Branching Strategies to Improve Regularity of Crew Schedules in Ex-Urban

Public Transit

Leena SuhlUniversity of Paderborn, Germany

joint work with Ingmar Steinzen and Natalia Kliewer

International GraduateSchool of DynamicIntelligent Systems

ATMOS 2007 – Nov. 16, 2007Page: 2

Outline

• Introduction

• Ex-urban vehicle and crew scheduling problem– Problem definition

– Irregular timetables

• Solution Approach– Column Generation with Lagrangian relaxation

– Distance measure

– modified Ryan/Foster branching rule

– Local Branching

• Computational results

ATMOS 2007 – Nov. 16, 2007Page: 3

Introduction

lines / service network

timetable of one line

service trip: 21:45 -- 22:00 from Westerntor to Liethstaudamm

ATMOS 2007 – Nov. 16, 2007Page: 4

Introduction

crew scheduling

timetabling

vehicle scheduling

crew rostering

line+frequency planning

timetable/service trips

vehicle blocks/tasks

crew duties

crew rosters

labour regulations

relief points

ATMOS 2007 – Nov. 16, 2007Page: 5

Multi-Depot Vehicle Scheduling Problem (MDVSP)

• Given: set of service trips of a timetable

• Task: find an assignment of trips to vehicles such that– Each trip is covered exactly once

– Each vehicle performs a feasible sequence of trips (vehicle block)

– Each sequence of trips starts and ends at the same depot

– (vehicle capital and operational) costs are minimized

block 1

block 2

block 3

D1 D1

D1 D1

D2 D2

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Crew Scheduling Problem (CSP)

• Given: set of tasks– From vehicle blocks and relief points (sequential CSP)

– From timetable and relief points (integrated CSP)

• Task: assign tasks to crew duties at minimum cost

such that– Each task is covered (exactly) once

– Each duty starts/ends at the same depot

– Each duty satifies (complex) governmental and in-house regulations

block 1

block 2

D1 D1

D1 D1break

ATMOS 2007 – Nov. 16, 2007Page: 7

Crew Scheduling Problem (CSP)

break

piece of work 1 piece of work 2

duty

trip

deadhead

relief point

task 1 task 4

piece of work-related

duty-relatedconstraints

ATMOS 2007 – Nov. 16, 2007Page: 8

Crew Scheduling Problem (CSP)

• Minimize total crew costs

• Constraints– Cover all tasks of vehicle schedule (sequential)

– Cover all tasks of timetable (independent)

I set of all tasks

K set of all feasible duties

K(i) set of all duties covering task iset partitioning orset coveringformulation possible

ATMOS 2007 – Nov. 16, 2007Page: 9

Ex-urban Vehicle and Crew Scheduling Problem

(VCSP)

• Given: set of service trips of a timetable and set of

relief points

• Task: find a set of vehicle blocks and crew duties such

that– Vehicle and crew schedule are feasible

– Vehicle and crew schedule are mutually compatible

– Sum of vehicle and crew costs is minimized

• Only few relief points in ex-urban settings

• Assumption: All relief points in depot (typical for ex-

urban settings)

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Irregular Timetables

• Timetable consists of– regular (daily) trips

– irregular trips (e.g. to school or plants): about 1-5% of all trips

• similar situation: timetable modifications

• similar and regular crew schedules– easier to manage in crew rostering phase

– less error-prone for drivers

regular trips

trips day Atrips day B

ATMOS 2007 – Nov. 16, 2007Page: 11

Irregular Timetables

• Naive approach: plan all periods sequentially, but

• Modifications of timetable have a strong impact on regularity of

vehicle and crew scheduling solutions

instance: Monheim (423 trips)

timetable Monday timetable Tuesday2% of trips different

vehicle schedule vehicle schedule

crew schedule crew schedule

66% of vehicle blocks different

100% of crew duties different

crew schedule crew schedule93% of crew duties different

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• No literature on irregular timetables in public transport

• Simple heuristics from practice– Solve problem with all trips of periods

– Solve problem with regular and irregular trips of periods separately

Irregular Timetables

fix (regular) duties C: set of remaining (unfixed) tasks

small problems

many deadheads, high costs

large problems

low regularity

trade-off

))\)(()\)(((CSP BBAABA

)(CSP BA

)(CSP BA

)(CSP C

ATMOS 2007 – Nov. 16, 2007Page: 13

Outline

• Introduction

• Ex-urban vehicle and crew scheduling problem– Problem definition

– Irregular timetables

• Solution Approach– Column Generation with Lagrangian relaxation

– Distance measure

– modified Ryan/Foster branching rule

– Local Branching

• Computational results

ATMOS 2007 – Nov. 16, 2007Page: 14

Solution approach

Construct feasible vehicle schedule (pieces of work correspond to service trips)

