Research ArticleAn Integrative Approach with Sequential Game to Real-TimeGate Assignment under CDM Mechanism
Jun-qiang Liu Ma-lan Zhang Peng-chao Chen Ji-wei Xie and Hong-fu Zuo
College of Civil Aviation Nanjing University of Aeronautics and Astronautics Nanjing 210016 China
Correspondence should be addressed to Jun-qiang Liu liujunqiangnuaaeducn
Received 26 December 2013 Revised 7 May 2014 Accepted 8 May 2014 Published 1 June 2014
Academic Editor Hu Shao
Copyright copy 2014 Jun-qiang Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper focuses on real-time airport gate assignment problem when small-scale or medium- to large-scale flight delays occurTaking into account the collaborative decision making (CDM) of the airlines and the airport as well as the interests of multiagent(airlines airports and passengers) especially those influenced by flight banks slot assignment and gate assignment are integratedintomixed set programming (MSP) and a real-time gate assignmentmodel is built and solved throughMSP coupledwith sequentialgame By this approach the delay costs of multiagent can be minimized simultaneously the fuel consumption of each airline canbe basically equalized the computation time can be significantly saved by sequential game most importantly the collaborationof the airlines and the airport is achieved so that the transferring cost caused by the delay of flight banks can be decreased asmuch as possible A case study on small-scale flight delays verifies that the proposed approach is economical robust timesavingand collaborative A comparison of the traditional staged method and the proposed approach under medium- to large-scale flightdelays proves that the integrative method is much more economical and timesaving than the traditional staged method
1 Introduction
According to the statistical data of Chinarsquos civilian aviation in2011 [1] the existing 175 airports include 3 4F-class airports30 4E-class airports 40 4D-class airports and 85 4C-classairports The number of civilian airports is expected toincrease up to 244 by 2020 Meanwhile the reconstructionand expansion of most major airports will be performedto meet the increasing demands In this case the resourcemanagement of these airports will turn out to be increasinglycomplex Therefore a great number of researches have beenpresented to solve the problem of airport resource manage-ment gate assignment included Gate assignment problemcan be divided into two categories preassignment and real-time assignment
In the aspect of preassignment Bolat developed a gateassignment model that is aimed at minimizing the idle timeof the gates both in 1996 and 2000 meanwhile heuristicalgorithm combined with branch-and-bound technique wasdesigned for solution [2] In 2001 Bolat devised a lin-earization representation of the same objective and replaced
heuristic algorithmwith genetic algorithm [3] In 2003 You etal [4] were devoted to the study of genetic algorithm whichpromoted themultiobjective algorithm for the preassignmentof airport gates In 2005 Lim and Wang constructed adifferent model that is aimed at reducing the number ofconflicts during aircraft operation and the model was solvedby combining tabu search with local search in heuristicalgorithm [5] In 2011 Li et al proposed an assignmentmodelto obtain the maximum comprehensive efficiency of airlinesairports and passengers besides the solution algorithm wasdeveloped based on greedy tabu search [6]
In the aspect of real-time assignment Cheng designeda gate scheduling knowledge base system [7] by utilizingmathematical programming techniques in order to supportpreassignment and real-time assignment In later studiesthey developed two other models to establish a gate assign-ment expert system network-based model [8] and rule-based reactive model [9] In 2010 Li devised an assignmentmodel that was aimed at maximizing the sum of all the flighteigenvalues (aircraft type flight inboundoutbound time andthe number of passengers) and the gate eigenvalues (idle time
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 143501 13 pageshttpdxdoiorg1011552014143501
2 Mathematical Problems in Engineering
of the gates) [10] Furthermore Wei and Liu proposed anapproach to perform gate reassignment by minimizing thenumber of the flights assigned to the apron the disturbancecaused by the assignment and the walking distance ofpassengers meanwhile a heuristic algorithm combining tabusearch was adopted for optimization [11]
The existing research shows that the studies on gate pre-assignment have evolved and matured over time However acomprehensive consideration on the interests of multiagentnamely airports airlines and passengers has not been givento real-time gate assignment
Additionally flight bank [12] a decisive role in deter-mining the satisfaction of passengers and the benefits ofairlines is generally considered in slot assignment rather thangate assignment Nevertheless the gate assignment will alsobe influenced by flight banks because transfer passengerswould have to wait longer if flight banks are delayed whichincreases the delay cost of gate assignment Flight bank isthe product of the collaborative decision making (CDM) [13]for airports airlines and air traffic control center (ATCC)In CDM mechanism airports airlines and ATCC shouldwork collaboratively to control the entire operation costHowever almost all kinds of assignment algorithms suchas gate assignment aircraft sorting and flight schedulingare performed separately which goes against the originalintentions of CDM mechanism In the existing real-timegate assignment [13] airports are given fixed schedule of thedelayed flights Airlines are not informed of the possible gateassignment schemes and not all the information of possibleslot rearrangements are delivered to the airport as a resultthe flight banks may be delayed to a great extent To solvethis problem the CDM should be implemented accuratelyATCC provides updated slots to airlines airlines choose theslot assignment schemes corresponding to the optimal gateassignment (based on minimum cost principle) finally theoptimal gate assignment is carried out by the airport based onthe provided slot assignment schemesThe innovation takinginto account the impacts of flight banks while achievingCDM is capable of guaranteeing reasonable slot assignmentand gate assignment thereby saving delay cost for multi-agent (airlines airports and passengers) significantly con-necting flight banks smoothly and transferring passengersconveniently
In order to achieve slot assignment and gate assignmentat the same time integrative recovery strategy [14] is consid-ered as a solution Generally traditional integrated recoverystrategies such as MSP are time-consuming because thepermutation and combination of the optional objects willbecome enormous with the increasing of the scale of theproblem to be solved However sequential game has beenapplied in resource allocation [15 16] and is proven tobe timesaving by deleting the infeasible permutation andcombination of the objects from the computation processBesides sequential game also guarantees that the benefits ofthe related game players aremaximizedTherefore sequentialgame is applied innovatively in real-time gate assignment fortimesaving computation and effective cost control
Consequently this paper proposes a novel model takinginto account the benefits of multiagent the impact of flight
banks CDM mechanism fairness principle and sequentialgame between the airlines in order to achieve the followinggoals Firstly the total delay cost including the walking costof passengers the taxiing cost of aircraft the idle cost ofgates and the waiting cost of transfer passengers should beminimized Secondly the taxiing fuel for aircraft of eachairline should be balanced Thirdly the flight banks con-taining delayed flights should be connected effectively withminimum adverse impacts on the interests of the airlinesLastly unreasonable solutions should be excluded in the firstplace thereby saving computation time and resources Tomake the model work MSP combined with sequential gameis designed for modeling and solving
The remainder of this paper is organized as follows Thereal-time gate assignment model is established in Section 2In Section 3 the detailed steps of the solution algorithm areprovided In Section 4 two cases (small-scale flight delaysand medium- to large-scale flight delays) are given andthe experimental results are analyzed accordingly Finallyconclusions are drawn in Section 5
2 Real-Time Gate Assignment Model
21 Variable Definition of the Model The definitions of thevariables in the model are given as follows
(1) 119873 denotes the set of flights and119872 denotes the set ofgates 119894 119896 isin 119873 119895 isin 119872 0 lt 119872 lt 119873
(2) 119882 denotes the set of flight banks119908 isin 119882119882119898denotes
the set of flight banks for airline 119898119882119898isin 119882 119905
1198981199081is
the scheduled arrival time of flight bank 119908 for airline119898 and 119905
1198981199082is the actual arrival time of flight bank 119908
for airline119898 1199051198981199082
ge 1199051198981199081
(3) 119876 is the set of aircraft types 119860 is the set of airlines119899 isin 119876119898 isin 119860119876
119894denotes the type of flight 119894 expressed
in numbers and the bigger 119876119894is the larger flight 119894 is
(4) 119878 is the set of slots 119904 isin 119878 119878119898is the set of slots for airline
119898 when airline 119898 exchanges slots with other airlinesto cut the waiting time of transfer passengers 119878
119898isin 119878
(5) 119883119894119895is a zero-one variable and119883
119894119895equals 1 when flight
119894 is assigned to gate 119895 else119883119894119895equals 0 119884
119898119908is a zero-
one variable and 119884119898119908
equals 1 when flight bank 119908belongs to airline119898 else 119884
119898119908equals 0
(6) 119865119894is the fuel consumption of flight 119894 per minute
(7) 1198791119894and 1198792
119894denote the taxiing time of flight 119894 before
and after the reassignment respectively
(8) 119875119894is the number of passengers in flight 119894
(9) 1198711119894and1198712
119894denote thewalking time for the passengers
of flight 119894 before and after the reassignment respec-tively
(10) 1198681119895and 1198682
119895denote the idle time of gate 119895 before and
after the reassignment respectively
(11) 119903119898119908
is the number of transfer passengers in flight bank119908 of airline119898
Mathematical Problems in Engineering 3
(12) 1198621 is the price of jet fuel per kilogram 1198622 is thewalking cost of each passenger per minute 1198623
119895is the
cost of the idle time for gate 119895perminute and1198624 is thewaiting cost for each transfer passenger per minute
(13) 119877119894119895is the time when flight 119894 arrives at gate 119895 119871
119894119895is the
time when flight 119894 departs from gate 119895 and 119877119896119895is the
time when flight 119896 arrives at gate 119895 119877119896119895gt 119871119894119895
(14) 119866119895denotes the type of gate 119895 expressed in numbers
and the bigger 119866119895is the larger gate 119895 is
(15) Δ119879 is the buffer time between any two consecutiveflights assigned to the same gate
(16) 119868119887119895denotes the beginning of the idle time for gate 119895
and 119868119890119895denotes the end of the idle time for gate 119895
(17) 119891119878119898
is the waiting cost of the transfer passengers forairline 119898 which is less than the waiting cost of thetransfer passengers before the slot exchange of airline119898
(18) 119883119878119898
is a zero-one variable119883119878119898
equals 1 when 119878119898is the
slots combination for airline119898 after the exchange else119883119878119898
equals 0(19) Ω
119898is the set of slot combinations which makes the
waiting time of transfer passengers for airline 119898 lessafter the slot exchange
(20) Δ119870119899119898
is the increased fuel consumption of type 119899aircraft belonging to airline119898 119911 is the number of air-lines so 1119911 is the expected proportion correspond-ing to the ideal situation of complete equalization forall airlines
22 Objectives
(1) Minimize the Increased Total Cost The total cost involvesfuel cost walking cost of passengers idle cost of gates andwaiting cost of transfer passengers To achieve the goal ofminimizing the increased total cost caused by flight delaysthe following four values need to be minimized at the sametime taxiing time of aircraft walking time of passengers idletime of gates and waiting time of transfer passengers Theobjective function is formulated as follows
119865119894(1198792119894minus 1198791119894) 1198621 denotes the increased taxiing cost of flight
119894 119875119894(1198712119894minus 1198711119894) 1198622 denotes the incremental walking cost for
passengers of flight 119894 (1198682119895minus1198681119895) 1198623119895denotes the change of the
idle cost for gate 119895 119903119898119908(1199051198981199082
minus1199051198981199081) 1198624 denotes the increased
waiting cost of transfer passengers in flight bank 119908 of airline119898
(2) Minimize the Increased Taxiing Time of Aircraft Since fuelcost accounts for about 30 of the total operation cost forairlines fuel cost decrease will make a huge difference in thecost control of airlines Fuel cost reduction can be achievedby minimizing the incremental taxiing time of all the aircraftaccording to
(3) Minimize the Increased Walking Time of PassengersThe reduction of walking distance or time improves thesatisfaction of passengers The purpose of minimizing thewalking time of passengers can be realized by
(4) Minimize the Increased Idle Time of Gates As airport gatesare the core resources of an airport improving the utilizationrate of the idle gates attributes to better airport operationTo realize this goal the idle time of airport gates can beminimized by
min 1198914= sum
119895isin119872
(1198682119895minus 1198681119895) (4)
(5) Minimize the Increased Waiting Time of Transfer Passen-gers Flight bank has been widely applied in hub airports sothat the transferring efficiency of passengers the utilizationrate of airport resources and the operational effectiveness ofairlines can be improved Generally flight bank is capableof connecting flights effectively thus minimizing the waitingtime of transfer passengers However delayed flight bankswill lead to a series of problems such as increased waitingtime of passengers and increased operation cost The objec-tive function that is aimed at reducing the increased waitingtime caused by the delay of flight banks is given by
As slot 119904 is randomly assigned to flight 119894 and the numberof slots is equal to the number of flights | sum
119904isin119878119905119904minus sum119894isin119873119905119894| is
4 Mathematical Problems in Engineering
a constant Hence the increased waiting time of transfer pas-sengers does not necessarily depend on the slot assignmentbut the delay of flight banks
Theorem 2 (gate assignment hinges on the slot assignment)Gate assignment depends on the arrival time of flights and theidle time of available gates therefore slot assignment plays adecisive role in gate assignment
According to Theorems 1 and 2 the slot assignment fordelayed flights should be implemented with minimum delayof flight banks so that the optimal gate assignment withminimum delay cost can be producedThis is why flight bankis involved in the objective of minimizing the waiting cost oftransfer passengers in gate assignment
(6) Minimize the Waiting Cost by Optimizing the Slot Assign-ment through Non-Zero-Sum Sequential Game The slotsassigned to airlines are exchangeable so airlines can reducethe waiting time of transfer passengers by exchanging theslots with each other The process of slot exchange can beachieved through non-zero-sum sequential game betweenairlines In sequential game all the airlines are aware of theirprevious policy or selection and have to make their currentdecisions according to their tradeoff of future possibilitiesZero-sum sequential game refers to the situation that theincome of one side is equal to the loss of the other side Asour subject is about controlling the loss caused by the flightdelays for all the related airlines non-zero-sum sequentialgame theory [15] is adopted in this application The modelof non-zero-sum sequential game is given by
119866 = 119860 (119878119898)119898isin119860
(120588119898)119898isin119860
119875 (120588) (7)
