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International Graduate School of Dynamic Intelligent Systems. Branching Strategies to Improve Regularity of Crew Schedules in Ex-Urban Public Transit. Leena Suhl University of Paderborn, Germany joint work with Ingmar Steinzen and Natalia Kliewer. Outline. Introduction

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Branching strategies to improve regularity of crew schedules in ex urban public transit

International Graduate

School of Dynamic

Intelligent Systems

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

Leena Suhl

University of Paderborn, Germany

joint work with Ingmar Steinzen and Natalia Kliewer


Outline
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


Introduction
Introduction

lines / service network

timetable of one line

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


Introduction1

line+frequency planning

timetabling

timetable/service trips

vehicle scheduling

vehicle blocks/tasks

crew scheduling

crew duties

crew rostering

crew rosters

Introduction

relief points

labour regulations


Multi depot vehicle scheduling problem mdvsp

D1

D1

block 1

D1

D1

block 2

D2

D2

block 3

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


Crew scheduling problem csp

D1

D1

block 1

break

D1

D1

block 2

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


Crew scheduling problem csp1

trip

piece of work-related

duty-related

deadhead

constraints

relief point

Crew Scheduling Problem (CSP)

duty

piece of work 2

piece of work 1

break

task 1

task 4


Crew scheduling problem csp2
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 i

set partitioning orset coveringformulation possible


Ex urban vehicle and crew scheduling problem vcsp
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)


Irregular timetables
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 A

trips day B


Irregular timetables1

instance: Monheim (423 trips)

2% of trips different

timetable Monday

timetable Tuesday

66% of vehicle blocks different

vehicle schedule

vehicle schedule

100% of crew duties different

crew schedule

crew schedule

93% of crew duties different

crew schedule

crew schedule

Irregular Timetables

  • Naive approach: plan all periods sequentially, but

  • Modifications of timetable have a strong impact on regularity of vehicle and crew scheduling solutions


Irregular timetables2

fix (regular) duties 

C: set of remaining (unfixed) tasks

trade-off

large problems

low regularity

small problems

many deadheads, high costs

Irregular Timetables

  • 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


Outline1
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


Solution approach

Column generation in combination with Lagrangean relaxation

duties= initial column set

while duties≠  and no termination criteria satisfied

Add duties to master

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

Volume Algorithm

Delete duties with high positive reduced costs

Partial Pricing with Dynamic Programming Algorithm

duties = Generate new negative reduced cost columns

Find integer solution

Solution approach

crew scheduling

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

vehicle scheduling


Network models for a decomposed pricing problem

Space

Time

Piece generation network

Network Models for a Decomposed Pricing Problem

pieces of work

pieces of work

connection-based duty generation network

(Freling et al. 1997, 2003)

aggregated time-space duty generation network

(Steinzen et al. 2006)

network size: O(#tasks2)

network size: O(#tasks4)


Guided ip branch and bound search
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

test set from Huisman, abort search after 2500 optima

set partitioning, independent crew scheduling, variable costs


Distance measure for crew duties

service trips

service trips

ti

si

timetable A

timetable B

2

6

9

14

21

56

duties Gi

duties Hi

1

2

3

4

5

1

2

3

4

5

2

6

84

9

24

56

irregular trip

Distance measure for crew duties

crew schedule

G

crew schedule

H

trip chain

T1={2,6,9}

Reference solution


Follow on branching
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)


Follow on branching to create regular crew schedules

Initialize set Sk of trip chains Ti with

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

Branch on

trip chain (r,s) with

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

Yes

No

Initialize

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

and

branch on Ti Skmax with max(f(Ti))

FOR2

Sk=?

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


Local branching
Local Branching

  • Strategic local search heuristic controls „tactical“ MIP solver

  • Local branching cuts equal Hamming distance

    with L0={kK: xk’=1}

  • Exact solution approach


Local branching to create regular crew schedules
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+dkwith

      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


Outline2
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


Computational results
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


Computational results column generation
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


Computational results regularity of crew schedules
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 much for your attention

International Graduate

School of Dynamic

Intelligent Systems

Thank you very muchfor your attention


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