Robust Airline Crew Pairing Optimization
Diego Klabjan
Sergei Chebalov
University of Illinois at Urbana-Champaign
NEXTOR-FAA-INFORMS
Robust Crew Pairing
Optimization
Acknowledgment
Introduction
Move-up crews
• NSF funded the project
• Collaboration with
– Sabre Inc., Southlake, TX
Models
Experiments
Conclusion
2
Robust Crew Pairing
Optimization
Flight Delays
Introduction
2,300,000
2,200,000
Move-up crews
2,100,000
Models
2,000,000
1,900,000
Experiments
1,800,000
Conclusion
1,700,000
1995
1996
1997
1998
1999
• Aviation Week & Space Technology, September
2000
3
Robust Crew Pairing
Optimization
Summer 2000 Collapse
Introduction
Move-up crews
• 11% increase
• Flight delays
– 1.7% delayed flights in 1995
– 2.3% delayed flights in 1999
Models
Experiments
Conclusion
• Summer 2000 the worst ever
• In Summer 2000 Northwest the best on
time performance
– 75% on time arrival rate
4
Robust Crew Pairing
Optimization
Sources of Delays
Introduction
Move-up crews
Models
Experiments
Conclusion
5
• Weather, congestion
• Unscheduled
maintenance
• Secondary delays
– crew not available
– plane not available
– passengers not available
Robust Crew Pairing
Optimization
Improve Performance
Introduction
Move-up crews
• FAA
– ATM
– CDM
• Airlines
Models
– Recovery procedures
Experiments
Conclusion
• Integrated recovery
• Aircraft recovery
• Crew recovery
– Robust solutions
• Robust aircraft routing
• Robust crew scheduling
6
Robust Crew Pairing
Optimization
Crew Pairing
Introduction
Move-up crews
Models
Experiments
• Input: A schedule of a fleet
• Objective: Find a set of crew itineraries (pairings)
that partition all of the legs such that the airline
incurs the least cost.
ATL
Conclusion
MIA
JAX
8:00
17:00
Monday
7
10:00
13:00
Tuesday
Robust Crew Pairing
Optimization
Crew Pairing Model
Introduction
Move-up crews
Models
Experiments
Conclusion
8
• Minimize crew cost
• To every flight assign a unique pairing
• Side constraints
– Manpower constraints
– Other constraints
Robust Crew Pairing
Optimization
Robust Crew Pairing
Introduction
Move-up crews
• A. Schaefer et. al. (2000)
– Replace pairing cost with expected pairing cost
• Pairings with long connection times
Models
Experiments
Conclusion
9
– Stochastic approach by J. Yen and J. Birge
(2000)
– Deterministic variant by M. Ehrgott and D.
Ryan (2001)
Robust Crew Pairing
Optimization
Why Robustness?
Introduction
Move-up crews
Models
Experiments
Conclusion
• `Excess cost/flying time’ for large fleets below 1%
• Solutions use many tight, short connections.
– Such connections are very vulnerable to disruptions.
• 1% relative excess cost in planning for large
fleets translates into 4% to 8% in operations.
• Solutions
– Better recovery procedures
– Robust solutions
10
Robust Crew Pairing
Optimization
Move-up Crews
Introduction
Move-up crews
Models
Experiments
Conclusion
11
• Crews that are ready to cover a different flight.
– At least
• Ready to fly
• Same crew base
• Same number of days till the end of the pairing
• Potentially in operations it yields crew swapping.