where119860 denotes the set of airlines 119878119898is the set of all optional
slot series for airline 119898 forall119898 isin 119860 120588119898
is the realizationprobability of 119878
119898 119875(120588) denotes the expected revenue matrix
Theorem 3 In a sequential game any realization probabilitypoints to a behavior strategy
Proof of Theorem 3 120588119898(119904119898) = prod
120572isin119904119898
120573119898(120572) where 120588
119898(119904119898) is
the realization probability for airline 119898 to obtain slot series119904119898(119904119898isin 119878119898) 120573119898is the probability distribution of 119879(ℎ
119898) and
119879(ℎ119898) is the set of optional policies under information set ℎ
119898
for airline119898Therefore any realization probability comes from a cor-
responding behavior strategyThe set of behavior sequences for airline 119898 on informa-
tion set ℎ119898is denoted by 120582(ℎ
119898) 120574 is an expansion of the
behavior sequences denoted by 120582(ℎ119898)120574 (120582(ℎ
119898)120574 = 120582(ℎ
119898)cup120574)
and the realization probability of 120582(ℎ119898) can be denoted by
120588119898(120582(ℎ119898))
Consequently the behavior 120574 on information set ℎ119898can
be confirmed by
120573119898(120574) =
120588119898(120582 (ℎ119898) 120574)
120588119898(120582 (ℎ119898)) (8)
where 120588119898(120582(ℎ119898)) gt 0 and 120573
119898(120574) can be any value when
120588119898(120582(ℎ119898)) = 0
Suppose that 120573 = (1205731 1205732 120573
119902) where 120573 is the behavior
strategy set for all airlines 120588 = (1205881 1205882 120588
119902) where 120588 is
the corresponding realization probability and 1205880denotes the
realization probability of the virtual player nature usually afixed value and 119878 = 119878
0times 1198781times sdot sdot sdot times 119878
119902 where 119878 denotes
sequence space and 119904 = (1199040 1199041 119904
119902) isin 119878 where 119904 is a set of
slot series The expected revenue function can be expressedby
119875 (120588) = sum
119904isin119878
119875 (119904)
119902
prod
119898=0
120588119898(119904119898) (9)
where 119875(119904) = 119875(119888) 119875(119888) is the revenue of implementing 119904towards some end note 119888 andprod119902
119898=0120588119898(119904119898) is the realization
probability of approaching 119888According to the above description when the revenue
119875(120588) for all airlines is maximized the transferring cost forall the airlines can be achieved and the objective function isgiven by
min 1198916= sum
119898isin119860
sum
119878119898isinΩ119898
119891119878119898
119883119878119898
(10)
where 119891119878119898
is the value of the objective function 1198915for airline
119898 when 119878119898is the slots combination after the exchange
(7) Balance the Increased Fuel Consumption for Each AirlineFairness principle requires that the fuel consumption causedby flight delays should be averaged for airlines to bearHowever airlines are of different scales and aircraft are ofdifferent types so the average fuel consumption should bemade from the aspects of both aircraft and airlines Thisgoal is achieved by averaging the proportion of the fuelconsumption change for a certain aircraft type belonging toa certain airline to the fuel consumption change for a certainaircraft type of all airlines The objective is represented by
is the proportion of the fuel con-sumption change of type 119899 aircraft belonging to airline 119898 tothe fuel consumption change of all airlinesrsquo type 119899 aircraft
23 Integrative Assignment Model According to the aboveanalysis the real-time gate assignment model based on theprinciple of minimum delay cost for multiagent can beexpressed as follows
min 119891 = min 1198911 1198912 1198913 1198914 1198915 1198916 1198917 (12)
where (12) is the objective function Equation (13) meansevery flight is assigned to one and only one gate Equation(14) is the corresponding relationship between flight and gateEquation (15) means each slot combination is adopted byone and only one airline Equation (16) is the correspondingrelationship of slot combinations and airlines Equation (17)enforces that the type of the gate where the aircraft is assignedshould match the type of the aircraft Equation (18) stipulatesthat the idle time of the gate should be longer than buffertime for the sake of safety Equation (19) indicates that thebeginning of the idle time for any gate should be earlier thanthe arrival time of the flight which will be assigned to thegate and the end of the idle time should be later than thedeparture time of the flight Equation (20) refers to validityconstraint
Compared with the traditional staged model (the slotassignment is produced before the gate assignment) theadvantages of the proposed model are presented as follows(1) the sequential game helps to obtain better slot combi-nations for all the airlines and (2) the CDM mechanismcontributes to generating gate reassignment with less delaycost of multiagent due to the collaboration of airlines andairports
To solve the multiobjective optimization problem(MOOP) [17] the objectives are sorted in order ofpriority because all the objectives cannot be optimizedsimultaneously As the service concept is becoming moreand more important the waiting time and waiting time ofpassengers are given the highest priority The second highestpriority is the taxiing time of aircraft because the fuel costis the direct operation cost of airlines Following the taxiingtime of aircraft is the implicit idle cost of the airport Fuelequalization is the lowest priority because the slot exchangebetween airlines also contributes to the fairness principlewhen the fuel consumption is not equalized at the verybeginning
3 Solution Algorithm
Mixed set programming (MSP) [18ndash20] is a logic reasoningalgorithm based on first-order logic and set reasoning InMSP set operations quantifiers Boolean logic logic func-tions datetime reasoning and numerical constraints areintegrated in one system the reasoning on numeric typessuch as reals and integers is expanded to global reasoning
over mixed domains of set types such as Booleans andreferences Most importantly MSP makes the modeling andsolving for constraint satisfaction problems (CSP) realizableThe so-called set programming here is to systematicallyintegrating set reasoning and operational research algorithmestablishing a rigorous and complete set theoretical for-mulation based on set variables and solving the model byset reasoning algorithm instead of simply combining setnotations with set variables and set constraints The MSPadopted in this paper involves three major parts detailed asfollows
Part 1 (carry out optional slot assignment schemes) Con-sider the following
Step 1 Sort the flight banks that have not been finished attime 119905 by ascending order of scheduled arrival time expressedas 119861(t) = 119887
1 1198872 119887
119896 where 119861 denotes flight bank set and
119896 is the serial number of flight bank
Step 2 Let 119906119896equal the number of flights in flight bank 119896the
actual arrival time of flight bank 119896 Then define the closetime of flight bank 119896 corresponding to the maximum 119906
119896as
1199051 1199052 as the close time of the next flight bank when it has notbeen finished at time 119905 and can be finished at time 1199052 and120591 = min(1199051 1199052)
Step 3 Assign the flights of the flight bank corresponding tothe maximum 119906
119896to the time slots before 120591 by the order of
scheduled arrival time If the number of delayed flights is 119899then 119899 slot assignment plans will be generated
Step 4 Repeat Steps 1 2 and 3 for the rest of the flight banksuntil all the flights are reassignedwith slots It should be notedthat when Part 1 is implemented a number of slot assignmentschemes are produced
Part 2 (optimize the slot assignment schemes throughnon-zero-sum sequential game) Consider the following
Step 1 Input the information needed to implement non-zero-sum sequential game namely airlines delayed flights andprovided slots
Step 2 Implement non-zero-sum sequential game for all theairlines and calculate the delay cost of the airlines accordingto the optional slot assignment schemes The non-zero-sumsequential game between airlines is implemented accordingto the following
(1) In order to cut the delay cost of airline A exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline A with the 119895st (119895 = 1 2 119898) combinationof flight and slot of airline B and the exchange meetthe requirements of heuristic rules
(2) Repeat (1) till all the combinations of flights and slotsof airline A are exchanged with that of airline B
(3) In order to cut the delay cost of airline B exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline B with the 119895st (119895 = 1 2 119898) combination
6 Mathematical Problems in Engineering
of flight and slot of airline A and the exchange meetthe requirements of heuristic rules
(4) Repeat (3) till all the combinations of flights and slotsof airline B are exchanged with that of airline A
(5) Combine the above results and the optional slotexchange schemes are obtained
(6) Divide the optional slot exchange schemes into twocategories equilibrium schemes and nonequilibriumschemes In equilibrium schemes three kinds of situ-ations are included a win-win situation for airlines Aand B airline A wins and airline B loses and airlineA loses and airline B wins Nonequilibrium schemesrefers to the schemes that make both airlines losethereby should be deleted
(7) Calculate the value of 1198916according to the equilibrium
schemes
Step 3 Repeat Step 2 till the slot assignment scheme corre-sponding to the minimum delay cost for all the airlines isgenerated
Part 3 (carry out the optimal gate assignment scheme)Consider the following
Step 1 Read the preassignment results of all the flights andobtain the time periods of the available airport gates
Step 2 For the delayed flights without subsequent flightbanks keep their gate assignment as far as possible
Step 3 For the delayed flights with subsequent flight banksgo to Step 4
Step 4 Select out the flights which are not delayed butassigned to the gates affected by the delayed flights and theselection is made on flights which arrive within the span of50 minutes around the scheduled arrival time of the delayedflights according to [8] Combine these selected flights withthe flights in Step 3 a new flight set is produced Then gateassignment for the flights of this flight set can be carried outbased on the principle of minimum delay cost according tothe types of the flights and the available gates
Step 5 Combine the assignment results of Steps 2ndash4 thenthe real-time gate assignment set is obtained The real-timegate assignment set includes three parts gates assigned to thedelayed flights with subsequent flight banks gates assigned tothe delayed flights without subsequent flight banks and gatesassigned to the flights which are not delayed but affected bythe reassignment of the delayed flights
To design a solving strategy with preferable performanceheuristic rules are organically integrated in the algorithms Bythis method on one hand the constraints in the optimizationmodel can be strictly satisfied to ensure the feasibility of thesolution on the other hand the search process can be flexiblycontrolled The heuristic rules are given as follows
(1) when the scheduled serial number of the flight bankfor some delayed flights is 119896 then the actual serialnumber of the flight bank should be no less than 119896
(2) when airlines exchange their time slots with eachother the serial numbers of the corresponding flightbanks should be as similar as possible
The above three parts as well as the heuristic rules areimplanted into depth first search algorithm (DFS) [14] so thatthe integrative research for slots assignment and gate assign-ment can be performed As a result the slot assignment isoptimized to be consistent with the optimal gate assignmentwhich satisfies the multiobjective set previously
In traditional staged algorithm the cooperation of air-lines is not taken into account so the slots for delayed flightsof airline A can only be adjusted within airline A insteadof airline B and the slots for delayed flights of airline Bcan only be adjusted within airline B instead of airline AAs a result the gate assignment may cause losses for bothairlines
Compared with the traditional staged algorithm the pro-posed integrative algorithm generates the following advan-tages (1) the slots are exchangeable between the airlines sothe transferring cost of airlines can be decreased as muchas possible (2) the slot assignment and gate assignment areintegrated into theMSPwhich supports integrativemodelingand solving so CDM mechanism for the airlines and theairport can be well achieved (3) based on the softwarePOEM an integrative MSP method which supports non-zero-sum sequential game is designed so the gate assignmentcan be generated much more effectively
4 Experimental Results
For integrative modeling and solving the software POEM[14] is taken into application In order to support sequentialgame a game class is added into the program Four partsare included in the game class players (airlines) actions (slotexchanges) costs of the players (delay cost of the airlines afterthe slot exchanges) and total cost of the sequential gameAdditionally rule class equilibrium class nonequilibriumclass and result class are designed to run the program Theconstraints on the behavior of all the players (rules for slotexchange) are defined in rule class the equilibrium charac-teristics for sequential game is included in the equilibriumclass the nonequilibrium characteristics for sequential gameis included in the nonequilibrium class the schemes andthe corresponding delay cost for each airline are generatedand stored in the result class By applying those classes inPOEM the sequential game for the airlines in Part 2 can beperformed Parts 1 and 3 are achieved by the original functionof the software POEM
The environment where the experiment is carried out isrepresented as follows (1) CPU Intel(R) Core(TM) i7-3770CPU 340GHz (2) RAM 800GB (3) system type x86-based PC (4) system manufacturer Dell Inc (5) OS nameMicrosoftWindows 7 (6)OS version 617601 Service Pack 1Build 7601
Mathematical Problems in Engineering 7
Table 1 Flight information
Flightnumber
Arrivaltime
Departuretime
Aircrafttype Passenger Flight
bank Airline
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
of the gates) [10] Furthermore Wei and Liu proposed anapproach to perform gate reassignment by minimizing thenumber of the flights assigned to the apron the disturbancecaused by the assignment and the walking distance ofpassengers meanwhile a heuristic algorithm combining tabusearch was adopted for optimization [11]
The existing research shows that the studies on gate pre-assignment have evolved and matured over time However acomprehensive consideration on the interests of multiagentnamely airports airlines and passengers has not been givento real-time gate assignment
Additionally flight bank [12] a decisive role in deter-mining the satisfaction of passengers and the benefits ofairlines is generally considered in slot assignment rather thangate assignment Nevertheless the gate assignment will alsobe influenced by flight banks because transfer passengerswould have to wait longer if flight banks are delayed whichincreases the delay cost of gate assignment Flight bank isthe product of the collaborative decision making (CDM) [13]for airports airlines and air traffic control center (ATCC)In CDM mechanism airports airlines and ATCC shouldwork collaboratively to control the entire operation costHowever almost all kinds of assignment algorithms suchas gate assignment aircraft sorting and flight schedulingare performed separately which goes against the originalintentions of CDM mechanism In the existing real-timegate assignment [13] airports are given fixed schedule of thedelayed flights Airlines are not informed of the possible gateassignment schemes and not all the information of possibleslot rearrangements are delivered to the airport as a resultthe flight banks may be delayed to a great extent To solvethis problem the CDM should be implemented accuratelyATCC provides updated slots to airlines airlines choose theslot assignment schemes corresponding to the optimal gateassignment (based on minimum cost principle) finally theoptimal gate assignment is carried out by the airport based onthe provided slot assignment schemesThe innovation takinginto account the impacts of flight banks while achievingCDM is capable of guaranteeing reasonable slot assignmentand gate assignment thereby saving delay cost for multi-agent (airlines airports and passengers) significantly con-necting flight banks smoothly and transferring passengersconveniently
