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Airline schedule development continues to remain one of the most challenging planning activity for any airline. An airline schedule comprises of a list of flights and specifies the origin, destination, scheduled departure, and arrival time of each flight in the airline's network. A critical component of the schedule development activity is the choice of flight block-times, which depend on several factors. Many airlines decide schedule block-times based on fixed percentiles of block-time distributions built from historical data, however, such techniques have not resulted in significantly improved on-time performance of the schedule during operations. Thus, from a passenger's perspective, the service level guarantee of an airline's network continues to be low. We first define two service level metrics for an airline schedule. The first one is similar to the on-time performance measure of the U.S. Department of Transportation and we define it as the flight service level. The second metric, called the network service level, is geared towards completion of passenger itineraries. We then develop a stochastic integer programming formulation that optimally perturbs a given schedule to maximize expected profit while ensuring the two service levels. We also develop a variant of this model that maximizes service levels while achieving desired network profitability. To solve these models we develop an efficient algorithm that guarantees optimality. Through extensive computational experiments, using real-world data, we demonstrate that our models and algorithms are efficient and achieve the desired trade-off between service level and profitability.
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TRANSPORTATION SCIENCE
Robust Airline Scheduling under Block Time
Uncertainty
Milind Sohoni
Indian School of Business, Hyderabad, India, milind sohoni@isb.edu
Yu-Ching Lee
University of Illinois at Urbana-Champaign, Urbana, IL, ylee77@uiuc.edu
Diego Klabjan
Northwestern University, Evanston, IL, d-klabjan@northwestern.edu
Airline schedule development continues to remain one of the most challenging planning activity for any
airline. An airline schedule comprises of a list of flights and specifies the origin, destination, scheduled
departure, and arrival time of each flight in the airline’s network. A critical component of the schedule
development activity is the choice of flight block-times, which depend on several factors. Many airlines
decide schedule block-times based on fixed percentiles of block-time distributions built from historical data,
however, such techniques have not resulted in significantly improved on-time performance of the schedule
during operations. Thus, from a passenger’s perspective, the service level guarantee of an airline’s network
continues to be low. We first define two service level metrics for an airline schedule. The first one is similar
to the on-time performance measure of the U.S. Department of Transportation and we define it as the flight
service level. The second metric, called the network service level, is geared towards completion of passenger
itineraries. We then develop a stochastic integer programming formulation that optimally perturbs a given
schedule to maximize expected profit while ensuring the two service levels. We also develop a variant of this
model that maximizes service levels while achieving desired network profitability. To solve these models we
develop an efficient algorithm that guarantees optimality. Through extensive computational experiments,
using real-world data, we demonstrate that our models and algorithms are efficient and achieve the desired
trade-off between service level and profitability.
Key words : robust scheduling, stochastic optimization, airline planning
1. Introduction
In a recent article (Associated Press 2007) The Associated Press reported that the U.S. airline
industry’s on-time performance (OTP) through the first eleven months of 2007, was the second
1
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
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worst on record. According to the U.S. Department of Transportation, a flight is delayed if it arrives
at its destination gate 15 minutes or more after its scheduled arrival time. Even in the previous
year, i.e., 2006, statistics showed that there were 823,030 arrival delays out of a total of 3,805,313
commercial flights operated by all the major U.S. carriers (Bureau of Transportation Statistics
2009). Flight delays and cancelations have been attributed to several causes some of which include
weather conditions, airport congestion, national air-space congestion, aircraft maintenance related
issues, and more recently airline security related services. Consequently, such delays lower service
reliability and adversely affect a commuter’s travel experience.
While some of the causes of delays, such as weather conditions, are beyond the control of the
airlines, previous research shows that some causes of delays are attributable to the network and
schedule design decisions of an airline. For example, while an airline develops its hub-and-spoke
network, it typically does not account for the congestion externality imposed on other carriers
operating out of the same hub stations. In a recent paper, Mayer and Sinai (2003a) empirically
demonstrate that the gains from hubbing activities offset the costs incurred by flight delays and
congestions. In such cases, congestion pricing at certain capacity constrained airports, may be a
solution to elevate the problem. In a companion paper, Mayer and Sinai (2003b) also hypothesize
that wage cost minimization and aircraft utilization maximization result in airlines flying with very
tight schedules. Such objectives are typical in most airline planning systems, which are designed to
achieve cost efficient resource utilization. Schedule planning models do not address the following
two important issues. First, they do not include passenger-centric service reliability measures in
the schedule development process. Second, the schedules ignore block-time uncertainty (variance)
and hence fail to capture robustness measures. In this paper we address these issues by devel-
oping schedule planning models that incorporate both, passenger centric metrics and block-time
uncertainty, in the planning process.
Airline schedule development continues to remain one of the most challenging planning activity
for any airline. An airline schedule comprises of a list of flights and specifies the origin, destination,
scheduled departure, and arrival time of each flight in the airline’s network. A critical component of
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 3
the schedule development activity is the choice of flight block-times. A flight block-time is defined
as the total elapsed time between the time an aircraft pushes back from its departure gate and
arrives at its destination gate. The block-time comprises of several components including taxi-out
time, enroute time, and taxi-in time. Each of these components is subject to different causes of
delay and the total block-time delay is the sum of all individual component delays. Since airline
schedules must be published well in advance of the actual day of operation, block-times, for all
the flights in the schedule, are typically decided using historical information of similar flights
operated in the past. The Department of Transportation OTP metric is computed against these
published flight block-times. Most airline operations are compared based on their OTP rankings and
hence airlines perceive their OTP as an important operational measure of their schedule reliability.
However, research indicates that airlines fail to adequately adjust block-times and typically do
not incorporate uncertainty in their planned schedules. Since most planned resource costs, such as
aircraft and crew utilization costs, depend on the cumulative hours in a schedule, airlines face a key
trade-off decision between adjusting (increasing) flight block-times to improve schedule reliability
and incurring additional planned costs. Using data made available by the Bureau of Transportation,
Deshpande and Arikan (2009) argue that airlines systematically “under-schedule” flights, i.e., the
amount of block-time allocated for a ight is less than the average block-time expected for the
flight. Conversations with planners at a large U.S. carrier suggested that airlines do not judiciously
allocate block-times to scheduled flights to balance costs versus operational benefits. Typically,
planners use ad-hoc techniques to either lower or raise block-times across the entire flight network in
the hope of increasing OTP. Results in Deshpande and Arikan (2009) also corroborate these findings
and indicate that airlines do not maintain consistent service levels by adjusting their schedules
based on the time of the day, origin airport congestion, and destination airport congestion.
Planning for uncertainty in the schedule building process becomes necessary not just to improve
OTP rankings but also to improve passenger service levels. As stated earlier, the goal of this paper
is to develop a robust optimization approach to schedule planning by specifically incorporating
passenger centric goals and block-time uncertainty in the planning models. The key trade-off in
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
4
such a process is between higher service levels achieved through increasing (and better allocation
of) flight block-times and higher planned costs (i.e., lower planned profits). In this paper we develop
a model that re-times (perturbs) a proposed flight schedule by considering block-time probability
distributions. First, we explicitly define notions of passenger and network service levels. Then we
develop a model that maximizes the expected profit while guaranteeing minimum service levels.
