A Rapid Solving Method to Large Airline Disruption Problems Caused by Airports Closure

Wu, Zhengtian; GAO, QING; LI, BENCHI; DANG, CHUANGYIN; HU, FUYUAN

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Publication details:Wu, Z., GAO, QING., LI, BENCHI., DANG, CHUANGYIN., & HU, FUYUAN. (2017). A Rapid Solving Method toLarge Airline Disruption Problems Caused by Airports Closure. IEEE Access, 5, 26545-26555. [8107483].https://doi.org/10.1109/ACCESS.2017.2773534

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SPECIAL SECTION ON RECENT ADVANCES IN FAULT DIAGNOSIS AND FAULT-TOLERANTCONTROL OF AEROSPACE ENGINEERING SYSTEMS

Received October 18, 2017, accepted November 11, 2017, date of publication November 14, 2017,date of current version December 22, 2017.

Digital Object Identifier 10.1109/ACCESS.2017.2773534

A Rapid Solving Method to Large AirlineDisruption Problems Causedby Airports ClosureZHENGTIAN WU 1,4, QING GAO 3, BENCHI LI2, CHUANGYIN DANG2, AND FUYUAN HU11School of Electronic and Information Engineering, Suzhou University of Science and Technology, Suzhou 215000, China2Department of Systems Engineering and Engineering Management, City University of Hong Kong, Hong Kong3Institute for Automatic Control and Complex Systems (AKS), Faculty of Engineering, University of Duisburg-Essen, 47057 Duisburg, Germany4Virtual Reality Key Laboratory of Intelligent Interaction and Application Technology of Suzhou, Suzhou 215000, China

Zhengtian Wu (wzht8@mail.usts.edu.cn); Qing Gao (gaoqing@iim.ac.cn)

This work was supported in part by RGC of the Government of Hong Kong under Grant 11301014, in part by the National Nature ScienceFoundation of China under Grant 71771161, Grant 71471091, Grant 71271119, Grant 61672371, Grant 51375323, and Grant 61472267, inpart by the Natural Science fund for colleges and universities in Jiangsu Province under Grant 17KJD110008, in part by the Key Researchand Development Plan of Jiangsu Province under Grant BE2017663, in part by the Foundation of Key Laboratory in Science andTechnology Development Project of Suzhou under Grant SZS201609, in part by the Alexander von Humboldt Foundation of Germany, andin part by the Jiangsu Provincial Department of Housing and Urban-Rural Development under Grant 2017ZD253.

ABSTRACT Airline disruption is universal phenomenon that block the originally scheduled flights, and theslow recovery scheduling causes a lot of losses to both the airline companies and the passengers. However, itis very difficult to generate a new recovery scheduling in a reasonable time by traditional methods, especiallyfor large dimension airline disruption problem. To deal with this problem rapidly, the airline disruptionproblem will be reformulated as integer programmings and a distributed network, which is based on thefixed-point iterative method for integer programming, will be developed in this paper. In response to theairport closure, an airplane reschedule is constructed by these feasible flight lines. In the implementation ofdistributed network, a certain number of independent segments are obtained by dividing the solution space.As a feature of this fixed-point method, the number of partial feasible flight lines, which have to be calculatedfor finding an optimized airplane reschedule, is much fewer compared with the number needed by CPLEXCP Optimizer. This is the first distributed integer programming method to large airline disruption problemscaused by airports closure. In the numerical results, the proposed distributed approach shows promising,especially for solving the large dimension airline disruption problem.

INDEX TERMS Airline disruption management, irregular operation, airports closure, integer programming,distributed computation.

I. INTRODUCTIONGenerally, a disruption situation which is one of the majorchallenges that the airline industry currently faces stems froma local urgency event, for example an airport closure, a flightdelay, or s an aircraft maintenance has problem, although theairlines put considerable effort into scheduling. The disrup-tions of one flight always have an important impact on thefollowing flights, so lots of flight cancelations and delays willbe generated because of the former delay [1]. The Europeanairline has an in-depth investigation about the flights delayand finds that even bad weather condition causes only about3.6% flight delay, however, the spread of the delayed flightscreate up to 15.1% delay [2]. In other words, it is more impor-tant to formulate the airline disruption problem and solve

the formulation for rescheduling rapidly than to research thesources of disruptions.

