Improving airline operational performance through schedule perturbation
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Ann Oper Res
DOI 10.1007/s10479-006-0003-1
Improving airline operational performance
through schedule perturbation
Andrew J. Schaefer · George L. Nemhauser
C Springer Science + Business Media, LLC 2006
Abstract Schedule development is typically the first phase of the airline planning process.
We present a framework for perturbing scheduled departure and arrival times after a crew
schedule has been found. We characterize perturbations that keep a schedule legal while not
increasing the planned cost of the crew schedule. We show that when random delays occur
in operations, the expected cost can be reduced and the on-time performance improved.
Computational results are reported for two real fleets and a large number of crew schedules.
Keywords Airline crew scheduling . Uncertain operations . Schedule development .
Integer programming
AMS Classification: 90B06
Airlines operate under uncertain conditions. Weather, congestion, and mechanical break-
downs are examples of why flights may not operate as planned. The operational performance
of airlines is becoming worse. In the United States, one flight in four is delayed during the
summer, and air traffic is expected to double by 2015 (Anonymous, 2001). At large airports,
nearly half of all flights are delayed (Anonymous, 2001). On-time statistics are of great im-
portance to airlines. The Bureau of Transportation Statistics (BTS, 1998) defines a flight to
be on-time if it arrives no later than 15 minutes after its scheduled arrival time and publishes
rankings of airline on-time performance. These ratings can be used for marketing purposes
and a good on-time performance may lead to greater customer satisfaction.
Airline schedules can affect their own on-time performance. In the airline planning process,
the first step is schedule development. During the schedule development phase an airline
determines when and where it will fly. It also determines the scheduled block time of each
flight, that is, the planned duration of each flight, or leg. For the same origin and destination
A. J. Schaefer ()
Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA 15261
e-mail: schaefer@ie.pitt.edu
G. L. Nemhauser
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332
SpringerAnn Oper Res this time can vary between airlines. It can also vary by time of day for a given airline. For instance, an airline might allocate a greater block time to a flight that departs or arrives during the busiest part of the day. Schedule development typically takes place at least one year before a flight departs. After schedule development, the airline solves the fleet assignment problem, which assigns each leg to a fleet, or different aircraft type. The routing problem determines the sequence in which each plane will fly legs. After these problems have been solved the crew scheduling problem determines a set of crew trips, or pairings, that partition the set of legs to be flown. Traditionally, these problems have been considered sequentially, with little or no feedback from later stages to earlier stages. However, some work has been done on integrating the various stages of airline planning. Lohatepanont and Barnhart (2004) integrate schedule development and fleet assignment to increase the number of flights that may be flown. Klabjan et al. (2002) combine crew scheduling and routing with time windows to increase the number of feasible pairings, thus reducing the planned cost of the resulting crew schedule. Rexing, Barnhart and Krishnamurthy (2000) propose a fleeting model with time windows. Cohn and Barnhart (2003) incorporate maintenance considerations into crew scheduling. Since on-time performance is defined relative to scheduled block time, an airline can improve its on-time performance by increasing the scheduled block times of legs. However, the planned flying time of a crew schedule is a lower bound on its planned cost, which is in turn a lower bound on its operational cost (Schaefer et al., 2004). Crew costs are second only to fuel costs for airlines, and increasing the planned flying time could cause crew costs to increase. This paper introduces a method for determining schedule perturbations that improve on-time performance without increasing crew costs. We propose a new approach that perturbs the original flight schedule to improve the operational performance of a given crew schedule. The perturbation is made in such a way that the crew schedule and the routing remain feasible. We show that such a perturbation will lead to a performance in operations which, under certain conditions, is likely to be better than, and is at least as good as, the performance of the crew schedule under the original schedule. Our computational experiments indicate that a crew schedule can have a noticeable improvement in operational cost and on-time percentage with the perturbed flight schedule. In Section 1 we review how pairing feasibility and costs are determined. This description is needed in Section 2 where we discuss schedule perturbations. The effect of schedule perturbation in airline operations is considered in Section 3. We provide computational results in Section 4 and give conclusions in Section 5. 1. Feasibility and crew pairing costs Because a pilot typically may fly only one fleet, the crew scheduling problem is separable by fleet. A crew flies a set of consecutive flight legs, called a duty, that follow certain regulations and contractual restrictions. The time between two consecutive legs within a duty, or sit time, must be at least a minimum amount if a crew changes planes. The elapsed time of a duty is the number of minutes the duty lasts, including a briefing period before the first leg and a debriefing period after the last leg. A pairing, or crew trip, is a sequence of duties, each separated by a rest period which must exceed a minimum length. The time away from base (TAFB) of a pairing is the number of minutes from the beginning to the end of the pairing. Pairings flown within the U.S. must adhere to both FAA and contractual rules. For instance, to prevent crew fatigue the “8-in-24” rule requires compensatory rest for a crew that is scheduled to fly more than 8 hours within Springer
Ann Oper Res
any 24 hour period (FAA, 1999). Although we focus on the airline industry in the United
States, similar regulations exist in other countries.
