulations with the same population size

pop119894= sp1119894 sp119899119904

119894 1 le 119894 le 119872

sp119895119894= 119904119894119889

1198951

119894 1199041198941198891198952

119894 119904119894119889

119895ps119894

1 le 119895 le 119899119904

119904119894119889119895119896

119894= (1199031015840

1198951198961 1205911015840

1198951198961 1199031015840

1198951198962 1205911015840

1198951198962 119903

1015840

119895119896(119873119899119904) 1205911015840

119895119896(119873119899119904))

1 le 119895 le ps

(8)

where 119904119894119889 denotes the individual of each subpopulation andsp119895119894indicates the subpopulation 119895 of population 119894

(2) Subpopulation Optimization Another critical point isoptimization for each subpopulation In this paper thewell-knownmultiobjective evolutionary algorithmNSGA2 isemployed by each subpopulation [26] Besides differentialevolution (DE) [27] is used in the framework to generate newsolutions because it is a simple yet effective algorithm forglobal optimization DE is a randomized parallel searchingalgorithm It begins with a random population according tospecific rules for example selection crossover and muta-tion An optimized resolution is reached by retaining goodindividuals and discarding bad individuals Compared withother optimization algorithms DE has the advantages inglobal optimization as well as easy operation Its operators aredescribed below

Suppose that 119891(119909) is the objective function and the goalis to minimize it

In mutation if current chromosome is 119909119894119866 then choosethree different chromosomes from current generation popu-lation named 1199091199031119866 1199091199032119866 and 1199091199033119866 The mutation operator isdefined by

119881119894119866+1 = 1199091199031119866 + 119865 sdot (1199091199032119866 minus 1199091199033119866) 119865 isin [0 2] (9)

where119865 is a parameter which decides the scale ofmutation Anew chromosome is generated by the crossover of 119881119894119866+1 and119909119894119866 as follows

119880119894119866+1 = (1199061119894119866+1 1199062119894119866+1 119906119873119894119866+1)

119906119895119894119866+1 = V119895119894119866+1 rand (119895) le 119862119877 or 119895 = rnbr (119894)119909119895119894119866 else

(10)

where rnbr(119894) is a random integer number between 1 and119873 which ensures that 119880119894119866+1 gets at least one componentfrom V119894119866+1 component from119881119894119866+1 and rand(119895) is a uniformlydistributed random number between 0 and 1

After the evaluation of each chromosome the chro-mosomes of next generation are chosen according to thefollowing rule

119909119894119866+1 = 119906119894119866+1 119891 (119906119894119866+1) le 119891 (119909119894119866)

119909119894119866 else(11)

34 Dynamic Adjustment Operator Based on Solution Con-gestion Degree As the optimization generation increases


54

55

71 53

77

34

1413

1215 16

17

3231

3938

37

10

20

11

19 18

246773 72 21 22 23 25

6263

47

4246

5657

58

36 353045

4344

40 414948

262728

134

2

56759

60 6165 64

89

74

75

76

50 5152

33

66

70

29

6869

(a) (b)

Figure 1 (a) Airspace sectors in China (b) Flights operation in China

the population diversity decreases rapidly for more solutionscongested at a local searching space [28 29] On the one handtoo crowded solutions will cause premature convergenceOn the other hand some local searching space with sparsesolutions is not explored enough andneedsmore attention forbetter solutions Hence the more evenly the nondominatedsolutions distribute the better the optimization is In thispaper we propose a dynamic adjustment operator to improvethe distribution of solutions and maintain the populationdiversity based on the congestion degree of solutions

(1) Congestion Degree of the Nondominated Solutions in theArchive The distance between solution 119894 and 119895 in archive isdefined by

119889119894119895 =1

2radic(119875

fit[1]119894

minus 119875fit[1]119895

)2

+ (119875fit[2]119894

minus 119875fit[2]119895

)2

(12)

where 119901fit[1]119894

and 119901fit[2]119894

denote the value of the first and thesecond objective functions of solution 119894 respectively Besidesthe average distance of all solution pairs in the archive isdescribed by

119904 =2

119899 (119899 minus 1)

119894=119899

sum

119894=1

119895=119899

sum

119895=119894+1

119889119894119895 (13)

where 119899 is the number of the solutions in archiveThen the congestion degree of a nondominated solution

can be expressed as

119889119894 =1

119899 minus 1

119895=119899

sum

119895=1119895 =119894

120575119894119895 (14)

where

120575119894119895 = 1 if 119889119894119895 le 120574 sdot 119904

0 if 119889119894119895 gt 120574 sdot 119904(15)

where 120574 is a parameter which can be predefined We can seethat the larger 119889119894 is the closer to other solutions solution 119894 is

(2) Update the Archive The solutions in the archive will beconstantly updated during the evolution of the subpopu-lations When the number of the nondominated solutionsexceeds the maximum size of the archive the solutions 119894 willbe moved out from the archive in a probability

