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A new linear programming approach and genetic algorithm for solving airline boarding problem

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A new linear programming approach and genetic algorithm for solvingairline boarding problem
Majid Soolaki
a
, Iraj Mahdavi
a,
⇑
, Nezam Mahdavi-Amiri
b
, Reza Hassanzadeh
c
, Aydin Aghajani
a
a
Department of Industrial Engineering, Mazandaran University of Science and Technology, Tabarsi Street, Babol 47166-95635, Iran
b
Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
c
Department of Industrial Engineering, Allameh Mohaddes Noori Institute, Noor, Iran
a r t i c l e i n f o
Article history:
Received 17 December 2010Received in revised form 28 October 2011Accepted 13 November 2011Available online 20 November 2011
Keywords:
OR in airlinesMixed integer linear programmingTransportationBoarding strategyGenetic algorithm
a b s t r a c t
Theairlineindustryisunderintensecompetitiontosimultaneouslyincreaseefﬁciencyandsatisfaction for passengers and proﬁtability and internal system beneﬁt for itself. Theboarding process is one way to achieve these objectives as it tends itself to adaptivechanges. Inordertoincreasetheﬂyingtimeofaplane,commercialairlinestrytominimizethe boarding time, which is one of the most lengthy parts of a plane’s turn time. To reduceboarding time, it is thus necessary to minimize the number of interferences between pas-sengers by controlling the order in which they get onto the plane through a boarding pol-icy. Here, we determine the passenger boarding problem and examine the different kindsof passenger boarding strategies and boarding interferences in a single aisle aircraft. Weoffer a new integer linear programming approach to reduce the passenger boarding time.A genetic algorithm is used to solve this problem. Numerical results show effectiveness of the proposed algorithm.
2011 Elsevier Inc. All rights reserved.
1. Introduction
Airlines start generating revenues by utilizing and ﬂying their aircrafts, and of course they do not generate any revenuewhiletheaircraftsaresittingontheground.Asaresult,atraditionalmetricusedbycommercial airlinestomeasuretheefﬁ-ciency of their operations is airplane turnaround time. Usually, turnaround time is measured by the time between an air-plane’s arrival and its departure [1–3].Some factors inﬂuencing turnaround time include passenger deplaning, baggage unloading, fueling, cargo unloading,airplane maintenance, cargo loading, baggage loading, and passenger boarding. Therefore, airlines ﬂying short-haul ﬂightstypically select airports (within the same region) with low air/ground congestions [4]. Airlines have little control overpassenger-boarding time because they have limited control over passengers. Therefore, while airlines want to speed upthe passengers boarding airplanes, they are cautious in making changes to increase operational efﬁciency.The boarding process of passenger aircraft has been an issue since the inception of the airline industry; however, it hasbeensteadilyincreasinginimportancesincethelate1970s.Theprocessofairplaneboardingisexperienceddailybymillionsofpassengersworldwide.Airlineshaveadoptedavarietyofboardingstrategiesinthehopeofreducingthegateturn-aroundtimeforairplanes. Toexert controlovertheboardingprocess, airlinesassignpassengerstoboardinggroupsor zones, callingeach boarding group to board in order (see Table 1). The deﬁciencies with respect to this aspect of ﬂying has annoyed manytravelers andsparkeda debateamongindustry professionals andtravelers alike as to howtorectify thesituation, eachwithhis/her own interest in mind (customers want ease/efﬁciency and airlines want efﬁciency/proﬁtability). Consequently,
0307-904X/$ - see front matter
2011 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2011.11.030
⇑
Corresponding author.