In order to achieve slot assignment and gate assignmentat the same time integrative recovery strategy [14] is consid-ered as a solution Generally traditional integrated recoverystrategies such as MSP are time-consuming because thepermutation and combination of the optional objects willbecome enormous with the increasing of the scale of theproblem to be solved However sequential game has beenapplied in resource allocation [15 16] and is proven tobe timesaving by deleting the infeasible permutation andcombination of the objects from the computation processBesides sequential game also guarantees that the benefits ofthe related game players aremaximizedTherefore sequentialgame is applied innovatively in real-time gate assignment fortimesaving computation and effective cost control
Consequently this paper proposes a novel model takinginto account the benefits of multiagent the impact of flight
banks CDM mechanism fairness principle and sequentialgame between the airlines in order to achieve the followinggoals Firstly the total delay cost including the walking costof passengers the taxiing cost of aircraft the idle cost ofgates and the waiting cost of transfer passengers should beminimized Secondly the taxiing fuel for aircraft of eachairline should be balanced Thirdly the flight banks con-taining delayed flights should be connected effectively withminimum adverse impacts on the interests of the airlinesLastly unreasonable solutions should be excluded in the firstplace thereby saving computation time and resources Tomake the model work MSP combined with sequential gameis designed for modeling and solving
The remainder of this paper is organized as follows Thereal-time gate assignment model is established in Section 2In Section 3 the detailed steps of the solution algorithm areprovided In Section 4 two cases (small-scale flight delaysand medium- to large-scale flight delays) are given andthe experimental results are analyzed accordingly Finallyconclusions are drawn in Section 5
2 Real-Time Gate Assignment Model
21 Variable Definition of the Model The definitions of thevariables in the model are given as follows
(1) 119873 denotes the set of flights and119872 denotes the set ofgates 119894 119896 isin 119873 119895 isin 119872 0 lt 119872 lt 119873
(2) 119882 denotes the set of flight banks119908 isin 119882119882119898denotes
the set of flight banks for airline 119898119882119898isin 119882 119905
1198981199081is
the scheduled arrival time of flight bank 119908 for airline119898 and 119905
1198981199082is the actual arrival time of flight bank 119908
for airline119898 1199051198981199082
ge 1199051198981199081
(3) 119876 is the set of aircraft types 119860 is the set of airlines119899 isin 119876119898 isin 119860119876
119894denotes the type of flight 119894 expressed
in numbers and the bigger 119876119894is the larger flight 119894 is
(4) 119878 is the set of slots 119904 isin 119878 119878119898is the set of slots for airline
119898 when airline 119898 exchanges slots with other airlinesto cut the waiting time of transfer passengers 119878
119898isin 119878
(5) 119883119894119895is a zero-one variable and119883
119894119895equals 1 when flight
119894 is assigned to gate 119895 else119883119894119895equals 0 119884
119898119908is a zero-
one variable and 119884119898119908
equals 1 when flight bank 119908belongs to airline119898 else 119884
119898119908equals 0
(6) 119865119894is the fuel consumption of flight 119894 per minute
(7) 1198791119894and 1198792
119894denote the taxiing time of flight 119894 before
and after the reassignment respectively
(8) 119875119894is the number of passengers in flight 119894
(9) 1198711119894and1198712
119894denote thewalking time for the passengers
of flight 119894 before and after the reassignment respec-tively
(10) 1198681119895and 1198682
119895denote the idle time of gate 119895 before and
after the reassignment respectively
(11) 119903119898119908
is the number of transfer passengers in flight bank119908 of airline119898
Mathematical Problems in Engineering 3
(12) 1198621 is the price of jet fuel per kilogram 1198622 is thewalking cost of each passenger per minute 1198623
119895is the
cost of the idle time for gate 119895perminute and1198624 is thewaiting cost for each transfer passenger per minute
(13) 119877119894119895is the time when flight 119894 arrives at gate 119895 119871
119894119895is the
time when flight 119894 departs from gate 119895 and 119877119896119895is the
time when flight 119896 arrives at gate 119895 119877119896119895gt 119871119894119895
(14) 119866119895denotes the type of gate 119895 expressed in numbers
and the bigger 119866119895is the larger gate 119895 is
(15) Δ119879 is the buffer time between any two consecutiveflights assigned to the same gate
(16) 119868119887119895denotes the beginning of the idle time for gate 119895
and 119868119890119895denotes the end of the idle time for gate 119895
(17) 119891119878119898
is the waiting cost of the transfer passengers forairline 119898 which is less than the waiting cost of thetransfer passengers before the slot exchange of airline119898
(18) 119883119878119898
is a zero-one variable119883119878119898
equals 1 when 119878119898is the
slots combination for airline119898 after the exchange else119883119878119898
equals 0(19) Ω
119898is the set of slot combinations which makes the
waiting time of transfer passengers for airline 119898 lessafter the slot exchange
(20) Δ119870119899119898
is the increased fuel consumption of type 119899aircraft belonging to airline119898 119911 is the number of air-lines so 1119911 is the expected proportion correspond-ing to the ideal situation of complete equalization forall airlines
22 Objectives
(1) Minimize the Increased Total Cost The total cost involvesfuel cost walking cost of passengers idle cost of gates andwaiting cost of transfer passengers To achieve the goal ofminimizing the increased total cost caused by flight delaysthe following four values need to be minimized at the sametime taxiing time of aircraft walking time of passengers idletime of gates and waiting time of transfer passengers Theobjective function is formulated as follows
119865119894(1198792119894minus 1198791119894) 1198621 denotes the increased taxiing cost of flight
119894 119875119894(1198712119894minus 1198711119894) 1198622 denotes the incremental walking cost for
passengers of flight 119894 (1198682119895minus1198681119895) 1198623119895denotes the change of the
idle cost for gate 119895 119903119898119908(1199051198981199082
minus1199051198981199081) 1198624 denotes the increased
waiting cost of transfer passengers in flight bank 119908 of airline119898
(2) Minimize the Increased Taxiing Time of Aircraft Since fuelcost accounts for about 30 of the total operation cost forairlines fuel cost decrease will make a huge difference in thecost control of airlines Fuel cost reduction can be achievedby minimizing the incremental taxiing time of all the aircraftaccording to
(3) Minimize the Increased Walking Time of PassengersThe reduction of walking distance or time improves thesatisfaction of passengers The purpose of minimizing thewalking time of passengers can be realized by
(4) Minimize the Increased Idle Time of Gates As airport gatesare the core resources of an airport improving the utilizationrate of the idle gates attributes to better airport operationTo realize this goal the idle time of airport gates can beminimized by
min 1198914= sum
119895isin119872
(1198682119895minus 1198681119895) (4)
(5) Minimize the Increased Waiting Time of Transfer Passen-gers Flight bank has been widely applied in hub airports sothat the transferring efficiency of passengers the utilizationrate of airport resources and the operational effectiveness ofairlines can be improved Generally flight bank is capableof connecting flights effectively thus minimizing the waitingtime of transfer passengers However delayed flight bankswill lead to a series of problems such as increased waitingtime of passengers and increased operation cost The objec-tive function that is aimed at reducing the increased waitingtime caused by the delay of flight banks is given by
As slot 119904 is randomly assigned to flight 119894 and the numberof slots is equal to the number of flights | sum
119904isin119878119905119904minus sum119894isin119873119905119894| is
4 Mathematical Problems in Engineering
a constant Hence the increased waiting time of transfer pas-sengers does not necessarily depend on the slot assignmentbut the delay of flight banks
Theorem 2 (gate assignment hinges on the slot assignment)Gate assignment depends on the arrival time of flights and theidle time of available gates therefore slot assignment plays adecisive role in gate assignment
According to Theorems 1 and 2 the slot assignment fordelayed flights should be implemented with minimum delayof flight banks so that the optimal gate assignment withminimum delay cost can be producedThis is why flight bankis involved in the objective of minimizing the waiting cost oftransfer passengers in gate assignment
(6) Minimize the Waiting Cost by Optimizing the Slot Assign-ment through Non-Zero-Sum Sequential Game The slotsassigned to airlines are exchangeable so airlines can reducethe waiting time of transfer passengers by exchanging theslots with each other The process of slot exchange can beachieved through non-zero-sum sequential game betweenairlines In sequential game all the airlines are aware of theirprevious policy or selection and have to make their currentdecisions according to their tradeoff of future possibilitiesZero-sum sequential game refers to the situation that theincome of one side is equal to the loss of the other side Asour subject is about controlling the loss caused by the flightdelays for all the related airlines non-zero-sum sequentialgame theory [15] is adopted in this application The modelof non-zero-sum sequential game is given by
119866 = 119860 (119878119898)119898isin119860
(120588119898)119898isin119860
119875 (120588) (7)
where119860 denotes the set of airlines 119878119898is the set of all optional
slot series for airline 119898 forall119898 isin 119860 120588119898
is the realizationprobability of 119878
119898 119875(120588) denotes the expected revenue matrix
Theorem 3 In a sequential game any realization probabilitypoints to a behavior strategy
Proof of Theorem 3 120588119898(119904119898) = prod
120572isin119904119898
120573119898(120572) where 120588
119898(119904119898) is
the realization probability for airline 119898 to obtain slot series119904119898(119904119898isin 119878119898) 120573119898is the probability distribution of 119879(ℎ
119898) and
119879(ℎ119898) is the set of optional policies under information set ℎ
119898
for airline119898Therefore any realization probability comes from a cor-
responding behavior strategyThe set of behavior sequences for airline 119898 on informa-
tion set ℎ119898is denoted by 120582(ℎ
119898) 120574 is an expansion of the
behavior sequences denoted by 120582(ℎ119898)120574 (120582(ℎ
119898)120574 = 120582(ℎ
119898)cup120574)
and the realization probability of 120582(ℎ119898) can be denoted by
120588119898(120582(ℎ119898))
Consequently the behavior 120574 on information set ℎ119898can
be confirmed by
120573119898(120574) =
120588119898(120582 (ℎ119898) 120574)
120588119898(120582 (ℎ119898)) (8)
where 120588119898(120582(ℎ119898)) gt 0 and 120573
119898(120574) can be any value when
120588119898(120582(ℎ119898)) = 0
Suppose that 120573 = (1205731 1205732 120573
119902) where 120573 is the behavior
strategy set for all airlines 120588 = (1205881 1205882 120588
119902) where 120588 is
the corresponding realization probability and 1205880denotes the
realization probability of the virtual player nature usually afixed value and 119878 = 119878
0times 1198781times sdot sdot sdot times 119878
119902 where 119878 denotes
sequence space and 119904 = (1199040 1199041 119904
119902) isin 119878 where 119904 is a set of
slot series The expected revenue function can be expressedby
119875 (120588) = sum
119904isin119878
119875 (119904)
119902
prod
119898=0
120588119898(119904119898) (9)
where 119875(119904) = 119875(119888) 119875(119888) is the revenue of implementing 119904towards some end note 119888 andprod119902
119898=0120588119898(119904119898) is the realization
probability of approaching 119888According to the above description when the revenue
119875(120588) for all airlines is maximized the transferring cost forall the airlines can be achieved and the objective function isgiven by
min 1198916= sum
119898isin119860
sum
119878119898isinΩ119898
119891119878119898
119883119878119898
(10)
where 119891119878119898
is the value of the objective function 1198915for airline
119898 when 119878119898is the slots combination after the exchange
(7) Balance the Increased Fuel Consumption for Each AirlineFairness principle requires that the fuel consumption causedby flight delays should be averaged for airlines to bearHowever airlines are of different scales and aircraft are ofdifferent types so the average fuel consumption should bemade from the aspects of both aircraft and airlines Thisgoal is achieved by averaging the proportion of the fuelconsumption change for a certain aircraft type belonging toa certain airline to the fuel consumption change for a certainaircraft type of all airlines The objective is represented by
is the proportion of the fuel con-sumption change of type 119899 aircraft belonging to airline 119898 tothe fuel consumption change of all airlinesrsquo type 119899 aircraft
23 Integrative Assignment Model According to the aboveanalysis the real-time gate assignment model based on theprinciple of minimum delay cost for multiagent can beexpressed as follows
min 119891 = min 1198911 1198912 1198913 1198914 1198915 1198916 1198917 (12)
where (12) is the objective function Equation (13) meansevery flight is assigned to one and only one gate Equation(14) is the corresponding relationship between flight and gateEquation (15) means each slot combination is adopted byone and only one airline Equation (16) is the correspondingrelationship of slot combinations and airlines Equation (17)enforces that the type of the gate where the aircraft is assignedshould match the type of the aircraft Equation (18) stipulatesthat the idle time of the gate should be longer than buffertime for the sake of safety Equation (19) indicates that thebeginning of the idle time for any gate should be earlier thanthe arrival time of the flight which will be assigned to thegate and the end of the idle time should be later than thedeparture time of the flight Equation (20) refers to validityconstraint
Compared with the traditional staged model (the slotassignment is produced before the gate assignment) theadvantages of the proposed model are presented as follows(1) the sequential game helps to obtain better slot combi-nations for all the airlines and (2) the CDM mechanismcontributes to generating gate reassignment with less delaycost of multiagent due to the collaboration of airlines andairports
To solve the multiobjective optimization problem(MOOP) [17] the objectives are sorted in order ofpriority because all the objectives cannot be optimizedsimultaneously As the service concept is becoming moreand more important the waiting time and waiting time ofpassengers are given the highest priority The second highestpriority is the taxiing time of aircraft because the fuel costis the direct operation cost of airlines Following the taxiingtime of aircraft is the implicit idle cost of the airport Fuelequalization is the lowest priority because the slot exchangebetween airlines also contributes to the fairness principlewhen the fuel consumption is not equalized at the verybeginning
3 Solution Algorithm
Mixed set programming (MSP) [18ndash20] is a logic reasoningalgorithm based on first-order logic and set reasoning InMSP set operations quantifiers Boolean logic logic func-tions datetime reasoning and numerical constraints areintegrated in one system the reasoning on numeric typessuch as reals and integers is expanded to global reasoning
