Research Article A Schedule Optimization Model on Multirunway … · 2019. 7. 31. · schedule (RBS) was proposed. Vossen et al. [ ] described a new allocation procedure based on

Research ArticleA Schedule Optimization Model on Multirunway Based onAnt Colony Algorithm

Yu Jiang Zhaolong Xu Xinxing Xu Zhihua Liao and Yuxiao Luo

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

Correspondence should be addressed to Yu Jiang jiangyu07nuaaeducn

Received 11 April 2014 Revised 7 July 2014 Accepted 18 July 2014 Published 28 September 2014

Academic Editor Hu Shao

Copyright copy 2014 Yu Jiang et alThis 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

In order tomake full use of the slot of runway reduce flight delay and ensure fairness among airlines a schedule optimizationmodelfor arrival-departure flights is established in the paper The total delay cost and fairness among airlines are two objective functionsThe ant colony algorithm is adopted to solve this problem and the result is more efficient and reasonable when compared withFCFS (first come first served) strategy Optimization results show that the flight delay and fair deviation are decreased by 4222and 3864 respectively Therefore the optimization model makes great significance in reducing flight delay and improving thefairness among all airlines

1 Introduction

With the rapid development of the Chinese civil aviationindustry the number of flights increases sharply and hubairports change their single runway to multirunway The airtraffic control managers put first come first served (FCFS)to use to schedule the arrival-departure flights which canlead to the waste of air resources and make the terminalarea congestionmore seriousTherefore the conflict betweendemand and supply is more and more sharp Hu and Paolo[1] Meanwhile with the implementation of collaborativedecision making (CDM) mechanism in airport resourcesmanagement flight scheduling problem should account fornot only flight delay but also equity among airlines A moreefficient and reasonable model or algorithm is needed toapproach the problem Therefore the air traffic congestionwill be alleviated and the total operation cost of airlines willbe reduced by approaching the scheduling and optimizationof arrival-departure flights

In recent years the capacity of most airports cannotsatisfy the rapid increase demand because of the increasein number of flights and severe weather At present grounddelay procedure (GDP) is the main method to be used toapproach the contradiction between capacity supply anddemand [2] With the improvement of airport resource

management technique a modified GDP strategy collabo-rative ground delay program (CDM-GDP) has been imple-mented in some hub airports The modified program canimprove the utilization efficiency of airport resources observ-ably [3] The basic process of CDM-GD is that the air trafficcontrol departments allocate landing slots to airlines thenthe airlines can merge and cancel the flights according tothe allocated information of slots and feed the adjustmentinformation back to the air traffic control management andthen the air traffic control departments make the final deci-sion after receiving the feedback information from airlines[4] Many scholars at home and abroad have done someresearch in CDM-GDP Hoffman et al [5] discussed severaltools applied to CDM system and a new algorithm ration-by-schedule (RBS) was proposed Vossen et al [6] described anew allocation procedure based on FCFS in CDM strategyto schedule the arrival-departure flights They established aequity allocation mechanism but the efficiency of algorithmshould be improved Mukherjee and Hansen [7] put forwarda dynamic stochastic integer programming (IP) model forthe airport ground holding problem but the equity wasignored for small airlines Ball et al [8] presented a newration-by-distance (RBD) algorithm showing that the equityand efficiency were improved at a certain extent Hu and Su[9] modeled the ground-holding management system which

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 368208 11 pageshttpdxdoiorg1011552014368208

2 Mathematical Problems in Engineering

provides the theoretical basis and method for the actualtraffic management but they ignored the influence of limitedcapacity for taking off flightsMa et al [10] designed a schemeof approaching queue and optimization schedule of flightsBut the satisfaction function is just a local optimizationand it is too complicated to make an adjustment in thesimulation Zhou et al [11] proposed an effectiveness-fairness(E-E) standard aimed at arrival slot time allocated methodby analyzing and simulating the traffic flow model in CDM-GDP The simulation results showed that the single prioritydecreased the total delay cost more efficiently than the doublepriority and the fair factors were also taken into accountZhou et al [12] proposed an evaluation function which wasused to evaluate the priority of delay cost coefficient on thebasis of existing slot allocation algorithmThemethodused inthis paper was more flexible and effective compared with thetraditional method on the total delay cost and equity but itlacked the flexibility in slot time allocation for arrival flightsZhang and Hu [13] proposed a multiobjective optimizationmodel based on the principles of effectiveness-efficiency-equity trade-offs But the research lacked further researchinto the slot time reassignment in CDM GDP based on thereal information of aircraft Zhan et al [14] adopted theAnt Colony Algorithm (ACA) and receding horizon control(RHC) to optimize the scheduling from the robustness andeffectiveness of the queue model on arrival flights But theinstability of the algorithm should be improved and the realscheduling of multirunway should be taken into accountAndrea DrsquoAriano et al [15] studied the problem of flightsorting at congested airportsThe research regarded the flightscheduling problem as an extension of workshop schedulingproblem but the optimized results were suboptimal fea-sible solutions Helmke et al [16] presented an integratedapproach to solve mixed-mode runway scheduling problemby using mixed-integer program techniques However fur-ther researches into the flight scheduling problem undermultirunway with mixed operation were required Samaet al [17] presented the real-time scheduling flight in orderto reschedule the flight based on receding horizon controlstrategy and conflict detection However the optimizationapproaches requiring frequent retiming and rerouting inconsecutive time horizons decreased the scheduling robust-ness Hancerliogullari et al [18] researched into the aircraftsequencing problem (ASP) under multirunway with mixedoperation mode They put greedy algorithm to simulate themodel However the fairness among airlines was not takeninto consideration

All in all though the optimization results of mostresearches at home and abroad satisfied the scheduling offlight the studies pay little attention to real-time flight in-formation Most of the studies consider collaborative grounddelays program of approach flight without analyzing depar-ture flight In fact if the delay of departure flight is dealtwith unreasonably it can lead to unfairness among arrival-departure flights and the increase of flight delay In the papera multiobjective optimization model is established based onmultirunway arrival-departure flight The maximum delayof departure flight is limited to ensure the fairness between

arrival-departure flights according to real-time flight infor-mation Meanwhile a fair runway slot allocation mechanismis establishedwith the objective ofminimizing the cost causedby airline delays As Ant Colony Algorithm has uniqueadvantages in continuous dynamic optimization the paperintroduces it to simulate and validate the model with theexpectation of reducing the loss of airline delays improvingrunway utilization and ensuring the fairness among airlines

2 Model

21 Description The queue of arrival-departure flights inmultirunway airport is a continuous dynamic process andit changes with the real-time information of flights Theschedule optimization of arrival-departure flight in multi-runway airport can be described as follows within a timewindow a number of flights belonging to different airlinesare waiting for landing or taking off The managers inairport shouldmake a reasonable allocation schedule (such asarrival-departure time sequence and operation runway) forall flights to minimize the total delay cost in the study periodunder the condition of safe operation of flight and airportresources and to balance the cost of total delay among airlinesThepaper selects the research time period in rush hour in hubairport to study the airport surface operation Aftermodelingand simulation the results can be applied in themanagementof airport surface operation in any type of airports