119901119894 =119889119894120572

sum119899

119895=1119889119895120572 (16)

where 120572 is a positive regulation factor Equation (16) indicatesthat the higher 119889119894 is the higher probability it will be movedout from the archive

4 Experimental Studies

41 Database and Experimental Setup The national routenetwork of China consists of 1706 airway segments 940waypoints and 150 airports Note that the takeoff andlanding phases of flights are truncated within a given radius(usually 10NM) around airports The traffic around airportsis managed with specific procedures by the terminal controlarea (TCA) control services in these zones The airspaceis divided into many sectors and Figure 1(a) shows thesectored airspace in China The air traffic data was extractedfrom flight schedule database (FSD) of the summer in2009 released by Civil Aviation Administration of China(CAAC) In order to better describe the difference betweenthe algorithmsrsquo performances we consider two scenarios 960flights (the busiest one hour) and 1664 flights (the busiestthree hours)

The parameters are set as follows the number of popula-tions119872 = 5 120601 = 09 120593 = 01 119903 = 03 120574 = 125 and 119899119904 = 10Themutation probability and the crossover probability of DEare 015 and 085

The algorithms such as our proposed method MOGAMOEAD [30] and cooperative coevolution based algorithm


15 2 25 3

2

3

4

5

6

7

8

9

10

11

12

Tota

l del

ay

MOGAMOEAD

CCMAPEA

Airspace congestion (n = 960)times10

6

times106

(a)

1 15 2 250

05

1

15

2

25

3

Tota

l del

ay

MOGAMOEAD

CCMAPEA

Airspace congestion (n = 1664) times107

times107

(b)

Figure 2 Comparison of different algorithms for 960 flights (a) and 1664 flights (b)

Table 1 Parameters of the experiments

Parameters Description MOGA MOEAD CCMA PEA

ps Populationsize 100 100 100 20 lowast 5

(IM = 5)

max gen Maxgeneration 150 150 150 150

119901119888Crossoverprobability 09 09 09 09

119901119898Mutationprobability 01 01 mdash mdash

in this work were implemented in C++ and the simulationswere performed on a server with an E5620 24GHzCPUwith12GB RAM For each algorithm the results were collectedand analyzed on the basis of 15 independent runs Besidesthe proposed approach was realized by multithreaded pro-gramming Then the optimization of all islands and allsubcomponents of each population can proceed separatelyand simultaneously which can reduce the computation time

The parameters used in all experiments are listed inTable 1

In order to evaluate the performance of the solutionsobtained by each of the algorithms three typical metricsare adopted the convergence metric (120574) [26] the spreadmetric (Δ) [31] and the hypervolume metric 119868119867 [32 33] 120574suggests the average Euclidean distance from the obtainednondominated solution set to the actual Pareto front Notethat it is difficult to find the actual Pareto front for most real-world optimization problems so we use the best solutionset obtained by these algorithms in 15 runs Δ indicates thediversity of solutions along the Pareto front 119868119867 can evaluatethe convergence and the extent of spread of the solutions

Table 2 Comparison of different algorithms for 960 flights (119868119867 120574Δ)

Algorithm 119868119867 120574 Δ

MOGA 11156119890 + 13 66865119890 + 06 10078MOEAD 19054119890 + 13 18715119890 + 06 12554CCMA 31758119890 + 13 33631119890 + 05 10540PEA 32575e + 13 31237e + 04 10047

simultaneously without the real Pareto front The smaller thefirst two indexes are the better the algorithm is On the con-trary the larger the third index is the better the algorithm is

42 Comparison with the Existing Methods In order to testthe effectiveness of the proposed multi-island PEA frame-work in this part we will compare it with some existingalgorithms including the classical multiobjective geneticalgorithm (MOGA) multiobjective evolutionary algorithmbased on decomposition (MOEAD) and a CC-based mul-tiobjective algorithm (CCMA)

Tables 2 and 3 show the average value of 119868119867 119868119863 andΔ over 15 independent runs of the algorithms for the twoscenarios respectively In each row of the table the best valueis highlighted in boldface It can be seen from the tables thatPEA outperforms the other three algorithms in terms of 119868119867119868119863 and Δ Moreover when the number of flights increasesPEA performs much better It can be concluded that PEA hassuperiority to solve this large-scale problem

Figure 2 shows the nondominated solutions obtained bythe four algorithms Specifically the nondominated solu-tions of each algorithm were obtained over 15 runs FromFigure 2 it can be concluded that PEA performs the bestbecause its solutions can dominate those obtained by other


13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO

Airspace congestion (n = 1664)

times106

times107

(b)

Figure 3 Comparison of PEA and PEA without DAO for 960 flights (a) and 1664 flights (b)