This model allows imposition of minimum service levels. Second, we develop a variant of this model
that maximizes service levels while achieving required profitability. While the optimization models
are complex, we also develop computational procedures, based on cut generation techniques, to
efficiently solve these models. To this end, this paper also has a methodological contribution to the
development of computationally efficient procedures. We provide extensive computational experi-
ments, using real airline data from a large U.S. carrier, that validate our model and demonstrate
potentially large operational gains for an airline. Overall network reliability is also improved.
The contributions of this paper are at several levels. First, to the best of our knowledge, this paper
is an initial attempt at developing a comprehensive and holistic model that includes block-time
uncertainty in developing robust schedules. Second, through chance constraints, we explicitly model
block-time distributions allowing us to incorporate operational uncertainty in the schedule planning
process. This makes the resultant schedule robust. We also incorporate network service levels, which
probabilistically model passenger connections. Third, we propose a new cut generation algorithm
to solve these stochastic binary integer programming models and establish its convergence. The
analysis is non-trivial since the feasible region of the original problem is non-convex and first a
linearization is required. Upon linearization, the resulting (modified) model is infinite dimensional
with infinitely many constraints. Thus, our algorithmic procedure and optimal convergence result
generalizes previously established convergence results for (1) semi-infinite linear programs with
finitely many variables but infinitely many constraints, and (2) infinite dimensional problems with
finitely many constraints and infinite number of variables.
Overall, this research is in line with the growing literature on linking operational variability
(and hence costs) to planning models. For example, research in robust fleeting (Rosenberger et al.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 5
2004); robust aircraft routing (Lan et al. 2003), robust crew scheduling (Shebalov and Klabjan
2006), and the robust approach to passengers rerouting in disruption management (Karow 2003)
show this emerging trend. Another growing area is the development of simulation systems of airline
operations, e.g., SimAir by Rosenberger et al. (2002) and MEANS by Bly et al. (2003). These
systems play a crucial role in evaluating and comparing the performance of different schedules. This
paper also contributes to several techniques developed in the airline schedule planning literature.
In general, airlines, though plagued with low profitability margins, airspace and airport congestion,
and high capital and operating costs are heavy users of mathematical optimization techniques
(Dobson and Lederer 1993, Lohatepanont and Barnhart 2004, Barnhart et al. 2003). Barnhart and
Cohn (2004) and Klabjan (2005) provide an extensive review of OR models used in airline schedule
planning. There is other literature in the domain of stochastic scheduling that is also related to
our work (see Portougal and Trietsch 2001). However, existing literature in stochastic scheduling
ignores the need to achieve high customer service level.
The rest of the paper is organized as follows. First in §2 we develop the two optimization models
for schedule perturbation. Next, in §3 we discuss issues related to the computational tractability of
these models and develop the solution methodology and optimal algorithms. We provide extensive
computational experiments in §4. Finally, in §5 we conclude the paper. Additionally, we provide
a complete set of results of all the other computational experiments in an online appendix (see
Sohoni et al. 2008).
2. Model Description
As discussed earlier, our goal is to develop a model to perturb the incumbent flight schedule to
improve the service levels provided to the end consumers. Perturbing a flight schedule implies
adjusting the scheduled departure times of flights1in the network within an allowable time window.
Soon after determining the flight schedule, the airlines determine capacity assignments (fleeting)
and assign generic aircraft to routes. The latter, in the literature, is referred to as the aircraft routing
1Throughout this paper we use the terms “flight” and “leg” interchangeably.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
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problem. It is after these processes that we consider the issue of schedule re-timing (perturbation)
to fine tune block-times and improve robustness of the schedule with respect to the service level
metrics defined later. While perturbing the incumbent schedule, however, we must guarantee that
the resulting schedule continues to remain feasible with respect to the aircraft turns built of the
incumbent schedule. Every flight in the incumbent schedule is assigned to exactly one aircraft. An
aircraft turn is essentially a pair of consecutive flights flown by the same aircraft. We assume that
the set of turns associated with the incumbent schedule is known a priori.
A passenger travel plan, or itinerary, may comprise of multiple flight legs. Broadly defined, a
fare class is the price an airline charges to book a passenger in a particular booking class. Airline
seats are divided into several booking classes. Next, we define the important modeling notation and
parameters.
N: The set of all flights (legs) in the airlines flight network,
B: the total available planned budget (depends on the total block-time across all flights),
O: the set of all passenger itineraries,
T: the complete set of aircraft turns,
F: the set of all fare classes,
αi: the origin station of flight i,
βi: the destination station of flight i,
mij : minimum passenger connection time between two flights iand j,
tij : minimum turn-time between flights iand jon the aircraft rotation,
Dof : expected demand for itinerary oand fare class f ,
[li, ui] : the allowable departure time-window for flight i,
ci: the per time unit cost incurred for flight i, which includes unit costs
corresponding to crew pay and aircraft utilization,
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 7
Bif : booking limit for fare class fon flight i,
rof : the average fare of itinerary oand fare class f ,
ds
i: the previously scheduled departure time of flight iin the incumbent schedule,
ei: the penalty for deviating from the preferred departure time of flight i, and
δ: the Department of Transportation OTP measure for flight delay
(typically 15 minutes after the scheduled arrival time).
Next, we define the decision variables of the model:
di: the published departure time of flight i,
ai: the published arrival time of flight i,
Xof : demand of itinerary oand fare-class fsatisfied, and
zij : binary variable indicating if the passenger connection between flights iand jis feasible.
We define dand ato be the set of departure and arrival times respectively. The only random
variables in the model are the block-times and are denoted by Yit where trepresents the departure
time of flight i. We assume that these are continuous random variables. The relation between
a flight’s departure time, arrival time, and the corresponding block-time is as follows: Ai=di+
Yidi, where Aiis the actual random arrival time of flight i. The probability density function of
a flight’s block-time is represented by pi(·, t) since it might depend on the departure time t. The
cumulative density function is assumed to have finite support [δi
l, δi
u]. To reduce the complexity of
our computational experiments we assume the following.
Assumption 1. The expected demand Dof for an itinerary does not vary significantly for rea-
sonable deviations in departure time.
Given that we disallow large perturbations of the departure time by controlling the time window
[li, ui] for every flight iin the network, it is reasonable to assume the following:
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
8
Assumption 2. For each flight iwe require that, pi(·, t) = pi(·), i.e., the pdf of the block-time
distribution does not depend on the departure time.
A flight jis said to follow-on flight iif passengers of flight ican connect to flight j. The set of
all passenger connections for flight idepends on the arrival time of flight iand departure times of
possible connecting flights. We define the connection set for flight ias follows.
Definition 1. The connection set for flight i, with respect to the incumbent schedule, is defined
as
Ci(d, a) = {jN:djaimij &βi=αj}.(1)
Building on the definition of Ci(d, a) we define a modified connection set ¯
Ci, which denotes the
largest set of possible connections for flight iunder any departure and arrival time adjustment.
For example, ¯
Cican be the set of all flights originating at station αi, or we can further refine the
set as
¯
Ci={jN:βj=αiand can connect to iregardless of re-timing}
=jN:βj=αi, ujli+δi
lmij .(2)
The advantage of using set ¯
Ciinstead of the original connection set Ci(d, a) is that, for any flight i
the latter set is non-stationary, i.e., as the departure time of flight ichanges, the flights in the set
may change. Thus it depends on the decision variables. As we show later, this poses a modeling
and optimization challenge since we cannot guarantee a convex feasible region.
