m i t i n t e r n a t i o n a l c e n t e r f o r a i r t r a n s p o r t a t i o n new approaches...
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M I T I n t e r n a t i o n a l C e n t e r f o r A i r T r a n s p o r t a t i o nM I T I n t e r n a t i o n a l C e n t e r f o r A i r T r a n s p o r t a t i o n
MIT MIT ICAT ICAT
New Approaches to Improving the Robustness of Airline Schedules
Prof. John-Paul ClarkeDepartment of Aeronautics & Astronautics
Massachusetts Institute of Technology
MIT MIT ICAT ICAT Outline
Background & Motivation
Robust Maintenance Routing
Flight Schedule Re-Timing
Degradable Airline Schedule
Conclusions
MIT MIT ICAT ICAT
Schedule Design
Crew Scheduling
Fleet Assignment
Maintenance Routing
Airline Schedule Planning Process
Most existing airline schedule planning methods assume that aircraft, crews, and passengers will operate as planned
MIT MIT ICAT ICAT 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
MIT MIT ICAT ICAT 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
MIT MIT ICAT ICAT 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
MIT MIT ICAT ICAT 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 MIT ICAT ICAT 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 MIT ICAT 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 routingsTotal 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
MIT MIT ICAT ICAT 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
MTT f2
f1’
f2’
MIT MIT ICAT ICAT Propagated v. Independent Delay
Flight delay may be divided into two categories: Propagated delay
Caused by inbound aircraft delay – function of routing20-30% of total delay (Continental Airlines)
Independent delayCaused 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 MIT ICAT ICAT Definitions
i
j
Slack Min Turn Time
PDT
PAT
ADT
AAT
PD IAD
TAD
j’
i’i’’ PD IDD
TDD
ijjj
ijjj
ijiij
ijij
ijij
PDTADIAD
PDTDDIDD
SlackTADPD
MTTPTTSlack
PATPDTPTT
−=
−=
−=
−=
−=
)0,(max
Planned Turn Time
MIT MIT ICAT ICAT Illustration of the Idea
f1
MTT
f2
f3
f1’
f4
MTT
Original routing
f3’
New routing
f1
MTT
f2
f3
f1’
f4
MTT
MIT MIT ICAT 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 routingPD 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
MIT MIT ICAT ICAT String Based Formulation
Ssx
Ggy
Nypxr
Fiyyx
Fiyyx
Fixa
ts
xpdE
s
g
Gggg
Ssss
aiaiSs
s
didiSs
s
sSs
is
ss
sji
sij
i
i
∈∀∈
∈∀≥
≤+
∈∀=+−−
∈∀=+−
∈∀=
∑∑
∑
∑
∑
∑ ∑
∈∈
+−
∈
+−
∈
∈
∈
−
+
},1,0{
,0
,0
,0
,1
:.
))((min
,,
,,
),(
MIT MIT ICAT ICAT
∑ ∑∑ ∑∑ ∑∈∈∈
×=×=×s sji
sijs
s sji
sijs
s sji
sijs pdExpdExpdxE )][(min][min)]([min
),(),(),(
Solution Approach
Random variables (PD) can be replaced by their mean
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
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
branch-and-bound tree using column generation
IP solution A special branching strategy: branching on follow-ons (Ryan and Foster 1981, Barnhart et
al. 1998)
)))()/ln(
(1()(2
2
1δθ
δθ
mem
pdE +−
Φ−=
MIT MIT ICAT ICAT Computational Results
Test Networks
July 2000 data Model Routes Aug 2000 data
Propagated delays (August 2000)
Model Building and Validation
MIT MIT ICAT ICAT Results - Delays
Total delays and on-time performance
Passenger misconnects
MIT MIT ICAT ICAT 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 MIT ICAT 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 MIT ICAT ICAT Definitions
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
Slack MCT
PCT
PDTAATPAT ADT
ACT
MIT MIT ICAT ICAT
Illustration of the Idea
Airport A
Airport B
Airport C
Airport D
2f
3f
1f
Suppose 100 passengers in flight f2 will connect to f3
∗
2f
P (misconnect)= 0.3, E(disrupted pax) = 30
P(misconnect)=0.1,E(disrupted pax) =10
Expected disrupted passengers reduced: 20
MIT MIT ICAT ICAT 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
Schedule Design
Crew Scheduling
Fleet Assignment
Maintenance Routing
MIT MIT ICAT ICAT 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.
1,if 2,if 3,if
1,jf 2,jf 3,jf
1,1,jix
3,2,jix
MIT MIT ICAT ICAT Connection-Based Formulations
Theorem 1: The second set of constraints
are redundant and can be relaxed
Theorem 2: The integrality of the connection
variables can be relaxed
{ }{ } nif
miCjnix
jCimjfx
iCjnifx
jix
if
ts
DPxEMin
ni
ji
mjn
ji
nim
ji
m nji
nni
mjnijiji
mn
mn
mn
mn
mnmn
,;1,0
;),(,,;1,0
);(,,,
);(,,,
;,,1
;,1
..
,
,
,,
,,
,
,
,,,,,
∀∈
∈∀∈
∈∀=
∈∀=
∀=
∀=
⎥⎦
⎤⎢⎣
⎡×
+
−
+
∑
∑
∑∑
∑
∑
• Formulation I: CFSR
MIT MIT ICAT ICAT Alternative Formulations
• Formulation II: ACFSR
{ } nif
miCjnix
miCjniffx
if
ts
DPxEMin
ni
ji
mjniji
nni
mjnijiji
mn
mn
mnmn
,;1,0
;),(,,;10
;),(,,,1
;,1
..
