Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 854364 9 pageshttpdxdoiorg1011552013854364

Research ArticleA Collaborative Optimization Model for Ground Taxi Based onAircraft Priority

Yu Jiang Zhihua Liao and Honghai Zhang

College of Civil Aviation Nanjing University of Aeronautics and Astronautics Nanjing Jiangsu 210016 China

Correspondence should be addressed to Zhihua Liao lzh0909nuaa163com

Received 3 July 2013 Revised 9 October 2013 Accepted 11 October 2013

Academic Editor John Gunnar Carlsson

Copyright copy 2013 Yu Jiang et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Large hub airports have gradually become the ldquobottleneckrdquo of the air transport network To alleviate the ldquobottleneckrdquo effectoptimizing the taxi scheduling is one of the solutions This paper establishes a scheduling optimization model by introducingpriority of aircraft under collaborative decision-making mechanism and a genetic algorithm is designed to verify the schedulingmodel by simulating Optimization results show that the reliability of the model and the adjusted genetic algorithm have a highefficiency The taxiing time decreases by 226 when compared with an empirical method and the flights with higher priorities areassigned better taxi routes It has great significance in reducing flight delays and cost of operation

1 Introduction

Civil aviation transportation industry inChina has developedinto an important period of rapid growth in recent yearsFrom 2006 to 2012 the average growth rate of total turnovervolume is 124 The contradiction between rapid develop-ment of air transport and supply of transport infrastructurehas become increasingly acute and airport has graduallybecome the ldquobottleneckrdquo of air transport network More andmore attentions are paid to airport scene resource schedul-ing especially the runway and taxiway system resourcescheduling problem The rate of utilization of taxiway will beimproved and the available capacity will also be increased byusing scheduling optimization technology Meanwhile it willbe more conductive to achieve fair efficient operation andease the contradiction between traffic flow and the availablecapacity by introducing aircraft priority under collaborativedecision-makingmechanismThe study from the perspectiveof systemwill reduce flight delays and fuel consumptionwhiletaxiing in the whole The airport scene resources schedulingoptimization problem has already become one of the hottopics in the study of domestic and foreign scholars now

In the study of foreign scholars Gotteland and Durand[1] presented an optimization model with safety separationand runway capacity constraints and set taxi time minimum

as the optimization objectiveThey solved themodel by usinggenetic algorithmThemodel did not take the aircraft priorityand taxi waiting problem caused by confliction into accountMarın [2] established a time-space network to describeconfliction and congestion during taxiing A network flowmodel is used to optimize the scheduling with real datafrom Madrid-Barajas airport But the constraint conditionsin the model are too complex Ravizza et al [3] put forwarda stand holding model and its objective function was tominimize taxi time and fuel consumption Anderson andMilutinovi [4] introduced deviation probability to controlthe security constraints and an uncertainty-based mixedinteger linear programming model was established Butthis article ignored the deadlock which may be caused bysome conflicts Clare and Richards [5] studied schedulingoptimization problem on runway and taxiway by a MILPmodel The interaction effect between arrival and departureaircrafts was considered Keith et al [6] presented a MILPmodel based on conflict-free In fact it allowed for someconflictions and a holding strategy made the result moreoptimal Burgain et al [7] analyzed departure aircrafts in con-gested airports by using queuing optimization Collaborativedecision-making concept was introduced in this paper Anaircraft taxi time estimate technique was studied in paper [8]which was based on fuzzy rule system Some papers [9ndash13]

2 Mathematical Problems in Engineering

just presented other new optimization methods for examplecellular automata model Some papers [14ndash17] presented newmethods to analyze and solve the problem for examplelinear method statistical analysis quantitative analysis andempirical methods for example iteration plan In thesestudies though they proposed different kinds of models theconstraints of taxi rules were the same and the substance ofoptimization objective was similar With the implementationof collaborative decision-making (CDM) mechanism thecodecision results from multiorganizations (such as ATCairlines and airports) must be taken into consideration whenscheduling in busy airports A normal way to make fairdecisions is to introduce some priorities The priority canguarantee benefit of all parties and make taxi schedule moresmooth The flight priorities are usually determined by thetype of flight aircraft type or the airlines they belonged toIn this paper the flight priorities are added into constrainsdirectly and we will not discuss how they are calculated Thecomputational complexity of accurate computation is higherthan intelligence algorithm But intelligence algorithm maybe unstable because the computation result this time may benot the same as the next timeThe efficiency of algorithm stillcan be improved

In the study of domestic scholars a mixed integerprogramming model algorithm is proposed by Zhang et al[18] The paper optimizes taxi time under the conditionsof basic safety separation and conflict-free Apparently thismethod cannot verify whether the taxi paths are optimalor not You and Han [19] present a multiagent model Theaircrafts invariably look for the shortest path from currentnode to the destination node in the process of optimizationon a simulation platform So the waiting-taxiing balanceproblem still exists Wang et al [20] put forward a dynamicpath algorithm based on conflict-free This approach caneffectively avoid conflicts but the result may not be optimalAn optimization scheduling algorithm based on genetic algo-rithm is studied by Liu et al [21] From these studies threepoints are summarized (1)The waiting-taxiing optimizationis not enough (2)Genetic algorithm has advantages in large-scale scheduling problems but the efficiency of the algorithmcan still be improved (3) On the CDM platform everyaircraft is given a specific priority and the taxi scheduling willreduce flight delays on the whole

Taking all the elements in the taxi scheduling into con-sideration the paper sets the total taxi time minimum as theoptimization goal The basic safety separation is consideredand the aircraft priority and taxiway-waiting strategy areintroduced A linear programming model is established anda genetic algorithm is designed to simulate The methodcan not only improve the algorithm efficiency but also getscheduling path directly

2 Modelling

21 Description and Analysis Taxi scheduling optimizationcan be defined as the work in which each aircraft is given aspecific path on a certain taxiway network structure withoutdeadlock conflict and make the total taxi time minimal

Table 1 Minimum safety separation standards (unit m)

After FrontHeavy Medium Light

Heavy 300 200 100Medium 300 200 100Light 300 200 100

The taxi system in the airport is composed of runwaypassageway taxiway and parking apron For a departureflight after finishing the work in an assigned stand such ascleaning on-off passengers catering and fuelling the aircraftwill wait for controllerrsquos command The air traffic control(ATC) in the towerwill give commands about taxi path aswellas take-off runway and entrance The aircraft will be pushedout and begins to taxi on taxiway In general more than twoaircrafts taxi on the taxiway at the same time a basic safetyseparation between aircrafts is required According to theaircraft operation management manual a minimum safetyseparation is regulated between different types of aircraft(including heavy medium-size and light aircraft) Table 1gives the minimum safety separation between different typesof aircrafts

During taxiing the pilot can keep safety separation withthe following aircraft by adjusting aircraft speed Only oneaircraft is allowed to pass the same node at one time andother aircrafts are required to wait to ensure safety Whentwo aircrafts need to taxi on the same segment of taxiwayfrom different nodes one aircraft must hold and wait atthe entrance node if the minimum safety separation is notsatisfied If an aircraft arrives at the assigned runway entranceit can enter runway and take off when ATC allows

For the arrival flight the aircraft enters taxiway fromassigned runway and exits according to ATC instructionsThe taxi path and stand are assigned before the aircraft enterstaxi system The taxi is over when the aircraft arrives at thestand Figure 1 shows the whole operation flow of aircraft inairport

Taxi paths in taxi system are very complex and allaircrafts must keep the basic safety separation so conflictstend to occur during taxiing In order to make the modelnot too complex rear-end conflict and intersectional conflictare classified as node-conflict Deadlock conflict is classifiedas edge-conflict So there are two types of conflicts node-conflict and edge-conflict

Node-conflict happens when two or more than twoaircrafts taxi through a common node without keeping theminimum safety separation (see Figure 2)

Edge-conflict happens when two or more than twoaircrafts taxi through the common segment but with oppositedirection One aircraft must hold and wait at the entrancenode if the minimum safety separation is not satisfied (seeFigure 3)

For most aircrafts they cannot taxi backward onceedge-conflict happens the common segment will come toa deadlock So edge-conflict can also be called deadlockconflict In general preventive measures must be taken ifedge-conflict is likely to happen for example one aircraft is

Mathematical Problems in Engineering 3

Arrival

Arr-flight

Dep-flightRunway

Take-off

Landing

Departure

Taxiway Apron

Taxiing

Taxiing

Push back

Enter

Airport ATC Airport flight

Stand Terminal

Figure 1 The whole operation flow in the airport

Holdandwait

Minimumsafety

separation

Figure 2 Node-conflict

Hold and wait

Edge-conflict

Figure 3 Edge-conflict

not allowed to taxi through until the common segment is notin use if the aircraft is estimated to arrive later than anotheror aircrafts are given different priorities and only the aircraftwith a higher priority can taxi through the common segmentat one time

Some scholars put forward a dynamic path algorithmbased on conflict-free They want to avoid all the conflictsThe dynamic path algorithm can find the shortest path in realtime by a sliding time windowThough the method canmakeoptimal decision sometimes it is more optimal if a taxiway-waiting strategy is taken Obviously taxiway-waiting strategyshould be taken into consideration In this paper node-conflict is allowed because aircrafts can resolve conflictseasily and this type of conflict has little effect on taxiwaysystem Edge-conflict should be avoided as far as possiblePart of taxi system or even the whole taxi system may cometo a deadlock once edge-conflict happens Besides solving adeadlock is costly Therefore an edge-conflict constraint isadded to the model to avoid this type of conflicts as far aspossible

22 Model Assumption The paper mainly studies taxi opti-mization between stand and runway passagewayThemethodof how aircrafts choose taxi path and how to avoid conflicts isanalyzedThe objective of path choice is tomake the total taxipath length minimum but the conflicts are also consideredduring taxiing In general aircrafts can avoid conflicts byadjusting taxi speed or waiting at an intersectional node Ifan aircraft reduces its speed to keep the safety separationit means that the aircraft will arrive at the next node latercompared with normal condition The time difference ofarrival can be equivalent to waiting time at the destinationnode So the objective function is to make the total time costof all aircrafts minimumThree assumptions are made for themodel based on the above analysis

(1) Generally all the aircrafts taxi at the same maximumspeed

(2) When it is likely to conflict aircrafts can adjust speedrapidly So the acceleration is ignored

(3) When node-conflict happens hold and wait strategyis always efficient whether the aircraft is large or small

