Airline Fleet Assignment
Cynthia Barnhart
16.75 Airline Management
Outline:
–
–
–
–
–
Problem Definition and Objective
Fleet Assignment Network Representation
Fleet Assignment Models and Algorithms
Extension of Fleet Assignment to Schedule Design
Conclusions
3/9/2006
16.75
1
Airline Schedule Planning
3/9/2006
Schedule Design
Select optimal set of flight legs
in a schedule
Fleet Assignment
AAssign
flight specifies
aircraft types
origin,todestination,
flight legs
such thatand
contribution
departureistime
maximized
Aircraft Routing
Route individual aircraft honoring
Contribution = Revenue - Costs
maintenance restrictions
Crew Scheduling
Assign crew (pilots and/or flight
attendants) to flight legs
16.75
2
Fleet Assignment
Problem Definition
• Given:
¾Flight Schedule: a set of (daily) flight legs;
¾Aircraft fleet: consisting of different fleet types;
¾Passenger demand pattern;
¾Revenue and operating cost data;
• Find:
A feasible fleet assignment, i.e. a mapping from flight
legs to fleet types that maximizes
profit = total revenue – total operating costs.
3/9/2006
16.75
3
Fleet Assignment
• Class Exercise: Flight Network
BOS
Boston Logan
ORD
Chicago O’Hare
3/9/2006
CL50x
(2 flights)
CL33x
(3 flights)
CL55x
(2 flights)
LGA
16.75
CL30x
(3 flights)
New York LaGuardia
4
Fleet Assignment
• Class Exercise: Flight Schedule, Fares, &
Demand
Flight #
From
To
Dept Time Arr Time
Fare
CL301
CL302
CL303
CL331
CL332
CL333
CL501
CL502
CL551
CL552
3/9/2006
LGA
LGA
LGA
BOS
BOS
BOS
LGA
LGA
ORD
ORD
BOS
BOS
BOS
LGA
LGA
LGA
ORD
ORD
LGA
LGA
(EST)
1000
1100
1800
0700
1030
1800
1100
1500
0700
0830
16.75
(EST)
1100
1200
1900
0800
1130
1900
1400
1800
1000
1130
[$]
150
150
150
150
150
150
400
400
400
400
Demand
[passengers]
250
250
100
150
300
150
150
200
200
150
5
Fleet Assignment
• Class Exercise: Fleet Information
Fleet type
DC-9
B737
A300
3/9/2006
Number of
aircraft owned
1
2
2
Capacity
[seats]
120
150
250
16.75
Per flight operating cost [$000]
LGA - BOS
LGA – ORD
10
15
12
17
15
20
6
Fleet Assignment
• Class Exercise:
Find:
An assignment of fleet types to the
flights in this network that
maximizes net profit.
3/9/2006
16.75
7
Fleet Assignment
‰Evaluating assignment profits:
cl , f := fare l × min( Dl , Cap f ) − OpCost l , f
where:
cl , f : profitability of assigning fleet type f to flight leg l ;
farel : fare of flight leg l ;
Dl : demand of flight leg l ;
Cap f : capacity of fleet type f ;
OpCostl , f : operating cost of assigning fleet type f to flight leg l.
