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Journal of Applied Mathematics
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Research Article
An Irregular Flight Scheduling Model and Algorithm under
the Uncertainty Theory
Deyi Mou and Wanlin Zhao
Institute of Mathematics for Applications, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Deyi Mou; deyimou@hotmail.com
Received June ; Revised August ; Accepted August
Academic Editor: Zhiwei Gao
Copyright © D. Mou and W. Zhao. is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
e ight scheduling is a real-time optimization problem. Whenever the schedule is disrupted, it will not only cause inconvenience
to passenger, but also bring about a large amount of operational losses to airlines. Especially in case an irregular ight happens, the
event is unanticipated frequently. In order to obtain an optimal policy in airline operations, this paper presents a model in which
the total delay minutes of passengers are considered as the optimization objective through reassigning eets in response to the
irregular ights and which takes into account available resources and the estimated cost of airlines. Owing to the uncertainty of the
problem and insucient data in the decision-making procedure, the traditional modeling tool (probability theory) is abandoned,
the uncertainty theory is applied to address the issues, and an uncertain programming model is developed with the chance
constraint. is paper also constructs a solution method to solve the model based on the classical Hungarian algorithm under
uncertain conditions. Numerical example illustrates that the model and its algorithm are feasible to deal with the issue of irregular
ight recovery.
1. Introduction
Aschedulewehavemadeisfrequentlycomplexanddynamic,
and uncertainty ttingly characters its intrinsic nature. Vari-
ous unanticipated events will disrupt the system and make the
schedule deviate from its intended course, even make it infea-
sible; furthermore, they will bring about a large quantity of
losses. en, we apply a method of disruption management to
cope with it, to reach our goals while minimizing all the neg-
ative impact caused by disruptions and to get back on track
in a timely manner while eectively using available resources.
e disruption management refers to the real time dynamic
revision of an operational plan when a disruption occurs.
is is especially important in situations where an operational
plan has to be published in advance, and its execution is
subject to severe random disruptions. When a published
operational plan is revised, there will be some deviation cost
associated with the transition from the original plan to the
new plan. To reduce such deviation cost, it is essential to take
them into account when generating the new plan. Disruption
management is a real-time practice and oen requires a quick
solution when a disruption occurs. e original planning
problem usually is regarded as a one-time eort, so it is prac-
tically acceptable if generating an optimal operational plan
takes a dozen of minutes or hours, or even longer. However,
when a disruption occurs, it is critical to immediately provide
a resolution to the responsible personnel. erefore, real-time
optimization techniques are very important.
In the area of transportation network, the schedule is
not frequently executed according to the original plan; the
system is oen disturbed because of uncertainties, time
delays, stochastic perturbations and so on. It is dicult to
deal with the situations. So, the complex dynamic systems are
raisedbyGaoetal.[] in many varieties, including the areas
of transportation networks, energy generation, storage and
distribution, ecosystems, gene regulation and health delivery,
safety and security systems, and telecommunications. ey
also present various mathematical methods and techniques
to discuss the issues.
Airlines spend a great deal of eorts developing ight
schedules for each of their eets, and the daily operations of
an airline are strictly based on a predetermined ight sched-
ule. So, we know how important the eet assignment is. But
there are many uncertain factors having eect on the ights,
Journal of Applied Mathematics
such as bad weather, aircra failure, and airline trac control.
When the ight schedule is disrupted, a deviation from the
ight schedule causes not only inconvenience to passenger,
butalsoalargeamountofoperationalcostbecausetheairline
has to make a new ight schedule that should satisfy all above
constraints. So, the recovery of irregular ights faced by all
airlines in the world is important and dicult to solve.
In USA, Delta Airline summed up , irregular ights,
which aected ,, passengers and caused million
of losses (not including satisfaction losses of passengers).
