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The rapid development of airlines, has made airports busier and more complicated. The assignment of scheduled to available gates is a major issue for daily airline operations. We consider the over-constrained airport gate assignment problem(AGAP) where the number of flights exceeds the number of available gates, and where the objectives are to minimize the number of ungated flights and the total walking distance or connection times. The procedures used in this project are to create a mathematical model formulation to identify decision variables to identify, constraints and objective functions. In addition, we will consider in the AGAP the size of each gate in the terminal and also the towing process for the aircraft. We will use a greedy algorithm to solve the problem. The greedy algorithm minimizes ungated flights while providing initial feasible solutions that allow flexibility in seeking good solutions, especially in case when flight schedules are dense in time. Experiments conducts give good results.
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Communications of the Korean Statistical Society
2011, Vol. 18, No. 2, 257–266
DOI: 10.5351/CKSS.2011.18.2.257
An Airline Scheduling Model and Solution Algorithms
Ahmed Thanyan AL-Sultan1,a, Fumio Ishiokab, Koji Kuriharaa
aGraduate School of Environmental Science, Okayama University
bSchool of Law, Okayama University
Abstract
The rapid development of airlines, has made airports busier and more complicated. The assignment of
scheduled to available gates is a major issue for daily airline operations. We consider the over-constrained airport
gate assignment problem(AGAP) where the number of flights exceeds the number of available gates, and where
the objectives are to minimize the number of ungated flights and the total walking distance or connection times.
The procedures used in this project are to create a mathematical model formulation to identify decision variablesto
identify, constraints and objective functions. In addition, we will consider in the AGAP the size of each gate in
the terminal and also the towing process for the aircraft. We will use a greedy algorithm to solve the problem.
The greedy algorithm minimizes ungated flights while providing initial feasible solutions that allow flexibility in
seeking good solutions, especially in case when flight schedules are dense in time. Experiments conducts give
good results.
Keywords: Operations research, mathematical programming, greedy algorithm, 0–1 integer pro-
gramming, Monte Carlo simulation.
1. Introduction
The problem of assigning gates to flight arrivals and archers is an important decision problem in daily
operations at major airports all over the world. Strong competition between airlines and the increasing
demands of passengers for increased comfort has made the measures of quality in their decisions at
an airport as important performance indices of airport management. This is why the mathematical
modeling of this problem and the application of Operations Research(OR) methods to solve those
models have been studied widely in OR literature. The common characteristics of busy international
airports usually involve serving a large number of dierent airlines, a large number of flights over day,
and accommodating various types of planes.
Much work has centered on the gate assigning problem with the objective of minimizing distance
cost (or variants of this). One of the first attempts to use quantitative means to minimize intra-terminal
travel into a design process was given by Braaksma and Shortreed (1971). The assignment of aircraft
to gates that minimize travel distances, is an easily motivated and understood problem but a dicult
one to solve. The total passenger walking distance is based on passenger embarkation and disem-
barkation volumes, transfer passenger volumes, gate to gate distances, check in to gate distances and
aircraft to gate assignments. In the gate assignment problem, the cost associated with the placing
of an aircraft at a gate depends on the distances from key facilities as well as the relations between
these facilities. The basic gate assignment problem is quadratic assignment problem as shown to be
NP-hard in Obata (1979). Babic et al. (1984) formulated the gate assignment problem as linear 0–1
1Corresponding author: Professor, Graduate School of Environmental Science, Okayama University, 3-1-1 Tsushima-
naka Okayama 700-8530, Japan. E-mail: alsultan@ems.okayama-u.ac.jp
258 Ahmed Thanyan AL-Sultan, Fumio Ishioka, Koji Kurihara
IP. A branch and bound algorithm is used to find the optimal solution where transfer passengers are
not considered. Haghani and Chen (1998) proposed an integer programming formulation of the gate
assignment problem and heuristic solution procedure for solving the problem. The multiple objective