3/9/2006
16.75
8
Fleet Assignment
• Class Exercise: Evaluating assignment
profitabilities…
Profitability [$000 per day]
Flight #
CL301
CL302
CL303
CL331
CL332
CL333
CL501
CL502
CL551
CL552
3/9/2006
DC-9
B737
8
8
5
8
8
8
33
33
33
33
10.5
10.5
3
10.5
10.5
10.5
43
43
43
43
16.75
A300
22.5
22.5
0
7.5
22.5
7.5
40
60
60
40
9
Fleet Assignment
‰Time-Line Network:
CL332
BOS
CL333
CL303
CL302
CL331
CL301
LGA
CL551
CL501
CL552
CL502
ORD
0800
1000
3/9/2006
1200
1400
16.75
1600
timescale
10
Fleet Assignment
‰Optimal solution:
CL332
CL331
CL301
B737
B737
CL552
Revenue = $428,500
Cost = $148,000
3/9/2006
DC9
00 CL302
3
A
DC9
A300
CL551
A300
CL333
CL303
B737
B737
A300
CL501
CL502
Leg
∆capacity
CL301
CL302
CL303
CL331
CL332
CL333
CL501
CL502
CL551
CL552
-130
0
+50
0
-50
-30
0
+50
+50
0
Profit = $280,500
16.75
11
Fleet Assignment
‰Time-Line Network:
Flight arc
Ground arc
Airport A
yv − v yv +
wl,f
Airport B
count line
3/9/2006
16.75
12
Notations
• Decision Variables
– fk,i equals 1 if fleet type k is assigned to flight leg i, and 0
otherwise
– yk,o,t is the number of aircraft of fleet type k, on the ground
at station o, and time t
• Parameters
– Ck,i is the cost of assigning fleet k to flight leg i
– Nk is the number of available aircraft of fleet type k
– tn is the “count time”
• Sets
–
–
–
–
L is the set of all flight legs i
K is the set of all fleet types k
O is the set of all stations o
CL(k) is the set of all flight arcs for fleet type k crossing the
count time
3/9/2006
16.75
13
Fleet Assignment Model (FAM)
Min ∑ ∑ ck,i fk,i
k∈K i∈L
Subject to:
∑ fk,i = 1
∀i∈ L
fk,i = 0
∀k, o,t
k∈
∈K
K
yk,o,t − +
∑
i∈I (k ,o,t )
fk,i − yk,o,t + −
∑
i∈O(k ,o,t )
∑ yk,o,tn + ∑ fk,i ≤ Nk ∀k ∈ K
o∈O
i∈CL(k )
fk,i ∈{0,1}
yk,o,t ≥ 0
Hane et al. (1995), Abara (1989), and Jacobs, Smith and
Johnson (2000)
3/9/2006
16.75
14
Constraints
• Cover Constraints
– Each flight must be assigned to exactly one
fleet
• Balance Constraints
– Number of aircraft of a fleet type arriving at
a station must equal the number of aircraft
of that fleet type departing
• Aircraft Count Constraints
– Number of aircraft of a fleet type used
cannot exceed the number available
3/9/2006
16.75
15
Objective Function
• For each fleet - flight combination: Cost ≡
Operating cost + Spill cost –Recaptured
revenue
• Operating cost associated with assigning a fleet
type k to a flight leg j is relatively
straightforward to compute
– Can capture range restrictions, noise restrictions,
water restrictions, etc. by assigning “infinite” costs
• Spill cost for flight leg j and fleet assignment k =
average revenue per passenger on j * MAX(0,
unconstrained demand for j – number of seats on
k)
– Unclear how to compute revenue for flight legs, given
revenue is associated with itineraries
• Recaptured revenue
– Revenue from passengers16.75
that are recaptured back to
the airline after being spilled from another flight leg
3/9/2006
16
FAM Example: Spill
A
B
Demand = 100
Fare = $100
Fleet Type
Capacity
Revenue
Spill
Cost
Op. Cost
Assignment
Cost
Contribution
i
80
$8,000
$2,000
$5,000
$7,000
$3,000
ii
100
$10,000
$0
$6,000
$6,000
$4,000
iii
120
$10,000
$0
$7,000
$7,000
$3,000
iv
150
$10,000
$0
$8,000
$8,000
$2,000
3/9/2006
16.75
17
FAM Example: Recapture
100 seats
100 seats
A
B
( 80, $200 )
( 90, $250 )
( 50, $400 )
9AM
10AM
C
20
30 X 0.3 = 9 recaptured passengers
29
( 20, $400 )
( Demand, Fare )
Recapture Rate
3/9/2006
16.75
18
Fleet Assignment
‰
A few observations on FAM:
Nodes can be consolidated to reduce model
size;
¾ Fleet-specific time-line networks are possible;
¾ Fleet assignment not aircraft assignment!
¾ Note that feasibility of FAM implies that
aircraft rotations exist (takes only a little bit
of thinking);
¾ However, these rotations might not be
maintenance feasible…
¾
3/9/2006
16.75
19
Fleet Assignment
‰
Solvability:
FAM can be solved using standard branchand-bound software;
¾ Solution times are FAST, thanks to FAM’s
small LP gaps…
¾
3/9/2006
16.75
20
Fleet Assignment
Computational Sample: 2,044 flight legs, 9 fleet types
Problem size
# of columns
# of rows
# of non-zero entries
Strength of formulation
Root node LP solution
Best IP solution
Difference
Solution time [sec]
3/9/2006
16.75
18,487
7,827
50,034
21,401,658
21,401,622
36
974
21
Fleet Assignment
• FAM suffers from a significant drawback
in its modeling of the revenue side…
• Passengers book itineraries not flight
legs…
• Capacity decisions on one leg will affect
passenger spill on other legs…
• This phenomenon is known as network
effects.