Each irregular ight caused losses up to more than ,
on average (not including losses to the passengers because of
delays). In China, the domestic three major airlines executed
,, ights, but there were , irregular ights, and
therateofirregularightwas.%in.Inthesameyear,
the domestic airlines of small and medium sized had ,
ights, and , were irregular ights; the rate of irregular
ight was up to .%. Hence, the irregular ights brought
a large amount of losses to airlines and inconvenience to
passenger in actual life.
Due to the complexity of irregular ight scheduling prob-
lem, it is impossible for airlines to optimize existing resources
relying on experience. e problem of searching fast and
ecient algorithm and soware has not been solved [].
en Teodorovi´
candGuberini
´
c[] proposed a branch and
bound method to minimize total delay minutes of passengers.
Teodorobic and Stojkovic [] raised dynamic programming
model based on principle of lexicographic optimization,
in order to minimize the cancelled ights and the total
delay minutes of passengers. Jarrah et al. []introducedtwo
separate models to minimize delay minutes and cancellations,
respectively, solved by critical path method (CPD). us, the
models were not able to consider the trade-o between delay
and cancellation. Gang []constructedatwo-commodities
network ow model without solving method. Yan and Yang
[] formulated a model to minimize the duration of time
in which the ight scheduling was disrupted. Arg¨
uello et al.
[,] presented resource allocation path ow model for eet
assignment; the model had clear ideas describing the essence
of eet assignment, but it was dicult to solve it. e period
network optimization model about ight operations recovery
could be found in Bard et al. [], and it transformed the
aircra routing problem to a network ow model depend-
ing on discrete-time. Bratu and Barnhart []introduced
a model of ight recovery and algorithm, when irregular
ight happened, considering simultaneously aircra, crew,
and passengers, to decide whether to cancel the ight or
not; the aim was to minimize the sum of total operation
cost of interconnection, cost of passengers, and canceling
cost. e objective function wished to search the trade-o
point between each cost. e essence of above models was to
construct ight leg and cancelled ight or not and select the
minimum cost of the program. But how to generate feasible
ight routes and calculate the cost of each route were dicult.
ere was not a paper which found the exact optimal solution
by solving model directly for airlines up to now.
During the period of irregular ights, we cannot optimize
all situations, and the reasons resulting to irregular ights are
frequently uncertainty. For the decision making of uncertain
problem, Kouvelis and Yu [] described robust discrete
optimization to deal with decision making in environment of
signicant data uncertainty. Matsveichuk et al. [–]dealt
with the ow-shop minimum-length scheduling problem
with jobs processed on two machines when processing time
is uncertain. Reference [] presented minimal (maximal)
cardinality of a -solution generated by Johnson’s algorithm
tosolvetheproblemabove;however,formostgeneralizations
of the two-machine ow-shop problem, the existence of
polynomial algorithms is unlikely. For the duration of each
irregular ight, we cannot analyze it without enough data,
and it is not feasible to be dealt with by using stochastic pro-
gramming. But we can invite experts to give the approximate
duration of delay time and its uncertainty distribution. en,
we can apply uncertainty theory with a great premise. Here
we will apply uncertainty theory to component model with
uncertain programming and provide a stepwise algorithm for
themodel.Inthispaper,wesearchforminimizingthetotal
delay minutes of passengers under the constraint of estimated
cost. Next, this paper gives introduction of uncertainty theory
and model, the method of constructing the model, the
algorithm of solving model, and numerical example for the
model. At last, we provide some future directions.
2. Preliminaries
In this section, some basic denitions are introduced, and
the arithmetic operations of uncertain theory which needed
throughout this paper are presented.
Denition 1 (Liu []). Let Γbe a nonempty set and La-
algebra over Γ. Each element Λ∈Lis called an event. e
set function Mis called an uncertain measure if it satises the
following four axioms.