A crew schedule is a set of pairings that partitions the legs to be flown by a single fleet.
The crew scheduling problem is usually modeled as a set partitioning problem:
{min cx : Ax = 1, x binary} (1)
where c j is the cost of pairing j, and ai j , the i jth component of A, is 1 if pairing j flies
leg i, and 0 otherwise. A survey of recent advances in deterministic airline crew scheduling
is given in Barnhart et al. (2002).
When crew schedules are found, pairings are not yet assigned to particular pilots, and so
the cost of a crew schedule is not expressed in monetary terms, but in minutes of pay and
credit. Since pilot salaries differ, determining the monetary cost of a crew schedule is only
possible once pairings have been assigned to particular pilots. The flight-time-credit (FTC)
of a duty is the difference between its total cost in minutes of pay and credit and the total
block time expressed as a percentage of the total block time of the duty. A similar measure
exists for pairings and crew schedules.
The method for calculating the planned cost of a crew schedule varies by airline. We give
an example of one method, using the notation from Schaefer et al. (2004). Let FTC(·) denote
the planned FTC of any duty, pairing or crew schedule. Let q be any pairing consisting of
duties d1 , . . . , dk . For any leg li , let dep (li ) be the scheduled departure time of leg li in minutes
and let arr (li ) be its scheduled arrival time in minutes. These times are relative to the start of
the pairing, so that the beginning of duty di+1 is greater than the end of duty di . Let block (li )
be the planned block time of leg li in minutes, defined by block (li ) = arr (li ) − dep (li ). Let
brief be the length of the pilot briefing period prior to every duty and debrief be the length of
the pilot debriefing period after every duty. The parameters brief and debrief are constants and
are in minutes. For a given duty di , let ls and lt be its first and last leg, respectively, and define
the planned elapsed time of the duty as elapse (di ) = arr (lt ) − dep (ls ) + brief + debrief .
Let re < 1 be a fraction representing the rate of pay for elapsed time in terms of minutes
of pay and credit. Let mgd be the minimum guarantee for a duty, which is given in minutes
of pay and credit. The planned duty cost of duty d is expressed in minutes of pay and credit
and is given by
b (d) = max block (li ), re × elapse(d), mgd . (2)
i∈d
Let la and lb be the first and last legs, respectively, of a pairing q. The planned time away
from base of pairing q is the total number of minutes that elapse during the pairing given
by TAFB (q) = arr (lb ) − dep (la ) + brief + debrief . Let rt < 1 be a fraction representing
the rate of pay of time away from base, and let mg p be a minimum guarantee per duty in a
pairing. Then the planned pairing cost of pairing q is given by
k
cq = max b(di ), rt × TAFB(q), mg p × k . (3)
i=1
Vance et al. (1997) use values of re = 47 , mgd = 0, rt = 27 , and mg p = 300.
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The planned FTC of pairing q is defined by
cq − j∈q block (l j )
F T C (q) = × 100. (4)
j∈q block (l j )
Let c (C) be the planned cost of a crew schedule C consisting of pairings q1 , . . . , q|C| , given
by
c (C) = cq . (5)
q∈C
Let block(C) be the total scheduled block time of all legs in the flight schedule. The planned
FTC of crew schedule C is then
c(C) − block(C)
F T C(C) = × 100. (6)
block(C)
2. Perturbing flight schedules
For each leg li ∈ L, let xi and yi be nonnegative variables that determine the new departure
and arrival times of leg li under the perturbed schedule L + (x, y), so that leg l has a scheduled
departure time of dep (li ) − xi and a scheduled arrival time of arr (li ) + yi .