22 Assumptions

(1) The parallel runways studied in the paper operateindependently

(2) In the research time period the runway capacitycannot meet the demands of flights

(3) All the arrival flights do not delay when they are in thetake-off airport and they can arrive at the destinationterminal aerospace studied on time

(4) The basic information (such as flight plans and otherinformation of all flights) within the studied period isknown

(5) Each arrival-departure flight can only be assigned toone time slot in the studied period

23 Definition

119865119860 Set of arrival flight 119865119860 = 119891119860

1 119891119860

2 119891

119860

119898

119865119863 Set of departure flight 119865119863 = 119891119863

1 119891119863

2 119891

119863

119899

119865 Set of arrival-departure flights 119865119860 cup 119865119863 = 119865119867 Set of airlines119867 = 119867

1 1198672 119867

119901

119878 Set of arrival-departure flights slots 119878 = 1199041 1199042

119904119882

119877 Set of runways in the airport 119877 = 1 2 119903

119891119903119905119894= 1 if flight 119894 takes off or lands on runway 1199030 else

119891119894

119867119886

Flight 119894 belongs to airline119867119886

Mathematical Problems in Engineering 3

119904119891119860119894

119867119886

119892 = 1 if flight slot 119904

119892is assigned to an arrival flight 119891119860119894

119867119886

0 else

119904119891119863119894

119867119886

ℎ= 1 if 119904

ℎis assigned to adeparture flight 119891119863119894

119867119886

0 else

119862 Total flight delay cost119862119867119886

Total flight delay cost of airline119867119886

119875119862 The sum of absolute deviation of flight delay cost120572119903= 1 if any flight lands on runway 1199030 else

120573119903= 1 if any flight takes off on runway 1199030 else

119874119877119879119903119891119887

The end time of flight 119891119887taking off from or

landing on runway 119903 after optimization119878119879119891119903119891119888

The original time of flight119891119888taking off from or

landing on at runway 119903 after optimization119878119860

119903119887119888 The minimum safety interval of continuous

landing on runway 119903119878119863


taking off on runway 119903119878119860119863

119903119887119888 The minimum time interval when a departure

flight follows an arrival flight on runway 119903119878119863119860

119903119887119888 The minimum time interval when an arrival

flight follows a departure flight on runway 119903119879119869119879119884max The maximum delay time when an arrivalflight lands in advance compared to scheduled time119879119869119885119884max The maximum delay time when an arrivalflight lands later than the scheduled time119879119871119879119884maxThemaximumdelay timewhen a departureflight takes off in advance compared to the scheduledtime119879119871119885119884maxThemaximumdelay timewhen a departureflight takes off later than the scheduled time119864119879119891119860119894

119867119886

The estimated arrival time of flight 119891119860119894which

belongs to119867119886

119864119879119891119863119895

119867119886

The estimated departure time of flight 119891119863119895

which belongs to119867119886

119878119879119891119860119894

119867119886

The actual arrival time of flight 119891119860119894belonging

to119867119886after optimization schedule

119878119879119891119863119895

119867119886

The actual departure time of flight 119891119863119895belong-

ing to119867119886after optimization schedule

119862119860119894

119867119886

The unit time delay cost of arrival flight 119891119860119894

belonging to119867119886after optimization schedule

119862119863119895

119867119886

The unit time delay cost of departure flight 119891119863119895


24 Objective Function Twoobjectives are taken into consid-eration in the paper total delay cost and the fairness amongairlines The total delay cost of all flights is used to reducethe light delay and improve the utilization of runway Thefairness is used to balance the equity among all the airlinesand to protect the benefit of small airlines So amultiobjectivefunction based on it is modeled

241 The Objective Function of Delay Cost Different wakevortex separations between different arrival-departure flightsare different according to the types of aircraft Thereforewe can improve the capacity of runway and reduce totaldelay time by adjusting the arrival-departure order of allflights ICAO aircraft wake turbulence separation criteria arespecified in Table 1

The optimized target of delay cost in the paper is tominimize the total delay of all arrival-departure flights whichis based on improving the capacity of runway The objectivefunction of delay cost can be described as follows

min119862

= min119903

sum

119903=1

119898

sum

119894=1

119899

sum

119895=1

119901

sum

119886=1

[119862119860119894

119867119886

(119878119879119891119860119894

119867119886

minus 119864119879119891119860119894

119867119886

) 120572119903119904119891119860119894

119867119886

119892

+ 119862119863119895

119867119886

(119878119879119891119863119895

119867119886

minus 119864119879119891119863119895

119867119886

) 120573119903119904119891119863119894

119867119886

ℎ]

(1)

242The Objective Function of Fairness Delay cost is relatedto the aircraft type In general large airlines are preferred tosmall aircraft types If the research only takes the delay costas a single function it is likely to lead to serious unfairnessamong airlines especially to small airlines Therefore abso-lute deviation of delay cost is introduced to ensure the fairnessamong airlines

Definition of standard flight assume some type of flightto be a standard flight and all other flights can be transformedinto it according to aircraft type and delay cost For exampleif we take a large aircraft as a standard flight and the numberequals 1 then a light aircraft may be transformed as 06 anda heavy aircraft as 18 If a standard flight is denoted by119891119861 120582119894119867119886

is defined as the number of standard flights aftertransformation from flight 119894 then the relation expression canbe demonstrated as follows

120582119894

119867119886

=

119891119894

119867119886

119891119861

(2)

In order to ensure fairness among airlines the researchersfirst transform all flights belonging to different airlines intostandard flights Then the researchers can calculate the aver-age delay cost of standard flight by using the total delay costof all flights In the same way the researcher can get averagedelay cost of each airline and the total absolute deviationof delay cost It is obvious that the lower the total absolutedeviation is themore fairness we can balance among airlinesThe fairness optimization objective function of airlines is asfollows

min119875119862 =

119901

sum

119886=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

119862

sum119901

119886=1sum119867119886

120582119894

119867119886

minus

119862119867119886

sum119867119886

120582119894

119867119886

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)

where 119862119867119886

sum119867119886

120582119894

119867119886

is the average delay cost of flightsbelonging to airline119867

119886 119862sum119901

119886=1sum119867119886

120582119894

119867119886

is the average delaycost of all flights


Table 1 Aircraft wake turbulence separation criteria of ICAO

Type of flightTrailing

The minimum time intervals The minimum distance intervalkmSmall Large Heavy Small Large Heavy

LeadingSlight 98 74 74 6 6 6Large 138 74 74 10 6 6Heavy 167 114 94 12 10 8

25 Constraints Consider119882

sum

119892=1

119904119891119860119894

119867119886

119892 = 1 (4)

An arrival flight occupies one slot resource and each slotresource can be assigned to a certain arrival flight Constraint(4) is the constraint of slot resource allocation for arrivalflights

