Robust Flight’ Scheduling in a Colombian Domestic Airline
Alejandro Cadavid Tobón, Pablo Andrés Maya Duque and Juan Guillermo Villegas
Departamento de Ingenieria Industrial, Facultad de Ingeniería, Universidad de Antioquia, Medellín, Colombia
Keywords: Flight Scheduling Design, Robustness, Profitability, Applied Operations Research, Timetabling.
Abstract: Air traffic has been grown rapidly, increasing the airlines’ competition, generating complex planning
problems for airlines and major customers’ demands. Airlines’ profitability is highly influenced by its
planners ability to face these challenges and build efficient schedules. In this paper, we developed a bi-
objective optimization model for the timetabling problem of a Colombian domestic airline. Preliminary
results show an increase of 12% respect to the current profitability of the airline.
1 INTRODUCTION
According to statistics from Colombian Civil
Aviation Authority (Aeronautica Civil, 2013), the
Colombian market for domestic air passengers
increased by 21.7%, corresponding to 1.49 million
of passengers, over the previous year in the first five
months of 2013. In turn, the load factor of the
market increased from 75.3% to 77.0% in the same
period. Besides, international passengers exhibit a
similar trend. By May of 2013, the number of
international passengers increased in about 403,000
passengers compared to the same period of 2012.
This traffic increase, the strong competition among
airlines and passenger demand for better services,
have created complex planning problems for
airlines, which require new models and solution
methods (Dorndorf et al., 2007).
All this has led to the airlines to spend
considerable time in a complex decision process
called airline planning (Cadarsoa and Marín, 2011).
This process seeks to produce an operational
program and it is composed of the following five
stages: fleet planning, flights’ network planning,
revenue management, crew scheduling and planning
of airport resources (Lohatepanont, 2002).
This paper focuses on a problem that arises in the
flights’ network planning stage. It begins about 12
months and lasts about 9 months before the
deployment of the program (Lohatepanont, 2002).
This stage comprises several subproblems since it
has been deemed untreatable because of its
computational complexity. Therefore, several
subproblems are optimized sequentially and the
output of one is taken as the input of the next one
(Papadakos, 2009). These subproblems are named:
schedule design, fleet assignment, maintenance
programming of aircrafts and sometimes also
include crew scheduling (Barnhart et al., 2003) .
Within the flights’ network planning, the
schedule design subproblem addresses the most
important decisions for an airline. These decisions
determine the profit of the airline because they
define which markets operate, including cities,
routes, frequencies and hours to be offered in the
day (Weide, 2009). Usually, the schedule design
problem is divided into two steps: the frequency
planning and the development of the timetables
(Cadarsoa and Marín, 2011). This paper focuses on
the latter step, i.e., the development of the
timetables.
In frequency planning, planners determine the
appropriate number of frequencies for a market
(Lohatepanont, 2002). Increases in the frequency of
departures on a route, commonly improve
convenience for customers and in turn the airline can
benefit from increased traffic and associated
revenues, provided that this increase is accompanied
by a market study to ensure that the operation is
profitable (Belobaba et al., 2009) .
After, the frequency planning process, the next
step is the development of the routes (also known as
timetable development), where the planners decide
the day and hour in which each flight will be
offered. The result of the timetable development is a
list of flights, with dates and departure and arrival
times, called basic programming (or itinerary)
(Rabetanety et al., 2006) .
156
Cadavid Tobón A., Andrés Maya Duque P. and Guillermo Villegas J..
Robust Flight’ Scheduling in a Colombian Domestic Airline.
DOI: 10.5220/0005220401560161
In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 156-161
ISBN: 978-989-758-075-8
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
Airlines commonly design their schedules under
the premise that all flights arrive and depart on the
planned hours. This scenario is rarely met, leading
the airline to incur in additional costs (Lan, 2003).
An efficient flight scheduling can contribute to
increase the level of service and customer
satisfaction. Under these ideas, the quality of a
scheduling is measured by its level of robustness
(Bian et al., 2005). The robustness of a schedule can
be defined as the ability to start all the flights
scheduled on time despite of the delays in their
predecessors. To achieve this goal, a schedule has to
include some firewalls or time windows without
programmed flights such that they can absorb the
flights delays through the day and the following
flights can depart on time.
This paper addresses the timetabling problem in
a Colombian domestic airline scheduling design.
Currently, the airline constructs its schedules based
on its planners’ expertise and lacks clearly defined
robustness measures. In this paper, we propose a bi-
objective optimization model and a solution method
designed to achieve optimal schedules that increase
profitability and take into account service measures
such as robustness.
The remainder of this paper is organized as
follows: Section 2 presents a brief literature review.
Section 3 introduces the proposed bi-objective
optimization problem. Section 4 summarizes the
results of preliminary computational results. Finally,
Section 5 gives conclusions and outlines future
research possibilities.
