ARRIVAL/ M ATHEON Periodic Timetabling for Networks Fall School 2006 Christian Liebchen EU Research Program ARRIVAL DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Contents 1 Interfaces of Periodic Timetabling 2 Graph Model for Periodic Timetables 3 IP Models for Periodic Timetabling 4 Integral Cycle Bases 5 Exercises 6 Conclusions ARRIVAL/ M ATHEON Fall School 2006 Page 1 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Contents ⊲ 1 Interfaces of Periodic Timetabling 2 Graph Model for Periodic Timetables 3 IP Models for Periodic Timetabling 4 Integral Cycle Bases 5 Exercises 6 Conclusions ARRIVAL/ M ATHEON Fall School 2006 Page 1 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes • Network Design Timetabling Within // . . . where to build the tracks? The Planning Process • Line Planning // incl. frequencies, stop policies of Railway Companies (cf. yesterday) • Timetabling • Vehicle Scheduling (cf. Bornd¨ orfer et al., Huisman et al., Desrosiers et al.) • Duty Scheduling • Crew Rostering • Operations/Delay Management (cf. Sch¨ obel et al., Clausen et al., Mellouli et al.) . . . and also • Fare System Design (cf. Bornd¨ orfer, Pfetsch, and Neumann, Sch¨ obel et al.) ARRIVAL/ M ATHEON Fall School 2006 Page 2 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Subtasks of • Definition of “Coordinated Groups of Lines” Timetabling (cf. Pagourtsis et al.) ARRIVAL/ M ATHEON Fall School 2006 Page 3 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Subtasks of • Definition of “Coordinated Groups of Lines” Timetabling (cf. Pagourtsis et al.) • Computation of “Basic Hourly Patterns” (BUP) → Periodic Timetabling ֒ ARRIVAL/ M ATHEON Fall School 2006 Page 3 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Subtasks of • Definition of “Coordinated Groups of Lines” Timetabling (cf. Pagourtsis et al.) • Computation of “Basic Hourly Patterns” (BUP) → Periodic Timetabling ֒ • Selection of first and last trips of Rush Hour Pe- riod, Weak Traffic Period, Night Traffic, etc., occa- sionally plus some trips in-between (cf. Leung et al.) ARRIVAL/ M ATHEON Fall School 2006 Page 3 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Subtasks of • Definition of “Coordinated Groups of Lines” Timetabling (cf. Pagourtsis et al.) • Computation of “Basic Hourly Patterns” (BUP) → Periodic Timetabling ֒ • Selection of first and last trips of Rush Hour Pe- riod, Weak Traffic Period, Night Traffic, etc., occa- sionally plus some trips in-between (cf. Leung et al.) • Introduce special trips (e.g. for pupils) ARRIVAL/ M ATHEON Fall School 2006 Page 3 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Subtasks of • Definition of “Coordinated Groups of Lines” Timetabling (cf. Pagourtsis et al.) • Computation of “Basic Hourly Patterns” (BUP) → Periodic Timetabling ֒ • Selection of first and last trips of Rush Hour Pe- riod, Weak Traffic Period, Night Traffic, etc., occa- sionally plus some trips in-between (cf. Leung et al.) • Introduce special trips (e.g. for pupils) Alternatively • Schedule Trips Individually (cf. Toth et al., Ingolotti et al., Leung et al.) ARRIVAL/ M ATHEON Fall School 2006 Page 3 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Periodicity Timetable Station RE7 RE7 RE7 Zossen 15:06 16:06 17:06 Dabendorf 15:08 16:08 17:08 Airport SXF 15:30 16:30 17:30 Berlin Hbf 15:59 16:59 17:59 Berlin Zoo 16:07 17:07 18:07 ARRIVAL/ M ATHEON Fall School 2006 Page 4 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Periodicity Timetable BUP Station RE7 RE7 RE7 Station RE7 Zossen 15:06 16:06 17:06 Zossen xx:06 Dabendorf 15:08 16:08 17:08 Dabendorf xx:08 Airport SXF 15:30 16:30 17:30 Airport SXF xx:30 