Numerous literatures reports different novel approachesfor solving this tough problem, i.e. airline disruption prob-lems caused by airports closure. The paper [3] identified thedemerits of scheduling at Heathrow airport because of a shortairport closure. In this closure, system disruptions assessedconsist of flight rerouting, flight delays, flight cancellations,and flight diversions to alternate airports. Additionally,both the influence on fuel consumption and CO2 emis-sions were analyzed employing the Reorganized Air Traf-fic Control Mathematical Simulator Plus simulation modeland the Advanced Emission Model tool. Then, the relation-ship between severe weather events associated with climate

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change and airport operations disruption can be assessed bythe results and it helps provide base for finding methods todeal with airline disruption problems originated from air-ports closure. The research [4] analyzed the effectiveness oftourism-oriented airports to arrange departing passengers incase of an unforeseen airport closure due to weather events,terrorist attacks and industrial actions and conducted a casestudy on the most occupied tourism-oriented island airportin Europe, i.e. Palma de Mallorca Airport (PMI). As a flashpoint, the paper considered the full passenger itineraries byusing air passenger demand data which help to model anairline recovery process in event of an airport closure. Forthe sake of reducing the impact on stranded tourists, theanalysis on policy development is concentrated. The novelmethodological process include generating a baseline traveldataset, simulating all the 19 closure scenarios and sequen-tially relocating the influenced passengers. A framework [5],[6] which is based on a basic model established as a time-space network has been developed to help control sched-ule perturbations caused by the temporary airports closure.Those models contribute to developing many strategic net-work models for scheduling and are translated to either purenetwork flow problems or network flow problems whichare then computed employing the network simplex methodand a Lagrangian relaxation-based algorithm, respectively.A duty-based formulation [7] is proposed to deal with thecrew recovery problem. The approach can reduce the problemsize by resolving the disruption within every duty period andhence shortening recovery horizons. In the branch-and-price-based solution method, the master problem is expressed in aset covering formulation and the pricing problem is expressedin a resource constrained shortest path. The novel methodwas examined on numerous scenarios ranging from singleflight delay to airport closure. The conclusions indicate thecomputational cost is acceptable for operational environment.The paper [8] employs a Method of Inequality-based Multi-objective Genetic Algorithm (MMGA) in which a traditionalGenetic Algorithm (GA) and a multi-objective optimizationmethod are combined to generate an efficient multi-fleetaircraft routing algorithm for the schedule disruption of shortflights. The combination enables MMGA to address multi-ple objectives simultaneously and explore the best solution.Although Operations Research (OR) techniques expressedby a precise mathematical model is traditionally employedto deal with the airline schedule disruption managementproblem, it is difficult to define a precise model for airlineoperations including numerous factors. However, in the men-tioned paper, it is efficient to recover the perturbation relyingon Multi-objective Optimization Airline Disruption Manage-ment by GA and it is confirmed by the experimental results.Furthermore, the results in the manuscript demonstrate thatthe method can be extended as a real-time decision supportmethod for relatively practical complex airline operations.

A local search heuristic method [9], which has the abilityto find suitable revised flight schedules in a certain goodquality within 10 seconds, has been developed. However, the

examples which are used in the paper are generated ran-domly rather than practical airline schedules. An optimizationmodel [10] contained by both a set partitioning problem and aroute generation procedure has been presented to rescheduleflight routes by minimizing an objective function that is aboutthe cancelation costs. A mixed integer multi-commodity flowmodel [11] with side constraints has proposed. Under theDantzig-Wolfe decomposition thismodel becomes a set pack-ing model. There are two instances tested in this paper, andthe larger instance costs much more time than the smallerone. A two-steps framework [12] has been developed toconstruct a real-time schedule. An introduction to airline dis-ruption management and experiences from project Descarteshas been given [13]. A preliminary idea of changing airlinedisruption problem into the integer programming has beenintroduced in the paper [14]. Further research about thisidea has been developed by the paper [15], [16] and [17].To obtain more theoretical descriptions on airline disruptionscheduling, one can refer to the review paper [18]–[22] andbooks [23], [24], [25], [26], [27], [28]. For the potentialmethods which can be used to airline disruptionmanagement,one can consult to the papers [29]–[35].

As is known, lots of losses have made due to the slowrecovery scheduling. However, in the process of recoveryscheduling it is very hard to find all the feasible flight linesin a reasonable time, especially for the large airline dis-ruption problem. Only partial feasible flight lines are nec-essary. In this research, the airline disruption problem isdivided into two subproblems: one is feasible flight linesgeneration problem and the other is airplanes rescheduleproblem. After dividing the solution space into several seg-ments, a distributed network which is based on Dang and Ye’salgorithm [36] is developed to obtain feasible flight lines.When the large airplanes reschedule problem is solved bypartial feasible flight lines, the comparisons have been made.According to the literature reviews, Lius method [8] has thebest performance. Therefore, we only compare our methodwith Lius method, as well as our former Dang’s method.