We now define the Service Level, SLi, of any flight iN.
Definition 2. Service level SLiis the probability that passengers from flight ican connect to
any follow-on flight included in the set Ci(d, a), i.e.,
SLi= Pr [Ai+mij djfor every jCi(d, a)] .(3)
Observe that from definition 2 it follows that S Li= Pr[Yidimin
jCi(d,a){djdimij }]. The Network
Service Level (NSL) is defined as follows.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 9
Definition 3. The N SL is defined as the minimum service level across all the flights in the
airline’s network, i.e.,
NSL = min
iSLi.(4)
Finally, the Flight Service Level (F SL), also referred to as the OTP, is defined as follows.
Definition 4. The F S L is the probability that a particular flight is not delayed based on the
Department of Transportation acceptable arrival delay measure δ, i.e.,
F SLi= Pr[Yi,diaidi+δ].(5)
Lastly, for notational convenience, we denote the fact that flight jfollows flight iin an itinerary
oOby ij. Next, we describe the two optimization models.
2.1. Maximizing Operational Profits
We first consider the case when an airline must maintain a minimum F SL,γf, over all flights in
the network and simultaneously guarantee a minimum NSL of γn. The profit maximizing model
(PMM) reads:
(PMM) max X
o,f
rof Xof X
iN
ei|dids
i| − X
iN
ci(aidi) (6)
Pr [Yididjdimij]γniN , j Ci(d, a) (7)
Pr[Yi,diaidi+δ]γfiN(8)
X
iN
ci(aidi)B(9)
Xof Dof oO, f F(10)
X
oO,io
Xof ≤ Bif iN, f F(11)
X
o,f,jo,ij
Xof ¯
Kij zij iN, j ¯
Ci(12)
djaimij zij K(1 zij)iN , j ¯
Ci(13)
djaitij 0 (i, j)T(14)
lidiuiiN(15)
zij ∈ {0,1}, d, a unrestricted.(16)
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
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The first term in the objective function (6) corresponds to the net revenue due to satisfied
itinerary demand, the second term is the net penalty due to deviation from preferred departure
time (departure time specified in the incumbent schedule), and the third term represents the total
operational cost. Constraint (7) ensures that the minimum N SL is at least as large as the desired
value of γn. It is not difficult to observe that NSL γnif and only if constraint (7) is satisfied.
Constraint (8) guarantees that the minimal F SL is at least γf. Constraint (9) restricts the total
network operating cost incurred and constraint (10) restricts the fare-class itinerary demand to the
maximum allowable. Since every flight iwithin an itinerary ocan carry at most Bif of a particular
fare-class f, constraint (11) ensures that the booking limit constraint on each flight is satisfied.
Constraints (12) and (13) ensure that we only account for those itineraries whose flight sequence
is legal with respect to the minimum passenger connection time. Here ¯
Kij =X
f
Bif +X
f
Bjf . The
constant Kis the length of the time horizon, i.e., K= max
iNuimin
iNli+ max
iNδi
u. Constraint (14)
guarantees that the pre-determined aircraft turns are preserved and hence the aircraft routing
solution always remains feasible. Finally, constraint (15) bounds the departure time adjustment
for every flight and the constraint (16) restrict the choice of zij to be binary.
In §3, we discuss issues regarding the computational tractability of the optimization model
PMM. One peculiarity of P MM is immediately observable; the constraint set in (7) depends on
the decision variables.
2.2. Maximizing Service Level
An alternate goal could be to maximize the service level across the entire flight network. However,
the airline may only be willing to do so provided it maintains minimum operational profitability.
In this case the optimization model differs from the P M M model described earlier, i.e., γfand γn
are no longer parameters but are decision variables. Furthermore, the profit objective in P M M is
now a constraint. We impose that the minimum operational profit must be at least Uounits. The
service level maximizing model (SLMM) reads:
(SLMM) max wfγf+wnγn(17)
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 11
Pr [Yididjdimij ]γn0iN, j Ci(d, a) (18)
Pr[Yi,diaidi+δ]γf0iN(19)
X
iN
ci(aidi)B(20)
Xof Dof oO, f F(21)
X
oO,io
Xof ≤ Bif iN, f F(22)
X
o,f,jo,ij
Xof ¯
Kij zij iN, j ¯
Ci(23)
djaimij zij K(1 zij)iN , j ¯
Ci(24)
djaitij 0 (i, j)T(25)
X
o,f
rof Xof X
iN
ei|dids
i| − X
iN
ci(aidi)Uo(26)
lidiuiiN(27)
zij ∈ {0,1}, d, a unrestricted.(28)
The objective function (17) is a weighted sum of the minimal N SL and F SL quantities where wf
and wnare the weights corresponding to the F S L and NSL, respectively. All the other constraints
are similar to those described in P MM . The only additional constraint is (26) which ensures that
any solution makes an expected operational profit of at least Uo.
3. Solution Methodology
In this section we discuss issues regarding computational complexity and tractability of the models
discussed in §2. More importantly, we exhibit two algorithms for solving P M M and SLM M .
In the model P M M constraints (7) and (8) are non-linear. This makes the model difficult to
solve computationally. Similarly, in model SLM M constraints (18) and (19) are non-linear. Addi-
tionally, objective function (6) and constraint (26) contain the absolute value function, however,
it is straightforward to linearize these terms. A technical assumption regarding the block-time
distribution allows us to simplify the model and reduce its computational complexity.
Assumption 3. The block-time distributions are log-concave and stationary with respect to the
departure time.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
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Through extensive empirical studies using Bureau of Transportation Statistics data, Deshpande
and Arikan (2009) estimate the best distribution fit for observed truncated block-times across
several US airlines. Specifically, they use log-Normal and log-Laplace distributions. While the log-
Normal distribution provides a reasonable fit, they show that the log-Laplace distribution is better.
It is noteworthy that both of these cumulative distribution functions are log-concave (Bagnoli
and Bergstrom 2005) and thus satisfy Assumption 3. The Laplace distribution is defined by two
parameters: γ, a location parameter, and b, a scale parameter where the mean equals to γand
the variance is 2b2. The probability density function of the Laplace(γ , b) distribution is f(x|γ , b) =
1
2bexp |xγ|
b. Assumption 3 allows us to simplify the complicating chance constraints (8) and
(19) into convex constraints. Given that we assume the block-time distribution is independent of
the departure time we drop the departure time subscript, i.e., Yidi=Yi. Constraints (7) and (8)
are transformed as follows.
log (Pr[Yidjdimij ]) log γniN, j Ci(d, a) (29)
log (Pr[Yiaidi+δ]) log γfiN. (30)
It is known that the feasible set of constraint (30) is convex due to log-concavity (see, e.g., Birge
and Louveaux 1997). Unfortunately, constraints in (29) are not convex since their index depends
on dand a. This fact poses a significant algorithmic and computational challenge. To devise an
efficient solution strategy we first develop a linear approximation scheme to these constraints in
§3.1. The resulting mixed-integer model has an infinite number of variables and constraints. We
then describe a cut generation algorithm that generates these linear constraints as needed and
builds an optimal solution to the models.
3.1. Model Reformulation
Our goal in this section is to develop a linear formulation to the two models, P M M and SLM M.
Recollect that the N SL constraints given by equation (29) are non-convex. To circumvent this
issue, we construct a linear approximation for the NSL constraint over a stationary set of linear
functions as follows. The added advantage of doing so is that the reformulation allows us to develop
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 13
an algorithm, similar to the Bender’s cut generation algorithm (Birge and Louveaux 1997), to solve
the model.