Increasing the block time of a leg may cause a pairing to become illegal or may reduce
the number of possible passenger connections. Possible reasons for this include:
r Minimum sits or minimum rests may be violated.
r The elapsed time of one of its duties may exceed the maximum amount.
r The flying time of one of its duties may exceed the maximum amount.
r A crew schedule may violate 8-in-24 rules.
r The routing may become infeasible because a plane may have insufficient time between
legs.
r Passenger itineraries may become infeasible.
|L|
Definition 1. Consider any vectors x and y in R+ . Then (x, y) is an admissible schedule
perturbation for flight schedule L if the routing remains feasible under schedule L + (x, y).
Let I be the set of all pairs of legs which appear consecutively in a passenger itinerary;
that is, if (i, j) ∈ I, then leg li immediately precedes leg l j in at least one passenger itinerary.
Let MINPASSCONNECT be the minimum time passengers need to connect.
Definition 2. A schedule perturbation (x, y) preserves all passenger itineraries if the differ-
ence between the perturbed departure time of leg l j and the perturbed arrival time of leg li is
at least MINPASSCONNECT for all (i, j) ∈ I.
Definition 3. Let C be the set of all crew schedules feasible for flight schedule L. For any
admissible schedule perturbation (x, y), let C(x, y) be the subset of all crew schedules C ∈ C
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such that crew schedule C obeys all legality rules, such as minimum sits, minimum rests,
and 8-in-24 rules, under flight schedule L + (x, y).
For any admissible schedule perturbation (x, y) and any crew schedule C ∈ C(x, y) let
c(x,y) (C) be its planned cost under the schedule perturbation. Using this notation, the planned
cost of schedule C under the unperturbed schedule is c(0,0) (C).
Proposition 1. Consider any admissible schedule perturbations (x, y) and (x̂, ŷ) where
(x, y) ≤ (x̂, ŷ), and any crew schedule C ∈ C(x̂, ŷ). Then c(x,y) (C) ≤ c(x̂, ŷ) (C).
Proof: Increasing the block times will not decrease the time away from base of a pairing,
the cost of any duty, and will not affect the minimum guarantee.
As a result of Proposition 1 we have
Proposition 2. Let C ∗ solve the crew planning problem for flight schedule L. Let x ≥ 0 and
y ≥ 0 be such that
1. C ∗ ∈ C(x, y), and
2. c(x,y) (C ∗ ) = c(0,0) (C ∗ ).
Then C ∗ solves the crew planning problem for the perturbed schedule L + (x, y).
In this paper we consider only those admissible schedule perturbations that do not increase
the planned cost of the crew schedule.
Definition 4. An admissible schedule perturbation is called feasible if it does not increase
the planned cost of the crew schedule.
An interesting question is to find an admissible schedule perturbation that maximizes
expected on-time performance while ensuring that the planned crew cost does not increase.
However, the solution to this problem is not straightforward, since the expected on-time per-
formance of a perturbation must be evaluated by a simulation, such as SimAir (Rosenberger
et al., 2002). Similarly, finding an admissible schedule perturbation that minimizes the ex-
pected crew cost is difficult.
Let D be the set of duties in crew schedule C. For any pairing q ∈ C, consider any two legs
in q flown consecutively by the crew, say li and l j . We call the legs li and l j crew-consecutive.
Let φi be the minimum amount of time needed to keep the crew connection between legs li
and l j legal. Then any perturbation must satisfy
yi + x j ≤ dep (l j ) − arr (li ) − φi , ∀ crew-consecutive legs li and l j . (7)
Similarly, plane connections must be preserved. Let li and l j be any two legs which are
flown consecutively by a plane. We call the legs li and l j plane-consecutive. Let ψi be
the minimum time needed to keep the plane connection between legs li and l j legal. Any
perturbation must satisfy
yi + x j ≤ dep (l j ) − arr (li ) − ψi , ∀ plane-consecutive legs li and l j . (8)
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Let MAXELAPSE be the maximum amount of elapsed time for each duty. For any duty d
let ls be its first leg, and let lt be its last leg. Then any perturbation must satisfy
xs + yt ≤ MAXELAPSE − brief − debrief − arr (lt ) + dep (ls ), ∀ d ∈ D. (9)
Let MAXFLY be the maximum amount of flying time for each duty. Then any perturbation
must satisfy
(xi + yi ) ≤ MAXFLY − (arr (li ) − dep (li )) ∀ d ∈ D. (10)
i∈d i∈d
Any feasible schedule perturbation must satisfy more complex rules, such as the so-called
“8-in-24” planning rules. We give constraints that eliminate the simplest form of 8-in-24
violations. For any 24-hour window W in a pairing q, let d(W ) and a(W ) be those legs
departing and arriving in window W , respectively. Let λ(W ) and κ(W ) be the last leg before
and the first leg after the rest following window W , respectively. Let ν(W ) be the number of
extra flying minutes permitted in window W before a planning 8-in-24 violation occurs, and
let γ (W ) be the number of extra minutes permitted in the compensatory rest period before a
planning 8-in-24 violation occurs. Then any schedule perturbation must satisfy
xi + yi ≤ ν(W ) for all 24-hour windows W , (11)
i∈d(W ) i∈a(W )
and
xκ (W ) + yλ (W ) ≤ γ (W ) for all 24-hour windows W . (12)
Let z d be the amount that the cost of duty d increases because of these perturbations.
