Branching Strategies to Improve Regularity of Crew Schedules in ExUrban Public Transit

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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

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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

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Introduction
lines / service network
timetable of one line
service trip 2145 -- 2200 from Westerntor to
Liethstaudamm
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Introduction
relief points
labour regulations
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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

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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

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Crew Scheduling Problem (CSP)
duty
piece of work 2
piece of work 1
break
task 1
task 4
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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
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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 A
trips day B
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Irregular Timetables

• Naive approach plan all periods sequentially,
  but
• Modifications of timetable have a strong impact
  on regularity of vehicle and crew scheduling
  solutions

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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

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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

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Solution approach
crew scheduling
Construct feasible vehicle schedule (pieces of
work correspond to service trips)
vehicle scheduling
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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)
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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
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Distance measure for crew duties
crew schedule G
crew schedule H
trip chain T12,6,9
Reference solution
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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)

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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

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Local Branching

• Strategic local search heuristic controls
  tactical MIP solver
• Local branching cuts equal Hamming distance
• with L0k?K xk1
• Exact solution approach

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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

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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

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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

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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
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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
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Thank you very muchfor your attention