In this paper, the airline disruption problems have beenformulated to the integer programming. Therefore, the com-putational complexity of this approach is the same with theinteger programming. As is known, the integer problem isNP-hardand it is very difficult to solve by traditional methods,especially for large dimension problem. Theoretically, Dangand Ye’s methods are very suitable for solving this kind ofproblems. What is more, a distributed implementation ofDang and Ye’s methods have also been developed to cal-culate the obtained integer formulation. The flight scheduleinstances used in this paper comes from the paper [8]. Com-parisons between the solutions, which are obtained by partialfeasible flight lines which are solved by Dang and Ye’s algo-rithm and those obtained by Liu’s method [8] for airplanesreschedule problems, will be provided in this manuscript.After more relatively partial feasible flight lines by Dang andYe’s algorithm are provided, final solutions which have lesstotal delay time than the solutions generated by Liu’s method

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can be shown. Therefore, one can see that the Dang and Ye’salgorithm obviously outperform the other methods.

The rest organization of this paper can be described asfollows. Section II gives the introduction of the mathematicalformula of this airline disruption problem and dividing theproblem into two kinds of subproblems. As a new methodfor integer programming, Dang and Ye’s iterative method isproposed in Section III. Section IV develops a distributedcomputation implementation to find feasible flight lines.In SectionV, performance comparisons and some discussionsare given. Section VI offers a conclusion of this this research.

II. PROBLEM FORMULATIONIn this paper, one can consider a situation that a thunder-storm happens, and it causes that one or more airports aretemporarily closed. All flights, which are served by theclosed airports, are suspended until the airports reopen again.A minimum impact from the thunderstorm can be obtainedafter reschedule of the airline by reassigning of all the flightlegs to all the airplanes. In order to obtain an airplanesreschedule which can minimize deviation from the originalschedule and the loss of disruption, the procedure of solvingthis airline disruption problem is divided into two subprob-lems: one is feasible flight lines generation problem and theother is airplanes reschedule problem.

The notations used in the formulation can be introduced asfollows.

Indices

i, j, k the indices of flightt the index station

Sets

F the set of all the flightsS the set of all the stations

Parameterstdi duration time of a flight itti turnaround time for a flight iTj total time from the departure of flight j to the

departure curfew time of a stationdit = 1, if flight i leaves the station t; = 0, if notait = 1, if flight i reaches the station t; = 0, if notestijk = 1, if flight k’s destination station and flight j’s

original station are the same station, and flight k’soriginal station and flight j’s destination station arethe same station; = −1, if flight i’s destination stationand flight k’s original station are the same station,or flight i’s destination station and flightk’s destination station are the same station,or flight i’s original station and flight k’soriginal station are the same station, or flight i’soriginal station and flight k’s destination station arethe same station; = 0, otherwise

enjt = −1, if flight j’s destination station is the station t ,or flight j’s original station is the station t; = 0,otherwise

Variables

xi = 1 if the i which is flight leg is contained amonga feasible flight line; = 0, if not

st = 1 if the feasible flight line’s starting pointis the station t; = 0, if not

kt = 1 if the feasible flight line’s destination point is thestation t; = 0, if not

The mathematical formulation of feasible flight line gener-ation

∃xi, st , kt ∈ {0, 1}, ∀i ∈ F, ∀t ∈ S, (1a)

subject to

(limitation on flight time)Card(F)−1∑

i=j

xi(tdi + tti) ≤ Tj, ∀j ∈ F, (1b)

(node conservation)∑i∈F

xiait −∑i∈F

xidit + st − kt = 0, ∀t ∈ S, (1c)

(flow from source node)∑i∈F

xidit ≥ st , ∀t ∈ S, (1d)

(flow to sink node)∑i∈F

xiait ≥ kt , ∀t ∈ S, (1e)

(source node cover)∑t∈S

st = 1, (1f)

(sink node cover)∑t∈S

kt = 1, (1g)

(subroute elimination)∑i∈F

xiestijk +∑t∈S

stenjt +∑t∈S

ktenjt ≥ 1, (1h)

∀j ∈ F, ∀k ∈ (j+ 1, · · · ,Card(F)− 1). (1i)

More details of the formulation and the feasible transfor-mation pseudo code can be found in the previous work [16].However, each of these two paper focuses on different dis-ruption. This manuscript focuses on disruption problemscaused by airports closure, while the other paper paysmore attention on disruption problem caused by ground-ings. What is more, the new method has obvious advantageon large dimension disruption problems caused by airportsclosure.

