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A Second Stage Network Recourse Problem in Stochastic

Airline Crew Scheduling

Joyce W. YenUniversity of Michigan

John R. BirgeNorthwestern University

INFORMSSan Antonio, TX

Nov. 5, 2000

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Presentation Overview

Problem Motivation: Stochastic Airline Crew Scheduling

Stochastic Programming Formulation Branching Algorithm Large Problem Computational Results Conclusion and Future Directions

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What is airline crew scheduling?

Begin with a set of flights that need to be flown by airline crews

Crews have restrictions for working hours Want to generate trip itineraries and select a

minimum cost set of trip itineraries for crews to fly that will cover every flight

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Why study airline crew scheduling?

Airlines want to increase profits Crew costs second largest cost next to fuel

costs– costs run in the billions of dollars for large

carriers Key to lowering crew costs

– Optimizing crew schedules and crew assignments

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General Formulation

let ai,j = 1 if pairing j covers flight i

let xk = 1 if pairing k is selected to be in the solution

binary

subject to

min

x

bAx

xcT

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Problems with Current Strategies

Crew schedules optimized only to be “deoptimized” in the actual implementation because of changes to the schedules

Problem solved as a linear program– LP’s optimized at extreme points

push solutions to brink of infeasibility

Unable to respond to changes– Companies spending millions to find ways to better react to

schedule disruptions

Pairings considered individually

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Disruptions and Schedules

Different schedules will react differently to disruptions

Schedules can perpetuate the problem Want to find a way to identify and select

schedules that are less disrupted by changes to the flight schedule

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Stochastic Crew Scheduling

Integrate information about the short range (recovery) problem into the long range (traditional crew scheduling) problem

Objective: Include expected cost of disruptions in the model formulation

Capture interaction effects of planes, crews, and schedule recovery

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Problem FormulationFor each disruption , we have different recourse.

The overall problem becomes:

where

is the expected value of future actions due to disruptionsin the original schedule

Q( ) ( , ) ( )x Q x P d

integer

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tosubject

)( minimize

x

x

bAx

xxc z T

Q

10

: delay scenario j : flight k : round trip itinerary

Recourse Formulation

subject to:• consistency constraints for flight arrival and departure

times

• consistency constraints for plane predecessor requirements

• consistency constraints for crew predecessor requirements

time_dep(j) - [ time_arr(crew_pred(j,k))-crew_gnd(j,k)]xk,j 0 **

segments

min),(j

penaltyxQ delay(j)

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General form of Recourse LP

Note cross product terms This form, in general, is not convex in x Traditional SIP solution methods don’t work

0

)()( ..

min),(

y

hyGxWts

yqxQT

T

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General Idea for Algorithm eliminate delays caused by crews changing planes flight pair ranks based on switch delay value

– higher value means more costly delay

branch on pairings with particular plane changes– identifies costly decisions and disallows or forces such

decisions

Note: branching is on a large set of variables — not just a single one

Algorithm terminates with an optimal solution Algorithm can be augmented onto existing SPP solvers

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Large Problem Results

Air New Zealand delay data from 1997 Create model for disruptions based on data 3 test problem

– 9 planes, 61 flights– 10 planes, 66 flights– 11 planes, 79 flights

Disruption time distribution: lognormal and gamma with maximum value 200 minutes

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Assumptions State-of-the-Art pairing generator and set

partitioning solver available No cancellations Crews still able to fly schedule when flight

delayed– no time restriction problems

Plane ground time constant, crew ground time constant

Static plane assignments Maximum delay time bounded

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10 Planes Results

100 scenarios Penalty = 100 Daily schedule No limit on number of crews scheduled Run length: 184 iterations, 367 nodes

created – computational time limit

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10-plane, Penalty = 100, Optimal Costs

0

1000

2000

3000

4000

5000

6000

0 50 100 150 200 250

no. nodes created

cTx

valu

e

24500

25000

25500

26000

26500

27000

27500

28000

28500

29000

29500

cTx+

E[Q

(x,w

)] v

alu

e

z cTx

10th best

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Observations from Example

Quick savings early in the algorithm– small changes to cTx value

– larger changes to the recourse value and objective value

Can get reasonably good solution after very few iterations– current optimal obtained after 214 nodes created (total

nodes created = 367)

– within 5% of best after 10 nodes created

– within 2% of best after 26 nodes created

– within 1% of best after 109 nodes created

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Plane Cnx for 10 Planes, Penalty = 100

0

20

40

60

80

100

120

140

160

0 5 10 15 20

i-th best solution

No

. Pla

ne

Ch

ang

es

0

20

40

60

80

100

120

140

160

Pla

ne

Cn

x T

ime

Median Plane Cnx Time No. Plane Changes Mean Plane Cnx Time

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Effect of Penalty Parameter

Penalty parameter reflection of how much disruption is worth

As increase penalty parameter, track the trade-off of disruption costs and crew costs

How to determine penalty value– penalty = capacity of plane

– penalty = # passengers / # crew

– penalty = cost / minute of disruption

– etc.

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10 Plane Penalty Variation

-25.0%

-20.0%

-15.0%

-10.0%

-5.0%

0.0%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0%

% change in cTx*

% c

ha

ng

e i

n E

[Q(x

*,w

)]Penalty = 0

Penalty = 1000

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10 Plane with Penalty around 100

-30.0%

-20.0%

-10.0%

0.0%

10.0%

20.0%

30.0%

40.0%

0 20 40 60 80 100 120 140

Penalty Value

% c

hang

e fr

om in

itial

so

ln

z* cTx* E[Q(x*,w)]

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General Comments How much value disruptions reflected in penalty

value– trade-off between disruption cost and crew cost

Can stop at any time and use algorithm to solve for desired degree of accuracy or efficiency– gap between LB and UB small enough

– computational budget exhausted

Can add a heuristic to pick a solution that has a value within those bounds

Reach point where impossible to eliminate more disruption costs

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Challenges Algorithm

– subproblems may be slow However, can find good solution quickly

– bounds may not be tight higher the penalty the less effective bounds are

Data– Need for lots of data– Must carefully evaluate relationships and

correlations– problem size

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Future Research

Direct column generation for pairings using delay costs to avoid costly plane changes– difficulty is delays depend on all the crew

assignments jointly– traditional column generation does one trip at a

time– improvement:generate collection of columns

that work well together

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Future Research

Investigate alternative data models Explore alternative recourse models Apply method to general network design

problems– telecommunications networks

– supply chain

– job shops