over mixed domains of set types such as Booleans andreferences Most importantly MSP makes the modeling andsolving for constraint satisfaction problems (CSP) realizableThe so-called set programming here is to systematicallyintegrating set reasoning and operational research algorithmestablishing a rigorous and complete set theoretical for-mulation based on set variables and solving the model byset reasoning algorithm instead of simply combining setnotations with set variables and set constraints The MSPadopted in this paper involves three major parts detailed asfollows
Part 1 (carry out optional slot assignment schemes) Con-sider the following
Step 1 Sort the flight banks that have not been finished attime 119905 by ascending order of scheduled arrival time expressedas 119861(t) = 119887
1 1198872 119887
119896 where 119861 denotes flight bank set and
119896 is the serial number of flight bank
Step 2 Let 119906119896equal the number of flights in flight bank 119896the
actual arrival time of flight bank 119896 Then define the closetime of flight bank 119896 corresponding to the maximum 119906
119896as
1199051 1199052 as the close time of the next flight bank when it has notbeen finished at time 119905 and can be finished at time 1199052 and120591 = min(1199051 1199052)
Step 3 Assign the flights of the flight bank corresponding tothe maximum 119906
119896to the time slots before 120591 by the order of
scheduled arrival time If the number of delayed flights is 119899then 119899 slot assignment plans will be generated
Step 4 Repeat Steps 1 2 and 3 for the rest of the flight banksuntil all the flights are reassignedwith slots It should be notedthat when Part 1 is implemented a number of slot assignmentschemes are produced
Part 2 (optimize the slot assignment schemes throughnon-zero-sum sequential game) Consider the following
Step 1 Input the information needed to implement non-zero-sum sequential game namely airlines delayed flights andprovided slots
Step 2 Implement non-zero-sum sequential game for all theairlines and calculate the delay cost of the airlines accordingto the optional slot assignment schemes The non-zero-sumsequential game between airlines is implemented accordingto the following
(1) In order to cut the delay cost of airline A exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline A with the 119895st (119895 = 1 2 119898) combinationof flight and slot of airline B and the exchange meetthe requirements of heuristic rules
(2) Repeat (1) till all the combinations of flights and slotsof airline A are exchanged with that of airline B
(3) In order to cut the delay cost of airline B exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline B with the 119895st (119895 = 1 2 119898) combination
6 Mathematical Problems in Engineering
of flight and slot of airline A and the exchange meetthe requirements of heuristic rules
(4) Repeat (3) till all the combinations of flights and slotsof airline B are exchanged with that of airline A
(5) Combine the above results and the optional slotexchange schemes are obtained
(6) Divide the optional slot exchange schemes into twocategories equilibrium schemes and nonequilibriumschemes In equilibrium schemes three kinds of situ-ations are included a win-win situation for airlines Aand B airline A wins and airline B loses and airlineA loses and airline B wins Nonequilibrium schemesrefers to the schemes that make both airlines losethereby should be deleted
(7) Calculate the value of 1198916according to the equilibrium
schemes
Step 3 Repeat Step 2 till the slot assignment scheme corre-sponding to the minimum delay cost for all the airlines isgenerated
Part 3 (carry out the optimal gate assignment scheme)Consider the following
Step 1 Read the preassignment results of all the flights andobtain the time periods of the available airport gates
Step 2 For the delayed flights without subsequent flightbanks keep their gate assignment as far as possible
Step 3 For the delayed flights with subsequent flight banksgo to Step 4
Step 4 Select out the flights which are not delayed butassigned to the gates affected by the delayed flights and theselection is made on flights which arrive within the span of50 minutes around the scheduled arrival time of the delayedflights according to [8] Combine these selected flights withthe flights in Step 3 a new flight set is produced Then gateassignment for the flights of this flight set can be carried outbased on the principle of minimum delay cost according tothe types of the flights and the available gates
Step 5 Combine the assignment results of Steps 2ndash4 thenthe real-time gate assignment set is obtained The real-timegate assignment set includes three parts gates assigned to thedelayed flights with subsequent flight banks gates assigned tothe delayed flights without subsequent flight banks and gatesassigned to the flights which are not delayed but affected bythe reassignment of the delayed flights
To design a solving strategy with preferable performanceheuristic rules are organically integrated in the algorithms Bythis method on one hand the constraints in the optimizationmodel can be strictly satisfied to ensure the feasibility of thesolution on the other hand the search process can be flexiblycontrolled The heuristic rules are given as follows
(1) when the scheduled serial number of the flight bankfor some delayed flights is 119896 then the actual serialnumber of the flight bank should be no less than 119896
(2) when airlines exchange their time slots with eachother the serial numbers of the corresponding flightbanks should be as similar as possible
The above three parts as well as the heuristic rules areimplanted into depth first search algorithm (DFS) [14] so thatthe integrative research for slots assignment and gate assign-ment can be performed As a result the slot assignment isoptimized to be consistent with the optimal gate assignmentwhich satisfies the multiobjective set previously
In traditional staged algorithm the cooperation of air-lines is not taken into account so the slots for delayed flightsof airline A can only be adjusted within airline A insteadof airline B and the slots for delayed flights of airline Bcan only be adjusted within airline B instead of airline AAs a result the gate assignment may cause losses for bothairlines
Compared with the traditional staged algorithm the pro-posed integrative algorithm generates the following advan-tages (1) the slots are exchangeable between the airlines sothe transferring cost of airlines can be decreased as muchas possible (2) the slot assignment and gate assignment areintegrated into theMSPwhich supports integrativemodelingand solving so CDM mechanism for the airlines and theairport can be well achieved (3) based on the softwarePOEM an integrative MSP method which supports non-zero-sum sequential game is designed so the gate assignmentcan be generated much more effectively
4 Experimental Results
For integrative modeling and solving the software POEM[14] is taken into application In order to support sequentialgame a game class is added into the program Four partsare included in the game class players (airlines) actions (slotexchanges) costs of the players (delay cost of the airlines afterthe slot exchanges) and total cost of the sequential gameAdditionally rule class equilibrium class nonequilibriumclass and result class are designed to run the program Theconstraints on the behavior of all the players (rules for slotexchange) are defined in rule class the equilibrium charac-teristics for sequential game is included in the equilibriumclass the nonequilibrium characteristics for sequential gameis included in the nonequilibrium class the schemes andthe corresponding delay cost for each airline are generatedand stored in the result class By applying those classes inPOEM the sequential game for the airlines in Part 2 can beperformed Parts 1 and 3 are achieved by the original functionof the software POEM
The environment where the experiment is carried out isrepresented as follows (1) CPU Intel(R) Core(TM) i7-3770CPU 340GHz (2) RAM 800GB (3) system type x86-based PC (4) system manufacturer Dell Inc (5) OS nameMicrosoftWindows 7 (6)OS version 617601 Service Pack 1Build 7601
Mathematical Problems in Engineering 7
Table 1 Flight information
Flightnumber
Arrivaltime
Departuretime
Aircrafttype Passenger Flight
bank Airline
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
(12) 1198621 is the price of jet fuel per kilogram 1198622 is thewalking cost of each passenger per minute 1198623
119895is the
cost of the idle time for gate 119895perminute and1198624 is thewaiting cost for each transfer passenger per minute
(13) 119877119894119895is the time when flight 119894 arrives at gate 119895 119871
119894119895is the
time when flight 119894 departs from gate 119895 and 119877119896119895is the
time when flight 119896 arrives at gate 119895 119877119896119895gt 119871119894119895
(14) 119866119895denotes the type of gate 119895 expressed in numbers
and the bigger 119866119895is the larger gate 119895 is
(15) Δ119879 is the buffer time between any two consecutiveflights assigned to the same gate
(16) 119868119887119895denotes the beginning of the idle time for gate 119895
and 119868119890119895denotes the end of the idle time for gate 119895
(17) 119891119878119898
is the waiting cost of the transfer passengers forairline 119898 which is less than the waiting cost of thetransfer passengers before the slot exchange of airline119898
(18) 119883119878119898
is a zero-one variable119883119878119898
equals 1 when 119878119898is the
slots combination for airline119898 after the exchange else119883119878119898
equals 0(19) Ω
119898is the set of slot combinations which makes the
waiting time of transfer passengers for airline 119898 lessafter the slot exchange
(20) Δ119870119899119898
is the increased fuel consumption of type 119899aircraft belonging to airline119898 119911 is the number of air-lines so 1119911 is the expected proportion correspond-ing to the ideal situation of complete equalization forall airlines
22 Objectives
(1) Minimize the Increased Total Cost The total cost involvesfuel cost walking cost of passengers idle cost of gates andwaiting cost of transfer passengers To achieve the goal ofminimizing the increased total cost caused by flight delaysthe following four values need to be minimized at the sametime taxiing time of aircraft walking time of passengers idletime of gates and waiting time of transfer passengers Theobjective function is formulated as follows
119865119894(1198792119894minus 1198791119894) 1198621 denotes the increased taxiing cost of flight
119894 119875119894(1198712119894minus 1198711119894) 1198622 denotes the incremental walking cost for
passengers of flight 119894 (1198682119895minus1198681119895) 1198623119895denotes the change of the
idle cost for gate 119895 119903119898119908(1199051198981199082
minus1199051198981199081) 1198624 denotes the increased
waiting cost of transfer passengers in flight bank 119908 of airline119898
(2) Minimize the Increased Taxiing Time of Aircraft Since fuelcost accounts for about 30 of the total operation cost forairlines fuel cost decrease will make a huge difference in thecost control of airlines Fuel cost reduction can be achievedby minimizing the incremental taxiing time of all the aircraftaccording to
(3) Minimize the Increased Walking Time of PassengersThe reduction of walking distance or time improves thesatisfaction of passengers The purpose of minimizing thewalking time of passengers can be realized by
(4) Minimize the Increased Idle Time of Gates As airport gatesare the core resources of an airport improving the utilizationrate of the idle gates attributes to better airport operationTo realize this goal the idle time of airport gates can beminimized by
min 1198914= sum
119895isin119872
(1198682119895minus 1198681119895) (4)
(5) Minimize the Increased Waiting Time of Transfer Passen-gers Flight bank has been widely applied in hub airports sothat the transferring efficiency of passengers the utilizationrate of airport resources and the operational effectiveness ofairlines can be improved Generally flight bank is capableof connecting flights effectively thus minimizing the waitingtime of transfer passengers However delayed flight bankswill lead to a series of problems such as increased waitingtime of passengers and increased operation cost The objec-tive function that is aimed at reducing the increased waitingtime caused by the delay of flight banks is given by
As slot 119904 is randomly assigned to flight 119894 and the numberof slots is equal to the number of flights | sum
119904isin119878119905119904minus sum119894isin119873119905119894| is
4 Mathematical Problems in Engineering
a constant Hence the increased waiting time of transfer pas-sengers does not necessarily depend on the slot assignmentbut the delay of flight banks
Theorem 2 (gate assignment hinges on the slot assignment)Gate assignment depends on the arrival time of flights and theidle time of available gates therefore slot assignment plays adecisive role in gate assignment
According to Theorems 1 and 2 the slot assignment fordelayed flights should be implemented with minimum delayof flight banks so that the optimal gate assignment withminimum delay cost can be producedThis is why flight bankis involved in the objective of minimizing the waiting cost oftransfer passengers in gate assignment
(6) Minimize the Waiting Cost by Optimizing the Slot Assign-ment through Non-Zero-Sum Sequential Game The slotsassigned to airlines are exchangeable so airlines can reducethe waiting time of transfer passengers by exchanging theslots with each other The process of slot exchange can beachieved through non-zero-sum sequential game betweenairlines In sequential game all the airlines are aware of theirprevious policy or selection and have to make their currentdecisions according to their tradeoff of future possibilitiesZero-sum sequential game refers to the situation that theincome of one side is equal to the loss of the other side Asour subject is about controlling the loss caused by the flightdelays for all the related airlines non-zero-sum sequentialgame theory [15] is adopted in this application The modelof non-zero-sum sequential game is given by
119866 = 119860 (119878119898)119898isin119860
(120588119898)119898isin119860
119875 (120588) (7)
where119860 denotes the set of airlines 119878119898is the set of all optional
slot series for airline 119898 forall119898 isin 119860 120588119898
is the realizationprobability of 119878
119898 119875(120588) denotes the expected revenue matrix
Theorem 3 In a sequential game any realization probabilitypoints to a behavior strategy
Proof of Theorem 3 120588119898(119904119898) = prod
120572isin119904119898
120573119898(120572) where 120588
119898(119904119898) is
the realization probability for airline 119898 to obtain slot series119904119898(119904119898isin 119878119898) 120573119898is the probability distribution of 119879(ℎ
119898) and
119879(ℎ119898) is the set of optional policies under information set ℎ
119898
for airline119898Therefore any realization probability comes from a cor-
responding behavior strategyThe set of behavior sequences for airline 119898 on informa-
tion set ℎ119898is denoted by 120582(ℎ
119898) 120574 is an expansion of the
behavior sequences denoted by 120582(ℎ119898)120574 (120582(ℎ
119898)120574 = 120582(ℎ
119898)cup120574)
and the realization probability of 120582(ℎ119898) can be denoted by
120588119898(120582(ℎ119898))
Consequently the behavior 120574 on information set ℎ119898can
be confirmed by
120573119898(120574) =
120588119898(120582 (ℎ119898) 120574)
120588119898(120582 (ℎ119898)) (8)
where 120588119898(120582(ℎ119898)) gt 0 and 120573
119898(120574) can be any value when
120588119898(120582(ℎ119898)) = 0
Suppose that 120573 = (1205731 1205732 120573
119902) where 120573 is the behavior
strategy set for all airlines 120588 = (1205881 1205882 120588
119902) where 120588 is
the corresponding realization probability and 1205880denotes the
realization probability of the virtual player nature usually afixed value and 119878 = 119878
0times 1198781times sdot sdot sdot times 119878
119902 where 119878 denotes
sequence space and 119904 = (1199040 1199041 119904
119902) isin 119878 where 119904 is a set of
slot series The expected revenue function can be expressedby
119875 (120588) = sum
119904isin119878
119875 (119904)
119902
prod
119898=0
120588119898(119904119898) (9)
where 119875(119904) = 119875(119888) 119875(119888) is the revenue of implementing 119904towards some end note 119888 andprod119902
