Group construction for cabin crew comparing constraint programming with branch price
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Group Construction for Cabin Crew Comparing Constraint Programming with Branch&Price. Presentation at SweConsNet 2005 Jesper Hansen Carmen Systems AB [email protected] In cooperation with Tomas Lidén. Outline. Introduction and Problem Formulation

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Group construction for cabin crew comparing constraint programming with branch price

Group Construction for Cabin CrewComparing Constraint Programming with Branch&Price

Presentation at SweConsNet 2005

Jesper Hansen

Carmen Systems AB

[email protected]

In cooperation with Tomas Lidén


  • Introduction and Problem Formulation

  • Constraint Programming (CP) version

  • Column Generation (CG) version

  • Computational results

  • Conclusions

Airline crew scheduling





Set of Groups

  • Buddy Bids

  • Incompatibles

  • Experience level

Due to:


Group repr.

Copy roster

to all in group


“drop-out” pairings

  • Days off

  • Preassignments

  • Historic values

  • Work reduction

Airline Crew Scheduling

Minimize exceptions after copy!

Anonymous tour-of-duties

starting and ending at home-base





Introduction, Problem Formulation


  • Problem: Construct “homogenous” groupsJan-Nov: Minimum number of groups at min. costDec: Maximum number of groups at min. cost

  • Application: Developed during 2001 for IberiaCP version in production since 2002CG version delivered 2004

  • Dimensions: 300-1200 crew in each problem,Giving 30-200 groups (3-13 persons per group)

Introduction, Problem Formulation

Factors to handle

Sequence must be surrounded by 3 untouched days.

{8 10 11 12}

{6 10 11 12}

Factors to handle


Introduction, Problem Formulation

Cp model with three step solving

Initial solution + global refinement

Global search for cheaper solution

Local search for cheaper solution

CP Model with three-step solving

  • One customized constraint – NoIllegalPattern()

  • Two global constraints:

    • Use all buddy bids, SubsetEq()

    • No overlap, AllNullIntersect()

Constraint Programming Version

Cg model with three step solving

Buddy bids

Other subgroups

Construct one IP feasible solution.

Generate initial set of columns.

Set-up initial problem matrix


Generate col’s with negative reduced cost.

Constraint branching (pairs of subgroups).

Branch & Price

CG Model with three-step solving

A column corresponds to one legal group alternative

Solving LP relaxation – called the restricted master problem.

Branch & Price to get integer solutions.

Column Generation Version


Generation tree


  • Very large tree

  • Depth First Search with pruning for:

    • Cost: When LB on reduced cost shows we can’t improve LP value (possible since duals are ordered)

    • Pattern: When needed additions for NoIllegalPattern() not among remaining subgroups

  • Search modifications:

    • Return to top for better coverage

    • Limited nr of backtracks (BBS)

    • Time limits etc.

Sg = { .. , .. , .. , .. , .. , .. , .. , .. }

Duals= { + , .. , + , 0 , .. , 0 , - , .. }

Column Generation Version

Branch price
Branch & Price


Ban col’s with only one of the two sg’s


Ban col’s with both sg’s

  • Constraint branching (Ryan, Foster) on pairs of subgroups.

  • Perform pricing in both branches.

  • Price according to all branching decisions

  • Fathom without complete enumeration of generation tree.

Branching tree

Column Generation Version

Solution progress
Solution progress

  • CP

    • starts bad and decreases evenly

  • CG

    • starts good and decreases slowly

Computational results

Normal instances jan nov
Normal Instances (Jan-Nov)

  • CG produces better quality

    • Less number of groups

  • Small fleets solved faster, but large ones slower

Tmax = 60 min

* First/Best/Total

Computational results

December instances
December Instances

  • CG produces better quality

    • Except when crew shortage

    • Always good day32 matching

  • Small fleets reasonably fast, large ones slow

  • Large fleets tuned for generating less and branching more

Tmax = 180 min

Computational results

Problem properties
Problem Properties

  • Optimization problem

    • Many constraints but few are global

    • Weak domain reduction

    • Hence not a feasibility problem

  • Challenges

    • Set Partitioning with no natural ordering. Instead multiple factors with no correlation

    • Non-additive cost factors

    • Weak domain reduction from constraints


Cp versus cg

Ease of modeling

Quick development

Several unexplored techniques:

Mirrored formulation

Tabu search or other meta-heuristic

Search heuristic highly adopted to problem

Better quality

Better or similar performance

Advanced techniques and careful tuning needed

CP versus CG


Possible future directions
Possible future directions

  • Reformulation of cost factors

  • Hybrid formulations

  • CP based pricing


End thank you for your attention questions


Thank you for your attention!