Periodic Timetabling for Networks Fall School 2006 Christian - - PowerPoint PPT Presentation

ARRIVAL/MATHEON Fall School 2006 Periodic Timetabling for Networks

Christian Liebchen

EU Research Program ARRIVAL DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Contents

ARRIVAL/MATHEON Fall School 2006 Page 1 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Contents

ARRIVAL/MATHEON Fall School 2006 Page 1 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

⊲

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

Timetabling Within The Planning Process

• f Railway Companies

ARRIVAL/MATHEON Fall School 2006 Page 2 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Network Design

// . . . where to build the tracks?

• Line Planning

// incl. frequencies, stop policies

(cf. yesterday)

• Timetabling
• Vehicle Scheduling

(cf. Bornd¨

• rfer et al., Huisman et al., Desrosiers et al.)
• Duty Scheduling
• Crew Rostering
• Operations/Delay Management

(cf. Sch¨

• bel et al., Clausen et al., Mellouli et al.)

. . . and also

• Fare System Design

(cf. Bornd¨

• rfer, Pfetsch, and Neumann, Sch¨
• bel et al.)

Subtasks of Timetabling

ARRIVAL/MATHEON Fall School 2006 Page 3 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition
• f

“Coordinated Groups

• f

Lines”

(cf. Pagourtsis et al.)

Subtasks of Timetabling

ARRIVAL/MATHEON Fall School 2006 Page 3 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition
• f

“Coordinated Groups

• f

Lines”

(cf. Pagourtsis et al.)

• Computation of “Basic Hourly Patterns” (BUP)

֒ → Periodic Timetabling

Subtasks of Timetabling

ARRIVAL/MATHEON Fall School 2006 Page 3 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition
• f

“Coordinated Groups

• f

Lines”

(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.)

Subtasks of Timetabling

ARRIVAL/MATHEON Fall School 2006 Page 3 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition
• f

“Coordinated Groups

• f

Lines”

(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)

Subtasks of Timetabling

ARRIVAL/MATHEON Fall School 2006 Page 3 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition
• f

“Coordinated Groups

• f

Lines”

(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.)

Periodicity

ARRIVAL/MATHEON Fall School 2006 Page 4 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Periodicity

ARRIVAL/MATHEON Fall School 2006 Page 4 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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 BUP Station RE7 Zossen xx:06 Dabendorf xx:08 Airport SXF xx:30 Berlin Hbf xx:59 Berlin Zoo xx:07

Contents

ARRIVAL/MATHEON Fall School 2006 Page 5 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

⊲

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

Contents

ARRIVAL/MATHEON Fall School 2006 Page 5 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

PERIODIC EVENT SCHEDULING PROBLEM (PESP)

ARRIVAL/MATHEON Fall School 2006 Page 6 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Introduced by Serafini & Ukovich (1989)
• Model each arrival and departure (“event”) of

any directed line at any station in the network as an individual vertex!

PERIODIC EVENT SCHEDULING PROBLEM (PESP)

ARRIVAL/MATHEON Fall School 2006 Page 6 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Introduced by Serafini & Ukovich (1989)
• Model each arrival and departure (“event”) of

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 πv within the period time T ,

πv ∈ [0, T).

PERIODIC EVENT SCHEDULING PROBLEM (PESP)

ARRIVAL/MATHEON Fall School 2006 Page 6 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Introduced by Serafini & Ukovich (1989)
• Model each arrival and departure (“event”) of

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 πv within the period time T ,

πv ∈ [0, T).

• For that the values π fit together, we impose re-

strictions on the time durations between pairs of

• events. . .

Computing Modulo the Period Time

ARRIVAL/MATHEON Fall School 2006 Page 7 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• The time duration from event v to event w is

πw − πv.

Computing Modulo the Period Time

ARRIVAL/MATHEON Fall School 2006 Page 7 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• The time duration from event v to event w is

πw − πv.

• Problem

ARRIVAL of line RE7 at Berlin Hbf at minute 59 and departure at minute 00 imply negative dwell time!

Computing Modulo the Period Time

ARRIVAL/MATHEON Fall School 2006 Page 7 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• The time duration from event v to event w is

πw − πv.

• 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:

(πw − πv) mod T.

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 8 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• To ensure the time duration from event v to

event w to be in [ℓa, ua], we require

(πw − πv − ℓa) mod T ≤ ua − ℓa

and introduce an arc a = (v, w).

• We use πw − πv ∈ [ℓa, ua]T as a shorthand.