Recollect that the distributions have a finite support Ki= [δi
l, δi
u]. Now, for every flight i, we
define a function gi(x) as
gi(x) = log Pr[Yix], x ∈ Ki.(31)
Since Yiis log-concave, gi(x) is concave, see, e.g. Birge and Louveaux (1997). To build an outer
linear approximation to equation (31), we consider a set of linear functions, Uik, defined over
interval Ki. We show the form of these linear functions later. For now, using these linear functions
we rewrite gi(x) as follows (this is a known fact in convex analysis):
gi(x) = min
k∈Ki
Uik(x).(32)
Using equation (32) we now reformulate the N SL constraint as
gi(djdimij )log γniN, j Ci(d, a).(33)
The above equation can be rewritten as
zij gi(djdimij)log γniN, j ¯
Ci,(34)
where ¯
Ciis defined by equation (2). Observe that log γn0 and thus inequality (34) holds if zij = 0.
If zij = 1, then jCi(d, a) and thus gi(djdimij )log γnmust hold, which is guaranteed by
constraint (34). Thus, constraint (29) is equivalent to
zij min
k∈Ki
Uik(djdimij )log γniN, j ¯
Ci.(35)
It is noteworthy that in (35) if djdimij 0, then zij = 0 and hence we need not worry about
negative arguments, i.e., we restrict our attention to positive values only.
We now characterize the functions Uik (x). Given the probability density function pi(·) for block-
time Yi, we can write these functions as
Uik(x) = pi(k)
Zk
0
pi(t)dt
(xk) + log Zk
0
pi(t)dt. (36)
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
14
To this end, notice that Uik(δi
lmij )<0 and Uik(δi
lmij )Uik(x) for all xδi
lmij (see Figure
1). This is the tangent of gi(x) at the point k∈ Ki. It is known that a concave function is the
minimum of its tangents and thus equation (32) holds. We still need to linearize constraints (35).
i
l
G
Figure 1 Linearization of constraints (35).
We now define additional continuous decision variables, sijk , for all iN,j¯
Ci, and k∈ Ki.
Constraint (35) can then be replaced by the following set of linear constraints:
sijk log γniN, j ¯
Ci, k ∈ Ki(37)
zij Uik(δi
lmij )sijk 0iN, j ¯
Ci, k ∈ Ki(38)
(1 zij )Uik(δi
lmij ) + sijk Uik (djdimij )iN, j ¯
Ci, k ∈ Ki.(39)
If zij = 0, then (38) implies that sijk = 0 and thus (37) holds. In this case, (39) also holds since
Uik(δi
lmij )Uik(djdimij ). On the other hand, if zij = 1, then we can assume that sijk =
min{0, Uik(djdimij )}and thus (37) holds if and only if Uik (djdimij )log γn.
Similarly, the F SL constraint given by equation (30) is equivalent to
min
k∈Ki
Uik(aidi+δ)log γfiN. (40)
It is clear that (40) is equivalent to
Uik(aidi+δ)log γfiN , k ∈ Ki.(41)
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 15
It is evident that the number of constraints in (37) - (39) and (41) is extremely large. Incorpo-
rating these constraints and variables a priori into the model is impossible. Hence, we must develop
an iterative cut generation algorithm that generates relevant inequalities at each iteration as the
solution progresses. A further complicating factor is the fact that we have an infinite number of
sijk variables (uncountably many).
As discussed earlier, in addition to the above service level constraints, the term X
iN
ei|dids
i|
is also a non-linear term in the objective function (6). However, this term can be linearized using
standard techniques and hence we do not discuss this linearization technique in detail. Next, in §
3.2, we describe the cut generation algorithm for the profit maximizing model P M M .
3.2. The Cut Generation Algorithm for P M M
Based on the constraint linearization procedure described earlier, in this section we develop a
constraint generation algorithm to solve our optimization model P M M .
We begin by ignoring the NSL and F S L constraints, i.e., constraints (35) and (40). Recollect
that we replace the original constraints (7) and (8) with these new constraints. In addition the term
X
iN
ei|dids
i|is linearized in the objective function. We refer to the resulting model, without these
constraints, as the restricted profit maximizing model RP M M . We initialize our algorithm with
RP M M . Let h0 denote an iteration step of the proposed algorithm. Further, let Zh=hzh
ij i,
dh=hdh
ii, and ah=hah
iidenote an optimal solution at the beginning of iteration h, i.e., after
solving RP M M. At every iteration let ¯
S(n)denote the set of new NSL constraints generated
and let ¯
S(f)denote the set of additional F SL constraints generated. Let Sdenote the set of
combined NSL and F S L constraints added to the restricted problem RP M M. Each time an
NSL constraint is generated, the corresponding svariable is also introduced into RP M M .
We list the steps of our constraint and variable generation algorithm in Algorithm 1. In Step 3 of
the algorithm we gather all the current passenger connections. Since kij is the function argument
in the right-hand side of (39), we need to consider tangents at this particular point (see Figure 1).
Flight jiis the index with the maximum violation in (37). Step 4.2, in Algorithm 1, introduces the
new svariable and adds the corresponding constraints (37) - (39).
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
16
Algorithm 1 Algorithm for solving P M M
Step 1: Initialize h= 1, S =and let RP M M consist of objective function (6) and constraints (9)-(16).
Step 2: Optimize RP M M with constraints in S. Let Zh, dh, ahbe the corresponding optimal solution.
Step 3: Build updated connection sets, i.e., for each flight iNcollect
Si=jN:zh
ij = 1.
Step 4: Check for the set of most violated N SL constraints. Set ¯
S(n)← ∅ and kij =dh
jdh
imij . For
each flight iN
1. Find
ji= arg max
jSilog γnUikij (kij ).
2. If log γnUikiji(kiji)>0, then define a variable si,ji,kiji, and generate constraints using (37) - (39) with
j=ji, k =kiji. Add these constraints to ¯
S(n).
Step 5: Check for the set of violated F S L constraints. Set ¯
S(f)← ∅ and ¯
ki=ah
idh
i+δ. For each flight
iN
1. If log γfUi,¯
ki(¯
ki)>0, generate a constraint using (41) with k=¯
ki, and add it to ¯
S(f).
Step 6: If ¯
S(f)¯
S(n)=,terminate;
Step 7: Set S ← S ¯
S(n)¯
S(f), h h+ 1, go to Step 2.
Next, in Theorem 1 we show that Algorithm 1 is guaranteed to converge to an optimal solution.
Theorem 1. There is a subsequence {hq}qsuch that d
i= lim
q→∞ dhq
i,a
i= lim
q→∞ ahq
ifor every
flight iNis an optimal solution to P M M .
Proof. See the appendix, §6, for the proof. 2
It is noteworthy that the proof of Theorem 1 also exhibits an optimal Z,s, and X. As a
result, Algorithm 1 converges to an optimal solution for model P M M . Furthermore, it is worth
emphasizing that the analysis is not trivial. As stated earlier the feasible region of the original
problem is non-convex, but, the linearization procedure allows us to circumvent this issue. However,
upon linearization, the modified model is infinite dimensional with infinitely many constraints.