1(normality). M{Γ}=1;
2(monotonicity). M{Λ1}≤M{Λ2}when-
ever Λ1⊂Λ2;
3(self-duality). M{Λ}+M{Λ𝑐}=1for any
event Λ;
4(countable subadditivity). For every count-
able sequence of events {Λ𝑖},wehave
M∞
𝑖=1Λ𝑖≤∞
𝑖=1
MΛ𝑖. ()
Denition 2 (Liu []). Let Γbe a nonempty set, La-
algebra over Γ,andMan uncertain measure. en the triple
(Γ,L,M)is called on uncertainty space.
Denition 3 (Liu []). An uncertain variable 𝜉is a measur-
able function from the uncertainty space (Γ,L,M)to the set
of real numbers; that is, for any Borel set Bof real numbers,
the set {𝜉∈B}=∈Γ|∈B()
is an event.
Journal of Applied Mathematics
For a sequence of uncertain variables 1,2,...,𝑛and a
measurable function ,Liu[]provedthat
𝜉=1,2,...,𝑛()
dened as ()=(1(),2(),...,𝑛()),forall∈Γis
also an uncertain variable. In order to describe an uncertain
variable, a concept of uncertainty distribution is introduced
as follows.
Denition 4 (Liu []). e uncertainty distribution Φof an
uncertain variable 𝜉is dened by
Φ(x)=M{𝜉≤x}()
for any real number x.
Peng and Li []provedthatafunctionΦ:R→[0,1]
is an uncertainty distribution if and only if it is a monotone
increasing function unless Φ()≡0or Φ()≡1.einverse
function Φ−1 is called the inverse uncertainty distribution of
𝜉. Inverse uncertainty distribution is an important tool in the
operation of uncertain variables.
eorem 5 (Liu []). Let 1,2,...,𝑛be independent
uncertain variables with regular uncertainty distributions
Φ1,Φ2,...,Φ𝑛,respectively.If(1,2,...,𝑛)is an increas-
ing function with respect to 1,2,...,𝑚and decreasing with
respect to 𝑚+1,𝑚+2,...,𝑛,then
𝜉=1,2,...,𝑛()
is an uncertain variable with inverse uncertainty distribution
Ψ−1 ()=Φ−1
1(),Φ−1
2(),...,Φ−1
𝑚(),
Φ−1
𝑚+1 (1−),...,Φ−1
𝑛(1−). ()
Expected value is the average of an uncertain variable in
the sense of uncertain measure. It is an important index to
rank uncertain variables.
Denition 6 (Liu []). Let 𝜉be an uncertain variable. en
the expected value of 𝜉is dened by
[𝜉]=∞
0
M{𝜉≥}−0
−∞
M{𝜉≤} ()
provided that at least one of the two integrals is nite.
In order to calculate the expected value via inverse
uncertainty distribution, Liu []provedthat
[𝜉]=1
0Φ−1
1(),...,Φ−1
𝑚(),
Φ−1
𝑚+1 (1−),...,Φ−1
𝑛(1−) ()
under the condition described in eorem . Generally, the
expected value operator has no linearity property for
arbitrary uncertain variables. But, for independent uncertain
variables 𝜉and 𝜂with nite expected values, we have
𝜉+𝜂=[𝜉]+𝜂()
for any real numbers and .
eorem 7 (Liu []). Assume the objective function (x,1,
2,...,𝑛)is strictly increasing with respect to 1,2,...,𝑚
and strictly decreasing with respect to 𝑚+1,𝑚+2,...,𝑛.If1,
2,...,𝑛are independent uncertain variables with uncertainty
distribution Φ1,Φ2,...,Φ𝑛, respectively, then the expected
objective function x,1,2,...,𝑛 ()
is equal to
1
0x,Φ−1
1(),...,Φ−1
𝑚(),
Φ−1
𝑚+1 (1−),...,Φ−1
𝑛(1−). ()
eorem 8 (Liu []). Assume (x,1,2,...,𝑛)is strictly
increasing with respect to 1,2,...,𝑚and strictly decreasing
with respect to 𝑚+1,𝑚+2,...,𝑛,and𝑗(x,1,2,...,𝑛)are
strictly increasing with respect to 1,2,...,𝑘and strictly
decreasing with respect to 𝑘+1,𝑘+2,...,𝑛,for=1,2,...,.