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 8 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• To ensure the time duration from event v to

event w to be in [ℓa, ua], we require

(πw − πv − ℓa) mod T ≤ ua − ℓa

and introduce an arc a = (v, w).

• We use πw − πv ∈ [ℓa, ua]T as a shorthand.
• Without loss of generality we may assume. . .
• ℓa ∈ [0, T)
• ua − ℓa ∈ [0, T)

PERIODIC EVENT SCHEDULING PROBLEM (PESP)

ARRIVAL/MATHEON Fall School 2006 Page 9 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

T -PESP

Given Directed graph D = (V, A), arc vectors ℓ and u Question Either find a node potential vec- tor π ∈ [0, T)V such that πw − πv ∈ [ℓa, ua]T , ∀a = (v, w) ∈ A,

• r decide that none exists

PERIODIC EVENT SCHEDULING PROBLEM (PESP)

ARRIVAL/MATHEON Fall School 2006 Page 9 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

T -PESP

Given Directed graph D = (V, A), arc vectors ℓ and u Question Either find a node potential vec- tor π ∈ [0, T)V such that πw − πv ∈ [ℓa, ua]T , ∀a = (v, w) ∈ A,

• r decide that none exists
• T -PESP generalizes T -VERTEX COLORING

֒ → is NP-complete

PERIODIC EVENT SCHEDULING PROBLEM (PESP)

ARRIVAL/MATHEON Fall School 2006 Page 9 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

T -PESP

Given Directed graph D = (V, A), arc vectors ℓ and u Question Either find a node potential vec- tor π ∈ [0, T)V such that πw − πv ∈ [ℓa, ua]T , ∀a = (v, w) ∈ A,

• r decide that none exists
• T -PESP generalizes T -VERTEX COLORING

֒ → is NP-complete

• Maximizing the number of constraints that can be

satisfied by a vector π is MAXSNP-hard

֒ → existence of PTAS unlikely

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

֒ → transfer, e.g. [5, 12]60

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

֒ → transfer, e.g. [5, 12]60

• Departures of two different lines from the same

station

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

֒ → transfer, e.g. [5, 12]60

• Departures of two different lines from the same

station

֒ → (minimum) headway constraints, e.g. [4, 56]60

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

֒ → transfer, e.g. [5, 12]60

• Departures of two different lines from the same

station

֒ → (minimum) headway constraints, e.g. [4, 56]60

• Single-track safety constraints. . .

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

֒ → transfer, e.g. [5, 12]60

• Departures of two different lines from the same

station

֒ → (minimum) headway constraints, e.g. [4, 56]60

• Single-track safety constraints. . .
• Disjunctive Constraints. . .

PESP Constraints

ARRIVAL/MATHEON Fall School 2006 Page 10 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Modeling Examples

• ARRIVAL and departure of the same directed line

in the same station

֒ → stop activity, e.g. [1, 3]60

• ARRIVAL and departure of different lines in the

same station

֒ → transfer, e.g. [5, 12]60

• Departures of two different lines from the same

station

֒ → (minimum) headway constraints, e.g. [4, 56]60

• Single-track safety constraints. . .
• Disjunctive Constraints. . .
• Aspects of Line Planning, Vehicle Scheduling

(Exercises)

Introducing a Linear Objective Function

ARRIVAL/MATHEON Fall School 2006 Page 11 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Penalize with a linear coefficient ca the time du-

ration that exceeds the minimum time duration ℓa that was defined for the arc a

• The linear objective function reads
• a=(v,w)∈A

ca · ((πw − πv − ℓa) mod T)

Introducing a Linear Objective Function

ARRIVAL/MATHEON Fall School 2006 Page 11 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Penalize with a linear coefficient ca the time du-

ration that exceeds the minimum time duration ℓa that was defined for the arc a

• The linear objective function reads
• a=(v,w)∈A

ca · ((πw − πv − ℓa) mod T)

• This enables us to define soft constraints within

the PESP. . .

Contents

ARRIVAL/MATHEON Fall School 2006 Page 12 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Contents

ARRIVAL/MATHEON Fall School 2006 Page 12 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 13 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• For each event v, introduce a time variable πv ∈

[0, T)

• We have to translate

πw − πv ∈ [ℓa, ua]T, a = (v, w) ∈ A

into the language of INTEGER PROGRAMMING

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 13 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• For each event v, introduce a time variable πv ∈

[0, T)

• We have to translate

πw − πv ∈ [ℓa, ua]T, a = (v, w) ∈ A

into the language of INTEGER PROGRAMMING

• This is equivalent to the existence of some auxil-

iary integer value pa such that

ℓa ≤ πw − πv + Tpa ≤ ua . . .