Algorithm 119868119867 120574 Δ


algorithms Besides it can be seen that MOGA has the worstperformance and CCMA performs better than MOEADbut MOEAD has good performance in terms of diversity

From the experimental results we find that PEAperformsbetter than the other three methods for the two scenarios Itadopts an effective multi-island parallel evolution frameworkwhich can improve the optimization capability Besides theone-way ringmigration strategy can further avoid prematureIn addition this paper introduces the cooperative coevolu-tionary (CC) into each population optimization via divid-ing the high-dimensional problem into low-dimensionalsubcomponents The subcomponents work cooperatively toobtain better solutionsWith the help of parallel computationthe computation is just about 30 minutes which is feasible forthe ANFA problem

43 Investigation of the Effectiveness of the Dynamic Adjust-ment Operator In the previous section the first set ofexperiments has justified the superiority of PEA to existingmethods The next experiment is designed to further investi-gate whether the dynamic adjustment operator (DAO) basedon solution congestion degree contributes to the success ofPEA


Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050

Table 5 Comparison of different algorithms for 1164 flights (119868119867 119868119863

Δ)

Algorithm 119868119867 119868119863 Δ

PEA without DAO 73890119890 + 6 34212119890 + 05 09395PEA 13107e + 7 12861e + 05 07091

Tables 4 and 5 show the results obtained by PEA andPEA without DAO in terms of the values of the metrics over15 independent runs of the algorithms when the number offlights is 996 and 1664 respectively It is seen from the tablesthat PEA always outperforms the other algorithm in terms of119868119867 119868119863 and Δ

Furthermore like the first experiment the nondominatedsolutions of the algorithms are shown in Figure 3 It showsthat PEAperformsmuch better than the othermethod and itsnondominated solutions can dominate the solutions obtainedby PEA without DAO For the scenario of 1664 flights PEAhas the most nondominated solutions and spreads nicely inthe objective space

As the optimization generation increases more solutionswill be congested in some local searching space which maycause premature convergenceThe dynamic adjustment oper-ator based on the congestion degree can effectively improvethe distribution of solutions by inheriting the nondominated


solutions with low congestion degree in a high probabilityHence it can avoid decreasing the diversity of all populations

5 Conclusion and Future Work

With the aim of reducing the airspace congestion and theflights delay simultaneously this paper formulates the airwaynetwork flow assignment (ANFA) problem into a multiob-jective optimizationmodel and presents a newmultiobjectiveoptimization framework to solve it Firstly an effective multi-island parallel evolution algorithm is employed to solve theproblem by multiple evolution populations Besides one-way ring migration topology is applied to improve theefficiency of the cooperation of populations by exchangingindividuals among populations Secondly the multiobjectiveevolutionary algorithm NSGA2 is used to optimize eachpopulation In addition a cooperative coevolution algorithmis adapted to improve the optimization capability by dividingthe ANFA problem into several low-dimensional biobjectiveoptimization problems Finally in order to maintain thediversity of solutions and avoid prematurity a dynamicadjustment operator based on solution congestion degree isspecifically designed Simulation results using the real trafficdata from the China air route network and daily flight plansdemonstrate that the proposed approach can improve thesolution quality effectively showing superiority to the exist-ing approaches such as the multiobjective genetic algorithmthe well-known multiobjective evolutionary algorithm basedon decomposition and a CC-based multiobjective algorithmas well as other parallel evolution algorithms with differentmigration topology For future research the ANFA problemwith the influence of severe weather will be considered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by the National High TechnologyResearch and Development Program of China (Grant no2011AA110101) the National Basic Research Program ofChina (Grant no 2010CB731805) and the Foundation forInnovative Research Groups of the National Natural ScienceFoundation of China (Grant no 61221061)

References

[1] W Liu and I Hwang ldquoProbabilistic trajectory prediction andconflict detection for air traffic controlrdquo Journal of GuidanceControl and Dynamics vol 34 no 6 pp 1779ndash1789 2011

[2] W B Du Z X Wu and K Q Cai ldquoEffective usage ofshortest paths promotes transportation efficiency on scale-freenetworksrdquo Physica A Statistical Mechanics and Its Applicationsvol 392 no 17 pp 3505ndash3512 2013

[3] X-M Guan X-J Zhang D-F Sun and S-T Tang ldquoThe effectof queueing strategy on network trafficrdquo Communications inTheoretical Physics vol 60 no 4 pp 496ndash502 2013

[4] S Oussedik and D Delahaye ldquoReduction of air traffic conges-tion by genetic algorithmsrdquo in Proceedings of the 5th Interna-tional Conference on Parallel Problem Solving fromNature 1998

[5] PWei Y Cao and D Sun ldquoTotal unimodularity and decompo-sitionmethod for large-scale air traffic cell transmissionmodelrdquoTransportation Research Part BMethodological vol 53 pp 1ndash162013

