1. 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
International Graduate
School of Dynamic
Intelligent Systems

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

3. lines / service network
Introduction
lines / service network
timetable of one line
service trip: 21: :00 from Westerntor to Liethstaudamm

4. line+frequency planning
Introduction
crew scheduling
timetabling
vehicle scheduling
crew rostering
line+frequency planning
timetable/service trips
vehicle blocks/tasks
relief points
crew duties
labour regulations
crew rosters

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 D2

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
break

7. Crew Scheduling Problem (CSP)
duty
piece of work 1
piece of work 2
break
task 1
task 4
trip
deadhead
relief point
piece of work-related
duty-related
constraints

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 i
set partitioning or set covering formulation possible

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)

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 A
trips day B

11. instance: Monheim (423 trips)
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 Tuesday
2% of trips different
crew schedule
93% of crew duties different
vehicle schedule
crew schedule
66% of vehicle blocks different
100% of crew duties different

12. 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
 fix (regular) duties 
C: set of remaining (unfixed) tasks
small problems
many deadheads, high costs
large problems
low regularity
trade-off

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

14. Solution approach crew scheduling vehicle scheduling Volume 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
Volume Algorithm
Partial Pricing with Dynamic Programming Algorithm
Construct feasible vehicle schedule (pieces of work correspond to service trips)
vehicle scheduling

15. Network Models for a Decomposed Pricing Problem
Piece generation network
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)
Time
Space
network size: O(#tasks4)
network size: O(#tasks2)

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
1807
2046
test set from Huisman, abort search after 2500 optima
set partitioning, independent crew scheduling, variable costs

17. Distance measure for crew duties2 6 9 14 21 56 84 24
service trips si ti
timetable A
crew schedule G
crew schedule H
timetable B 1 2 3 4 5 …
duties Gi 1 2 3 4 5 …
duties Hi
trip chain
T1={2,6,9}
irregular trip
Reference solution

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)

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=?
Initialize
Skmax={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

20. 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

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

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

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 ~ 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

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

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

26. Thank you very much for your attention
International Graduate
School of Dynamic
Intelligent Systems