Robust Airline Crew Pairing Optimization

1. Robust Airline Crew Pairing Optimization Diego Klabjan Sergei Chebalov University of Illinois at Urbana-Champaign

2. Acknowledgment • NSF funded the project • Collaboration with • Sabre Inc., Southlake, TX

3. Flight Delays • Aviation Week & Space Technology, September 2000

4. Summer 2000 Collapse • 11% increase • Flight delays • 1.7% delayed flights in 1995 • 2.3% delayed flights in 1999 • Summer 2000 the worst ever • In Summer 2000 Northwest the best on time performance • 75% on time arrival rate

5. Sources of Delays • Weather, congestion • Unscheduled maintenance • Secondary delays • crew not available • plane not available • passengers not available

6. Improve Performance • FAA • ATM • CDM • Airlines • Recovery procedures • Integrated recovery • Aircraft recovery • Crew recovery • Robust solutions • Robust aircraft routing • Robust crew scheduling

7. ATL MIA JAX 8:00 17:00 10:00 13:00 Monday Tuesday Crew Pairing • 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.

8. Crew Pairing Model • Minimize crew cost • To every flight assign a unique pairing • Side constraints • Manpower constraints • Other constraints

9. Robust Crew Pairing • A. Schaefer et. al. (2000) • Replace pairing cost with expected pairing cost • Pairings with long connection times • Stochastic approach by J. Yen and J. Birge (2000) • Deterministic variant by M. Ehrgott and D. Ryan (2001)

10. Why Robustness? • `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

11. Move-up Crews • 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

12. disrupted crew schedule move-up crew j’ i’ `min sit’or `min rest’ ready time j i disrupted flight Move-up Crews

13. Objectives • 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

14. Model • 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)

15. Objective Function • 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 cost of move-up crews · z(leg i, crew base cb, day d)

16. Constraints • 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

17. Constraints • 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

18. Small M • 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.

19. Mathematical Model

20. Enhancements • Both schedules produce an objective value of 2 • The bottom one preferable Pick me!

21. Enhancement • Additional variables • v = the number of available move-up crews • Objective function • Additional constraints

22. Enhancements • Both schedules produce the same objective value • The top one covers one disruption • The bottom one covers two disruptions Pick me!

23. Enhancement • It can be done • Objective function • Additional constraints

24. Methodology • 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

25. Lagrangian Relaxation • Relaxations • Relax (P) • Relax (Q) • Relax (P) and (Q) • Result: It does not matter! • They all yield the same relaxation.

26. Computational Experiments • 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

27. Computational Experiments • 10,000 pairings • Increasing M increases the number of move-up crews.

28. Computational Experiments

29. Crew Schedule Evaluation • What do the previous tables convey? Not much! • Are these crew schedules really robust? • It is a sound concept! • Agree? • Evaluate them

30. Simulation? • On our wish list • Simulation available but unable to integrate it with the crew recovery module • Instead

31. Crew Schedule Evaluation • 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

32. Disruption Scenario Generation • 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

33. What are we Comparing? • Robust crew schedules (cost+robustness) vs. traditional crew schedules (cost) • + robust wins • - traditional wins

34. Robust vs. Traditional

35. And The Winner Is: I leave it up to you!

36. Yearly Estimate • Savings around $1.5 million • Includes larger cost on ‘regular’ days • Not counting savings of deadheading, reserved crews, and cancellations • Savings per fleet

37. What is Next? • An airline to use this approach • Science fiction • Perform within an alliance • Increase passenger demand

38. Final Remark Diego, your approach won’t work as airlines put in place a DSS for crew recovery. • DSS used only for major disruptions • 10 a year • Minor disruptions recovered manually

39. Thank you for your attention!