[6] P Vranas D Bertsimas and A Odoni ldquoDynamic ground-holding policies for a network of airportsrdquo TransportationScience vol 28 pp 275ndash291 1994

[7] M Ball and G Lulli ldquoGround delay programs optimizingover included flight set based on distancerdquo Air Traffic ControlQuarter vol 12 pp 1ndash25 2004

[8] N Barnier and P Brisset ldquoGraph coloring for air traffic flowmanagementrdquo in Proceedings of the CPAIOR pp 1ndash15 2002

[9] AM Abad and J-P B Clarke ldquoUsing tactical flight level alloca-tion to alleviate airspace corridor congestionrdquo in Proceedings ofthe AIAA 4th Aviation Technology Integration and OperationsForum (ATIO rsquo04) pp 1ndash9 September 2004

[10] D Bertsimas and S Patterson ldquoThe air traffic flowmanagementproblem with enroute capacitiesrdquo Operations Research vol 46no 3 pp 406ndash422 1998

[11] D Bertsimas G Lulli and A Odoni ldquoAn integer optimizationapproach to large-scale air traffic flowmanagementrdquoOperationsResearch vol 59 no 1 pp 211ndash227 2011

[12] D Delahaye and A Odoni ldquoAirspace congestion smoothingby stochastic optimizationrdquo in Proceedings of the EvolutionaryProgramming Conference 1997

[13] L Huaxian Z Yanbo C Kaiquan and P Qingge ldquoRouteNetwork Flow Assignment in the new generation of aviationby cooperative co-evolutionrdquo in Proceedings of the IEEE 5thInternational Conference on Cybernetics and Intelligent Systems(CIS rsquo11) pp 175ndash180 September 2011

[14] A M Bayen R L Raffard and C J Tomlin ldquoAdjoint-basedcontrol of a new Eulerian network model of air traffic flowrdquoIEEE Transactions on Control Systems Technology vol 14 no5 pp 804ndash818 2006

[15] Z Zheng Z Guo Y-N Zhu and X-Y Zhang ldquoA criticalchains based distributed multi-project scheduling approachrdquoNeurocomputing vol 143 pp 282ndash293 2014

[16] D Daniel S Oussedik and P Stephane ldquoAirspace congestionsmoothing bymulti-objective genetic algorithmrdquo inProceedingsof the 20th Annual ACM Symposium on Applied Computing pp907ndash912 March 2005

[17] Y Zhu Z Zheng X Zhang and K Cai ldquoThe r-interdictionmedian problem with probabilistic protection and its solutionalgorithmrdquo Computers amp Operations Research vol 40 no 1 pp451ndash462 2013

[18] C Liu W-B Du and W-X Wang ldquoParticle swarm optimiza-tionwith scale-free interactionsrdquoPLoSONE vol 9 no 5 ArticleID e97822 2014

[19] X Zhang Z Zheng Y Zhu and K Y Cai ldquoProtection issuesfor supply systems involving random attacksrdquo Computers andOperations Research vol 43 no 1 pp 137ndash156 2014

[20] J Tang M H Lim and Y S Ong ldquoDiversity-adaptive parallelmemetic algorithm for solving large scale combinatorial opti-mization problemsrdquo Soft Computing vol 11 no 9 pp 873ndash8882007

[21] J Tang M H Lim Y S Ong and M J Er ldquoStudy ofmigration topology in island model parallel hybrid-GA forlarge scale quadratic assignment problemsrdquo inProceedings of the


8th International Conference on Control Automation Roboticsand Vision (ICARCV rsquo04) pp 2286ndash2291 Kunming ChinaDecember 2004

[22] J Tang M H Lim and Y S Ong ldquoParallel memetic algorithmwith selective local search for large scale quadratic assignmentrdquoJournal of Innovative Computing Information and Control vol2 pp 1399ndash1416 2006

[23] Z Yang K Tang and X Yao ldquoLarge scale evolutionary opti-mization using cooperative coevolutionrdquo Information Sciencesvol 178 no 15 pp 2985ndash2999 2008

[24] M Potter and K D Jong ldquoA cooperative coevolutionaryapproach to function optimizationrdquo in Proceedings of the 3rdConference on Parallel Problem Solving from Nature pp 249ndash257 1994

[25] F Zhu and S-U Guan ldquoCooperative co-evolution of GA-based classifiers based on input decompositionrdquo EngineeringApplications of Artificial Intelligence vol 21 no 8 pp 1360ndash13692008

[26] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fastand elitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2 pp 182ndash197 2002

[27] R Storn and K Price ldquoDifferential evolution-a simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[28] W Liu Z Zheng and K-Y Cai ldquoBi-level programmingbased real-time path planning for unmanned aerial vehiclesrdquoKnowledge-Based Systems vol 44 pp 34ndash47 2013