model for the gate assignments were proposed in Yan and Huo (2001). Where the model is formulated
as a multiple objective 0–1 integer program. Network model (Yan and Chang, 1998) and simulation
models (Cheng, 1998a, b) were also proposed to formulate the problem. Since the gate assignment
problem is NP-hard, various heuristic approaches have been suggested by search, e.g. Haghnani and
Chen (1998). Proposed a heuristic that assigns successive flights parking at the same gate when there
is no overlapping, flights are assigned based on the shortest walking distance coecients. Xu and
Bailey (2001) provide a Tabu search meta-heuristic to solve the problem. The algorithm exploits the
special properties of dierent types of neighborhood moves, and creates highly eective candidate list
strategies. The work of Yan et al. (2008) considered stochastic disturbances in the daily passenger de-
mand that occur in actual operations. They established a stochastic-demand flight scheduling model,
SDFSM. Two heuristic algorithms, based on arc-based and route based strategies, were developed to
solve the SDFSM. In addition, previous work (Ding et el., 2004) has considered the over constrained
gate assignment problem which addressed both the objectives of minimizing the number of ungated
aircraft while minimizing low total walking distance. In the work of AL-Sultan et el. (2010), some of
the assumptions has been added and changed for a previous work of Ding et el. (2004) such as consid-
ering in the airport gate assignment problem the size of each gate in the terminal and also the towing
process for the aircraft; however, the data collection was for one day only. In the current project, the
data collection is for one week. In addition, analysis will be added for the buer time that is the time
that locks the aircraft gate after departure. To the best of our knowledge, no previous work has con-
sidered the over constrained gate assignment problem. In particular, no previous work has addressed
both the objectives of minimizing the number of ungated aircraft while minimizing the total walking
distance. In addition, we inserted the terminal gate sizes, the towing process, and aircraft capacity.
The Airport gate assignment problem(AGAP) seeks to a find feasible flight to gate assignments so
that the number of the flights that need be assigned to the apron and total passenger connection times,
as can be proxies for walking distances, are minimized. In this paper, we discuss a greedy algorithm
that minimizes the number of flights not assigned to gates.
We will apply our model to Kuwait International Airport(KIA) that has become busier after apply-
ing for the “Open Skies” policy that applies to passenger and cargo operations, forms an essential part
of the Kuwait government’s latest initiative to promote the state as a major center for financial, com-
mercial and economic activities in the Gulf Region. KIA already serves more than 50 airlines currently
connected to Kuwait with over 100 international destinations; in addition, there exists considerable
room for expansion. In this project, we will use actual aircraft’s arrival and departure schedules. The
number of passengers for each aircraft will be generated randomly using the Monte Carlo method.
2. The Nature of KIA’s Gate Terminals
Kuwait international airport has one terminal that has ten gates for the aircraft. The gate numbers are
1, 2, 3, 4, 5, 21, 22, 24, 25 and 26. Figure 1 describes the shape of the terminal. Next to the terminal
there are stands (aprons) for the ungated flights, there are several locations for aircraft stands such as
the cargo flight stand area, the VIP or privet flight stand area, and the regular stand area which is used
mostly for the ungated commercial aircraft.
Table 1 presents the actual schedule (daily movement) for a sample of the arrival and departure
flights for a specific day. The schedule contents:
An Airline Scheduling Model and Solution Algorithms 259
PASSPORT
CONTROL
GATE1
GATE2
GATE3
GATE4
GATE5 GATE26
GATE25
GATE24
GATE22
GATE21
APRON
Transfer by bus
Figure 1:
KIA’s gates terminal
Table 1: Daily movement
A/L A/C FLTArrival Incoming FLTDepature Outgoing
From time Gate To time Gate
JAI 737 574 COK 0040 25 573 COK 0140 25
RJA 319 5256 AMM 0045 3 5257 AMM 0130 3
MEA 320 408 BEY 0110 21 409 BEY 0200 21
THY 737 1172 IST 0115 26 1173 IST 0215 26
MSR 737 614 CAI 0145 22 615 CAI 0245 22
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JZR 320 489 DAM 2125 4 188 DXB 2220 4
JZR 320 415 BEY 2200 5 636 ALP 2320 5
JZR 320 407 DXB/BAH 2210 3 502 LXR 2315 3
JZR 320 185 DXB 2240 25 516 HRG 2325 25
A/L (airline)
A/C (type of aircraft)
FLT(flight number)
Arrival from /departure to
GATE (Gate number for arrival /departure flight)
From the collected data:
We have 756 arrival flights and 754 departure flights.