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 13 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• For each event v, introduce a time variable πv ∈

[0, T)

• We have to translate

πw − πv ∈ [ℓa, ua]T, a = (v, w) ∈ A

into the language of INTEGER PROGRAMMING

• This is equivalent to the existence of some auxil-

iary integer value pa such that

ℓa ≤ πw − πv + Tpa ≤ ua . . .

• We may restrict pa to {0, 1, 2} — in the case of

ua ≤ T it even suffices to declare pa binary. . .

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 14 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min

• a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

0 ≤ πw − πv − ℓa + Tpa ≤ ua − ℓa, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 14 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min

• a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

0 ≤ πw − πv − ℓa + Tpa ≤ ua − ℓa, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

• What about the LP-relaxation?

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 14 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min

• a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

0 ≤ πw − πv − ℓa + Tpa ≤ ua − ℓa, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

• What about the LP-relaxation?

֒ → π ≡ 0 and pa = ℓa

T ∈ [0, 1) is an optimum

solution of objective value zero!

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 14 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min

• a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

0 ≤ πw − πv − ℓa + Tpa ≤ ua − ℓa, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

• What about the LP-relaxation?

֒ → π ≡ 0 and pa = ℓa

T ∈ [0, 1) is an optimum

solution of objective value zero!

• Adding valid inequalities is essential!

An Event-Based IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 14 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min

• a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

0 ≤ πw − πv − ℓa + Tpa ≤ ua − ℓa, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

• What about the LP-relaxation?

֒ → π ≡ 0 and pa = ℓa

T ∈ [0, 1) is an optimum

solution of objective value zero!

• Adding valid inequalities is essential!
• By the way: w.l.o.g. πv ∈ {0, . . . , T − 1}. . .

Cycle Periodicity Property

ARRIVAL/MATHEON Fall School 2006 Page 15 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

ℓa ≤ πw − πv + Tpa ≤ ua, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

Observation Let C be an oriented circuit having forward arcs C+ and backward arcs C−. For every feasible solu- tion (π, p), summing up the time durations πw−πv+

Tpa around C provides

• a∈C+(πw − πv + Tpa) −

a∈C−(πw − πv + Tpa)

= T ·

• a∈C+ pa −

a∈C− pa

• ∈ TZ.

(“cycle periodicity property”)

Valid Inequalities

ARRIVAL/MATHEON Fall School 2006 Page 16 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

ℓa ≤ πw − πv + Tpa ≤ ua, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

Lemma Summing up the bounds ℓ and u provides us with the following valid inequalities

• 1

T

• a∈C+ ℓa −

a∈C− ua

• ≤
• a∈C+ pa −

a∈C− pa,

• a∈C+ pa −

a∈C− pa ≤

• 1

T

• a∈C+ ua −

a∈C− ℓa

• .

(“cycle inequalities”)

Towards an Alternative IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 17 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

ℓa ≤ πw − πv + Tpa ≤ ua, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

Goals

• Reduce number of integer variables while keeping

strong bounds

Towards an Alternative IP Formulation

ARRIVAL/MATHEON Fall School 2006 Page 17 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · (πw − πv − ℓa + Tpa)

s.t.

ℓa ≤ πw − πv + Tpa ≤ ua, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2}

Goals

• Reduce number of integer variables while keeping

strong bounds

• Encode time information in variables having less

symmetries, e.g. if for a pair

• f

node potential vari- ables (πv, πw) the values (6, 9) are advanta- geous, so will be the values (8, 1) — though look- ing pretty different

Periodic Tensions

ARRIVAL/MATHEON Fall School 2006 Page 18 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Given a feasible solution (π, p), we consider

xa := πw − πv + Tpa, ∀a = (v, w) ∈ A.

(“periodic tension”)

Periodic Tensions

ARRIVAL/MATHEON Fall School 2006 Page 18 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Given a feasible solution (π, p), we consider

xa := πw − πv + Tpa, ∀a = (v, w) ∈ A.

(“periodic tension”)

• Simply replacing (π, p) with x is not enough for

the IP formulation

(integrality information would just disappear!)

Periodic Tensions

ARRIVAL/MATHEON Fall School 2006 Page 18 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Given a feasible solution (π, p), we consider

xa := πw − πv + Tpa, ∀a = (v, w) ∈ A.