Recall that ls and lt are the first and last leg of duty d, respectively.
z d = max (arr (li ) − dep (li ) + xi + yi , re × (elapse (d) + xs + yt ), mgd − bd ,
i∈d
so that
zd − (xi + yi ) ≥ (arr (li ) − dep (li )) − bd , ∀ d ∈ D, (13)
i∈d i∈d
z d ≥ mgd − bd , ∀ d ∈ D, (14)
and
z d − re × (xs + yt ) ≥ re × (elapse(d)) − bd , ∀d ∈ D. (15)
To ensure that no pairing cost increases, two types of constraints are sufficient. The sum of
the new duty costs may not exceed the pairing cost, and the time away from base component
of the pairing cost may not exceed the original planned pairing cost. Thus any block shift
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which does not increase the cost of the pairings must satisfy
z d ≤ cq − bd , ∀ q ∈ C, (16)
d∈q d∈q
and
rt (xa + yb ) ≤ cq − rt × TAFB (q), ∀ q ∈ C, (17)
where a and b are the first and last leg, respectively, in pairing q.
The pairing minimum guarantee is not affected by block time shifts.
If every passenger connection is maintained, a block-time shift must satisfy
yi + x j ≤ dep (l j ) − arr (li ) − MINPASSCONNECT ∀ (i, j) ∈ I. (18)
We define the schedule perturbation polytope (SP) by
SP = x, y ∈ R|L| , z ∈ R|D| | subject to (7)–(18), x, y, z ≥ 0 . (19)
In practice, flights are scheduled to depart or arrive on the minute. Similar results hold if
the x and y variables are restricted to be integers, except that this would make optimization
more difficult.
3. Schedule perturbation under push-back recovery
Operational decisions occur when an airline plan needs to be modified. Airline operations have
been studied very little from a research point of view relative to airline planning. Recovery
is the process by which airlines react to disruptions. Planes and passengers may be rerouted,
legs may delayed or cancelled, and crews may fly different pairings. Lettovský, Johnson
and Nemhauser (2000) provide a framework for an optimization-based approach for crew
recovery. To evaluate different plans and recovery policies in operations, (Rosenberger et al.,
2002) developed SimAir, a simulation of airline operations. SimAir aggregates sources of
delays into two categories: ground delays and block-time delays. A block-time delay occurs
between the operational departure of a leg and its operational arrival. Any other delay is a
ground-time delay. Schaefer et al. (2004) consider the problem of finding crew schedules
that perform well in operations. They give a method for calculating the operational cost of a
crew schedule. This method is essentially the same as calculating the planned cost of a crew
schedule, except that operational departure and arrival times replace the planned times in the
formulae. Furthermore, the operational cost of a pairing (and hence a crew schedule) must
be at least its planned cost.
Initially, we assume that planes are always available. Consider push-back recovery. This
policy delays the departure of each flight until both the scheduled departure time has occurred
and the crew is available. For any pairing q ∈ C consisting of k legs, say l1 , . . . , lk , let
γ1 , . . . , γk be nonnegative random variables where γi indicates how late leg i departs relative
to the scheduled departure time of leg i; that is, the random departure time of leg li is given
by dep(li ) + γi . Let the random variable ωi denote the block-time error on leg li . The block-
time error distribution ωi is relative to the deterministic block time, so that a realization of
ωi = 0 would mean that leg li had a random block time equal to its deterministic block time.
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We assume that the block-time and ground-time error distributions are independent of the
time of day.