Algorithm 1 and Theorem 1 generalize the convergence results achieved with (1) semi-infinite linear
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 17
programs with finitely many variables but infinitely many constraints, and (2) infinite dimensional
problems with finitely many constraints and infinite number of variables.
The computational time to convergence could still be an issue. While we cannot guarantee a
bound on the computational time, in §4 we demonstrate that the algorithm converges within
reasonable CPU time through extensive computational experiments using real airline data. Next,
in §3.3 we develop an approximate algorithm to solve SLM M .
3.3. Cut Generation Algorithm for SLMM
The algorithm to solve P M M can be modified to approximately solve the alternate model SLM M .
Notice that constraints (37) - (39), and constraint (41) are also valid for SLM M ; they replace
constraints (18) and (19). To enable a complete linear transformation we define variables ζn= log γn
and ζf= log γf. Thus, constraints (37) - (39) and (41) transform as follows:
sijk ζniN , j ¯
Ci, k ∈ Ki(42)
zij Uik(δi
lmij )sijk 0iN, j ¯
Ci, k ∈ Ki(43)
(1 zij )Uik(δi
lmij ) + sijk Uik (djdimij )iN, j ¯
Ci, k ∈ Ki(44)
Uik(aidi+δ)ζfiN , k ∈ Ki.(45)
We change the objective function of model SLMM using ζnand ζf, however, the objective func-
tion is now a non-linear function, i.e., wfexp{ζf}+wnexp{ζn}. Unfortunately, this is a maximiza-
tion problem of a convex function and thus is not easily amendable to computational tractability. To
simplify the computational procedure we use the first-order linear approximation of exp{x}= 1+x
which transforms the objective function to
max wfζf+wnζn.(46)
The new objective is an approximation of the original problem. Thus, any optimal solution to the
transformed objective function may not result in an optimal solution to the original problem. How-
ever, the linear approximation allows us to solve for the service levels efficiently. We demonstrate
this using several computational experiments in §4.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
18
In addition to the above service level constraints, the term X
iN
ei|dids
i|, in constraint (26), is
also non-linear. Just as in the case of P M M , this term can be linearized using standard techniques
and hence we do not discuss it in detail here.
As with the solution methodology for P MM , to begin, we ignore the N S L and F S L con-
straints, i.e., constraints (42)-(45). We refer to the resulting model, without these constraints, as
the restricted service level maximizing model RSLMM. In addition, the objective function is
replaced by (46). We initialize our algorithm with RSLM M . Let h0 denote an iteration
step of the proposed algorithm. Further, let Zh=hzh
ij i,dh=hdh
ii, and ah=hah
ii,ζh
f, and
ζh
ndenote the optimal solution at the beginning of iteration h. At every iteration let ¯
S(n)denote
the set of new N SL constraints generated and ¯
S(f)denote the set of additional F SL constraints
generated. Let Sdenote the set of combined N SL and F SL constraints added to the restricted
problem RSLM M . We list the steps used to solve SLMM in Algorithm 2. The steps are similar
to those in Algorithm 1.
Similar to Theorem 1 it is easy to verify that Algorithm 2 is guaranteed to converge to an
optimal solution for the approximate model of SLM M with the objective function (46). We state
this result as a corollary to Theorem 1 without proof.
Corollary 1. Algorithm 2 converges to an optimal solution of the approximate model SLM M
with the objective function (46).
Next, we describe the computational experiments.
4. Computational Experiments
In this section we describe a series of computational experiments using real airline data. The goal of
these experiments is twofold. The primary goal is to study the efficiency of the optimization models
P M M and SLM M to solve the robust scheduling problem. A secondary goal is to study the trade-
off an airline faces between higher passenger service levels (as defined by the N SL and F SL) and
the possible degradation in profit using the models described earlier. To this end, we implemented
the algorithms described in §3.2 and §3.3 to solve 5 airline network instances. We first describe
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 19
Algorithm 2 Algorithm for solving S LM M
Step 1: Initialize h= 1, S =and let RSLM M consist of ob jective function (46) and constraints (20)-
(28).
Step 2: Optimize RSLM M with constraints in S. Let Zh, dh, ah
i, ζh
fand ζh
nbe the corresponding
optimal solution.
Step 3: Build updated connection sets, i.e., for each flight iNcollect
Si=jN:zh
ij = 1.
Step 4: Check for the set of most violated N SL constraints. Set ¯
S(n)← ∅ and kij =dh
jdh
imij . For
each flight iN
1. Find
ji= arg max
jSiζh
nUikij (kij ).
2. If ζh
nUikiji(kiji)>0, then define a variable si,ji,kijiand generate constraints using (42) - (44) with
j=ji, k =kiji. Add them to ¯
S(n).
Step 5: Check for the set of violated F S L constraints. Set ¯
S(f)← ∅ and ¯
ki=ah
idh
i+δ. For each flight
iN,
1. If ζfUi,¯
ki(¯
ki)>0, generate a constraint using (45) with k=¯
ki, and add it to ¯
S(f).
Step 6: If ¯
S(f)¯
S(n)=,terminate;
Step 7: Set S ← S ¯
S(n)¯
S(f), h h+ 1, go to Step 2.
the characteristics of these network instances in Table 1. Due to confidentiality issues we report
only the underlying ranges. Instance 1 is the largest network covering all the fleets. Instances 3,
Instance Flights Stations Itineraries # Fleets
1 1500 85 50,000 5
2 450 75 30,000 2
3 850 80 45,000 3
4 1000 70 30,000 3
5 850 80 35,000 2
Table 1 Characteristics of network instances for the computational experiments.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
20
4, and 5 are relatively medium sized networks and instance 2 is the smallest network. In Table
1, the largest network consists of 5 fleets with varying capacities. The set of itineraries consists
of itineraries with up to 4 flights. All the networks are hub-and-spoke and the largest network
has 5 hubs. Given that there are a large number of flights in each network, we do not report the
block-time statistics for individual flights in these networks. To obtain block time distributions, we
analyzed the realized block-times over two consecutive years. We concluded that arrivals are never
earlier than 30 minutes before the scheduled arrival time based on the incumbent schedule, however
flights can be significantly late. As a result, we assume that the block-times follow a truncated
Normal distribution with a lower limit of 30 minutes before the scheduled arrival time and no
upper limit. The truncated-Normal distribution satisfies Assumption 3 (Bagnoli and Bergstrom
2005). Obviously, our computational results depend on the form of the assumed distribution and we
acknowledge the limitations of the results discussed in the section. However, as mentioned earlier,
our main goal is to demonstrate that the solution methodology performs well on the real-world
data. Additionally, our models allow planners to study important tradeoffs faced by an airline while
increasing schedule reliability. Fine tuning the distributions would definitely provide more accurate
results. The means of the block-time distributions vary from 36 minutes to 387 minutes and the
variances range from 24.9 to 595.3.
All the problem instances were solved on an Intel Xeon 3.2 GHz dual core server running Redhat’s
4.1 version of the Linux operating system. The cut generation algorithm, Algorithm 1, and its
variant for the SLM M model, Algorithm 2, were developed using the g++ compiler, version 4.1.
The mixed integer programming instances were solved using ILOG CPLEX version 10.1 and the
models were developed using the ILOG Concert library, version 2.3.