3/9/2006
16.75
22
Fleet Assignment Model (FAM)
Min ∑ ∑ ck,i fk,i
k∈K i∈L
Subject to:
∑ fk,i = 1
∀i∈ L
fk,i = 0
∀k, o,t
k∈K
yk,o,t − +
∑
i∈I (k ,o,t )
fk,i − yk,o,t + −
∑
i∈O(k ,o,t )
∑ yShortcoming:
k ,o,tn + ∑ fk ,i ≤ Nk ∀k ∈ K
Major
o∈O
i∈CL(k )
FAM assumes leg independence
fk,i ∈{0,1} yk,o,t ≥ 0
Hane et al. (1995), Abara (1989), and Jacobs, Smith and
Johnson (2000)
3/9/2006
16.75
23
FAM Example: Network Effects
A
B
( 80, $200 )
C
( 90, $250 )
( 50, $400 )
( Demand, Fare )
Fleet Type
Capacity
Spill Cost
i
80
?
ii
100
?
iii
120
?
iv
150
$0
3/9/2006
16.75
Leg Interdependence
Network Effects
24
Spill Cost Computation and
Underlying Assumption
• Given:
– Spill cost for flight leg j and fleet assignment k
= average revenue per passenger on j *
MAX(0, unconstrained demand for j –
number of seats on k)
• Implication:
– A passenger might be spilled from some, but
not all, of the flight legs in his/ her itinerary
3/9/2006
16.75
25
An Iterative Approach
Fleet
Assignment
Solver
• FAM Solver
Problem
Modification
– Basic Fam
• Possibly with minor modifications
Spill
Calculation
• Spill Calculation:
– Simulations
– Passenger mix model
• Problem Modification:
– Objective cost coefficient update
3/9/2006
16.75
26
Passenger Mix
• Passenger Mix Model (PMIX)
– Kniker (1998)
– Given a fixed, fleeted schedule, unconstrained
passenger demands by itinerary (requests), and
recapture rates find maximum revenue for
passengers on each flight leg
PMIX
3/9/2006
Network Effects and Recapture
16.75
27
Problem Modification
• Based on differences in expected spill from
FAM and the Spill Calculator, we modify
the FAM problem
– Update objective cost coefficients
• Cost coefficient update, many heuristics
possible
3/9/2006
16.75
28
FAM Spill Calculation
Heuristics
• Fare Allocation
– Full fare - the full fare is assigned to each leg of the
itinerary
– Partial fare - the fare divided by the number of legs is
assigned to each leg of the itinerary
– Shared fare - the fare divided by the number of
capacitated legs is assigned to each capacitated leg in
the itinerary
• Spill Cost for each variable
– Representative Fare
• A “spill fare” is calculated; each passenger spilled results in a loss of
revenue equal to the spill fare
– Integration
• Sort each itinerary by fare, spill costs are sum of x lowest fare
passengers, where x = max{0, demand - capacity}
3/9/2006
16.75
29
An Illustrative Example
Fleet Type
A
B
Market
X-Y
Seats
100
200
Itinerary
1
flight 1
X
Average
Fare
$200
flight 2
Y
Z
No. of
Pax
75
Y-Z
2
$225
150
X-Z
1-2
$300
75
Partial Alloc.
Full Alloc.
Actual Opt.
Spill
Spill
Spill Spilled Pax
A-A
$30,000
$38,125
31,875
50 X-Z, 75 Y-Z
A-B
$11,250
$15,625
12,500
25 X-Z, 25 X-Y
B-A
$22,500
$28,125
28,125
125 Y-Z
B-B
$3,750
$5,625
5,625
Fleet Assign.
Fl. 1- Fl. 2
3/9/2006
16.75
25 Y-Z
30
Spill Calculation: Results
• For a 3 fleet, 226 flights problem:
– The best representative fare solution results in
a gap with the optimal solution of $2,600/day
– Using a shared fare scheme and integration
approach, we found a solution with an $8/day
gap.
