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ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING CADARSO, Luis; MARÍN, Ángel
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ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING
Luis Cadarso, Universidad Politécnica de Madrid, [email protected]
Ángel Marín, Universidad Politécnica de Madrid, [email protected]
ABSTRACT
In scheduled air transportation, airline profitability is influenced by the airline's ability to
construct flight schedules. To produce operational schedules, airlines engage in a complex
decision-making process, referred to as airline schedule planning. Because it is impossible to
simultaneously solve the entire airline schedule planning problem, the decisions required
have historically been separated and optimized in a sequential manner.
We propose a multi-objective integrated robust approach for the schedule design phase,
considering the passenger behaviour, deciding jointly flight frequencies and timetable. The
objectives are passengers' satisfaction and operator costs. We try to fix the timetable
ensuring that enough time is available to perform passengers' connections, making the
system robust avoiding misconnected passengers. Some test networks are solved in order to
demonstrate the achieved robustness and choose an appropriate objective.
Keywords: robustness, airline schedule design, multi-objective.
INTRODUCTION
Commercial aviation operations are supported by what is probably the most complex
transportation system and possibly the most complex man-made system in the world
(Barnhart and Cohn, 2004). Airports represent the nodes on which the system is built.
Aircraft represent the very valuable assets that provide the basic transportation service.
Passengers demand transportation between origins and destinations, and request specific
travel times. Crews operate the aircraft and provide service to passengers. These entities are
coordinated through a flight schedule, comprised of flight legs between airports.
In order to produce operational schedules, airlines engage in a complex decision-making
process, referred to as airline schedule planning. Most of the time the schedule planning
starts from an existing schedule. Then, changes are introduced to the existing schedule to
reflect changing demands and environment; this is referred to as schedule development.
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However, we will suppose that a coldstart is needed, that is, we must create the schedule
planning from scratch.
There are three major components in the schedule development step. The first step, the
schedule design, is arguably the most complicated step of all. This step is the one we will
treat in this work. The purpose of the second step of schedule development, fleet
assignment, is to assign available aircraft types to flight legs such that seating capacity on an
assigned aircraft matches closely with flight demand and such that costs are minimized. In
base to these flows the network is decomposed into sub networks, each one associated only
with a particular fleet type. Given these sub networks, the assignment of individual aircraft to
flight legs is done in the aircraft maintenance routing step, the third step. Crew scheduling
involves the process of identifying sequences of flight legs and assigning both the cockpit
and cabin crews to these sequences.
The final goal in airline scheduling is to integrate all phases into a single one. Integrated
models would optimize schedules, capacities, pricing and seat inventory. However
integrating the planning phases is a big challenge: dynamics and competitive behaviours,
organizational coordination...
Traditionally the schedule design has been decomposed into two sequential steps. The first,
the frequency planning, in which planners determine the appropriate service frequency in a
market; and the second one, the timetable development, in which planners place the
proposed services throughout the day, subject to network consideration and other
constraints.
Airline schedule design, including how to determine a network's type, flight frequencies and
timetable for each flight leg, is a prerequisite for any airline's operational planning such as
fleet assignment, routing and crew assignment. Network design is heavily important since the
chosen network type, flight frequency and timetable directly influence the operating
effectiveness of the airline and the quality of service provided to passengers.
Designing an airline network is an extremely complex task due to the huge number of
variables affecting the design, i.e. passenger demand, ground facilities and capacity, the
competence, etc. These issues are not always easily modelled and usually result in huge
models.
The most important issue is probably the demand forecasting. Thus, accurately forecasting
the future passenger demand on each market is of priority concern in the planning and
design of an airline network. However, accumulating a large number of data with good
statistical distribution to develop conventional statistical forecasting models is a challenging
task. Besides, the uncertainty in other input data also complicates the design of the airline
network. For example, a situation frequently arises in which we cannot know the operational
costs for possible new routes in the schedule that have not been performed before. Airlines
try to generate the lowest possible operating costs and achieve a higher load factor, while
passengers concern about flight frequencies, nonstop flights, and in case of stops minimum
stop time.
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In case of intermediate stops in itineraries, passengers must perform a flight connection. In
order to accomplish this connection an undetermined time is needed by passengers. Airlines
usually design itineraries trying to make big enough connection time. However, this issue
makes passengers to be dissatisfied. In order to avoid this situation, a new robustness
criterion is introduced. Every intermediate stop will always have a minimum connection time;
however, this time will not be enough in some situations. In order to avoid misconnected
passengers, a penalty based on statistical data is proposed. In this way, expected
misconnected passengers will be penalized, but also accounting for passengers’
dissatisfaction.