23 Objective Function Usuallymore than two aircrafts needan assigned taxi path at the same time The path schedulingcan be evaluated by the length of path the type of conflictsthe time of conflict and the degree of conflict The total timecost of all aircrafts reflects partly the path scheduling So theoptimization objective in this paper is to make the total timecost of all aircrafts minimum Consider

Min119879 = sum(119878119899119894

start119899119894

end

V119894

+

endsum119899=start

119879119891119899119894+

endminus1sum119899119909=start

119879119888119899119909119899119909+1

) (1)

24 Constraints Variable119891119894119895119899

is used to detect whether node-conflict happens and it must satisfy the following constraint

119891119894119895119899=

110038161003816100381610038161003816119905119899119894minus 119905119899119895

10038161003816100381610038161003816le 1199050

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (2)

Variable119908119894119895is used to compare priority between aircraft 119894 and

119895 and it must satisfy the following constraint

119908119894119895=

1 119901119894ge 119901119895

0 others119894 = 119895 (3)

4 Mathematical Problems in Engineering

Variable 119909119894119895119899

is used to detect the sequencing of aircraft 119894 and119895 and it must satisfy the following constraint

119909119894119895119899=

1 119905119899119894le 119905119899119895

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (4)

If node-conflict happens at node 119899 the hold andwait time119879119891119899119894

must satisfy the following constraint

119879119891119899119894= 119891119894119895119899(1 minus 119908

119894119895) (1199050+ 119905119899119895minus 119905119899119894) forall119899 isin 119877

119894cap 119877119895 119894 = 119895

(5)

If edge-conflict happens at edge (119898 119899) the hold andwait time119879119888119898119899119894

must satisfy the following constraint

119879119888119899119909119899119909+1119894

= (1 minus 119908119894119895)(

119878119899119909119899119909+1

V119895

+ 119905119899119909119895minus 119905119899119909+1119894

)

forall (119899119909 119899119909+1) isin 119877119894cap 119877119895 119894 = 119895

(6)

119878119898119899 the path length from node119898 to node 119899 must satisfy the

following constraint

119878119898119899=

119899

sum119909=119898

119878119899119909119899119909+1

(7)

The time of arrival at node 119899119909 119905119899119909119894 must satisfy the following

condition

119905119899119909119894=1198781198991119899119909

V119894

+

119909

sumstart119879119891119899119909119894

+

119909minus1

sumstart119879119888119899119909119899119909+1119894

+ 119879119894 (8)

In addition to the above basic constraints there are four otherconstraints during taxiing

Theminimum safety time interval constraint is as follows

119905119899119909119894minus 119905119899119909119895ge 1199050 forall119899

119909isin 119877119894cap 119877119895 119894 = 119895 (9)

Theminimum safety time interval at intersectional node is asfollows

119905119899119909119895ge 119909119894119895119899119909(119905119899119909119894+ 1199050) forall119894 119895 isin 119865 119894 = 119895 (10)

Rear-end conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909 119899119909+1) isin 119877119895

(11)

Deadlock conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909+1 119899119909) isin 119877119895

(12)

3 Design and Genetic Algorithm

31 Coding In order to make the results evident segmentedreal-coded method is chosen in the algorithm Every gene inchromosome stands for corresponding node in taxi path Inthis way every chromosome can signify multipaths in a moredirect way For example supposing that all departure aircraftstaxi from node 5 to node 1 and arrival aircrafts are on thecontrary the path codes of two departure aircrafts and twoarrival aircrafts can be expressed as follows

5 3 5 1 02 0 3014 052410541 (13)

The numbers 1ndash5 stand for the node numbers of taxi path and0 is a pad character

32 Population Initialization Population initialization of tra-ditional genetic algorithm is completely random to someextent Supposing that there is a taxiway with 119899 nodes 119898aircrafts need to be assigned a path from node 119860 to node 119861There are 119903 feasible taxi paths between 119860 and 119861 Then theprobability of obtaining a feasible path is 119903119899119898119899 For 119903 is farsmaller than 119899119898119899 this method is ineffective in populationinitialization

In order to improve the efficiency of algorithm a traversalalgorithm is used to compute all of the feasible paths betweenstart node and destination node The initial chromosomeis produced by selecting feasible paths randomly In thisway the initial chromosome is a set of feasible solution Sothe population initialization is based on the set of feasiblesolution in this method Apparently the change will beconductive to the implementation of genetic operations and

impel the population evolve rapidly The efficiency of thealgorithm is improved

33 Crossover Operation The paper chooses segmented real-coded method so multipoint matched crossover method isthe best choiceThemultipoint crossovermeans thatmultiplegenes implement crossover operation at the same timeSuppose that there are two parent chromosomes parent1 andparent2

7 5

6 2 13 0 8

102 60409

7549

Parent1

Parent2(14)

Mathematical Problems in Engineering 5

T12T11T10

T9

T8T7

T6

T5

T4T3T2

T1

45 75

89

85

854

4

5

2

3 4 6

9

875

20191817161413121110 15 21

39

24 2329 28 27 26 25 22

38

33 32 31 30

1

3637 3534Runway

Taxiway

Stand

40 41 42 43

Figure 4 The taxiway network

Three genes in parent1 and parent2 are matched andthey are nodes 2 9 and 4 New generation is produced aftercrossover operation

7 5

6 2 03 1 8

012 75409

6049

New-generation1

New-generation2

(15)

Multipoint matched crossover method can prevent somesuperior chromosomes from being destroyed to some extentMeanwhile this method can ensure that the new generationis still a feasible path and prevents population size reducingsharply

34 Fitness Function Suppose that all the aircrafts taxi at thesame speed then the total time cost of aircraft can easily beconverted to the length of path (hold and wait time is alsoconverted to path length) So optimization objective is tomake the total length of path minimumThe fitness functioncan be divided into three parts the actual path length node-conflict path length edge-conflict path length The actualpath length is the sum of every edge length in its taxi path Ifthe estimated time of two aircrafts which arrive at a commonnode is dissatisfied with 119905

0 one aircraft must adjust speed or

hold and wait at the common node It leads to the extensionof its total taxi timeThe extended time of taxi is the differencevalue between theoretical taxi time and actual taxi timeApparently the extended taxi time is converted into node-conflict path length The computing method of edge-conflictpath length is the same as node-conflict path length

When the time is converted into path length the fitnessfunction can be expressed as follows

119891fitness =120572

119878route length + 120573 lowast 119878119881-length + 120574 lowast 119878119864-length (16)

In the formula 120572 120573 and 120574 are unknown parameters119878route length is the actual path length 119878

119881-length is node-conflictpath length 119878

119864-length is edge-conflict path length

Table 2 Flight scheduling information

Flight no ETD Type Priority Stand DA1 MU5178 1035 320 4 T1 D2 CZ3118 1035 330 45 T2 D3 CZ6218 1035 330 45 T7 D4 MU2078 1035 320 4 T8 D5 CA1605 1035 737 5 T5 D6 CA1802 1035 738 35 T4 A7 MF8115 1035 737 3 T9 A8 HU7196 1035 734 25 T12 A9 GS6574 1035 319 2 T11 A

4 Simulation Verification

In order to verify whether the algorithm is feasible andeffective or not one large hub airport in China is chosenas an example We consider flight scheduling at the sametime in a rush hour According to actual operation taxiroutes commonly used are limited So we choose one runwaywith partial taxiways as an example Figure 4 shows thenetwork of the scene A virtual node is introduced in theexample The virtual node can transform the problem intopath problem between any two nodes Besides the lengthbetween virtual node and stand node can be regarded as thewaiting time of push back The initial value of the lengthis 0 An adjusted genetic algorithm is programmed in C++programming language on VC++ 60 platform

The minimum safety time interval is set as 4 units (theconversion of 300 meters) The average speed of aircraft isset as 36Kmh which means that the minimum safety timeinterval is 30 seconds According to the flight scheduling inthis airport 9 flights need to be scheduled at 1035 am Thespecific flight information is listed in Table 2

The flight priorities are associated with the type of flightaircraft type and the airlines they belonged to The flightpriorities are given directly in scheduling information table

6 Mathematical Problems in Engineering

Table 3 Scheduling by experience

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 T11

41 Experience Scheduling When the ground controllerassigns paths by experience method some routes are pre-ferred and waiting phenomenon is universal Assume theground controller takes FCFS (first come first service) strat-egy and assigns routes with experience One scheduling maybe like the following in Table 3

The distribution of arrival time at each node is shown inFigure 5 It is easy to find that conflict happens at node 10 1112 13 and 33 theoretically

In the actual operation some aircrafts wait at node inorder to avoid conflictionThis increases the whole taxi timeMore information about the experience scheduling is listed inTable 4 The average length of taxi route is 23663m and theaverage waiting time is about 283 s The actual average taxitime is about 2756 s and confliction happens 6 timesThoughthe priority of CA1605 is higher than others it conflicts withMU2078 and CZ6218 at nodes 3 and 12 As a result CA1605waits 94 s in the whole It is obvious that the scheduling canstill be improved

The distribution of actual arrival time at each node isshown in Figure 6 All the flights arrive at each nodewith timeinterval no smaller than the minimum safety time intervalThe whole taxi time is increased by 255 s

42 Genetic Algorithm Scheduling The population size is setas 20 crossover probability is 0618 andmutation probabilityis 0025Themaximum iteration is 100The initial populationshows that the sum of fitness is 10126 and the average fitnessis 506 The maximum fitness is 513 and the shortest path is27300mThe best assigned routes in the initial population arelisted in Table 5

To solve the problem with genetic algorithm we pro-grammed in C++ programming language on VC++ 60platform and work in a computer with dual core processorof Inter(R) Core(TM) i3 and 2G RAM After 100 iterationsthe program output the resultsThe solving time is about 12 sTable 6 is about the optimized results

The optimized population shows that the sum of fitness is13157 and the average fitness is 658 The maximum fitness is662 and the shortest path is 21297m

More information is shown in Table 7The average lengthof taxi route is 23663m and the average taxi time is about2693 s (decreased by 103 compared with an experiencevalue of 5min) Confliction happens 5 times and the whole

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900

103958

103906103836

103928 103936

103800103830 103831

103730103700

103731 103734

103630103600

103657

103530103500

Tim

e

103500

103600

Figure 5 The distribution of arrival time at each node

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900 103913

103958 103943103928

103800103830 103843

103831

103730103700

103738 103734

103630103600 103600

103657

103530103500 103500

Tim

e

Figure 6 The distribution of actual arrival time at each node

waiting time is about 199 s with an average of 221 s Thewaiting time of CA1605 is only 4 s and this is mostly becauseof its high priority

