1.206J/16.77J/ESD.215J
Airline Schedule Planning
Cynthia Barnhart
Spring 2003
1.963/1.206J/16.77J/ESD.215J The
Schedule Design Problem
• Outline
– Problem Definition and Objective
– Schedule Design with Constant Market Share
– Schedule Design with Variable Market Share
– Schedule Design Solution Algorithm
– Results
– Next Steps
– A Look to the Future in Airline Schedule Optimization
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Airline Schedule Planning
Schedule Design
Select optimal set of flight legs
in a schedule
Fleet Assignment
Assign aircraft types to flight legs
such that contribution is maximized
Aircraft Routing
Crew Scheduling
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Objectives
• Given origin-destination demands and
fares, fleet composition and size, fleet
operating characteristics and costs
• Find the revenue maximizing flight
schedule
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Schedule Design: Fixed Flight
Network, Flexible Schedule
Approach
• Fleet assignment model with time windows
– Allows flights to be re-timed slightly (plus/
minus 10 minutes) to allow for improved
utilization of aircraft and improved capacity
assignments
Initial step in integrating flight schedule
design and fleet assignment decisions
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Schedule Design: Optional
Flights, Flexible Schedule
Approach
• Fleet assignment with “optional” flight legs
– Additional flight legs representing varying flight
departure times
– Additional flight legs representing new flights
– Option to eliminate existing flights from future
flight network
Incremental Schedule Design
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Integrated, Incremental Schedule
Design and Fleet Assignment Models
Base Schedule
Deletion Candidates
Mandatory Flight List
Addition Candidates
Optional Flight List
Master Flight List
Select optimal set of flight legs from master flight list
Assign fleet types to flight legs
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Demand and Supply Interactions
Market Share
450
Market Share
410
Market Share
300
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A
100
150
100
100
B
100
190
A
120
40
B
100
200
A
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150
B
Non-Linear
Interactions
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Schedule Design: Constant
Market Share Model
• Constant market share model
–Integrated Schedule Design and
Fleet Assignment Model (ISDFAM)
–Utilize recapture mechanism to
adjust demand approximately
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ISD-FAM: Example
Market Share
450
Market Share
450
A
A
100
150
100
100
100
150
100
B
B
100 + recap1
150 + recap2
A 100 + recap3 B
100
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ISD-FAM Formulation
r
r
~ f 
Min   c
  ( farep  bp farer )t p
k,i k ,i
kK iL
pP rP
 fk ,i  1
Subject to:
kK

k K
yk ,o,t  

iI (k ,o,t )
fk ,i  yk ,o,t  
iO(k ,o,t )
 yk ,o,tn 
oO


iCL(k )
f k ,i  1
i  LO
fk ,i  0
k, o, t
fk ,i  Nk
p r
p r r
 fk,iSEATS
k    i t p    i bpt p  Qi
k
rP pP
i  LF
k  K
iL
rP pP
 t p  Dp
r
pP
rP
trp  0
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fk,i 0,1
yk,o,t  0
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ISD-FAM Formulation
r
r
~ f 
Min   c
  ( farep  bp farer )t p
k,i k ,i
kK iL
pP rP
 fk ,i  1
i  LF
 f 1
Selection
i  LO
Subject to:
Flight
yk ,o,t  

iI (k ,o,t )
kK
k K
fk ,i  yk ,o,t  
FAM
iO(k ,o,t )
 yk ,o,tn 
oO


iCL(k )
k ,i
fk ,i  0
fk ,i  Nk
p r
p r r
 fk,iSEATS
k    i t p    i bpt p  Qi
k
PMM
rP pP
k  K
iL
rP pP
trp  0
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k, o, t
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 t p  Dp
r
pP
rP
fk,i 0,1
yk,o,t  0
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ISD-FAM Formulation
r
r
~ f 
Min   c
  ( farep  bp farer )t p
k,i k ,i
kK iL
pP rP
 fk ,i  1
i  LF
 f 1
Flight Selection
Schedule
Design
i  LO
Subject to:
kK
k K
yk ,o,t  

