MIT
ICAT
MIT
International
Center
for
Air
Transportation
New Approaches to Improving the
Robustness of Airline Schedules
Prof. John-Paul Clarke
Department of Aeronautics & Astronautics
Massachusetts Institute of Technology
MIT
ICAT
Outline
 Background & Motivation
 Robust Maintenance Routing
 Flight Schedule Re-Timing
 Degradable Airline Schedule
 Conclusions
MIT
ICAT
Airline Schedule Planning Process
Schedule Design
Fleet Assignment
Maintenance Routing
Crew Scheduling
 Most existing airline schedule planning methods
assume that aircraft, crews, and passengers will
operate as planned
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Airline Operations
 Bad weather reduces airport capacity
 Airlines cancel or delay flights to reduce demand
 Delays propagate through the network
 Airlines must reschedule aircraft/crew and re-
accommodate passengers
 Passengers are not satisfied
 They are delayed
 They have no control over their delay
 All passengers on a given aircraft are delayed equally regardless
of fare class
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Delays & Cancellations
 Trend (1995-1999)
 Significant increase (100%) in flights delayed more than 45 min
 Significant increase (500%) in the number of cancelled flights
 Year 2000
 30% of flights delayed
 3.5% of flights (approx. 140,000) cancelled
 Future:
 Delays and cancellations may increase dramatically  more
frequent and serious schedule disruptions and revenue loss
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Passenger Disruptions
 Flight delays and cancellations often cause
passenger schedule disruptions
 26 million passengers (4% of passengers) disrupted
 65% of disruptions caused by missed connections
 Very long delays for disrupted passengers
 Average delay for disrupted passengers is approx. 4 hours
(versus 15 min delay for non-disrupted passengers)
 Significant revenue loss - approx. $4 Billion /year
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Robustness
 Need schedules that are robust (insensitive) to
delays and cancellations
 Definitions of robustness
 Minimize cost (expected/worst case deviations from optimal)
 Minimize aircraft/passenger delays and disruptions
 Easy to recover (aircraft, crew, passenger)
 Isolate disruptions and reduce the downstream impact
 Two ways to provide robustness
 Re-optimize schedule after disruptions occur (operation stage)
 Build robustness into the schedules (planning stage)
MIT
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Outline
 Background & Motivation
 Robust Maintenance Routing
 Graduate Student: Shan Lan
 Joint work with Prof. Cindy Barnhart
 Flight Schedule Re-Timing
 Degradable Airline Schedule
 Conclusions
MIT
ICAT
Robust Maintenance Routing
 Objective
 Reduce the propagation of delays by combining flight segments
in optimal (from the point of view of follow-on delays)
maintenance routings
 Total delay for a route is uniquely determined by routing
 Solution Approach
 Derive distributions from historical data for delay introduced into a
route by an airport
 Formulate and solve maintenance routing model that minimizes
the propagation of delays subject to maintenance feasibility
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Delay Propagation
 Arrival delay may cause departure delay for the next
flight that is using the same aircraft if there is not
enough slack between these two flights
 Delay propagation may cause schedule, passenger
and crew disruptions for downstream flights
(especially at hubs)
f1
f1’
MTT
f2
f2’
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Propagated v. Independent Delay
 Flight delay may be divided into two categories:
 Propagated delay
 Caused by inbound aircraft delay – function of routing
 20-30% of total delay (Continental Airlines)
 Independent delay
 Caused by other factors – not a function of routing
 Appropriately allocated slack can reduce propagated
delay
 Add slack where advantageous
 Reduce slack where less needed
MIT
ICAT
Definitions
i
TDD
i’’
i’
PD
PDT
Slack
IDD
ADT
Min Turn Time
Planned Turn Time
j’
j
PAT
PTTij  PDT j  PATi
Slack ij  PTTij  MTT
PDij  max( TADi  Slack ij ,0)
IDD j  TDD j  PDij
IAD j  TAD j  PDij
AAT
PD
IAD
TAD
MIT
ICAT
Illustration of the Idea
f1
f1’
MTT
f2
MTT
f3
f3’
f4
Original routing
f1
f1’
MTT
f2
MTT
New routing
f3
f4
MIT
ICAT
Modeling Issues
 Difficult to use leg-based models to track the delay
propagation
 One variable (string) for each aircraft route between
two maintenance events (Barnhart, et al. 1998)
 A string: a sequence of connected flights that begins and ends at
maintenance stations
 Delay propagation for each route can be determined
 Need to determine delays for each feasible route
 Most of the feasible routes haven’t been realized yet
 PD and TAD are a function of routing
 PD and TAD for these routes can’t be found in the historical data
 IAD is not a function of routing and can be calculated by tracking
the route of each individual aircraft in the historical data
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String Based Formulation
min E ( (
s