The other subproblem is formulated as a resource assign-ment problem which is very hard to solve as introduced in theresearch [37], [38]. However, after a certain number of partialfeasible flight lines of the formulation (1) obtained using thedistributed fixed-point method in Section III, this airplanesrescheduled problem could be worked out easily.

The formulation for the airline rescheduled subproblemcan be introduced as follows. To simplify the problem, theassumption is adopted that all airplanes are derived from thesame fleet.

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The notations, which are used in the formulation, can bedescribed as follows.

Indices

i the indices of flightj the indices of routet the index of station

Sets

F the set of all flightsS the set of all stationsP the set of all feasible flight lines

Parameters

xij = 1, if i, which stands for a flight, is contained in thefeasible flight line j; = 0, if not

stj = 1, if the feasible flight line j’s starting point is thestation t; = 0, if not

ktj = 1, if the feasible flight line j’s destination point isthe station t; = 0, if not

dtj the feasible flight line j’s delay timeht number of airplane needed to leave the starting

point tgt number of airplane needed to arrive at the sink

point tTN the sum number of all airplanes

Variables

yj = 1, if the feasible flight line j is rescheduled to anairplane; = 0, if not

zi = 1 if the flight i is canceled; = 0, if not

The mathematical formulation of airplane reschedule

minimize∑j∈P

dtjyj (2a)

subject to(flight cover)∑

j∈P

xijyj + zi = 1, ∀i ∈ F, (2b)

(airplanes balance of source stations)∑j∈P

stjyj = ht , ∀t ∈ S, (2c)

(airplanes balance of sink stations)∑j∈P

ktjyj = gt , ∀t ∈ S, (2d)

(total available airplanes)∑j∈P

yj = TN , (2e)

(binary assignment)yj ∈ {0, 1}, ∀j ∈ P, (2f)

(binary assignment)zi ∈ {0, 1}, ∀i ∈ F . (2g)

For the details of the explanation for this formulation, onecan also reference the paper [16], while all the variables are

on the conditions of airports closure. Traditionally, CPLEXOptimizers Concert Technology is applied to calculate theformulation (2) of this kind, whose solutions are combinedwith feasible flight lines. Sometimes more than one optimumsolutions exist. However, the number of preliminary flightlines are different among the optima. One can see that themore preliminary flight lines contained in a combination ofthe feasible flight lines, the better. This is because morepreliminary flight lines in the combination imply that thereschedule has less deviation from the original schedule.In this paper a distributed fixed-point iterative integer pro-gramming is developed to obtain the combination whichcontains the maximum number of preliminary flight lines.

III. METHODOLOGYAs known, the CPLEX can solve both the formulation (1)and the formulation (2). However, for the airline disrup-tion problem with large dimension it is impossible to findall the formulation (1)’s feasible flight lines using CPLEXCP Optimizer tool. Let’s explain this through an instance.In Section V, an instance includes 140 legs of flight. If eachflight line’s length is limited between 10 and 13 legs offlight in the original arrangement, then the upper limit num-ber of feasible flight lines of the formulation (1) will beC10140 + C

11140 + C

12140 + C

13140 = 7.98× 1017. The upper time-

limit of finding all the feasible flight lines of the formula-tion (1) for this instance through CPLEX CP Optimizer toolcan be obtained by a simple calculation. Generally, it costsabout 21762ms to obtain 100000 feasible flight lines usingthe CPLEX CP Optimizer tool, and the average computingtime to obtain a feasible flight line is more than 0.21762msbecause more time will be cost to obtain a feasible flightline when the search tree goes deeper. In order to findall the feasible flight lines which is about 7.98 × 1017, itwill cost more than 173623605779542.113s which is equalabout 5505568.423years in the worst case using CPLEX CPOptimizer.

To solve the above issue, a distributed implementationof Dang and Ye’s algorithm is developed. This algorithmhas a good performance to find feasible flight lines throughdividing the solution space of the formulation (1) into acertain number of segments. This distributed method workswell even when the dimension of the problem is very large.This Dang and Ye’s algorithm, which has been developedin the research [36], is the improvement of Dang’s iterativemethod [39]. Numerical calculation has show that the for-mer method requires fewer number of linear programmingand costs less time to find a integer solution than the latermethod. Firstly, a simple example will be given to illustrateDang and Ye’s algorithm.

