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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
ATMOS 2007 – Nov. 16, 2007Page: 6
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)
ATMOS 2007 – Nov. 16, 2007Page: 10
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
ATMOS 2007 – Nov. 16, 2007Page: 12
• 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