2 LITERATURE REVIEW
Since the late 50’s, operations research has played a
fundamental role on helping airline industry to
sustain high rates of growth. Thus, over 100 airlines
and air transport associations created the Air Group
of Operations Research Societies (AGIFORS) in
1961(Barnhart et al., 2003) . Within the list of air
transport issues in which operations research has
contributed through optimization and stochastic
models are: airline fleet planning, maintenance
planning, decision support tools for managing air
operations, classical problems of flights scheduling
and crews assigning, revenue management, flights
performance management, among others (Barnhart
et al., 2003) .
Regarding the frequency of scheduling, the
literature distinguishes between daily, weekly and
dated - problems. The first one assumes that the
schedule repeats every day with the same flights in
each of them. The second one assumes that
scheduling repeats weekly and the flight may vary
on some days of the week. And the third one
considers that there are no restrictions about the
replication of flights for different days (Weide,
2009). In this work we address a daily scheduling
problem.
Table 1: Solution Techniques for Flights Scheduling
Models.
Solution
techniques
Works
Lag
r
angian
relaxation
(Chen et al. 2010)(Yan &
Tseng 2002)(Sherali et al.
2009)
N
etwork
techniques
(Stojkovi & Solomon
2002)(Tan
g
et al. 2008)
Heuristics (Tang et al. 2008)(Kim &
Kim 2011)(Babic et al.
2011)(Yan et al. 2008)(Clarke
1998)(Weide 2009)
Metaheuristics (Jungai & Hongjun 2012)
(Kim & Barnhart 2007)(Lee et
al. 2007)(Burke et al. 2010)
Colums
Generation
(Barnhart et al.
1998)(Papadakos 2009)
Benders
descomposition
(Mercier et al.
2005)(Papadakos 2009)
Commonly, flights scheduling models are large scale
in terms of the number of variables requiring
solution methodologies that decompose and reduce
the size of the problem. Usually, these problems
have been solved by column generation, branch-and-
cut or branch-and-price algorithms, lagrangian
relaxation and Benders decomposition (Weide et al.,
2008). However, as Table 1 illustrates, approximate
techniques such as heuristics and metaheuristics
have also been used.
The aforementioned techniques have been
applied in several real world problems. For instance,
(Kim and Kim, 2011) considered the planning of
operations in a military aviation unit. They deal with
the problem of assigning flight missions to aircraft
and schedule those tasks. The authors developed
heuristic algorithms to reduce the time required to
complete all missions, they conclude that the
heuristics reach near optimal solutions in reasonable
computational times. Similarly, (Cadarsoa and
Marín, 2011) presented a robust approach that
integrates frequency planning and timetable
development in a simple model in order to build
economic solutions. Their model was implemented
at IBERIA airline.
RobustFlight'SchedulinginaColombianDomesticAirline
157
Several robustness measures have been studied, the
most common are: the probability that a flight can
connect to any next flight (Sohoni et al., 2011), the
probability of having misconnected passengers
(Sohoni et al., 2011), the deviations of optimal
departing hours, the minimization of the costs of
deviations from the optimal scheduling, the
minimization of flight delays, and the capacity to
recover the operation after a delay (Lan, 2003).
The reviewed literature reveals that the
simultaneous consideration of schedule robustness
and usual airlines targets is seldom studied.
However, (Sohoni et al., 2011) proposed an integer
programming formulation that perturbs optimally a
given schedule in order to maximize expected profits
while maintaining service levels.
In the Colombian context, flights schedules are
generally built based only on the planners or
managers expertise. Additionally, in the airline
under study there is a lack of a solution method that
targets the schedule robustness and take into
account, simultaneously, the usual airlines goal of
maximizing its profitability. This paper aims at
address these two issues.
This paper aims to fill this gap through the
construction of a bi-objective optimization model
and solution method, which maximize airline
profitability, considering the robustness of the flight
schedule.
3 PROBLEM DEFINITION
Through this section, the term timetable or itinerary
is defined as the final configuration of the schedule,
which provides the time at which each route will be
offered.
Given a set 1,2,…, of cities to be
connected the frequency
represents the number of
required flights connecting the origin city ∈ and
the destination city ∈. The set 1,2,…,
corresponds to the time slots (i.e., hours available to
operate the flights), and
, ∈ ) represents
the duration of each flight leg.
3.1 Problem Representation and
Notation
The flight-scheduling problem is represented
through a space-time graph, as it is shown in Figure
1. A vertical move in this graph represents a travel
between cities, while horizontal moves represent a
temporal movements between time slots Using this
representation, a feasible flight is represented by the
arc that joins the origin city ∈ and the
destination city ∈ , taking off at hour ∈ and
landing at hour ∈ . The profitability of an arc
is defined as the revenue generated by operating a
flight at a given departure hour. Figure 1 shows the
arc representing a flight from city C to city E,
starting at time 1 and ending at 5. Note that only
feasible arcs are included in this representation (i.e.,
arcs where
=
Figure 1: Space-time graph.
Considering the feasible set of arcs the notation used
to formulate the problem is as follows:
Sets
Set of cities that makes up the origin and
destination of a leg
Set of available aircraft
Set of day’s hours
Set of feasible arcs
Parameters
Frequency for leg , , ∈
Profitability of arc ∈
Origin of leg ∈
Destination of leg ∈
Start time of leg ∈
Ending time of leg ∈
Maximum allowed arrival time of an aircraft
to its base ∈
Number of aircraft at base ∈
A non-negative real number ∈0,1
Maximum profitability
Maximum robustness
Some assumptions were made for the construction of
the following optimization model: (i) The rotation
time required to prepare an aircraft for the next flight
is included into the length of time for each flight arc.