A polytope P will be defined to explain this method. LetP = {x ∈ R2

|Ax ≤ b} with

A =

−17 26 5−3 −3

and b =

−847

(3)

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and let a lattice D(P) = {x ∈ Z2|x l ≤ x ≤ xu} which

can be described in Figure 1, the problem intend to find ax(x ∈ P

⋂D(P)) or prove that there is no such point exists.

The idea to calculate this problem is to state an increasing-mapping. This mapping is defined form the lattice into itself.

FIGURE 1. Explanation of polytope P and lattice.

All the integer solutions, which are outside the polytope P,are mapped to the first integer point in this polytope. This firstinteger point is not bigger than them which are in the lexico-graphical order or x l . All integer solutions, which are insidethe P, are fixed points by this procedure. After a certain num-ber of iterations, this procedure either proves no such integerpoint exists or obtains an integer solution in P for any giveninitial integer point. After a simple improvement, all the inte-ger solutions in the polytope could be found sequentially [36].This method has been widely used in many applications, suchas market split problem [40], Nash equilibria problem [41]and so on. The research [36], [39] has given the details ofthis method. It is stated that the more preliminary flight linescontained in a combination of the feasible flight lines, thebetter. One feature of the Dang and Ye’s algorithm for integerprogramming is that the integer solution near the startingpoint will be found firstly. This feature could be explainedby the Figure 1. Under this feature, more preliminary flightlines contained in a combination of the feasible flight lineswill be obtained.

At the beginning of this distributed implementation, thesolution space is split up to a certain number of parts, and thecomputing integer points is conducted simultaneously in eachpart by Dang and Ye’s algorithm. Since the divided parts areself-governed to each other, the computing procedure for inte-ger points in each segment could be executed simultaneouslyin each computer and processor. Two division methods havebeen developed to solve airline disruption problems [16].

FIGURE 2. Nearly average division method.

One division method, which have a good performance, isdividing the solution space into a certain number of segmentswhich contain nearly equal integer points inD(P). For a largedimension problem, if a sufficient number of computers andprocessors are provided, the solution space could be dividedinto subsegments small enough, each of subsegments canbe solved in a reasonable time. If the computing of somesubsegment lasts too long or there is no more processors canbe given, another division method has been developed. Thiscutting method is designed especially for airline disruptionproblems. In this method, the preliminary flight lines aretaken as the initial points, subsegments are divided by 2 boundpoints which are among two consecutive initial points by thelexicographical order. Then the cluster is stated around eachinitial point in each segment for computing, and a parameteris applied to control the feasible flight lines’ number. Thedetails of the two division methods have been presented inthe previous work [16].

IV. IMPLEMENTATION OF THE DISTRIBUTED NETWORKEach subsegment is divided through the split method inthe paper [16] is self-governed to each other. Therefore, thecomputing in each part could be done synchronously from thestart point to the end point by each independently computingprocessor. In order to take greater advantage of hardware, thecommunication network among the individual computers isdeveloped by Message Passing Interface (MPI), the a paral-lel computing among the processors in a same computer isconducted by OpenMP. The flow diagram of the distributedcomputing process can be described in Figure 3.

The pseudo code of developing the distributed computingnetwork by OpenMP and MPI can be described as Figure 4.One of a function of MPI called ‘‘MPI_Comm_rank’’ stands

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FIGURE 3. Distributed network process.

for the rank of a individual computer which begins from0 and is kept as a variable computerid . A independent filewill be created as configuration file which contains all thecomputer’s names. The master computer which takes chargeof controlling the procedure of all computing is defined inthe first part of this configuration file, and the master com-puter’s computerid is equal to 0. The other slave ones areassigned a certain number from one to the sum number ofslave computers. This configuration file has to be copied intoeach participating computer, as well as the execution file ofthe distributed computing programming. All execution filesin every computer have to be executed synchronously usingMPI order. At the beginning of execution procedure, the num-ber of processors in each computer feedbacks to a variableSystemInfo.dwNumberOfProcessors. Firstly the number ofeach slave’s processors is sent from the individual slave tothe master computer, then the sum number of processors ofall the computers can be obtained. In the master computer, thesolution space is divided by one of the two division methodsas mentioned in the previous work [16] on account of thesum number of processors. After then, each part’s beginningpoint xsi and the last point xei are sent to the correspondingslave computer from master computer. If a slave computerhas more than one processors, there is a parallel computationwill be carried out by OpenMP through a compiler directive‘‘#pragma omp parallel’’ which will split up the loop itera-tions. If the number of parts assigned to a computer is equal

to the number the computer’s processors, computing on allthe parts could be done synchronously too. If the number ofparts assigned to a computer is more than the number of thiscomputer’s processors, computing on the surplus parts couldbegin at the moment of any of its processors completes itswork. As soon as the computing on all the parts in a individualcomputers conpletes, the feasible flight lines obtained will betransmited to the master computer in order to calculate theformulation (2).