State of Art
Generally, the approach is to cast these problems as network design models. In the past,
there have been efforts to improve the profitability of the schedule. Simpson (1966) presents
a computerized schedule construction system that begins by generating demand using a
gravity model, then solves the frequency planning problem and, finally, constructs a flight
schedule and solves vehicle optimization problem upon that schedule. Chan (1972) provides
a framework for designing airline flight schedules covering route generation and route
selection. The previous presented work was performed before deregulation of the passenger
air transportation industry in 1978.
Soumis, Ferland and Rousseau (1980) consider the problem of selecting passengers that will
fly on their desired itinerary with the objective of minimizing spill costs. No recaptures are
considered. Flight schedules are optimized by adding and dropping flights. When flights are
added or dropped, their heuristic recalculates demands only in markets with significant
amounts of traffic. Then, the passenger selection problem is resolved. Their method can be
viewed as an enumeration of all possible combinations of flight additions and deletions.
Marín and Salmeron (1996) apply the frequency planning to rail freight transportation. The
formulation of the rail freight transportation design model is based on the modelling of the
physical network, the services and the demand. They study the problem with the help of non-
convex optimization models which they solved using heuristic methods to obtain the solution
for realistic networks. Marín, Barbas and Gallo (1999) propose a model where the timetable
was developed from the frequency planning. The objective is in general to minimize the total
passenger delay cost.
Armacost et al. (2002) describe a new approach for solving the express shipment service
network design problem. Under a restricted version of the problem, they transform
conventional formulations to a new formulation using what they term composite variables.
The formulation relies on two key ideas: first, they capture aircraft routes with a single
variable, and second, package flows are implicitly built into the new variables, the composite
variables.
Lately, researchers have focused on determining incremental changes to flight schedules,
producing a new schedule by applying a limited number of changes to the existing schedule.
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Lohatepanont and Barnhart (2004), in their incremental optimization approach select flight
legs to include in the flight schedule and simultaneously optimize aircraft assignments to
these flight legs. Garcia (2004) extends the previous model and proposes a combination
between it and a decision time window model.
Lan, Clarke and Barnhart (2006) consider passengers who miss their flight legs due to
insufficient connection time. They develop a new approach to minimize passenger
misconnections by retiming the departure times of flight legs within a small time window.
They present computational results using data from a major U.S. airline and showing that
misconnected passengers can be reduced without significantly increasing operational costs.
Kim and Barnhart (2007) consider the problem of designing the flight schedule for a charter
airline. Exploiting the network structure of the problem, they develop exact and approximate
models and solutions, and compare their results using data provided by an airline. They
show that the heuristic approach is capable of generating good solutions very quickly.
Jiang and Barnhart (2009) propose a dynamic scheduling approach that reoptimizes
elements of the flight schedule during the passenger booking process. They recognize that
demand forecast quality for a particular departure date improves as it approaches; thus they
redesign the flight schedule at regular intervals, using information from both revealed
booking data and improved forecasts.
Our contributions in this paper include the following:
1. As market demand may be stimulated as a result of changes in the flight schedule,
airlines try purposefully to design schedules to capture the largest demands, so we
include the demand and supply interaction in the context of airline schedule design.
2. Passengers' itineraries; we have the possibility of representing misconnected
passengers due to lack of time to perform intermediate stops. Robustness is
introduced avoiding misconnected passengers.
3. Passengers' recapture; as far as we include passengers' itineraries we have the
chance of including recapture in a realistic way.
4. Airports capacities for arrivals and departures.
We have developed a new integrated robust model to solve the schedule design problem in
one unique step. As a proof of the model we have done some computational experience.
Synopsis
The paper is organized as follows: in section 2, we consider the supply modeling, in which
the used time expanded graph is introduced. Section 3 explains the passengers behavior
that we have considered, that is, the demand modeling. In section 4, the robustness criterium
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is introduced. In the following section the robust airline scheduling model is introduced.
Section 5 proposes the multiobjective optimization. And finally, in section 6 the computational
experience is described. Conclusions and references are included in the followings sections.
MODELING SUPPLY
The objective of schedule design is to develop a schedule, defining an origin, a destination, a
departure time, and an arrival time for each service to accommodate passenger demands
given available resources. Given the estimated demand for travel, an airline wishes to
determine the flight schedule which maximizes its profit while taking into account the
satisfaction of its customers. In this system, two components interact: the aircraft flow in the
physical network, and passengers using flights to travel. The most important transport supply
data include, among others, flight costs, travel time, capacities, etc.; variables include flight
frequencies and its timetable.