The optimized distribution of actual arrival time at eachnode is shown in Figure 7 All flights which arrive at eachnode satisfy the minimum time interval The whole taxi timeis increased by 199 s

Genetic evolution process is shown in Figure 8 It can beclearly seen how the population average fitness changes Asthe initial chromosome is produced by selecting feasible pathsrandomly in the process of evolution the average fitness isclose to the optimal solution after 37 iterations The averagefitness is stable after 65 iterations the maximum averagefitness is about 658

43 Comparisons Two methods are used to analyze theproblem the results are listed in Table 8

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

just presented other new optimization methods for examplecellular automata model Some papers [14ndash17] presented newmethods to analyze and solve the problem for examplelinear method statistical analysis quantitative analysis andempirical methods for example iteration plan In thesestudies though they proposed different kinds of models theconstraints of taxi rules were the same and the substance ofoptimization objective was similar With the implementationof collaborative decision-making (CDM) mechanism thecodecision results from multiorganizations (such as ATCairlines and airports) must be taken into consideration whenscheduling in busy airports A normal way to make fairdecisions is to introduce some priorities The priority canguarantee benefit of all parties and make taxi schedule moresmooth The flight priorities are usually determined by thetype of flight aircraft type or the airlines they belonged toIn this paper the flight priorities are added into constrainsdirectly and we will not discuss how they are calculated Thecomputational complexity of accurate computation is higherthan intelligence algorithm But intelligence algorithm maybe unstable because the computation result this time may benot the same as the next timeThe efficiency of algorithm stillcan be improved

In the study of domestic scholars a mixed integerprogramming model algorithm is proposed by Zhang et al[18] The paper optimizes taxi time under the conditionsof basic safety separation and conflict-free Apparently thismethod cannot verify whether the taxi paths are optimalor not You and Han [19] present a multiagent model Theaircrafts invariably look for the shortest path from currentnode to the destination node in the process of optimizationon a simulation platform So the waiting-taxiing balanceproblem still exists Wang et al [20] put forward a dynamicpath algorithm based on conflict-free This approach caneffectively avoid conflicts but the result may not be optimalAn optimization scheduling algorithm based on genetic algo-rithm is studied by Liu et al [21] From these studies threepoints are summarized (1)The waiting-taxiing optimizationis not enough (2)Genetic algorithm has advantages in large-scale scheduling problems but the efficiency of the algorithmcan still be improved (3) On the CDM platform everyaircraft is given a specific priority and the taxi scheduling willreduce flight delays on the whole

Taking all the elements in the taxi scheduling into con-sideration the paper sets the total taxi time minimum as theoptimization goal The basic safety separation is consideredand the aircraft priority and taxiway-waiting strategy areintroduced A linear programming model is established anda genetic algorithm is designed to simulate The methodcan not only improve the algorithm efficiency but also getscheduling path directly

2 Modelling

21 Description and Analysis Taxi scheduling optimizationcan be defined as the work in which each aircraft is given aspecific path on a certain taxiway network structure withoutdeadlock conflict and make the total taxi time minimal

Table 1 Minimum safety separation standards (unit m)

After FrontHeavy Medium Light

Heavy 300 200 100Medium 300 200 100Light 300 200 100

The taxi system in the airport is composed of runwaypassageway taxiway and parking apron For a departureflight after finishing the work in an assigned stand such ascleaning on-off passengers catering and fuelling the aircraftwill wait for controllerrsquos command The air traffic control(ATC) in the towerwill give commands about taxi path aswellas take-off runway and entrance The aircraft will be pushedout and begins to taxi on taxiway In general more than twoaircrafts taxi on the taxiway at the same time a basic safetyseparation between aircrafts is required According to theaircraft operation management manual a minimum safetyseparation is regulated between different types of aircraft(including heavy medium-size and light aircraft) Table 1gives the minimum safety separation between different typesof aircrafts

During taxiing the pilot can keep safety separation withthe following aircraft by adjusting aircraft speed Only oneaircraft is allowed to pass the same node at one time andother aircrafts are required to wait to ensure safety Whentwo aircrafts need to taxi on the same segment of taxiwayfrom different nodes one aircraft must hold and wait atthe entrance node if the minimum safety separation is notsatisfied If an aircraft arrives at the assigned runway entranceit can enter runway and take off when ATC allows

For the arrival flight the aircraft enters taxiway fromassigned runway and exits according to ATC instructionsThe taxi path and stand are assigned before the aircraft enterstaxi system The taxi is over when the aircraft arrives at thestand Figure 1 shows the whole operation flow of aircraft inairport

Taxi paths in taxi system are very complex and allaircrafts must keep the basic safety separation so conflictstend to occur during taxiing In order to make the modelnot too complex rear-end conflict and intersectional conflictare classified as node-conflict Deadlock conflict is classifiedas edge-conflict So there are two types of conflicts node-conflict and edge-conflict

Node-conflict happens when two or more than twoaircrafts taxi through a common node without keeping theminimum safety separation (see Figure 2)

Edge-conflict happens when two or more than twoaircrafts taxi through the common segment but with oppositedirection One aircraft must hold and wait at the entrancenode if the minimum safety separation is not satisfied (seeFigure 3)

For most aircrafts they cannot taxi backward onceedge-conflict happens the common segment will come toa deadlock So edge-conflict can also be called deadlockconflict In general preventive measures must be taken ifedge-conflict is likely to happen for example one aircraft is

Mathematical Problems in Engineering 3

Arrival

Arr-flight

Dep-flightRunway

Take-off

Landing

Departure

Taxiway Apron

Taxiing

Taxiing

Push back

Enter

Airport ATC Airport flight

Stand Terminal

Figure 1 The whole operation flow in the airport

Holdandwait

Minimumsafety

separation

Figure 2 Node-conflict

Hold and wait

Edge-conflict

Figure 3 Edge-conflict

not allowed to taxi through until the common segment is notin use if the aircraft is estimated to arrive later than anotheror aircrafts are given different priorities and only the aircraftwith a higher priority can taxi through the common segmentat one time

Some scholars put forward a dynamic path algorithmbased on conflict-free They want to avoid all the conflictsThe dynamic path algorithm can find the shortest path in realtime by a sliding time windowThough the method canmakeoptimal decision sometimes it is more optimal if a taxiway-waiting strategy is taken Obviously taxiway-waiting strategyshould be taken into consideration In this paper node-conflict is allowed because aircrafts can resolve conflictseasily and this type of conflict has little effect on taxiwaysystem Edge-conflict should be avoided as far as possiblePart of taxi system or even the whole taxi system may cometo a deadlock once edge-conflict happens Besides solving adeadlock is costly Therefore an edge-conflict constraint isadded to the model to avoid this type of conflicts as far aspossible

22 Model Assumption The paper mainly studies taxi opti-mization between stand and runway passagewayThemethodof how aircrafts choose taxi path and how to avoid conflicts isanalyzedThe objective of path choice is tomake the total taxipath length minimum but the conflicts are also consideredduring taxiing In general aircrafts can avoid conflicts byadjusting taxi speed or waiting at an intersectional node Ifan aircraft reduces its speed to keep the safety separationit means that the aircraft will arrive at the next node latercompared with normal condition The time difference ofarrival can be equivalent to waiting time at the destinationnode So the objective function is to make the total time costof all aircrafts minimumThree assumptions are made for themodel based on the above analysis

(1) Generally all the aircrafts taxi at the same maximumspeed

(2) When it is likely to conflict aircrafts can adjust speedrapidly So the acceleration is ignored

(3) When node-conflict happens hold and wait strategyis always efficient whether the aircraft is large or small

23 Objective Function Usuallymore than two aircrafts needan assigned taxi path at the same time The path schedulingcan be evaluated by the length of path the type of conflictsthe time of conflict and the degree of conflict The total timecost of all aircrafts reflects partly the path scheduling So theoptimization objective in this paper is to make the total timecost of all aircrafts minimum Consider

Min119879 = sum(119878119899119894

start119899119894

end

V119894

+

endsum119899=start

119879119891119899119894+

endminus1sum119899119909=start

119879119888119899119909119899119909+1

) (1)

24 Constraints Variable119891119894119895119899

is used to detect whether node-conflict happens and it must satisfy the following constraint

119891119894119895119899=

110038161003816100381610038161003816119905119899119894minus 119905119899119895

10038161003816100381610038161003816le 1199050

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (2)

Variable119908119894119895is used to compare priority between aircraft 119894 and

119895 and it must satisfy the following constraint

119908119894119895=

1 119901119894ge 119901119895

0 others119894 = 119895 (3)

4 Mathematical Problems in Engineering

Variable 119909119894119895119899

is used to detect the sequencing of aircraft 119894 and119895 and it must satisfy the following constraint

119909119894119895119899=

1 119905119899119894le 119905119899119895

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (4)

If node-conflict happens at node 119899 the hold andwait time119879119891119899119894

must satisfy the following constraint

119879119891119899119894= 119891119894119895119899(1 minus 119908

119894119895) (1199050+ 119905119899119895minus 119905119899119894) forall119899 isin 119877

119894cap 119877119895 119894 = 119895

(5)

If edge-conflict happens at edge (119898 119899) the hold andwait time119879119888119898119899119894

must satisfy the following constraint

119879119888119899119909119899119909+1119894

= (1 minus 119908119894119895)(

119878119899119909119899119909+1

V119895

+ 119905119899119909119895minus 119905119899119909+1119894

)

forall (119899119909 119899119909+1) isin 119877119894cap 119877119895 119894 = 119895

(6)

119878119898119899 the path length from node119898 to node 119899 must satisfy the

following constraint

119878119898119899=

119899

sum119909=119898

119878119899119909119899119909+1

(7)

The time of arrival at node 119899119909 119905119899119909119894 must satisfy the following

condition

119905119899119909119894=1198781198991119899119909

V119894

+

119909

sumstart119879119891119899119909119894

+

119909minus1

sumstart119879119888119899119909119899119909+1119894

+ 119879119894 (8)

In addition to the above basic constraints there are four otherconstraints during taxiing

Theminimum safety time interval constraint is as follows

119905119899119909119894minus 119905119899119909119895ge 1199050 forall119899

119909isin 119877119894cap 119877119895 119894 = 119895 (9)