In the accompanying online appendix (Sohoni et al. 2008) we list all the detailed computational
results of all the instances described in Table 1. In this paper, however, to demonstrate the efficiency
of our models and algorithm, we only summarize some of the performance metrics of P M M , for all
the 5 instances, in Table 2. A priori, after adjusting for a few outliers, the NSL of the incumbent
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 21
schedule is 0.4 and the F S L is 0.6. Essentially, we ignored 10 flights with very low service levels
to compute the N S L and 5 flights to compute the F SL.
For the first set of experiments the F S L was held constant at 0.8 and the N SL level was allowed
to vary from 0.8 to 0.95 in increments of 0.1. We denote this set of experiments as Fixed-F SL. In
the next set of experiments the N SL was held constant and the F SL was allowed to vary from 0.8
to 0.95 in increments of 0.1. We denote this set of experiments as Fixed-N SL. All these instances
were solved to optimality. We report the maximum CPU time, maximum number of iterations,
and maximum number of cuts generated using Algorithms 1 and 2 in Table 2. The results show
Network Instance
Metric 1 2 3 4 5
Fixed-F SL (max CPU secs) 3,842 167 952 838 801
Fixed-F SL (max iterations) 27 10 17 12 13
Fixed-F SL (max cuts added) 980 227 767 315 428
Fixed-NSL (max CPU secs) 3,921 187 921 873 792
Fixed-NSL (max iterations) 24 8 13 8 10
Fixed-NSL (max cuts added) 987 243 793 343 437
Table 2 Algorithm performance for model P M M .
that P M M performs reasonably well on all networks, especially, considering the fact that schedule
development is performed several months prior to the day of operations and airlines do not mind
spending additional computation time. Furthermore, the results also indicate that both algorithms
converge within a few iterations.
For the remainder of the computational experiments we restrict our attention to instance 1
because it is the largest network. We discuss these experiments in §4.1 and §4.2. As mentioned
earlier, results for all other instances can be found in Sohoni et al. (2008).
4.1. The P M M Model
The first set of experiments are for model P M M . Through several experiments we demonstrate
the trade-off between higher service levels and planned profit. For all these experiments we restrict
the flight departure times to be adjusted within 60 minutes of those specified in the incumbent
schedule. Furthermore, the penalty for adjusting the departure time is assumed to be the same for
all the flights iNand is held at 1, i.e. ei=ej= 1 for all i, j N.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
22
Effect on Profit. First, in Figure 2 we show how the profit (objective function) varies as the
NSL is varied for different F SL levels. The profit is computed as a percentage of the planned
profit of the incumbent schedule. In general the profits decrease as N SL increases. Since the N SL
and F SL of the incumbent schedule are lower than those considered, these profits are also lower.
We vary the NSL from 0.8 to 0.95. With lower F SL the decrease in profit is less pronounced as
the NSL increases. To achieve extremely high N SL and F SL, substantial profit decrease must be
tolerated. Next, in Figure 3, we show how the profit varies as the F SL is varied for different NSL
 
 
 
 
Figure 2 Effect of N SL on the profit level (percent of incumbent schedule profit).
levels. In this case too the profit levels decrease. Again, the profit is computed as a percentage
of the planned profit of the incumbent schedule. Similar to the NSL experiments, we vary the
F SL from 0.8 to 0.95. Finally, Figure 4 summarizes the reduction in profit, as a percentage of the
planned profit of the incumbent schedule, as both the NSL and F S L vary from 0.8 to 0.95. It
is noteworthy that at very high service levels the reduction in profit is 13%. However, the airline
may be willing to consider a lesser degradation in profit to still achieve substantial improvement
in F SL and N SL. We observe that the profit decreases almost linearly with respect to the F SL
and NSL. This is confirmed also in Figures 2 and 3.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 23
 
 
 
 
Figure 3 Effect of F SL on the profit level (percent of incumbent schedule profit).
Figure 4 Effect of F SL and NS L on the profit level (percent of incumbent schedule profit).
Number of Departures Changed and Passenger Connections for the P M M Model.
In this set of experiments we vary the N SL and F SL between 0.8 and 0.95 and study the number
of departure times and passenger connections affected. Figure 5 shows the effect of varying the
NSL and Figure 6 shows the effect of varying F S L. In the former the F SL is fixed at 0.8 and
in the latter set of experiments the NSL is fixed at 0.8. In both these figures the solid line with
block markers represents the number of passenger connections achievable and the dotted line with
diamond markers represents the number of flight departures affected. In Figure 5 the connections
steadily decrease from 4,186 to 3,725 while the number of departures increases (as shown by the thin
dark trend-line) from 171 to 273 as the N SL changes. Similarly, in Figure 6 the connections vary
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
24

Figure 5 Effect of N SL on the departures changed and passenger connections.
Figure 6 Effect of F SL on the departures changed and passenger connections.
between 4,197 to 4,055 and the total departures adjusted fluctuate between 164 and 200. There is
not a clear trend in how the F S L affects the number of departures adjusted, however, as indicated
by the thin dark trend-line the connections show a decreasing trend. Intuitively, to achieve high
service levels, flexibility is required and thus more departure time changes are expected. Figure 5
confirms this, while it is not evident from Figure 6. The NSL captures passenger connections. If
an airline operates a single flight, the NS L is 100%. We expect that as the NSL is increased, the
number of passenger connections should decrease (clearly at the expense of diminishing profit).
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 25
This intuition is confirmed by both figures.
Effect of Deviation Penalty for the P M M Model. Here we vary the penalty from the
departure time in the incumbent schedule (ei, i N). In these experiments, however, we do not
discriminate between flights, i.e., we assume ei=ejfor every i, j N. First, in Figure 7 we plot
   
 
Figure 7 Effect of departure deviation penalty on profit and passenger connections.
the profit (as a percentage of the profit of the incumbent schedule) and the number of passenger
connections changed as the deviation penalty is varied from 0 to 200. The thick-line represents the
profit level and the dashed-line represents the connections affected. The N SL and F SL are held
at 0.8 for all these experiments. The profit, as well as the connections, show a decreasing trend
with respect to the deviation penalty. This is expected since higher deviation penalties imply more
costly perturbations and thus the trade-off between schedule changes and profit sways towards
schedule changes.
4.2. The SLM M Model.
Here, we report the experimental results for the SLM M model. For these set of experiments we
define the relative weight between the NSL and F SL as ω=wn
wf. The first set of experiments for
the SLM M model study the effect of ωon the N SL and F SL achieved. The minimum profit level
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
26
(for constraint (26)) is restricted to 100% of the profit achieved with the incumbent schedule. Thus
we do not allow any decrease in profitability. Figure 8 shows the results of these experiments. The
 ʘ
Figure 8 Effect of ωon N SL and F SL.
dotted-line represents the F S L and the solid-line represents the N SL. Since the objective function
in Algorithm 2 is an approximation to the two service levels, given a solution, we have to compute
the service levels from the schedule obtained. The experimental results, with the departure time
window fixed at 30 minutes, show that F SL drops initially while the NSL increases as ωincreases.
Interestingly, though, both these service levels stabilize beyond a weight level and remain almost
constant even for very large values of ω. Such a behavior is expected since ωcaptures the trade-off
between the two service levels. From Figure 8 we can also observe the maximum possible service
levels with the same profits.