pd ijs ) xs )
( i , j )s
s.t :
a
is
sS
xs  1, i  F


x

y

y
 s i ,d i ,d  0, i  F
sS i



x

y

y
 s i ,a i ,a  0, i  F
sS i
r x
sS
s
s
  pg yg  N
gG
y g  0, g  G
xs  {0,1}, s  S
MIT
ICAT

Solution Approach
Random variables (PD) can be replaced by their mean
min E[ xs  (
s

 pd
( i , j )s
s
ij
)]  min
x
s
 E[
s
 pd
( i , j )s
s
ij
]  min
x
s
s
(
 E[ pd
( i , j )s
s
ij
])
Distribution of Total Arrival Delay
 Possible distributions analyzed: Normal, Exponential, Gamma, Weibull, Lognormal,
etc. lognormal distribution is the best fit

A closed form of expected value function
1 2

ln(  / m)
E ( pd )  (1  (
))(  me 2 )

 Mixed-integer program with a huge number of 0-1 variables
 Branch-and-price
 Branch-and-Bound with a linear programming relaxation solved at each node of the branchand-bound tree using column generation
 IP solution
 A special branching strategy: branching on follow-ons (Ryan and Foster 1981, Barnhart et
al. 1998)
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Computational Results
 Test Networks

Model Building and Validation
July 2000 data

Model
Routes
Propagated delays (August 2000)
Aug 2000 data
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Results - Delays
 Total delays and on-time performance

Passenger misconnects
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Outline
 Background & Motivation
 Robust Maintenance Routing
 Flight Schedule Re-Timing
 Graduate Student: Shan Lan
 Joint work with Prof. Cindy Barnhart
 Degradable Airline Schedule
 Conclusions
MIT
ICAT
Flight Schedule Re-Timing
 Objective
 Reduce the number of passenger misconnections by adjusting
departure times so that passenger connection times are
correlated with the likelihood of a missed connection (disruption)
 Add connection slack where it is need most
 Solution Approach
 Derive distributions from historical data for number of passengers
disrupted for each connection
 Formulate and solve re-timing model that minimizes the number
of disrupted passengers
MIT
ICAT
Definitions
ACT
PAT
AAT PDT ADT
MCT
Slack
PCT

AAT = Actual Arrival Time

ACT = Actual Connection Time

ADT = Actual Departure Time

MCT = Minimum Connection Time

PAT = Planned Arrival Time

PCT = Planned Connection Time

PDT = Planned Departure Time
MIT
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Illustration of the Idea
Suppose 100 passengers in flight f2 will connect to f3
Airport A
P (misconnect)= 0.3,
E(disrupted pax) = 30
f1
Airport B
f

f2
2
Airport C
f3
Airport D
 Expected disrupted passengers reduced: 20
P(misconnect)=0.1,
E(disrupted pax) =10
MIT
ICAT
Schedule
Design
Fleet
Assignment
Maintenance
Routing
Crew
Scheduling
Implementation Options
 Passenger disruption depends on
flight delays, a function of fleeting and
routing
 Before maintenance routing problem
 Delay propagation not considered
 New fleeting and routing solution may cause
delay to propagate in a different way 
change the number of disrupted passengers
 After maintenance routing problem
 Delay propagation considered
 Need to enable the current fleeting and routing
solution
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Connection-Based Formulation
 Objective
 minimize the expected total number of passenger misconnects
 Constraints:
 For each flight, exactly one copy will be selected.
 For each connection, exactly one copy will be selected and this
selected copy must connect the selected flight-leg copies.
 The current fleeting and routing solution cannot be altered.
f i ,1
fi,2
f i ,3
xij, 2,3
xij,1,1
f j ,1
f j,2
f j ,3
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Connection-Based Formulations
• Formulation I: CFSR