We now summarize briefly SimAir’s model of push-back recovery. Let ξi ≥ 0 denote the
ground-time delay of leg li . For 1 ≤ i ≤ k − 1, let θi denote the scheduled length of the sit
after leg li if it is not the last leg in its duty, and let θi denote the scheduled length of the rest
after leg li if leg li is the last leg in its duty. In other words,
θi = dep (l j ) − arr (li ), (li , l j ) crew − consecutive. (20)
Let the parameter φi denote the minimum connection time after leg i if leg i is not the last
leg in its duty, and let φi denote the minimum rest amount if it is the last leg in its duty.
We assume that there are no delays prior to the start of each pairing and that each pairing
begins on time. The operation (·)+ is defined by (·)+ = max(·, 0).
We first establish a formula for the lateness of each leg under push-back recovery.
Proposition 3. The lateness of each leg under push-back recovery is given by
γ1 = ξ1 , and γi = (γi−1 + ωi−1 + φi−1 − θi−1 )+ + ξi (21)
for i = 2, . . . , k.
The proof is given in Schaefer (2000).
|L|
Let x and y be vectors in R+ such that crew schedule C remains feasible for flight
schedule L + (x, y).
Define
r the perturbed scheduled departure time, dep (li ) = dep (li ) − xi ;
r the perturbed scheduled arrival time, arr (li ) = arr (li ) + yi ;
r the perturbed random block-time error, ω = ωi − xi − yi ; and
r the difference between the unperturbed scheduled departure time of leg i + 1 and the
i
perturbed arrival time of leg i, θi = dep (li+1 ) − arr (li ) = θi − yi .
Let the random variable γi ≥ 0 be such that under the perturbed schedule, leg li departs
at time dep (li ) + γi . We now show that for perturbed schedules, the lateness is no greater
than under the unperturbed schedules, and is sometimes less.
Theorem 4. The lateness under any perturbed schedule is no greater than the lateness under
the original unperturbed schedule.
Proof: We show that γi ≥ γi for all legs li by induction. Consider any sample paths
ω1 , . . . , ωk of block-time errors and ξ1 , . . . , ξk of ground-time delays. Consider the per-
formance of crew schedule C under schedule L + (x, y). The first leg of the pairing li is
scheduled to start at time dep (li ), which is −x1 minutes late relative to dep (li ), the original
scheduled departure time. After a ground-time delay of ξ1 is observed,
γ1 = −x1 + ξ1 , (22)
and so γ1 ≥ γ1 since γ1 = ξ1 from Proposition 3 and x1 ≥ 0.
When leg i − 1 departs, it is γi−1 minutes late relative to the original departure time.
Suppose that γi−1 ≥ γi−1 . The leg experiences a block-time error of ωi−1 and is available to fly
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again after φi−1 minutes. Hence the crew is available at time γi−1 + ωi−1 + φi−1 + arr (li−1 ).
Relative to the original scheduled departure time of leg li , the crew is
γi−1 + ωi−1 + φi−1 − θi−1 (23)
minutes late. Relative to the perturbed scheduled departure time of leg li , the crew is
γi−1 + ωi−1 + φi−1 − θi−1 + xi (24)
minutes late. Leg li may not depart before its newly scheduled departure time, so its new
scheduled departure time is
(γi−1 + ωi−1 + φi−1 − θi−1 + xi )+ (25)
minutes later than dep (li ), the perturbed scheduled departure time of leg li . Relative to dep(li ),
the original scheduled departure time of leg li , this time is
(γi−1 + ωi−1 + φi−1 − θi−1 + x i )+ − x i (26)
minutes late since dep (li ) = dep(li ) − xi . After the crew experiences a ground delay of ξi ,
its departure relative to the original scheduled departure time of leg li , dep(li ), is
γi = (γi−1
+ ωi−1
+ φi−1 − θi−1 + xi )+ − xi + ξi . (27)
Substituting for ωi−1 and θi−1 in Eq. (27),
γi = (γi−1
+ (ωi−1 − xi − yi ) + φi−1 − (θi−1 − yi ) + xi )+ − xi + ξi
= (γi−1 + ωi−1 + φi−1 − θi−1 )+ − xi + ξi . (28)
Recall from Proposition 3,
γi = (γi−1 + ωi−1 + φi−1 − θi−1 )+ + ξi .