(“periodic tension”)

• Simply replacing (π, p) with x is not enough for

the IP formulation

(integrality information would just disappear!)

• Values x should “at least” satisfy the cycle peri-
• dicity property for all oriented circuits of D

Periodic Tensions

ARRIVAL/MATHEON Fall School 2006 Page 18 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Given a feasible solution (π, p), we consider

xa := πw − πv + Tpa, ∀a = (v, w) ∈ A.

(“periodic tension”)

• Simply replacing (π, p) with x is not enough for

the IP formulation

(integrality information would just disappear!)

• Values x should “at least” satisfy the cycle peri-
• dicity property for all oriented circuits of D
• Proposition

If a vector x satisfies the cycle periodicity for all

• riented circuits of D, then it is a periodic tension,

i.e. we may reconstruct a solution (π, p)

(Exercise)

An Other IP for PESP

ARRIVAL/MATHEON Fall School 2006 Page 19 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min

• a=(v,w)∈A

ca · (πw − πv + Tpa)

s.t.

ℓa ≤ πw − πv + Tpa ≤ ua, ∀a ∈ A πv ∈ [0, T) pa ∈ {0, 1, 2} min

• a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A

• a∈C+ xa −

a∈C− xa = TzC, ∀C circuit in D

zC ∈ Z

Properties of the Tension Formulation

ARRIVAL/MATHEON Fall School 2006 Page 20 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ∈ Z

Properties of the Tension Formulation

ARRIVAL/MATHEON Fall School 2006 Page 20 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ∈ Z

• Small values for the xa might always be preferred,

Properties of the Tension Formulation

ARRIVAL/MATHEON Fall School 2006 Page 20 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ∈ Z

• Small values for the xa might always be preferred,

but exponentially many constraints and integer variables

Properties of the Tension Formulation

ARRIVAL/MATHEON Fall School 2006 Page 20 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ∈ Z

• Small values for the xa might always be preferred,

but exponentially many constraints and integer variables

• TODOs
• Identify bounds for integer variables zC

Properties of the Tension Formulation

ARRIVAL/MATHEON Fall School 2006 Page 20 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ∈ Z

• Small values for the xa might always be preferred,

but exponentially many constraints and integer variables

• TODOs
• Identify bounds for integer variables zC
• Identify polynomially many circuits on which we

ensure the cycle periodicity property explicitly — and which imply it for all the other circuits

Bounds on Integer Variables

ARRIVAL/MATHEON Fall School 2006 Page 21 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ≤ zC ≤ zC, ∀C circuit in D zC ∈ Z

• Bounds (z, z) on integer variables can be derived

from cycle inequalities

Bounds on Integer Variables

ARRIVAL/MATHEON Fall School 2006 Page 21 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ≤ zC ≤ zC, ∀C circuit in D zC ∈ Z

• Bounds (z, z) on integer variables can be derived

from cycle inequalities

• The number of possible values for zC is

zC − zC ≈ 1 T ·

• a∈C

(ua − ℓa)

Bounds on Integer Variables

ARRIVAL/MATHEON Fall School 2006 Page 21 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ≤ zC ≤ zC, ∀C circuit in D zC ∈ Z

• Bounds (z, z) on integer variables can be derived

from cycle inequalities

• The number of possible values for zC is

zC − zC ≈ 1 T ·

• a∈C

(ua − ℓa)

• ֒

→ seek for short circuits with respect to undi-

rected edge weights ua − ℓa

Contents

ARRIVAL/MATHEON Fall School 2006 Page 22 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Contents

ARRIVAL/MATHEON Fall School 2006 Page 22 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Cycle Bases of Graphs

ARRIVAL/MATHEON Fall School 2006 Page 23 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Incidence Vector of an oriented circuit C

γ(C)a :=        1,

if a ∈ C+,

−1,

if a ∈ C−,

0,

if a ∈ C+ ˙

∪ C−.

• Cycle Space C(D) of a directed graph D

C(D) :=

span({γ(C) | C oriented circuit in D})

⊂ QA

• Cycle Basis B of D
• dim(C(D)) = m − n + 1

// cyclomatic number

ARRIVAL/MATHEON Fall School 2006 Page 24 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Examples of Cycle Bases

Integral Cycle Bases

ARRIVAL/MATHEON Fall School 2006 Page 25 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition

A cycle basis B of a directed graph D is integral, if every oriented circuit of D can be written as an integer linear combination of the elements of B.