E-mail address:
irajarash@rediffmail.com (I. Mahdavi).Applied Mathematical Modelling 36 (2012) 4060–4072
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
airlines are currently struggling with how to more effectively address the boarding-process problem. The airline industryrecognizes that the boarding process is a signiﬁcant cost, yet little published research exists in the literature to addressthe ways to improve the process. Table 1 presents a summary of the boarding processes used by major US airlines.Marelli et al. [5] described a simulation-based analysis performed for Boeing 757. They designed the passenger enplane/deplane simulation (PEDS) to test different boarding strategies and different interior conﬁgurations on a Boeing 757 airplane.PEDS showed that by boarding outside-in, that is, window seats ﬁrst, middle seats second, and aisle seats last, airlines couldreduce boarding times signiﬁcantly.Van Landeghem and Beuselinck [2] conducted another simulation-based study on airplane boarding showing that thefastest way to get people on an airplane would be to board them individually by their row and seat number through callingeach one of the passengers individually to board the aircraft. In their study, they analyzed many alternative boarding pat-terns. One pattern that seemed practical and efﬁcient was boarding passengers by half-row, that is, by splitting each rowinto a starboard-side group and a port-side group and then boarding the half-rows one by one. Prior to executing their sim-ulation model, they assumed the arrival rate of passengers to be continuous, recognizing that ‘‘in reality, the passenger ar-rival rate is determined by the gating operations’’. Their model also took into account that 60% of all passengers carried onone bag, 30% carried on two bags, and only 10% carried on three bags. In a recent shift in airline security policy, airlines nowplace limits on the number, size, and weight of carry-on luggages. The standard allowance typically includes one baggage-type item and one personal item, e.g., purse or briefcase. However, US domestic airlines have been generally liberal in enforc-ing these limits.Ferrari and Nagel [6] expounded upon the ‘‘bin occupancy model’’ of Van Landeghem and Beuselinck [2], to determine the
amount of time attributed to storing carry-on luggage. The model calculated the time associated with storing carry-on lug-gages by evaluating the number of bags already in the bin plus the number of bags being carried by each passenger. Theydetermined that boarding outside-in or individually by seat were the two best methods, a ﬁnding concurrent with VanLandeghem and Beueslinck (2002). In their study, a sensitivity analysis was conducted to determine the robustness of thevarious boarding strategies under the effect of three recognizable disturbances: early and/or late passengers, aircraft dimen-sions, and the occupancy level of the plane. This sensitivity study is known as the ‘‘average worst case’’ boarding time model[6]. Calculations determined that those boarding strategies which yielded good performance ﬁgures also yielded well ‘‘aver-age worst case’’ boarding times and vice versa.While traditional computer-based simulation studies are good tools for testing the performance of already identiﬁedalternatives, they do not provide efﬁcient mechanisms for constructing the most promising alternatives. For this reason, herewe offer to use analytical models to analyze the problem. Surprisingly, in the airline industry, having a rich background inapplying operations research techniques, we found only simulation-based solutions for analyzing and improving passengerairplaneboarding.One exceptionis a study byBachmatet al. [7] thatapproachesthe airplaneboardingproblemfroma phys-icist’s point of view. They constructed a model based on spacetime geometry and random matrix theory that captures theasymptotic behavior of airplane boarding. They were able to interpret their ﬁndings to provide an explanation for whythe different boarding strategies performed the way they did.An analytical approachto aircraft boarding strategy was given by Van den Briel et al. [8]. They modeled the aircraft board-ing strategy using a non-linear assignment model with quadratic and cubic terms. The model attempts to minimize the totalinterferences among the passengers. The non-linear problem, being NP-hard, was solved and veriﬁed using simulation mod-eling. The ﬁnal recommended ‘‘reverse pyramid’’ boarding strategy was implemented at America West Airline.A recent analytical approach to aircraft boarding strategy is due to Massoud Bazargan [9]. He models the aircraft boardingstrategy using a mixed integer linear programming approach to generate efﬁcient boarding strategies.
Table 1
Summary of the boarding processes used by major US airlines.
