Vehicle Circulation and the Hungarian Method

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Vehicle Circulation and the Hungarian Method. Martin Grötschel joint work with Ralf Borndörfer Andreas Löbel Celebration Day of the 50th Anniversary of the Hungarian Method Budapest, October 31, 2005. About the assignment problem.

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### Vehicle Circulation andthe Hungarian Method

Martin Grötschel

joint work with

Ralf Borndörfer Andreas Löbel

Celebration Day of the 50th Anniversary of the Hungarian MethodBudapest, October 31, 2005

• The assignment problem is a mathematical problem. Mathematicians have spent an awful lot of time to create “real-life interpretations” that look like applications to “prove” that it is useful. And hence, the Hungarian Method is of no practical value.
• The “truth”, in fact is the other way around. Practitioners have “tuned” their applied problems in order to be able to employ the Hungarian Method.

Martin Grötschel

Contents
• What is vehicle circulation/scheduling?
• Single depot vehicle scheduling
• Multiple depot vehicle scheduling
• Extensions

Martin Grötschel

Contents
• What is vehicle circulation/scheduling?
• Single depot vehicle scheduling
• Multiple depot vehicle scheduling
• Extensions

Martin Grötschel

The ZIB Transportation Team,including former members:

Public Transport:

Ralf BorndörferFridolin KlostermeierChristian KüttnerAndreas LöbelSascha LukacMarc PfetschThomas SchlechteSteffen Weider

Online Transportation:

Norbert AscheuerPhilipp FrieseSven O. KrumkeDiana PoensgenJörg RambauLuis Miguel TorresAndreas TuchschererTjark Vredeveld

plus several master students

Martin Grötschel

Cost Recovery

Fares

Construction Costs

Network Topology

Velocities

Lines

Service Level

Frequencies

Connections

Timetable

Sensitivity

Rotations

Relief Points

Duties

Duty Mix

Rostering

Fairness

Crew Assignment

Disruptions

Operations Control

Planning in Public Transport(Product, Project, Planned)

multidepartmental

Departments

multidepotwise

Depots

multiple line groups

Line Groups

multiple lines

Lines

multiple rotations

Rotations

VS-OPT2

B15

IS-OPT

Martin Grötschel

APD

DS-OPT

VS-OPT

BS-OPT

AN-OPT

B1

B3

B1

The ZIB Transportation Teamspin-off companies

Intranetz:

Fridolin Klostermeier

Christian Küttner

Norbert Ascheuer

LBW:

Ralf Borndörfer

Andreas Löbel

Steffen Weider

Martin Grötschel

What is vehicle circulation/scheduling?
• We are given a transportation system in a region.
• It is subdivided by carrier/vehicle types (busses, trams, subways, planes, ships…).
• For each carrier type, a (daily, weekly, or monthly,..) timetable (the scheduled/timetabled trips) is given.
• Task: Assign the available vehicles to the scheduled trips of the timetable such that some objective function is optimized and a (usually large) system of side constraints is satisfied.

Martin Grötschel

• Each vehicle (usually) has a home base. In colloquial language this is called its depot. Transportation professionals have to be more precise. A depot consists of all indistinguishable vehicles that have their home base in the same physical location.
• In most cases, a vehicle leaves its depot in the “morning” and returns to its depot in the “evening” of the planning period. Thus, every vehicle “circulates” along a tour of the region.
• The vehicle circulation problem is hence the task to find, for each available vehicle and for the given planning horizon, a tour such that all scheduled/timetabled trips are covered by exactly one tour and some objective is optimized and certain side constraints respected.