119882

sum

ℎ=1

119904119891119863119894

119867119886

ℎ= 1 (5)

Similarly a departure flight occupies one slot resource andeach slot resource can be assigned to a certain departureflight Constraint (5) is the constraint of slot resource allo-cation for departure flights

119891119903119905119894le 1 (6)

For the safety of flight operation each flight can only occupyone runway and only one or none aircraft can occupy therunway at the same time A constraint of slot resourceallocation of runways is expressed as constraint (6)

sum119891119903119905119894le 119903

120572119903+ 120573119903le 1 120572

119903isin 0 1 120573119903 isin 0 1

(7)

According to constraint (6) at a certain time the numberof flights occupying runways will not be greater than thenumber of runways Meanwhile in order to ensure safety in acertain slot only one flight is arriving or leaving at runway119903 The runway resource controlling constraint is shown asconstraint (7)

119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (8)

To ensure the safety of arrival flights the actual arrivaltime should meet the maximum delay time (119879119871119879119884max and119879119871119885119884max) after optimization Constraint (8) is to ensure thetime of arrival flight

119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (9)

Constraint (9) is to ensure the time of departure flightExtending the departure flight delay can decrease the servicelevel in most airports and it is necessary to limit the amount

of delay time Like the constraint for arrival flights theactual departure time should meet the maximum delay time(119879119869119879119884max and 119879119869119885119884max) after optimization Consider

119874119877119879119903119891119887

minus 119878119879119891119903119891119888

ge

119878119860

119903119887119888continue arrivals

119878119863

119903119887119888continue departures

119878119860119863

119903119887119888departure follows arrival

119878119863119860

119903119887119888arrival follows departure

(10)

The minimum safety interval in different conditions which isrelated to the order of arrival-departure flight should be takeninto account Constraint (10) is the minimum flight safetyinterval constraint of runway 119903

3 Ant Colony Algorithm Design

Ant Colony Algorithm (ACA) is a metaheuristic algorithmwhich uses a heuristic method to search for the spacethat may be related to feasible solutions In ACA the antcan select the path comprehensively based on pheromonesand heuristic factors of the environment It can release thepheromones after traveling the path of the network In thealgorithm an individual ant can identify and release all thepheromones All pheromones from the ant colony are used tocomplete the whole and complex optimization process ACAhas the characteristics of self-organization and distributedcomputingTherefore it is able to make a global search It caneffectively avoid local solutions to an extentMeanwhile ACAcan get the optimized solution faster than other traditionalalgorithms

31 Algorithm Description

311 Single Runway Flight Scheduling of ACA Single runwayscheduling problem can be transformed to a TSP which takeseach flight as a node and the interval time between flightsas the path length of nodes The problem can be solved bytraditional ACA and the results are satisfactory The modelis built as follows the node 119891

119894in the network is the element

of flight set 119865 and the distance 119891119894119895between nodes 119891

119894and 119891

119895is

the time interval When the algorithm begins the ant 119896 headsfrom a virtual starting node 119891

0 The starting node is set up to

ensure that all the ants in ACA can start at the same nodeTheant 119896 traverses all the nodes of network so a flight sequenceis obtained We can calculate wait time and delay cost in thequeue according to the sequenceThe ACA for single runwayflight scheduling is shown in Figure 1


f0

f1 f2 f3

f4 f5 f6

Figure 1 The ACA for single runway schedule model

312 Multirunway Flight Scheduling of ACA In order toadapt to the multirunway flight scheduling model the ACAfor single runwaymodel should bemodified In themultirun-way flight scheduling model a node 119891

119894may contain several

subnodes 119903119899isin 119877 The distance 119897

119894119898119895119899

between the subnode 119903119898

of 119891119894and the subnode 119903

119899of node 119891

119895is the minimum safety

interval The ant 119896 heads from a virtual starting node 1198910and

travels all nodes of the network When the ant arrives at anode 119891

119894 it selects a subnode 119903

119899to get a sequence contained

runway number The ACA for multirunway flight schedulingis shown in Figure 2

313 The State Transition Equation The amount of informa-tion on each path and the heuristic information can decidethe transition direction of ant 119896 (119896 = 1 2 119898) It recordsthe traveled nodes by search table tabu

119896(119896 = 1 2 119898)

119901119896

119894119898119895119899

(119905) is denoted as the state transition probability of ant 119896changing direction from the subnode 119903

119898to the subnode 119903

119899at

time 119905 Formula (11) is as follows

119901119896

119894119898119895119899

(119905) =

[120591119894119898119895119899

(119905)]120572

lowast [1205781198951015840

119899119895119899

(119905)]120573

sum120581isinallowed

119896

(sum119899isin119877

[120591119894119898120581119899

(119905)]120572

lowast sum119899isin119877

[1205781205811015840

119899120581119899

(119905)]120573

)

if 120581 isin allowed119896

0 otherwise

(11)

120591119894119898119895119899

(119905) is the pheromone concentration on path 119897119894119898119895119899

at time119905 allowed

119896= 119865 minus tabu

119896 means that the ant 119896 can choose

the node which is the node that never traversed next step120572 is the pheromone heuristic factor which decides how thepheromones have impact on path choosing 120573 is the expectedheuristic factor which decides the degree of attention ofvisibility when ants make a choice

1205781198951015840

119899119895119899

(119905) is an expected factor which is evaluated as

1205781198951015840

119899119895119899

(119905) =1

1199031198951015840

119899119895119899

(12)

where 1199031198951015840

119899119895119899

is theminimum safety interval of the flight119891119895and

former flight 1198911015840119895

314 The Updating Strategy of Pheromone Updating phe-romone of all nodes is needed when all the iterations arecompletedWith the increasing of pheromone concentrationthe residual pheromone evaporates in proportion In order toget better optimization results only the best ant can releasepheromone of iteration Therefore the pheromone updatingcan be adjusted as the following rules

120591119894119898119895119899

(119905 + 1) = 120588120591119894119898119895119899

(119905) + Δ120591best119894119898119895119899

Δ120591119894119898119895119899

=

119876

119891 (119896best) 119894 isin 119896

best

0 119894 notin 119896best

(13)

where 120588 is the volatilization coefficient of pheromone 119876 isthe amount of pheromone Δ120591

119894is the total incremental of

pheromone in this circulation of node 119894

32The Design of ACA The design of ACA in the simulationis as follows

Step 1 Set parameters We set 120572 = 2 120573 = 15 120588 = 07 and119876 = 120000

Step 2 Get the flight information We get flight information(including the type of flight the estimated time of arrival ordeparture and so forth) and other known data by reading thefiles

Step 3 Initialize the pheromone and expectations of paths ofsolution space and empty the tabu list

Step 4 Set119873119862 = 0 (119873119862 is the iteration) Generate the initial119896 ants on the virtual node 119891

0

Step 5 Ants select a node orderly according to formula (11)

Step 6 If all ants complete a traversal turn to Step 7otherwise turn to Step 5

Step 7 If the searching results of all ants meet the con-straints then reduce the pheromone increment when updat-ing pheromone

