service network design: applications in transportation and logistics professor cynthia barnhart...
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Service Network Design: Applications in Transportation
and Logistics
Professor Cynthia BarnhartProfessor Cynthia Barnhart
Center for Transportation and LogisticsCenter for Transportation and Logistics
Operations Research CenterOperations Research Center
Massachusetts Institute of TechnologyMassachusetts Institute of Technology
September 8, 2002September 8, 2002
Institute for Mathematics and its Applications
2
Problem Definition Service network design problems in transportation Service network design problems in transportation
and logistics, subject to limited resources and and logistics, subject to limited resources and variable service demandsvariable service demands Determine the cost minimizing or profit Determine the cost minimizing or profit
maximizing set of services and their schedulesmaximizing set of services and their schedules What is the best location and size of terminals such What is the best location and size of terminals such
that overall costs are minimized? that overall costs are minimized? What is the best fleet composition and size such that What is the best fleet composition and size such that
service requirements are met and profits are service requirements are met and profits are maximized?maximized?
Institute for Mathematics and its Applications
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Service Network Design Applications ExamplesExamples
Determining the set of flights and their Determining the set of flights and their schedules for an airlineschedules for an airline
Determining the routing and scheduling of Determining the routing and scheduling of tractors and trailers in a trucking operationtractors and trailers in a trucking operation
Jointly determining the aircraft flights, ground Jointly determining the aircraft flights, ground vehicle and package routes and schedules for vehicle and package routes and schedules for time-sensitive package deliverytime-sensitive package delivery
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Research on Service Network Design Rich history of research on network design applicationsRich history of research on network design applications
Network DesignNetwork Design Balakrishnan et al (1996); Desaulniers, et al (1994); Gendron, Crainic Balakrishnan et al (1996); Desaulniers, et al (1994); Gendron, Crainic
and Frangioni (1999); Gendron and Crainic (1995); Kim and Barnhart and Frangioni (1999); Gendron and Crainic (1995); Kim and Barnhart (1999); Magnanti (1981); Magnanti and Wong (1984); Minoux (1989)(1999); Magnanti (1981); Magnanti and Wong (1984); Minoux (1989)
Freight Transportation Service Network DesignFreight Transportation Service Network Design Armacost, Barnhart and Ware (2002); Crainic and Rousseau (1986); Armacost, Barnhart and Ware (2002); Crainic and Rousseau (1986);
Crainic (2000); Farvolden and Powell (1994); Lamar, Sheffi and Crainic (2000); Farvolden and Powell (1994); Lamar, Sheffi and Powell (1990); Newton (1996); Ziarati, et al (1995) Powell (1990); Newton (1996); Ziarati, et al (1995)
Fleet Routing and SchedulingFleet Routing and Scheduling Appelgren (1969, 1971); Desaulniers, et al (1997); Desrosiers, et al Appelgren (1969, 1971); Desaulniers, et al (1997); Desrosiers, et al
(1995); Dumas, Desrosiers, Soumis (1991); Leung, Magnanti and (1995); Dumas, Desrosiers, Soumis (1991); Leung, Magnanti and Singhal (1990); Ribeiro and Soumis (1994) Singhal (1990); Ribeiro and Soumis (1994)
Institute for Mathematics and its Applications
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Challenges Service network design problems in transportation and Service network design problems in transportation and
logistics are characterized bylogistics are characterized by Costly resources, tightly constrainedCostly resources, tightly constrained Many highly inter-connected decisionsMany highly inter-connected decisions Large-scale networks involving time Large-scale networks involving time and and spacespace Integrality requirementsIntegrality requirements Fixed costsFixed costs
Associated with sets of design decisions, not a single Associated with sets of design decisions, not a single design decisiondesign decision
Huge Huge mathematical programsmathematical programs Notoriously weak linear programming relaxationsNotoriously weak linear programming relaxations
Both models and algorithms are Both models and algorithms are critical to tractabilitycritical to tractability
Institute for Mathematics and its Applications
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Designing Service Networks for Time-Definite Parcel Delivery
Problem Description and BackgroundProblem Description and Background Designing the Air NetworkDesigning the Air Network
Optimization-based approachOptimization-based approach Case StudyCase Study
Research conducted jointly with Prof. Andrew Armacost, USAFA
Institute for Mathematics and its Applications
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Problem Overview
Gateway
HubGround centers
Pickup Route
Delivery RouteH
pickup linkdelivery linkfeeder/ground link
2
1
3
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UPS Air Network Overview
AircraftAircraft 168 available for Next-Day Air operations168 available for Next-Day Air operations 727, 747, 757, 767, DC8, A300727, 747, 757, 767, DC8, A300
101 domestic air “gateways”101 domestic air “gateways” 7 hubs (Ontario, DFW, Rockford, Louisville, 7 hubs (Ontario, DFW, Rockford, Louisville,
Columbia, Philadelphia, Hartford)Columbia, Philadelphia, Hartford) Over one million packages nightlyOver one million packages nightly
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Research Question
What aircraft routes and schedules provide What aircraft routes and schedules provide on-time service for all packages while on-time service for all packages while minimizing total costs?minimizing total costs?
