New Approaches to Improving the Robustness of Airline Schedules

1. New Approaches to Improving the Robustness of Airline Schedules Prof. John-Paul ClarkeDepartment of Aeronautics & AstronauticsMassachusetts Institute of Technology

2. Outline • Background & Motivation • Robust Maintenance Routing • Flight Schedule Re-Timing • Degradable Airline Schedule • Conclusions

3. Schedule Design Fleet Assignment Maintenance Routing Crew Scheduling Airline Schedule Planning Process • Most existing airline schedule planning methods assume that aircraft, crews, and passengers will operate as planned

4. Airline Operations • Bad weather reduces airport capacity • Airlines cancel or delay flights to reduce demand • Delays propagate through the network • Airlines must reschedule aircraft/crew and re-accommodate passengers • Passengers are not satisfied • They are delayed • They have no control over their delay • All passengers on a given aircraft are delayed equally regardless of fare class

5. Delays & Cancellations • Trend (1995-1999) • Significant increase (100%) in flights delayed more than 45 min • Significant increase (500%) in the number of cancelled flights • Year 2000 • 30% of flights delayed • 3.5% of flights (approx. 140,000) cancelled • Future: • Delays and cancellations may increase dramatically  more frequent and serious schedule disruptions and revenue loss

6. Passenger Disruptions • Flight delays and cancellations often cause passenger schedule disruptions • 26 million passengers (4% of passengers) disrupted • 65% of disruptions caused by missed connections • Very long delays for disrupted passengers • Average delay for disrupted passengers is approx. 4 hours (versus 15 min delay for non-disrupted passengers) • Significant revenue loss - approx. $4 Billion /year

7. Robustness • Need schedules that are robust (insensitive) to delays and cancellations • Definitions of robustness • Minimize cost (expected/worst case deviations from optimal) • Minimize aircraft/passenger delays and disruptions • Easy to recover (aircraft, crew, passenger) • Isolate disruptions and reduce the downstream impact • Two ways to provide robustness • Re-optimize schedule after disruptions occur (operation stage) • Build robustness into the schedules (planning stage)

8. Outline • Background & Motivation • Robust Maintenance Routing • Graduate Student: Shan Lan • Joint work with Prof. Cindy Barnhart • Flight Schedule Re-Timing • Degradable Airline Schedule • Conclusions

9. Robust Maintenance Routing • Objective • Reduce the propagation of delays by combining flight segments in optimal (from the point of view of follow-on delays) maintenance routings • Total delay for a route is uniquely determined by routing • Solution Approach • Derive distributions from historical data for delay introduced into a route by an airport • Formulate and solve maintenance routing model that minimizes the propagation of delays subject to maintenance feasibility

10. f1’ f2’ Delay Propagation • Arrival delay may cause departure delay for the next flight that is using the same aircraft if there is not enough slack between these two flights • Delay propagation may cause schedule, passenger and crew disruptions for downstream flights (especially at hubs) f1 MTT f2

11. Propagated v. Independent Delay • Flight delay may be divided into two categories: • Propagated delay • Caused by inbound aircraft delay – function of routing • 20-30% of total delay (Continental Airlines) • Independent delay • Caused by other factors – not a function of routing • Appropriately allocated slack can reduce propagated delay • Add slack where advantageous • Reduce slack where less needed

12. Definitions TDD i’’ i i’ PD IDD PDT ADT Slack Min Turn Time j’ Planned Turn Time j PAT AAT PD IAD TAD

13. f1’ f3’ f1’ MTT f1 f2 MTT f3 f4 New routing Illustration of the Idea MTT f1 f2 MTT f3 f4 Original routing

14. Modeling Issues • Difficult to use leg-based models to track the delay propagation • One variable (string) for each aircraft route between two maintenance events (Barnhart, et al. 1998) • A string: a sequence of connected flights that begins and ends at maintenance stations • Delay propagation for each route can be determined • Need to determine delays for each feasible route • Most of the feasible routes haven’t been realized yet • PD and TAD are a function of routing • PD and TAD for these routes can’t be found in the historical data • IAD is not a function of routing and can be calculated by tracking the route of each individual aircraft in the historical data