In order to make more processors can work together toenhance the efficiency of computing, the number of subseg-ments is matching to the number of processors,. On one handthe efficiency of the proposed distributed computation is high.On the other hand, the feasible flight lines found using thedivision approach in the paper [16] outperform those perfor-mance found using CPLEX CP Optimizer when the feasibleflight lines are used for solving the formulation (2). In addi-tion to this, solutions using feasible flight routes calculated byDang and Ye’s algorithm through the second division methodin paper [16] have much less total delay time. Numericalresult comparisons are given in Section V.

V. NUMERICAL RESULTSIn this paper, there are two flight schedule instances which arefrom Taiwanese Domestic Airlines. The instance are MD-90fleet and DH-8 fleet which are firstly used in the paper [8].The constraints and assumptions used in paper [8] are also

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FIGURE 4. Distributed computing network pseudo code.

adopted in this paper. MD-90 fleet makes up of 7 airplanes inthe same fleet that serve 70 flights between 6 cities a day, andit works in a hub-and-spoke system. DH-8 fleet consists of12 airplanes in the same fleet which serve 140 flights between11 cities a day, and it is operated in a combined point-to-point and hub-and-spoke system. Details of the two schedule

instances can be obtained in the paper [8]. This paper intendsto get a feasible solution which has less total delay time whenthe stations TSA and TXG are restarted after a 60 minutestemporary closure happens.

In this paper, a distributed computing network is built byMPI to find the feasible flight lines formulation (1). Thisnetwork hardware makes up of 3 different individual com-puters. One computer has 16 threads and the other two have2 threads. The C++ is used for coding all programs, andthe CPLEX Concert Technology with 12.6.1 version is usedfor solving the linear programming which has to be solvedin Dang and Ye’s algorithm. Firstly, the solution space ofthe formulation (1) will be divided into several parts throughthe initial seeds cluster division method, and the feasibleflight lines of the formulation (1) are obtained by CPLEXCP Optimizer tool and [36]’s algorithm separately in everyparts. All the feasible flight lines are applied to generate asolution of the formulation (2) with the situation that stationsTSA and TXG are temporarily closed for one hour becauseof a thunderstorm. CPLEXOptimizers’s Concert Technologyis applied to solve the formulation (2). After the optimalvalues of the formulation (2) has been calculated through thepartial feasible flight lines found by CPLEX CP Optimizertool and Dang’s algorithm respectively, a comparison is pre-sented. After the optimal values of the formulation (2) hasbeen calculated through the partial feasible flight lines foundby Dang and Ye’s algorithm and Liu’s method respectively,another comparison will be given too.

In the paper [14], Figure 5(a) shows the capabilities of thepartial feasible flight lines of the formulation (1) obtainedthrough CPLEX CP Optimizer and Dang’s algorithm to solvethe formulation (2). When the cluster size is increasing, thetotal delay time of a solution solving by Dang’s algorithm isdecreasing in MD-90 fleet. In other words, the more partialfeasible flight lines obtained by Dang’s algorithm, the lesstotal delay time will be got. However, it is not the same forthe method of CPLEX CP Optimizer. For the DH-8 fleet, thesimilar result as the MD-90 could be obtained. For explana-tion of this result, one can consult the Figure 5(b).

Because Dang and Ye’s algorithm is the improvement ofDang’s iterative method, the Dang and Ye’s algorithm cer-tainly performance better than the CPLEX CP Optimizer.Both of the two methods could be the core for integer pro-gramming in the distributed computing network. The numer-ical comparison of these twomethods will be given in Table 3.

The solutions, which calculated by the partial feasibleflight lines obtained by Dang and Ye’s algorithm in somecluster sizes and a pareto optimal set found in [8], is presentedin Table 1 and Table 2. The solutions of MD-90 fleet areshown in the Table 1 while the solutions of DH-8 fleet arepresented in theTable 2. A multi-objective genetic approachhas been developed in the research [8] to the airline disrup-tion problem. In the paper [8], each population stands foran airline schedule which is consisted of the flight line ofeach airplane and the size of population is 100. The paretooptimal sets in Liu’s method is got after 50000 generations

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TABLE 1. Comparison between Dang and Ye’s algorithm with Liu’s method for MD-90 fleet.