Theminimum safety time interval at intersectional node is asfollows

119905119899119909119895ge 119909119894119895119899119909(119905119899119909119894+ 1199050) forall119894 119895 isin 119865 119894 = 119895 (10)

Rear-end conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909 119899119909+1) isin 119877119895

(11)

Deadlock conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909+1 119899119909) isin 119877119895

(12)

3 Design and Genetic Algorithm

31 Coding In order to make the results evident segmentedreal-coded method is chosen in the algorithm Every gene inchromosome stands for corresponding node in taxi path Inthis way every chromosome can signify multipaths in a moredirect way For example supposing that all departure aircraftstaxi from node 5 to node 1 and arrival aircrafts are on thecontrary the path codes of two departure aircrafts and twoarrival aircrafts can be expressed as follows

5 3 5 1 02 0 3014 052410541 (13)

The numbers 1ndash5 stand for the node numbers of taxi path and0 is a pad character

32 Population Initialization Population initialization of tra-ditional genetic algorithm is completely random to someextent Supposing that there is a taxiway with 119899 nodes 119898aircrafts need to be assigned a path from node 119860 to node 119861There are 119903 feasible taxi paths between 119860 and 119861 Then theprobability of obtaining a feasible path is 119903119899119898119899 For 119903 is farsmaller than 119899119898119899 this method is ineffective in populationinitialization

In order to improve the efficiency of algorithm a traversalalgorithm is used to compute all of the feasible paths betweenstart node and destination node The initial chromosomeis produced by selecting feasible paths randomly In thisway the initial chromosome is a set of feasible solution Sothe population initialization is based on the set of feasiblesolution in this method Apparently the change will beconductive to the implementation of genetic operations and

impel the population evolve rapidly The efficiency of thealgorithm is improved

33 Crossover Operation The paper chooses segmented real-coded method so multipoint matched crossover method isthe best choiceThemultipoint crossovermeans thatmultiplegenes implement crossover operation at the same timeSuppose that there are two parent chromosomes parent1 andparent2

7 5

6 2 13 0 8

102 60409

7549

Parent1

Parent2(14)

Mathematical Problems in Engineering 5

T12T11T10

T9

T8T7

T6

T5

T4T3T2

T1

45 75

89

85

854

4

5

2

3 4 6

9

875

20191817161413121110 15 21

39

24 2329 28 27 26 25 22

38

33 32 31 30

1

3637 3534Runway

Taxiway

Stand

40 41 42 43

Figure 4 The taxiway network

Three genes in parent1 and parent2 are matched andthey are nodes 2 9 and 4 New generation is produced aftercrossover operation

7 5

6 2 03 1 8

012 75409

6049

New-generation1

New-generation2

(15)

Multipoint matched crossover method can prevent somesuperior chromosomes from being destroyed to some extentMeanwhile this method can ensure that the new generationis still a feasible path and prevents population size reducingsharply

34 Fitness Function Suppose that all the aircrafts taxi at thesame speed then the total time cost of aircraft can easily beconverted to the length of path (hold and wait time is alsoconverted to path length) So optimization objective is tomake the total length of path minimumThe fitness functioncan be divided into three parts the actual path length node-conflict path length edge-conflict path length The actualpath length is the sum of every edge length in its taxi path Ifthe estimated time of two aircrafts which arrive at a commonnode is dissatisfied with 119905

0 one aircraft must adjust speed or

hold and wait at the common node It leads to the extensionof its total taxi timeThe extended time of taxi is the differencevalue between theoretical taxi time and actual taxi timeApparently the extended taxi time is converted into node-conflict path length The computing method of edge-conflictpath length is the same as node-conflict path length

When the time is converted into path length the fitnessfunction can be expressed as follows

119891fitness =120572

119878route length + 120573 lowast 119878119881-length + 120574 lowast 119878119864-length (16)

In the formula 120572 120573 and 120574 are unknown parameters119878route length is the actual path length 119878

119881-length is node-conflictpath length 119878

119864-length is edge-conflict path length

Table 2 Flight scheduling information

Flight no ETD Type Priority Stand DA1 MU5178 1035 320 4 T1 D2 CZ3118 1035 330 45 T2 D3 CZ6218 1035 330 45 T7 D4 MU2078 1035 320 4 T8 D5 CA1605 1035 737 5 T5 D6 CA1802 1035 738 35 T4 A7 MF8115 1035 737 3 T9 A8 HU7196 1035 734 25 T12 A9 GS6574 1035 319 2 T11 A

4 Simulation Verification

In order to verify whether the algorithm is feasible andeffective or not one large hub airport in China is chosenas an example We consider flight scheduling at the sametime in a rush hour According to actual operation taxiroutes commonly used are limited So we choose one runwaywith partial taxiways as an example Figure 4 shows thenetwork of the scene A virtual node is introduced in theexample The virtual node can transform the problem intopath problem between any two nodes Besides the lengthbetween virtual node and stand node can be regarded as thewaiting time of push back The initial value of the lengthis 0 An adjusted genetic algorithm is programmed in C++programming language on VC++ 60 platform

The minimum safety time interval is set as 4 units (theconversion of 300 meters) The average speed of aircraft isset as 36Kmh which means that the minimum safety timeinterval is 30 seconds According to the flight scheduling inthis airport 9 flights need to be scheduled at 1035 am Thespecific flight information is listed in Table 2

The flight priorities are associated with the type of flightaircraft type and the airlines they belonged to The flightpriorities are given directly in scheduling information table

6 Mathematical Problems in Engineering

Table 3 Scheduling by experience

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 T11

41 Experience Scheduling When the ground controllerassigns paths by experience method some routes are pre-ferred and waiting phenomenon is universal Assume theground controller takes FCFS (first come first service) strat-egy and assigns routes with experience One scheduling maybe like the following in Table 3

The distribution of arrival time at each node is shown inFigure 5 It is easy to find that conflict happens at node 10 1112 13 and 33 theoretically

In the actual operation some aircrafts wait at node inorder to avoid conflictionThis increases the whole taxi timeMore information about the experience scheduling is listed inTable 4 The average length of taxi route is 23663m and theaverage waiting time is about 283 s The actual average taxitime is about 2756 s and confliction happens 6 timesThoughthe priority of CA1605 is higher than others it conflicts withMU2078 and CZ6218 at nodes 3 and 12 As a result CA1605waits 94 s in the whole It is obvious that the scheduling canstill be improved

The distribution of actual arrival time at each node isshown in Figure 6 All the flights arrive at each nodewith timeinterval no smaller than the minimum safety time intervalThe whole taxi time is increased by 255 s

42 Genetic Algorithm Scheduling The population size is setas 20 crossover probability is 0618 andmutation probabilityis 0025Themaximum iteration is 100The initial populationshows that the sum of fitness is 10126 and the average fitnessis 506 The maximum fitness is 513 and the shortest path is27300mThe best assigned routes in the initial population arelisted in Table 5

To solve the problem with genetic algorithm we pro-grammed in C++ programming language on VC++ 60platform and work in a computer with dual core processorof Inter(R) Core(TM) i3 and 2G RAM After 100 iterationsthe program output the resultsThe solving time is about 12 sTable 6 is about the optimized results

The optimized population shows that the sum of fitness is13157 and the average fitness is 658 The maximum fitness is662 and the shortest path is 21297m

More information is shown in Table 7The average lengthof taxi route is 23663m and the average taxi time is about2693 s (decreased by 103 compared with an experiencevalue of 5min) Confliction happens 5 times and the whole

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900

103958

103906103836

103928 103936

103800103830 103831

103730103700

103731 103734

103630103600

103657

103530103500

Tim

e

103500

103600

Figure 5 The distribution of arrival time at each node

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900 103913

103958 103943103928

103800103830 103843

103831

103730103700

103738 103734

103630103600 103600

103657

103530103500 103500

Tim

e

Figure 6 The distribution of actual arrival time at each node

waiting time is about 199 s with an average of 221 s Thewaiting time of CA1605 is only 4 s and this is mostly becauseof its high priority

The optimized distribution of actual arrival time at eachnode is shown in Figure 7 All flights which arrive at eachnode satisfy the minimum time interval The whole taxi timeis increased by 199 s

Genetic evolution process is shown in Figure 8 It can beclearly seen how the population average fitness changes Asthe initial chromosome is produced by selecting feasible pathsrandomly in the process of evolution the average fitness isclose to the optimal solution after 37 iterations The averagefitness is stable after 65 iterations the maximum averagefitness is about 658

43 Comparisons Two methods are used to analyze theproblem the results are listed in Table 8

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Arrival

Arr-flight

Dep-flightRunway

Take-off

Landing

Departure

Taxiway Apron

Taxiing

Taxiing

Push back

Enter

Airport ATC Airport flight

Stand Terminal

Figure 1 The whole operation flow in the airport

Holdandwait

Minimumsafety

separation

Figure 2 Node-conflict

Hold and wait

Edge-conflict

Figure 3 Edge-conflict

not allowed to taxi through until the common segment is notin use if the aircraft is estimated to arrive later than anotheror aircrafts are given different priorities and only the aircraftwith a higher priority can taxi through the common segmentat one time

Some scholars put forward a dynamic path algorithmbased on conflict-free They want to avoid all the conflictsThe dynamic path algorithm can find the shortest path in realtime by a sliding time windowThough the method canmakeoptimal decision sometimes it is more optimal if a taxiway-waiting strategy is taken Obviously taxiway-waiting strategyshould be taken into consideration In this paper node-conflict is allowed because aircrafts can resolve conflictseasily and this type of conflict has little effect on taxiwaysystem Edge-conflict should be avoided as far as possiblePart of taxi system or even the whole taxi system may cometo a deadlock once edge-conflict happens Besides solving adeadlock is costly Therefore an edge-conflict constraint isadded to the model to avoid this type of conflicts as far aspossible

22 Model Assumption The paper mainly studies taxi opti-mization between stand and runway passagewayThemethodof how aircrafts choose taxi path and how to avoid conflicts isanalyzedThe objective of path choice is tomake the total taxipath length minimum but the conflicts are also consideredduring taxiing In general aircrafts can avoid conflicts byadjusting taxi speed or waiting at an intersectional node Ifan aircraft reduces its speed to keep the safety separationit means that the aircraft will arrive at the next node latercompared with normal condition The time difference ofarrival can be equivalent to waiting time at the destinationnode So the objective function is to make the total time costof all aircrafts minimumThree assumptions are made for themodel based on the above analysis