• Choice of flight cancellation
Robust Crew Pairing
Optimization
Move-up Crews
Introduction
disrupted crew schedule
move-up crew
Move-up crews
i’
Models
`min sit’ or `min rest’
Experiments
ready
time
Conclusion
j
i
disrupted flight
12
j’
Robust Crew Pairing
Optimization
Objectives
Introduction
Move-up crews
Models
Experiments
Conclusion
13
•
•
•
•
Low crew cost
Maximize the number of move-up crews
Trade-off
Maximize the number of move-up crews
subject to
crew cost ≤ (1+r)Q
Robust Crew Pairing
Optimization
Model
Introduction
Move-up crews
Models
Experiments
Conclusion
• Given a flight, a crew base, and a day count
– set of pairings (P )
• covering this flight
• originating at a given crew base
• a given number of days till the end,
– set of pairings (R ) that yield a move crew to this flight
• Variables
– Pairing variables
– Number of move up crews (z)
14
Robust Crew Pairing
Optimization
Objective Function
Introduction
Move-up crews
Models
Experiments
Conclusion
 cost of move-up crews · z(leg i, crew base cb, day d)
all legs
• More move-up crews for strategically
important flights
• Flights toward the end of the pairing more
move-up crews
• Maximize the number of move-up crews
– Cost one for all flights
15
Robust Crew Pairing
Optimization
Constraints
Introduction
Move-up crews
Models
Experiments
Conclusion
16
• Standard partitioning constraints
sum of pairings covering the leg = 1
• Count move-up crews for every leg
departing from a hub
sum of all pairings that yield a move-up crew
≥ move-up crew count variable z
Robust Crew Pairing
Optimization
Constraints
Introduction
Move-up crews
Models
Experiments
Conclusion
• Undesirable
–N
move-up crews for one flight
– Zero move-up crews for many flights
• Evenly distribute move-up crews
M · sum of all pairings covering the leg ≥
move-up crew count variable z
17
Robust Crew Pairing
Optimization
Small M
Introduction
Move-up crews
Models
Experiments
Conclusion
18
• Objective value of 20
– 20 different legs with a single move-up crew
– 1 leg with 20 move-up crews
• Limit the maximum number of counted
move-up crews per leg
– M is this limit.
Robust Crew Pairing
Optimization
Mathematical Model
Introduction
max  z
Move-up crews
y 1
every flight
z  My
Models
z y
Q
Experiments
Conclusion
19
every flight, crew base,day
(P)
every flight, crew base, day
(Q)
P
y binary
Robust Crew Pairing
Optimization
Enhancements
Introduction
Move-up crews
Models
Experiments
• Both schedules
produce an objective
value of 2
• The bottom one
preferable
Conclusion
Pick me!
20
Robust Crew Pairing
Optimization
Enhancement
Introduction
Move-up crews
Models
Experiments
• Additional variables
– v = the number of available move-up crews
• Objective function
 z  f v
Conclusion
• Additional constraints
21
Robust Crew Pairing
Optimization
Enhancements
Introduction
Move-up crews
Models
Experiments
Conclusion
• Both schedules
produce the same
objective value
• The top one covers
one disruption
• The bottom one
covers two disruptions
Pick me!
22
Robust Crew Pairing
Optimization
Enhancement
Introduction
Move-up crews
Models
• It can be done
• Objective function
 z  f v  gu
Experiments
• Additional constraints
Conclusion
23
Robust Crew Pairing
Optimization
Methodology
Introduction
Move-up crews
Models
Experiments
Conclusion
24
• Select a small subset of pairings
– First solve traditional crew pairing problem.
– Pick columns with low reduced cost at the root
node.
• Maximize the number of move-up crews
– Over selected pairings
– Keep cost in control
Robust Crew Pairing
Optimization
Lagrangian Relaxation
Introduction
Move-up crews
Models
Experiments
Conclusion
25
• Relaxations
– Relax (P)
– Relax (Q)
– Relax (P) and (Q)
• Result: It does not matter!
• They all yield the same relaxation.
Robust Crew Pairing
Optimization
Computational Experiments
Introduction
Move-up crews
Models
Experiments
Conclusion
26
•
•
•
•
United Airlines A320
Daily problem with 123 legs
3 move-up crews by just minimizing cost
Do not know how to find an optimal solution
Robust Crew Pairing
Optimization
Computational Experiments
Introduction
Move-up crews
Models
Experiments
Conclusion
27
M
1 2 3 5 10 100
no. move-up crews 11 18 21 28 32 32
• 10,000 pairings
• Increasing M increases the number of move-up
crews.