260 Ahmed Thanyan AL-Sultan, Fumio Ishioka, Koji Kurihara
Table 2: Types of aircraft that could be assigned to each gate in the terminal
Gate1 Gate2 Gate3 Gate4 Gate5 Gate21 Gate22 Gate24 Gate25 Gate26
310 ⃝ ⃝ ⃝ ⃝ ⃝
319 ⃝ ⃝ ⃝ ⃝ ⃝
320 ⃝ ⃝ ⃝ ⃝ ⃝
321 ⃝ ⃝ ⃝ ⃝ ⃝
300 ⃝ ⃝ ⃝ ⃝ ⃝
330 ⃝ ⃝ ⃝ ⃝
340 ⃝ ⃝ ⃝ ⃝
727 ⃝ ⃝ ⃝ ⃝ ⃝
737 ⃝ ⃝ ⃝ ⃝ ⃝
747 ⃝ ⃝
767 ⃝ ⃝ ⃝ ⃝
777 ⃝ ⃝ ⃝ ⃝
DC10 ⃝ ⃝ ⃝ ⃝
MD90 ⃝ ⃝ ⃝ ⃝
From the arrival flights, we have 212 flights ungated (which means that 28.04% from the arrival
flights are ungated).
From the departure flights, we have 30 flights ungated(which means that 25.60% from the depar-
ture flights are ungated).
Table 2 presents the types of aircraft that could be assigned to each gate in the terminal. We can notice
that Gate 1 is the smallest gate in the terminal since it has the smallest number of types of aircraft
which can be assigned to it. On the other hand, Gate 2, Gate 4, Gate 5, Gate 21 and Gate 22 are
considered the largest gates in the terminal since they could be use by any type of aircraft.
3. Problem Description and Model Formulation
In this project, we consider the airport gate assignment problem(AGAP), where the number of flights
exceeds the number of gates available. Our objective is to minimize the number of ungated flights
or minimize the number of flights assigned to the apron and the total walking distance or connection
times. We will consider the size of each gate in the terminal. We represent ωiset of gates that can
be assigned to flight. According to KIA ocials, the towing process for the aircraft will be applied if
this aircraft is scheduled to a specific gate for more than 6 hours. The towing process means pulling
oan aircraft from the terminal gate to the aircraft stand area. The same process will be used to pull
oan aircraft from the stand area to the terminal gate for departure. According to the KIA ocials,
the towing process takes approximately one hour. One of the reasons for using the towing process is
if a flight is scheduled to use a gate for more than 6 hours, we pull othis aircraft from the assigned
gate after one hour from its arrival to give an opportunity to other aircraft to use this gate. In addition,
we will add a buer time that is added between two continuous flights assigned to the same gate. The
time interval locked for particular aircraft is equal to [ai,di+α]. According to KIA ocials, the buer
time is one hour for each flight.
3.1. Identify decision variables
Notations:
N: Represent set of flights arriving to/departing from the airport.
M: Represents set of gates available at the airport.
An Airline Scheduling Model and Solution Algorithms 261
n: Total number of flights.
m: Total number of gates.
ai: Arrival time of flight i.
di: Departure time of flight i.
fi,j: Number of passengers transferring from flight ito flight j.
wk,l: Walking distance for passengers from Gate kto Gate l.
α: Buer time that locks the gate before aircraft’s arrival and after its departure (0–1 hour).
ωi: Represents set of gates that can be assigned to flight i.
DF: the dierence between the departure and the arrival time (6 hours).
TP: The towing process (1 hour).
Additionally, we will make use of two dummy gates. Gate 0 represents the entrance or exit of the
airport, and gate m+1 represents the apron where flights arrive when no gates are available. Hence,
wk,0represents the walking distance between Gate kand the airport entrance or exit, and f0,irepresents
the number of originating departure passengers of flight i;fi,0represents number of the disembarking
arrival passengers of flight i. So wm+1,krepresent the walking distance between the apron and Gate k
(usually significantly larger than the distance among dierent gates).
The binary variables
yi,k={1,if flight iis assigned to Gate k(0 <km+1),
0,otherwise.
The following constraint must be satisfied:
(i,j),kωikωj,
yi,k=yj,k=1(k,m+1) Implies ai>djaj>di.
This condition disallows any two flights to be scheduled to the same gate simultaneously (except if
they are scheduled to the apron).
3.2. Constraints and objective function
Our objective is to minimize the number of flights assigned to the apron and the total walking distance.