(ii) Two flights with the same destination cannot
take off at the same time from the same city of
origin. (iii) It is possible to have more than one
ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems
158
plane parked at the same time in any city. (iv) An
aircraft returns to its base at the end of the day. (v)
The fleet is homogenous, i.e., all airline aircrafts are
of the same type.
3.2 Mathematical Model
The flight-scheduling problem was formulated as a
bi-objective optimization model where the two
objectives are profitability and robustness. These
objectives were combined in a single objective using
a weighting method after scaling their magnitudes to
make them comparable.
∗
∗
∈
1
∗
∈|
(1)
∈
|
∀ ∈ ,
∈
(2)
∈
|
∀ ∈
(3)
∈
|
∀ ∈
(4)
∈
|
∈
|
∀ ∈ ; ∈ | 1,
(5)
∈
0,1
∀|
∈
(6)
∈ ∀|
∈
(7)
The first term of the objective function (1) aims at
maximizing the profitability of the timetable, while
the second part seeks to maximize the scheduling
robustness by creating time windows with aircraft
parked at some cities to absorb the delay of previous
flights. The constraint set (2) ensures that the
number of frequencies defined for each leg is met.
Constraints (3) guarantee that at the beginning of the
day the number of planes that leave each city is
equal to the number of planes located in each base.
Constraints (4) requires that to each base city arrive
the total number of aircrafts corresponding to them
at the maximum arrival time
. Constraints (5)
maintain the balance between the number of planes
that goes in and that comes out at each node in the
graph. Finally, constraints (6) and (7) define the
nature of the decision variables.
Notice that decision variables take binary values
for the cases in which the origin and destination
cities are different. This is due to the assumption that
two flights with the same destination cannot take off
at the same time in a given origin city. On the other
hand, decision variables can take integer values for
the cases in which these cities are the same. This
was necessary in order to model the fact that one or
more aircrafts can be parked in the same city during
a given period of time. In that case, the arc
corresponding to the same origin and destination
would take the value of the number of aircrafts
parked in that city.
4 COMPUTATIONAL
EXPERIMENTS
The model (1)-(7) was implemented using Xpress
7.5 and Gurobi 5.6.2 and all the computational
experiments were run in a computer with an Intel
Corei3-2350M processor running under Windows 8
at 2.30GHz with 6GB of RAM.
The data to run the model was gathered from a
Colombian domestic airline. Based on this data, we
created a realistic instance corresponding to their
flight-scheduling decision. From this instance, we
created other 11 instances that correspond to
interesting scenarios that the company may face in
the near future. The convention used for naming the
instances is such that, the first two digits indicate the
number of aircrafts included, the following two
digits represent the number of legs to be scheduled
Table 2: Results for instances.
Instance
Profitabili
ty Robustness O.F
Time
(s)
043210 575 236 811 1.3
043411 604 194 798 2.2
043410 600 201 801 1.8
053812 760 347 1107 5.4
054213 964 215 1179 6.2
054613 993 132 1125 6.1
065014 1100 269 1369 3.8
065215 1134 222 1356 10.7
065415 1146 186 1332 35.5
076014 1273 307 1580 4.3
075813 1280 349 1629 1.1
044211 944 33 977 5.6
RobustFlight'SchedulinginaColombianDomesticAirline
159
Figure 2: Solution obtained.
and the last two digits stand for the number of cities
to be connected. These instances are available from
the author upon request. Table 2 shows the results
obtained for each of these instances.
For the realistic instance 044211, Figure 2 shows
the structure of the solution obtained. Each line
shows the path of each aircraft (flight legs flown for
each aircraft).
To explore the trade-off between objectives we
changed systematically ( ={0,0.05,…,0.95,1.0})
to approximate the efficient frontier for the bi-
objective flight-scheduling problem. The trade-off
between the two objectives using different weights,
for the more realistic instance, is shown in Figure 3.
Moreover, in this instance, and where the two
objectives have the same importance, the model
obtains a good solution in terms of both profitability
and robustness since this model found a solution that
improves the daily average airline profitability by
12%. While in terms of robustness the solution
presents 155 minutes of firewalls to recover the
operation if case of some flight delays.
In terms of computing time, taking into account
that this type of decisions is revised every year and
Figure 3: Trade-off between objectives.
since the solution for each value of takes less than
6 seconds, these running times seem reasonable.
5 CONCLUSIONS
The computational experiments shows that is
possible to solve exactly the timetabling problem
related to the flight scheduling of a Colombian
domestic airline. Moreover, the efficient frontier
obtained with a weighting method reveals that there
exist a trade-off between profitability and
robustness. However, the robustness measure used
in this paper has some limitations, since it does not
take into account aspects like the durations of the
firewalls and their spread through the day that are
important in terms of the quality of the delays
absorption. However, we are already working on
improving it.
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