Consider two cases. If
γi−1 + ωi−1 + φi−1 − θi−1 ≥ 0
then
γi−1 + ωi−1 + φi−1 − θi−1 ≥ γi−1 + ωi−1 + φi−1 − θi−1
since γi−1 ≥ γi−1 by the induction hypothesis. Since xi ≥ 0, in this case, γi ≥ γi . Otherwise,
suppose
γi−1 + ωi−1 + φi−1 − θi−1 < 0.
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Then
γi−1 + ωi−1 + φi−1 − θi−1 < 0
since γi−1 ≥ γi−1 by the induction hypothesis. In this case, γi = ξi and γi = −xi + ξi . Since
xi ≥ 0, γi ≥ γi .
Corollary 5. Comparing two perturbations, the larger the perturbation, the smaller the
lateness.
Corollary 6. Smaller lateness implies better on-time performance.
We now consider how perturbations affect crew costs in operations. Although, as explained
below, it is possible that operational crew costs might increase in operations under a perturbed
schedule, as long as no duties may begin early, operational crew costs will not increase under
a perturbed schedule.
It is possible that a perturbation in the polyhedron SP could increase the expected op-
erational cost of a schedule. Consider a pairing q containing duty d, which is not the first
or last duty in pairing q. Suppose that the planned FTC of pairing q is small, and that the
planned FTC of duty d is 0. Let (x, y, z) ∈ SP be such that the departure of the first leg of d
is perturbed. Consider a sample path in which a large ground-time delay is observed prior to
the last leg in duty d. If this delay is sufficiently large, the operational cost of duty d will be
based on its operational elapsed time rather than its operational flying time. In such a case,
since the initial leg of duty d departs earlier under the perturbed schedule than under the
original schedule, the operational cost of duty d is larger under the perturbed schedule than
the original schedule. Since the planned FTC of pairing q was small, it is possible that under
the perturbed schedule pairing q could have a larger operational cost than under the original
schedule.
Let L be the set of legs that begin a duty. Suppose no legs in L may depart early, i.e.
consider
SP = (x, y, z) ∈ SP xi = 0 . (29)
i∈L
Corollary 7. If no duty begins early, then larger perturbations lead to smaller operational
crew costs.
Proof: Consider any crew schedule C and suppose (x, y, z) ∈ SP and (x̂, ŷ, ẑ) ∈ SP , where
x̂ ≥ x and ŷ ≥ y. Then, we will show that under push-back recovery, the operational cost
of C under L + (x̂, ŷ) is no greater than the operational cost of C under L + (x, y). Recall
that the operational cost of a pairing is the maximum of four factors: the planned cost of the
pairing, the sum of the operational duty costs, the minimum guarantee, and the operational
TAFB multiplied by rt . Since (x, y, z) ∈ SP and (x̂, ŷ, ẑ) ∈ SP , the planned cost of C under
both L + (x̂, ŷ) and L + (x, y) is the same as the planned cost of C under L. From Theorem
4, for any sample path of delays the operational departure and arrival times of C under
L + (x̂, ŷ) are no later than the operational departure times of C under L + (x, y). Since
i∈L x i = 0, the operational elapsed time of each duty and operational TAFB will be no
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greater under schedule perturbation L + (x̂, ŷ). The operational flying time will be the same:
the planned flying time plus the sum of all block-time errors over the sample path. As a result,
the operational duty costs will be no greater under the perturbed schedule. Of the four factors
determining the operational cost of a pairing, two remain unchanged under the perturbation,
and two are no greater. Therefore the perturbation will not increase the cost of any pairing.
Since the operational cost of a crew schedule is the sum of the operational pairing costs of
the pairings that comprise it, the result holds.
4. Computational results
We considered two fleets provided to us by a major domestic carrier. Fleet F1 has nearly
120 daily legs, and Fleet F2 has over 440 daily legs. We considered 97 crew schedules for
Fleet F1 and 58 for Fleet F2. The average deterministic FTC for the Fleet F1 crew schedules
is 2.57, and the average deterministic FTC for the Fleet F2 crew schedules is 3.07. For both
fleets, these crew schedules consisted of the deterministic schedule, and crew schedules found
by methods described in Schaefer et al. (2004). These additional crew schedules were found
by solving a set partitioning problem with a different objective function than that used by the
standard crew scheduling model described in Section 1.
As a result of Corollary 7, the perturbation that minimizes operational costs is a maximal
perturbation, that is, it is on the boundary of SP . Unfortunately, it is not clear how to find a
best perturbation. We will consider those perturbations that are extreme points of SP .