(1) Generally all the aircrafts taxi at the same maximumspeed

(2) When it is likely to conflict aircrafts can adjust speedrapidly So the acceleration is ignored

(3) When node-conflict happens hold and wait strategyis always efficient whether the aircraft is large or small

23 Objective Function Usuallymore than two aircrafts needan assigned taxi path at the same time The path schedulingcan be evaluated by the length of path the type of conflictsthe time of conflict and the degree of conflict The total timecost of all aircrafts reflects partly the path scheduling So theoptimization objective in this paper is to make the total timecost of all aircrafts minimum Consider

Min119879 = sum(119878119899119894

start119899119894

end

V119894

+

endsum119899=start

119879119891119899119894+

endminus1sum119899119909=start

119879119888119899119909119899119909+1

) (1)

24 Constraints Variable119891119894119895119899

is used to detect whether node-conflict happens and it must satisfy the following constraint

119891119894119895119899=

110038161003816100381610038161003816119905119899119894minus 119905119899119895

10038161003816100381610038161003816le 1199050

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (2)

Variable119908119894119895is used to compare priority between aircraft 119894 and

119895 and it must satisfy the following constraint

119908119894119895=

1 119901119894ge 119901119895

0 others119894 = 119895 (3)

4 Mathematical Problems in Engineering

Variable 119909119894119895119899

is used to detect the sequencing of aircraft 119894 and119895 and it must satisfy the following constraint

119909119894119895119899=

1 119905119899119894le 119905119899119895

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (4)

If node-conflict happens at node 119899 the hold andwait time119879119891119899119894

must satisfy the following constraint

119879119891119899119894= 119891119894119895119899(1 minus 119908

119894119895) (1199050+ 119905119899119895minus 119905119899119894) forall119899 isin 119877

119894cap 119877119895 119894 = 119895

(5)

If edge-conflict happens at edge (119898 119899) the hold andwait time119879119888119898119899119894

must satisfy the following constraint

119879119888119899119909119899119909+1119894

= (1 minus 119908119894119895)(

119878119899119909119899119909+1

V119895

+ 119905119899119909119895minus 119905119899119909+1119894

)

forall (119899119909 119899119909+1) isin 119877119894cap 119877119895 119894 = 119895

(6)

119878119898119899 the path length from node119898 to node 119899 must satisfy the

following constraint

119878119898119899=

119899

sum119909=119898

119878119899119909119899119909+1

(7)

The time of arrival at node 119899119909 119905119899119909119894 must satisfy the following

condition

119905119899119909119894=1198781198991119899119909

V119894

+

119909

sumstart119879119891119899119909119894

+

119909minus1

sumstart119879119888119899119909119899119909+1119894

+ 119879119894 (8)

In addition to the above basic constraints there are four otherconstraints during taxiing

Theminimum safety time interval constraint is as follows

119905119899119909119894minus 119905119899119909119895ge 1199050 forall119899

119909isin 119877119894cap 119877119895 119894 = 119895 (9)

Theminimum safety time interval at intersectional node is asfollows

119905119899119909119895ge 119909119894119895119899119909(119905119899119909119894+ 1199050) forall119894 119895 isin 119865 119894 = 119895 (10)

Rear-end conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909 119899119909+1) isin 119877119895

(11)

Deadlock conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909+1 119899119909) isin 119877119895

(12)

3 Design and Genetic Algorithm

31 Coding In order to make the results evident segmentedreal-coded method is chosen in the algorithm Every gene inchromosome stands for corresponding node in taxi path Inthis way every chromosome can signify multipaths in a moredirect way For example supposing that all departure aircraftstaxi from node 5 to node 1 and arrival aircrafts are on thecontrary the path codes of two departure aircrafts and twoarrival aircrafts can be expressed as follows

5 3 5 1 02 0 3014 052410541 (13)

The numbers 1ndash5 stand for the node numbers of taxi path and0 is a pad character

32 Population Initialization Population initialization of tra-ditional genetic algorithm is completely random to someextent Supposing that there is a taxiway with 119899 nodes 119898aircrafts need to be assigned a path from node 119860 to node 119861There are 119903 feasible taxi paths between 119860 and 119861 Then theprobability of obtaining a feasible path is 119903119899119898119899 For 119903 is farsmaller than 119899119898119899 this method is ineffective in populationinitialization

In order to improve the efficiency of algorithm a traversalalgorithm is used to compute all of the feasible paths betweenstart node and destination node The initial chromosomeis produced by selecting feasible paths randomly In thisway the initial chromosome is a set of feasible solution Sothe population initialization is based on the set of feasiblesolution in this method Apparently the change will beconductive to the implementation of genetic operations and

impel the population evolve rapidly The efficiency of thealgorithm is improved

33 Crossover Operation The paper chooses segmented real-coded method so multipoint matched crossover method isthe best choiceThemultipoint crossovermeans thatmultiplegenes implement crossover operation at the same timeSuppose that there are two parent chromosomes parent1 andparent2

7 5

6 2 13 0 8

102 60409

7549

Parent1

Parent2(14)

Mathematical Problems in Engineering 5

T12T11T10

T9

T8T7

T6

T5

T4T3T2

T1

45 75

89

85

854

4

5

2

3 4 6

9

875

20191817161413121110 15 21

39

24 2329 28 27 26 25 22

38

33 32 31 30

1

3637 3534Runway

Taxiway

Stand

40 41 42 43

Figure 4 The taxiway network

Three genes in parent1 and parent2 are matched andthey are nodes 2 9 and 4 New generation is produced aftercrossover operation

7 5

6 2 03 1 8

012 75409

6049

New-generation1

New-generation2

(15)

Multipoint matched crossover method can prevent somesuperior chromosomes from being destroyed to some extentMeanwhile this method can ensure that the new generationis still a feasible path and prevents population size reducingsharply

34 Fitness Function Suppose that all the aircrafts taxi at thesame speed then the total time cost of aircraft can easily beconverted to the length of path (hold and wait time is alsoconverted to path length) So optimization objective is tomake the total length of path minimumThe fitness functioncan be divided into three parts the actual path length node-conflict path length edge-conflict path length The actualpath length is the sum of every edge length in its taxi path Ifthe estimated time of two aircrafts which arrive at a commonnode is dissatisfied with 119905

0 one aircraft must adjust speed or

hold and wait at the common node It leads to the extensionof its total taxi timeThe extended time of taxi is the differencevalue between theoretical taxi time and actual taxi timeApparently the extended taxi time is converted into node-conflict path length The computing method of edge-conflictpath length is the same as node-conflict path length

When the time is converted into path length the fitnessfunction can be expressed as follows

119891fitness =120572

119878route length + 120573 lowast 119878119881-length + 120574 lowast 119878119864-length (16)

In the formula 120572 120573 and 120574 are unknown parameters119878route length is the actual path length 119878

119881-length is node-conflictpath length 119878

119864-length is edge-conflict path length

Table 2 Flight scheduling information

Flight no ETD Type Priority Stand DA1 MU5178 1035 320 4 T1 D2 CZ3118 1035 330 45 T2 D3 CZ6218 1035 330 45 T7 D4 MU2078 1035 320 4 T8 D5 CA1605 1035 737 5 T5 D6 CA1802 1035 738 35 T4 A7 MF8115 1035 737 3 T9 A8 HU7196 1035 734 25 T12 A9 GS6574 1035 319 2 T11 A

4 Simulation Verification

In order to verify whether the algorithm is feasible andeffective or not one large hub airport in China is chosenas an example We consider flight scheduling at the sametime in a rush hour According to actual operation taxiroutes commonly used are limited So we choose one runwaywith partial taxiways as an example Figure 4 shows thenetwork of the scene A virtual node is introduced in theexample The virtual node can transform the problem intopath problem between any two nodes Besides the lengthbetween virtual node and stand node can be regarded as thewaiting time of push back The initial value of the lengthis 0 An adjusted genetic algorithm is programmed in C++programming language on VC++ 60 platform

The minimum safety time interval is set as 4 units (theconversion of 300 meters) The average speed of aircraft isset as 36Kmh which means that the minimum safety timeinterval is 30 seconds According to the flight scheduling inthis airport 9 flights need to be scheduled at 1035 am Thespecific flight information is listed in Table 2

The flight priorities are associated with the type of flightaircraft type and the airlines they belonged to The flightpriorities are given directly in scheduling information table

6 Mathematical Problems in Engineering

Table 3 Scheduling by experience

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 T11

41 Experience Scheduling When the ground controllerassigns paths by experience method some routes are pre-ferred and waiting phenomenon is universal Assume theground controller takes FCFS (first come first service) strat-egy and assigns routes with experience One scheduling maybe like the following in Table 3

The distribution of arrival time at each node is shown inFigure 5 It is easy to find that conflict happens at node 10 1112 13 and 33 theoretically

In the actual operation some aircrafts wait at node inorder to avoid conflictionThis increases the whole taxi timeMore information about the experience scheduling is listed inTable 4 The average length of taxi route is 23663m and theaverage waiting time is about 283 s The actual average taxitime is about 2756 s and confliction happens 6 timesThoughthe priority of CA1605 is higher than others it conflicts withMU2078 and CZ6218 at nodes 3 and 12 As a result CA1605waits 94 s in the whole It is obvious that the scheduling canstill be improved

The distribution of actual arrival time at each node isshown in Figure 6 All the flights arrive at each nodewith timeinterval no smaller than the minimum safety time intervalThe whole taxi time is increased by 255 s

42 Genetic Algorithm Scheduling The population size is setas 20 crossover probability is 0618 andmutation probabilityis 0025Themaximum iteration is 100The initial populationshows that the sum of fitness is 10126 and the average fitnessis 506 The maximum fitness is 513 and the shortest path is27300mThe best assigned routes in the initial population arelisted in Table 5

To solve the problem with genetic algorithm we pro-grammed in C++ programming language on VC++ 60platform and work in a computer with dual core processorof Inter(R) Core(TM) i3 and 2G RAM After 100 iterationsthe program output the resultsThe solving time is about 12 sTable 6 is about the optimized results

The optimized population shows that the sum of fitness is13157 and the average fitness is 658 The maximum fitness is662 and the shortest path is 21297m

More information is shown in Table 7The average lengthof taxi route is 23663m and the average taxi time is about2693 s (decreased by 103 compared with an experiencevalue of 5min) Confliction happens 5 times and the whole