If 1,2,...,𝑛are independent uncertain variables with
uncertainty distributions Φ1,Φ2,...,Φ𝑛, respectively, then the
uncertain programming
min
𝑥x,1,2,...,𝑛,
s.t. M𝑗x,1,2,...,𝑛≤≥𝑗, =1,2,...,,
()
is equivalent to the crisp mathematical programming
min
𝑥1
0x,Φ−1
1(),...,Φ−1
𝑚(),
Φ𝑚+1−1 (1−),...,Φ−1
𝑛(1−),
s.t. 𝑗x,Φ−1
1𝑗,...,Φ−1
𝑘𝑗,Φ−1
𝑘+1 1−𝑗,...,
Φ−1
𝑛1−𝑗≤, =1,2,...,. ()
3. Problem Description
For airlines, irregular ights are not expected but they are
inevitable frequently owing to objective factors in practice.
So, each airline has its own methods to deal with the issue.
At present, the irregular ight recovery is a complex and
huge project to airlines. Generally speaking, the Airline
Journal of Applied Mathematics
Flights
Cancel
Homeplate
Make decisions for flights aected
From other
eld
Preliminary
scheme
End
Real-time situation
Adjust flight
schedule or not
Ye s
Ye s
Ye s
Ye s
Delays happened
No
No
No
No Adjust or
not
Cancel or
not
postpone
Homeplate or
not
Ferry
F : Flow chart of procedure.
Operational Control (AOC) of each airline takes the
following procedure for dealing with the irregular ights.
Step . Maintenance Control Center (MCC) reports delays to
AOC.
Step . AOC gets detailed information from relevant depart-
ments.
Step . AOC gives the expected minutes of ight delay
according to the information.
Step . AOC makes a decision whether the rst delay ight
cancels or postpones.
Step . Making decisions to the following ights.
Step . AOC always pays attention to real-time situations and
whether the decision will be to remake or not based on real-
time changes.
e ow chart of procedure is shown in Figure .From
this ow chart, we can get that it is dicult to make a suitable
and reasonable schedule. In this paper, we resolve the irregu-
lar ight problem about eet reassignment based on the ow
chart. When irregular ights happen, the durations of delay
time only are predicted through the data given by experts,
in order to help the decision makers to reassign eets better.
During the period of delay time, the airlines can postpone
the ights, minimize the delay minutes or delay cost by
reassigning and canceling ights, or minimize the total delay
minutes under the constraint of estimated cost by reassigning
ights; in a word, the aim is to serve the passengers better.
Tobia s [ ] presented a tabu search and a simulated
annealing approach to the ight perturbation problem and
used a tree-search algorithm to nd new schedules for
airlines,anditcouldbesuccessfullyusedtosolvethe
ight perturbation problem. Gao et al. []putforwarda
greedy simulated annealing algorithm which integrated the
characteristics of GRASP and simulated annealing algorithm,
and the algorithm was able to solve large-scale irregular ight
schedule recovery. Xiuli []introducedastepwise-delay
algorithm to research irregular ight problem. She solved the
problem from the delay cost and delay minutes, respectively.
In the paper, the delay minutes and delay cost were set down
as constant values. From comparing the two methods, Xiuli
got the result that to construct the model with delay minutes
obtained successful results.
But there are not enough data of irregular ights to
analyze; the data is dicult to deal with under the stochastic
condition. In actual situations, we cannot obtain enough data
to analyze, so, in this paper, we treat the duration of delay
time of aircra as an uncertain variable during the period
of irregular ights happening, and its distribution is given by
experts. Under the constraints of estimated cost and aircra
assignment model, the objective function is to minimize the
total delay minutes of passengers relying on irregular ights.
enwesolvethemodelandgetthesolution.
