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...

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)



Robust Maintenance Routing Graduate Student: Shan Lan Joint work with Prof. Cindy Barnhart



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




Flight Schedule Re-Timing Graduate Student: Shan Lan Joint work with Prof. Cindy Barnhart


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


Assume 30 minute minimum connecting time




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)





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






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

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...

Documents

passengers passengers

delay flights

passenger schedule disruptions

serious schedule disruptions

stage slide

year slide

maintenance feasibility

planned slide