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
• Cant 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
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
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
• Ax 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