,
,
,,,
,
,,,,,
∀∈
∈∀≤≤
∈∀−+≥
∀=
⎥⎦
⎤⎢⎣
⎡×
+
+
∑
∑
• Formulation III: DCFSR
{ }.),(,,,10
;,,1,0
;,,)(
;,,)(
;,1
..
,
,
,)(
,
,)(
,
,
,,,,,
miCjnix
nif
mjfjCx
nifiCx
if
ts
DPxEMin
mn
mn
mn
mnmn
ji
ni
mjjCi n
ji
niiCj m
ji
nni
mjnijiji
+
−
∈
+
∈
∈∀≤≤
∀∈
∀=
∀=
∀=
⎥⎦
⎤⎢⎣
⎡×
∑ ∑
∑ ∑
∑
∑
−
+
MIT MIT ICAT ICAT 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.
MIT MIT ICAT ICAT
Solution Approach
Random variables can be replaced by their mean
Distribution of
Branch-and-Price
[ ] [ ]∑∑∑ ×=×=⎥⎦
⎤⎢⎣
⎡×
mjnijiji
mjnijiji
mjnijiji mnmnmnmnmnmn
DPExDPxEDPxE,,,
,,,,,
,,,,,
,,
mn jiDP ,
p
pcDP ji
ji mn −⎩⎨⎧
=1 prob with
prob with
,0,,
,
( )MCTAATADT
p
nm ij <− prob --
by determined becan Probility
MIT MIT ICAT ICAT
July 2000 data RAMR Routes
Aug 2000 data FSRSchedule
CFSR
ACFSR
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
Strength of the formulations
MIT MIT ICAT ICAT Computational Results
Assume 30 minute minimum connecting time
Assume 25 minute minimum connecting time
Assume 20 minute minimum connecting time
MIT MIT ICAT ICAT 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 MIT ICAT ICAT Outline
Background & Motivation
Robust Maintenance Routing
Flight Schedule Re-Timing
Degradable Airline Schedule Graduate Student: Laura Kang
Conclusions
MIT MIT ICAT ICAT 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
MIT MIT ICAT ICAT Implementation Options
fleet assignment
aircraft routing
crew scheduling
schedule design
DASD-ARM (Route-based Formulation)
DAS D-FAM
D-SPM (Flight-based Formulation)DAS
MIT MIT ICAT ICAT 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%
MIT MIT ICAT ICAT 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 MIT ICAT ICAT Notation
Indices r route f itinerary ij flight k layer (k=1 … K) γij
f 1 if flight ij is in itinerary f, 0 otherwise
Decision variables yr
k 1 if route r is in layer k, 0 otherwise zf
k 1 if itinerary f is in layer k, 0 otherwise xij
k 1 if flight ij is in layer k, 0 otherwise
Parameters vf
k revenue for itinerary f is placed in layer k Ch capacity at hub h in bad weather Sk fraction of layer k ar number of flights in route r ar
h number of flights departing at hub h in route r ACN number of aircraft
MIT MIT ICAT ICAT Flight-based Formulation
MIT MIT ICAT ICAT Route-based Formulation
MIT MIT ICAT ICAT 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
MIT MIT ICAT ICAT Greedy Flight-Leg Pairing
100 1010 100
MIT MIT ICAT ICAT Swapping Search
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
route i
route j
i1 i2 i3 i4 i5 i6
j1 j2 j3 j4 j5 j6
i1 i2 i3 i4 i5 i6
j1 j2 j3 j4 j5 j6
i1 i2 i3 i4 i5 i6
j1 j2 j3 j4 j5 j6
i1 i2
i3 i4
i5 i6
j1 j2
j3 j4 j5
j6
MIT MIT ICAT ICAT 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
MIT MIT ICAT ICAT
D-ARM w/Heuristics 8,123,060
D-SPM 8,667,632
IP Objective Function Value
Current routing 6,492,895
Upper bound for route-based DAS
Lower bound for route-based DAS
MIT MIT ICAT ICAT
D-ARM w/Heuristics 9,057,750
D-SPM 9,624,460
Protected Revenue
Current routing 7,302,040
74.5%
70.1%
56.6%
MIT MIT ICAT ICAT
D-ARM w/Heuristics 43,051
D-SPM 44,984
Protected Passengers
Current routing 37,587
67.5%
64.6%
56.4%
MIT MIT ICAT ICAT Simulation Results - Good Weather
Pr(delay > 15)
Pr(delay >0)
Average delay
Layer 1
6 min
0.37
0.14
Layer 2
6 min
0.48
0.17
Current Routing
6 min
0.42
0.16
MIT MIT ICAT ICAT Simulation Results - Bad Weather
Pr(delay > 15)
Pr(delay >0)
Average delay
Layer 1
13 min
0.52
0.30
Layer 2
25 min
0.69
0.48
Current Routing
17 min
0.61
0.37
MIT MIT ICAT ICAT Outline
Background & Motivation
Robust Maintenance Routing
Flight Schedule Re-Timing
Degradable Airline Schedule
Conclusions
MIT MIT ICAT 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