The mathematical formulation can be expressed as follow:
Minimize
n
i=1
yi,m+1,(3.1)
Minimize
n
i=1
n
j=1
m+1
k=1
m+1
l=1
fi,jwk,lyi,kyj,l+
n
i=1
f0,iw0,i+
n
i=1
fi,0wi,0.(3.2)
262 Ahmed Thanyan AL-Sultan, Fumio Ishioka, Koji Kurihara
Equation (3.1) refers to the first objective that minimizes the number of flights assigned to the apron.
And Equation (3.2) refers to the second objective which is minimizes the total walking distance. We
will call the value of Equation (3.2) the walking distance cost. The constraints:
1. Ensures that every flight must be assigned to one and only one gate or assigned to the apron.
kωi
yi,k=1,(i,1in).
Where ωirepresent set of gates that can be assigned to flight i.
2. Each flight’s departure time is later than its arrival time.
ai<di,(i,1in).
3. Two flights schedule cannot overlap if they are assigned to the same gate.
yi,kyj,k(djai)(diaj)0,(i,j,kωikωj,1i,jn,k,m+1),yi,k∈ {0,1}.
4. Algorithm and Data Generation
To solve the AGAP, we will use greedy algorithm which uses a heuristic methods for minimizing the
number of flights assigned to the apron. The basic details of the algorithm are as follow:
1. Sort the flights according to the departure time di(1 in). Let gk(1 km) represents the
earliest available time (actually the departure time of last flight) of Gate k. Set gk=1 for all k.
2. For each flight i
- Find Gate ksuch that gk<aiand gkis maximized; and kωi.
- If such kexists, assign flight ito Gate k, update gk=di.
- If kdoes not exist, assign flight ito the apron.
3. Output the result.
Note that in step 2 before assigning flight ito Gate k, we will check if diai>DF =6 hours. If the
answer is yes, we will divide this flight into two flights and apply the towing process which will take
TP =1 hour. The first flight’s time interval becomes [ai,ai+1] and the second flight’s time interval
become [di1,di]. This means if a flight is scheduled to use a gate for more than 6 hours, we pull
othis aircraft from the assigned gate after one hour from its arrival to give an opportunity to other
aircraft to use this gate. Then for this flight’s departure, if we find a gate, we will assign this flight to
this gate one hour before its departure.
Proof of the correctness of the greedy algorithm: By induction, assume we have found the opti-
mum solution after scheduling flight iby the greedy algorithm. Now by this, we will assign flight f
to Gate k. But the optimal solution is to drop flight fand assign f(f>f) to Gate k. Hence we can
always replace fby fto make our greedy solution no worse than the optimal solution. There are two
cases we should consider:
1. If k=k, since we sort the flight by departure time, dfdf. We have gkg
k. As we considered
the earliest available time of the gates, we find the greedy solution is better or at least equal to the
optimal solution.
An Airline Scheduling Model and Solution Algorithms 263
Gate k
Gate k’
Gate k
Gate k’
Flight f’
Flight f
...................................
..........
...................................
..........
g’_k
g’_k’
g_k
g_k’
Replace Flight f’ by Flight f
Figure 2:
The correctness of the greedy algorithm
Table 3: Walking distance
GATE DISTANCE (in units)
1,21 2
2,22 1
3,24 4
4,25 5
5,26 6
APRON 8
2. If k,k, we find that gkg
kand gkg
k, since we choose the maximum gkin the greedy
solution. The Figure 2 illustrates this.
The arrival time and the departure time for flight iare actual data from the schedule department
at Kuwait international airport. But other data should be generated or assumed to apply our model.
First for the walking distance, we assume that the distance measure between two gates which are next
to each other is 1 unit. For example, if one passenger arrived at Gate 25 his walking distance to the
passport control is 5 units (The distance measure is known as Manhattan Distance). Table 3 represents
a summary for the assumed walking distance from a specific gate to the passport control.
We can use Table 3 to assume the walking distance from Gate kto the airport entrance or exit
(wk,0) or the walking distance from the airport entrance or exit to Gate k(w0,k). For the transferring
passengers, the walking distance from Gate kto Gate l(wk,l) is randomly generated in the interval
[1,8]. Now for the incoming passengers fi,0and the departure passengers f0,iare randomly generated
from dierent interval sizes depending on the type of the aircraft. Table 4 represents the scenarios to
generate the incoming and the departure passengers.