Step 8 Calculate the target value of all ants and record thebest ant solutions

Step 9 Update the pheromone of each node according toformula (13)

Step 10 If 119873119862 lt 119873119862max without stagnating delete the antand set119873119862 rarr 119873119862+1 then reset the data and turn to Step 5otherwise output the optimal results and the calculation isover


r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

f0

f1 f2 f3

f4 f5 f6

Figure 2 The ACA for multirunway schedule model

4 Simulation and Vertification

In the paper we putCprogram to use to simulate flight sched-ule problem on multirunway with mixed operation modeThe core algorithm of the program is the ACA design Inorder to achieve the objectives of delay cost and fairness wefirst take the delay cost as the objective andwe take fairness asa constraint and we can get the flight sequence with minimaldelay cost After that we set fairness as the objective and thedelay cost as a constraint andwe get a flight sequencewith thebest fairness Finally the initial flight sequence and these twoflight sequences are compared A peak hour is selected froma certain large airport of China and the authors chose flightsin the busiest 15 minutes from that peak hour Two parallelrunways run independently There are 38 flights (belongingto 7 airlines) to be scheduled After optimization the initialflight information and optimized results are shown inTable 2

In the paper some real operation data is selected from acertain hub airport and ACA is designed to solve the modelAs shown in Table 2 flight delay is serious due to unreason-able slot assignment it can lead to unfair competition amongairlines before optimizationThe total delay cost of three typesof flight sequences is listed in Table 3 and Figure 3

The initial sequence is the initial flight sequence beforeoptimization The minimal delay cost and fairness amongairlines are solved according to object function (1) andfunction (2)Theminimal delay sequence means that we takethe objective function (1) as the main objective function andobjective function (2) as a constraint in optimizationThebestfairness sequence means that we set objective function (2) asthemain objective function and the objective function (1) as aconstraint in optimizationThen the paper contrasts the threeflight data among airlines

The detailed information of delay cost of standard flightis listed in Table 4 and Figure 4 After transforming flightsinto standard flights we can compare delay cost and fairnessdirectly

FromFigures 3 and 4 we can draw the conclusion that thesequence of minimal delay cost can decrease the delay costobviously after optimization The sequence of best fairnesscan improve the fairness among all the airlines obviously Butfrom Figure 4 we can see that the fairness among 7 airlinesdecreases when delay cost is minimal the delay cost of 7airlines improves obviously when the fairness is best

In order to reduce the delay cost of airlines and increasethe fairness among airlines we view the minimum delay

020000400006000080000

100000

Dela

y co

stCN

Y

Airlines

The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness

minus20000

minus40000

H1 H2 H3 H4 H5 H7H6

Figure 3 The delay cost comparison of airlines on three flightsequences

12000

Stan

dard

flig

hts d

elay

cost

CNY

02000400060008000

10000

Airlines


minus2000

minus4000

H1 H2 H3 H4 H5 H7H6

minus6000

minus8000

Figure 4 The delay cost comparison of airlines standard flights onthree flight sequences

cost as objective and control the range of flight delay costvariation The simulation steps are as follows

(1) First the fairness is not taken into account and we getthe minimum delay cost We statistic the total delayand the delay cost deviation

(2) Then limit the range of airline flight delay costdeviation by a large number of data We get thesimulated data of total delay and delay cost deviationof flights in the cases119875119862 lt 50000 119875119862 lt 30000 119875119862 lt

25000 119875119862 lt 20000 119875119862 lt 15000 respectively(3) Finally we make an analysis of simulation data and

study the relationship between the delay cost andfairness The relation curve of delay cost and delaycost deviation is shown in Figure 5

As shown in Figure 5 there is a trend relationshipbetween delay cost and fairness When the delay costdecreases the fairness among airlines is not satisfied Reduc-ing the delay deviation of flights may lead to the increase intotal delay cost In order to make balance of the relationshipbetween the delay cost and fairness and make a better


Table 2 The initial flight delay data

Flightnumber Airline Type Unit delay cost

of departureEstimated

time of departureActual

time of departure Departure runway The delay cost ofactual departure

F001 H1 S 12 00000 00000 0 0F002 H2 S 11 00000 00136 0 1056F003 H2 L 62 00000 00242 0 10044F004 H3 M 24 00000 00000 1 0F005 H4 M 25 00500 00420 0 minus100F006 H4 S 14 00500 00829 1 2926F007 H5 M 24 00500 00935 1 660F008 H6 M 26 00500 00842 0 5772F009 H6 L 57 01000 01741 0 26277F010 H3 M 24 01000 01907 1 13128F011 H1 M 23 01000 02023 1 14329F012 H2 L 62 01000 01859 0 33418F013 H7 M 27 01000 02037 0 17199F014 H5 S 13 01500 02511 1 7943F015 H2 M 26 01500 02515 0 1599F016 H7 M 27 01500 02617 1 18279F017 H3 S 1 01500 02715 0 735F018 H1 L 51 01500 02733 1 38403Flightnumber Airline Type Unit delay cost

of approachEstimated

time of approachActual

time of approach Approach runway The delay cost ofactual approach

F019 H7 L 626 00000 00114 1 46324F020 H7 S 256 00112 00401 1 43264F021 H7 M 428 00232 00515 1 69764F022 H5 L 632 00356 00416 0 1264F023 H4 M 415 00402 00629 1 61005F024 H6 M 416 00431 00610 0 41184F025 H7 S 256 00506 01153 1 104192F026 H3 M 42 00534 00956 0 11004F027 H7 M 428 00542 01110 0 140384F028 H1 M 424 00654 01307 1 158152F029 H7 L 626 00701 01224 0 202198F030 H7 L 626 00734 01421 1 254782F031 H4 M 415 00803 01615 1 20418F032 H3 S 247 00843 01511 0 95836F033 H5 L 632 00912 01729 1 314104F034 H3 M 42 00951 01625 0 16548F035 H6 M 416 01013 02137 1 284544F036 H1 M 424 01014 02151 0 295528F037 H2 L 63 01106 02251 1 44415F038 H1 M 424 01253 02409 0 16900

Table 3 Comparison on delay cost of flights from three flight sequences

Airline sequence H1 H2 H3 H4 H5 H6 H7

Total delay costInitial 675412 504658 391834 271761 341287 360703 896386Minimal delay minus11009 minus49006 211321 minus178375 64506 minus321706 88220Best fairness 447344 497198 398215 366143 405456 375004 809474


Table 4 Comparison on delay cost of standard flights from three flight sequences


Total delay costInitial 1125687 72094 753527 754892 656321 745369 845647Minimal delay minus183483 minus700086 4063865 minus495486 12405 minus670221 8322642Best fairness 7455733 710282 7657981 1017064 779723 7812583 7636547Note H1 to H7 refer to standard flights in Table 4