• By simply modifying the basic spill model,
significant gains can be achieved
3/9/2006
16.75
31
Itinerary-Based Fleet
Assignment
• Impossible to estimate airline profit
exactly using link-based costs
• Enhance basic fleet assignment model
to include passenger flow decision
variables
– Associate operating costs with fleet
assignment variables
– Associate revenues with passenger flow
variables
3/9/2006
16.75
32
Itinerary-based Fleet
Assignment Definition
• Given
– a fixed schedule,
– number of available aircraft of different types,
– unconstrained passenger demands by
itinerary, and
– recapture rates,
Find maximum contribution
ODFAM
3/9/2006
Network effects
16.75
33
Itinerary-Based FAM (IFAM)
M in ∑
∑
k ∈ K i∈ L
c%k , i f k , i +
∑∑
p∈ P r ∈ P
( fare p − b pr fare r ) t rp
∑ f k ,i = 1
S u b je c t to :
k∈K
∀i ∈ L
Fleet Assignment
FAM
y k ,o ,t − +
∑
f k ,i = 0
∀ k , o,t
f k ,i ≤ N k
∀k ∈ K
+ ∑ ∑ δ i p t rp − ∑ ∑ δ i p b rp t rp ≥ Q i
∀i ∈ L
i∈ I ( k , o , t )
f k ,i − y k ,o ,t + −
i∈ O ( k , o , t )
∑ y k ,o ,t n +
o∈O
∑ f k , i SEATS
∑
∑
i∈ CL ( k )
ConsistentPMM
Spill + ∑Recapture
k
k
r ∈ P p∈ P
r ∈ P p∈ P
r∈ P
t rp ≥ 0
t rp ≤ D p
f k , i ∈ {0 ,1}
∀p ∈ P
y k , o ,t ≥ 0
Kniker (1998)
3/9/2006
16.75
34
Itinerary-Based FAM (IFAM)
M in ∑
∑
k ∈ K i∈ L
c%k , i f k , i +
∑∑
p∈ P r ∈ P
( fare p − b pr fare r ) t rp 3
∑ f k ,i = 1
∀i ∈ L
f k ,i = 0
∀ k , o,t
f k ,i ≤ N k
∀k ∈ K
+ ∑ ∑ δ i p t rp − ∑ ∑ δ i p b rp t rp ≥ Q i
∀i ∈ L
S u b je c t to :
k∈K
y k ,o ,t − +
∑
i∈ I ( k , o , t )
f k ,i − y k ,o ,t + −
i∈ O ( k , o , t )
∑ y k ,o ,t n +
o∈O
1
∑ f k , i SEATS
k
k
r ∈ P p∈ P
∑
∑
i∈ CL ( k )
r ∈ P p∈ P
2
r
∑ tp ≤ Dp
∀p ∈ P
r∈ P
t rp ≥ 0
f k , i ∈ {0 ,1}
y k , o ,t ≥ 0
Kniker (1998)
3/9/2006
16.75
35
Column and Constraint
Generation
Original RMP
1
3
5
2
7
4
6
8
3/9/2006
16.75
36
Implementation Details
• Computer
• RMP constraint
matrix size
– Workstation class
computer
– 2 GB RAM
– CPLEX 6.5
– ~77,000 columns
– ~11,000 rows
• Final size
• Full size schedule
–
–
–
–
– ~86,000 columns
– ~19,800 rows
~2,000 legs
~76,000 itineraries
~21,000 markets
9 fleet types
• Solution time
– LP: > 1.5 hours
– IP: > 4 hours
88% Saving from Row Generation
> 95% Saving from Column Generation
3/9/2006
16.75
37
Fleet Assignment
Computational Sample:
Problem size
# of columns
# of rows
# of non-zero entries
Strength of formulation
Root node LP solution
Best IP solution
Difference
Solution time [sec]
Contribution [$/day]
3/9/2006
2,044 flight legs, 9 fleet types
FAM
IFAM
18,487
7,827
50,034
77,284
10,905
128,864
21,401,658
21,401,622
36
974
21,178,815
21,302,460
21,066,811
235,649
>100,000
21,066,811
16.75
38
IFAM Contributions
• Annual improvements over basic FAM
– Network Effects: ~$30 million
– Recapture: ~$70 million
• These estimates are upper bounds on
achievable improvements
3/9/2006
16.75
39
Subnetwork-Based FAM
• IFAM has limited opportunity for
expansion to include schedule design
decisions
– Fractionality of solution to LP relaxation is a
big issue
• Need alternative fleet assignment kernel
– Capture network effects
Modeling
– Maintains tractability Accuracy
IFAM
?