The network is built considering the airports associated to the demand to be met. It is formed
by the airports or nodes and all the feasible airway or sections alternatives linking them. The
airports are defined by the operations that can be performed within them. The sections are
the links between the airports. Each section is characterized by an origin airport and a
destination airport. Each section has other technical characteristics as the section time and
the capacity of the airplane assigned to it. The time is given by an average speed and the
capacity by number of passengers that can be moved in each airplane using the section.
In this way, we will consider that the time is discretized by partitioning the planning period, T,
into intervals of equal length with starting points 0,1,...,T-1. The intervals' length will be taken
as the time unit. As an example, if the period T corresponds to one week, and if the intervals'
length is of one hour, then time 0 corresponds to 0:00 a.m. of Monday and time T
corresponds to 12:00 p.m. of Sunday. The number of periods is 168.
When a section is flown it will be called as flight leg; a flight leg is defined by an origin,
destination and a departure time, that is, a flight leg is defined by the pair (s,t), where s is an
element of the sections' set, S, and t in {0,1,...,T-1} is the departure time from the origin of S.
The set of all possible flight legs is I = S x {0,1,...,T-1}.
The proposed supply model is based on the definition of the time graph G= (K,A), where the
nodes are:
K= {(k,t) / k K is an airport and t {0,1,...,T-1},
and the arcs are: A=A1 A2; with
A1= {(k,t),(k',t')} / s S and i I, where i={k,k',sts,t=t'-t} / k,k' K and t,t' {0,1,...,T-1}
A2= {(k,t),(k,t+CT)} / t,t+CT {0,1,...,T-1}.
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The arcs in A1 correspond to flight legs, i.e. the possible physical movements of passengers
between two airports, sts,t represents that movement's time duration; those in A2 correspond
to connection times where CT is the connection time for a flight connection. Note that a
section instance or flight leg (s,t) defines in G a path going from the node (k,t) corresponding
to the initial airport at time t, to the node (k',t') corresponding to the terminal airport, and (t'-t)
is the section time duration.
Since the type of flight network that an airline uses has a dominant impact on many of the
planning problems, we will mention the common network types. Flight network is an informal
name for the geographical network created by the flights operated by an airline timetable.
There are three types (Lohatepanont and Barnhart, 2004) of airline networks: linear
networks, point-to-point networks and hub-and-spoke networks.
The network types described above are the pure definitions. In reality it is seldom the case
that an airline has a pure point-to-point or hub-and-spoke network. Most airlines have some
sort of hub, housing their main maintenance facilities and crew headquarters. And in most
hub-and-spoke networks multiple hubs exist, as well as some direct flights between outlying
airports.
Supply interacts directly with demand and vice versa. Hub-and-spoke networks illustrate
demand and supply interactions. To see this, consider removing a flight leg arriving or
departing at a hub airport. The removal of a flight from a hub can have serious effects on
passengers in many markets throughout the network. The issue is that the removed flight
does not only carry local passengers from the flight's origin city to the flight's destination city,
it also carries a significant number of passengers from many other markets that have that
flight leg on their itineraries.
In order to avoid interactions between flights of the same airline, that is, the competence, a
time separation is introduced for departure times of the same sections. Thus, when a section
departs, it cannot depart again until some separation time has been spent. In this way,
competition for the uptake of demand is avoided.
MODELING DEMAND
For this work the demand is characterized by the origin, p, and destination, q, airports. Each
pair (p,q) is mentioned as the market w. For each pair w the demand of passengers dw is
assumed fixed and known datum. This demand will vary in time, that is, it is a dynamic
demand, as we can see in Figure 1. For each demand, the passengers from origin to
destination are considered in all possible itineraries or routes r, that may be classified by pair
w as Rw.
In schedule design for a given airline, we are interested in its unconstrained market demand,
that is, the maximum demand the airline is able to capture. Unconstrained market demand is
allocated to itineraries or passenger routes, sequences of connecting flight legs, in each
market to determine unconstrained itinerary or route demand. The demand is unconstrained
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because the quantity of interest is measured without taking into account capacity restrictions.
In the proposed model, the unconstrained market demand is assigned to passengers' routes
taking into account the capacity; in this way we obtain the constrained passengers' route or
itinerary demand.
Each itinerary or route is defined by a set of sections that connect different airports. It can be
composed of one section or more than one including in this last case intermediate stops at
different airports. Considering large number of sections for the network, the number of
possible passenger itineraries grows exponentially. Different proportions of the same market
demand can be routed by different ways in order to use as well as possible the network
capacity.