Major US airlines Boarding method Boarding strategyAmerican airlines Traditional block method By groups, starting at the rear of the aircraft and moving forward, about 1/5of the rows at a timeContinental airlines Traditional by-row method By rows, starting at the rear of the aircraft and moving forward, about 1/4 of aircraft at a timeDelta airlines Non-traditional method By zones, starting with the back few rows, followed by the middle and thenfront sections, then back to a rear sectionNorthwest airlines Random boarding method Passengers line up and take their assigned seat in no particular orderSouthwest airlines Open seating method Passengers are assigned a group and boarding number based on check-intimes. After group is called, passengers take a position next to the columnrepresenting their number and proceed onto the aircraft. Passengers choosetheir own seats once onboardUnited airlines Open seating method Passengers are assigned a group and boarding number based on check-intimes. After group is called, passengers take a position next to the columnrepresenting their number and proceed onto the aircraft. Passengers choosetheir own seats once onboardAmerica West Reversed pyramid Window seats ﬁrst, followed by middle, then aisle and loading diagonally
M. Soolaki et al./Applied Mathematical Modelling 36 (2012) 4060–4072
4061
Our focus here is to develop a mathematical model to capture some passenger behaviors in boarding the aircraft.Although the models make several simplifying assumptions, they provide a good analysis of the factors affecting the board-ing process. All other effects, such as the time used by passengers standing up to retrieve an item, sittings in the wrong seats,or late arrivals, or passing by passengers already seated are initially treated as negligible. Our model does not include theeffects of the clustering of passengers into companions or families and other effects of human nature. Some airlines (espe-cially low fare ones) do not assign seats at all. This free seating is not regarded to be acceptable by traditional passengers, andis not included in our study. It is likely to increase boarding time, as this system is comparable to being random, but therewill be additional delays, especially for the late passengers looking for available free seats.Here, we present a new mixed integer linear programming approach to generate new boarding strategies. The mathe-matical model attempts to minimize the total interferences among the passengers which are the major causes of boardingdelays, subject to operational and side constraints. The model provides the ﬂexibility to optimize the boarding times forvarious aircrafts with different seat capacities. We then propose a mathematical model for a single aisle Airbus-320 aircraft.This speciﬁc type of aircraft is selected because it is one of the most common types of aircrafts adopted by many airlines.Another reason is that it enables us to provide a comparison between the performance of our offered model and those re-ported by the non-linear model developed by Van den Briel et al. [8], and the linear model developed by Massoud Bazargan[9].The rest of this paper is organized as follows. Section 2 deﬁnes interferences and how they are formulated in our model.Section 3 provides the problem description. Section 4 deﬁnes constrains of interferences and how they are formulated in ourmodel. Section 5 proposes the mixed linear integer model. Then, in Section 6, a genetic algorithm is proposed for solving thisproblem. In Section 7, we examine the parameters and solutions generated by the mathematical model. Finally, the conclu-sions and directions for further research are provided in Section 8.
2. Interferences
Boarding interference is deﬁned as an instance of a passenger blocking another passenger’s access to his (or her) seat. Wedeﬁne two types of interferences: seat interferences and aisle interferences. Seat interferences occur when passengers seatedclose to the aisle block other passengers seated in the same row. Aisle interferences occur when passengers stowing luggagein overhead bins block other passengers’ access to seats.We intend to develop a model to minimize expected boarding interferences. The decision is to assign each passengerboarding the airplane to a boarding group, to minimize boarding interferences. The objective function includes all the dif-ferent interferences that could possibly occur during boarding. Each of these interferences has a certain penalty, and thesum of all the penalties corresponding to a particular seat assignment determines the objective value.