SFAM
FAM
Tractability
3/9/2006
16.75
40
Basic Concept
• Isolate network effects
– Spill occurs only on constrained legs
5
Potentially
Constrained
Flight Leg
3
Unconstrained
Flight Leg
Potentially
Binding
Itinerary
6
9
1
7
4
2
Non-Binding
Itinerary
8
“
“
3/9/2006
SFAM
IFAM
FAM
< 30% of total legs are potentially constrained
< 6% of total itineraries are potentially binding
16.75
41
Modeling Challenges
• Utilize composite variables (Armacost, 2000;
Barnhart, Farahat and Lohatepanont, 2001)
1
1
2
3
4
5
6
7
8
9
A
A
A
B
B
B
C
C
C
A
B
C
A
B
C
A
B
C
3 Fleet Types: A, B, and C
“ Challenges
“ Efficient
3/9/2006
column enumeration
16.75
42
SFAM Formulation
m
M S ηΠS
Min∑∑( CΠmS ) ( fΠmS )
n
m=1 n=1
n
m
M S ηΠS
∑∑(δ ) ( f )
Subject to:
m=1 n=1
yk ,o,t− +
∑
m
M S ηΠS
m
κ
(
∑∑ Π
i∈I ( k ,o,t ) m=1 n=1
S
)
k ,i
n
( fΠmS ) − yk,o,t+ −
n
m
M S ηΠS
m
κ
(
∑∑ Π
=1
∀i∈ L
= 0 ∀k, o,t
∑ ∑∑(γ ) ( f )
≤ Nk ∀k ∈ K
S
i∈O(k ,o,t ) m=1 n=1
o∈A
m
ΠS n
) (f )
∑
∑ yk,o,tn +
m i
ΠS n
m
M S ηΠS
i∈CL( k ) m=1 n=1
( f ) ∈{0,1}
m
ΠS n
k ,i
n
m k
ΠS n
m
ΠS n
m
ΠS n
yk,o,t ≥ 0
FAM solution algorithm applies
3/9/2006
16.75
43
SFAM Results
• Testing performed on full size schedules
– Runtimes similar to FAM, much faster than
itinerary-based approaches
• Tight LP relaxations
– SFAM achieve improved solutions relative to
FAM and itinerary-based approach
• SFAM has potential for further
integration or extension
– Time windows, stochastic demand, schedule
design
3/9/2006
16.75
44
Caveats
2. Deterministic
Demand
A
B
( 80, $200 )
4. Optimal
9AM
Control of Paxs
10AM
C
( 70, $250 )
( 50, $400 )
20
3. Demand
Forecast Errors
30 X 0.3 = 9 recaptured passengers
1. Recapture
Rate
( 20, $400
)
Errors
( Demand, Fare )
Recapture Rate
3/9/2006
16.75
45
Recapture Rate Sensitivity
Specified
Recapture Rate
IFAM
Fleeting Decision
PMix flows
passengers on
fleeted schedule
assuming full
knowledge of
passenger choices
Solve
SolvePMix
PMix
with
withvaried
varied
recapture
recapturerates
rates
Operating Cost
Estimated
Revenue
Fleeting Contribution
3/9/2006
16.75
46
Recapture Rate Sensitivity
Improvement over
Basic FAM ($/day)
Recapture Rate Sensitivity
8,000
7,000
6,000
5,000
4,000
3,000
2,000
1,000
0
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
Recapture Rate Multiplier ( δ)
Sensitivity of IFAM
Improvement gained from network effects alone
Improvement gained from network effects and recapture
3/9/2006
16.75
47
IFAM Sensitivity Analysis
• Simulations
“
FAM
or
IFAM
Average
Demand
Simulate 500
realizations of
demand based on
Poisson distributions
Fleeting Decision
Demand
Variations
Passenger
Passenger
Allocation
Allocation
Simulation
Simulation
Realizations
Operating Cost
Estimated
Revenue
Fleeting Contribution
3/9/2006
16.75
48
IFAM vs. FAM
Demand
Variations
Demand Stochasticity
Realizations
Demand deviation ~14%
FAM
Problem 1N-3A
Revenue
Operating Cost
Contribution
Problem 2N-3A
Revenue
Operating Cost