Due to the necessity of a symmetric flight schedule, empty flights may appear. In order to
avoid them an average demand is required by the airline for flight legs. This requisite could
become in disrupted passengers in the real world that we are not taking into account. This is
because enforcing a minimum demand, passengers willing to travel at determined time may
be obeyed to do it at a different time. In order to avoid this, we also introduce the demand-
supply interaction. The demand-supply interaction is represented by the possibility of not
attending all the demand. In this way, non-profitable demand will be neglected.
Figure 1 – Market demand disaggregation
To represent passengers' preferences we use market disaggregation, that is, for each market
demand we separate it in blocks of demand requiring approximately the same average
departure time. These average times can be obtained from passengers' surveys. The market
disaggregation is made as in Figure 1. For a specific market, we separate it in groups with
the same required average time atw.
Passengers' dissatisfaction costs represent the difference between their required departure
time and the one assigned by the model, and intermediate stops. When these costs are high
or there is not enough capacity, they may be lost to the system or recaptured in other
compatible market; this possibility is known as passenger recapture. With partial recapture,
only a percentage of passengers will accept travel on an alternative itinerary, and that
percentage depends on the desired and the offered alternative itinerary.
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ROBUSTNESS
As mentioned before, robustness is introduced through passengers in itineraries with more
than one flight, where a connection is mandated. Adding more slack for connection can be
good for connecting passengers, but can result in reduced productivity of the fleet; the
challenge then is to determine where to add this slack so as to maximize the benefit to
passengers without getting worse the network operation (Lan, Clarke and Barnhart (2006)).
Every connection is characterized by the minimum time required to perform it. This time
varies from airport to airport and it can also vary in the same airport along the day. If a
passenger is not able to perform the connection due to lack of time, the passenger will be
misconnected.
In this way, in itineraries with more than one flight, every passenger is mandated a minimum
connection time (MCT) for flight connections. However, this time will not be always enough to
perform the connection due to congested airports for example, and passengers will be lost to
the system in the real world.
We assume that the number of disrupted passengers depends on the available time to
perform the flight connection. Once flights' arrival (AT) and departure (DT) times are known,
the available connection time (CT) is also known. From airlines historical data, disrupted
passengers number variability with connection time might be known and, consequently its
number may be calculated for each flight connection.
Assigning a statistical distribution to misconnected passengers, the probability of getting
misconnected passengers depending on connection time can be calculated. For our test
networks the exponential distribution has been chosen; misconnected passengers will
decrease exponentially as the available connection time increases. Its probability distribution
is as follows:
, where depends on the itinerary connection characteristics and is chosen adjusting the
probability distribution to historical data; it is supposed that once the connection
characteristics are known (airport and time at which it is performed), the assigned gates will
be probably known due to historical availability. ECT represents the available excess
connection time, that is, the available time exceeding the minimum connection time
(ECT=CT-MCT). In this way, given the available excess connection time (ECT), the
probability of having misconnected passengers ( ) is:
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Once misconnected passengers are known, they must be removed from the remaining flights
of the itinerary, so extra capacity arises in those flights making possible to accommodate
other passengers in it in case of disrupted passengers.
AIRLINE SCHEDULING MODEL
In the proposed model we will assume some issues. We consider the passengers transfer
possibility, that is, for every passenger itinerary we will consider the possibility of
intermediate stops in the flight. We will consider flights composed of up to two flight legs or
one intermediate stop; this issue is mandated for most passengers in hub and spoke
networks but not in point to point networks.
We will suppose that the unconstrained demand number for each market disaggregation is
known; to obtain the actual attended demand we include the demand and supply interaction.
In this way, the attended demand will be function of the capacity assigned for each flight leg.
We will not enforce the entire demand satisfaction. The neglected demand is penalized in the
objective function. However, we do enforce the demand maintaining in its entire flight, that is,
in the two flights legs of its itineraries, for example.
In supply's side, we include airports capacities, that is, we enforce that the airplanes arrivals
and departures must satisfy the airport's runway capacity. Each section has a determined
capacity for each condition (for example, weather conditions), so the airplanes number at
each period must be limited. In this way, we also include the section time duration
dependency on time departure; this is due to the possibility of busy airports, bad weather
conditions, etc. In the supply aspect the most important issue is the fleet size; it will
determine the flight legs that may be performed in the planning period, and consequently, the
attended demand.
Finally, we suppose that the schedule will be periodic, that is, the schedule will repeat after
the planning period ends. For this purpose, we must take care about airplanes location at the
end of the planning period. Its location must be the necessary one to repeat the schedule. In
this way, we will enforce for each airport to have the same number of arrivals and
departures.
As we have said above the schedule design is comprised of two steps: the frequency
planning and the timetable development. Historically, this process has been done
sequentially, that is, first the frequency planning problem is solved, and then, with
frequencies as inputs, the timetable is developed. In this work we define the Robust Airline
Scheduling Model (RASM) which solves both at once.