4. Model Development
In actual situations, the factors which cause irregular ights
are uncontrolled frequently. Especially when the weather is
bad and aircra failure happens, the durations of bad weather
and aircra failure cannot be predicted exactly because of
lacking enough data under stochastic condition. In this
model, we consider the delay time as an uncertain variable,
and its uncertainty distribution is given by experts.
Firstly, we introduce the following notations to present
the mathematical formulation throughout the remainder of
this paper.
Indices, Sets, and Parameters
: index for set of ights
: index for set of airports
: index for set of types of aircras
:setoftypesofaircras
:setofights
: set of airports
: set of available aircras
𝑖:readytimeofaircra
𝑑
𝑓: planning departure time of ight
𝑎
𝑓: planning arrival time of ight
𝑓
𝑖: uncertain delay time of ight executed by type
of aircra, 𝑓
𝑖∼𝑖(),∈
𝑏
𝑓,𝑒
𝑓: reservation number of business and economy
class in ight
Journal of Applied Mathematics
𝑏
𝑓,𝑒
𝑓: fare of a ticket of business and economy class
in ight
𝑓: total reservations of ight
: disappointment rate of passengers
V:lossesvalueofpassengersperminute
𝑓
𝑖: delay loss of ight executed by aircra ,𝑓
𝑖=
𝑓𝑓
𝑖,when𝑓
𝑖>0,or=
𝑓: cost of canceling ight
:airportusagechargeperminute
: cost of depreciation of aircra per minute
𝑓
𝑖: delay cost of ight executed by aircra
𝑓
0: estimated cost of ight
𝑓=(𝑓
𝑖),0=(𝑓
0)
0: condence level
:weightofcostofcancellingight
𝑓
𝑖: binary variable, = if ight is executed by
aircra and otherwise
𝑓: binary variable, = if ight is canceled and
otherwise.
Xiuli []presentedtwomodelstodealwiththeirregular
ight. eir objective functions were to minimize delay
minutes and total delay cost:
min
𝑓∈𝐹
𝑖∈𝐴𝑓
𝑖𝑓
𝑖+
𝑓∈𝐹𝑓𝑓,
min 𝑓x,𝑓. ()
In our paper, we construct the model under the basis
ofthetwomodels,andintegratethemtoone.Next,based
on the analysis of the decision making process, we integrate
the estimated cost and the total delay minutes of passengers
model and propose the following model:
min
𝑓∈𝐹
𝑖∈𝐴𝑓
𝑖𝑓
𝑖+
𝑓∈𝐹𝑓𝑓
,()
s.t.M𝑓x,𝑓<0≥0,()
𝑖∈𝐴𝑓
𝑖+𝑓=1, ∀∈,,∈, ()
𝑓
𝑖=𝑓
𝑖𝑒
𝑓×𝑒
𝑓+𝑏
𝑓×𝑏
𝑓×
++𝑓
𝑖+𝑓𝑓,
𝑓=1−𝑓
𝑖,()
𝑓∈𝐹𝑓
𝑖≤1, ∀∈, ()
𝑓
𝑖∈{0,1},
𝑓∈{0,1}.()
In the model, () is the objective function of minimizing
the total delay minutes of passengers, the former is the delay
minutes depending on the irregular ights, the latter is the
equivalent delay minutes relying on ights cancelled; ()is
the constraint of estimated cost; () states a ight is either
own once by an aircra or canceled; ()isthedelaycostof
ight executed by aircra ;() assigns no more than one
aircra to execute ight ;and() is the integer constraint
of -.
e model is to minimize the total delay minutes of
passengers under the estimated cost. Generally speaking, we
need to recur to intelligent algorithm to solve the model; it is
ahugeprojecttogetitssolution[].