Min E   xin , jm  DPin , jm 
i , n , j , m

s.t.
f
i ,n
 1, i;
n
 xin , jm  1, i, j;
m
n
x
in , j m
 f i ,n , i, n, j  C  (i );
m
x
in , j m
 f j ,m , j , m, i  C  ( j );
n
xin , jm  0,1; i, n, j  C  (i ), m;
f i ,n  0,1; i, n
 Theorem 1:
 The second set of constraints
are redundant and can be
relaxed
 Theorem 2:
 The integrality of the connection
variables can be relaxed
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Alternative Formulations
• Formulation II: ACFSR


Min E   xin , jm  DPin , jm 
i , n , j , m

s.t.
f
i ,n
 1, i;
 xin , jm  C  (i) f i ,n , i, n;
jC  ( i ) m
 x
iC  ( j ) n


Min E   xin , jm  DPin , jm 
i , n , j , m

s.t.
f
i ,n
 1, i;
n
n

• Formulation III: DCFSR
in , j m
 C  ( j ) f j ,m , j , m;
f i ,n  0,1, i, n;
0  xin , jm  1, i, n, j  C  (i ), m.
xin , jm  f i ,n  f j ,m  1, i, n, j  C  (i ), m;
0  xin , jm  1; i, n, j  C  (i ), m;
f i ,n  0,1; i, n
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More Model Properties
 Theorem 3:
 ACFSR model is equivalent to CFSR model
 Theorem 4:
 DCFSR model is equivalent to CFSR model
 Theorem 5:
 The LP relaxation of CFSR model is at least as strong as that of
ACFSR, and can be strictly stronger.
 Theorem 6:
 The LP relaxation of CFSR model is at least as strong as that of
DCFSR, and can be strictly stronger.
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Solution Approach
 Random variables can be replaced by their mean


E   xin , jm  DP in , jm    E xin , jm  DP in , jm   xin , jm  E DP in , jm
i ,n, j ,m
i , n , j , m
 i ,n, j ,m

 Distribution of DPi , j
n
DPin , jm


m
ci , j , with prob p

 0, with prob 1  p
Probility p can be determined by

- - prob ADT jm  AATin  MCT
 Branch-and-Price


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Computational Results
 Network
 We use the same four networks, but add all flights together and form one
network with total 278 flights.
 Model Building and Validation
July 2000 data
RAMR
Routes
Aug 2000 data
Schedule
FSR
 Strength of the formulations
ACFSR
CFSR
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Computational Results
 Assume 30 minute minimum connecting time
 Assume 25 minute minimum connecting time
 Assume 20 minute minimum connecting time
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Computational Results

Number of copies

Estimated reduction in total passenger delays: (30 minutes MCT)
 20% (30 minute time window), 16% (20 minute time window), 10% (10 minute time
window)
MIT
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Outline
 Background & Motivation
 Robust Maintenance Routing
 Flight Schedule Re-Timing
 Degradable Airline Schedule
 Graduate Student: Laura Kang
 Conclusions
MIT
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Degradable Airline Schedule
 Objective
 Develop airline schedule that is robust, i.e. delays are isolated
 Provide priority (and thus reliability) for each flight
 Improve customer satisfaction by giving passengers an accurate
expectation of the level of service
 Provide basis for revenue management and ATC auctions
 Solution Approach
 Partition schedule into smaller independent prioritized schedules
(layers) subject to operational feasibility
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Implementation Options
schedule design
fleet assignment
aircraft routing
crew scheduling
DAS
D-SPM
(Flight-based Formulation)
DAS
D-FAM
DAS
D-ARM
(Route-based Formulation)
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IP Model
 Prioritize layers based on revenue (e.g. group highest
revenue flights together in most reliable layer)
 Revenue is “protected” if all flight legs in an itinerary
are in a “protected” layer
 IP model maximizes the total protected revenue
subject to feasibility constraints
 Prototype 2 layers implementation
 Layer 1: 60% (protected layer)
 Layer 2: 40%
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Model Statistics
 1,134 flight legs
 274 aircraft
 1,744 itineraries (8% of total)
 Single flight leg: 1,130
 2 flight legs: 613
 3 flight legs: 1
 53,091 passengers (80% of total)
 $10,839,340 revenue (84% of total)
MIT
ICAT