Martin Grötschel

• Minimize the number of vehicles that are necessary to cover all scheduled trips.
• Minimize the cost of the deadhead trips.(Deadhead trips are moves of a vehicle without passengers; a move can be just a break where the vehicle keeps waiting in a parking lot.)
• A combination of these two.
• Interlining
• Turns
• Pull-in pull-out trips

Martin Grötschel

Leuthardt Survey(Leuthardt 1998, Kostenstrukturen von Stadt-, Überland- und Reisebussen, DER NAHVERKEHR 6/98, pp. 19-23.)

annual cost: 150 – 250 thousand US dollars per bus

Martin Grötschel

Vehicle Scheduling in Berlin

The transportation research group at ZIB has produced software with which the

• busses
• street cars, and
• subways

in Berlin have been scheduled.

A film shows some of the problems of bus scheduling:

Martin Grötschel

Vihicle Circulation Film

Martin Grötschel

Some Users

Martin Grötschel

Contents
• What is vehicle circulation/scheduling?
• Single depot vehicle scheduling
• Multiple depot vehicle scheduling
• Extensions

Martin Grötschel

Single Depot Vehicle Scheduling(Assignment Model)

depot (in the morning)

D

D

1

2

1

2

with starting time and location

timetabled trip

1

1

2

2

with ending time and location

4 timetabled trips

4

4

3

3

• A single depot:
• one location
• one bus type

3

3

4

4

depot (in the evening)

D

D

Martin Grötschel

Single Depot Vehicle Scheduling(Assignment Model)

D

1

D

D

2

3

D

4

1

2

1

2

3

D

D

1

2

4

1

1

2

2

D

1

D

2

3

4

D

1

D

2

3

4

4

4

3

3

3

3

3

D

D

1

2

4

4

4

3

D

D

1

2

4

D

D

4 timetabled trips plus

1 blue and 1 red

bus circulation

2 depot nodes foreach available bus

The assignment model of thesingle depot vehicle circulation problem

Martin Grötschel

But
• In the seventies the available computers were not able to solve large size assignment problems due to time and space problems.
• The Hungarian method was the algorithm of choice. There was nothing better.

Martin Grötschel

Problem Specific Size Reduction:HOT = Hamburger OptimierungsTechnik

HOT only looked atpeak times (about 7 a.m) and made heuristic (manual = interactive) choices toreduce the problem size.

~ 1975 beginning of code development

~ 2003 last installations replaced

Martin Grötschel

All other companies did basically the same;but it is hard to find out what they really did.

Martin Grötschel

Surprise
• Due to expertise and practical experience, the HOT specialists were able to come up with very good (and often almost optimal) solutions when “number of busses” was the major objective.

Martin Grötschel

Single Depot Vehicle Scheduling

1

2

3

4

3

6

7

3

3

7

8

10

1

2

3

• The Assignment Problem
• Input: 3 Buses, 3 trips, costs
• Output: cost minimal assignment

Buses

Solution

Cost = 20

Trips

Martin Grötschel

Single Depot Vehicle Scheduling

1

2

3

4

3

6

7

3

3

7

8

10

1

2

3

• The Greedy-Heuristik
• heuretikos (gr.): inventiveheuriskein (gr.): to find

Buses

Solution

Cost = 17

Trips

Martin Grötschel

Single Depot Vehicle Scheduling

1

2

3

4

3

6

7

3

3

7

8

10

1

2

3

• The Greedy-Heuristik
• heuretikos (gr.): inventiveheuriskein (gr.): to find

Busses

Solution

Cost = 16

Trips

Martin Grötschel

The "Primal Problem"

Minimum Cost

Assignment

The "Dual Problem"

Maximum Sales Revenues

Single Depot Vehicle Scheduling

5

4

0

4

3

6

7

3

3

7

8

10

7

8

9

Buses

Optimum

Cost = 15

Trips

Martin Grötschel

Graph Theoretic Model

Integer Programming Model

Linear Programming Relaxation

Mathematical Models(Assignment Problem)