Contribution
3/9/2006
IFAM
Difference (IFAM-FAM)
$
$
$
4,858,089
2,020,959
2,837,130
$
$
$
4,918,691
2,021,300
2,897,391
$
$
$
60,602
341
60,261
$
$
$
3,526,622
2,255,254
1,271,368
$
$
$
3,513,996
2,234,172
1,279,823
$
$
$
(12,626)
(21,082)
8,455
$/day
16.75
49
Demand
Variations
IFAM vs. FAM
Demand Stochasticity
Realizations
Forecast Errors
Data Quality Issue
Demand deviation ~15%
Contributions
($ million/day)
Model Sensitivity to Demand Forecast Errors
$1.80
$1.70
$1.60
$1.50
$1.40
$1.30
$1.20
$1.10
$1.00
-5% -4% -3% -2% -1%
0
+1% +2% +3% +4% +5%
Simulated Demands (% of Forecasted Demand)
FAM
3/9/2006
16.75
IFAM
50
Extending Fleet
Assignment Models to
Include “Incremental”
Schedule Design…
3/9/2006
16.75
51
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
3/9/2006
16.75
52
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
¾Initial step in integrating flight schedule
design and fleet assignment decisions
3/9/2006
16.75
53
Example: Results
Aircraft Utilization
Do time windows allow a reduction
in the number of required aircraft?
TW = 0
P1
P2
3/9/2006
a/c req'd
365
428
TW = 20
cost
28,261,302
29,000,175
a/c req'd
363
426
16.75
cost
28,114,913
28,965,409
54
Results
• Time windows can provide significant cost
savings, as well as a potential for freeing
aircraft
– $50 million in operating costs alone for one
U.S. airline
3/9/2006
16.75
55
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
¾Incremental Schedule Design
3/9/2006
16.75
56
Demand and Supply100Interactions
Market Share
450
Market Share
410
Market Share
150
100
100
A
100
190
A
120
40
B
100
200
A
150
B
300
3/9/2006
B
16.75
Non-Linear
Interactions
57
Formulation
∑ c~k , i f k , i + ∑
Min ∑
k ∈ K i∈ L
∑ ( fare
p
p∈ P r∈ P
f k ,i = 1
∀ i ∈ LF
∑ f ≤1
Flight Selection
Schedule
Design
∀ i ∈ LO
∑
S u b je c t to :
k∈K
k ,i
k∈ K
y k ,o ,t − +
∑
f k ,i − y k ,o ,t + −
f k ,i = 0
∀ k ,o,t
f k ,i ≤ N k
∀k ∈ K
p r
p r r
∑ δ i t p − ∑ ∑ δ i b pt p ≥ Qi
∀i ∈ L
∑
FAM
Fleet Assignment
Spill +PMM
Recapture
i∈ I ( k , o , t )
i∈ O ( k , o , t )
∑ y k ,o ,t n +
o∈ O
∑ f k , i SEATS
k
− b rp fare r ) t rp
k
+ ∑
r∈ P p∈ P
∑
i ∈ CL ( k )
r∈ P p∈ P
r
∑ tp ≤ Dp
∀p ∈ P
r∈ P
t rp ≥ 0
f k , i ∈ {0 ,1}
y k ,o ,t ≥ 0
Lohatepanont, M. and Barnhart, Cynthia, “Airline Schedule Planning: Integrated Models
and Algorithms for Schedule Design and Fleet Assignment,” Transportation Science,
future.
3/9/2006
16.75
58
Schedule Design: Results
• Demand and supply interactions
– Tractability potentially a big issue
• Resulting schedules operate fewer flights
– Lower operating costs
– Fewer aircraft required
• Order of magnitude impact: ~$100 - $350
million improvement annually for variable
market demand
– Rough estimates: sensitive to quality of data, spill and
recapture assumptions, demand forecasts and
stochasticity
– Comparison to planners’ schedules
– Excludes benefits from saved aircraft
3/9/2006
16.75
59