119898=0120588119898(119904119898) is the realization
probability of approaching 119888According to the above description when the revenue
119875(120588) for all airlines is maximized the transferring cost forall the airlines can be achieved and the objective function isgiven by
min 1198916= sum
119898isin119860
sum
119878119898isinΩ119898
119891119878119898
119883119878119898
(10)
where 119891119878119898
is the value of the objective function 1198915for airline
119898 when 119878119898is the slots combination after the exchange
(7) Balance the Increased Fuel Consumption for Each AirlineFairness principle requires that the fuel consumption causedby flight delays should be averaged for airlines to bearHowever airlines are of different scales and aircraft are ofdifferent types so the average fuel consumption should bemade from the aspects of both aircraft and airlines Thisgoal is achieved by averaging the proportion of the fuelconsumption change for a certain aircraft type belonging toa certain airline to the fuel consumption change for a certainaircraft type of all airlines The objective is represented by
is the proportion of the fuel con-sumption change of type 119899 aircraft belonging to airline 119898 tothe fuel consumption change of all airlinesrsquo type 119899 aircraft
23 Integrative Assignment Model According to the aboveanalysis the real-time gate assignment model based on theprinciple of minimum delay cost for multiagent can beexpressed as follows
min 119891 = min 1198911 1198912 1198913 1198914 1198915 1198916 1198917 (12)
where (12) is the objective function Equation (13) meansevery flight is assigned to one and only one gate Equation(14) is the corresponding relationship between flight and gateEquation (15) means each slot combination is adopted byone and only one airline Equation (16) is the correspondingrelationship of slot combinations and airlines Equation (17)enforces that the type of the gate where the aircraft is assignedshould match the type of the aircraft Equation (18) stipulatesthat the idle time of the gate should be longer than buffertime for the sake of safety Equation (19) indicates that thebeginning of the idle time for any gate should be earlier thanthe arrival time of the flight which will be assigned to thegate and the end of the idle time should be later than thedeparture time of the flight Equation (20) refers to validityconstraint
Compared with the traditional staged model (the slotassignment is produced before the gate assignment) theadvantages of the proposed model are presented as follows(1) the sequential game helps to obtain better slot combi-nations for all the airlines and (2) the CDM mechanismcontributes to generating gate reassignment with less delaycost of multiagent due to the collaboration of airlines andairports
To solve the multiobjective optimization problem(MOOP) [17] the objectives are sorted in order ofpriority because all the objectives cannot be optimizedsimultaneously As the service concept is becoming moreand more important the waiting time and waiting time ofpassengers are given the highest priority The second highestpriority is the taxiing time of aircraft because the fuel costis the direct operation cost of airlines Following the taxiingtime of aircraft is the implicit idle cost of the airport Fuelequalization is the lowest priority because the slot exchangebetween airlines also contributes to the fairness principlewhen the fuel consumption is not equalized at the verybeginning
3 Solution Algorithm
Mixed set programming (MSP) [18ndash20] is a logic reasoningalgorithm based on first-order logic and set reasoning InMSP set operations quantifiers Boolean logic logic func-tions datetime reasoning and numerical constraints areintegrated in one system the reasoning on numeric typessuch as reals and integers is expanded to global reasoning
over mixed domains of set types such as Booleans andreferences Most importantly MSP makes the modeling andsolving for constraint satisfaction problems (CSP) realizableThe so-called set programming here is to systematicallyintegrating set reasoning and operational research algorithmestablishing a rigorous and complete set theoretical for-mulation based on set variables and solving the model byset reasoning algorithm instead of simply combining setnotations with set variables and set constraints The MSPadopted in this paper involves three major parts detailed asfollows
Part 1 (carry out optional slot assignment schemes) Con-sider the following
Step 1 Sort the flight banks that have not been finished attime 119905 by ascending order of scheduled arrival time expressedas 119861(t) = 119887
1 1198872 119887
119896 where 119861 denotes flight bank set and
119896 is the serial number of flight bank
Step 2 Let 119906119896equal the number of flights in flight bank 119896the
actual arrival time of flight bank 119896 Then define the closetime of flight bank 119896 corresponding to the maximum 119906
119896as
1199051 1199052 as the close time of the next flight bank when it has notbeen finished at time 119905 and can be finished at time 1199052 and120591 = min(1199051 1199052)
Step 3 Assign the flights of the flight bank corresponding tothe maximum 119906
119896to the time slots before 120591 by the order of
scheduled arrival time If the number of delayed flights is 119899then 119899 slot assignment plans will be generated
Step 4 Repeat Steps 1 2 and 3 for the rest of the flight banksuntil all the flights are reassignedwith slots It should be notedthat when Part 1 is implemented a number of slot assignmentschemes are produced
Part 2 (optimize the slot assignment schemes throughnon-zero-sum sequential game) Consider the following
Step 1 Input the information needed to implement non-zero-sum sequential game namely airlines delayed flights andprovided slots
Step 2 Implement non-zero-sum sequential game for all theairlines and calculate the delay cost of the airlines accordingto the optional slot assignment schemes The non-zero-sumsequential game between airlines is implemented accordingto the following
(1) In order to cut the delay cost of airline A exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline A with the 119895st (119895 = 1 2 119898) combinationof flight and slot of airline B and the exchange meetthe requirements of heuristic rules
(2) Repeat (1) till all the combinations of flights and slotsof airline A are exchanged with that of airline B
(3) In order to cut the delay cost of airline B exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline B with the 119895st (119895 = 1 2 119898) combination
6 Mathematical Problems in Engineering
of flight and slot of airline A and the exchange meetthe requirements of heuristic rules
(4) Repeat (3) till all the combinations of flights and slotsof airline B are exchanged with that of airline A
(5) Combine the above results and the optional slotexchange schemes are obtained
(6) Divide the optional slot exchange schemes into twocategories equilibrium schemes and nonequilibriumschemes In equilibrium schemes three kinds of situ-ations are included a win-win situation for airlines Aand B airline A wins and airline B loses and airlineA loses and airline B wins Nonequilibrium schemesrefers to the schemes that make both airlines losethereby should be deleted
(7) Calculate the value of 1198916according to the equilibrium
schemes
Step 3 Repeat Step 2 till the slot assignment scheme corre-sponding to the minimum delay cost for all the airlines isgenerated
Part 3 (carry out the optimal gate assignment scheme)Consider the following
Step 1 Read the preassignment results of all the flights andobtain the time periods of the available airport gates
Step 2 For the delayed flights without subsequent flightbanks keep their gate assignment as far as possible
Step 3 For the delayed flights with subsequent flight banksgo to Step 4
Step 4 Select out the flights which are not delayed butassigned to the gates affected by the delayed flights and theselection is made on flights which arrive within the span of50 minutes around the scheduled arrival time of the delayedflights according to [8] Combine these selected flights withthe flights in Step 3 a new flight set is produced Then gateassignment for the flights of this flight set can be carried outbased on the principle of minimum delay cost according tothe types of the flights and the available gates
Step 5 Combine the assignment results of Steps 2ndash4 thenthe real-time gate assignment set is obtained The real-timegate assignment set includes three parts gates assigned to thedelayed flights with subsequent flight banks gates assigned tothe delayed flights without subsequent flight banks and gatesassigned to the flights which are not delayed but affected bythe reassignment of the delayed flights
To design a solving strategy with preferable performanceheuristic rules are organically integrated in the algorithms Bythis method on one hand the constraints in the optimizationmodel can be strictly satisfied to ensure the feasibility of thesolution on the other hand the search process can be flexiblycontrolled The heuristic rules are given as follows
(1) when the scheduled serial number of the flight bankfor some delayed flights is 119896 then the actual serialnumber of the flight bank should be no less than 119896
(2) when airlines exchange their time slots with eachother the serial numbers of the corresponding flightbanks should be as similar as possible
The above three parts as well as the heuristic rules areimplanted into depth first search algorithm (DFS) [14] so thatthe integrative research for slots assignment and gate assign-ment can be performed As a result the slot assignment isoptimized to be consistent with the optimal gate assignmentwhich satisfies the multiobjective set previously
In traditional staged algorithm the cooperation of air-lines is not taken into account so the slots for delayed flightsof airline A can only be adjusted within airline A insteadof airline B and the slots for delayed flights of airline Bcan only be adjusted within airline B instead of airline AAs a result the gate assignment may cause losses for bothairlines
Compared with the traditional staged algorithm the pro-posed integrative algorithm generates the following advan-tages (1) the slots are exchangeable between the airlines sothe transferring cost of airlines can be decreased as muchas possible (2) the slot assignment and gate assignment areintegrated into theMSPwhich supports integrativemodelingand solving so CDM mechanism for the airlines and theairport can be well achieved (3) based on the softwarePOEM an integrative MSP method which supports non-zero-sum sequential game is designed so the gate assignmentcan be generated much more effectively
4 Experimental Results
For integrative modeling and solving the software POEM[14] is taken into application In order to support sequentialgame a game class is added into the program Four partsare included in the game class players (airlines) actions (slotexchanges) costs of the players (delay cost of the airlines afterthe slot exchanges) and total cost of the sequential gameAdditionally rule class equilibrium class nonequilibriumclass and result class are designed to run the program Theconstraints on the behavior of all the players (rules for slotexchange) are defined in rule class the equilibrium charac-teristics for sequential game is included in the equilibriumclass the nonequilibrium characteristics for sequential gameis included in the nonequilibrium class the schemes andthe corresponding delay cost for each airline are generatedand stored in the result class By applying those classes inPOEM the sequential game for the airlines in Part 2 can beperformed Parts 1 and 3 are achieved by the original functionof the software POEM
The environment where the experiment is carried out isrepresented as follows (1) CPU Intel(R) Core(TM) i7-3770CPU 340GHz (2) RAM 800GB (3) system type x86-based PC (4) system manufacturer Dell Inc (5) OS nameMicrosoftWindows 7 (6)OS version 617601 Service Pack 1Build 7601
Mathematical Problems in Engineering 7
Table 1 Flight information
Flightnumber
Arrivaltime
Departuretime
Aircrafttype Passenger Flight
bank Airline
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
a constant Hence the increased waiting time of transfer pas-sengers does not necessarily depend on the slot assignmentbut the delay of flight banks
Theorem 2 (gate assignment hinges on the slot assignment)Gate assignment depends on the arrival time of flights and theidle time of available gates therefore slot assignment plays adecisive role in gate assignment
According to Theorems 1 and 2 the slot assignment fordelayed flights should be implemented with minimum delayof flight banks so that the optimal gate assignment withminimum delay cost can be producedThis is why flight bankis involved in the objective of minimizing the waiting cost oftransfer passengers in gate assignment
(6) Minimize the Waiting Cost by Optimizing the Slot Assign-ment through Non-Zero-Sum Sequential Game The slotsassigned to airlines are exchangeable so airlines can reducethe waiting time of transfer passengers by exchanging theslots with each other The process of slot exchange can beachieved through non-zero-sum sequential game betweenairlines In sequential game all the airlines are aware of theirprevious policy or selection and have to make their currentdecisions according to their tradeoff of future possibilitiesZero-sum sequential game refers to the situation that theincome of one side is equal to the loss of the other side Asour subject is about controlling the loss caused by the flightdelays for all the related airlines non-zero-sum sequentialgame theory [15] is adopted in this application The modelof non-zero-sum sequential game is given by
119866 = 119860 (119878119898)119898isin119860
(120588119898)119898isin119860
119875 (120588) (7)
where119860 denotes the set of airlines 119878119898is the set of all optional
slot series for airline 119898 forall119898 isin 119860 120588119898
is the realizationprobability of 119878
119898 119875(120588) denotes the expected revenue matrix
Theorem 3 In a sequential game any realization probabilitypoints to a behavior strategy
Proof of Theorem 3 120588119898(119904119898) = prod
120572isin119904119898
120573119898(120572) where 120588
119898(119904119898) is
the realization probability for airline 119898 to obtain slot series119904119898(119904119898isin 119878119898) 120573119898is the probability distribution of 119879(ℎ
119898) and
119879(ℎ119898) is the set of optional policies under information set ℎ
119898
for airline119898Therefore any realization probability comes from a cor-
responding behavior strategyThe set of behavior sequences for airline 119898 on informa-
tion set ℎ119898is denoted by 120582(ℎ
119898) 120574 is an expansion of the
behavior sequences denoted by 120582(ℎ119898)120574 (120582(ℎ
119898)120574 = 120582(ℎ
119898)cup120574)
and the realization probability of 120582(ℎ119898) can be denoted by
120588119898(120582(ℎ119898))
Consequently the behavior 120574 on information set ℎ119898can
be confirmed by
120573119898(120574) =
120588119898(120582 (ℎ119898) 120574)
120588119898(120582 (ℎ119898)) (8)
where 120588119898(120582(ℎ119898)) gt 0 and 120573
119898(120574) can be any value when
120588119898(120582(ℎ119898)) = 0
Suppose that 120573 = (1205731 1205732 120573
119902) where 120573 is the behavior
strategy set for all airlines 120588 = (1205881 1205882 120588
119902) where 120588 is
the corresponding realization probability and 1205880denotes the
realization probability of the virtual player nature usually afixed value and 119878 = 119878
0times 1198781times sdot sdot sdot times 119878
119902 where 119878 denotes
sequence space and 119904 = (1199040 1199041 119904
119902) isin 119878 where 119904 is a set of
slot series The expected revenue function can be expressedby
119875 (120588) = sum
119904isin119878
119875 (119904)
119902
prod
119898=0
120588119898(119904119898) (9)
where 119875(119904) = 119875(119888) 119875(119888) is the revenue of implementing 119904towards some end note 119888 andprod119902
119898=0120588119898(119904119898) is the realization
probability of approaching 119888According to the above description when the revenue
119875(120588) for all airlines is maximized the transferring cost forall the airlines can be achieved and the objective function isgiven by
min 1198916= sum
119898isin119860
sum
119878119898isinΩ119898
