vehicle scheduling problems

Ralf Borndörfer DFG Research Center MATHEON “Mathematics for key technologies”Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)Löbel, Borndörfer & Weider GbR (LBW)

[email protected] http://www.zib.de/borndoerfer

Martin GrötschelBeijing Block Course

“Combinatorial Optimization at Work”

September 25 – October 6, 2006

09M1 LectureVehicle Scheduling Problems

Ralf Borndörfer

2

BVG (Berlin)

Ralf Borndörfer

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bus costs (DM) urban % regional

73.5 195,000

30,000

12,900

10,000

18,000

5,000

18,000

288,900

7.4

3.2

2.9

4.7

1.0

7.1

100.0

crew

depreciation

calc. interest

materials

fuel

repairs

other 34,000 7.2

total

%

349,600 67.5

35,400 10.4

15,300 4.5

14,000 3.5

22,200 6.2

5,000 1.7

475,500 100.0

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

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Vehicle Utilization

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"Camel Curve"

68

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Interlining

too short to turntoo long to wait

best choice

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Sea Freight(Koopmans 1965, 7 sources, 7 sinks, all sea links)

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Optimal Allocation of Scarce Resources(Nobel Price in Economics 1975)

Leonid V. Kantorovich Tjalling C. Koopmans

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Single Depot Vehicle Scheduling

The Assignment ProblemInput: 3 Buses, 3 trips, costs

Output: cost minimal assignment

1 2 3

34

33

76 7

8 10

1 2 3

SolutionCost = 20

Buses

Trips

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SolutionCost = 17


The Greedy-Heuristikheuretikos (gr.): inventiveheuriskein (gr.): to find

1 2 3

34

33

76 7

8 10

1 2 3

Buses

Trips

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The Greedy-Heuristikheuretikos (gr.): inventiveheuriskein (gr.): to find

1 2 3

34

33

76 7

8 10

1 2 3

Buses

Trips

SolutionCost = 16

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OptimumCost = 15


The "Primal Problem"Minimum Cost

Assignment

The "Dual Problem"Maximum Sales Revenues

"Shadow Prices"

5 4 0

34

33

76 7

8 10

7 8 9

Buses

Trips

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Mathematical Models(Assignment Problem)

Graph Theoretic Model Integer Programming Model

Linear Programming Relaxation

1 2 3

34

33

76 7

8 10

1 2

11 12 13

21 22 23

31 32 33

11 12 13

21 22 23

31 32 33

11 21 31

12 22 32

13 23 33

11 33

11 33

min 3 3 43 7 67 8 10

s.t. 111111

, , 0, , {0,1}

+ ++ + ++ + +

+ + =+ + =+ + =+ + =+ + =+ + =

≥∈

……

x x xx x xx x xx x xx x xx x xx x xx x xx x xx xx x

3

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"Shadow Prices"

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The „Successive Shortest Path“ Algorithm

0 0 033

443

3

33 7

7

66

77 8

81010

0 0 0

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The „Successive Shortest Path“-Algorithm

0 0 033

443

3

33 7

7

66

77 8

81010

0 0 0

00 0 0

000

Boundcost = 15Partial sol.cost = 0

Buses

Trips

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0 0 030

403

0

30 7

4

62

74 8

5106

0 0 0

00 0 0

000

+3+3

+0 +0 +0Bound

cost = 10Partial sol.cost = 3

+4

Buses

Trips

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0 0 030

40-3

0

30 7

4

62

74 8

5106

3 3 4

00 0 0

000 Buses

Trips


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0 0 030

40-3

0

30 7

4

62

74 8

5106

3 3 4

00 0 0

000

+0+0

+0 +0 +0

+0

Buses

Trips


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0 0 0-30

403

0

-30 7

4

62

74 8

5106

3 3 4

00 0 0

000 Buses

Trips


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0 0 0-30

403

0

-30 7

4

62

74 8

5106

3 3 4

00 0 0

000

+5+4

+5 +4 +0

+5

Buses

Trips

Bound= 15

= 15

costPartial sol.cost

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5 4 030

-403

1

-30 7

3

61

70 -8

0101

7 8 9

00 0 0

000 Buses

Trips


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The „Successive Shortest Path“ AlgorithmPath Search