[29] Z Zheng L Shumin G Ze and Z Yueni ldquoResource-constraintmulti-project scheduling with priorities and uncertain activitydurationsrdquo International Journal of Computational IntelligenceSystems vol 6 no 3 pp 530ndash547 2013

[30] Q Zhang and H Li ldquoMOEAD a multiobjective evolutionaryalgorithm based on decompositionrdquo IEEE Transactions onEvolutionary Computation vol 11 no 6 pp 712ndash731 2007

[31] E Zitzler L Thiele M Laumanns C M Fonseca and VG Da Fonseca ldquoPerformance assessment of multiobjectiveoptimizers an analysis and reviewrdquo IEEE Transactions onEvolutionary Computation vol 7 no 2 pp 117ndash132 2003

[32] M Fleischer ldquoThe measure of pareto optima applications tomulti-objectivemetaheuristicsrdquo inEvolutionaryMulti-CriterionOptimization Proceedings of the 2nd International ConferenceEMO 2003 Faro Portugal April 8ndash11 2003 vol 2632 ofLectureNotes in Computer Science pp 519ndash533 Springer BerlinGermany 2003

[33] F Wilcoxon ldquoIndividual comparisons by ranking methodsrdquoBiometrics Bulletin vol 1 pp 80ndash83 1945

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



multiple objectives which are all nondifferentiable or evennoncontinuous

Due to the importance of the ANFA problem it hasdrawn a mass of attention of researchers [5] Earlier workhas focused on the slot-time allocation known as groundholding in a single ormultiple airport setting in which delayspropagate through the network [6 7] Besides the flight levelallocation has been considered to the flights in this problem[8 9] In addition Bertsimas et al [10 11] considered thetime slots and routes assignment simultaneously through anefficient deterministic approach From a stochastic optimiza-tion point of view Delahaye and Odoni proposed a geneticalgorithm to solve the problem [12] However it often fallsinto the local optimum because of the large-scale decisionvariables More recently a cooperative coevolution (CC)algorithm was introduced into the resolution of the ANFAproblem [13] It adopted the divide-and-conquer strategy todivide the complex problem into several low-dimensionalsubproblems which becomes easier to deal with

So far these works took the minimization of the airspacecongestion or the flight delay as the sole objective [14 15]However it might be more appropriate to consider bothairspace congestion and extra flight cost and try to seeka good trade-off between them Hence Daniel et al [16]formulated the problem into a biobjective optimization prob-lem and solved it with the multiobjective genetic algorithm(MOGA) However the application of existingmultiobjectiveevolutionary algorithms (MOEAs) to the ANFA problemmight lead to poor performance due to the scale of the prob-lem To the best of our knowledge fewworks could effectivelysolve complicatedmultiobjective optimization problemswiththousands of variables

With the consideration of the reduction of the airspacecongestion and the flight delay simultaneously this paperformulates the ANFA problem as a multiobjective optimiza-tion model Then an efficient multiobjective optimizationframework is presented to solve it This framework employsthe parallel computation and population diversity adjustmenttechniques to improve the optimization capability Firstlyan effective multi-island parallel evolution algorithm (PEA)with multiple evolution populations is adopted Besidesone-way ring migration topology is applied to exchangeindividuals among populations to improve the efficiency ofthe cooperation of populations Secondly the multiobjectiveevolutionary algorithm NSGA2 is used to optimize eachpopulation In addition a cooperative coevolution algorithmis adapted to divide the ANFA problem into several low-dimensional biobjective optimization problems which areeasier to deal with Finally in order to maintain the diversityof solutions and avoid prematurity a dynamic adjustmentoperator based on solution congestion degree is specifi-cally designed It can greatly improve the distribution ofthe nondominated solutions in the archive and maintainthe population diversity by inheriting the nondominatedsolutions with low congestion degree in a high probabilitySimulation results using the real traffic data from Chinaair route network and daily flight plans demonstrate thatthe proposed approach can improve the solution qualityeffectively showing superiority to the existing approaches

such as the multiobjective genetic algorithm the well-knownmultiobjective evolutionary algorithm based on decomposi-tion (MOEAD) and a CC-based multiobjective algorithmas well as other parallel evolution algorithms with differentmigration topology

The rest of this paper is organized as follows Section 2introduces the formulation of the investigated biobjectiveANFA problem Section 3 describes the proposed multiob-jective optimization framework in detail Experimental studyis presented in Section 4 to evaluate the effectiveness ofour algorithm Finally Section 5 concludes this paper anddiscusses directions for further research

2 Problem Formulation

The airway network can be modeled as a directed graphincluding the waypoints as nodes and segments as edgesEach flight can be considered as a single commodity onthe network with a defined pair of origin-destination nodesBesides each flight has its predefined departure time slots andthe optional routes In this paper with the consideration ofsafety and efficiency the ANFA problem is formulated as abiobjective problem to reduce the airspace congestion and thetotal flight delay simultaneously