Institute for Mathematics and its Applications
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UPS Air Network Overview
Delivery Routes
Pickup Routes
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Problem Formulation
Select the minimum cost routes, fleet assignments, Select the minimum cost routes, fleet assignments, and package flowsand package flows
Subject to:Subject to: Fleet size restrictionsFleet size restrictions Landing restrictionsLanding restrictions Hub sort capacitiesHub sort capacities Aircraft capacitiesAircraft capacities Aircraft balance at all locationsAircraft balance at all locations Pickup and delivery time requirementsPickup and delivery time requirements
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k2K
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kij +
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f 2F
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r2R f
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r
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8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
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X
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±hij x
kij · eh h 2 H (7)
X
r2R f
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fr = 0 i 2 N; f 2 F (8)
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f 2F
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r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
Express Shipment ServiceNetwork Design Problem
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kij +
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r2R f
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subject to:
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8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
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i 2 N; k 2 K (6)
X
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(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
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X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
minX
k2K
X
(i;j )2A
ckij x
kij +
X
f 2F
X
r2R f
dfr yf
r
subject to:
X
k2K
xkij ·
X
f 2F
X
r2R f
±f rij uf
r yfr (i; j ) 2 A (5)
X
j :(i;j )2A
xkij ¡
X
j :(j ;i)2A
xkj i =
8>>><
>>>:
bk if i = O(k)
¡ bk if i = D(k)
0 otherwise
i 2 N; k 2 K (6)
X
k2K
X
(i;j )2A
±hij x
kij · eh h 2 H (7)
X
r2R f
¯ ri y
fr = 0 i 2 N; f 2 F (8)
X
r2R f
yfr · nf f 2 F (9)
X
f 2F
X
r2R f
±rhyf
r · ah h 2 H (10)
xkij ¸ 0 (i; j ) 2 A; k 2 K (11)
yfr 2 Z+ r 2 Rf ; f 2 F (12)
6
Institute for Mathematics and its Applications
13
The Size Challenge
Conventional modelConventional model Number of variables exceeds one Number of variables exceeds one
billionbillion Number of constraints exceeds Number of constraints exceeds
200,000200,000
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Column and Cut Generation
Constraint MatrixConstraint Matrix
variables in thevariables in theoptimal solutionoptimal solution
variables not consideredvariables not considered
billions of variablesbillions of variables
Hu
nd
red
s o
f H
un
dre
ds
of
tho
usa
nd
sth
ou
san
ds
of
o
f
co
nst
rain
ts c
on
stra
ints
additionalconstraints added
constraints not considered
additionaladditionalvariablesvariablesconsideredconsidered
Institute for Mathematics and its Applications
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Algorithms for Huge Integer Programs: Branch-and-Price-and-Cut
Determines Optimal Solutions to Huge Integer Determines Optimal Solutions to Huge Integer ProgramsPrograms Combines Branch-and-Bound with Column Combines Branch-and-Bound with Column
Generation and Cut Generation to solve the Generation and Cut Generation to solve the LP’sLP’s
x1 = 1 x1 = 1 x1=0x1=0
x2=1x2=1 x2=0x2=0 x3=1x3=1 x3=0x3=0
x4=1x4=1 x4=0x4=0
Institute for Mathematics and its Applications
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The Integrality Challenge
Initial feasible solution about triple Initial feasible solution about triple the best boundthe best bound Multiple day runtimes to achieve Multiple day runtimes to achieve
first feasible solutionfirst feasible solution
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Resolution of Challenges
Algorithms are not enoughAlgorithms are not enough Key to successful solution of these Key to successful solution of these
very large-scale problems are the very large-scale problems are the models themselvesmodels themselves
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Alternative Formulations
A given problem may have different A given problem may have different formulations that are all logically equivalent formulations that are all logically equivalent yet differ significantly from a yet differ significantly from a computational point of viewcomputational point of view
This has motivated the study of systematic This has motivated the study of systematic
procedures for generating and solving procedures for generating and solving alternativealternative formulations formulations
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Reformulation: Key IdeasAircraft Route VariablesAircraft Route Variables
3000y1 + 8000y2 6000
Capacity-demand:
Cover:
g hdemand =6000
capacity =3000
capacity =8000
Cover:
Composite Variables
g hdemand =6000
capacity =6000
capacity =8000
y1 + y2 1y3+ y2 1
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Strength Results: Single Hub Example
ESSND ARMRows 53 34Cols 67 42NZ 274 255LP Solution 10663 28474IP Solution 28474 28474B&B Nodes 781 1LP-IP Gap 167% 0%