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900

103958

103906103836

103928 103936

103800103830 103831

103730103700

103731 103734

103630103600

103657

103530103500

Tim

e

103500

103600

Figure 5 The distribution of arrival time at each node

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900 103913

103958 103943103928

103800103830 103843

103831

103730103700

103738 103734

103630103600 103600

103657

103530103500 103500

Tim

e

Figure 6 The distribution of actual arrival time at each node

waiting time is about 199 s with an average of 221 s Thewaiting time of CA1605 is only 4 s and this is mostly becauseof its high priority

The optimized distribution of actual arrival time at eachnode is shown in Figure 7 All flights which arrive at eachnode satisfy the minimum time interval The whole taxi timeis increased by 199 s

Genetic evolution process is shown in Figure 8 It can beclearly seen how the population average fitness changes Asthe initial chromosome is produced by selecting feasible pathsrandomly in the process of evolution the average fitness isclose to the optimal solution after 37 iterations The averagefitness is stable after 65 iterations the maximum averagefitness is about 658

43 Comparisons Two methods are used to analyze theproblem the results are listed in Table 8

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Variable 119909119894119895119899

is used to detect the sequencing of aircraft 119894 and119895 and it must satisfy the following constraint

119909119894119895119899=

1 119905119899119894le 119905119899119895

0 othersforall119899 isin 119877

119894cap 119877119895 119894 = 119895 (4)

If node-conflict happens at node 119899 the hold andwait time119879119891119899119894

must satisfy the following constraint

119879119891119899119894= 119891119894119895119899(1 minus 119908

119894119895) (1199050+ 119905119899119895minus 119905119899119894) forall119899 isin 119877

119894cap 119877119895 119894 = 119895

(5)

If edge-conflict happens at edge (119898 119899) the hold andwait time119879119888119898119899119894

must satisfy the following constraint

119879119888119899119909119899119909+1119894

= (1 minus 119908119894119895)(

119878119899119909119899119909+1

V119895

+ 119905119899119909119895minus 119905119899119909+1119894

)

forall (119899119909 119899119909+1) isin 119877119894cap 119877119895 119894 = 119895

(6)

119878119898119899 the path length from node119898 to node 119899 must satisfy the

following constraint

119878119898119899=

119899

sum119909=119898

119878119899119909119899119909+1

(7)

The time of arrival at node 119899119909 119905119899119909119894 must satisfy the following

condition

119905119899119909119894=1198781198991119899119909

V119894

+

119909

sumstart119879119891119899119909119894

+

119909minus1

sumstart119879119888119899119909119899119909+1119894

+ 119879119894 (8)

In addition to the above basic constraints there are four otherconstraints during taxiing

Theminimum safety time interval constraint is as follows

119905119899119909119894minus 119905119899119909119895ge 1199050 forall119899

119909isin 119877119894cap 119877119895 119894 = 119895 (9)

Theminimum safety time interval at intersectional node is asfollows

119905119899119909119895ge 119909119894119895119899119909(119905119899119909119894+ 1199050) forall119894 119895 isin 119865 119894 = 119895 (10)

Rear-end conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909 119899119909+1) isin 119877119895

(11)

Deadlock conflict constraint is as follows

119909119894119895119899119909

minus 119909119894119895119899119909+1

= 0 forall119894 119895 isin 119865 119894 = 119895

forall (119899119909 119899119909+1) isin 119877119894 forall (119899

119909+1 119899119909) isin 119877119895

(12)

3 Design and Genetic Algorithm

31 Coding In order to make the results evident segmentedreal-coded method is chosen in the algorithm Every gene inchromosome stands for corresponding node in taxi path Inthis way every chromosome can signify multipaths in a moredirect way For example supposing that all departure aircraftstaxi from node 5 to node 1 and arrival aircrafts are on thecontrary the path codes of two departure aircrafts and twoarrival aircrafts can be expressed as follows

5 3 5 1 02 0 3014 052410541 (13)

The numbers 1ndash5 stand for the node numbers of taxi path and0 is a pad character

32 Population Initialization Population initialization of tra-ditional genetic algorithm is completely random to someextent Supposing that there is a taxiway with 119899 nodes 119898aircrafts need to be assigned a path from node 119860 to node 119861There are 119903 feasible taxi paths between 119860 and 119861 Then theprobability of obtaining a feasible path is 119903119899119898119899 For 119903 is farsmaller than 119899119898119899 this method is ineffective in populationinitialization

In order to improve the efficiency of algorithm a traversalalgorithm is used to compute all of the feasible paths betweenstart node and destination node The initial chromosomeis produced by selecting feasible paths randomly In thisway the initial chromosome is a set of feasible solution Sothe population initialization is based on the set of feasiblesolution in this method Apparently the change will beconductive to the implementation of genetic operations and

impel the population evolve rapidly The efficiency of thealgorithm is improved

33 Crossover Operation The paper chooses segmented real-coded method so multipoint matched crossover method isthe best choiceThemultipoint crossovermeans thatmultiplegenes implement crossover operation at the same timeSuppose that there are two parent chromosomes parent1 andparent2

7 5

6 2 13 0 8

102 60409

7549

Parent1

Parent2(14)

Mathematical Problems in Engineering 5

T12T11T10

T9

T8T7

T6

T5

T4T3T2

T1

45 75

89

85

854

4

5

2

3 4 6

9

875

20191817161413121110 15 21

39

24 2329 28 27 26 25 22

38

33 32 31 30

1

3637 3534Runway

Taxiway

Stand

40 41 42 43

Figure 4 The taxiway network

Three genes in parent1 and parent2 are matched andthey are nodes 2 9 and 4 New generation is produced aftercrossover operation

7 5

6 2 03 1 8

012 75409

6049

New-generation1

New-generation2

(15)

Multipoint matched crossover method can prevent somesuperior chromosomes from being destroyed to some extentMeanwhile this method can ensure that the new generationis still a feasible path and prevents population size reducingsharply

34 Fitness Function Suppose that all the aircrafts taxi at thesame speed then the total time cost of aircraft can easily beconverted to the length of path (hold and wait time is alsoconverted to path length) So optimization objective is tomake the total length of path minimumThe fitness functioncan be divided into three parts the actual path length node-conflict path length edge-conflict path length The actualpath length is the sum of every edge length in its taxi path Ifthe estimated time of two aircrafts which arrive at a commonnode is dissatisfied with 119905

0 one aircraft must adjust speed or

hold and wait at the common node It leads to the extensionof its total taxi timeThe extended time of taxi is the differencevalue between theoretical taxi time and actual taxi timeApparently the extended taxi time is converted into node-conflict path length The computing method of edge-conflictpath length is the same as node-conflict path length

When the time is converted into path length the fitnessfunction can be expressed as follows

119891fitness =120572

119878route length + 120573 lowast 119878119881-length + 120574 lowast 119878119864-length (16)

In the formula 120572 120573 and 120574 are unknown parameters119878route length is the actual path length 119878

119881-length is node-conflictpath length 119878

119864-length is edge-conflict path length

Table 2 Flight scheduling information

Flight no ETD Type Priority Stand DA1 MU5178 1035 320 4 T1 D2 CZ3118 1035 330 45 T2 D3 CZ6218 1035 330 45 T7 D4 MU2078 1035 320 4 T8 D5 CA1605 1035 737 5 T5 D6 CA1802 1035 738 35 T4 A7 MF8115 1035 737 3 T9 A8 HU7196 1035 734 25 T12 A9 GS6574 1035 319 2 T11 A

4 Simulation Verification

In order to verify whether the algorithm is feasible andeffective or not one large hub airport in China is chosenas an example We consider flight scheduling at the sametime in a rush hour According to actual operation taxiroutes commonly used are limited So we choose one runwaywith partial taxiways as an example Figure 4 shows thenetwork of the scene A virtual node is introduced in theexample The virtual node can transform the problem intopath problem between any two nodes Besides the lengthbetween virtual node and stand node can be regarded as thewaiting time of push back The initial value of the lengthis 0 An adjusted genetic algorithm is programmed in C++programming language on VC++ 60 platform

The minimum safety time interval is set as 4 units (theconversion of 300 meters) The average speed of aircraft isset as 36Kmh which means that the minimum safety timeinterval is 30 seconds According to the flight scheduling inthis airport 9 flights need to be scheduled at 1035 am Thespecific flight information is listed in Table 2

The flight priorities are associated with the type of flightaircraft type and the airlines they belonged to The flightpriorities are given directly in scheduling information table

6 Mathematical Problems in Engineering

Table 3 Scheduling by experience

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 T11

41 Experience Scheduling When the ground controllerassigns paths by experience method some routes are pre-ferred and waiting phenomenon is universal Assume theground controller takes FCFS (first come first service) strat-egy and assigns routes with experience One scheduling maybe like the following in Table 3

The distribution of arrival time at each node is shown inFigure 5 It is easy to find that conflict happens at node 10 1112 13 and 33 theoretically

In the actual operation some aircrafts wait at node inorder to avoid conflictionThis increases the whole taxi timeMore information about the experience scheduling is listed inTable 4 The average length of taxi route is 23663m and theaverage waiting time is about 283 s The actual average taxitime is about 2756 s and confliction happens 6 timesThoughthe priority of CA1605 is higher than others it conflicts withMU2078 and CZ6218 at nodes 3 and 12 As a result CA1605waits 94 s in the whole It is obvious that the scheduling canstill be improved

The distribution of actual arrival time at each node isshown in Figure 6 All the flights arrive at each nodewith timeinterval no smaller than the minimum safety time intervalThe whole taxi time is increased by 255 s

42 Genetic Algorithm Scheduling The population size is setas 20 crossover probability is 0618 andmutation probabilityis 0025Themaximum iteration is 100The initial populationshows that the sum of fitness is 10126 and the average fitnessis 506 The maximum fitness is 513 and the shortest path is27300mThe best assigned routes in the initial population arelisted in Table 5

To solve the problem with genetic algorithm we pro-grammed in C++ programming language on VC++ 60platform and work in a computer with dual core processorof Inter(R) Core(TM) i3 and 2G RAM After 100 iterationsthe program output the resultsThe solving time is about 12 sTable 6 is about the optimized results

The optimized population shows that the sum of fitness is13157 and the average fitness is 658 The maximum fitness is662 and the shortest path is 21297m

More information is shown in Table 7The average lengthof taxi route is 23663m and the average taxi time is about2693 s (decreased by 103 compared with an experiencevalue of 5min) Confliction happens 5 times and the whole