Let (x ∗ , y ∗ , z ∗ ) be a solution to the problem
max (αl xl + βl yl ) (x, y, z) ∈ SP . (30)
l∈L
The crew schedule was then evaluated under perturbation L + (x ∗ , y ∗ ). When solving the
optimization problems we did not explicitly add all possible constraints preventing 8-in-24
planning violations. Rather, we solved the model without these constraints, and checked to
see if the resulting schedules violated any such rules. If so, a cut would be added preventing
such a violation. However, this never occurred in the approximately 150 crew schedules that
we considered.
Table 1 Fleet F1 without
passenger connection constraints Average Average
operational FTC operational on-time
No perturbation 4.38 73.8%
Equal weight 4.36 75.7%
Departure preferred 4.34 75.5%
Table 2 Fleet F1 with passenger
connection constraints Average Average
operational FTC operational on-time
No perturbation 4.38 73.8%
Equal weight 4.37 74.7%
Departure preferred 4.35 74.8%
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Table 3 Fleet F2 without
passenger connection constraints Average Average
operational FTC operational on-time
No perturbation 5.95 72.4%
Equal weight 5.92 74.1%
Departure preferred 5.91 74.2%
Table 4 Fleet F2 with passenger
connection constraints Average Average
operational FTC operational on-time
No perturbation 5.95 72.4%
Equal weight 5.93 73.1%
Departure preferred 5.92 73.3%
Fig. 1 The on-time percentage for randomly generated perturbations
We simulated each of the crew schedules using different methods of perturbing the sched-
ule. These methods corresponded to different values of α and β as described in Section 3.
The control method set α j = β j = 0 for all j, and is denoted by “No Perturbation.” A sec-
ond method set α j = β j > 0 for all j, and is denoted by “Equal Weight”. Since operational
costs are reduced more by earlier departure than by later arrivals, perturbing departure times
may be preferable to perturbing arrival times. Motivated by this possibility, a third method,
denoted by “Departure Preferred”, set α j = 2β j > 0 for all j. Some of the models included
no passenger connection constraints, while others included all such constraints. The follow-
ing tables compare the various methods in terms of their expected operational FTC and the
expected on-time performance for both fleets.
Both methods offered improvements over the unperturbed method, and both give better
results when passenger connection constraints are not present. Of course, not all passenger
connections are equally valuable, since some may be used infrequently and others, such
as those feeding passengers to international flights, may be more valuable. Therefore, only
preserving strategic connections may yield better results than preserving every possible
SpringerAnn Oper Res
connection. Also, legs flown by different fleets may impose other passenger connection
constraints.
We performed a more in-depth study of one of the crew schedules, the deterministic so-
lution to Fleet F1. The expected on-time performance for the unperturbed schedule was
74.4%. We randomly generated positive objective coefficients and then solved problem (30)
using the new objective coefficients for α and β. We did this multiple times to create 49
new perturbations. The resulting perturbations were extreme points of SP . Figure 1 illus-
trates the expected on-time performance for these perturbations. It is interesting to note that
all of the perturbations had a significantly better on-time percentage than the unperturbed
schedule.
5. Conclusions
Traditionally, airlines have planned hierarchically. For instance, the flight schedule is made
many months before the crew scheduling problem is solved. Under push-back recovery, our
computational results demonstrate that making schedule perturbations after the crew schedule
has been found may improve operational crew costs as well as on-time performance. This can
be done while maintaining the feasibility of the routing and without increasing the planned
cost of the crew schedule. When passenger itineraries are preserved, the improvement in
average operational FTCs is reduced, but there is still an improvement. Most airline on-time
performances are between 65% and 80%, and in many cases airlines differ by less than
1–2%. Therefore, our method could often improve an airline’s on-time performance ranking.
Because these perturbations ensure that the resulting schedule remains feasible and that the
crew cost will not increase, the method improves operational performance under push-back
recovery without any adverse effects. It remains to be seen if these results apply under more
realistic recovery policies.
Acknowledgments This research was sponsored by grants from United Airlines Research and Development
and National Science Foundation grant DMI-0085723. The authors would like to thank the editors and three
anonymous referees, as well as Srini Ramaswamy of United Airlines, Eric Gelman and Jay Rosenberger
of American Airlines, and Dave Goldsman, Ellis Johnson and Anton Kleywegt of Georgia Tech for their
suggestions and insight. The computational experiments were made using hardware donated by the Intel
Corporation and optimization software provided by ILOG.
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