2

1

3

4

7

3

6

3

3

10

8

7

3

2

1

Martin Grötschel

Single Depot Vehicle Scheduling

0

0

0

4

4

3

3

6

6

7

7

3

3

3

3

7

7

8

8

10

10

0

0

0

• The „Successive Shortest Path“ Algorithm

Martin Grötschel

Single Depot Vehicle Scheduling

0

0

0

0

0

0 0

0

0

0

4

4

3

3

6

6

7

7

3

3

3

3

7

7

8

8

10

10

0

0

0

• The „Successive Shortest Path“-Algorithm

Buses

Bound

cost = 15

Partial sol.

cost = 0

Trips

Martin Grötschel

Single Depot Vehicle Scheduling

0

0

0

0

0

0 0

0

0

0

4

0

3

0

6

2

7

4

3

0

3

0

7

4

8

5

10

6

0

0

0

• The „Successive Shortest Path“ Algorithm

Buses

+0

+0

+0

Bound

cost = 10

Partial sol.

cost = 3

Trips

+3

+3

+4

Martin Grötschel

Single Depot Vehicle Scheduling
• The „Successive Shortest Path“ Algorithm

0

0

0

0

0

0 0

Buses

0

0

0

Bound

cost = 10

Partial sol.

cost = 3

4

0

3

0

6

2

7

4

3

0

-3

0

7

4

8

5

10

6

Trips

3

3

4

Martin Grötschel

Single Depot Vehicle Scheduling
• The „Successive Shortest Path“ Algorithm

0

0

0

0

0

0 0

Buses

0

+0

+0

0

0

+0

Bound

cost = 10

Partial sol.

cost = 6

4

0

3

0

6

2

7

4

3

0

-3

0

7

4

8

5

10

6

Trips

3

+0

3

+0

4

+0

Martin Grötschel

Single Depot Vehicle Scheduling
• The „Successive Shortest Path“ Algorithm

0

0

0

0

0

0 0

Buses

0

0

0

Bound

cost = 10

Partial sol.

cost = 6

4

0

-3

0

6

2

7

4

-3

0

3

0

7

4

8

5

10

6

Trips

3

3

4

Martin Grötschel

Single Depot Vehicle Scheduling
• The „Successive Shortest Path“ Algorithm

0

0

0

0

0

0 0

Buses

0

+4

+5

0

0

+0

Bound

cost = 15

Partial sol.

cost = 15

4

0

-3

0

6

2

7

4

-3

0

3

0

7

4

8

5

10

6

Trips

3

+4

3

+5

4

+5

Martin Grötschel

Single Depot Vehicle Scheduling
• The „Successive Shortest Path“ Algorithm

0

0

0

0

0

0 0

Buses

5

4

0

Bound

cost = 15

Partial sol.

cost = 15

-4

0

-3

0

6

1

7

0

3

0

3

1

7

3

-8

0

10

1

Trips

7

8

9

Martin Grötschel

Single Depot Vehicle Scheduling

5

4

0

4

0

3

0

6

1

7

0

3

0

3

1

7

3

8

0

10

1

7

8

9

• The „Successive Shortest Path“ Algorithm
• Path Search
• Solution + Proof
• Efficient

Buses

Bound

cost = 15

Solution

cost = 15

Trips

Martin Grötschel

SPEC

Andreas Löbel

Martin Grötschel

Contents
• What is vehicle circulation/scheduling?
• Single depot vehicle scheduling
• Multiple depot vehicle scheduling
• Extensions

Martin Grötschel

Vehicle Scheduling
• InputTimetabled and deadhead tripsVehicle types and depot capacitiesVehicle costs (fixed and variable)
• OutputVehicle rotations
• ProblemCompute rotations to cover all timetabled trips
• GoalsMinimize number of vehiclesMinimize operation costsMinimize line hopping etc.