But, it will be much easier under the uncertainty theory
to deal with the problem. Note that 𝑓(x,𝑓)is strictly
increasing with respect to 𝑓
𝑖(=1,2,...,),and𝑓
1,𝑓
2,...,𝑓
𝑛
are independent uncertain variables with uncertainty distri-
butions 𝑓
𝑖∼𝑖()(=1,2,...,), respectively. According
to eorem ,theabovemodelwillbeequivalenttothe
following deterministic model:
min
𝑓∈𝐹
𝑖∈𝐴 𝑓
𝑖𝑓1
0−1
𝑖𝑓
𝑖,
+
𝑓∈𝐹𝑓𝑓
,
s.t.Φ−1 x,0<0,
𝑖∈𝐴𝑑𝑖
𝑓𝑎𝑖
𝑓𝑓
𝑖+𝑓=1, ∀∈,,∈,
𝑓
𝑖=𝑓
𝑖𝑒
𝑓×𝑒
𝑓+𝑏
𝑓×𝑏
𝑓×
+𝑓
𝑖+𝑓𝑓, 𝑓=1−𝑓
𝑖,
𝑓∈𝐹𝑓
𝑖≤1, ∀∈,
𝑓
𝑖∈{0,1},
𝑓∈{0,1}.
()
In this model, the objective function is to minimize the
expected total delay minutes of passengers, and the estimated
cost is treated as a chance constraint, where
Φ−1 (x,)=−1
𝑖𝑓
𝑖,. ()
5. Solution Method and Complexity
5.1. Solution Method. In the model, there is an objective
function, but it contains two kinds of decision variables 𝑓
𝑖
and 𝑓.Tosolvethemodel,wemakeuseofastepwise-delay
algorithm.eprocedureofsolutionisasfollows.
Step 1. Basedontheinformationpostponed,gettingthe
timetable of ights as .
Journal of Applied Mathematics
T : Estimated cost.
0/RMB
xx
xx
xx
xx
xx
xx
xx
Total: ,
Step 2. Sorting the delay ights depending on original depar-
ture time from the timetable of delay ight, and searching the
rst airport where delay happened. During the delay period,
we retrieve the serial number of aircras through the airport,
andnotethemdowninthetable.
Step 3. Finding available aircras in the delay airport, a time
permutation table is built via the constraints, Φ−1(x,)<
0. e delay minutes are replaced by [𝑓
𝑖];thenwecanget
the following: 1∗ 2∗ ⋅⋅⋅∗
=𝑓
𝑖=1
2
.
.
.
𝑛
1
12
1⋅⋅⋅𝑛
1
1
22
2⋅⋅⋅𝑛
2
.
.
..
.
..
.
..
.
.
1
𝑛2
𝑛⋅⋅⋅𝑛
𝑛
.()
Step 4. For , we use Hungarian algorithm to reassign avail-
able aircra and get the new timetable:
= 1∗ 2∗ ⋅⋅⋅∗
1
2
.
.
.
𝑛10.
.
.
000.
.
.
1⋅⋅⋅
⋅⋅⋅
.
.
.
⋅⋅⋅ 00.
.
.
0.()
Step 5. Renewing the , the relevant aircra assignment
is replaced by the consequence from Step ;thenturnto
Step , and steps are repeated until there are no delay ights
or optimal ights. en the results are put out.
5.2. Complexity. For one delay airport, we use Hungarian
algorithm to reassign eets in that the complexity is (2).
When the number of delay airports is ,thealgorithmwill
be iterated in each airport, so that the total complexity is
(2)whichisapolynomial.Soitisafeasiblemethodin
applications.
6. Illustrative Example and
Computational Result
In order to test the model and the solution algorithms
applied in the actual situation, we perform numerical tests
based on the domestic operation department with reasonable
assumptions.