The next set of experiments vary the departure time window between which the flight departure
times are varied. Again, we study the effect on the service levels achievable while still assuring
100% of the original schedule’s profit. Figure 9 plots both these curves. The dotted-line represents
the F SL and the solid-line represents the N SL. As the departure time window is expanded, the
NSL improves while the F SL marginally drops. This affirms that it is harder to increase the NSL
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 27
than the F S L. To increase the NSL, further flexibility has to be provided such as increased time
windows. Figure 9 provides a clear trade-off with respect to the permissible schedule change and
 
Figure 9 Effect of departure time window on N SL and F SL (ω= 0.7).
the two service levels without reducing profitability. For example, if the departure time windows
are 10 minutes (which may not be always acceptable, e.g., due to competition and implications
on demand), then a N S L of about 58% and a F S L of about 79% is achievable at the same profit
level. This is a substantial improvement over the original values of 40% and 60% for the NSL and
F SL, respectively. For these experiments ω= 0.7, which is a value of ωwhere the service levels are
stable.
The final set of experiments study the trade-off between the service levels and the profit level.
The profit is computed as a percentage of the profit for the incumbent schedule when ω= 0.7 and
the departure time window is set to 30 minutes. Figure 10 shows the variation in NSL,F S L, and
the value ωNSL +F S L as the profit is reduced. At 100% profit, N SL = 0.68 and F SL = 0.68.
However, as the profit percentage is reduced, N S L increases only slightly, i.e., up to 0.71 at 90
% profit level, while F SL increases substantially to 0.91. Note that a different trade-off would be
assessed if ωis changed. Thus, the airline may gain on service levels by slightly adjusting the profit.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
28
 
   
Figure 10 Effect of reduction in profit level on N S L and F SL (ω= 0.7).
5. Discussion
In this paper we developed two models that incorporate uncertainty associated with block-times
into the schedule development process. It is an initial attempt in developing a comprehensive
and holistic model for incorporating block-time uncertainty in schedule planning. We explicitly
model time distributions through chance constraints and hence the resulting schedule is robust
with respect to the operational on-time performance measure. We also incorporate network service
levels, which probabilistically model passenger connections. The new cut generation algorithm and
linearization technique proposed are novel in the sense that the convergence result generalizes
previously established results with (1) semi-infinite linear programs with finitely many variables but
infinitely many constraints, and (2) infinite dimensional problems with finitely many constraints
and infinite number of variables.
The benefits of our approach are two fold: (1) airlines could adjust the schedule to increase
operational reliability, and (2) passengers could be guaranteed higher service levels. There are
potentially other indirect benefits of adjusting the schedule by incorporating block-time uncer-
tainty. For example, the schedule recovery cost due to a disruption during actual operations could
be reduced because the planned block-times allow additional flexibility. However, we have not
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 29
specifically included such additional benefits in the models presented. Through extensive compu-
tational experiments we demonstrate the efficiency of our algorithms and models in trading off
between profitability and service level guarantees. The algorithms perform well in achieving this
trade-off and provide airline schedule planners the ability to decide on acceptable reduction in
profitability to achieve desired passenger service levels.
There are several possible modifications and enhancements to the models described in this paper.
First, the dependency of block-time distributions on the departure time can be included, if such
information is readily available. This also allows for wider time windows to vary the departure
times of scheduled flights and capture “time-of-the-day” effects related to block-time distributions.
However, the resulting model is more complicated than those described by the P M M and S LM M
because it requires the introduction of additional binary variables and several additional constraints
to model the choice of the appropriate departure-time dependent block-time distribution. The
key concept is to discretize each time window and assign a specific block-time distribution to
each subinterval. Standard modeling techniques using piecewise linear functions capture these
assignments.
Second, in the current model we assume that the block-times follow continuous log-concave
distributions. It is possible that there may be a discrete jump in the actual block-time, i.e., the
block-time distributions follow a discrete log-concave distribution. While our model can be con-
sidered as an approximation to the discrete case, incorporating discrete distributions may not
guarantee convergence, unlike the case discussed in this paper with continuous distributions.
Third, to keep our analysis tractable and focus on the issue of schedule reliability, our model
does not incorporate the trade-off between the local and through passengers. While constraints,
(14) in the PMM model and (25) in the SLMM model, enforce the fact that the original pas-
senger connections (itineraries) are feasible with any schedule perturbation, any solution to our
model provides a lower bound to the potential revenue (and profit) achievable, if such a profitable
trade-off (substitution of demand) were to be specifically included in these optimization models.
Additionally, such data would also have to be captured. These constraints also enforce that the
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
30
aircraft routing solution to the incumbent solution continues to remain feasible under any schedule
perturbations. In a more general setting it is possible to relax these constraints and allow larger
perturbations of the schedule by embedding fleet assignment and aircraft routing constraints. In
this case several additional passenger connections may also become feasible as the departure times
are perturbed. Such a model would be an extension of our model and significantly harder to solve.
To address the issue of time dependent demand distributions for local and through passengers,
one possible way is to construct multiple copies of the same flight, each with its own specific
demand. This would necessitate the inclusion of additional constraints enforcing that exactly one
of these copies is chosen as well as the connections of aircraft rotations and passenger flows remain
feasible.
Fourth, in this paper, we focus on perturbing the schedule by including block-time uncertainty.
However, as the block-times are varied, and the departure times are perturbed, the ground-times are
automatically adjusted. It is possible, that airlines would want to trade-off between the allocation
of ground-times and block-times to perturb the schedule. Currently, we do not explicitly model
ground-time constraints because the form of the distributions for the ground-time variables is not
known. Our solution methodology could be extended if these distributions are log-concave. It is
possible that, in the current model, we could capture the change in the objective function due to
increase or decrease in ground-times. This could be done by including some cost/profit associated
with perturbing ground-times in the objective function. For example, if Gij denotes the cost/profit
associated with increasing the ground-time between flights iand jin an aircraft’s rotation, we
could include the terms X
(i,j)T
Gij (djaj) in the objective function. This, of course, would not
qualitatively change our main insights.
Finally, in the current setting we do not distinguish between various markets an airline serves
(i.e., different portions of the network). It is possible to incorporate different service levels for
different markets and use similar models, as described in this paper, to perturb the schedule and
set suitable block-times. In the current setting we only guarantee a minimum service level for the
entire network.
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 31
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6. Appendix
Proof of Theorem 1.
Recollect that hdenotes the iteration index in Algorithm 1. Given that Zhonly has a finite
number of different values, there exists a subsequence such that Zh1=Zh2=Zh3=···=Z.
Furthermore, since for every flight iN,dhq
i[li, ui], there exists a convergent subsequence in
{dhq
i}q. From constraint (8) it follows that dhq
iahq
ifor every iN. Additionally, constraint
(9) ensures that the subsequence {ahq
i}qis upper bounded. Thus, we conclude that there exists a
convergent subsequence in {ahq
i}q.
Let us denote the values of sij k in the optimal solution to RP M M by sh. We now assume
that if zh
ij = 1, then sh
ijk = min 0, Uik dh
jdh
imij  for every kKi, i N, and j¯
Ci. If
this is not the case, we can easily increase sh
ijk to satisfy this property without affecting feasibility.
Furthermore, observe that for every itinerary oand fare-class fcombination we have 0 Xhq
of
Dof . Therefore, there must be a convergent subsequence in nXhq
of oq. Here Xhdenotes the optimal
itinerary fare-class demand values.
From the above set of arguments and due to a finite number of flight legs, there is a subsequence
where all the departure and arrival times converge in addition to the itinerary fare-class demand
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty 33
values. For ease of notation we denote this subsequence by {hq}q. Let dand abe the sets of
departure and arrival times of flights, respectively, as defined in the statement of the theorem.