vehiclecirculationsrotationsblocksschedules

Martin Grötschel

Graph Theoretic Model

pull-out trips

pull-in trips

timetabled trips

Martin Grötschel

Example: Regensburg

Map deleted

Martin Grötschel

Vehicle Scheduling

Depot capacities: soft upper limits

Fleet minimum: pull-in trips

No line changes: interlining trips

Peaks: pull-in/pull-out trips

Turning: turns

• Precise control at point, time, or trip
• Changes of vehicles, lines, modes, turning, etc.
• Automatic generation of pull-in/pull-out trips
• Depot capacities (soft)

Martin Grötschel

Timelines

d

d

Needs Fow model

assignment model will be too large

Martin Grötschel

Theoretical Results
• Observation: The LP relaxation of the Multicommodity Flow Problem does in general not produce integeral solutions.
• Theorem: The Multicommodity Flow Problem is NP-hard.
• Theorem (Tardos et. al.): There are pseudo-polynomial time approximation algorithms to solve the LP-relaxation of Multicommodity Flow Problems which are faster than general LP methods.

Martin Grötschel

Lagrangean Relaxation

f3

x2

x1

f2

f4

P(A,b)P(B,d)

f

P(A,b)

f1

x3

x4

Martin Grötschel

Bundle Method(Kiwiel [1990], Helmberg [2000])

2

1

3

• Max

X polyhedral (piecewise linear)

f

Martin Grötschel

Primal Approximation
• Theorem

converges to a point

Martin Grötschel

(1)

(2)

(3)

Martin Grötschel

Bundle Method(IVU41 838,500 x 3,570, 10.5 NNEs per column)

450

400

350

300

250

200

bundle

volume

barrier

cascent

150

100

50

20

40

60

80

100

120

140

sec

Martin Grötschel

Lagrangean Relaxation I

Martin Grötschel

Lagrangean Relaxation I
• Subproblem: Min-Cost-Flow (single-depot)

Martin Grötschel

Lagrangean Relaxation II

Martin Grötschel

Lagrangean Relaxation II
• Subproblem: Several independent Min-Cost-Flows (single-depot)

Martin Grötschel

Heuristics
• Cluster First – Schedule Second
• "Nearest-depot" heuristic
• Lagrange Relaxation II + tie breaker
• Schedule First – Cluster Second
• Lagrange relaxation I
• Schedule – Cluster – Reschedule
• Schedule: Lagrange relaxation I
• Cluster: Look at paths
• Solve a final min-cost flow
• Plus tabu search

Martin Grötschel

Computational Results

Martin Grötschel

Vehicle Utilization

Martin Grötschel

"Camel Curve"

68

Martin Grötschel

Interlining

too short to turn

too long to wait

best choice

Martin Grötschel

Umlaufoptimierung

Umlaufoptimierung mit MICROBUS 2

Erzielte Einsparungen durch die Umlaufoptimierung:

Im Busbereich wurden bei einer Gesamtzahl von knapp über 200 Fahrzeugen 5 Busse eingespart.

Im Bahnbereich wurden aufgrund der fehlenden Leerfahrt- und Überholmöglichkeiten keine Fahrzeuge eingespart.

Slide of SWB

Martin Grötschel

Heiko Klotzbücher

4

26.02.2002

Vehicle Scheduling at ZIB

Martin Grötschel

Vehicle Scheduling at ZIB
• Ralf Borndörfer
• Andreas Löbel

Ramifications:

• Corinna Bönisch
• Ines Spenke
• Steffen Weider

Martin Grötschel

BVG (Berlin)

Martin Grötschel

Contents
• What is vehicle circulation/scheduling?
• Single depot vehicle scheduling
• Multiple depot vehicle scheduling
• Extensions

Martin Grötschel

Discussion/Extensions
• Properties
• Exploiting all degrees of freedom
• Vehicle mix
• Extensions
• Trip shifting  current work
• Multiperiod scheduling
• Periodic schedules
• Assimilation
• Balanced depot exchange
• Maintenance constraints
• Integration
• Vehicle and duty scheduling  current work
• Timetabling
• Line planning

Martin Grötschel

Trip Shifting

Martin Grötschel

The END