The following notation is introduced to explain RASM:
Sets:
periods' set.
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sections' set. Each section is defined by an origin, , and a destination,
: markets' set. We now define the markets by the origin, , destination, , and the
average departure time .
airports' set.
: itineraries' set.
itineraries' set composed of more than one section.
: itineraries' set attending market .
: markets' set attended by itinerary .
: itineraries' set containing section as first section.
: itineraries' set containing section as second section.
: sections' set arriving at airport .
: sections' set departing from airport .
compatible markets for passenger recapture.
: feasible departure time set for the second
flight leg in itineraries with more than one flight leg.
Parameters:
: operating cost in section instance .
: passengers' dissatisfaction in market using itinerary at period.
: passengers' dissatisfaction due to transhipments times in itinerary with more than
one section, being the second one
: cost per disrupted passenger from market .
recapture rate from market to . Its value depends on markets times.
: cost per disrupted passenger in itinerary due to lack of time to perform transhipments.
: passenger capacity in each section .
: maximum airplane arrival capacity of airport at each time period .
: maximum airplane departure capacity of airport at each time period .
: maximum airplane capacity in each section and period .
: minimum separation time between two consecutive departures of section instances (in
periods).
: passenger demand for each market .
: section instance trip time. We include the section trip time duration dependent on
departure time; this is due, i.e., to congested airports or weather conditions which may obey
to slow down the airplane.
: relative time to the planning period.
: 1, if flight leg ( ) is flying at period time.
: minimum connection time for each itinerary departing at time period .
: minimum average demand required by the airline in section .
: fleet size.
: likelihood that passengers from ( ) will be misconnected in flight connection with
( ).
: it is a real parameter. It represents the time needed to make operative a flight schedule,
accounting for network flows. Its value depends on the network size and the fleet diversity
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that will be assigned. Its value grows with both variables, network size and fleet diversity. Its
value ranges from 2.25 to 3.75.
Variables:
: positive variable. Passengers in itinerary and market departing at period.
: positive variable. Passengers using itinerary at period , and the second
section , at period . This auxiliary variable represents passengers using a flight leg after a
transfer or transhipment.
: positive variable. The airline tries to disrupt passengers from market to market .
: =1, if section departs at period; 0, otherwise.
The RASM is defined by the following objective functions and constraints:
Objective Functions Coefficients
As we have seen above, we have two different objective functions: one measures the
passengers costs and the other one the operator costs.
Passengers' Costs
Passengers costs are composed of the dissatisfaction. This term measures the difference
between the required average departure time and the actual departure time assigned by the
model. The more the difference is, the more the penalization is. However, we can suppose
that little differences will give little dissatisfaction. In this way, we decide to use a quadratic
function for the time penalization. This penalization is as follows:
The time dissatisfaction will be null if the assigned departure time is equal to the required
average departure time. However, the overall dissatisfaction might not be null, for example, if
the route is composed of more than one flight leg. To the previous formulation we add the
following term (4), representing the transhipment time:
The constants K1 and K2 may be calibrated through passengers surveys and transform the
time units into costs units.
Operator's Costs
Disrupted passengers are passengers that the company does not attend due to lack of
capacity or high dissatisfaction costs. In this way, we could think that these costs are related
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to passengers because they do not travel. However, we can consider them as spill costs,
that is, lost revenue. This costs can be computed as the distance the passengers would go if
they were attended by the spill cost. But as we do not have the distance that passengers in
each market would go until the model assigns them to routes, we can consider an average
distance for each market. In this way, the spill costs may be computed as the product
between and the average distance in a market , where is the revenue for
available seat-kilometre. However, for disrupted passengers due to lack of time to perform
the transhipment, the distance they would have flown is well known and these spill costs
may be calculated as the route's distance by .
Operating costs are the costs the company incurs due to the operation of flight legs. We
include the costs related to each section length, , and those related to the departure or
arrival time, for example slot costs. We compute these costs as:
, where is the cost for available seat-kilometre, and captures the departure time, ,
costs modifications.
We try to minimize the number of disrupted passengers due to misconnections. In this way
we try to introduce the robustness criteria defined above. The expected misconnected
passengers ( ) will be as follows:
Objective Functions
The first objective ( ) function (7) accounts for passengers costs: the first term is the
dissatisfaction cost with departure time and, the second one is the dissatisfaction with
intermediate stops. The second objective ( function (8) accounts for operator costs, that
is, the first term represents operating costs, the second one incurred costs due to disrupted
passengers, that is, spill costs, and the last one, costs due to lack of time to perform
transhipments.