Robust Crew Pairing
Optimization
Computational Experiments
Introduction
Move-up crews
r
no. move-up crews
0 0.08 0.12 0.36 0.76
4
5
8
17
18
Models
Experiments
Conclusion
 cp y p  (1  r)Q
p
28
Robust Crew Pairing
Optimization
Crew Schedule Evaluation
Introduction
Move-up crews
• What do the previous
tables convey?
Models
Experiments
Conclusion
• Are these crew schedules really robust?
– It is a sound concept!
– Agree?
• Evaluate them
29
Robust Crew Pairing
Optimization
Simulation?
Introduction
Move-up crews
Models
Experiments
Conclusion
30
• On our wish list
• Simulation available but unable to integrate
it with the crew recovery module
• Instead
Robust Crew Pairing
Optimization
Crew Schedule Evaluation
Introduction
Move-up crews
Models
Experiments
Conclusion
31
• Generate disruptions
– Reduced capacity at a hub
– Random block time distributions
• For each disruption run crew recovery
• Crew recovery module
– Solves an optimization model
– Change pairings only within a 24 hour time
window
– No crew swapping in advance
Robust Crew Pairing
Optimization
Disruption Scenario Generation
Introduction
Move-up crews
Models
Experiments
Conclusion
• Random block times
– Distributions obtained from United
• Disruptions at hubs
– Reduced capacity
– Shut down a hub for one hour
• Numbers are averages over
– Disruptions at each hub
– 10 scenarios for each hub
32
Robust Crew Pairing
Optimization
What are we Comparing?
Introduction
Move-up crews
Models
Experiments
Conclusion
33
• Robust crew schedules (cost+robustness)
vs. traditional crew schedules (cost)
• + robust wins
• - traditional wins
Robust Crew Pairing
Optimization
Robust vs. Traditional
Introduction
Move-up crews
Models
M=1
Cost ($)
28,996
r =1.5%
7,515
r =3.4%
33,711
r =∞
traditional
No. No. Res. No. Uncov.
Dhds Crews
Legs
26
15
3
35
18
-8
31
19
-7
FTC
6.80%
8.90%
11.04%
5.30%
No. Move-up
Crews
13
16
16
3
FTC
6.80%
8.90%
11.23%
5.30%
No. Move-up
Crews
16
21
18
3
Experiments
Conclusion
34
M=2
Cost ($)
43,042
r =1.5%
6,030
r =3.4%
-28,975
r =∞
traditional
No. No. Res. No. Uncov.
Dhds Crews
Legs
31
21
1
28
18
-3
22
13
5
Robust Crew Pairing
Optimization
And The Winner Is:
Introduction
Move-up crews
Models
Experiments
r =1.5%,M=1
r =1.5%,M=2
No. No. Res.
Cost ($) Dhds Crews
28,996
26
15
43,042
31
21
Conclusion
I leave it up to you!
35
No. Uncov.
Legs
3
1
Robust Crew Pairing
Optimization
Yearly Estimate
Introduction
Move-up crews
Models
Experiments
Conclusion
36
• Savings around $1.5 million
– Includes larger cost on ‘regular’ days
• Not counting savings of deadheading,
reserved crews, and cancellations
• Savings per fleet
Robust Crew Pairing
Optimization
What is Next?
Introduction
Move-up crews
Models
Experiments
Conclusion
37
• An airline to use this approach
• Science fiction
– Perform within an alliance
– Increase passenger demand
Robust Crew Pairing
Optimization
Final Remark
Introduction
Diego, your approach
won’t work as airlines put
in place a DSS for
crew recovery.
Move-up crews
Models
Experiments
Conclusion
• DSS used only for major disruptions
– 10 a year
• Minor disruptions recovered manually
38
Robust Crew Pairing
Optimization
Introduction
Move-up crews
Models
Experiments
Conclusion
39