fk ,i  yk ,o,t  

k ,i
fk ,i  0
FAM
Fleet Assignment
Spill +PMM
Recapture
iI (k ,o,t )
iO(k ,o,t )
 yk ,o,tn 
oO

iCL(k )
fk ,i  Nk
p r
p r r
 fk,iSEATS
k    i t p    i bpt p  Qi
k
rP pP
k, o, t
k  K
iL
rP pP
 t p  Dp
r
pP
rP
trp  0
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fk,i 0,1
yk,o,t  0
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Schedule Design: Variable
Market Share Model
• Variable market share model
–Extended Schedule Design and
Fleet Assignment Model (ESDFAM)
–Utilize demand correction term to
adjust demand explicitly
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ESD-FAM: Demand Correction
Market Share
450
A
B
100
100+ 0
150
190+ 40
Market Share
410
A
Demand Correction Terms
Data Quality Issue
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100
150
100
100
100
120+ 20
40
B
100
150+40+40
-30
A
B
2nd degree
correction
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150
80
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ESD-FAM Formulation
Min  ck ,i f k ,i   ( farep  bpr farer )t pr 
kK iL
pP rP

p
fare
D

fare

D

O

q q
p
q  1  Z q 
pP: p q
qP 

f
Subject to:
k ,i
1
i  LF
k ,i
1
i  LO
f k ,i  0
k , o, t
kK
f
kK
yk ,o,t  

iI ( k ,o ,t )
f k ,i  yk ,o,t  
y
oO

p P q P
O
p
i
k , o ,t n

iO ( k ,o ,t )


iCL ( k )
f k ,i  N k
 Dqp 1  Z q    f k ,i SEATS k     i p t pr     i p b pr t pr  Qi
k K
r  P p P
k  K
i L
r  P p P
 D 1 Z   t
p
q
qPO
q
rP
r
p
 Dp
Z q   f k ,i  0
p P
i  L  q 
kK
Zq 
fk ,i 0,1
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Zq 0,1
t rp  0
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 f
iL ( q ) kK
k ,i
 1  Nq q  PO
yk ,o,t  0
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ESD-FAM Formulation
Min  ck ,i f k ,i   ( farep  bpr farer )t pr 
kK iL
pP rP

p
fare
D

fare

D

O

q q
p
q  1  Z q 
pP: p q
qP 

f
Subject to:
k ,i
1
i  LF
k ,i
1
i  LO
f k ,i  0
k , o, t
kK
f
ISD-FAM
kK
yk ,o,t  

iI ( k ,o ,t )
f k ,i  yk ,o,t  
y
oO

p P q P
O
p
i
k , o ,t n

iO ( k ,o ,t )


iCL ( k )
f k ,i  N k
 Dqp 1  Z q    f k ,i SEATS k     i p t pr     i p b pr t pr  Qi
k K
r  P p P
k  K
i L
r  P p P
 D 1 Z   t
p
q
qPO
q
rP
r
p
 Dp
Z q   f k ,i  0
p P
i  L  q 
Market Share Adjustment

kK
Zq 
fk ,i 0,1
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Zq 0,1
t rp  0
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iL ( q ) kK
f k ,i  1  Nq q  PO
yk ,o,t  0
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ESD-FAM Formulation




Constant

Market Share
Min
kK iL
ck ,i f k ,i 
pP rP
( farep  bpr farer )t pr 


farep Dqp  1  Zq 
 fareq Dq 
pP: p q
qPO 

Subject to:
f k ,i  1
kK
f
1
i  LO
f k ,i  0
k , o, t
yk ,o,t  
iI ( k ,o ,t )
f k ,i  yk ,o,t  
oO
p P q P
O
p
i
k ,i


Schedule
Design
ISD-FAM


& Fleet Assgn.
kK

i  LF
iO ( k ,o ,t )
yk ,o,tn 
iCL ( k )
f k ,i  N k
 Dqp 1  Z q    f k ,i SEATS k     i p t pr     i p b pr t pr  Qi
k K
r  P p P
k  K
i L
r  P p P
 D 1 Z   t
p
q
qPO
q
rP
r
p
 Dp
Z q   f k ,i  0
p P
i  L  q 
Market
Share
Adjustment