Solution + Proof

Efficient


5 4 030

403

1

30 7

3

61

70 8

0101

7 8 9

Buses

Trips

Boundcost = 15Solution

cost = 15

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D D 1 2 3

D D 1 2 3

4

4

1

1

2

2

3

3

D

D

4

4

Single Depot Vehicle Scheduling(Assignment Model)

1

1

2

2

3

3

D

D

4

4

D D 1 2 3

D D 1 2 3

4

4

D D 1 2 3

D D 1 2 3

4

4

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SPEC

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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.

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Graph Theoretic Model

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Example: Regensburg

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Vehicle Scheduling

Definition + cost of deadhead tripsPrecise control at point, time, or tripChanges of vehicles, lines, modes, turning, etc.Automatic generation of pull-out-pull-in tripsMaintencance of all possible deadhead trips

Depot capacities (soft)

Fleet minimum: pull-in tripsNo line changes: interlining tripsTurning: turnsPeaks: pull-in/pull-out tripsDepot capacities: soft upper limits

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Timelines

d

d

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Integer Programming Model(Multicommodity Flow Problem)

0

min

0 , Vehicle flow

0 Aggregated flow

1 Timetabled trips

Depot capacities

{0,1} , Deadhead trips

− = ∀

= ∀

= ∀

≤ ∀

∈ ∀

−

∑∑

∑ ∑

∑∑ ∑∑

∑∑

∑

d dij ij

d ij

d dij jk

i kd dij jk

d i d kdij

d id

j dj

dij

c x

x x j d

x x j

x j

x d

x ij d

κ

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Theoretical Results

Observation: The LP relaxation of theMulticommodity Flow Problem is in general notinteger.

Theorem: The Multicommodity Flow Problem isNP-hard.

Theorem (Tardos et. al.): There are pseudo-polynomial time approximation algorithms to solve the LP-relaxation of Multicommodity FlowProblems which are faster than general LP methods.

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Lagrangean Relaxation

min

0

=

=

≥

T

bBx dx

c xAx

min (

0

max )+ −

=

≥

T b

dx

c x Ax

Bxλ

λ

(max min )+ −Ti ii

bc x Axλ

λ( )max fλ

λ

=

==

x1

x2

x3x4

P(A,b)

f1

f2

f3

f4fP(A,b)∩P(B,d)

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

Max

X polyhedral (piecewise linear)

λ

f

λ1

1fλ

λ2

f̂

λ3

μ μ μλ λ= + −T T( ) ( )f c x b Ax

+ = − −2

1ˆ ˆargmax ( )

2k

k k kuf

λλ λ λ λ

∈=ˆ ( ) : min ( )

kk Jf fμμ

λ λ

∈= + −T T( ) : min ( )

x Xf c x b Axλ λ

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Primal Approximation

Theorem

∈⇒ ( )k k Nx converges to a point { }∈ = ∈: ,x x Ax b x X

T( ) ( )k k kf c x b Axλ λ= + −k̂f

z

f

λ

1kf +

1kλ +

+∈

= ∑1k

kJ

x xμ μμ

α

+∈

= + −∑11ˆ ( )

k

k kJ

b Axu μ μ

μλ λ α

0 ( )kb Ax k− → →∞

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Quadratic Subproblem

− −2ˆ ˆmax ( )

2k

k kuf λ λ λ(1)

μ

λ λ

λ μ

− −

≤ ∈

2ˆmax2

s.t. ( ), for all

kk

k

uv

v f J

(2)

μ μ μ μμ μ

μμ

μ

α λ α

α

α μ

∈ ∈

∈

− −

=

≤ ≤ ∈

∑ ∑

∑

21ˆmax ( ) ( )