There are rarely small numbers of passengers transferring from one flight to another flight. The
number of transfer passengers will increase if flight schedules are close, but not too close (At least 1
hour dierent). The number of transferring passengers from flight ito flight j(fi,j) is usually within a
certain interval, say [1,50].
5. Results
We implement R (statistical software) to solve the problem. The detailed of the results and analysis
are presented in the next section.
264 Ahmed Thanyan AL-Sultan, Fumio Ishioka, Koji Kurihara
Table 4: Incoming and the departure passenger’s generation
Type Data Generation
310 [180, 280]
319 [80, 126]
320 [80, 180]
321 [86, 186]
300 [235, 335]
330 [235, 335]
340 [195, 295]
727 [87, 187]
737 [89, 189]
747 [324, 424]
767 [88, 188]
777 [244, 344]
DC10 [270, 370]
MD90 [87, 187]
Table 5: The output of the result
A/L A/CArrival Gate Departure
From time To time
AXB 737 - - 2 TRV/CCJ 10
SYZ 320 - - 1 DAM 20
MSR 320 - - 4 LXR 30
LZB 320 - - 5 BOJ 40
IAC 320 - - 21 BOM/MAA 50
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RJA 310 RJA 2310 1 - -
KLM 330 KLM 2315 Apron - -
JZR 320 JZR 2350 3 - -
PIA 310 PIA 2355 22 - -
5.1. Result of objective 1
We will apply the greedy algorithm to obtain initial feasible solutions for the first objective function
(minimize the number of flights assigned to the apron). Table 5 represents a sample for the output of
the result. From the output, after applying the greedy algorithm, the total number of ungated flights
for α=0 is 85 for the incoming flights and 82 for the outgoing flights. The dierence between the
actual data and the output is 127 for the incoming flights and 111 for the outgoing flights. Note that the
number of the ungated flights for the actual data is 212 in the incoming flights and 193 in the outgoing
flights. Table 6 shows the output results for using the greedy algorithm and we have compared these
results with the actual data to get the saving percentage. We tried α=0, 10, 20, 30, 40 and 60 minutes.
The results give positive results until we let α=40 minutes. The saving percentage is positive for the
incoming flights but it becomes negative for outgoing flights.
5.2. Result of objective 2
After applying the greedy algorithm, we will use the previous output as shown in table to find the
solution for the second objective function (minimize the total walking distance). To do this, we must
generate random data for the number of passengers and walking distance as explained in the previous
section (Data generation). Table 7 represents the sample for the generated data for the number of
passengers and the walking distance.
Before applying the greedy algorithm to get an initial feasible solution for the first objective func-
An Airline Scheduling Model and Solution Algorithms 265
Table 6: Output analysis
α=0α=10 α=20
Incoming Outgoing Incoming Outgoing Incoming Outgoing
Greedy algorithm 85 82 112 113 135 136
Dierence from the actual data 127 111 100 80 77 57
Saving percentage 59.91% 57.51% 47.17% 41.45% 36.32% 29.53%
α=30 α=40 α=60
Incoming Outgoing Incoming Outgoing Incoming Outgoing
Greedy algorithm 168 168 196 196 243 244
Dierence from the actual data 44 25 16 331 51
Saving percentage 20.75% 12.95% 7.55% 1.55% 14.62% 26.42%
Table 7: Generated data for the number of passengers and the walking distance.
A/L A/CArrival f0,jGate wi,jfi,jfi,0
Departure
From time To time
AXB 737 - - 0 2 2 0 178 TRV/CCJ 10
SYZ 320 - - 0 1 1 0 137 DAM 20
MSR 320 - - 0 4 4 0 127 LXR 30
LZB 320 - - 0 5 5 0 99 BOJ 40
IAC 320 - - 0 21 6 0 140 BOM/MAA 50
DLH 330 - - 0 2 2 0 322 FRA 55
AFG 310 - - 0 1 1 0 236 KBL 100
KAC 320 BEY 5 80 4 4 0 0 - 105
RJA 319 AMM 45 107 5 5 44 97 AMM 130
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RJA 310 RJA 2310 212 1 2 0 0 - -
KLM 330 KLM 2315 327 Apron 8 0 0 - -
JZR 320 JZR 2350 154 3 4 0 0 - -
PIA 310 PIA 2355 246 22 1 0 0 - -
Table 8: The walking distance cost change comparison with the buer time α=0
α=10 α=20 α=30 α=40 α=60
The estimated walking distance cost 1362301 1426177 1497324 1576803 1710229
percentage change 7.42% 12.46% 18.07% 24.34% 34.86%
tion, we have estimated the walking distance cost for the actual data by generating random data for the
number of passengers and walking distance. The estimated cost was 1,509,752. After applying the
greedy algorithm and generating random data for the number of passengers and walking distance with
a buer time α=0, the walking distance costs was 1,268,182. This means that the walking distance
saving percentage is 16.00%. Table 8 represents the walking distance cost change comparison with
the buer time α=0.