Wesupposethattheairlinehasthehubairportof1;the
ights are 1to 2,1to 3,and2to 3. Its types of aircras are
310and 737. Assume that the delay minutes of irregular
ights are linear uncertain variables, and their distributions
are as follows:
1∼L(10:40,11:20),
2∼L(15:50,16:30),
3∼L(12:10,12:50),
4∼L(15:10,15:50),
5∼L(16:20,17:00).()
e disappointment rate of passengers =0.07𝑓
𝑖/60+
0.4. e delay minutes of passengers are as follows:
𝑓
𝑖=
0.3𝑓×𝑓
𝑖
60 𝑓
𝑖∈[0,60),
0.5𝑓×𝑓
𝑖
60 𝑓
𝑖∈[60,120),
0.7𝑓×𝑓
𝑖
60 𝑓
𝑖∈[120,240),
0.9𝑓×𝑓
𝑖
60 𝑓
𝑖∈[240,∞).
()
We assume that the constraint of estimated cost is shown
in Table .
At last, we assume that the timetable is shown in Table .
e predetermined condence level 0=0.9.
en, we use the algorithm and get the optimal solution
shown in Table .
Comparing Tables ,and ,wecangettheoptimal
solution through the model. From Tabl e ,wecanseethat
the cost is , RMB under the constraint of , RMB,
and the ight delay time is ten minutes less than Tab l e .
Based on Tables and , there is a great dierence in the
total delay minutes of passengers. Ta b l e is about . times as
many as Tab l e , and there is not a cancelled ight in Tab l e .
So the reassignment of Table is much better than Table .
e example shows that the model and algorithm can get a
method of ight recovery better.
7. Conclusions and Future Directions
e irregular ights always happen in actual situations; in
order to deal with the issue, in this paper, we developed a
model for eet reassignment based on uncertain program-
ming during the period of irregular ights and presented a
stepwise optimization method strategy based on Hungarian
Journal of Applied Mathematics
T : Timetable postponed.
𝑑
𝑓𝑎
𝑓𝑖(𝑓
𝑖)/minute 𝑏
𝑓/𝑏
𝑓𝑒
𝑓/𝑒
𝑓𝑑𝑖
𝑓𝑎𝑖
𝑓
/ xx : : : / / 12
/ xx : : : / / 21
/ xx : : 1 / / 13
/ xx : : 2 / / 31
/ xx : : 3 / / 12
/ xx : : 4 / / 21
/ xx : : : Canceled / / 13
/ xx : : : Canceled / / 31
/ xx : : : / / 31
/ xx : : 5 / / 12
Total delay minutes of ights: ; canceled: ; value of objective function: ,.
T : Timetable aer reassigning.
𝑑
𝑓𝑎
𝑓𝑖(𝑓
𝑖)/minute 𝑑𝑖
𝑓𝑎𝑖
𝑓
/ xx : : : 13
/ xx : : : 31
/ xx : : : 13
/ xx : : : 31
/ xx : : : 31
/ xx : : : 12
/ xx : : : 12
/ xx : : : 21
/ xx : : : 12
/ xx : : : 21
Total delay minutes of ights: ;, canceled: ; value of objective function: ,.
Total cost of ights delay: , RMB.
algorithm to solve the problem. Compared with the tradi-
tional model, we introduce an uncertain variable into the
model and construct it based on uncertain programming.
Weconsiderthedelayminutesasuncertainvariableswith
their uncertainty distributions given by experts. We con-
struct a stepwise optimization method based on Hungarian
algorithm to solve the model. From results of the numerical
example, the total delay minutes of passengers are declined
extensively, and we can get that the model and algorithm are
feasible to deal with the issue of irregular ights.
A major contribution of this paper is that we provide
a comprehensive framework for eet reassignment during
the period of irregular ights happen. Much work still needs
to be done to improve on the current framework. Partially,
we believe future research can be conducted on an integral
framework of eet reassignment and crew schedule recovery.
us approaching the irregular ights problem is more
systematical. Furthermore, considering actual situations, we
canalsoconstructanintegraluncertainandstochasticmodel;
thus, dealing with the issue of irregular ights is more
comprehensive.
Acknowledgments
is work was supported by the Fundamental Research
Funds for the Central Universities (Grant ZXHC).
e authors would like to thank the anonymous referees for
useful suggestions and comments on the earlier dra of this
paper.
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