Each flight (flight 119896) is associated with a pair of decisionvariables (119903119896 120591119896) where 119903119896 is a possible route and 120591119896 is afeasible departure time slot In addition they are subject tothe following

(1) 119903119896 isin Path119896 = 1199031015840

1 1199031015840

2 119903

1015840

max where Pathk is the setof all possible paths of flight 119896

(2) 119905min119896

le 120591119896 le 119905max119896

where 119905max119896

is the latest departuretime slot and 119905

min119896

is the earliest departure time slot

We can see that constraint (1) denotes a feasible route of eachflight from a defined route set while constraint (2) enforcesthe flight to depart at a predefined departure time slot

The first objective function is the minimization of theairspace congestion Here the workload of the sectors isused to indicate the airspace congestion which is defined asfollows

It is generally known that the workload of a sector mainlydepends on the monitoring workload and the coordinationworkload Hence the total workload of a sector 119878119896 at time 119905

can be roughly expressed by the following [16]

119882119905

119878119896= 119882119905

mo119878119896 + 119882119905

co119878119896 (1)

where 119882119905mo119878119896 is the monitoring workload and can be numer-ically estimated by

119882mo119904119896 = 1 + 119872

119905

119878119896minus 119862119905

119898119878119896 if 119872119905

119878119896gt 119862119905

119898119878119896

0 else(2)

where 119872119905119878119896is related to the number of aircraft in sector 119878119896 at

time 119905 and119862119905

119898119878119896is themonitoring critical capacity of the sector

at time 119905








ENDWHILEEND



119882co119904119896 = 1 + 119862119905

119878119896minus 119862119905

119888119878119896 if 119862119905

119878119896gt 119862119905

119888119878119896

0 else(3)

where 119862119905






min119860119862 =

119896=119899119878

sum

119896=1

((sum

119905isin119879

119882119905

119878119896)

1minus120593


119882119905

119878119896)

120593

) (4)



min EFC = sum

119894isin119865

(10038161003816100381610038161003816120591119894 minus 119905

prf119896

10038161003816100381610038161003816+ (119905119903119894

minus 119905119903prf119894

))2

(5)

where 119905119903119894 and 119905119903prf119894









pop119894= 119894119889119894V1198941 119894119889119894V1198942 119894119889119894V119894119901119904 1 le 119894 le 119872 (6)


119894119889119894V119894119895 = (1199031198941198951 1205911198941198951 1199031198941198952 1205911198941198952 119903119894119895119873 120591119894119895119873) 1 le 119895 le ps (7)







pop119894= sp1119894 sp119899119904

119894 1 le 119894 le 119872

sp119895119894= 119904119894119889

1198951

119894 1199041198941198891198952

119894 119904119894119889

119895ps119894

1 le 119895 le 119899119904

119904119894119889119895119896

119894= (1199031015840

1198951198961 1205911015840

1198951198961 1199031015840

1198951198962 1205911015840

1198951198962 119903

1015840

119895119896(119873119899119904) 1205911015840

119895119896(119873119899119904))

1 le 119895 le ps

(8)





119881119894119866+1 = 1199091199031119866 + 119865 sdot (1199091199032119866 minus 1199091199033119866) 119865 isin [0 2] (9)


119880119894119866+1 = (1199061119894119866+1 1199062119894119866+1 119906119873119894119866+1)


(10)



119909119894119866+1 = 119906119894119866+1 119891 (119906119894119866+1) le 119891 (119909119894119866)

119909119894119866 else(11)



54

55

71 53

77

34

1413

1215 16

17

3231

3938

37

10

20

11

19 18

246773 72 21 22 23 25

6263

47

4246

5657

58

36 353045

4344

40 414948

262728

134

2

56759

60 6165 64

89

74

75

76

50 5152

33

66

70

29

6869

(a) (b)




119889119894119895 =1

2radic(119875

fit[1]119894

minus 119875fit[1]119895

)2

+ (119875fit[2]119894

minus 119875fit[2]119895

)2

(12)

where 119901fit[1]119894

and 119901fit[2]119894


119904 =2

119899 (119899 minus 1)

119894=119899

sum

119894=1

119895=119899

sum

119895=119894+1

119889119894119895 (13)


can be expressed as

119889119894 =1

119899 minus 1

119895=119899

sum

119895=1119895 =119894

120575119894119895 (14)

where

120575119894119895 = 1 if 119889119894119895 le 120574 sdot 119904

0 if 119889119894119895 gt 120574 sdot 119904(15)



119901119894 =119889119894120572

sum119899

119895=1119889119895120572 (16)







15 2 25 3

2

3

4

5

6

7

8

9

10

11

12

Tota

l del

ay

MOGAMOEAD

CCMAPEA


6

times106

(a)