Integral Cycle Bases

ARRIVAL/MATHEON Fall School 2006 Page 25 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition

A cycle basis B of a directed graph D is integral, if every oriented circuit of D can be written as an integer linear combination of the elements of B.

• Theorem

A cycle basis B is integral, if and only if the m ×

(m − n + 1) arc-cycle incidence matrix has an (m − n + 1) × (m − n + 1) submatrix with

determinant ±1. . .

Integral Cycle Bases

ARRIVAL/MATHEON Fall School 2006 Page 25 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Definition

A cycle basis B of a directed graph D is integral, if every oriented circuit of D can be written as an integer linear combination of the elements of B.

• Theorem

A cycle basis B is integral, if and only if the m ×

(m − n + 1) arc-cycle incidence matrix has an (m − n + 1) × (m − n + 1) submatrix with

determinant ±1. . .

• Corollary

If an arc vector x satisfies the cycle periodicity property on some integral cycle basis, then it sat- isfies it for each circuit of D

An Other IP for PESP

ARRIVAL/MATHEON Fall School 2006 Page 26 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C circuit in D

zC ≤ zC ≤ zC, ∀C circuit in D zC ∈ Z min P

a=(v,w)∈A

ca · xa

s.t.

ℓa ≤ xa ≤ ua, ∀a ∈ A P

a∈C+ xa −

P

a∈C− xa = TzC, ∀C ∈ B

zC ≤ zC ≤ zC, ∀C ∈ B zC ∈ Z

B being an integral cycle basis of D

Benefit of Short Integral Cycle Bases

ARRIVAL/MATHEON Fall School 2006 Page 27 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Consider the instance of 10-PESP that is defined
• n this graph, where ℓ ≡ 7 and u ≡ 13

ARRIVAL/MATHEON Fall School 2006 Page 28 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

Minimum Cycle Basis Problems

O(n)

N P-h.

? ? ? 2-basis

strict fund TUM weak fund integral undirected directed

O(mAPSP) O(mnAPSP)

Deo, Krishnamoorthy, and Prabhu (1982), Kavitha, Mehlhorn, Michail, Paluch (2004), L. and Rizzi (2006), Hariharan, Kavitha, Mehlhorn (2006)

Contents

ARRIVAL/MATHEON Fall School 2006 Page 29 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Contents

ARRIVAL/MATHEON Fall School 2006 Page 29 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Exercises

ARRIVAL/MATHEON Fall School 2006 Page 30 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Taking precautions for vehicle scheduling within

the PESP. . .

• Relaxing earlier decisions that were made during

line planning within the PESP. . .

• Characterization of periodic tension vectors x. . .
• Non-integral cycle bases are inappropriate for pe-

riodic timetabling. . .

Hint: Here, every arc is contained in at least two circuits!

• A minimum cycle basis that is non-integral. . .
• Robustness aspects of timetabling. . .

Contents

ARRIVAL/MATHEON Fall School 2006 Page 31 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Contents

ARRIVAL/MATHEON Fall School 2006 Page 31 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

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

Conclusions

ARRIVAL/MATHEON Fall School 2006 Page 32 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Periodic timetabling is performed by most railway

companies

Conclusions

ARRIVAL/MATHEON Fall School 2006 Page 32 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Periodic timetabling is performed by most railway

companies

• The PERIODIC EVENT SCHEDULING PROBLEM

(PESP) is a very rich model for periodic timetabling

Conclusions

ARRIVAL/MATHEON Fall School 2006 Page 32 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Periodic timetabling is performed by most railway

companies

• The PERIODIC EVENT SCHEDULING PROBLEM

(PESP) is a very rich model for periodic timetabling

• The resulting IPs are small but hard to solve

Conclusions

ARRIVAL/MATHEON Fall School 2006 Page 32 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Periodic timetabling is performed by most railway

companies

• The PERIODIC EVENT SCHEDULING PROBLEM

(PESP) is a very rich model for periodic timetabling

• The resulting IPs are small but hard to solve
• Short integral cycle bases are of help

Conclusions

ARRIVAL/MATHEON Fall School 2006 Page 32 of 32

DFG Research Center MATHEON Mathematics for key technologies Modelling, simulation, and optimization of real-world processes

• Periodic timetabling is performed by most railway

companies

• The PERIODIC EVENT SCHEDULING PROBLEM

(PESP) is a very rich model for periodic timetabling

• The resulting IPs are small but hard to solve
• Short integral cycle bases are of help
• Mathematical Optimization has just entered the

practice of service design in public transport. . .