Furthermore, we also define, for every kKiand iN,j¯
Ci
s
ijk =0z
ij = 0
min 0, Uik d
jd
imij  z
ij = 1.(47)
It remains to be shown that d,a,Z, and sis an optimal solution to P M M , i.e., these values
satisfy constraints (9)-(16), constraints (37)-(39), and (41). It is easy to verify that constraints
(9)-(16) are satisfied because only a finite number of them exist. Hence, we first discuss constraints
(37)-(39). On closer observation it is easy to note that constraints (38) and (39) hold by definition.
Thus, we focus our attention on constraints (37).
To this end, let us fix a iNand j¯
Ci. We first show that
s
ijkh¯q
i
log γn(48)
for every ¯qand kh¯q
i=kijiin iteration h¯q.
First, let z
ij = 1 and q¯q+ 1. We have zhq
ij = 1 for every qand thus shq
ijkhq
i
Uikhq
idhq
jdhq
imij and log γnshq
ijkhq
i
. We conclude log γnUikhq
idhq
jdhq
imij . Since
these constraints are not removed from RP MM in later iterations, we have log γn
Uikh¯q
idhq
jdhq
imij . Since U’s are continuous, by taking the limit as q→ ∞, we obtain
log γnUikh¯q
id
jd
imij .
Since log γn0, we obtain
log γnmin 0, Uikh¯q
id
jd
imij =s
ijkh¯q
i
.(49)
Thus, we have proved(48).
We now consider
min
kKi
s
ijk = min
kKi0, Uik(d
jd
imij )=gi(d
jd
imij ).
Sohoni, Lee, and Klab jan: Robust Airline Scheduling under Block Time Uncertainty
34
Observe that we have
gi(d
jd
imij ) = gi(dhq
jdhq
imij ) + gi(d
jd
imij )gi(dhq
jdhq
imij )
=Uikhq
idhq
jdhq
imij +gi(d
jd
imij )gi(dhq
jdhq
imij ) (50)
Uikhq
idhq
jdhq
imij Uikhq
id
jd
imij + log γn
+gi(d
jd
imij )gi(dhq
jdhq
imij ) (51)
=pi(khq
i)
Rkhq
i
0pi(t)dt hdhq
jd
j+d
idhq
ii+ log γn
+gi(d
jd
imij )gi(dhq
jdhq
imij ).(52)
In the above, equation (50) follows from the fact that khq
imaximizes the violation of constraint
(37) (see Step 4.1 in Algorithm 1). Furthermore, (51) follows from (48) and (49). The last equality,
i.e. equation (52), follows from the definition of Uik .
Observe that the first term in equation (52) converges to 0 since pi(k)
Rk
0pi(t)dt is bounded for kKi
for all iN. Similarly, the last two terms also converge to 0 since giis a continuous function. Thus,
we conclude that
s
ijk log γnfor every kKi, i N. (53)
It is easy to verify that when z
ij = 0 constraint (37) holds. We conclude that (47) holds in general.
Using similar arguments it can be shown that constraint (41) also holds. Hence, Z,d,a, and
sis a feasible solution to P M M .
It remains to show optimality. Let V(Z, d, a, s) denote the objective value of the correspond-
ing solution. Further, notice that in each iteration h, the optimal value Vhof RP M M is an
upper bound on the global optimal value V, i.e., VVh. Thus, we must have
V(Z, d, a, s)VVhq=V(Zhq, dhq, ahq, shq).(54)
Since the objective function is continuous, by taking the limit, we obtain
lim
q→∞ V(Zhq, dhq, ahq, shq) = V(Z, d, a, s). From (54) we obtain V=V(Z, d, a, s). To
arrive at this we must also have {Xhq
of }qbe a convergent subsequence, which was assumed earlier.
Hence, we have completed the proof.
... Secondly, by constructing flexible timetable, in which the airline mangers can find multiple swapping opportunities for the aircraft [14]. Lastly, by developing a stochastic FSP model that considers the block time uncertainty [15]. In a recent study by Jiang and Barnhart [16], the robustness was presented in terms of the number of potential connected itinerary. ...
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Thesis
Recently, aviation and airline maintenance providers are of the most significant worldwide industries. This is shown by the enormous growth in the number of passengers, which was around 3.5 billion passengers in 2015, expecting an annual growth of 5%. To cope with this passenger growth, the number of aircraft is expected to increase from 24,579 in 2014 to 29,955 in 2022. As a result, the aircraft maintenance cost paid by airlines to maintenance providers is expected to increase from US $62.1 billion in 2014 to US $90 billion in 2024. Despite this pleasing economic situation for airlines and maintenance providers, many difficult challenges have been emerged during the planning and operating processes. One of the challenges facing airlines is how to build efficient routes for their aircraft, while respecting the operational maintenance restrictions. In this regard, aircraft maintenance routing problem (AMRP) is very significant for airlines, as it builds the routes for their aircraft and schedule their maintenance visits. On the other hand, for the maintenance providers, it is a great challenge to manage the workforce capacity required to serve the increased number of aircraft. Therefore, maintenance staffing problem (MSP) is recognized as an effective tool for maintenance providers, as it manages the workforce capacity required to serve the airlines' aircraft. In the existing research, on the focus of AMRP, most AMRP models consider one operational maintenance restriction, which is a single maintenance visit every four days and overlook the restrictions of the total cumulative flying time, the total number of takeoffs , the workforce capacity and the working hours of the maintenance providers. Consequently, the generated routes are not applicable in real practice due to their infeasibility. This motivates us to develop a model, in which all the aforementioned restrictions are considered in a single model. Therefore, the routes determined by this model can be implemented in reality. In addition, an efficient solution algorithm is proposed for solving the developed model. Meanwhile, one of the glaring facts in the literature is that AMRP and MSP are studied independently, and their interdependence have not been investigated. To fulfil this research gap, a leader-follower Stackelberg game (LFSG) model is developed to capture this interdependence. Moreover, a nested ant colony optimization-based algorithm is proposed as a solution method for the game theoretic model. Towards the goal of showing the superiority of the proposed model, we present a case study of LFSG for one major airline and four maintenance providers located in the Middle East. The results show significant cost savings for all players. Although the LFSG presents a formulation for a unique problem in the literature, it overlooks one important aspect, called the price competition among the maintenance providers. Indeed, this aspect has a direct influence on the AMRP, as it changes the routing plan constructed by airlines. In this connection, it is imperative to consider the price competition among the maintenance providers besides the interdependence between AMRP and MSP. For this purpose, a Stackelberg-Nash game model (SNGM) is proposed to capture the above-mentioned problem. In addition, an iterative game algorithm is developed in order to obtain the overall Nash equilibrium for the SNGM. To demonstrate the viability of the proposed model, we use the previous case study, in which its results reveal significant savings for airline and maintenance providers. The contribution of this thesis is threefold. Firstly, proposing a new scalable AMRP that considers all the operational maintenance restrictions along with developing an efficient solution algorithm to solve this model. Secondly, developing a coordinated decision support system based on game theory to capture the interdependence between AMRP of airlines and MSP of maintenance providers. Lastly, modeling the previous interdependence in the presence of the price competition among the maintenance providers, we develop a new model, called Stackelberg-Nash game model. Acknowledgments (if any)
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