Passengers Constraints
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Constraints (9) ensure the passenger demand allocation through available itineraries in the
network; they account for disrupted and recaptured passengers. Group of constraints (10)
ensures that passengers in two sections itineraries remain in their trip during the second
section; they also consider the average necessary time for performing transhipment between
two sections of the same passenger route; note that for the same market demand attended
in the same first flight leg, it can be satisfied through different second flight legs. Constraints
(11) ensure that each flight leg has an average demand mandated by the airline;
misconnected passengers are removed from the flight leg through the term .
Constraints (12) ensure that there are enough active sections or flight legs to satisfy the
passengers flow; the capacity in these active sections is a very important issue in the model.
Depending on this value, the schedule will strongly change. This value must be estimated
from demand models, airlines requirements, airports constraints, ect. Once again,
misconnected passengers are removed from the flight leg.
Flight Legs Constraints
Constraints (13) are section capacity constraints; they ensure that the number of aircraft in a
section at each period is lower than a maximum number; this capacity may depend on air
navigation systems and regulations. We must adequate the aircraft number for every period
of time to the allowed one. Group of constraints (14) ensures that the same flight leg does
not depart until a specified time has been spent; this time is the separation time between two
consecutive flight legs; in this way, competence between flights from the same airline is
avoided.
Airports Constraints
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Constraints (15)-(16) are airport capacity constraints that try to spare the departures and
arrivals at airports at each period; this is mandated by the available slots in the airport to land
or take off. Depending on the time, these slots may vary in costs. These costs are included in
the operating costs.
Fleet and Symmetry Constraints
Constraints (17) are the fleet capacity constraints; we must count the necessary aircraft to
perform the schedule and compare it to the available ones. Block of constraints (18) ensures
that the flight network is symmetric. In this way, the obtained schedule may be repeated
periodically.
Variable Dominion
Constraints (19)-(22) define the variable dominion. As the demand number is an average
value, passengers' variables can be defined as positive variables.
MULTIOBJECTIVE OPTIMIZATION
The ASM has been developed considering the case of multiple design objectives:
passengers' and operator's costs. The multiobjective optimization problems (MOP) are
generally solved by combining the multiple objectives into one scalar objective, whose
solution is a Pareto optimal point for the original MOP.
A standard technique in multiobjective optimization is to minimize a positively weighted
convex sum of the objectives. It is easy to prove that the minimiser of this combined function
is Pareto optimal (Ehrgott, (2005)). But, the problem is up to the user to choose appropriate
weights. Until recently, considerations of computational expense forced users to restrict
themselves to performing only one such minimization, considering just one set of weights
chosen with care. Nowadays, more ambitious approaches aim to minimize convex sums of
the objectives for various settings of the convex weights, therefore generating various points
in the Pareto set. Though computationally more expensive, this approach gives an idea of
the shape of the Pareto surface and provides the user with more information about the trade-
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off among the various objectives. In this, weighted aggregation approach, different objectives
are weighted and summed up to one single objective.
We can generally define the ASM as follows: , where is a
vector containing the problem variables, and is the feasible region of the problem.
With multiobjective optimization techniques the problem then becomes as:
.
A Pareto boundary can be found by assigning varying values to .
COMPUTATIONAL RESULTS
Introduction
Computational tests study the Pareto Optima curves showing different solutions by varying
objective coefficients. Then, an appropriate value is chosen for each case in order to
compare the achieved robustness in this new approach with a no robust one.
All of our computational experience is for tests cases proposed below. Three different
networks are studied, all of them Hub and Spoke (HS) networks. In Figure 2 the first network
HS1 is shown: it is characterized by one hub and three spokes; each spoke is connected to
the hub in both senses. The following network HS2 in Figure 3 has two different hubs and
eight outlaying airports. Finally, the third network HS3 is drawn in Figure 4.
For each network there are different markets and passenger routes. In network HS1 there
are 34 markets and 12 passengers routes; in network HS2 there are 270 markets and 58
passenger routes; and, in network HS3 there are 1260 different markets and 210 passenger
routes.
Our runs have been performed on a Personal Computer with an Intel Core2 Quad Q9950
CPU at 2.83 GHz and 8 GB of RAM, running under Windows 7 64Bit, and our programs have
been implemented in GAMS 23.2/Cplex 12.
Figure 2 –Air network HS1 Figure 3 – Air network HS2
ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING CADARSO, Luis; MARÍN, Ángel
16
Figure 4 –Air network HS3
The RASM size is shown in Table 1. The RASM number of discrete and continuous
variables, constraints and non-zero elements are given for the model in proposed networks.
