Some of my earlier research work on "clean sheet" scheduling for airlines. AIrline timetables are generally created by hand, and then mathematical models are used to optimize and tweak them. This was the first published work in the industry for diong the whole process in one system. Combines a number of different airline operations research models into a single framework.
When the AA / Sabre Research Group was created, one of the goals was a clean-sheet scheduling system. Bob Crandall told my boss that there no way it could be done for even three aircraft - so this is what I built.
3. Introduction
• Suppose you start from nothing
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–
–
–
Clean sheet of paper
Information on your competitors
Do everything - Fleet, timetable, FAM, YM & pricing
Can you even handle 3 aircraft?
• This is a big, ugly search space
– Non-linear
– Stochastic
– Discontinuous
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4. Introduction
• History of the problem
– We didn’t have an optimization model to do this
– Built a simple GA-based approach in late 1996
– Code fragments, ideas and discussions were a catalyst
for integrated planning models using more traditional
optimization techniques
• Why revisit it now?
– Computing resources are much greater
– Could be useful to others
– Interesting results
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5. Model Structure
Inputs
Plan
New Individuals
Population
of Marketing
Plans (Price
and Schedule)
GA
- reproduction
- mutation
Fittest
Individuals
(higher profit)
Profitability
Outputs
5
Pax Preference
Demand Model
Yield Mgmt.
Optimization
Spill
Model
6. Model - Inputs
• Passenger preferences
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–
–
–
–
–
• Schedule parameters
– Fleet type and number
– Aircraft capacity and
operating costs
– Block times / model
– Minimum turn times
Market segment
Base demand
Base price
Elasticity
Time of day preference
Service preference
• OA schedule and price
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7. Model - Genetic Algorithm
• General
– Each potential solution is represented as a bit-string
– The bit-string contains all information associated with
scheduling and pricing decisions
– Simulates evolution with survival-of-the-fittest
• Has been applied to diverse range of difficult
combinatorial problems in other industries
– Parametric design of aircraft and aircraft engines
– Job-shop scheduling
– Strategy discovery for multi-player games
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8. Model - Genetic Algorithm
• Initial population
– A set of potential solutions is generated randomly
• Reproduction
– Parents chosen randomly, weighted by profit
– New solutions are generated by combining elements
from 2 parents, using random crossover operations
– Mutation occurs randomly to bits within the offspring
– New individuals replace less profitable solutions
• Stopping criteria
– Based on a specified number of generations
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9. Model - Genetic Algorithm
Population
(about 100)
Select profitable plans
as parents
Crossover
Point
New
Individuals
Each bit string in the population
can be decoded as a complete
marketing plan (schedule and price)
Replace less
profitable plans
Mutation
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10. Model - GA Encoding
DECL;&C;CITY;NOM;DFW;ABQ;MCO;MSY;SAT;TUL
DECL;&W;WAIT;ORD;0;5;10;15;20;25;30;45;60;75;90;120;150;180;210
# Skeleton schedule to be substituted
BTRT;AA;F10;&C;&W;&C;&W;&C;&W;&C;&W;&C;&W;&C;&W
BTRT;AA;S80;&C;&W;&C;&W;&C;&W;&C;&W;&C;&W;&C;&W
# OA Flt Leg;Carrier;Orig;Dest;DepTime;ArrTime
FLEG;DL;ABQ;DFW;1030;1206;100;25
FLEG;DL;ABQ;DFW;1630;1805;100;25
# Market definitions
MSEG;ABQ;DFW;Y;105.184;102.163;0;LOGIT
MSEG;ABQ;DFW;M;105.184;81.73;0;LOGIT
MSEG;ABQ;DFW;Q;105.184;61.2975;0;LOGIT
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11. Model - GA Encoding
• What do planes do?
– Fly, wait, fly, wait, fly, wait, etc…
– Wait time is minimum ground time + additional
– Network is a collection of aircraft cycles
• GA replaces the parameter of the route string with
a value from the named define set
– DECL;&C;CITY;NOM;DFW;ABQ;MCO;MSY;SAT;TUL
DECL;&W;WAIT;ORD;0;5;10;15;20;25;…
BTRT;AA;F10;&C;&W;&C;&W;&C;&W;&C;&W;&C;&W;&C;&W
• One sample substitution
– BTRT;AA;F10;DFW;5;ABQ;15;MCO;30…
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12. Model - GA Encoding
• Heuristic to ensure that only feasible cycles are
created and used in the schedule
• Can create restricted subsets, i.e. don’t send
narrow body aircraft to Europe
– DECL;&C_S80;CITY;NOM;DFW;SAT;MIA;MCO...
DECL;&C_767;CITY;NOM;DFW;ORY;LHR;FRA...
BTRT;AA;767;&C_767;&W;&C_767;&W;&C_767;&W...