Volume Algorithm

Partial Pricing with Dynamic Programming Algorithm

Column generation in combination with Lagrangean relaxation

Compute dual multipliers by solving Lagrangean dual problem with current set of columns

while duties ≠ and no termination criteria satisfied

duties = initial column set

Delete duties with high positive reduced costs

duties = Generate new negative reduced cost columns

Add duties to master

Find integer solution

crew scheduling

vehicle scheduling

ATMOS 2007 – Nov. 16, 2007Page: 15

Network Models for a Decomposed Pricing Problem

Piece generation network

pieces of work

connection-based duty generation network

(Freling et al. 1997, 2003)

network size: O(#tasks4)

pieces of work

aggregated time-space duty generation network

(Steinzen et al. 2006)

Tim

e

Space

network size: O(#tasks2)

ATMOS 2007 – Nov. 16, 2007Page: 16

Guided IP Branch-and-Bound search

• Average number of different optima for ICSP

• Idea: guide IP solution method to „favorable“ solutions

(concerning distance to reference solution)– Follow-on branching

– Adaptive local branching

– Adaptive local branching with follow-on branching

tolerance

#trips #instances 0% 0,01%

80 10 1052 1115

100 9 723 945

160 9 1807 2046

test set from Huisman, abort search after 2500 optima

set partitioning, independent crew scheduling, variable costs

ATMOS 2007 – Nov. 16, 2007Page: 17

Distance measure for crew duties

trip chain

T1={2,6,9}

crew schedule

G

1

2

3

4

5

…

duties Gi

crew schedule

H

1

2

3

4

5

…

duties Hi

timetable A timetable B

2

6

9

14

21

56

2

6

84

9

24

56

service trips

si

service trips

ti

irregular trip

Reference solution

ATMOS 2007 – Nov. 16, 2007Page: 18

Follow-on Branching

• Ryan/Foster branching rule for fractional solution of a

set partitioning problem and two rows r and s

• Create two subproblems

• Choose r and s with max f(r,s)

• Follow-on branching: allow only consecutive tasks

(rows)

ATMOS 2007 – Nov. 16, 2007Page: 19

Follow-on branching to create regular crew schedules

• Follow-on branching strategies– DEF: Original– FOR1: Sequences from reference schedule– FOR2: Piece of work from reference schedule– FOR3: Maximum length sequence from reference

schedule

Initialize set Sk of trip chains Ti with

Sk={Ti: 0<f(Ti)<1}Sk=

?

InitializeSk

max={Ti:max(|Ti|)}and

branch on Ti Skmax with

max(f(Ti))

Branch on trip chain (r,s) with

0<f(r,s)<1 and max(f(r,s))

No

Yes

FOR2

ATMOS 2007 – Nov. 16, 2007Page: 20

Local Branching

• Strategic local search

heuristic controls „tactical“

MIP solver

• Local branching cuts equal

Hamming distance

with L0={kK: xk’=1}

• Exact solution approach

ATMOS 2007 – Nov. 16, 2007Page: 21

Local Branching to create regular crew schedules

• Use local branching to search subspaces that contain

„regular“ solutions first

• Initial solution

– modify cost function ck’ = ck+dk with

dk distance of duty to reference crew schedule

weight of distance

• Adapt neighbourhood size if necessary (time limit

exceeded)

• Optional: use follow-on branching in subproblem

ATMOS 2007 – Nov. 16, 2007Page: 22

Outline

• Introduction

• Ex-urban vehicle and crew scheduling problem– Problem definition

– Irregular timetables

• Solution Approach– Column Generation with Lagrangian relaxation

– Distance measure

– modified Ryan/Foster branching rule

– Local Branching

• Computational results

ATMOS 2007 – Nov. 16, 2007Page: 23

Computational Results

• Tests with both real-world and artificial data– Artificial data generated like Huisman (2004) with 320/400/640/800

trips (two instances each), relief points only in depots

– Real-world data with ~430 trips (German town with ~45.000 inh.)

– Irregular trips: 5% (artificial), 2-3% (real-world)

• Reference crew schedule is known for all instances

• All tests on Intel Pentium IV 2.2GHz/2 GB RAM with

CPLEX 9.1.3

• Limited branch-and-bound time to 2 hours

ATMOS 2007 – Nov. 16, 2007Page: 24

Computational Results(Column Generation)

irr% - percentage of irregular trips

cpu_ma – cpu time (sec) for the master problem

cpu_pr – cpu time (sec) for the pricing subproblem

ATMOS 2007 – Nov. 16, 2007Page: 25

Computational Results(Regularity of Crew Schedules)

prd% - percentage of duties (completely) preserved from reference crew schedule

prp% - percentage of trip sequences preserved from reference

avcl% - percentage of average trip sequence length preserved from reference

Thank you very muchfor your attention

International GraduateSchool of DynamicIntelligent Systems