Table 1 – RASM size
Network Variables Constraints Non-zero
elements Discrete Continuous
HS1 584 90596 4512 319372
HS2 1112 245043 10093 851423
HS3 2870 1023277 29246 3429380
Pareto Curves
In order to determine the best value for the multiobjective approach some experiments
have been carried out; one group of them for each presented network. Each group of
experiments consists of obtaining the Pareto curve for every network by varying the value.
However the detailed results are shown, that is, every term in the objective function is drawn
in the following figures depending on . In Figure 5 the results for the network HS1 are
shown, in Figure 6 for network HS2 and, finally in Figure 7 for network HS3. The blue line
represents the objective function values, the red one operating costs, the green line spill
costs, and, the purple one dissatisfaction costs. When it is necessary an additional line is
drawn in light blue colour, it represents modified spill costs. In the graphics, every marked
point represents a solution to the mixed integer model.
The value will be chosen accounting for dissatisfaction costs and spill costs. This is due to
the fact that operative costs remain almost constant. Thus, the point where dissatisfaction
and spill costs are equal will give the optimal value. In order to determine it correctly,
disrupted passengers must be accounted for ; at this point, only operator's costs are
ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING CADARSO, Luis; MARÍN, Ángel
17
taken into account, so the operator tries to attend all the possible demand without accounting
for dissatisfaction costs; thus, passengers not attended at this point will be probably
disrupted for every value. So, spill costs curve is moved downward a quantity equal to spill
costs at . The intersection between this modified spill costs curve and dissatisfaction
curve will give the optimal value.
Figure 5 –Air network HS1 computational results
Figure 6 –Air network HS2 computational results
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
0 0,2 0,4 0,6 0,8 1
Objective Function
Operative costs
Spill Costs
Dissatisfaction Costs
Modified Spill Costs
0
200000
400000
600000
800000
1000000
1200000
1400000
1600000
1800000
2000000
0 0,2 0,4 0,6 0,8 1
Objective Function
Operating Costs
Dissatisfaction Costs
Spill Costs
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
0 0,2 0,4 0,6 0,8 1
Objective Function
Operating Costs
Dissatisfaction Costs
Spill Costs
ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING CADARSO, Luis; MARÍN, Ángel
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Figure 7 –Air network HS3 computational results
Under the criterion explained above, an objective coefficient value is chosen. This value will
be used in the following subsection addressing robustness in the proposed study cases.
Although pareto curves have been drawn continuously, we have to note that only the marked
points correspond to real obtained solutions, and that there will not probably exist a different
solution for every value because the RASM is an integer mixed problem.
Robustness
As it was explained above, robustness is achieved through passengers that must perform
flight connections. In order to demonstrate that a more robust schedule is obtained using the
proposed approach, a comparison is made with a no robust Airline Scheduling Model (ASM).
The ASM is the same model explained above but removing robustness aspects, that is, the
objective function's term in (4) penalizing misconnected passengers, and the terms in
constraints (7) and (8) accounting for misconnected passengers in flight leg's minimum
demand and capacity, respectively.
In Table 2 misconnected passengers are compared for every network. For each network in
the first column, the optimal value that has been chosen appears in the second column. In
the third column the percentage of misconnected passengers is shown for the ASM; this
percentage is calculated with respect to the total number of attended passengers. In the last
column the percentage of misconnected passengers is shown for the robust case (RASM).
For every network the percentage is sensitively reduced.
Table 2 – Misconnected passengers
Network ASM RASM
HS1 0.19 5.8 0.16
HS2 0.25 13.4 0.48
HS3 0.38 14 0.30
Robustness is achieved through the reduction in misconnected passengers. However, this
reduction is not for free, it has a price: the robustness price. I order to analyse this concept,
objective function values are shown in Table 3 for every study case. In the robust case
(RASM), objective function's values are greater than the no robust one (ASM). However, this
increase may be due to the term of misconnected passengers that is not included in the no
robust case (ASM). To clarify this aspect information about some objective function terms' is
provided in Tables 4 and 5.
In Table 4 operating costs are shown. For HS1 study case, operating costs are greater in the
robust approach (RAMR). However for the rest of the study cases these costs are greater for
the no robust case (ASM). Consequently, it cannot be said that the price of robustness falls
on operating costs.
In Table 5 passengers' dissatisfaction costs are written for every study case. One must note
that these costs are always greater in the robust case (RASM), that is, in the robust
ROBUST PASSENGER ORIENTED AIRLINE SCHEDULING CADARSO, Luis; MARÍN, Ángel
19
approach where misconnected passengers number has been reduced; passengers'
dissatisfaction has been increased. This is due to the fact that in the robust approach
departure times are chosen accounting not only for passengers satisfaction and capacities
but also for misconnected passengers in itineraries with more than one flight leg. Thus, it can
be concluded that the price of robustness remains in passengers' satisfaction.