119891119878119898
119883119878119898
(10)
where 119891119878119898
is the value of the objective function 1198915for airline
119898 when 119878119898is the slots combination after the exchange
(7) Balance the Increased Fuel Consumption for Each AirlineFairness principle requires that the fuel consumption causedby flight delays should be averaged for airlines to bearHowever airlines are of different scales and aircraft are ofdifferent types so the average fuel consumption should bemade from the aspects of both aircraft and airlines Thisgoal is achieved by averaging the proportion of the fuelconsumption change for a certain aircraft type belonging toa certain airline to the fuel consumption change for a certainaircraft type of all airlines The objective is represented by
is the proportion of the fuel con-sumption change of type 119899 aircraft belonging to airline 119898 tothe fuel consumption change of all airlinesrsquo type 119899 aircraft
23 Integrative Assignment Model According to the aboveanalysis the real-time gate assignment model based on theprinciple of minimum delay cost for multiagent can beexpressed as follows
min 119891 = min 1198911 1198912 1198913 1198914 1198915 1198916 1198917 (12)
where (12) is the objective function Equation (13) meansevery flight is assigned to one and only one gate Equation(14) is the corresponding relationship between flight and gateEquation (15) means each slot combination is adopted byone and only one airline Equation (16) is the correspondingrelationship of slot combinations and airlines Equation (17)enforces that the type of the gate where the aircraft is assignedshould match the type of the aircraft Equation (18) stipulatesthat the idle time of the gate should be longer than buffertime for the sake of safety Equation (19) indicates that thebeginning of the idle time for any gate should be earlier thanthe arrival time of the flight which will be assigned to thegate and the end of the idle time should be later than thedeparture time of the flight Equation (20) refers to validityconstraint
Compared with the traditional staged model (the slotassignment is produced before the gate assignment) theadvantages of the proposed model are presented as follows(1) the sequential game helps to obtain better slot combi-nations for all the airlines and (2) the CDM mechanismcontributes to generating gate reassignment with less delaycost of multiagent due to the collaboration of airlines andairports
To solve the multiobjective optimization problem(MOOP) [17] the objectives are sorted in order ofpriority because all the objectives cannot be optimizedsimultaneously As the service concept is becoming moreand more important the waiting time and waiting time ofpassengers are given the highest priority The second highestpriority is the taxiing time of aircraft because the fuel costis the direct operation cost of airlines Following the taxiingtime of aircraft is the implicit idle cost of the airport Fuelequalization is the lowest priority because the slot exchangebetween airlines also contributes to the fairness principlewhen the fuel consumption is not equalized at the verybeginning
3 Solution Algorithm
Mixed set programming (MSP) [18ndash20] is a logic reasoningalgorithm based on first-order logic and set reasoning InMSP set operations quantifiers Boolean logic logic func-tions datetime reasoning and numerical constraints areintegrated in one system the reasoning on numeric typessuch as reals and integers is expanded to global reasoning
over mixed domains of set types such as Booleans andreferences Most importantly MSP makes the modeling andsolving for constraint satisfaction problems (CSP) realizableThe so-called set programming here is to systematicallyintegrating set reasoning and operational research algorithmestablishing a rigorous and complete set theoretical for-mulation based on set variables and solving the model byset reasoning algorithm instead of simply combining setnotations with set variables and set constraints The MSPadopted in this paper involves three major parts detailed asfollows
Part 1 (carry out optional slot assignment schemes) Con-sider the following
Step 1 Sort the flight banks that have not been finished attime 119905 by ascending order of scheduled arrival time expressedas 119861(t) = 119887
1 1198872 119887
119896 where 119861 denotes flight bank set and
119896 is the serial number of flight bank
Step 2 Let 119906119896equal the number of flights in flight bank 119896the
actual arrival time of flight bank 119896 Then define the closetime of flight bank 119896 corresponding to the maximum 119906
119896as
1199051 1199052 as the close time of the next flight bank when it has notbeen finished at time 119905 and can be finished at time 1199052 and120591 = min(1199051 1199052)
Step 3 Assign the flights of the flight bank corresponding tothe maximum 119906
119896to the time slots before 120591 by the order of
scheduled arrival time If the number of delayed flights is 119899then 119899 slot assignment plans will be generated
Step 4 Repeat Steps 1 2 and 3 for the rest of the flight banksuntil all the flights are reassignedwith slots It should be notedthat when Part 1 is implemented a number of slot assignmentschemes are produced
Part 2 (optimize the slot assignment schemes throughnon-zero-sum sequential game) Consider the following
Step 1 Input the information needed to implement non-zero-sum sequential game namely airlines delayed flights andprovided slots
Step 2 Implement non-zero-sum sequential game for all theairlines and calculate the delay cost of the airlines accordingto the optional slot assignment schemes The non-zero-sumsequential game between airlines is implemented accordingto the following
(1) In order to cut the delay cost of airline A exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline A with the 119895st (119895 = 1 2 119898) combinationof flight and slot of airline B and the exchange meetthe requirements of heuristic rules
(2) Repeat (1) till all the combinations of flights and slotsof airline A are exchanged with that of airline B
(3) In order to cut the delay cost of airline B exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline B with the 119895st (119895 = 1 2 119898) combination
6 Mathematical Problems in Engineering
of flight and slot of airline A and the exchange meetthe requirements of heuristic rules
(4) Repeat (3) till all the combinations of flights and slotsof airline B are exchanged with that of airline A
(5) Combine the above results and the optional slotexchange schemes are obtained
(6) Divide the optional slot exchange schemes into twocategories equilibrium schemes and nonequilibriumschemes In equilibrium schemes three kinds of situ-ations are included a win-win situation for airlines Aand B airline A wins and airline B loses and airlineA loses and airline B wins Nonequilibrium schemesrefers to the schemes that make both airlines losethereby should be deleted
(7) Calculate the value of 1198916according to the equilibrium
schemes
Step 3 Repeat Step 2 till the slot assignment scheme corre-sponding to the minimum delay cost for all the airlines isgenerated
Part 3 (carry out the optimal gate assignment scheme)Consider the following
Step 1 Read the preassignment results of all the flights andobtain the time periods of the available airport gates
Step 2 For the delayed flights without subsequent flightbanks keep their gate assignment as far as possible
Step 3 For the delayed flights with subsequent flight banksgo to Step 4
Step 4 Select out the flights which are not delayed butassigned to the gates affected by the delayed flights and theselection is made on flights which arrive within the span of50 minutes around the scheduled arrival time of the delayedflights according to [8] Combine these selected flights withthe flights in Step 3 a new flight set is produced Then gateassignment for the flights of this flight set can be carried outbased on the principle of minimum delay cost according tothe types of the flights and the available gates
Step 5 Combine the assignment results of Steps 2ndash4 thenthe real-time gate assignment set is obtained The real-timegate assignment set includes three parts gates assigned to thedelayed flights with subsequent flight banks gates assigned tothe delayed flights without subsequent flight banks and gatesassigned to the flights which are not delayed but affected bythe reassignment of the delayed flights
To design a solving strategy with preferable performanceheuristic rules are organically integrated in the algorithms Bythis method on one hand the constraints in the optimizationmodel can be strictly satisfied to ensure the feasibility of thesolution on the other hand the search process can be flexiblycontrolled The heuristic rules are given as follows
(1) when the scheduled serial number of the flight bankfor some delayed flights is 119896 then the actual serialnumber of the flight bank should be no less than 119896
(2) when airlines exchange their time slots with eachother the serial numbers of the corresponding flightbanks should be as similar as possible
The above three parts as well as the heuristic rules areimplanted into depth first search algorithm (DFS) [14] so thatthe integrative research for slots assignment and gate assign-ment can be performed As a result the slot assignment isoptimized to be consistent with the optimal gate assignmentwhich satisfies the multiobjective set previously
In traditional staged algorithm the cooperation of air-lines is not taken into account so the slots for delayed flightsof airline A can only be adjusted within airline A insteadof airline B and the slots for delayed flights of airline Bcan only be adjusted within airline B instead of airline AAs a result the gate assignment may cause losses for bothairlines
Compared with the traditional staged algorithm the pro-posed integrative algorithm generates the following advan-tages (1) the slots are exchangeable between the airlines sothe transferring cost of airlines can be decreased as muchas possible (2) the slot assignment and gate assignment areintegrated into theMSPwhich supports integrativemodelingand solving so CDM mechanism for the airlines and theairport can be well achieved (3) based on the softwarePOEM an integrative MSP method which supports non-zero-sum sequential game is designed so the gate assignmentcan be generated much more effectively
4 Experimental Results
For integrative modeling and solving the software POEM[14] is taken into application In order to support sequentialgame a game class is added into the program Four partsare included in the game class players (airlines) actions (slotexchanges) costs of the players (delay cost of the airlines afterthe slot exchanges) and total cost of the sequential gameAdditionally rule class equilibrium class nonequilibriumclass and result class are designed to run the program Theconstraints on the behavior of all the players (rules for slotexchange) are defined in rule class the equilibrium charac-teristics for sequential game is included in the equilibriumclass the nonequilibrium characteristics for sequential gameis included in the nonequilibrium class the schemes andthe corresponding delay cost for each airline are generatedand stored in the result class By applying those classes inPOEM the sequential game for the airlines in Part 2 can beperformed Parts 1 and 3 are achieved by the original functionof the software POEM
The environment where the experiment is carried out isrepresented as follows (1) CPU Intel(R) Core(TM) i7-3770CPU 340GHz (2) RAM 800GB (3) system type x86-based PC (4) system manufacturer Dell Inc (5) OS nameMicrosoftWindows 7 (6)OS version 617601 Service Pack 1Build 7601
Mathematical Problems in Engineering 7
Table 1 Flight information
Flightnumber
Arrivaltime
Departuretime
Aircrafttype Passenger Flight
bank Airline
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
where (12) is the objective function Equation (13) meansevery flight is assigned to one and only one gate Equation(14) is the corresponding relationship between flight and gateEquation (15) means each slot combination is adopted byone and only one airline Equation (16) is the correspondingrelationship of slot combinations and airlines Equation (17)enforces that the type of the gate where the aircraft is assignedshould match the type of the aircraft Equation (18) stipulatesthat the idle time of the gate should be longer than buffertime for the sake of safety Equation (19) indicates that thebeginning of the idle time for any gate should be earlier thanthe arrival time of the flight which will be assigned to thegate and the end of the idle time should be later than thedeparture time of the flight Equation (20) refers to validityconstraint
Compared with the traditional staged model (the slotassignment is produced before the gate assignment) theadvantages of the proposed model are presented as follows(1) the sequential game helps to obtain better slot combi-nations for all the airlines and (2) the CDM mechanismcontributes to generating gate reassignment with less delaycost of multiagent due to the collaboration of airlines andairports
To solve the multiobjective optimization problem(MOOP) [17] the objectives are sorted in order ofpriority because all the objectives cannot be optimizedsimultaneously As the service concept is becoming moreand more important the waiting time and waiting time ofpassengers are given the highest priority The second highestpriority is the taxiing time of aircraft because the fuel costis the direct operation cost of airlines Following the taxiingtime of aircraft is the implicit idle cost of the airport Fuelequalization is the lowest priority because the slot exchangebetween airlines also contributes to the fairness principlewhen the fuel consumption is not equalized at the verybeginning
3 Solution Algorithm
Mixed set programming (MSP) [18ndash20] is a logic reasoningalgorithm based on first-order logic and set reasoning InMSP set operations quantifiers Boolean logic logic func-tions datetime reasoning and numerical constraints areintegrated in one system the reasoning on numeric typessuch as reals and integers is expanded to global reasoning
over mixed domains of set types such as Booleans andreferences Most importantly MSP makes the modeling andsolving for constraint satisfaction problems (CSP) realizableThe so-called set programming here is to systematicallyintegrating set reasoning and operational research algorithmestablishing a rigorous and complete set theoretical for-mulation based on set variables and solving the model byset reasoning algorithm instead of simply combining setnotations with set variables and set constraints The MSPadopted in this paper involves three major parts detailed asfollows
Part 1 (carry out optional slot assignment schemes) Con-sider the following
Step 1 Sort the flight banks that have not been finished attime 119905 by ascending order of scheduled arrival time expressedas 119861(t) = 119887
1 1198872 119887
119896 where 119861 denotes flight bank set and
119896 is the serial number of flight bank
Step 2 Let 119906119896equal the number of flights in flight bank 119896the
actual arrival time of flight bank 119896 Then define the closetime of flight bank 119896 corresponding to the maximum 119906
119896as
1199051 1199052 as the close time of the next flight bank when it has notbeen finished at time 119905 and can be finished at time 1199052 and120591 = min(1199051 1199052)
Step 3 Assign the flights of the flight bank corresponding tothe maximum 119906
119896to the time slots before 120591 by the order of
scheduled arrival time If the number of delayed flights is 119899then 119899 slot assignment plans will be generated
Step 4 Repeat Steps 1 2 and 3 for the rest of the flight banksuntil all the flights are reassignedwith slots It should be notedthat when Part 1 is implemented a number of slot assignmentschemes are produced
Part 2 (optimize the slot assignment schemes throughnon-zero-sum sequential game) Consider the following
Step 1 Input the information needed to implement non-zero-sum sequential game namely airlines delayed flights andprovided slots
Step 2 Implement non-zero-sum sequential game for all theairlines and calculate the delay cost of the airlines accordingto the optional slot assignment schemes The non-zero-sumsequential game between airlines is implemented accordingto the following
