Air Transportation Service Design

Pamela H. VanceGoizueta Business School

Emory University

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

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

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

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

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

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)

The Airline Planning Process

The Airline Planning Proces

ScheduleSelection

FleetAssign.

Routing Crew Planning

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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

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

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

Fleet Assignment Models

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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

Network Effects

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Capacity: 100 200

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

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

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

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

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

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

Crew Planning

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Outline of Remainder of Talk Recent research in column generation

methods Combining phases of the crew pairing

solution process

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 cxA’x = 1x binary

– A’ contains only a subset of the possible columns (pairings) in A

– Identify new columns to add to A’ to improve the solution

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

Generating Good Pairings

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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

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

Hybrid Network

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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

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 1945ORD-DCA 2030 2315overnightDCA-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.

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

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 weekLet: 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

A Combined Model

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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

Problem Instances

Name 7-day 6-day 5-day Daily TotalFs 90 24 5 0 119Fm 273 64 5 38 380Fl 331 104 15 42 492

Computational Results

Name Method FTC Reg.Fs d/ e 9.2% 276

relax1 7.0% 120relax2 5.6% 127

Fm d/ e 4.2% 622relax1 3.8% 318relax2 2.5% 390

Fl d/ e 2.3% 1365relax1 2.2% 520relax2 2.0% 567

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?

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?

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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