Table 3 – Objective function
Network ASM RASM
HS1 0.19 777312.10 787009.78
HS2 0.25 1343476.21 1399941.26
HS3 0.38 2380931.84 2492877.56
Table 4 – Operating costs
Network ASM RASM
HS1 0.19 595554.06 602897.31
HS2 0.25 1197655.17 1174586.53
HS3 0.38 3199454.90 3156868.01
Table 5 – Dissatisfaction costs
Network ASM RASM
HS1 0.19 218988.91 219409.11
HS2 0.25 562065.80 591901.27
HS3 0.38 669697.06 772547.35
CONCLUSIONS
A new robust approach has been proposed to solve the airline scheduling problem, where
frequency and timetable problems are jointly solved. In addition, passengers' flows are
obtained through the different itineraries in the network.
Market demand and supply interaction have been included, making possible to stimulate
demand through flight schedule changes. Furthermore, passengers' partial recapture has
been included in a realistic way; this due to the fact that market demand is allocated to
itineraries. As far as itineraries are composed of more than one flight leg, intermediate stops
have been included, accounting for passengers dissatisfaction.
Airports' arrival and departure capacities have been included. In the presented approach we
suppose that these capacities (slots) are well known, and that they usually are associated to
determined gates. It also may be included the purchase of new slots, however, nowadays
this issue is a very difficult and time consuming task in some aiports.
Robustness has been introduced through itineraries with more than one flight leg. When an
intermediate stop must be performed, passengers need some undetermined time to
accomplish it. This undetermined time is captured through statistical distribution and it is
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introduced into the model to represent expected misconnected passengers. In this way, the
expected costs that the operator would incur due to misconnected passengers are reduced.
The model has been tested in three different networks. Computational results show how
robustness may be achieved. However, this robustness has a price. The robust approach
has been compared with a no robust approach showing the price of the achieved robustness.
Future research may integrate the airline scheduling problem and the fleet assignment
problem. In this way, the average values used in the proposed approach would be
substituted by real values depending on the assigned fleet type.
REFERENCES
Armacost, A., Barnhart, C. and Ware, K. (2002). Composite Variable Formulations for
Express Shipment Service Network Design. Transportation Science, 36, 1-20.
Barnhart, C. and Cohn, A. (2004). Airline Schedule Planning: Accomplishments and
Opportunities. Manufacturing & Service Operations Management, 6, 3-22.
Chan, Y. (1972). Route Network Improvement in Air Transportation Schedule Planning.
Flight Transportation Laboratory R72-3, Massachusetts Institute of Technology,
Cambridge, MA.
Garcia, FA. (2004). Integrated Optimization Model for Airline Schedule Design: Profit
Maximization and Issues of Access for Small Markets. Department of Civil and
Environmental Engineering and the Engineering Systems, Massachusetts Institute of
Technology, Cambridge, MA.
Ehrgott, M. (2005). Multicriteria Optimization. Springer.
Jiang, H. and Barnhart, C. (2009). Dynamic Airline Scheduling. Transportation Science, 43,
336-354.
Kim, D. and Barnhart, C. (2007). Flight Schedule Design for a Charter Airline. Computers &
Operations Research, 34, 1516-1531.
Lan, S., Clarke, J.P. and Barnhart, C. (2006). Planning for Robust Airline Operations:
Optimizing Aircraft Routings and Flight Departure Times to Minimize Passenger
Disruptions. Transportation Science, 40, 15-28.
Lohatepanont, M. and Barnhart, C. (2004). Airline Schedule Planning: Integrated Models and
Algorithms for Schedule Design and Fleet Assignment. Transportation Science, 38,
19-32.
Marín, A. and Salmerón, J. (1996). Tactical Design of Rail Freight Networks. Part I: Exact
and Heuristic Methods. European Journal of Operational Research, 90, 26-44.
Marín, A., Barbas, J. and Gallo, G. (1999). Railway Freight Scheduling Using Bender's
Decomposition. Working Paper, Universidad Politecnica de Madrid.
Simpson, RW. (1966). Computerized Schedule Construction for an Airline Transportation
System. Flight Transportation Laboratory, Massachusetts Institute of Technology,
Cambridge, MA.
Soumis, F., Ferland, JA. and Rousseau, JM. (1980). A Model for Large Scale Aircraft
Routing and Scheduling Problems. Transportation Research, 14, 191-201.