(1) In order to cut the delay cost of airline A exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline A with the 119895st (119895 = 1 2 119898) combinationof flight and slot of airline B and the exchange meetthe requirements of heuristic rules
(2) Repeat (1) till all the combinations of flights and slotsof airline A are exchanged with that of airline B
(3) In order to cut the delay cost of airline B exchangethe 119894st (119894 = 1 2 119899) combination of flight and slotof airline B with the 119895st (119895 = 1 2 119898) combination
6 Mathematical Problems in Engineering
of flight and slot of airline A and the exchange meetthe requirements of heuristic rules
(4) Repeat (3) till all the combinations of flights and slotsof airline B are exchanged with that of airline A
(5) Combine the above results and the optional slotexchange schemes are obtained
(6) Divide the optional slot exchange schemes into twocategories equilibrium schemes and nonequilibriumschemes In equilibrium schemes three kinds of situ-ations are included a win-win situation for airlines Aand B airline A wins and airline B loses and airlineA loses and airline B wins Nonequilibrium schemesrefers to the schemes that make both airlines losethereby should be deleted
(7) Calculate the value of 1198916according to the equilibrium
schemes
Step 3 Repeat Step 2 till the slot assignment scheme corre-sponding to the minimum delay cost for all the airlines isgenerated
Part 3 (carry out the optimal gate assignment scheme)Consider the following
Step 1 Read the preassignment results of all the flights andobtain the time periods of the available airport gates
Step 2 For the delayed flights without subsequent flightbanks keep their gate assignment as far as possible
Step 3 For the delayed flights with subsequent flight banksgo to Step 4
Step 4 Select out the flights which are not delayed butassigned to the gates affected by the delayed flights and theselection is made on flights which arrive within the span of50 minutes around the scheduled arrival time of the delayedflights according to [8] Combine these selected flights withthe flights in Step 3 a new flight set is produced Then gateassignment for the flights of this flight set can be carried outbased on the principle of minimum delay cost according tothe types of the flights and the available gates
Step 5 Combine the assignment results of Steps 2ndash4 thenthe real-time gate assignment set is obtained The real-timegate assignment set includes three parts gates assigned to thedelayed flights with subsequent flight banks gates assigned tothe delayed flights without subsequent flight banks and gatesassigned to the flights which are not delayed but affected bythe reassignment of the delayed flights
To design a solving strategy with preferable performanceheuristic rules are organically integrated in the algorithms Bythis method on one hand the constraints in the optimizationmodel can be strictly satisfied to ensure the feasibility of thesolution on the other hand the search process can be flexiblycontrolled The heuristic rules are given as follows
(1) when the scheduled serial number of the flight bankfor some delayed flights is 119896 then the actual serialnumber of the flight bank should be no less than 119896
(2) when airlines exchange their time slots with eachother the serial numbers of the corresponding flightbanks should be as similar as possible
The above three parts as well as the heuristic rules areimplanted into depth first search algorithm (DFS) [14] so thatthe integrative research for slots assignment and gate assign-ment can be performed As a result the slot assignment isoptimized to be consistent with the optimal gate assignmentwhich satisfies the multiobjective set previously
In traditional staged algorithm the cooperation of air-lines is not taken into account so the slots for delayed flightsof airline A can only be adjusted within airline A insteadof airline B and the slots for delayed flights of airline Bcan only be adjusted within airline B instead of airline AAs a result the gate assignment may cause losses for bothairlines
Compared with the traditional staged algorithm the pro-posed integrative algorithm generates the following advan-tages (1) the slots are exchangeable between the airlines sothe transferring cost of airlines can be decreased as muchas possible (2) the slot assignment and gate assignment areintegrated into theMSPwhich supports integrativemodelingand solving so CDM mechanism for the airlines and theairport can be well achieved (3) based on the softwarePOEM an integrative MSP method which supports non-zero-sum sequential game is designed so the gate assignmentcan be generated much more effectively
4 Experimental Results
For integrative modeling and solving the software POEM[14] is taken into application In order to support sequentialgame a game class is added into the program Four partsare included in the game class players (airlines) actions (slotexchanges) costs of the players (delay cost of the airlines afterthe slot exchanges) and total cost of the sequential gameAdditionally rule class equilibrium class nonequilibriumclass and result class are designed to run the program Theconstraints on the behavior of all the players (rules for slotexchange) are defined in rule class the equilibrium charac-teristics for sequential game is included in the equilibriumclass the nonequilibrium characteristics for sequential gameis included in the nonequilibrium class the schemes andthe corresponding delay cost for each airline are generatedand stored in the result class By applying those classes inPOEM the sequential game for the airlines in Part 2 can beperformed Parts 1 and 3 are achieved by the original functionof the software POEM
The environment where the experiment is carried out isrepresented as follows (1) CPU Intel(R) Core(TM) i7-3770CPU 340GHz (2) RAM 800GB (3) system type x86-based PC (4) system manufacturer Dell Inc (5) OS nameMicrosoftWindows 7 (6)OS version 617601 Service Pack 1Build 7601
Mathematical Problems in Engineering 7
Table 1 Flight information
Flightnumber
Arrivaltime
Departuretime
Aircrafttype Passenger Flight
bank Airline
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
of flight and slot of airline A and the exchange meetthe requirements of heuristic rules
(4) Repeat (3) till all the combinations of flights and slotsof airline B are exchanged with that of airline A
(5) Combine the above results and the optional slotexchange schemes are obtained
(6) Divide the optional slot exchange schemes into twocategories equilibrium schemes and nonequilibriumschemes In equilibrium schemes three kinds of situ-ations are included a win-win situation for airlines Aand B airline A wins and airline B loses and airlineA loses and airline B wins Nonequilibrium schemesrefers to the schemes that make both airlines losethereby should be deleted
(7) Calculate the value of 1198916according to the equilibrium
schemes
Step 3 Repeat Step 2 till the slot assignment scheme corre-sponding to the minimum delay cost for all the airlines isgenerated
Part 3 (carry out the optimal gate assignment scheme)Consider the following
Step 1 Read the preassignment results of all the flights andobtain the time periods of the available airport gates
Step 2 For the delayed flights without subsequent flightbanks keep their gate assignment as far as possible
Step 3 For the delayed flights with subsequent flight banksgo to Step 4
Step 4 Select out the flights which are not delayed butassigned to the gates affected by the delayed flights and theselection is made on flights which arrive within the span of50 minutes around the scheduled arrival time of the delayedflights according to [8] Combine these selected flights withthe flights in Step 3 a new flight set is produced Then gateassignment for the flights of this flight set can be carried outbased on the principle of minimum delay cost according tothe types of the flights and the available gates
Step 5 Combine the assignment results of Steps 2ndash4 thenthe real-time gate assignment set is obtained The real-timegate assignment set includes three parts gates assigned to thedelayed flights with subsequent flight banks gates assigned tothe delayed flights without subsequent flight banks and gatesassigned to the flights which are not delayed but affected bythe reassignment of the delayed flights
To design a solving strategy with preferable performanceheuristic rules are organically integrated in the algorithms Bythis method on one hand the constraints in the optimizationmodel can be strictly satisfied to ensure the feasibility of thesolution on the other hand the search process can be flexiblycontrolled The heuristic rules are given as follows
(1) when the scheduled serial number of the flight bankfor some delayed flights is 119896 then the actual serialnumber of the flight bank should be no less than 119896
(2) when airlines exchange their time slots with eachother the serial numbers of the corresponding flightbanks should be as similar as possible
The above three parts as well as the heuristic rules areimplanted into depth first search algorithm (DFS) [14] so thatthe integrative research for slots assignment and gate assign-ment can be performed As a result the slot assignment isoptimized to be consistent with the optimal gate assignmentwhich satisfies the multiobjective set previously
In traditional staged algorithm the cooperation of air-lines is not taken into account so the slots for delayed flightsof airline A can only be adjusted within airline A insteadof airline B and the slots for delayed flights of airline Bcan only be adjusted within airline B instead of airline AAs a result the gate assignment may cause losses for bothairlines
Compared with the traditional staged algorithm the pro-posed integrative algorithm generates the following advan-tages (1) the slots are exchangeable between the airlines sothe transferring cost of airlines can be decreased as muchas possible (2) the slot assignment and gate assignment areintegrated into theMSPwhich supports integrativemodelingand solving so CDM mechanism for the airlines and theairport can be well achieved (3) based on the softwarePOEM an integrative MSP method which supports non-zero-sum sequential game is designed so the gate assignmentcan be generated much more effectively
4 Experimental Results
For integrative modeling and solving the software POEM[14] is taken into application In order to support sequentialgame a game class is added into the program Four partsare included in the game class players (airlines) actions (slotexchanges) costs of the players (delay cost of the airlines afterthe slot exchanges) and total cost of the sequential gameAdditionally rule class equilibrium class nonequilibriumclass and result class are designed to run the program Theconstraints on the behavior of all the players (rules for slotexchange) are defined in rule class the equilibrium charac-teristics for sequential game is included in the equilibriumclass the nonequilibrium characteristics for sequential gameis included in the nonequilibrium class the schemes andthe corresponding delay cost for each airline are generatedand stored in the result class By applying those classes inPOEM the sequential game for the airlines in Part 2 can beperformed Parts 1 and 3 are achieved by the original functionof the software POEM
The environment where the experiment is carried out isrepresented as follows (1) CPU Intel(R) Core(TM) i7-3770CPU 340GHz (2) RAM 800GB (3) system type x86-based PC (4) system manufacturer Dell Inc (5) OS nameMicrosoftWindows 7 (6)OS version 617601 Service Pack 1Build 7601
Mathematical Problems in Engineering 7
Table 1 Flight information
Flightnumber
Arrivaltime
Departuretime
Aircrafttype Passenger Flight
bank Airline
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
1 920 1020 E 300 1 C2 930 1030 E 300 1 S3 935 1025 C 100 1 S4 940 1035 D 200 2 E5 940 1035 D 200 2 S6 940 1040 E 300 2 E7 940 1030 C 100 2 C8 940 1035 D 200 2 C9 940 1040 E 300 2 E10 945 1040 D 200 1 S11 945 1035 C 100 1 E12 945 1045 E 300 1 C13 945 1035 D 200 1 S14 950 1040 C 100 1 E15 955 1055 E 300 2 C16 955 1050 D 200 2 E17 1000 1100 E 300 3 E18 1000 1055 D 200 3 E19 1000 1100 E 300 3 C20 1000 1055 D 200 3 C21 1005 1055 C 100 2 S22 1005 1105 E 300 2 C23 1010 1100 C 100 2 C24 1010 1110 D 200 2 C25 1015 1115 E 300 2 C26 1015 1105 D 300 2 C27 1025 1115 D 200 2 S28 1025 1115 C 100 2 C29 1035 1135 E 300 2 S30 1040 1130 C 100 3 E31 1040 1140 E 300 3 S32 1045 1135 C 100 4 E33 1045 1145 D 200 4 S34 1045 1135 C 100 4 S35 1045 1140 D 200 4 E36 1050 1140 C 100 3 S37 1050 1140 C 100 3 C38 1050 1150 E 300 3 E39 1050 1140 C 100 3 E40 1055 1150 D 200 4 C41 1100 1200 E 300 4 S42 1100 1200 E 300 4 E
41 A Case Study on Small-Scale Flight Delays Thedata listedin Table 1 is from the 42 operational flights arriving from920 to 1100 at some major airport involving three airlinesand three types of aircraft The airlines are Air China (CA)
China Eastern (MU) andChina Southern (CZ) symbolicallydenoted by C E and S respectively The types of the aircraftare small medium and large symbolically denoted by C Dand E respectively
In Table 1 number 17 and number 37 are special flightsmeaning the gates should remain the same when flightdelays occur and real-time assignment is needed The gateinformation is listed in Table 2 35 gates involved
The provided arrival times for flights number 13 number17 and number 37 are 1005 1030 and 1110 As number 17 andnumber 37 are special flights the adjustment should be madeon flights arriving within the interval [950 1050] accordingto [8] In other words a part of the flights in flight bank 1 andflight bank 2 will be influenced by the delayThe original gateassignment is listed in Table 3
By utilizing the software POEM for the integrativemodel-ing and solving flights number 13 number 17 and number 37are delayed to arrive at 1005 1030 and 1110 respectively andthe real-time gate assignment is produced with results listedin Table 4
411 Economic Efficiency According to the practical opera-tion of most airlines the fuel consumptions for large aircraftmedium aircraft and small aircraft are 46 kilograms perminute 28 kilograms per minute and 12 kg kilograms perminute respectively The idle costs of large gates mediumgates and small gates are 6 CNY per minute 4 CNY perminute and 2 CNY per minute respectively In additionthe fuel price is 7 CNY per kilogram the walking cost ofpassengers is 3 CNY per minute and the waiting cost oftransfer passengers is 1 CNY per minute
The total cost is 301986 CNY in the preassignment while305560 CNY in the reassignment so it is increased by 3574CNY a small growth of 118 After the reassignment theincreases of all kinds of costs are given in Figure 1
Fuel cost is increased from 68306 CNY to 69860 CNYwith a growth of 228 and the fuel consumption increasedby the flight delays is equalized for airlines to bear illustratedin Figure 2 Walking cost is increased from 153000 CNY to154200 CNY with a growth of 078 Idle cost is decreasedfrom 17980 CNY to 17940 CNY with a drop of 022 Sincethe gates are of three types the increased costs of each type areminimized at the same time results represented in Figure 3Waiting cost is increased from 62640 CNY to 63600 CNYwith a growth of 153 The reason why the waiting cost isincreased with just a minor growth of 153 is that most ofthe flight banks are not delayed The increases of all coststurn out to be quite small after the reassignment thereforethe real-time assignment is acceptable Besides the increasedwaiting cost accounts for 26 of the total increased costwhich testifies that taking into account the waiting cost oftransfer passengers in the cost control is very necessary
Figure 2 shows that the increased fuel consumption ofeach type of aircraft is basically equalized for each airline sothe fairness principle is well abided by
It is demonstrated in Figure 3 that the idle cost of smallgates remains the same For medium gates the idle cost isincreased by 80 CNY and for large gates the idle cost is
8 Mathematical Problems in Engineering
1800
1600
1400
1200
1000
800
600
400
200
0
minus200
Cos
tCN
Y1554
1200
960
minus40
26
4
32
41
Fuel costWalking cost
Idle costWaiting cost
Fuel costWalking cost
Idle costWaiting cost
Figure 1 Increases of all kinds of costs
40
30
20
10
0
minus10
minus20
minus30
minus40
minus50
Fuel
bur
nkg
23184
23
336 35 336
minus45minus432minus45
Large Medium SmallAircraft type
CAMU
CZ
Figure 2 Balanced fuel consumption
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