3. Problem description
We assume thatthereis a correspondence between minimizingthe expected numberof passengerinterferences andmin-imizing the boarding time. We deﬁned just two types of interferences: seat interferences and aisle interferences. We attemptto minimize the total number of interferences subject to operational and side constraints. The model provides the ﬂexibilityto optimize the boarding times for various aircraft (e.g., Airbus-320 aircraft, Boeing 737) with different numerical values(e.g., seat capacities, number of boarding groups, weights for aisle and seat interferences). The model also allows us to exam-ine the impact of number of boarding groups and the speed that these groups are called on the overall adopted boardingstrategy. Here, we assume that for any passenger in group
k
(
k
> 1) boarding the aircraft, there is a fraction of passengersfrom the previous group (
k
1) still in the jet-way trying to reach to their rows and seats. We call this fraction
a
. When
a
is 0, no aisle interferences occur between groups. This occurs when a new group of passengers is called to board the aircraftand all passengers in the previous group are fully seated and when
a
is 1, maximum aisle interferences occur betweengroups. This occurs when a new group of passengers is called to board the aircraft and all passengers in the previous groupare in the jet-way. On the other extreme, when
a
is equal to 1, the time between calling groups to board is so short that thepassengers in each group line up behind the previous group in the aisle or jet-way. In our analysis, we examine various val-ues for
a
and examine the impact on boarding pattern and strategy.The assumptions for the proposed mathematical model are:
Boarding of passengers is applied to an Airbus-320 aircraft which has 23 seat rows with three seats in every side of theaisle.
Boarding of passengers is the same in two sides of the aisle.
All seats are fully occupied.
The passengers are boarded in groups 4, 5 and 6.
Upper and lower bounds for passengers of each group are ﬁnite.
A passenger is assigned only to one group and every seat is assigned to one group.
Boarding groups are called to board the aircraft one by one in a sequence.
Passengers are seated on seats in the order: window, middle, aisle.
4062
M. Soolaki et al./Applied Mathematical Modelling 36 (2012) 4060–4072
The notations, the objective functions, constraints and model properties are given next.
Setsi
= {1,2,3,
. . .
,
N
} index set of seat rows
j
= {
A
,
B
,
C
,
. . .
,
b
} index set of seat columns
k
= {1,2,
. . .
,
G
} index set of passenger groups
Models parametersN
= 23 number of airbus 320 rows
b
=
F
index of last seat column of airbus 320
G
e
{4,5,6} numerical values for passenger’s boarding groups
a
e
{0,0.1,0.3,0.5,0.7,0.9,1} fraction of passengers in the previous group still in the jet-way trying to reach their rowsand seatsVan Landeghem and Beuselinck [2] use triangular distributions (3, 3.6, 4.2) and (1.8, 2.4, 3) s in their models for seat and aisleinterferences respectively. Similar time parameters are used in the simulation studies by Ferrari and Nagel [1] and Bazargan[9]. We adopted the mean of these distributions to represent the penalties. These distributions have a mean of 3.6 and 2.4 forseat and aisle interferences respectively. Without loss of generality, we assign the same weight to aisle (aws, Tag) interfer-ences as follows:
p
1
= 3.6 seat interferences
p
2
= 2.4 aisle interferencesTypically, the airlines favor a balanced number of passengers among different groups and here, we assume boarding of pas-sengers is the same in two sides of the aisle. So, we set min
p
and max
p
on each side of the aisle for every group, to allow amaximum of 10% ﬂuctuations around the mean as follows:
min
p
¼
N
3
G
0
:
9
;
max
p
¼
N
3
G
1
:
1
;
min
p
= 21, max
p
= 24 for passenger group 4,min
p
= 15, max
p
= 18 for passenger group 5,min
p
= 12, max
p
= 15 for passenger group 6.
N
3 = all of the passengers on each side of the aisle.Similar to Van de Briel et al. [8] and Bazargan [9], we consider an Airbus-320 airplane with 26 rows. The ﬁrst three rows
(with 4 seats in each row) are assigned to ﬁrst/business class (group 1) and always board ﬁrst. In our model, we study allother passengers who are assigned to the other 23 rows (
N
= 23) and 4, 5, 6 groups (
G
= 4, 5, 6).
