Periodic Railway Timetabling with Event Flexibility G. Caimi, M. Fuchsberger, M. Laumanns, K. Schüpbach ETH Zurich ATMOS, Sevilla, 16 November 2007
© ETH Zürich
16 November 2007
Outline Approach to the train scheduling problem Timetable generation with PESP Event flexibility in timetables (FPESP) Results Conclusions
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
2
Generating train schedules
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Generating train schedules INPUT:
Global train service intention (GSI)
Aggregated and detailed track topology
Rolling stock with dynamic properties
• Train lines with stops and frequencies • Interconnections • Rolling stock
OUTPUT:
Conflict-free train schedule
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Two level approach Macro scheduling: Find a timetable that fulfills basic properties, such as trip times, connections and headways. Micro scheduling: Find locally a conflict free routing, fulfilling detailed safety requirements for a given macro schedule. 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Scheduling on the macro level Periodic Event Scheduling Problem
(Serafini Ukovich 1989)
Generates cyclic timetables or gives a proof of infeasibility
Train line has to be known a priori (from GSI) Simplified safety system: headway time
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Periodic Event Scheduling Problem (PESP) Events: train departures and arrivals 0 · πi < T Constraints: τi
(lij, uij)
τj
lij · πj - πi + T pij · uij
Periodicity: T Period jumps: pij, integer variables (binary for 0 · lij · uij < T) 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
Serafini and Ukovich 1989 7
PESP constraints Trip time
Dwell time
A
Departure in A
Headway time
Connection
B
Arrival at B
A (2,58)
A
(22,25)
B
(1,5)
B
(10,12)
C
(2,58) (28,33)
(3,8)
B
B
(18,21)
D 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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MIP formulations of PESP Cycle Periodicity1
Original
Problem is NP-complete Cycle Periodicity is more 16 November 2007
efficient1
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
1 Peeters
03 1 Liebchen 06 9
Outline Approach to the train scheduling problem Timetable generation with PESP Event flexibility in timetables (FPESP) Results Conclusions
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Two level interface Interaction of the macro and micro scheduling Find a feasible macro schedule
check
Find a conflict free routing
C. et al 2007 Kroon et al 1996 Ehrgott et al 2005
reject
Goal: Increase search space in micro scheduling Æ generate time slots instead of fixed times for the events at the macro level 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Recall PESP – Example 4
x = 48 [45,49]
x = 3 [2,4]
52
[2,6] x = 4
May be too restrictive! 1
[2,57]
56
x=5
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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FPESP: introduce event slots [4,5]
[45,49]
[2,4]
[2,6]
[2,57] [1,2]
16 November 2007
[51,53]
Each choice of event times in interval corresponds to a feasible schedule [55,57]
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Introduction of event slots Assign time slots (πli, πui) for the events instead of fixed times
t
All πi 2 (πli, πui) fulfill the PESP constraints Event flexibility δi = πui - πli Event flexibilities are dependent
δj δi i
δi + δj · γij for all constraints where the γij = uij – lij is the constraint interval length 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
δi
j
γij
δj 14
Flexible PESP model (FPESP) Event slots are modelled by constraint adaptation (l,u)
(l,u)
π
(l,u−δ)
(π,π+δ)
(l +δ,u)
(l +δ,u)
(l,u)
Fits into both MIP formulations Close to the original PESP model No additional integer variables 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Testcase Bi-objective problem of minimizing the total triptime and maximizing the flexibility
16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Test Case
Zug – Luzern – ArthGoldau Trains: intercity, local, cargo Service Intention from SBB timetable 2007 Matlab implementation with Mosek/Cplex solver 16 November 2007
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
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Test case: generated timetable
Muri 16 November 2007
Rotkreuz
Immensee
Arth-Goldau
G. Caimi | ETH Zurich | Periodic Railway Timetabling with Event Flexibility
Erstfeld 18
Trip time vs flexibility: Pareto line >0.5
MIXFLEX