2

s.t. 1

0 1, for all

k k

k

kJ J

J

k

f b Axu

J

(3)

⇔

⇔

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

50

100

150

200

250

300

350

400

20 40 60 80 100 120 140 sec

450

bundlevolumebarrier

cascent

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Lagrangean Relaxation I

0

min

0 , Vehicle flow

0 Aggregated flow

1 Timetabled trips

Depot capacities


− = ∀

= ∀

= ∀

≤ ∀

∈ ∀

−

∑∑

∑ ∑

∑∑ ∑∑

∑∑

∑

d dij ij

d ij

d dij jk

i kd dij jk

d i d kdij

d id

j dj

dij

c x

x x j d

x x j

x j

x d

x ij d

κ

Ralf Borndörfer

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Lagrangean Relaxation I

Subproblem: Min-Cost-Flow (single-depot)

0

min

Vehicle flow0 Aggregated flow

1 Timetabled trips

Depot capacities


max 0+

= ∀

= ∀

≤ ∀

∈ ∀

⎡ ⎤⎛ ⎞− −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

−

∑∑

∑∑ ∑∑

∑∑

∑

∑ ∑d dij ij

d ij

d dij jk

d i d kdij

d id

j dj

dij

d dij jk

i k

c x

x x j

x j

x d

x ij d

x xλ

λ

κ

Ralf Borndörfer

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Lagrangean Relaxation II

0

min

0 , Vehicle flow

0 Aggregated flow

1 Timetabled trips

Depot capacities


− = ∀

= ∀

= ∀

≤ ∀

∈ ∀

−

∑∑

∑ ∑

∑∑ ∑∑

∑∑

∑

d dij ij

d ij

d dij jk

i kd dij jk

d i d kdij

d id

j dj

dij

c x

x x j d

x x j

x j

x d

x ij d

κ

Ralf Borndörfer

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Lagrangean Relaxation II

Subproblem: Several independent Min-Cost-Flows (single-depot)

0

min

0 , Vehicle flow

0 Aggregated flow

Timetabled tripsDepot capacities


max 1+

− = ∀

= ∀

≤ ∀

∈ ∀

⎛ ⎞−⎜ ⎟

⎝ ⎠

−

∑∑

∑ ∑

∑∑ ∑∑

∑

∑∑d dij ij

d ij

d dij jk

i kd dij jk

d i d k

dj d

j

dij

dij

d i

c x

x x j d

x x j

x d

x ij d

xλ

λ

κ

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Heuristics

Cluster First – Schedule Second"Nearest-depot" heuristic

Lagrange Relaxation II + tie breaker

Schedule First – Cluster SecondLagrange relaxation I

Schedule – Cluster – RescheduleSchedule: Lagrange relaxation I

Cluster: Look at paths

Solve a final min-cost flow

Plus tabu search

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Lagrangean Relaxation Algorithm

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Computational Results

BVG HHA VHH

depots 10 14

40

timetabled trips 25,000 16,000 5,500

deadheads 70,000,000 15,100,000 10,000,000

cpu mins 200 50 28

vehicle types 44

10

19

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UmlaufoptimierungUmlaufoptimierung mit MICROBUS 2

26.02.2002 4Heiko Klotzbücher

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

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Discussion

PropertiesExploiting all degrees of freedom

Vehicle mix

ExtensionsTrip shifting → current work

Multiperiod scheduling

Periodic schedules

Assimilation

Balanced depot exchange

Maintenance constraints

IntegrationVehicle and duty scheduling → talk of M. Grötschel

Timetabling → next topic

Line planning → research

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Trip Shifting

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Integer Programming Model(Multicommodity Flow Problem)