Market Share Adjustment
kK
Zq 
fk ,i 0,1
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Zq 0,1
t rp  0
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iL ( q ) kK
f k ,i  1  Nq q  PO
yk ,o,t  0
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Solution Algorithm
START
Update modifiers
Solve I/ESD-FAM
Identify itineraries that
cause discrepancies
Contribution 1
Calculate new demand
for the resulting schedule
Obtain revenue estimates
from PMM
Contribution 2
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NO
Has the
stopping criteria
been met?
YES
STOP
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State Of The Practice/ Theory
Practice:
Theory:
• Most schedule decisions
made without optimization
• At least one major airline
uses Fleet Assignment with
Time Windows
• Implementation of
Incremental Schedule
Design approach underway
at a major airline
• Models and algorithms for
incremental schedule
design have been
developed and prototyped
• Validation in progress
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Computational Experiences
• ISD-FAM requires long runtimes and large
amounts of memory
– ~ 40 minutes on a workstation class computer
for medium size (800 legs) schedules
– ~ 20 hours on a 6-processor workstation,
running parallel CPLEX for full size (2,000 legs)
schedules
• ESD-FAM takes even longer runtimes and
exhausts the memory in some cases
– 40 mins (ISD-FAM) vs. 12 hrs (ESD-FAM) on
same medium size schedule
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Schedule Design: Results
• Demand and supply interactions
– ESD-FAM captures interactions more accurately
• Resulting schedules operate fewer flights
– Lower operating costs
– Fewer aircraft required
• ~$100 - $350 million improvement annually
– Compared to planners’ schedules
– Exclude benefits from saved aircraft
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Schedule Design Results
• Results are subject to several caveats
– Plans are often disrupted
– Competitors’ responses
– Underlying assumptions
•
•
•
•
Deterministic demand
Optimal control of passengers
Demand forecast
Recapture rates/Demand correction terms
Nonetheless, significant improvements are
achievable
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Potential for Improved Results
• Replace IFAM with SFAM
m
M S S
MinCmS   fmS 
n
m1 n1
n
m
M S S
m
 f 
11 i L

    f 
 0 k, o,t
Subject to:
S
m1 n1
yk ,o,t 
m
M S S
    f 
iI ( k ,o,t ) m1 n1
m k ,i
S n
m
S n
 yk ,o,t 
y
oA
m
M S S
iO( k ,o,t ) m1 n1
k ,o,tn

m
M S S
n
m k ,i
S n
m
S n
m
S n
    f 
iCL(k ) m1 n1
 f  0,1
m
S n
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m k
S n
m
S n
 Nk k  K
yk,o,t  0
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SFAM Basic Concept
• Isolate network effects
– Spill occurs only on constrained legs
5
Potentially
Constrained
Flight Leg
3
Unconstrained
Flight Leg
Potentially
Binding
Itinerary
Non- Binding
Itinerary
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6
9
1
7
4
2
SFAM
IFAM
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FAM
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A Look to the Future: Airline
Schedule Planning Integration
Schedule Design
Fleet Assignment
Aircraft Routing
Crew Scheduling
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Integrating crew
scheduling and
fleet assignment
models yields:
• Additional 3%
savings in total
operating, spill and
crew costs
•Fleeting costs
increase by about
1%
•Crew costs
decrease by about
7%
26
A Look to the Future: Real-time
Decision Making
• For a typical airline, about 10% of scheduled
revenue flights are affected by irregularities (like
inclement weather, maintenance problems, etc.)
• According to the New York Times, irregular
operations (due mostly to weather) result in more
than $440 million per year in lost revenue, crew
overtime pay, and passenger hospitality costs
 Increasing use and acceptance of optimizationbased decision support tools for operations recovery
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A Look to the Future: Robust
Scheduling
• Issue: Optimizing “plans” results in
minimized planned costs, not realized costs
– Optimized plans have little slack, resulting in
• Increased likelihood of plan “breakage” during
operations
• Fewer recovery options
• Challenge: Building “robust” plans that
achieve minimal realized costs
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