15. String Based Formulation

16. Solution Approach • Random variables (PD) can be replaced by their mean • Distribution of Total Arrival Delay • Possible distributions analyzed: Normal, Exponential, Gamma, Weibull, Lognormal, etc. lognormal distribution is the best fit • A closed form of expected value function • Mixed-integer program with a huge number of 0-1 variables • Branch-and-price • Branch-and-Bound with a linear programming relaxation solved at each node of the branch-and-bound tree using column generation • IP solution • A special branching strategy: branching on follow-ons (Ryan and Foster 1981, Barnhart et al. 1998)

17. July 2000 data Model Routes Aug 2000 data Computational Results • Test Networks • Model Building and Validation • Propagated delays (August 2000)

18. Results - Delays • Total delays and on-time performance • Passenger misconnects

19. Outline • Background & Motivation • Robust Maintenance Routing • Flight Schedule Re-Timing • Graduate Student: Shan Lan • Joint work with Prof. Cindy Barnhart • Degradable Airline Schedule • Conclusions

20. Flight Schedule Re-Timing • Objective • Reduce the number of passenger misconnections by adjusting departure times so that passenger connection times are correlated with the likelihood of a missed connection (disruption) • Add connection slack where it is need most • Solution Approach • Derive distributions from historical data for number of passengers disrupted for each connection • Formulate and solve re-timing model that minimizes the number of disrupted passengers

21. ACT PAT AAT PDT ADT MCT Slack PCT Definitions • AAT = Actual Arrival Time • ACT = Actual Connection Time • ADT = Actual Departure Time • MCT = Minimum Connection Time • PAT = Planned Arrival Time • PCT = Planned Connection Time • PDT = Planned Departure Time

22. P (misconnect)= 0.3, E(disrupted pax) = 30 P(misconnect)=0.1, E(disrupted pax) =10 Illustration of the Idea Suppose 100 passengers in flight f2 will connect to f3 Airport A Airport B Airport C Airport D  Expected disrupted passengers reduced: 20

23. Implementation Options Schedule Design • Passenger disruption depends on flight delays, a function of fleeting and routing • Before maintenance routing problem • Delay propagation not considered • New fleeting and routing solution may cause delay to propagate in a different way  change the number of disrupted passengers • After maintenance routing problem • Delay propagation considered • Need to enable the current fleeting and routing solution Fleet Assignment Maintenance Routing Crew Scheduling

24. Connection-Based Formulation • Objective • minimize the expected total number of passenger misconnects • Constraints: • For each flight, exactly one copy will be selected. • For each connection, exactly one copy will be selected and this selected copy must connect the selected flight-leg copies. • The current fleeting and routing solution cannot be altered.

25. Connection-Based Formulations • Theorem 1: • The second set of constraints are redundant and can be relaxed • Theorem 2: • The integrality of the connection variables can be relaxed • Formulation I: CFSR

26. Alternative Formulations • Formulation II: ACFSR • Formulation III: DCFSR

27. More Model Properties • Theorem 3: • ACFSR model is equivalent to CFSR model • Theorem 4: • DCFSR model is equivalent to CFSR model • Theorem 5: • The LP relaxation of CFSR model is at least as strong as that of ACFSR, and can be strictly stronger. • Theorem 6: • The LP relaxation of CFSR model is at least as strong as that of DCFSR, and can be strictly stronger.

28. Solution Approach • Random variables can be replaced by their mean • Distribution of • Branch-and-Price

29. July 2000 data RAMR Routes Aug 2000 data Schedule FSR ACFSR CFSR Computational Results • Network • We use the same four networks, but add all flights together and form one network with total 278 flights. • Model Building and Validation • Strength of the formulations

30. Computational Results • Assume 30 minute minimum connecting time • Assume 25 minute minimum connecting time • Assume 20 minute minimum connecting time

31. Computational Results • Number of copies • Estimated reduction in total passenger delays: (30 minutes MCT) • 20% (30 minute time window), 16% (20 minute time window), 10% (10 minute time window)

32. Outline • Background & Motivation • Robust Maintenance Routing • Flight Schedule Re-Timing • Degradable Airline Schedule • Graduate Student: Laura Kang • Conclusions

33. Degradable Airline Schedule • Objective • Develop airline schedule that is robust, i.e. delays are isolated • Provide priority (and thus reliability) for each flight • Improve customer satisfaction by giving passengers an accurate expectation of the level of service • Provide basis for revenue management and ATC auctions • Solution Approach • Partition schedule into smaller independent prioritized schedules (layers) subject to operational feasibility