Table 5 Comparison on total delay of five flight sequences for flights


Total delay costMinimal delay minus11009 minus49006 211321 minus178375 64506 minus321706 88220Optimization 1 193098 54311 17095 59725 82885 28083 582195Optimization 2 338326 487211 415183 174494 193361 331485 536024Optimization 3 166938 136825 203221 159769 144304 67438 338008Best fairness 447344 497198 398215 366143 405456 375004 809474

Table 6 Comparison on delay cost of five flight sequences for standard flights


Total delay costMinimal delay minus183483 minus700086 4063865 minus495486 12405 minus670221 8322642Optimization 1 32183 7758714 32875 1659028 1593942 5850625 5492406Optimization 2 5638767 6960157 7984288 4847056 3718481 6905938 505683Optimization 3 27823 1954643 3908096 4438028 2775077 1404958 3188755Best fairness 7455733 7102829 7657981 1017064 7797231 7812583 7636547Note H1 to H7 refer to standard flights in Table 6

sequence of flights we can make flight sequence by limitingthe delay deviation of flights So it not only reduces the totaldelay cost of airlines but also takes care of the fairness amongairlines

From the simulation results we can get a number offlight sequences by controlling the sum of absolute deviationof flight delay cost Five sequences (minimal delay costoptimization 1 optimization 2 optimization 3 and the bestfairness) to make a comparison of total delay cost of airlinesare shown in Table 5 and Figure 6 Table 6 and Figure 7 showthe same comparison by transforming flights into standardflights

The total delay cost and the fairness among airlineshave been improved obviously after optimization In actualoperation the decision makers can get several optimizedflight sequences by controlling the range of flight delay costdeviation and by selecting a preferred one according to real-time information

In Table 7 we list the optimal flight sequence in whichboth the delay cost and fairness are acceptableThe optimizedresults show that the total delay cost reduces greatly and thefairness is also acceptable when compared with the initialflight sequence The standard flight delay cost deviationof initial flight sequence and optimized flight sequence is

1 2 3 40

1

2

3

4

5

The total delay cost of flightCNY

The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104

Figure 5 The trend relationship between delay cost and fairness

calculated based on Table 7 The results are shown in Table 8and Figure 8 The histogram shows the contrast of flightdelay deviation between the initial flight sequence and theoptimized flight sequence From the histogram we can findthat the delay cost of airlines declines 4222 at least afteroptimization The sum of delay deviation declines 3864So the schedule model and solution have not only reducedthe total delay cost significantly but also ensured the fairnessamong all the airlines


Table 7 The comparison between initial flight delay cost and the optimized flight delay cost

Flightnumber Type Unit delay cost

of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway

Delay costof actualdeparture

OptimalDeparture

time

Departurerunway afteroptimization

Departure delay costafter optimization

H1F001 S 12 00000 00000 0 0 01830 0 1332H2F002 S 11 00000 00136 0 1056 00850 1 583H2F003 L 62 00000 00242 0 10044 01158 0 44516H3F004 M 24 00000 00000 1 0 00310 0 456H4F005 M 25 00500 00420 0 minus100 01454 0 1485H4F006 S 14 00500 00829 1 2926 02220 1 1456H5F007 M 24 00500 00935 1 660 01112 1 8928H6F008 M 26 00500 00842 0 5772 02730 1 3510H6F009 L 57 01000 01741 0 26277 02544 0 53808H3F010 M 24 01000 01907 1 13128 02616 1 23424H1F011 M 23 01000 02023 1 14329 02256 0 17848H2F012 L 62 01000 01859 0 33418 02410 0 5270H7F013 M 27 01000 02037 0 17199 01650 1 1107H5F014 S 13 01500 02511 1 7943 01908 1 3224H2F015 M 26 01500 02515 0 1599 01344 1 minus1976H7F016 M 27 01500 02617 1 18279 02502 1 16254H3F017 S 1 01500 02715 0 735 02142 0 402H1F018 L 51 01500 02733 1 38403 01024 0 minus14076


of approach

Estimatedtime ofapproach

Actualtime ofapproach

Approachrunway

Delay costof actualapproach

OptimalApproach

time

Approachrunway afteroptimization

Approach delay costafter optimization

H7F019 L 626 00000 00114 1 46324 00600 0 22536H7F020 S 256 00112 00401 1 43264 02356 1 349184H7F021 M 428 00232 00515 1 69764 00228 1 minus1712H5F022 L 632 00356 00416 0 1264 01316 0 35392H4F023 M 415 00402 00629 1 61005 00342 1 minus830H6F024 M 416 00431 00610 0 41184 00650 1 57824H7F025 S 256 00506 01153 1 104192 01544 1 163328H3F026 M 42 00534 00956 0 11004 00754 0 5880H7F027 M 428 00542 01110 0 140384 00000 1 minus146376H1F028 M 424 00654 01307 1 158152 01228 1 141616H7F029 L 626 00701 01224 0 202198 00000 0 minus263546H7F030 L 626 00734 01421 1 254782 00456 1 minus98908H4F031 M 415 00803 01615 1 20418 00956 1 46895H3F032 S 247 00843 01511 0 95836 02044 1 178087H5F033 L 632 00912 01729 1 314104 00426 0 minus180752H3F034 M 42 00951 01625 0 16548 00154 0 minus20034H6F035 M 416 01013 02137 1 284544 00908 0 minus2704H1F036 M 424 01014 02151 0 295528 00114 1 minus22896H2F037 L 63 01106 02251 1 44415 01610 0 19152H1F038 S 25 01253 02409 0 16900 02006 0 10825Total the total cost of the initial flight sequence delay is 3434465CNY the total cost of optimized flight sequence delay is 1026810CNY


Table 8 The total delay cost and deviation of standard flight

Airline sequence H1 H2 H3 H4 H5 H6 H7 DeviationTotal delay cost

Initial 112569 72094 75352 754892 656321 74537 84565 104565Optimized 27823 19546 39081 44380 27751 14050 31888 54870

0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6

Optimization 1Optimization 2Optimization 3

The sequence of minimal delay costThe sequence of best fairness

Figure 6 Comparison of the delay cost of airlines

0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines

Figure 7 Comparison of the delay of five flight sequences for airlinestandard flights

5 Conclusions

In the paper a mixed multirunway operation flight schedul-ing optimization model based on multiobjective is proposedTwo objectives are considered the total delay cost andfairness among airlines are two objective functionsThe ACAis introduced to solve the model The simulation resultsshow that the total delay cost decreases significantly and thefairness among airlines is also acceptable Meanwhile ACAused in the paper solves the model with great efficiency

0 PC

Initial sequenceOptimized sequence


Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5

Figure 8 Contrast between the total delay cost and deviation ofstandard flight

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China the Civil Aviation Administration ofChina (no U1333117) China Postdoctoral Science Foun-dation (no 2012M511275) and the Fundamental ResearchFunds for the Central Universities (no NS2013067)

References

[1] X Hu and E di Paolo ldquoBinary-representation-based geneticalgorithm for aircraft arrival sequencing and schedulingrdquo IEEETransactions on Intelligent Transportation Systems vol 9 no 2pp 301ndash310 2008