FIGURE 5. Comparison between Dang’s algorithm and CPLEX CPOptimizer. (a) MD-90 fleet. (b) DH-8 fleet.

of computing. Billions of feasible flight lines are solved in thecomputing optimum through this approach, at the same timefar fewer feasible flight lines obtained through Dang and Ye’salgorithm are required to get the same or even better solutionsthan those developed by Liu’s method [8].

The solution of MD-90 fleet instance can be see in Table 1,the partial feasible flight lines obtained by Dang and Ye’salgorithm with cluster size 20 can solve one of the paretooptimal sets found by Liu’s method. If the cluster size add upto 6000, the total delay time is 435 through the partial feasibleflight lines obtained through Dang and Ye’s algorithm. Thisresult is better than the one obtained by Liu’s method.

The total computational time of calculating the airlinedisruption problem should contain both the computationaltime of solving the formulation (1) and the computationaltime of solving the fomulation (2). Considering the originalschedules are usually determined some days before the exe-cution of schedules [18]. The partial feasible flight lines forformulation (1) could be began to obtained once the originalschedules are ready. There is no need to wait till a disruptionhappened that the feasible flight routes are started to generate.Once a disruption occurred, partial feasible flight lines getready to obtain a solution of formulation (2). So computingtime of the formulation (1) can be ignored in the total com-puting time of calculating these airline disruption problems.Only the computing time of the formulation (2) is counted.Therefore, for the computational time, the duration of thismethod in Table 1 and Table 2 is the computing time of theformulation (2), and it is much less than the duration of Liu’smethod which can obtain the solutions in minutes. It is veryimportant for practical application. If the dimension of theproblem adds and the number of slave computers increases,the advantages are more obvious.

Since Dang and Ye’s algorithm is the improvement ofDang’s iterative method, a comparison between the twomethod when they are used to solve the problem (1) is givenin Table 3. Partial feasible flight lines obtained using the twoapproaches are all the same. However, there are differences inthe cost of the computing time and the number of linear pro-gramming. The number of the linear programming which areused in Dang and Ye’s approach to find a feasible flight lineis about 1/6 of that used in Dang’s approach. The computingtime of Dang and Ye’s approach to obtain a feasible flight lineis about 1/4 of the computing time of Dang’s approach.

The initial points of Dang and Ye’s algorithm for genera-tion is the starting point of flight lines. So the feasible flightlines in the lexicographical order could be got sequentially byDang andYe’s algorithm. Because the variables xi’s in the for-mulation (1) stand for the legs of flight that are rescheduled toan increasing order of leaving time, the solutions found fromthe preliminary flight line earlier are not more deflected fromthe preliminary flight line than the ones found later. Besides,there is no feasible flight line again among any two feasibleflight lines found continuously in the lexicographical order.

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TABLE 2. Comparison between Dang and Ye’s algorithm with Liu’s method for DH-8 fleet.

TABLE 3. Comparison between Dang and Dang-Ye.

Changes of new feasible flight lines are alterations of the legsof flight in the last part of the primary flight lines at the first ofcomputing byDang andYe’s algorithm. The variations of legsof flight spread from the last part of the primary flight linesto the front part of flight lines with computing goes ahead.Thus, the feasible flight lines found later are more deflectedfrom the preliminary flight lines. From comparisons one cansee that most of the solutions which have less total delaytime of the formulation (2) are obtained through the feasibleflight lines which are around each preliminary flight line inthe lexicographical order. Solutions which has much less totaldelay time could be found through the feasible flight lineswhich are relatively far from each preliminary flight line inthe lexicographical order. However, these solutions are moredeflected from the original airplane schedule.With a properlycluster size, a solution of the formulation (2) can be obtainedwhich is very close to the optimum and less deflected fromthe preliminary flight lines by the partial feasible flight lineswhich are got by Dang and Ye’s algorithm together withthe initial seeds cluster division. If it is necessary, a set ofsolutions, which have different total delay time and otherproperties, could be obtained through the partial feasibleflight lines obtained using Dang andYe’s algorithm in variouscluster size for the users to make decision.

