Air Transportation Service Design

1. Air Transportation Service Design Pamela H. Vance Goizueta Business School Emory University

2. Outline • Current State of Practice in domestic (U.S.) passenger airlines • Schedule Development • Fleet Assignment • Routing • Crew Scheduling • Current active areas of research • Overview of research on service design issues • Focus on recent crew scheduling results

3. The Airline Planning Process • Flight Schedule Development • Given: • historical data on passenger OD demand • air traffic and airport restrictions • aggregate aircraft availability • Find: • departure/arrival times for each segment to maximize potential revenue • State of Practice • schedules are usually generated by marketing department with little or no input from operations

4. The Airline Planning Process • Fleet Assignment • Given: • Flight Schedule • Each flight covered exactly once by one fleet type • Number of Aircraft by Equipment Type • Can’t assign more aircraft than are available, for each type • FAA Maintenance Requirements • Turn Times by Fleet Type at each Station • Other Restrictions: Gate, Noise, Runway, etc. • Operating Costs, Spill and Recapture Costs, Total Potential Revenue of Flights, by Fleet Type

5. The Airline Planning Process • Fleet Assignment (cont.) • Find: • Cost minimizing (or profit maximizing) assignment of aircraft fleets to scheduled flights such that maintenance requirements are satisfied, conservation of flow (balance) of aircraft is achieved, and the number of aircraft used does not exceed the number available (in each fleet type) • State of Practice • IP models are used • Deterministic demand representation • Aggregate demand and fare class • Approximate spill and recapture representation

6. The Airline Planning Process • Aircraft routing • Given: • set of flight legs assigned to each aircraft type • through value associated with possible flight connections • Find a routing that: • provides sufficient maintenance opportunities • maximizes total through value • State of Practice • typically performed manually once fleet assignment and required throughs are set • required throughs may be implied by fleet assignment and/or required by marketing

7. The Airline Planning Process • Crew Planning • Given: • flight segments to be covered by a single fleet • aircraft turns • contractual/FAA work rules • Find: • minimum cost set of crew itineraries or pairings that covers each flight exactly once • State of Practice • use of large-scale IP models • problem is decomposed into several parts (more later)

8. The Airline Planning Process The Airline Planning Proces Schedule Selection Fleet Assign. Crew Planning Routing dep/arr times decomp. by fleet aircraft turns

9. Current State of Practice • Hierarchical approach to service design • Little or no feedback between stages in the process • organizationally, decisions may be the responsibility of different departments • Decisions at earlier stages may have significant effects on the quality of solutions at later stages

10. Opportunities for Improvement • Improvements in large-scale optimization may someday allow simultaneous solution of more than one part of the problem • Models that account for the interaction between stages or allow feedback between phases • Models that account for uncertainty in operations

11. Research Overview • Combined Fleeting and Schedule Selection • Fleeting with time windows • Desaulniers et al. (1997) • Rexing et al. (2000) • discretize time window • use multiple copies of each departure • Time windows can provide significant cost savings, as well as a potential for freeing aircraft • Incremental Schedule Design • Lohatepanont and Barnhart (1999) • Select flights from an expanded set of flight legs

12. Subject to: Fleet Assignment Models

13. Research Overview • Improved Fleet Assignment Models • Itinerary-based fleet assignment • Knicker (1998) • Compensate for network effects due to multi-leg itineraries • More accurately capture revenue by fare class • Iterates between solution of traditional Fleet Assignment Model and a Passenger Flow model to calculate revenue • Adjust cost coefficients to improve approximation

14. Network Effects 75 $300 75 $200 150 $225 X Y Z Request: 150 225 Capacity: 100 200

15. Research Overview • Combined Routing and Fleeting • Barnhart et al. (1998) • use maintenance to maintenance strings of flights • assign an aircraft type to a string rather than a single flight • Crew Scheduling before Routing • Klabjan et al (1999) • add plane count constraints to the crew scheduling problem • implies certain aircraft turns

16. Crew Planning • Definitions • duty period • pairing • Restrictions on legal pairings • FAA rules • minimum rest • maximum flying per duty • 8-in-24 • Contractual rules • max TAFB • max sit • Operational considerations • min sit

17. Crew Planning • Pairing cost structure • nonlinear and discontinuous • duty cost = maximum of: flying time, minimum guarantee, fraction of elapsed time • pairing cost = maximum of: duty cost, minimum per day, TAFB • flying time in schedule provides a lower bound • schedule quality is measured as % paid over flying time • each percentage point translates to millions annually for major domestic carriers

18. Crew Planning • Problem is formulated as a set partitioning problem min cx Ax = 1 x binary • A has one row for each flight in the schedule and one column for each potential pairing • Because of the hub-and-spoke network structure used by most U.S. carriers, the number of columns in A is HUGE so • column generation methods are used

19. Crew Planning • Typically crew planning problems are solved in phases • problem size may prohibit solving the entire weekly schedule for a single fleet • small problems may have a few hundred thousand possible pairings which large problems (500+ flights) may have billions of potential pairings • for operational reasons, airlines would prefer to maintain daily regularity of the pairings • weekly solutions contain many more different pairings which can create headaches for bidline generation or rostering purposes