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900

103958

103906103836

103928 103936

103800103830 103831

103730103700

103731 103734

103630103600

103657

103530103500

Tim

e

103500

103600

Figure 5 The distribution of arrival time at each node

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900 103913

103958 103943103928

103800103830 103843

103831

103730103700

103738 103734

103630103600 103600

103657

103530103500 103500

Tim

e

Figure 6 The distribution of actual arrival time at each node

waiting time is about 199 s with an average of 221 s Thewaiting time of CA1605 is only 4 s and this is mostly becauseof its high priority

The optimized distribution of actual arrival time at eachnode is shown in Figure 7 All flights which arrive at eachnode satisfy the minimum time interval The whole taxi timeis increased by 199 s

Genetic evolution process is shown in Figure 8 It can beclearly seen how the population average fitness changes Asthe initial chromosome is produced by selecting feasible pathsrandomly in the process of evolution the average fitness isclose to the optimal solution after 37 iterations The averagefitness is stable after 65 iterations the maximum averagefitness is about 658

43 Comparisons Two methods are used to analyze theproblem the results are listed in Table 8

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

T12T11T10

T9

T8T7

T6

T5

T4T3T2

T1

45 75

89

85

854

4

5

2

3 4 6

9

875

20191817161413121110 15 21

39

24 2329 28 27 26 25 22

38

33 32 31 30

1

3637 3534Runway

Taxiway

Stand

40 41 42 43

Figure 4 The taxiway network

Three genes in parent1 and parent2 are matched andthey are nodes 2 9 and 4 New generation is produced aftercrossover operation

7 5

6 2 03 1 8

012 75409

6049

New-generation1

New-generation2

(15)

Multipoint matched crossover method can prevent somesuperior chromosomes from being destroyed to some extentMeanwhile this method can ensure that the new generationis still a feasible path and prevents population size reducingsharply

34 Fitness Function Suppose that all the aircrafts taxi at thesame speed then the total time cost of aircraft can easily beconverted to the length of path (hold and wait time is alsoconverted to path length) So optimization objective is tomake the total length of path minimumThe fitness functioncan be divided into three parts the actual path length node-conflict path length edge-conflict path length The actualpath length is the sum of every edge length in its taxi path Ifthe estimated time of two aircrafts which arrive at a commonnode is dissatisfied with 119905

0 one aircraft must adjust speed or

hold and wait at the common node It leads to the extensionof its total taxi timeThe extended time of taxi is the differencevalue between theoretical taxi time and actual taxi timeApparently the extended taxi time is converted into node-conflict path length The computing method of edge-conflictpath length is the same as node-conflict path length

When the time is converted into path length the fitnessfunction can be expressed as follows

119891fitness =120572

119878route length + 120573 lowast 119878119881-length + 120574 lowast 119878119864-length (16)

In the formula 120572 120573 and 120574 are unknown parameters119878route length is the actual path length 119878

119881-length is node-conflictpath length 119878

119864-length is edge-conflict path length

Table 2 Flight scheduling information

Flight no ETD Type Priority Stand DA1 MU5178 1035 320 4 T1 D2 CZ3118 1035 330 45 T2 D3 CZ6218 1035 330 45 T7 D4 MU2078 1035 320 4 T8 D5 CA1605 1035 737 5 T5 D6 CA1802 1035 738 35 T4 A7 MF8115 1035 737 3 T9 A8 HU7196 1035 734 25 T12 A9 GS6574 1035 319 2 T11 A

4 Simulation Verification

In order to verify whether the algorithm is feasible andeffective or not one large hub airport in China is chosenas an example We consider flight scheduling at the sametime in a rush hour According to actual operation taxiroutes commonly used are limited So we choose one runwaywith partial taxiways as an example Figure 4 shows thenetwork of the scene A virtual node is introduced in theexample The virtual node can transform the problem intopath problem between any two nodes Besides the lengthbetween virtual node and stand node can be regarded as thewaiting time of push back The initial value of the lengthis 0 An adjusted genetic algorithm is programmed in C++programming language on VC++ 60 platform

The minimum safety time interval is set as 4 units (theconversion of 300 meters) The average speed of aircraft isset as 36Kmh which means that the minimum safety timeinterval is 30 seconds According to the flight scheduling inthis airport 9 flights need to be scheduled at 1035 am Thespecific flight information is listed in Table 2

The flight priorities are associated with the type of flightaircraft type and the airlines they belonged to The flightpriorities are given directly in scheduling information table

6 Mathematical Problems in Engineering

Table 3 Scheduling by experience

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 T11

41 Experience Scheduling When the ground controllerassigns paths by experience method some routes are pre-ferred and waiting phenomenon is universal Assume theground controller takes FCFS (first come first service) strat-egy and assigns routes with experience One scheduling maybe like the following in Table 3

The distribution of arrival time at each node is shown inFigure 5 It is easy to find that conflict happens at node 10 1112 13 and 33 theoretically

In the actual operation some aircrafts wait at node inorder to avoid conflictionThis increases the whole taxi timeMore information about the experience scheduling is listed inTable 4 The average length of taxi route is 23663m and theaverage waiting time is about 283 s The actual average taxitime is about 2756 s and confliction happens 6 timesThoughthe priority of CA1605 is higher than others it conflicts withMU2078 and CZ6218 at nodes 3 and 12 As a result CA1605waits 94 s in the whole It is obvious that the scheduling canstill be improved

The distribution of actual arrival time at each node isshown in Figure 6 All the flights arrive at each nodewith timeinterval no smaller than the minimum safety time intervalThe whole taxi time is increased by 255 s

42 Genetic Algorithm Scheduling The population size is setas 20 crossover probability is 0618 andmutation probabilityis 0025Themaximum iteration is 100The initial populationshows that the sum of fitness is 10126 and the average fitnessis 506 The maximum fitness is 513 and the shortest path is27300mThe best assigned routes in the initial population arelisted in Table 5

To solve the problem with genetic algorithm we pro-grammed in C++ programming language on VC++ 60platform and work in a computer with dual core processorof Inter(R) Core(TM) i3 and 2G RAM After 100 iterationsthe program output the resultsThe solving time is about 12 sTable 6 is about the optimized results

The optimized population shows that the sum of fitness is13157 and the average fitness is 658 The maximum fitness is662 and the shortest path is 21297m

More information is shown in Table 7The average lengthof taxi route is 23663m and the average taxi time is about2693 s (decreased by 103 compared with an experiencevalue of 5min) Confliction happens 5 times and the whole

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900

103958

103906103836

103928 103936

103800103830 103831

103730103700

103731 103734

103630103600

103657

103530103500

Tim

e

103500

103600

Figure 5 The distribution of arrival time at each node

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900 103913

103958 103943103928

103800103830 103843

103831

103730103700

103738 103734

103630103600 103600

103657

103530103500 103500

Tim

e

Figure 6 The distribution of actual arrival time at each node

waiting time is about 199 s with an average of 221 s Thewaiting time of CA1605 is only 4 s and this is mostly becauseof its high priority

The optimized distribution of actual arrival time at eachnode is shown in Figure 7 All flights which arrive at eachnode satisfy the minimum time interval The whole taxi timeis increased by 199 s

Genetic evolution process is shown in Figure 8 It can beclearly seen how the population average fitness changes Asthe initial chromosome is produced by selecting feasible pathsrandomly in the process of evolution the average fitness isclose to the optimal solution after 37 iterations The averagefitness is stable after 65 iterations the maximum averagefitness is about 658

43 Comparisons Two methods are used to analyze theproblem the results are listed in Table 8

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 3 Scheduling by experience

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 T11

41 Experience Scheduling When the ground controllerassigns paths by experience method some routes are pre-ferred and waiting phenomenon is universal Assume theground controller takes FCFS (first come first service) strat-egy and assigns routes with experience One scheduling maybe like the following in Table 3

The distribution of arrival time at each node is shown inFigure 5 It is easy to find that conflict happens at node 10 1112 13 and 33 theoretically

In the actual operation some aircrafts wait at node inorder to avoid conflictionThis increases the whole taxi timeMore information about the experience scheduling is listed inTable 4 The average length of taxi route is 23663m and theaverage waiting time is about 283 s The actual average taxitime is about 2756 s and confliction happens 6 timesThoughthe priority of CA1605 is higher than others it conflicts withMU2078 and CZ6218 at nodes 3 and 12 As a result CA1605waits 94 s in the whole It is obvious that the scheduling canstill be improved

The distribution of actual arrival time at each node isshown in Figure 6 All the flights arrive at each nodewith timeinterval no smaller than the minimum safety time intervalThe whole taxi time is increased by 255 s

42 Genetic Algorithm Scheduling The population size is setas 20 crossover probability is 0618 andmutation probabilityis 0025Themaximum iteration is 100The initial populationshows that the sum of fitness is 10126 and the average fitnessis 506 The maximum fitness is 513 and the shortest path is27300mThe best assigned routes in the initial population arelisted in Table 5

To solve the problem with genetic algorithm we pro-grammed in C++ programming language on VC++ 60platform and work in a computer with dual core processorof Inter(R) Core(TM) i3 and 2G RAM After 100 iterationsthe program output the resultsThe solving time is about 12 sTable 6 is about the optimized results

The optimized population shows that the sum of fitness is13157 and the average fitness is 658 The maximum fitness is662 and the shortest path is 21297m

More information is shown in Table 7The average lengthof taxi route is 23663m and the average taxi time is about2693 s (decreased by 103 compared with an experiencevalue of 5min) Confliction happens 5 times and the whole

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900

103958

103906103836

103928 103936

103800103830 103831

103730103700

103731 103734

103630103600

103657

103530103500

Tim

e

103500

103600

Figure 5 The distribution of arrival time at each node

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000

104058104030

103930103900 103913

103958 103943103928

103800103830 103843

103831

103730103700

103738 103734

103630103600 103600

103657

103530103500 103500

Tim

e

Figure 6 The distribution of actual arrival time at each node

waiting time is about 199 s with an average of 221 s Thewaiting time of CA1605 is only 4 s and this is mostly becauseof its high priority

The optimized distribution of actual arrival time at eachnode is shown in Figure 7 All flights which arrive at eachnode satisfy the minimum time interval The whole taxi timeis increased by 199 s

Genetic evolution process is shown in Figure 8 It can beclearly seen how the population average fitness changes Asthe initial chromosome is produced by selecting feasible pathsrandomly in the process of evolution the average fitness isclose to the optimal solution after 37 iterations The averagefitness is stable after 65 iterations the maximum averagefitness is about 658