Berlin Hbf 15:59 16:59 17:59 Berlin Hbf xx:59 Berlin Zoo 16:07 17:07 18:07 Berlin Zoo xx:07 ARRIVAL/ M ATHEON Fall School 2006 Page 4 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Contents ⊲ 1 Interfaces of Periodic Timetabling 2 Graph Model for Periodic Timetables 3 IP Models for Periodic Timetabling 4 Integral Cycle Bases 5 Exercises 6 Conclusions ARRIVAL/ M ATHEON Fall School 2006 Page 5 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Contents 1 Interfaces of Periodic Timetabling ⊲ 2 Graph Model for Periodic Timetables 3 IP Models for Periodic Timetabling 4 Integral Cycle Bases 5 Exercises 6 Conclusions ARRIVAL/ M ATHEON Fall School 2006 Page 5 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes P ERIODIC E VENT • Introduced by Serafini & Ukovich (1989) S CHEDULING P ROBLEM • Model each arrival and departure (“event”) of (P ESP ) any directed line at any station in the network as an individual vertex! ARRIVAL/ M ATHEON Fall School 2006 Page 6 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes P ERIODIC E VENT • Introduced by Serafini & Ukovich (1989) S CHEDULING P ROBLEM • Model each arrival and departure (“event”) of (P ESP ) any directed line at any station in the network as an individual vertex! • A periodic timetable π assigns to each vertex v a point in time π v within the period time T , π v ∈ [0 , T ) . ARRIVAL/ M ATHEON Fall School 2006 Page 6 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes P ERIODIC E VENT • Introduced by Serafini & Ukovich (1989) S CHEDULING P ROBLEM • Model each arrival and departure (“event”) of (P ESP ) any directed line at any station in the network as an individual vertex! • A periodic timetable π assigns to each vertex v a point in time π v within the period time T , π v ∈ [0 , T ) . • For that the values π fit together, we impose re- strictions on the time durations between pairs of events. . . ARRIVAL/ M ATHEON Fall School 2006 Page 6 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Computing Modulo • The time duration from event v to event w is the Period Time π w − π v . ARRIVAL/ M ATHEON Fall School 2006 Page 7 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Computing Modulo • The time duration from event v to event w is the Period Time π w − π v . • Problem ARRIVAL of line RE7 at Berlin Hbf at minute 59 and departure at minute 00 imply negative dwell time ! ARRIVAL/ M ATHEON Fall School 2006 Page 7 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes Computing Modulo • The time duration from event v to event w is the Period Time π w − π v . • Problem ARRIVAL of line RE7 at Berlin Hbf at minute 59 and departure at minute 00 imply negative dwell time ! • Solution Consider cyclic time difference by computing modulo the period time : ( π w − π v ) mod T. ARRIVAL/ M ATHEON Fall School 2006 Page 7 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes P ESP Constraints • To ensure the time duration from event v to event w to be in [ ℓ a , u a ] , we require ( π w − π v − ℓ a ) mod T ≤ u a − ℓ a and introduce an arc a = ( v, w ) . • We use π w − π v ∈ [ ℓ a , u a ] T as a shorthand. ARRIVAL/ M ATHEON Fall School 2006 Page 8 of 32
DFG Research Center M ATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes P ESP Constraints • To ensure the time duration from event v to event w to be in [ ℓ a , u a ] , we require ( π w − π v − ℓ a ) mod T ≤ u a − ℓ a and introduce an arc a = ( v, w ) . • We use π w − π v ∈ [ ℓ a , u a ] T as a shorthand. • Without loss of generality we may assume. . . • ℓ a ∈ [0 , T ) • u a − ℓ a ∈ [0 , T ) ARRIVAL/ M ATHEON Fall School 2006 Page 8 of 32
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