Model’s decision variables:
As indicated earlier, boarding of passengers is the same in two sides of the aisle. So, to keep the model simple at this stageand without loss of generality, we only present the model for left hand side of the aisle.
q
i
,
k
e
{0, 1, 2, 3} number of passengers of group
k
to be located in row
isw
i
,
k
within groups seat interferences
Tsw
sum of total within groups seat interferences
aw
1
i
,
k
within groups aisle interferences for row
i
and group
k
by passengers of row
i
1
aw
2
i
,
k
within groups aisle interferences for row
i
and group
k
by passengers of row
iaws
sum of total within groups aisle interferences in the same row
ab
i
,
k
between groups aisle interferences for row
i
and group
k
with the passengers in the previous group
k
1
abg
sum of total aisle interferences within groups with previous rows and aisle interferences betweengroups
Tag
sum of total within groups seat interferences
y
1
i
,
k
,
y
2
i
,
k
,
y
3
i
,
k
,
y
4
i
,
k
binary variables used for linearization
M
= 4000 a large positive number
Note
: The above variables are all assumed to be nonnegative numbers.
M. Soolaki et al./Applied Mathematical Modelling 36 (2012) 4060–4072
4063
4. Constraints of interferences
4.1. Within group seat interferences
This type of interference occurs among the passengers boarding in the same group. We assume that the sequence of pas-sengers boarding within a group is random. This type of occurs if at least 2 of the 3 seats in a row are occupied by passengersof the same group.For boarding, each seat has a special priority, so that, window seat (A) has a higher priority than middle seat (B) andmiddle seat has higher priority than aisle seat (C). Also, each passenger has his own priority in choosing her seat. If the pas-senger chooses the seat with the highest priority as her ﬁrst choice, we say that interference does not exist; otherwise, aninterference exists. For example, when there is two passengers of the same group in the jet-way and the ﬁrst passenger seatson the window seat, then no interference occurs for the second passenger, but if the ﬁrst passenger seats on the middle seatand the second passenger desires to have the window seat, then one interference occurs. Since the sequence in line israndom, in this case we consider the scope of the interference to be
12
.When there are three passengers of the same group in the jet-way, since the sequence in line is random, in this case theinterference is considered to be
3
þ
2
þ
1
þ
2
þ
0
þ
16
¼
32
. Table 2 shows within group seat interferences for various sequences.Table 3 shows amount of interferences for different quantities of decision variables. Since the interferences are on the twosides of the aisle, we multiply these interferences by 2.We use the following constraints to calculate seat interferences:
sw
i
;
k
P
ð
q
i
;
k
1
Þ
;
8
i
;
8
k
;
sw
i
;
k
P
3
ð
q
i
;
k
2
Þ
;
8
i
;
8
k
;
Tsw
¼
X
N i
¼
1
X
Gk
¼
1
sw
i
;
k
:
4.2. Within group aisle interferences in the same row
This type of interference occurs when a passenger blocks other passengers of the same row behind him/her in order to beseated. Table 4 shows different quantities of decision variables for this type of interference.In Table 4, the expected values of the interferences in each row corresponding to the number of passengers of each groupare listed. For example, when the number of passengers from the same group in one side of the aisle is 3, then the expectedvalue for this type of interference is 15 because the best situation for a passenger is to be in front of the other 5 passengerssince the expected interference for this passenger is 0 and in the worst situation is 5 and it is why he seat after the other 5passengers. Since the sequence of passengers boarding within a group is random and the number of passengers in each rowis 6, then the expected value is
0
þ
52
þ
0
þ
52
þ
0
þ
52
þ
0
þ
52
þ
0
þ
52
þ
0
þ
52
¼
15 .We use the following constrains to calculate within group aisle interferences:
Table 2
Within group seat interferences for various sequences.
Sequence of passengersin lineNumber of within group seatinterferences..A..B..C.. 3..A..C..B.. 2..B..C..A.. 1..B..A..C.. 2..C..B..A.. 0..C..A..B.. 1Sum 9
Table 3
Size of interferences for different quantities of decision variables.
q
i
,
k
sw
i
,
k
0 01 02 13 34064
M. Soolaki et al./Applied Mathematical Modelling 36 (2012) 4060–4072

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