43 Comparisons Two methods are used to analyze theproblem the results are listed in Table 8

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 4 The result of experience scheduling

Flight no Assigned routes Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 0 231CZ3118 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 30 261CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 3112 7 321CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 94 351CA1802 37rarr 30rarr 31rarr 12rarr 2 1950 90 286MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264Average 23663 283 2756

Table 5 The best flight scheduling in initial population

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 32rarr 33rarr 38 T1CZ3118 2rarr 12rarr 13rarr 30rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 29rarr 30rarr 31rarr 12rarr 11rarr 10rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 32rarr 33rarr 38 T5CA1802 37rarr 30rarr 31rarr 32rarr 11rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 28rarr 15rarr 14rarr 4 T9HU7196 37rarr 30rarr 13rarr 14rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 31rarr 12rarr 13rarr 14rarr 15rarr 5 T11

Table 6 Scheduling by genetic algorithm

Flight no Assigned routes StandMU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 T1CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 T2CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T7MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 T8CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 T5CA1802 37rarr 30rarr 13rarr 12rarr 2 T4MF8115 37rarr 30rarr 29rarr 14rarr 4 T9HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 T12GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 T11

Though the actual taxi lengths are equal in two meth-ods the confliction times and the whole waiting time aredecreased in genetic algorithm method The waiting timehas decreased by 56 s and the whole taxi time (waiting timeincluded) has decreased by 226 ForCA1605 has the highestpriority the optimized result shows that the waiting time is4 s The waiting time of CA1605 in the experience methodis 94 s So the important flights are guaranteed with betterroutes for their priorities

For the efficiency of different algorithm You and Han(2009) proposed a route optimization algorithm based onmultiagent That paper solves a scheduling problem with 3flights and 14 nodes The comparison of two algorithms islisted in Table 9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33Node number

MU5178

CZ3118CZ6218MU2078

CA1605

CA1802

MF8115HU7196

GS6574

104130104100

104000104030

103930103900

103800103830

103730103700103630103600103530103500

Tim

e

Figure 7 The optimized distribution of actual arrival time at eachnode

68

66

64

62

6

58

56

54

Aver

age fi

tnes

s

1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97103109

Interation

Figure 8 Genetic evolution process

The number of flights and nodes is less than that in thispaper It is no doubt that the scale in this paper ismuch biggerThe solving time of the genetic algorithm is about 12 s but themultiagent takes about 95 s

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 7 The optimized result

Flight no Assigned route Length Waiting (s) Actual time (s)MU5178 2rarr 12rarr 11rarr 10rarr 33rarr 38 2287 90 321CZ3118 2rarr 12rarr 31rarr 32rarr 33rarr 38 2287 0 231CZ6218 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2550 34 291MU2078 4rarr 14rarr 13rarr 12rarr 31rarr 32rarr 33rarr 38 3112 37 351CA1605 3rarr 13rarr 12rarr 11rarr 10rarr 33rarr 38 2250 4 261CA1802 37rarr 30rarr 13rarr 12rarr 2 1950 34 230MF8115 37rarr 30rarr 29rarr 14rarr 4 1537 0 211HU7196 37rarr 30rarr 29rarr 28rarr 15rarr 5 2662 0 264GS6574 37rarr 30rarr 29rarr 14rarr 15rarr 5 2662 0 264Average 23663 221 2693

Table 8 The comparison of two methods

Length (m) Conflict times Waiting (s) Whole taxi time (s)Experience method 21297 6 255 2480Genetic algorithm 21297 5 199 2424

Table 9 The comparison of two algorithms

Algorithm The scale of problem Solving time (s)Nodes Flights

Multiagent 14 3 95Genetic algorithm 43 9 12

From the optimized results and the comparison of differ-ent methods the study of scheduling problems in large hubairports makes great practical significance and the geneticalgorithm has great advantage in solving such big scale prob-lems From the economic view the fuel consumption will begreatly reduced by decreasing the total time cost of all aircraftsduring taxiing On the one hand conflicts happen rarely onthe other hand the operation cost will also be reduced inairlines From environmental protection point of view theaircraft engine emissions of nitrogen oxides are reduced andit is beneficial to reduce environmental pollution From theperspective of operation and management the use of newscheduling technology will help to improve work efficiencyand management level especially for large-scale schedulingproblems

5 Conclusions

The CDMmechanism will raise a higher requirement for theairport scene management level The use of a more efficientscheduling technology will help to make decision fairer thanexperience reduce flight delays in thewhole and decrease thecost of flight delay and fuel consumptionThe paper proposesa taxiing scheduling optimization model based on adjustedgenetic algorithm The results show that the algorithm isefficient

In fact aircraft taxiing speed is different and it is relatedto the aircraft type Taking aircraft taxiing speed into consid-eration we will get a more optimized result

Symbol Description

119866(119881 119864) Taxi network structure119881 Set of all nodes119864 Set of all edges119899119894start The start node of taxi 119894119899119894end The destination node of aircraft 119894119877 Set of feasible taxi path for all aircrafts

119877119894isin 119877

119899119894 Node in taxi network 119899

119894isin 119881

119877119894= 119899119894start 119899

119894

2 119899119894end

119865 Set of all aircrafts 119894 isin 119865119875 Set of aircraft priorities 119875

119894is the priority of

aircraft 119894119879119894 The release time of aircraft 119894

V119894 The taxi speed of aircraft 119894

119878119898119899 The edge length between node119898 and node 119899

119905119898119894 The time of arrival at node119898

1199050 The minimum safety time interval119879119891119899119894 The hold and wait time at node 119899 for

node-conflict119879119888119898119899119894

The hold and wait time at edge (119898 119899) foredge-conflict

119891119894119895119899 Node-conflict detection 0-1 variables

119908119894119895 Priority comparison 0-1 variables

119909119894119895119899 Arrival sequence detection 0-1 variables

Acknowledgments

This work was supported in National Natural Science Foun-dation of China and Civil Aviation Administration of China(no U1333117) China Postdoctoral Science Foundation (no2012M511275) and the Fundamental Research Fund forthe Central Universities (nos NS2013067 NN2012019 andNS2012115)

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

References

[1] J B Gotteland and N Durand ldquoGenetic algorithms appliedto airport ground traffic optimizationrdquo in Proceedings of theCongress on Evolutionary Computation (CEC rsquo03) vol 1 pp544ndash551 December 2003

[2] A G Marın ldquoAirport management taxi planningrdquo Annals ofOperations Research vol 143 no 1 pp 191ndash202 2006

[3] S Ravizza J A D Atkin and E K Burke ldquoA more realisticapproach for airport groundmovement optimisationwith standholdingrdquo Journal of Scheduling 2013

[4] R Anderson and D Milutinovi ldquoAn approach to optimizationof airport taxiway scheduling and traversal under uncertaintyrdquoProceedings of the Institution ofMechanical Engineers G vol 227no 2 pp 273ndash284 2013

[5] G L Clare and A G Richards ldquoOptimization of taxiway rout-ing and runway schedulingrdquo IEEE Transactions on IntelligentTransportation Systems vol 12 no 4 pp 1000ndash1013 2011

[6] G Keith J Tait and A Richards ldquoEfficient path optimizationwith terrain avoidancerdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 2940ndash2949 August2007

[7] P Burgain E Feron and J P Clarke ldquoCollaborative virtualqueue benefit analysis of a collaborative decision makingconcept applied to congested airport departure operationsrdquo AirTraffic Control Quarterly vol 17 no 2 pp 195ndash222 2009

[8] J Chen S Ravizza and J A D Atkin ldquoOn the utilisation offuzzy rule-based systems for taxi time estimations at airportsrdquoin Proceedings of the 11th Workshop on Algorithmic Approachesfor Transportation Modelling Optimization and Systems 2011

[9] J W Smeltink M J Soomer P R de Waal and R D vander Mei An Optimisation Model for Airport Taxi SchedulingElsevier Science 2004

[10] R Mori ldquoAircraft ground-taxiing model for congested airportusing cellular automatardquo IEEE Transactions on Intelligent Trans-portation Systems vol 14 no 1 pp 180ndash188 2013

[11] S Rathinam J Montoya and Y Jung ldquoAn optimization modelfor reducing aircraft taxi times at the Dallas Fort WorthInternational Airportrdquo in Proceedings of the 26th InternationalCongress of the Aeronautical Sciences (ICAS rsquo08) pp 14ndash19 2008

[12] J W Smeltink M J Sooner P R de Waal and R D van derMei ldquoAn Optimization Model for Airport Taxi Schedulingrdquo inProceedings of the INFORMS Annual Meeting (INFORMS rsquo04)Denver Colo USA 2004

[13] R Anderson and D Milutinovic Optimization of TaxiwayTraversal at Congested Airports American Institute of Aeronau-tics and Astronautics 2010

[14] P C Roling and H G Visser ldquoOptimal airport surface trafficplanning using mixed-integer linear programmingrdquo Interna-tional Journal of Aerospace Engineering vol 2008 Article ID732828 11 pages 2008

[15] C Lesire ldquoIterative planning of airport ground movementsrdquo inProceedings of the 4th International Conference on Research inAir Transportation pp 147ndash154 2010

[16] D B Rappaport P Yu K Griffin and C Daviau ldquoQuantita-tive analysis of uncertainty in airport surface operationsrdquo inProceedings of the AIAA Aviation Technology Integration andOperations Conference September 2009

[17] S Ravizza J A D Atkin M H Maathuis and E K BurkeldquoA combined statistical approach and groundmovement modelfor improving taxi time estimations at airportsrdquo Journal of theOperational Research Society vol 64 no 9 pp 1347ndash1360 2013

[18] Y Zhang M H Hu and Y J Wang ldquoThe ground skidding timein aeronef airport is excellent to turn pattern of searchrdquo Journalof Civil Aviation Flight University of China vol 17 no 5 pp 3ndash62006

[19] J You and S C Han ldquoApplication of MAS to airport surfaceroute optimizationrdquo Computer and Communications vol 26no 6 pp 61ndash64 2008

[20] Y Wang M Hu and W Su ldquoDynamic taxiway routingalgorithm based on conflict avoidancerdquo Journal of SouthwestJiaotong University vol 44 no 6 pp 933ndash939 2009

[21] Z Liu H Ge and F Qian ldquoAirport scheduling optimizationalgorithm based on genetic algorithmrdquo Journal of East ChinaUniversity of Science and Technology vol 34 no 3 pp 392ndash3942008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of