[2] MCWambsganss ldquoCollaborative decisionmaking in air trafficmanagementrdquo in New Concepts and Methods in Air TrafficManagement pp 1ndash15 Springer Berlin Germany 2001

[3] M Wambsganss ldquoCollaborative decision making throughdynamic information transferrdquo Air Traffic Control Quarterlyvol 4 no 2 pp 107ndash123 1997

[4] K Chang K Howard R Oiesen L Shisler M Tanino andM C Wambsganss ldquoEnhancements to the FAA ground-delayprogram under collaborative decision makingrdquo Interfaces vol31 no 1 pp 57ndash76 2001

[5] RHoffmanWHallM Ball A R Odoni andMWambsganssldquoCollaborative decisionmaking in air traffic flowmanagementrdquoNEXTOR Research Report RR-99-2 UC Berkley 1999


[6] T Vossen M Ball R Hoffman et al ldquoA general approach toequity in traffic flow management and its application to mit-igating exemption bias in ground delay programsrdquo Air TrafficControl Quarterly vol 11 no 4 pp 277ndash292 2003

[7] A Mukherjee and M Hansen ldquoA dynamic stochastic modelfor the single airport ground holding problemrdquo TransportationScience vol 41 no 4 pp 444ndash456 2007

[8] M O Ball R Hoffman and A Mukherjee ldquoGround delayprogram planning under uncertainty based on the ration-by-distance principlerdquo Transportation Science vol 44 no 1 pp 1ndash14 2010

[9] M H Hu and L G Su ldquoModeling research in air traffic flowcontrol systemrdquo Journal of Nanjing University Aeronautics ampAstronautics vol 32 no 5 pp 586ndash590 2000

[10] Z Ma D Cui and C Chen ldquoSequencing and optimal schedul-ing for approach control of air trafficrdquo Journal of TsinghuaUniversity vol 44 no 1 pp 122ndash125 2004

[11] Q Zhou J Zhang and X J Zhang ldquoEquity and effectivenessstudy of airport flow management modelrdquo China Science andTechnology Information vol 4 pp 126ndash116 2005

[12] Q Zhou X Zhang and Z Liu ldquoSlots allocation in CDMGDPrdquoJournal of BeijingUniversity of Aeronautics andAstronautics vol32 no 9 pp 1043ndash1045 2006

[13] H H Zhang and M H Hu ldquoMulti-objection optimizationallocation of aircraft landing slot in CDM GDPrdquo Journal ofSystems amp Management vol 18 no 3 pp 302ndash308 2009

[14] Z Zhan J Zhang Y Li et al ldquoAn efficient ant colony systembased on receding horizon control for the aircraft arrivalsequencing and scheduling problemrdquo IEEE Transactions onIntelligent Transportation Systems vol 11 no 2 pp 399ndash4122010

[15] A DAriano P DUrgolo D Pacciarelli and M Pranzo ldquoOpti-mal sequencing of aircrafts take-off and landing at a busyairportrdquo in Proceeding of the 13th International IEEE Conferenceon Intelligent Transportation Systems (ITSC 10) pp 1569ndash1574Funchal Portugal September 2010

[16] HHelmkeOGluchshenko AMartin A Peter S Pokutta andU Siebert ldquoOptimal mixed-mode runway schedulingmdashmixed-integer programming for ATC schedulingrdquo in Proceedings of theIEEEAIAA 30thDigital Avionics SystemsConference (DASC rsquo11)pp C41ndashC413 IEEE Seattle Wash USA October 2011

[17] M Sama A DrsquoAriano and D Pacciarelli ldquoOptimal aircrafttraffic flow management at a terminal control area duringdisturbancesrdquo ProcediamdashSocial and Behavioral Sciences vol 54pp 460ndash469 2012

[18] G Hancerliogullari G Rabadi A H Al-Salem and M Khar-beche ldquoGreedy algorithms and metaheuristics for a multiplerunway combined arrival-departure aircraft sequencing prob-lemrdquo Journal of Air Transport Management vol 32 pp 39ndash482013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


provides the theoretical basis and method for the actualtraffic management but they ignored the influence of limitedcapacity for taking off flightsMa et al [10] designed a schemeof approaching queue and optimization schedule of flightsBut the satisfaction function is just a local optimizationand it is too complicated to make an adjustment in thesimulation Zhou et al [11] proposed an effectiveness-fairness(E-E) standard aimed at arrival slot time allocated methodby analyzing and simulating the traffic flow model in CDM-GDP The simulation results showed that the single prioritydecreased the total delay cost more efficiently than the doublepriority and the fair factors were also taken into accountZhou et al [12] proposed an evaluation function which wasused to evaluate the priority of delay cost coefficient on thebasis of existing slot allocation algorithmThemethodused inthis paper was more flexible and effective compared with thetraditional method on the total delay cost and equity but itlacked the flexibility in slot time allocation for arrival flightsZhang and Hu [13] proposed a multiobjective optimizationmodel based on the principles of effectiveness-efficiency-equity trade-offs But the research lacked further researchinto the slot time reassignment in CDM GDP based on thereal information of aircraft Zhan et al [14] adopted theAnt Colony Algorithm (ACA) and receding horizon control(RHC) to optimize the scheduling from the robustness andeffectiveness of the queue model on arrival flights But theinstability of the algorithm should be improved and the realscheduling of multirunway should be taken into accountAndrea DrsquoAriano et al [15] studied the problem of flightsorting at congested airportsThe research regarded the flightscheduling problem as an extension of workshop schedulingproblem but the optimized results were suboptimal fea-sible solutions Helmke et al [16] presented an integratedapproach to solve mixed-mode runway scheduling problemby using mixed-integer program techniques However fur-ther researches into the flight scheduling problem undermultirunway with mixed operation were required Samaet al [17] presented the real-time scheduling flight in orderto reschedule the flight based on receding horizon controlstrategy and conflict detection However the optimizationapproaches requiring frequent retiming and rerouting inconsecutive time horizons decreased the scheduling robust-ness Hancerliogullari et al [18] researched into the aircraftsequencing problem (ASP) under multirunway with mixedoperation mode They put greedy algorithm to simulate themodel However the fairness among airlines was not takeninto consideration

All in all though the optimization results of mostresearches at home and abroad satisfied the scheduling offlight the studies pay little attention to real-time flight in-formation Most of the studies consider collaborative grounddelays program of approach flight without analyzing depar-ture flight In fact if the delay of departure flight is dealtwith unreasonably it can lead to unfairness among arrival-departure flights and the increase of flight delay In the papera multiobjective optimization model is established based onmultirunway arrival-departure flight The maximum delayof departure flight is limited to ensure the fairness between

arrival-departure flights according to real-time flight infor-mation Meanwhile a fair runway slot allocation mechanismis establishedwith the objective ofminimizing the cost causedby airline delays As Ant Colony Algorithm has uniqueadvantages in continuous dynamic optimization the paperintroduces it to simulate and validate the model with theexpectation of reducing the loss of airline delays improvingrunway utilization and ensuring the fairness among airlines