20. Crew Planning • Daily Problem • Given: • flights flown 4 or more times per week • Find: • low cost schedule assuming flights are flown every day • Exceptions • Given: • flights flown fewer than 4 times per week • broken pairings from the daily solution • Find: • low-cost weekly solution for this subset of flights • Transition • Provides pairings for monthly schedule changes

21. Crew Planning Daily Problem Exceptions Transition broken pairings broken pairings

22. Outline of Remainder of Talk • Recent research in column generation methods • Combining phases of the crew pairing solution process

23. Column Generation • Column generation is an approach for solving LPs with a large number of variables • basic concepts from sensitivity analysis are used to solve the LP to optimality without explicitly considering all the possible variable • Solve the linear programming relaxation of the crew scheduling problem min cx A’x = 1 x binary • A’ contains only a subset of the possible columns (pairings) in A • Identify new columns to add to A’ to improve the solution

24. Column Generation • Current state-of-the-art • multi-label shortest path methods (dynamic programming) on specially structured networks • duty networks • large number of arcs • one arc per duty • can be hundreds of connections per duty • Ex: 363 flights, 7838 duties, 1.65 M connections • fewer labels per path since duty rules are built in • flight networks • smaller number of arcs • one arc per flight • typically not more than 30 connections per flight • larger number of labels

25. Generating Good Pairings

26. Column Generation • enumeration and SPRINT Anbil et al. (1991) • feasible pairings are enumerated up-front and stored off-line • after solving the LP relaxation, run through the list and identify several thousand negative reduced cost columns to add to A’ • use specialized data structures (Hu and Johnson (1999)) • random enumeration and SPRINT • Klabjan et al. (1999) • even when specialized data structures are used the enumerated pairings may require too much memory • use randomly enumerated pairings rather than enumerating the full set • include a potential connection with probability p, p is a nonincreasing function of the connection time

27. Column Generation: On-going Research • Hybrid networks • Duty-flight network • create a departure and arrival node for each flight • Two types of arcs • duty arcs connect first and last flights in the duty period • overnight arcs connect flight arrivals to departures the next day • has the same number of connection arcs as the flight network • explicitly builds duty rules into the network

28. Hybrid Network Day 1 Day 2 f1 f2 f3 f4 Dep. Arr. Dep. Arr. Overnight Arcs Duty Arcs Duty Arcs

29. Column Generation: On-going Research • Another hybrid network • strings • a string might be a duty or portion of a duty • typically a string of flights between two “busy” places in the network • Ongoing work by Tina Shaw

30. Interaction Between Phases • Daily and Exceptions crew pairing • the exceptions problem is partially defined by broken pairings from the daily solution • Ex: Daily Pairing: LAX-ORD 1405 1945 ORD-DCA 2030 2315 overnight DCA-LAX 1410 1800 • the second leg is not flown on Saturday or Sunday and the third is not flown on Saturday • the copies of this daily pairing beginning on Friday, Saturday, and Sunday will all be broken. The remaining flights will end up in the exceptions problem.

31. Combined Daily and Exceptions Crew Pairing • Experience with daily problems has shown that there may be many near-optimal solutions • Current practice does not explicitly consider the number of daily pairings that will be broken when assessing the quality of the daily solution

32. A Combined Model • Klabjan et al. (1999) • Consider the special case where we wish to increase the number of daily pairings that can actually be flown 7 days per week Let: xi = 1 if all 7 copies of flight i are covered by daily pairings yp = 1 if pairing p is used in the solution • Two kinds of constraints • if xi = 1, we must cover the flight with a daily pairing • if xi = 0 or the flight is a less than 7-day flight, we must cover the flight with a dated pairing

33. A Combined Model

34. A Combined Model daily pairings dated pairings -1 -1 -1 ... 1 0 1 ... 7-day flights (not dated) =0 1 1 1 1 1 1 1 1 1 1 1 ... 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 all flights (dated) =1

35. Computational Challenges • If we included all possible columns in the previous special case, we would have as many pairings as the combined daily and weekly problems • This modeling idea can be extended by creating separate blocks depending on the number of consecutive times per week a flight repeats in the same pairing • Solve a relaxed problem where crew base to crew base paths are substituted for pairings in some of the blocks

36. Problem Instances

37. Computational Results

38. Insights into Schedule Regularity • Models are extremely large and impractical for planning use on all but small problems • Computational results show that there is potential to improve regularity and cost simultaneously • Open question: can we develop more tractable models that will enable reliable construction of more regular crew schedules?

39. Another Model for Crew Schedule Regularity • Use traditional weekly (dated) set partitioning model • Columns are now “super-pairings” • a super pairing may contain 1 or more copies of a daily pairing • Consider a daily pairing • suppose all flights operate 7 days per week • there are potential super pairings • is there a sensible way to control the combinatorial explosion?

40. Conclusion • Major opportunities for improvement in air transport service design • closer integration of stages of the planning process • improvements in model accuracy • advances in large-scale optimization • incorporation of stochasticity