• Encode price changes
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13. Model - Passenger Preference
• Both logit & QSI models are supported
– Selectable at the market-segment level
• Models take into account
– Time Of Day (TOD) utility, based on a passenger’s
preferred departure / arrival times and the difference
between these times and the actual schedule
– Price utility, taking into account a passenger’s
preference for a given fare amount and restrictions
– Service utility for aircraft type and direct vs. connecting
service
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14. Passenger Preference (cont.)
E Padj P0
Dadj D0 1
P0
Dadj
Adjusted market demand
D0
Base market demand
Padj
Adjusted price
P0
Base price
E
Elasticity
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15. Model - Yield Management
• Uses an O&D YM optimizer
– Don’t need a GA for this
max
s.t.
ˆ
as
ˆ
RTotalYM Rs ys f ( ys )dys as f ( ys )dy s
sSvc
0
ˆ
as
ˆ
as CAPj j
sS j
ˆ
as 0
s S
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16. Model - Spill
• Traffic is found by spilling demand based on
allocations, using standard spill model:
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–
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Trafficj = P(Dj<Xj)E(Dj|Dj<Xj) + P(Dj>Xj)Xj
Where:
Dj = demand for market/class j
Xj = yield management allocation for market/class j,
based on the sum of bid prices for itinerary j
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17. Model - Outputs
• Marketing Plan
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–
• Statistics
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–
–
–
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Prices
Timetable
Capacity allocation
Bid prices
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Profit
Revenue
Operating costs
Spill
Demand (OD and flight)
18. Results - Sample Problem
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•
•
•
•
Six cities
One hub
3 aircraft
30 O&Ds
One competitor
TUL
ABQ
DFW
SAT
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MCO
MSY
19. Results - Sample Output
ABQ
DFW
• Price variations
SAT
– + $20 on DFW-MCO
– + $10 on DFW-MSY
– Assumes competitors match
changes
TUL
MSY
MCO
SAT
MSY
SAT
MCO
• Statistics
– Profit is $47,327
– Without varying the price,
the model only makes
$44,302 (but with a
different schedule)
SAT
TUL
ABQ
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20. Results - Solution Quality
Algorithm Progress Over Time
50000
45000
40000
35000
Max
25000
Avg
20000
15000
10000
5000
100
89
78
67
56
45
34
23
12
0
1
Profit
30000
Generation
20
• Less profitable plans are
progressively eliminated
from the population.
Hence, the average profit
approaches the maximum
• Approximately 100
generations are required to
reach a plateau
21. Results - Solution Quality
• Running the model several times generates
different plans having little variance in
profitability
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Run 1: $47,985
Run 2: $48,995
Run 3: $47,837
Run 4: $48,291
Run 5: $48,290
• The ability to vary price affects the schedule that
is generated
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22. Results - Solution Quality
• Statistical model to determine solution quality.
– Can’t find globally optimal solution
– Confidence interval to estimate where the global
optimum is likely to be. Ref: Smith & Sucur, 1996
– We’re 95% confident we’re within 3% of optimal
Frequency
Model
results
Profitability
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Global Optimal
Confidence
interval
23. Results - Sensitivity Analysis
• Vary each of the input parameters
• Percent change in output for a 1% change in input
• Can be used to determine
– What happens to a schedule’s structure as the inputs are
varied
– The change in outputs, particularly profit, as a function
of inputs
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26. Results - Schedule Structure
• Varying the demand CV has a visible effect on schedule structure and
profitability. Specifically, at a higher CV, the schedule generation has
more emphasis on connections, fewer flights and is less profitable
ABQ
MSY
TUL
DFW
cv=0.4
lf=76%
profit=$25034
DFW
MCO
SAT
0830
SAT
ABQ
ABQ
SAT
SAT
MCO
ABQ
MCO
ABQ
cv=0.1
lf=93%
profit=$48303
SAT
ABQ
ABQ
TUL
1500
TUL
MSY
TUL
MCO
SAT
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SAT
SAT
MCO
27. Results - Scalability
• 98% of CPU used in the objective function
• 12 aircraft, with prices, took about 2 days
– Sun Super SPARC based system
– Some code not optimized (i.e. connection generator)
• What could we do today?
– CPU is 10x faster
– Can easily parallelize on MPP or networked machines
– Should be able to handle 50-100 aircraft
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28. Conclusion - IPM Family Tree
Simultaneous
YM & Pricing
While IPM never
made production,
fragments of code
and ideas generated
an explosion of
follow-on models
O&D FAM
FLITEWISE
Price Balance
Statistic
IPM
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Frequency
Model
29. Conclusion
• GA may not be practical for a large airline
– Runtime
– Some constraints not considered
• Potentially useful for
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High-level analysis
Long-term planning
New hub or alliance planning
Research
• Catalyst for integrated planning
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30. References
• Goldberg, D.E. [1989]. Genetic Algorithms in Search, Optimization,
and Machine Learning. Addison-Wesley.
• Smith, B.C., Sucur, M. [1996], Analysis of Solution Quality in
Schedule Planning, AGIFORS Symposium Atlanta GA
• Jacobs T.L., Ratliff R.M. & Smith B.C., [2000] Soaring with
Synchronized Systems, ORMS Today
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