Indices






r
f
ij
k
γijf
route
itinerary
flight
layer (k=1 … K)
1 if flight ij is in itinerary f, 0 otherwise
Decision variables
 y rk
 zfk
 xijk

Notation
1 if route r is in layer k, 0 otherwise
1 if itinerary f is in layer k, 0 otherwise
1 if flight ij is in layer k, 0 otherwise
Parameters






v fk
Ch
Sk
ar
a rh
ACN
revenue for itinerary f is placed in layer k
capacity at hub h in bad weather
fraction of layer k
number of flights in route r
number of flights departing at hub h in route r
number of aircraft
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Flight-based Formulation
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Route-based Formulation
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Greedy Flight-Leg Pairing
STEP 0: Fix connections for non-hub to non-hub flights
STEP 1: Pair flight segments at spoke airports using the revenue
paring with aircraft utilization heuristic
STEP 2: Combine paired flight segments from step 1 at hub
airports using the revenue paring with aircraft
utilization heuristic
STEP 3: Partition very long routes into several shorter routes
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100
10
Greedy Flight-Leg Pairing
10
100
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ICAT
Swapping Search
route i
i1
i2
j3i3
j4 i4
route j
j1
j2
j3
i3
j4
i4
j5
i5
i5
j5 j6
i6
j6
 Check swapping feasibility
 Check constraints satisfaction
 Check objective function improvement
 Assume revenue is protected proportionally to the number of flight
legs in the protected layer
 Swap
i6
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Tabu Search
STEP 0: start with initial solution x* from revenue paring heuristics
WHILE( number of iteration is less than N )
STEP 1: Swapping Search. If f(x) > f(x*), x*  x
STEP 2: Update Tabu list
 If a pair was in a tabu list for Y iterations, remove it from the tabu list
 Set X pairs which were swapped in the search in the tabu list
 Tabu search is sensitive to its parameters X, Y, N
 State-of-art decision for X, Y, N
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IP Objective Function Value
D-SPM
8,667,632
Upper bound for
route-based DAS
D-ARM w/Heuristics
8,123,060
Lower bound for
route-based DAS
Current routing
6,492,895
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Protected Revenue
D-SPM
9,624,460
74.5%
D-ARM w/Heuristics
9,057,750
70.1%
Current routing
7,302,040
56.6%
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Protected Passengers
D-SPM
44,984
67.5%
D-ARM w/Heuristics
43,051
64.6%
Current routing
37,587
56.4%
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Simulation Results - Good Weather
Layer 1
Layer 2
Current Routing
Average delay
6 min
6 min
6 min
Pr(delay >0)
0.37
0.48
0.42
Pr(delay > 15)
0.14
0.17
0.16
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ICAT
Simulation Results - Bad Weather
Layer 1
Layer 2
Current Routing
13 min
25 min
17 min
Pr(delay >0)
0.52
0.69
0.61
Pr(delay > 15)
0.30
0.48
0.37
Average delay
MIT
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Outline
 Background & Motivation
 Robust Maintenance Routing
 Flight Schedule Re-Timing
 Degradable Airline Schedule
 Conclusions
MIT
ICAT
Conclusions
 Robust Maintenance Routings provide:
 Airline schedule with reduced delay propagation
 Flight Schedule Re-Timing provides:
 Airline schedule with fewer passenger disruptions or missed
connections
 Degradable Airline Schedules provide:
 Airline schedule that isolates delays
 Tool for managing passengers’ expectation
 Potential revenue enhancement