VI. CONCLUSIONSIn this paper, it has developed a procedure for calculatingairline disruption problem. In this procedure, the problem hasbeen divided into two subproblems: one is solving feasibleflight lines generation subproblem and the other is airplanesreschedule subproblem. The main focus of this paper is thefeasible flight lines generation subproblem. If the dimensionof airline disruption problem is large, in a reasonable time itis impossible to calculate all the feasible flight lines. There-fore, the solution space of this subproblem has been dividedinto a certain number of segments which can be calculatedby distributed Dang and Ye’s algorithm simultaneously andindependently. From the computing experiments of two flightschedules given, it is shown that Dang and Ye’s algorithm

obtains much fewer partial feasible flight lines for formula-tion (2) to get a better solution than CPLEX CP Optimizer.Besides, solutions which have less total delay time than thosesolved by Liu’s method could be shown by relatively morefeasible flight lines provided. The solution of the formula-tion (2) is more and more close to the optimum of the formu-lation (2) if the cluster size is increasing. However, the flightlines of the solution are more deflected from the preliminaryflight lines. In a reasonable time, for large dimension airlinedisruption problems which are impossible to calculate all thefeasible solutions, a properly chosen cluster size could obtaina solution of the formulation (2) which is at least very closeto the optimum and less deflected from the preliminary flightlines by partial feasible flight lines which are got by Dang andYe’ method combined with the initial seeds cluster division.The numerical results have shown that this distributedmethodproposed is promising.

In the future work, disruptions caused by both airportclosures and airplane groundings will be considered together.One can believe that Dang and Ye’s algorithm will have agood performance in practice.

COMPETING INTERESTSThe authors declare that they have no competing interests.

AUTHOR’S CONTRIBUTIONSAll authors read and approved the final manuscript.

ACKNOWLEDGEMENTSThe authors are very grateful to the editor and reviewers fortheir valuable comments and suggestions. Zhengtian Wu andQing Gao contributed equally to this work.

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ZHENGTIAN WU received the B.Sc. degreein mechanical manufacturing and automationfrom the Hefei University of Technology, China,in 2008, and double Ph.D. degrees in opera-tions research from the University of Scienceand Technology of China and the City Univer-sity of Hong Kong in 2014. He is currently aLecturer with the Suzhou University of Scienceand Technology, Suzhou, China. His researchinterests include Nash equilibrium, mixed integer

programming, approximation algorithm, and distributed computation.

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QING GAO received the B.Sc. degree from theUniversity of Science and Technology of China,China, in 2008, and double Ph.D. degrees fromthe University of Science and Technology ofChina and the City University of Hong Kong in2014. He currently holds a post-doctoral posi-tion with the Institute for Automatic Controland Complex Systems (AKS), Faculty of Engi-neering, University of Duisburg-Essen, Duisburg,Germany. He has demonstrated outstanding

research track record in the area of system engineering. In the last fiveyears as a System Engineering Researcher in various roles, he have had over18 journal articles published in international high quality journals, such asthe IEEE TRANSACTIONS and AUTOMATICA, which all have high impact factorsand are highly ranked in the area of control theory and applications. So farhis publications have been cited by over 400 times by scholars from over tencountries, such as the U.K., the USA, China, Germany, and France, and hish-index on Google scholar is 11.

BENCHI LI received the B.Sc. degree in mechani-cal manufacturing and automation from the Uni-versity of Science and Technology of China,China, in 2007, and double Ph.D. degrees inoperations research from the University of Sci-ence and Technology of China and City Univer-sity of Hong Kong in 2013. He is currently aResearcher with China National Petroleum, China.His research interests include approximation algo-rithm and distributed computation.

CHUANGYIN DANG received the B.Sc. degree incomputational mathematics from Shanxi Univer-sity, China, in 1983, the M.Sc. degree in appliedmathematics from Xidian University, China,in 1986, and the Ph.D. degree (cum laude) inoperations research/economics from Tilburg Uni-versity, The Netherlands, in 1991. He is currently aProfessor with the Department of Systems Engi-neering and Engineering Management, City Uni-versity of Hong Kong, China. He has published

over 100 papers in journals and his research interests include computa-tional optimization, logistics, supply chain management, and modeling ineconomics.

FUYUAN HU is pursuing the Ph.D. degree withNorthwestern Polytechnical University, and is aVisiting Ph.D. Student with the City University ofHong Kong. He was a Post-Doctoral Researcherwith Vrije Universiteit Brussel, Belgium. He iscurrently an Associate Professor in computervision and machine learning with the Suzhou Uni-versity of Science and Technology. His researchinterests include graphical models, structuredlearning, and tracking.

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