6
2
1
5
3
4
6
2
1
5
3
4
6
Time-Space Representation of Plane/Package Movements
Institute for Mathematics and its Applications
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ARM vs UPS PlannersMinimizing Operating Cost for UPS
Improvement (reduction)Improvement (reduction) Operating cost: 6.96 %Operating cost: 6.96 % Number of Aircraft: 10.74 %Number of Aircraft: 10.74 % Aircraft ownership cost: 29.24 %Aircraft ownership cost: 29.24 % Total Cost: 24.45 %Total Cost: 24.45 %
Running timeRunning time Under 2 hoursUnder 2 hours
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Planners’ Solution
ARM vs. PlannersRoutes for One Fleet Type
Pickup Routes Delivery Routes
ARM Solution
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ARM SolutionNon-intuitive double-leg routes
Model takes advantage of timing requirements, especially in case of Model takes advantage of timing requirements, especially in case of A-3-1, which exploits time zone changesA-3-1, which exploits time zone changes
Model takes advantage of ramp transfers at gateways 4 and 5Model takes advantage of ramp transfers at gateways 4 and 5
1
2
A
4
3
6
5
B
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Conclusions
Solving large-scale service network Solving large-scale service network design problemsdesign problemsBlend art and scienceBlend art and scienceComposite variable modeling can Composite variable modeling can
often facilitateoften facilitateTractabilityTractabilityExtendibilityExtendibility
Institute for Mathematics and its Applications
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Research conducted jointly with Stephane Bratu
Service Network Design and Passenger Service in the Airline Industry
Problem Description and BackgroundProblem Description and Background AnalysisAnalysis Some Research FindingsSome Research Findings
Institute for Mathematics and its Applications
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Route individual aircraft honoringmaintenance restrictions
Assign aircraft types to flight legs such that contribution is maximized
Airline Schedule Planning
Schedule Design
Fleet Assignment
Aircraft Routing
Crew Scheduling
Select optimal set of flight legs in a schedule
Assign crew (pilots and/or flight attendants) to flight legs
Institute for Mathematics and its Applications
27
Some Simple Statistics …
Institute for Mathematics and its Applications
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30%
40%
50%
60%
70%
1995 1996 1997 1998 1999 200032000003300000340000035000003600000370000038000003900000400000041000004200000
% of flights arriving later thanscheduled
Total number of flights operated
Number and Percentage of Delayed Flights
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Average Delay Duration of Operated Flights
0
5
10
15
20
25
1995 1996 1997 1998 1999 2000
(Min
utes
)
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15-minute On-Time Performance
50%
60%
70%
80%
90%
100%
1995 1996 1997 1998 1999 2000
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Number of Delayed Flights
The delay distribution has shifted from short to long delays
Factor 1: Shift to Longer Flight Delays
Total Delay Minutes
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Factor 2: Hub-and-Spoke and Connecting Passengers
-5
0
5
10
15
20
25
30
35
(min
utes
)
Average Delay
Local
Connecting
Flight Delay
Flight delays underestimate passenger delays Key explanation lies in the connecting passengers
Average Passenger and Flight Delays
Institute for Mathematics and its Applications
34
Factor 3: Number of Canceled Flights and Cancellation Rates
0
20,000
40,000
60,000
80,000
100,000
120,000
140,000
160,000
1995 1996 1997 1998 1999 2000
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
3.5%
4.0%
Number of CanceledFlights
Cancellation Rate
Delay statistics do not consider cancellations
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Cancellation Rate: Southwest and the Other Majors Airlines
0%
2%
4%
6%
8%
10%
12%
1995 1996 1997 1998 1999 2000
Maximum Others
Average Others
Minimum Others
Southwest
Southwest has a lower cancellation rate than any other Major from 1995 to 2000 due in part to increases in cancellation rates at some congested hubs
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Hub Cancellation Rates
0%
1%
2%
3%
4%
5%
6%
7%
8%
ORDAA
ORDUA
EWRCO
DFWAA
SFOUA
ATLDL
EWRCO
MSPNW
1995
2000
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Research Findings: Service Network Design and Passenger Service DOT 15 minute on-time-performance is inadequateDOT 15 minute on-time-performance is inadequate There are a number of alternative “flight schedules” with There are a number of alternative “flight schedules” with
similar associated costs and profitability, but vastly similar associated costs and profitability, but vastly different associated passenger delaysdifferent associated passenger delays ServiceService network design needs to incorporate network design needs to incorporate service service
considerationsconsiderations Flight cancellations can reduce overall passenger delayFlight cancellations can reduce overall passenger delay
High load factors together with flight delays can result High load factors together with flight delays can result in excessive passenger delaysin excessive passenger delays
De-banking can result in much longer planned connection De-banking can result in much longer planned connection times, but only slightly longer connection times in actualitytimes, but only slightly longer connection times in actuality