6. Conclusion
In this paper, we considered the over constrained the AGAP(airport gate assignment problem) to
minimize the number of flights assigned to the apron while minimizing the total walking distances.
We provide a greedy algorithm that minimizes the number of flights not assigned to gates. This
algorithm can allocate the flights that will be ungated as well as provide an initial feasible solution
while putting in our considerations for the size of each gate in the terminal, the towing process for the
aircraft and the aircraft capacity. In addition, we added some analysis for the buer time which is the
266 Ahmed Thanyan AL-Sultan, Fumio Ishioka, Koji Kurihara
time that locks the aircraft gate after departure for to minimize the number of flights assigned to the
apron and minimizing the total walking distances by estimating the walking distance cost. In future
work, further research will be conducted in order to extend the report work. Such as gate utilization
(usage percentage) and aircraft’s delays forecasting for both arrival and departure flights.
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ResearchGate has not been able to resolve any citations for this publication.
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Modern airport terminal building designs that have finger piers and airbridges sometimes result in significant walking distances for passengers. For a given set of aircraft parking positions next to a passenger terminal buildingand a given set of aircraft (with associated numbers of arriving and departing passengers), a method to find aircraft stand assignment which minimizes average passenger walking distances is proposed. The method is based on a branchand- bound technique (backtracking version), slightly modified to accelerate computation. This type of optimization is to be applied only if the number of passengers varies enough between the aircraft in question. The optimum solution mainly affected the number of passengers who walk maximum distances (decreased this number) and minimum distances (increased this number), compared to random aircraft stand assignment.
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In this research we developed a network model that will help the airport authorities assign flights to gates both efficiently and effectively. The model was formulated as a multi-commodity network flow problem. An algorithm based on the Lagrangian relaxation, with subgradient methods, accompanied by a shortest path algorithm and a Lagrangian heuristic was developed to solve the problem. The model was tested using data from Chiang Chiek-Shek Airport.
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This paper proposes a multiple objective model to help airport authorities to efficiently and effectively solve gate assignment problems. The model is formulated as a multiple objective zero–one integer program. To efficiently solve large-scale problems in practice, we used the weighting method, the column generation approach, the simplex method and the branch-and-bound technique to develop a solution algorithm. To test how well the model may be applied in actual operations, a case study regarding the operation of Chiang Kai-Shek (CKS) Airport was performed. The results show that the model could be useful for actual operations.
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When setting a good flight schedule airlines not only have to consider their fleet supply and related operations, as well as market share, but also stochastic variations caused by daily passenger demands in actual operations. Most of the past research on short-term flight scheduling has used the average passenger demand as input to produce the final timetable and schedule, which means that daily passenger variations that occur in actual operations are neglected. To consider such stochastic disturbances we developed a stochastic-demand scheduling model. We employed arc-based and route-based strategies to develop two heuristic algorithms that can be used to solve the model. The test results, based on a major Taiwan airline’s operation, show the good performance of the model and the solution algorithms.
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Apron control, which is the control of aircraft to and from airport gates, is strongly characterised by its dynamic, ever changing nature. Simulation can be used as a powerful tool to evaluate the effectiveness and appropriateness of the proposal in an apron control activity and can potentially be a valuable tool to support resolving time and resource conflicts in apron control. This paper proposes a rule-based reactive model for the simulation of aircraft on airport gates. It attempts to bring together object-oriented simulations and knowledge-based systems to produce a new kind of model in which knowledge-based technologies can be more deeply structured within a simulator and the simulation can be entirely embedded in a knowledge-based system. Validation by experiment shows that the proposed model is workable.