,

, ,

,

0,

min

0 , , Vehicle flow

0 Aggregated flow

1 Timetabled trips

Depot capacities

{0,1} , , Deadhead trips

− = ∀

= ∀

= ∀

≤ ∀

∈ ∀

−

∑∑

∑ ∑

∑∑ ∑∑

∑∑

∑

d dij ij

d ij

d dij jk

i kd dij jk

d i d k

dij

d i

dj d

j

dij

c x

x x j d

x x j

x j

x d

x ij d

ε εε

ε ε

ε εε ε

ε

εε

ε

ε

κ

ε

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Discussion

PropertiesExploiting all degrees of freedom

Vehicle mix

ExtensionsTrip shifting → current work

Multiperiod scheduling

Periodic schedules

Assimilation

Balanced depot exchange

Maintenance constraints

IntegrationVehicle and duty scheduling → talk of M. Grötschel

Timetabling → next topic

Line planning → research

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Efficiency

50

60

70

80

90

100

55 60 65 70 75 80 85 90Efficiency of Vehicle Schedule [%]

Effic

ienc

y of

Dut

ySc

hedu

le [%

] 53,44% 58,00% 63,90%

A

B C

(A) Sequential Scheduling(B) Duty Oriented Vehicle Scheduling(C) Integrated Scheduling

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Regional Scenarios

Max. duty time 8:00

6:00

6:00 2:00

2:00 6:00

6:00 2:00

2:00

hh:mm deadhead trip (1:00)timetabled trip

6:00

6:00 2:00

2:00

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δδ

=

= ∀= ∀ ∈= ∀ ∈

≤

∈

+

=∈

−

T

out

in

T(ISP) min(1a)(1b)(1c

0, depots ( ( )) 1, \ { } ( ( )) 1, \ {)

(2a)}

{0,

1

{0,1}

(2b)(3) 0

,1}

D

m n

c xN

d y

A

x Dx v v V tx v v V

yBy b

x Dyy

s

Cx

⎧= ⎨⎩

1, if deadhead covers duty element :

0, elsedt

d tC

⎧= ⎨⎩

1, if duty d covers duty elements :

0, sonstdt

tD

Integer Programming Model

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Lagrangean Relaxation

Dual Problem: Find multipliers λ maximizing f

Primal Problem: Find fractional solution(x,y)∈[0,1]mxn fulfilling (1), (2), (3)

Note: Integrality relaxed

∈∈

=

=

−− +

=

T T

y fulfills (2) and[0,

T T

fulfills (1) and{0,1 1]}

( ) :

max

: ( ) : (

mam x (n ( ))

)

inm

V D

yx

x

c C d D y

f

f f

xλ

λ

λ

λ

λ

λ

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Trip Scheduling

InputLines with lengths and speed profilesPassenger volume, frequency, and regularity profilesVehicle types with speeds and capacitiesDepot capacities

OutputTrips and vehicle rotations for all lines

ProblemSchedule trips and vehicle rotations to transport passengers

GoalsMaximize trip regularityMinimize number of passengers left behindMinimize number of vehicles

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Graph Theoretic Model

6:306:31

7:30

8:30

9:30

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Integer Programming Model

00

0

min

0 , Vehicle flow

1 Timetabled trips

Frequencies

Pax volume

Depot capacities


∈

∈

− = ∀

≤ ∀

≥ ∀

≥ ∀

≤ ∀

∈ ∀

∑∑

∑ ∑

∑∑

∑∑

∑∑

∑

dj

d j

d dij jk

i kdij

d idij

d ij H

d dij

d ij H

dj d

j

dij

x

x x j d

x j

x H

x d H

x d

x ij d

f

κ

κ

Ralf Borndörfer

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Computational Results

1 2

lines 11 38

2

frequency 12 38

pax/line/hour > 1,000 > 1,000

vehicles ~ 300 ~ 700

hours 3 3

vehicle types 2

Ralf Borndörfer DFG Research Center MATHEON “Mathematics for key technologies”Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)Löbel, Borndörfer & Weider GbR (LBW)

[email protected] http://www.zib.de/borndoerfer

Martin GrötschelBeijing Block Course

“Combinatorial Optimization at Work”

September 25 – October 6, 2006

09M1 LectureVehicle Scheduling Problems

The End

vehicle scheduling problems

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