34. D-SPM (Flight-based Formulation) DAS DAS D-FAM DAS D-ARM (Route-based Formulation) Implementation Options schedule design fleet assignment aircraft routing crew scheduling

35. IP Model • Prioritize layers based on revenue (e.g. group highest revenue flights together in most reliable layer) • Revenue is “protected” if all flight legs in an itinerary are in a “protected” layer • IP model maximizes the total protected revenue subject to feasibility constraints • Prototype 2 layers implementation • Layer 1: 60% (protected layer) • Layer 2: 40%

36. Model Statistics • 1,134 flight legs • 274 aircraft • 1,744 itineraries (8% of total) • Single flight leg: 1,130 • 2 flight legs: 613 • 3 flight legs: 1 • 53,091 passengers (80% of total) • $10,839,340 revenue (84% of total)

37. Notation • Indices • r route • f itinerary • ij flight • k layer (k=1 … K) • γijf 1 if flight ij is in itinerary f, 0 otherwise • Decision variables • yrk 1 if route r is in layer k, 0 otherwise • zfk 1 if itinerary f is in layer k, 0 otherwise • xijk 1 if flight ij is in layer k, 0 otherwise • Parameters • vfk revenue for itinerary f is placed in layer k • Chcapacity at hub h in bad weather • Sk fraction of layer k • ar number of flights in route r • arh number of flights departing at hub h in route r • ACN number of aircraft

38. Flight-based Formulation

39. Route-based Formulation

40. Greedy Flight-Leg Pairing STEP 0: Fix connections for non-hub to non-hub flights STEP 1: Pair flight segments at spoke airports using the revenue paring with aircraft utilization heuristic STEP 2: Combine paired flight segments from step 1 at hub airports using the revenue paring with aircraft utilization heuristic STEP 3: Partition very long routes into several shorter routes

41. Greedy Flight-Leg Pairing 10 100 100 10

42. i1 i2 i3 i4 i5 i6 j1 j2 j3 j4 j5 j6 i1 i2 i3 i4 i5 i6 i1 i2 i3 i4 i5 i6 j3 j4 j5 i1 i2 i5 i6 j1 j2 j3 j4 j5 j6 j1 j2 j3 j4 j5 j6 j1 j2 i3 i4 j6 Swapping Search • Check swapping feasibility • Check constraints satisfaction • Check objective function improvement • Assume revenue is protected proportionally to the number of flight legs in the protected layer • Swap route i route j

43. Tabu Search STEP 0: start with initial solution x* from revenue paring heuristics WHILE( number of iteration is less than N ) STEP 1: Swapping Search. If f(x) > f(x*), x*  x STEP 2: Update Tabu list • If a pair was in a tabu list for Y iterations, remove it from the tabu list • Set X pairs which were swapped in the search in the tabu list • Tabu search is sensitive to its parameters X, Y, N • State-of-art decision for X, Y, N

44. IP Objective Function Value Upper bound for route-based DAS D-SPM 8,667,632 D-ARM w/Heuristics 8,123,060 Lower bound for route-based DAS Current routing 6,492,895

45. Protected Revenue D-SPM 9,624,460 74.5% D-ARM w/Heuristics 9,057,750 70.1% Current routing 7,302,040 56.6%

46. Protected Passengers D-SPM 44,984 67.5% D-ARM w/Heuristics 43,051 64.6% Current routing 37,587 56.4%

47. Simulation Results - Good Weather Layer 1 Layer 2 Current Routing Average delay 6 min 6 min 6 min Pr(delay >0) 0.37 0.48 0.42 Pr(delay > 15) 0.14 0.17 0.16

48. Simulation Results - Bad Weather Layer 1 Layer 2 Current Routing Average delay 13 min 25 min 17 min Pr(delay >0) 0.52 0.69 0.61 Pr(delay > 15) 0.30 0.48 0.37

49. Outline • Background & Motivation • Robust Maintenance Routing • Flight Schedule Re-Timing • Degradable Airline Schedule • Conclusions

50. Conclusions • Robust Maintenance Routings provide: • Airline schedule with reduced delay propagation • Flight Schedule Re-Timing provides: • Airline schedule with fewer passenger disruptions or missed connections • Degradable Airline Schedules provide: • Airline schedule that isolates delays • Tool for managing passengers’ expectation • Potential revenue enhancement