2 Model

21 Description The queue of arrival-departure flights inmultirunway airport is a continuous dynamic process andit changes with the real-time information of flights Theschedule optimization of arrival-departure flight in multi-runway airport can be described as follows within a timewindow a number of flights belonging to different airlinesare waiting for landing or taking off The managers inairport shouldmake a reasonable allocation schedule (such asarrival-departure time sequence and operation runway) forall flights to minimize the total delay cost in the study periodunder the condition of safe operation of flight and airportresources and to balance the cost of total delay among airlinesThepaper selects the research time period in rush hour in hubairport to study the airport surface operation Aftermodelingand simulation the results can be applied in themanagementof airport surface operation in any type of airports

22 Assumptions

(1) The parallel runways studied in the paper operateindependently

(2) In the research time period the runway capacitycannot meet the demands of flights

(3) All the arrival flights do not delay when they are in thetake-off airport and they can arrive at the destinationterminal aerospace studied on time

(4) The basic information (such as flight plans and otherinformation of all flights) within the studied period isknown

(5) Each arrival-departure flight can only be assigned toone time slot in the studied period

23 Definition

119865119860 Set of arrival flight 119865119860 = 119891119860

1 119891119860

2 119891

119860

119898

119865119863 Set of departure flight 119865119863 = 119891119863

1 119891119863

2 119891

119863

119899

119865 Set of arrival-departure flights 119865119860 cup 119865119863 = 119865119867 Set of airlines119867 = 119867

1 1198672 119867

119901

119878 Set of arrival-departure flights slots 119878 = 1199041 1199042

119904119882

119877 Set of runways in the airport 119877 = 1 2 119903

119891119903119905119894= 1 if flight 119894 takes off or lands on runway 1199030 else

119891119894

119867119886

Flight 119894 belongs to airline119867119886


119904119891119860119894

119867119886

119892 = 1 if flight slot 119904


119867119886

0 else

119904119891119863119894

119867119886

ℎ= 1 if 119904


119867119886

0 else





119874119877119879119903119891119887













119867119886



119864119879119891119863119895

119867119886



119878119879119891119860119894

119867119886



119878119879119891119863119895

119867119886



119862119860119894

119867119886



119862119863119895

119867119886






min119862

= min119903

sum

119903=1

119898

sum

119894=1

119899

sum

119895=1

119901

sum

119886=1

[119862119860119894

119867119886

(119878119879119891119860119894

119867119886

minus 119864119879119891119860119894

119867119886

) 120572119903119904119891119860119894

119867119886

119892

+ 119862119863119895

119867119886

(119878119879119891119863119895

119867119886

minus 119864119879119891119863119895

119867119886

) 120573119903119904119891119863119894

119867119886

ℎ]

(1)




120582119894

119867119886

=

119891119894

119867119886

119891119861

(2)


min119875119862 =

119901

sum

119886=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

119862

sum119901

119886=1sum119867119886

120582119894

119867119886

minus

119862119867119886

sum119867119886

120582119894

119867119886

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)

where 119862119867119886

sum119867119886

120582119894

119867119886


119886 119862sum119901

119886=1sum119867119886

120582119894

119867119886








sum

119892=1

119904119891119860119894

119867119886

119892 = 1 (4)


119882

sum

ℎ=1

119904119891119863119894

119867119886

ℎ= 1 (5)


119891119903119905119894le 1 (6)


sum119891119903119905119894le 119903

120572119903+ 120573119903le 1 120572

119903isin 0 1 120573119903 isin 0 1

(7)


119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (8)


119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (9)



119874119877119879119903119891119887

minus 119878119879119891119903119891119888

ge

119878119860


119878119863


119878119860119863


119878119863119860


(10)








119894and 119891

119895is





f0

f1 f2 f3

f4 f5 f6





119894119898119895119899



119899of node 119891








119896(119896 = 1 2 119898)

119901119896

119894119898119895119899



119899at


119901119896

119894119898119895119899

(119905) =

[120591119894119898119895119899

(119905)]120572

lowast [1205781198951015840

119899119895119899

(119905)]120573


119896

(sum119899isin119877

[120591119894119898120581119899

(119905)]120572


[1205781205811015840

119899120581119899

(119905)]120573

)


0 otherwise

(11)

120591119894119898119895119899



119896= 119865 minus tabu



1205781198951015840

119899119895119899


1205781198951015840

119899119895119899

(119905) =1

1199031198951015840

119899119895119899

(12)

where 1199031198951015840

119899119895119899


former flight 1198911015840119895


120591119894119898119895119899

(119905 + 1) = 120588120591119894119898119895119899

(119905) + Δ120591best119894119898119895119899

Δ120591119894119898119895119899

=

119876

119891 (119896best) 119894 isin 119896

best

0 119894 notin 119896best

(13)









0








r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

f0

f1 f2 f3

f4 f5 f6









020000400006000080000

100000

Dela

y co

stCN

Y

Airlines


minus20000

minus40000

H1 H2 H3 H4 H5 H7H6


12000

Stan

dard

flig

hts d

elay

cost

CNY

02000400060008000

10000

Airlines


minus2000

minus4000

H1 H2 H3 H4 H5 H7H6

minus6000

minus8000




































1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119904119891119860119894

119867119886

119892 = 1 if flight slot 119904


119867119886

0 else

119904119891119863119894

119867119886

ℎ= 1 if 119904


119867119886

0 else





119874119877119879119903119891119887













119867119886



119864119879119891119863119895

119867119886



119878119879119891119860119894

119867119886



119878119879119891119863119895

119867119886



119862119860119894

119867119886



119862119863119895

119867119886






min119862

= min119903

sum

119903=1

119898

sum

119894=1

119899

sum

119895=1

119901

sum

119886=1

[119862119860119894

119867119886

(119878119879119891119860119894

119867119886

minus 119864119879119891119860119894

119867119886

) 120572119903119904119891119860119894

119867119886

119892

+ 119862119863119895

119867119886

(119878119879119891119863119895

119867119886

minus 119864119879119891119863119895

119867119886

) 120573119903119904119891119863119894

119867119886

ℎ]

(1)




120582119894

119867119886

=

119891119894

119867119886

119891119861

(2)


min119875119862 =

119901

sum

119886=1

10038161003816100381610038161003816100381610038161003816100381610038161003816

119862

sum119901

119886=1sum119867119886

120582119894

119867119886

minus

119862119867119886

sum119867119886

120582119894

119867119886

10038161003816100381610038161003816100381610038161003816100381610038161003816

(3)

where 119862119867119886

sum119867119886

120582119894

119867119886


119886 119862sum119901

119886=1sum119867119886

120582119894

119867119886








sum

119892=1

119904119891119860119894

119867119886

119892 = 1 (4)


119882

sum

ℎ=1

119904119891119863119894

119867119886

ℎ= 1 (5)


119891119903119905119894le 1 (6)


sum119891119903119905119894le 119903

120572119903+ 120573119903le 1 120572

119903isin 0 1 120573119903 isin 0 1

(7)