1 15 2 250

05

1

15

2

25

3

Tota

l del

ay

MOGAMOEAD

CCMAPEA


times107

(b)





(IM = 5)








Algorithm 119868119867 120574 Δ







13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO


times106

times107

(b)



Algorithm 119868119867 120574 Δ






Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050


Δ)

Algorithm 119868119867 119868119863 Δ











Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















ENDWHILEEND



119882co119904119896 = 1 + 119862119905

119878119896minus 119862119905

119888119878119896 if 119862119905

119878119896gt 119862119905

119888119878119896

0 else(3)

where 119862119905






min119860119862 =

119896=119899119878

sum

119896=1

((sum

119905isin119879

119882119905

119878119896)

1minus120593


119882119905

119878119896)

120593

) (4)



min EFC = sum

119894isin119865

(10038161003816100381610038161003816120591119894 minus 119905

prf119896

10038161003816100381610038161003816+ (119905119903119894

minus 119905119903prf119894

))2

(5)

where 119905119903119894 and 119905119903prf119894









pop119894= 119894119889119894V1198941 119894119889119894V1198942 119894119889119894V119894119901119904 1 le 119894 le 119872 (6)


119894119889119894V119894119895 = (1199031198941198951 1205911198941198951 1199031198941198952 1205911198941198952 119903119894119895119873 120591119894119895119873) 1 le 119895 le ps (7)







pop119894= sp1119894 sp119899119904

119894 1 le 119894 le 119872

sp119895119894= 119904119894119889

1198951

119894 1199041198941198891198952

119894 119904119894119889

119895ps119894

1 le 119895 le 119899119904

119904119894119889119895119896

119894= (1199031015840

1198951198961 1205911015840

1198951198961 1199031015840

1198951198962 1205911015840

1198951198962 119903

1015840

119895119896(119873119899119904) 1205911015840

119895119896(119873119899119904))

1 le 119895 le ps

(8)





119881119894119866+1 = 1199091199031119866 + 119865 sdot (1199091199032119866 minus 1199091199033119866) 119865 isin [0 2] (9)


119880119894119866+1 = (1199061119894119866+1 1199062119894119866+1 119906119873119894119866+1)


(10)



119909119894119866+1 = 119906119894119866+1 119891 (119906119894119866+1) le 119891 (119909119894119866)

119909119894119866 else(11)



54

55

71 53

77

34

1413

1215 16

17

3231

3938

37

10

20

11

19 18

246773 72 21 22 23 25

6263

47

4246

5657

58

36 353045

4344

40 414948

262728

134

2

56759

60 6165 64

89

74

75

76

50 5152

33

66

70

29

6869

(a) (b)




119889119894119895 =1

2radic(119875

fit[1]119894

minus 119875fit[1]119895

)2

+ (119875fit[2]119894

minus 119875fit[2]119895

)2

(12)

where 119901fit[1]119894

and 119901fit[2]119894


119904 =2

119899 (119899 minus 1)

119894=119899

sum

119894=1

119895=119899

sum

119895=119894+1

119889119894119895 (13)


can be expressed as

119889119894 =1

119899 minus 1

119895=119899

sum

119895=1119895 =119894

120575119894119895 (14)

where

120575119894119895 = 1 if 119889119894119895 le 120574 sdot 119904

0 if 119889119894119895 gt 120574 sdot 119904(15)



119901119894 =119889119894120572

sum119899

119895=1119889119895120572 (16)







15 2 25 3

2

3

4

5

6

7

8

9

10

11

12

Tota

l del

ay

MOGAMOEAD

CCMAPEA


6

times106

(a)

1 15 2 250

05

1

15

2

25

3

Tota

l del

ay

MOGAMOEAD

CCMAPEA


times107

(b)





(IM = 5)








Algorithm 119868119867 120574 Δ







13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO


times106

times107

(b)



Algorithm 119868119867 120574 Δ






Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050


Δ)

Algorithm 119868119867 119868119863 Δ











Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












pop119894= 119894119889119894V1198941 119894119889119894V1198942 119894119889119894V119894119901119904 1 le 119894 le 119872 (6)


119894119889119894V119894119895 = (1199031198941198951 1205911198941198951 1199031198941198952 1205911198941198952 119903119894119895119873 120591119894119895119873) 1 le 119895 le ps (7)







pop119894= sp1119894 sp119899119904

119894 1 le 119894 le 119872

sp119895119894= 119904119894119889

1198951

119894 1199041198941198891198952

119894 119904119894119889

119895ps119894

1 le 119895 le 119899119904

119904119894119889119895119896

119894= (1199031015840

1198951198961 1205911015840

1198951198961 1199031015840

1198951198962 1205911015840

1198951198962 119903

1015840

119895119896(119873119899119904) 1205911015840

119895119896(119873119899119904))

1 le 119895 le ps

(8)