decreased by 120 CNY Although the idle cost of mediumgates grows the total cost of all gates turns out to bedecreased because the unit idle cost of large gates is morethan that of medium gates
Consequently in the circumstance of small-scale flightdelays the real-time gate assignment model proposed in thispaper is capable of achieving economic efficiency by adjustinga small number of gates
412 Robustness In fleet assignment [21 22] and fleet plan-ning [23 24] robustness has been widely applied but notin the research of gate assignment As a complex systemgate assignment should also be robust on one hand thegates influenced by the flight delays can be restored in shortterm on the other hand the disturbance brought by theadjustment of the gates can be restricted within a certain
Large Medium Small
12000
10000
8000
6000
4000
2000
0
Cos
tCN
Y
Airport gate type
1062010500
5860 5940
1500 1500
BeforeAfter
Figure 3 Idle costs for different types of gates
scale Theoretically the evaluation criteria of the robustnessfor gate assignment include the utilization of gates therecoverability of the affected gates and the service qualityfor passengers Gate assignment with good performanceis supposed to be generated with high utilization rate ofgates small-scale disturbance and convenient service forpassengers
Two major factors are considered to evaluate the robust-ness of the real-time assignment
(1) Maximum utilization rate of the gates involves userate and occupancy rate Use rate is equal to thenumber of engaged gates divided by the total numberof the gates occupancy rate is equal to the holdingtime of the gates divided by the available time of allthe available gates
Mathematical Problems in Engineering 9
Table 2 Gate information
Gatenumber
Gatetype
Walkingtimemin
Taxiingtimemin
Idle period
1 C 8 12 900sim1200
2 E 9 11 900sim1200
3 E 9 10 900sim1200
4 E 8 9 900sim1200
5 E 7 8 900sim1200
6 E 6 7 900sim1200
7 E 8 6 900sim1200
8 E 6 5 900sim1200
9 E 7 4 900sim1200
10 E 8 3 900sim1200
11 D 9 2 900sim1200
12 D 8 3 900sim1200
13 E 7 4 900sim1200
14 E 5 5 900sim1200
15 E 3 4 900sim1200
16 E 2 5 900sim1200
17 E 1 6 900sim1200
18 E 2 8 900sim1200
19 E 4 10 900sim1200
20 E 4 9 900sim1200
21 D 7 8 900sim1200
22 D 8 11 900sim1200
23 D 9 10 900sim1200
24 D 9 11 900sim1200
25 D 7 12 900sim1200
26 C 6 13 900sim1200
27 D 6 14 900sim1200
28 E 5 15 900sim1200
29 D 6 20 900sim1200
30 D 7 17 900sim1200
31 D 8 18 900sim1200
32 D 9 18 900sim1200
33 C 9 20 900sim1200
34 C 9 21 900sim1200
35 C 8 22 900sim1200
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
(2) Maximum service level for passengers means mini-mizing the growth of the walking time and waitingtime for passengers
Table 5 shows that the use rate is increased from 8857to 9143 with a growth of 286 the occupancy rate isincreased from 3675 to 3786 with a growth of 111walking time is increased from 51000 minutes to 51400
minutes with a growth of 078 waiting time is increasedfrom 62640 minutes to 63280 minutes with a growth of102 The flight delays indeed lower the service quality forpassengers by a small decrease however the utilization rateof the gates has risenwith a big growth Tomake a conclusionthe robustness of the real-time assignment scheme is welltestified
413 Timeliness and Collaboration The cost caused by flightdelays can be reduced as much as possible through CDMmechanism The following part of the case study is taken asan example to make a clear illustration on CDM
ATCC provides three slots (slot 1 [1005 1055] of flightbank 2 slot 2 [1030 1130] of flight bank 3 and slot 3[1110 1200] of flight bank 4) for the three delayed flights(A number 13 B number 37 and C number 17) to theairlines It should be noted that number 13 number 17 and
10 Mathematical Problems in Engineering
Table 5 Robustness evaluation
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
Utilization of gates Service for passengersUse rate () Occupancy rate () Walking time () Waiting time ()
Growth 286 111 078 102
number 37 are flights from three different airlines As theslots can be exchanged between any two airlines 6 differentslot assignment schemes will be produced theoreticallyaccordingly the corresponding gate assignment schemes willalso be different However if all the schemes are calculated inturn to locate the optimal one the computation process willbe very time-consuming and resource-wasting So non-zero-sum sequential game theory is utilized to exclude the infea-sible schemes and find out the cost-optimal slot assignmentscheme the process is illustrated in Figure 4 (applying (10))
Figure 4 shows that only two feasible schemes are carriedout from six optional choices meaning 23 of the actionsequences are excluded from the set of feasible solutionsthereby saving 23 of the computation time Suppose that therealization probability of each slot exchange plan is equalthen the loss caused by flight delays for each slot assignmentplan can be calculated by (11) In plan 1 slots 1 2 and 3 aredirectly assigned to flights A B and Crespectively causing atotal loss of 1600 CNY In plan 2 slots 1 2 and 3 are assignedto flights A C and B respectively causing a total loss of 960CNY
Traditionally the airlines will adopt plan 1 directly andthe slot assignment will be delivered to the airport withoutconsidering the related costs of the airport However thegate assignment scheme under this slot assignment is notthe optimal choice According to the calculation the gateassignment corresponding to plan 2 is better than the gateassignment under plan 1 Under plan 2 the operation costof both the airport and the airlines can be controlled moreeffectively meanwhile the satisfaction of the passengerscan be improved to a greater extent It can be concludedthat real-time gate assignment is produced based on theinformation of the delayed flights hence varying degreesof flight delays (slot reassignment) will lead to differentreal-time gate assignment schemes but only one is opti-mal when comparing the total costs of all the schemesIn turn the optimal gate assignment scheme can providea reference for airlines to reassign the delayed flights toupdated slots provided by ATCC In the whole process CDMbetween the airlines (and the airport) is effectively achievedthereby protecting the benefits of airlines airports andpassengers
42 A Case Study on Medium- to Large-Scale Flight DelaysIn the traditional staged method the slots are reassigned tothe flights without consideration ofminimizing the delay costbefore the gate assignment as a result the real-time gateassignment without CDM mechanism may not be optimalNevertheless in the integrative method presented in thispaper the slots can be interchanged between the airlines in
Table 6 Delay information before the slot exchange
the process of real-time gate assignment therefore the delaycost of the real-time gate assignment can be minimized asmuch as possible To present a significant comparison of theintegrative method and the traditional staged method large-scale flight delays are introduced into the case study Theinformation on delayed flights before slot exchange is listedin Table 6 and the information on delayed flights after slotexchange is listed in Table 7 The computation results aregiven in Table 8
As the equalization of fuel cost and the robustness ofthe gates almost remain the same the comparison is mainly
Mathematical Problems in Engineering 11
Table 8 Comparison of the results generated by traditional staged method and integrative method
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
Figure 4 Non-zero-sum sequential game for airlines
made on the costs and the computation time Since thenumber of the optional slot assignment schemes is 132 andthe number of the feasible schemes is 32 the computationtime of the integrative method should be around 14 of thecomputation time needed in the traditional staged methodTable 8 shows that the staged method is able to reassignthe gates appropriately but produces an increase of 354on the total cost and the computation process consumes1017 minutes However the integrative method just bringsan increase of 251 on the total cost and the computationprocess only takes 266 minutes Therefore the integrativemethod is superior to the traditional staged method inthe aspects of cost control and computation time control(2661017 is approximately equal to 14) not only in thecircumstance of the aforementioned small-scale flight delaysbut also under the situation of medium- to large-scale flightdelays
Figure 5 gives a visualized comparison of each cost itemfor the staged method and the integrative method The barstands for the difference of the increase produced by thosetwo methods and the greater the value is the more cost theintegrative method can cut Among these items the most
Fuel cost Walking cost Idle cost Waiting cost0
05
1
15
2
25
Items
Dec
reas
e gen
erat
ed b
y th
e
086
039061
239
Decrease of the total delay cost 103
inte
grat
ive m
etho
d (
)Figure 5 Decreases generated by the integrative method comparedwith the staged method
significant change caused by the integrative method is madein the waiting cost for transfer passengers which is cutby 239 and this is because (1) flight bank is taken intoaccount in the optimization (2) sequential game is appliedin the slot exchange between airlines and (3) gate assign-ment and slot assignment are implemented under the CDMmechanism
According to the comparison under the condition ofmedium to large scale flight delays the integrative methodproposed in this paper is much superior to the traditionalstaged method
43 Conclusions of the case Studies Based on the experimen-tal results of Sections 41 and 42 the conclusion are made asfollows
(1) As the approach proposed in this paper is practicalthe gate assignment problem is well solved mean-while all the constraints posed in the multiobjectivefunction are satisfied
(2) The approach is well applied in minimizing delay costunder the situations of both small-scale flight delaysand medium- to large-scale flight delays
(3) The interests of both airlines and airports are takeninto account which contributes to the application ofCDMmechanism
(4) The non-zero-sum sequential game excludes theinfeasible slot combinations so the computation timeof the approach is saved to a great extent In thefirst case (small-scale flight delays) as the feasibleslot assignment schemes account for 13 of the totalslot combinations the computation time is saved by
12 Mathematical Problems in Engineering
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
around 23 In the second case (medium- to large-flight delays) because the feasible slot assignmentschemes account for around 14 of the total slotcombinations the computation time is saved by about34 If the proportion of the feasible schemes issmaller the computation time can be saved muchmore significantly
5 Conclusions
This research focuses on the integrative approach withsequential game to the problem of real-time gate assign-ment The assignment model is formulated based on CDMmechanism and minimal delay cost principle for multiagentwhen flight delays occur meanwhile MSP combined withsequential game method is designed for calculationThe casestudies for both small-scale andmedium- to large-scale flightdelays verify the validity of the integrative method Firstly allkinds of costs areminimized better than the traditional stagedmethod especially the waiting cost of transfer passengersSecondly the increased fuel burn is basically balanced foreach airlineThirdly theCDMof the airlines and the airport iswell achieved Lastly more than half of the computation timeneeded in the traditional method is saved in the integrativemethod In summary due to the economic efficiency robust-ness collaboration and timeliness the integrative approachproposed in this paper is reasonable and feasible in restor-ing normal airport operation and guaranteeing regular airtransportation
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research is funded by the Key Program of NationalNatural Science Foundation of China (no 61232002 andno 60939003) China Postdoctoral Science Foundation (nos2012M521081 and 2013T60537) the Fundamental ResearchFunds for the Central Universities (no NS2014066) andPostdoctoral Science Foundation of Jiangsu Province (no1301107C)
References
[1] PlanningampDevelopmentDepartment of Civil AviationAdmin-istration of China 2011 Statistical Data on Civil Aviation ofChina China Civil Aviation Publishing House Beijing China2011
[2] A Bolat ldquoProcedures for providing robust gate assignments forarriving aircraftsrdquo European Journal of Operational Researchvol 120 no 1 pp 63ndash80 2000
[3] A Bolat ldquoModels and a genetic algorithm for static aircraft-gate assignment problemrdquo Journal of the Operational ResearchSociety vol 52 no 10 pp 1107ndash1120 2001
[4] J-J You C-M Ji and X Fu ldquoNew method for solving multi-objective problem based on genetic algorithmrdquo Journal ofHydraulic Engineering no 7 pp 64ndash69 2003
[5] A Lim and F Wang ldquoRobust airport gate assignmentrdquo inProceedings of the 17th IEEE International Conference on Toolswith Artificial Intelligence (ICTAI rsquo05) pp 74ndash81 November2005
[6] J-H Li J-F Zhu and Q Gao ldquoAirport gate assignment basedon Greedy Tabu Search algorithmrdquo Journal of TransportationSystems Engineering and Information Technology vol 11 no 4pp 173ndash179 2011
[7] Y Cheng ldquoNetwork-based simulation of aircraft at gates inairport terminalsrdquo Journal of Transportation Engineering vol124 no 2 pp 188ndash196 1998
[8] C Yu ldquoA knowledge-based airport gate assignment systemintegrated with mathematical programmingrdquo Computers andIndustrial Engineering vol 32 no 4 pp 837ndash852 1997
[9] Y Cheng ldquoA rule-based reactive model for the simulation ofaircraft on airport gatesrdquo Knowledge-Based Systems vol 10 no4 pp 225ndash236 1998
[10] W Li ldquoOptimized assignment of civil airport gaterdquo in Pro-ceedings of the International Conference on Intelligent SystemDesign and Engineering Application (ISDEA rsquo10) vol 2 pp 33ndash38 October 2010
[11] D X Wei and C Y Liu ldquoAirport gate reassignment problemrdquoJournal of Nanjing University of Aeronautics and Astronauticsvol 41 no 2 pp 257ndash261 2009
[12] W Li A Method to Construct Flight Bank for Hub AirportsNanjing University of Aeronautics and Astronautics College ofCivil Aviation Nanjing China 2010
[13] Q Gao J Yan and J-F Zhu ldquoAirlinesrsquo optimization decisionof slot allocation in CDMrdquo Journal of Transportation SystemsEngineering and Information Technology vol 11 no 5 pp 94ndash98 2011
[14] B Zhu J F Zhu and Q Gao ldquoConstraint programming modelof integrated recovery for aircraft and crewrdquo Journal of Trafficand Transportation Engineering vol 13 no 1 pp 77ndash83 2013
[15] D Gerardi and R B Myerson ldquoSequential equilibria inBayesian games with communicationrdquo Games and EconomicBehavior vol 60 no 1 pp 104ndash134 2007
[16] Z-J Li C-T Cheng F-X Huang and X Li ldquoSequential game-based resource allocation strategy in grid environmentrdquo Journalof Software vol 17 no 11 pp 2373ndash2383 2006
[17] H C Gomes F de Assis das Neves and M J F SouzaldquoMulti-objective metaheuristic algorithms for the resource-constrained project scheduling problem with precedence rela-tionsrdquo Computers amp Operations Research vol 44 pp 92ndash1042014
[18] J Yan T S Wu Q Gao and J F Zhu ldquoSlot switching modelof airlines under cooperative gamerdquo Journal of Traffic andTransportation Engineering vol 12 no 5 pp 85ndash90 2012
[19] J Y Zhou ldquoA note on mixed set programmingrdquo in Proceedingsof the IEEE The 7th International Symposium on OperationsResearch and Its Applications pp 131ndash140 2008
[20] J Zhou ldquoIntroduction to the constraint language NCLrdquo Journalof Logic Programming vol 45 no 1ndash3 pp 71ndash103 2000
[21] X H Zhu J F Zhu and Q Gao ldquoThe research on robust fleetassignment problem based on flight purityrdquo Forecasting vol 30no 1 pp 71ndash74 2011
[22] D Y Mou and Z X Zhang ldquoRobust fleet scheduling problembased on probability of flight delayrdquo Journal of Civil AviationUniversity of China vol 28 no 6 pp 35ndash39 2010
Mathematical Problems in Engineering 13
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
[23] Y Wang and H Sun ldquoHeuristic algorithm to incorporatingrobustness into airline fleet planningrdquo Systems EngineeringmdashTheory amp Practice vol 33 no 4 pp 963ndash970 2013
[24] H Sun P Zhang and Y Wang ldquoFleet planning approach basedon optimized fleet capacity allocation in airline networksrdquoJournal of Southwest Jiaotong University vol 45 no 1 pp 111ndash115 2010