119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (8)


119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (9)



119874119877119879119903119891119887

minus 119878119879119891119903119891119888

ge

119878119860


119878119863


119878119860119863


119878119863119860


(10)








119894and 119891

119895is





f0

f1 f2 f3

f4 f5 f6





119894119898119895119899



119899of node 119891








119896(119896 = 1 2 119898)

119901119896

119894119898119895119899



119899at


119901119896

119894119898119895119899

(119905) =

[120591119894119898119895119899

(119905)]120572

lowast [1205781198951015840

119899119895119899

(119905)]120573


119896

(sum119899isin119877

[120591119894119898120581119899

(119905)]120572


[1205781205811015840

119899120581119899

(119905)]120573

)


0 otherwise

(11)

120591119894119898119895119899



119896= 119865 minus tabu



1205781198951015840

119899119895119899


1205781198951015840

119899119895119899

(119905) =1

1199031198951015840

119899119895119899

(12)

where 1199031198951015840

119899119895119899


former flight 1198911015840119895


120591119894119898119895119899

(119905 + 1) = 120588120591119894119898119895119899

(119905) + Δ120591best119894119898119895119899

Δ120591119894119898119895119899

=

119876

119891 (119896best) 119894 isin 119896

best

0 119894 notin 119896best

(13)









0








r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

f0

f1 f2 f3

f4 f5 f6









020000400006000080000

100000

Dela

y co

stCN

Y

Airlines


minus20000

minus40000

H1 H2 H3 H4 H5 H7H6


12000

Stan

dard

flig

hts d

elay

cost

CNY

02000400060008000

10000

Airlines


minus2000

minus4000

H1 H2 H3 H4 H5 H7H6

minus6000

minus8000




































1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















sum

119892=1

119904119891119860119894

119867119886

119892 = 1 (4)


119882

sum

ℎ=1

119904119891119863119894

119867119886

ℎ= 1 (5)


119891119903119905119894le 1 (6)


sum119891119903119905119894le 119903

120572119903+ 120573119903le 1 120572

119903isin 0 1 120573119903 isin 0 1

(7)


119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (8)


119879119869119879119884max le 119878119879119891119867119886119863119895minus 119864119879119891

119867119886119863119895le 119879119869119885119884max (9)



119874119877119879119903119891119887

minus 119878119879119891119903119891119888

ge

119878119860


119878119863


119878119860119863


119878119863119860


(10)








119894and 119891

119895is





f0

f1 f2 f3

f4 f5 f6





119894119898119895119899



119899of node 119891








119896(119896 = 1 2 119898)

119901119896

119894119898119895119899



119899at


119901119896

119894119898119895119899

(119905) =

[120591119894119898119895119899

(119905)]120572

lowast [1205781198951015840

119899119895119899

(119905)]120573


119896

(sum119899isin119877

[120591119894119898120581119899

(119905)]120572


[1205781205811015840

119899120581119899

(119905)]120573

)


0 otherwise

(11)

120591119894119898119895119899



119896= 119865 minus tabu



1205781198951015840

119899119895119899


1205781198951015840

119899119895119899

(119905) =1

1199031198951015840

119899119895119899

(12)

where 1199031198951015840

119899119895119899


former flight 1198911015840119895


120591119894119898119895119899

(119905 + 1) = 120588120591119894119898119895119899

(119905) + Δ120591best119894119898119895119899

Δ120591119894119898119895119899

=

119876

119891 (119896best) 119894 isin 119896

best

0 119894 notin 119896best

(13)









0








r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

f0

f1 f2 f3

f4 f5 f6









020000400006000080000

100000

Dela

y co

stCN

Y

Airlines


minus20000

minus40000

H1 H2 H3 H4 H5 H7H6


12000

Stan

dard

flig

hts d

elay

cost

CNY

02000400060008000

10000

Airlines


minus2000

minus4000

H1 H2 H3 H4 H5 H7H6

minus6000

minus8000




































1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










f0

f1 f2 f3

f4 f5 f6





119894119898119895119899



119899of node 119891








119896(119896 = 1 2 119898)

119901119896

119894119898119895119899



119899at


119901119896

119894119898119895119899

(119905) =

[120591119894119898119895119899

(119905)]120572

lowast [1205781198951015840

119899119895119899

(119905)]120573


119896

(sum119899isin119877

[120591119894119898120581119899

(119905)]120572


[1205781205811015840

119899120581119899

(119905)]120573

)


0 otherwise

(11)

120591119894119898119895119899



119896= 119865 minus tabu



1205781198951015840

119899119895119899


1205781198951015840

119899119895119899

(119905) =1

1199031198951015840

119899119895119899

(12)

where 1199031198951015840

119899119895119899


former flight 1198911015840119895


120591119894119898119895119899

(119905 + 1) = 120588120591119894119898119895119899

(119905) + Δ120591best119894119898119895119899

Δ120591119894119898119895119899

=

119876

119891 (119896best) 119894 isin 119896

best

0 119894 notin 119896best

(13)









0








r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

f0

f1 f2 f3

f4 f5 f6









020000400006000080000

100000

Dela

y co

stCN

Y

Airlines


minus20000

minus40000

H1 H2 H3 H4 H5 H7H6


12000

Stan

dard

flig

hts d

elay

cost

CNY

02000400060008000

10000

Airlines


minus2000

minus4000

H1 H2 H3 H4 H5 H7H6

minus6000

minus8000




































1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

r1r2

f0

f1 f2 f3

f4 f5 f6









020000400006000080000

100000

Dela

y co

stCN

Y

Airlines


minus20000

minus40000

H1 H2 H3 H4 H5 H7H6


12000

Stan

dard

flig

hts d

elay

cost

CNY

02000400060008000

10000

Airlines


minus2000

minus4000

H1 H2 H3 H4 H5 H7H6

minus6000

minus8000




































1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























1 2 3 40

1

2

3

4

5


The s

um o

f abs

olut

e dev

iatio

n o

f flig

ht d

elay

cost

05 15 25 35 45times105

times104






of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












of departure

Estimatedtime of

departure

Actualtime of

departure

Departurerunway


OptimalDeparture

time





of approach



Approachrunway


OptimalApproach

time








0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0

20000

40000

60000

80000

100000

Dela

y co

stCN

Y

Airlinesminus20000

minus40000

H1 H2 H3 H4 H5 H7H6




0

5000

10000

15000

Stan

dard

flig

hts d

elay

cost

CNY



minus5000

minus10000

H1 H2 H3 H4 H5 H7H6

Airlines


5 Conclusions


0 PC



Airlines

H1 H2 H3 H4 H7H6

12

2

4

6

8

10

comparison

times103

Flig

hts d

elay

cost

of th

e sum

of

abso

lute

dev

iatio

n

H5




Acknowledgments


References



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Schedule Optimization Model on Multirunway … · 2019. 7. 31. · schedule (RBS) was proposed. Vossen et al. [ ] described a new allocation procedure based on