119881119894119866+1 = 1199091199031119866 + 119865 sdot (1199091199032119866 minus 1199091199033119866) 119865 isin [0 2] (9)


119880119894119866+1 = (1199061119894119866+1 1199062119894119866+1 119906119873119894119866+1)


(10)



119909119894119866+1 = 119906119894119866+1 119891 (119906119894119866+1) le 119891 (119909119894119866)

119909119894119866 else(11)



54

55

71 53

77

34

1413

1215 16

17

3231

3938

37

10

20

11

19 18

246773 72 21 22 23 25

6263

47

4246

5657

58

36 353045

4344

40 414948

262728

134

2

56759

60 6165 64

89

74

75

76

50 5152

33

66

70

29

6869

(a) (b)




119889119894119895 =1

2radic(119875

fit[1]119894

minus 119875fit[1]119895

)2

+ (119875fit[2]119894

minus 119875fit[2]119895

)2

(12)

where 119901fit[1]119894

and 119901fit[2]119894


119904 =2

119899 (119899 minus 1)

119894=119899

sum

119894=1

119895=119899

sum

119895=119894+1

119889119894119895 (13)


can be expressed as

119889119894 =1

119899 minus 1

119895=119899

sum

119895=1119895 =119894

120575119894119895 (14)

where

120575119894119895 = 1 if 119889119894119895 le 120574 sdot 119904

0 if 119889119894119895 gt 120574 sdot 119904(15)



119901119894 =119889119894120572

sum119899

119895=1119889119895120572 (16)







15 2 25 3

2

3

4

5

6

7

8

9

10

11

12

Tota

l del

ay

MOGAMOEAD

CCMAPEA


6

times106

(a)

1 15 2 250

05

1

15

2

25

3

Tota

l del

ay

MOGAMOEAD

CCMAPEA


times107

(b)





(IM = 5)








Algorithm 119868119867 120574 Δ







13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO


times106

times107

(b)



Algorithm 119868119867 120574 Δ






Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050


Δ)

Algorithm 119868119867 119868119863 Δ











Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










54

55

71 53

77

34

1413

1215 16

17

3231

3938

37

10

20

11

19 18

246773 72 21 22 23 25

6263

47

4246

5657

58

36 353045

4344

40 414948

262728

134

2

56759

60 6165 64

89

74

75

76

50 5152

33

66

70

29

6869

(a) (b)




119889119894119895 =1

2radic(119875

fit[1]119894

minus 119875fit[1]119895

)2

+ (119875fit[2]119894

minus 119875fit[2]119895

)2

(12)

where 119901fit[1]119894

and 119901fit[2]119894


119904 =2

119899 (119899 minus 1)

119894=119899

sum

119894=1

119895=119899

sum

119895=119894+1

119889119894119895 (13)


can be expressed as

119889119894 =1

119899 minus 1

119895=119899

sum

119895=1119895 =119894

120575119894119895 (14)

where

120575119894119895 = 1 if 119889119894119895 le 120574 sdot 119904

0 if 119889119894119895 gt 120574 sdot 119904(15)



119901119894 =119889119894120572

sum119899

119895=1119889119895120572 (16)







15 2 25 3

2

3

4

5

6

7

8

9

10

11

12

Tota

l del

ay

MOGAMOEAD

CCMAPEA


6

times106

(a)

1 15 2 250

05

1

15

2

25

3

Tota

l del

ay

MOGAMOEAD

CCMAPEA


times107

(b)





(IM = 5)








Algorithm 119868119867 120574 Δ







13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO


times106

times107

(b)



Algorithm 119868119867 120574 Δ






Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050


Δ)

Algorithm 119868119867 119868119863 Δ











Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










15 2 25 3

2

3

4

5

6

7

8

9

10

11

12

Tota

l del

ay

MOGAMOEAD

CCMAPEA


6

times106

(a)

1 15 2 250

05

1

15

2

25

3

Tota

l del

ay

MOGAMOEAD

CCMAPEA


times107

(b)





(IM = 5)








Algorithm 119868119867 120574 Δ







13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO


times106

times107

(b)



Algorithm 119868119867 120574 Δ






Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050


Δ)

Algorithm 119868119867 119868119863 Δ











Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










13 135 14 145 15

24

26

28

3

32

34

36

38

Tota

l del

ay


6

times106

PEAPEA without DAO

(a)

116 118 12 122 124 126 128 13 132

52

54

56

58

6

62

64

66

68

7

72

Tota

l del

ay

PEAPEA without DAO


times106

times107

(b)



Algorithm 119868119867 120574 Δ






Algorithm 119868119867 119868119863 Δ

PEA without DAO 92026119890 + 10 70467119890 + 04 12471

PEA 13937e + 11 33957e + 04 10050


Δ)

Algorithm 119868119867 119868119863 Δ











Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















Acknowledgments


References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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