Research Article A Schedule Optimization Model on Multirunway … · 2019. 7. 31. · schedule...
Transcript of Research Article A Schedule Optimization Model on Multirunway … · 2019. 7. 31. · schedule...
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
119903119887119888 The minimum safety interval of continuous
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
belonging to119867119886after optimization schedule
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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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
119903119887119888 The minimum safety interval of continuous
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
belonging to119867119886after optimization schedule
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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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
119903119887119888 The minimum safety interval of continuous
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
belonging to119867119886after optimization schedule
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
6 Mathematical Problems in Engineering
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
The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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
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
6 Mathematical Problems in Engineering
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
The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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
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
The initial flight sequenceThe sequence of minimal delay costThe sequence of best fairness
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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 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
8 Mathematical Problems in Engineering
Table 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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 4 Comparison on delay cost of standard flights from three flight sequences
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Airline sequence H1 H2 H3 H4 H5 H6 H7
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
Mathematical Problems in Engineering 9
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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
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
Flightnumber Type Unit delay cost
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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
10 Mathematical Problems in Engineering
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
Optimization 1Optimization 2Optimization 3
The sequence of minimal delay costThe sequence of best fairness
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
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
Mathematical Problems in Engineering 11
[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
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 11
[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
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