Vehicle Scheduling Problems - co-at-work.zib.deco-at-work.zib.de/berlin/download/CD/Talks/09M1-VSP-051012.pdfRalf Borndörfer DFG Research Center MATHEON “Mathematics for key technologies”

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

Ralf Borndörferjoint work with

Martin Grötschel Andreas Löbel Steffen Weider

Block Course at TU Berlin "Combinatorial Optimization at Work“

October 4 – 15, 2005

09M1Vehicle Scheduling Problems

Ralf Borndörfer

2

Outline

Introduction

Single Depot Vehicle Scheduling

Multiple Depot Vehicle Scheduling

Lagrangean Relaxation

ExtensionsTrip Shifting

Integrated Vehicle and Duty Scheduling

Trip Scheduling

Ralf Borndörfer

3

BVG (Berlin)

Ralf Borndörfer

4

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.)

Ralf Borndörfer

5

Vehicle Utilization

Ralf Borndörfer

6

"Camel Curve"

68

Ralf Borndörfer

7

Interlining

too short to turntoo long to wait

best choice

Ralf Borndörfer

8

Outline

Introduction






Trip Scheduling

Ralf Borndörfer

9

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

Ralf Borndörfer

10

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

Leonid V. Kantorovich Tjalling C. Koopmans

Ralf Borndörfer

11


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

Ralf Borndörfer

12

SolutionCost = 17


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

1 2 3

34

33

76 7

8 10

1 2 3

Buses

Trips

Ralf Borndörfer

13


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

1 2 3

34

33

76 7

8 10

1 2 3

Buses

Trips

SolutionCost = 16

Ralf Borndörfer

14

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

Ralf Borndörfer

15

Mathematical Models(Assignment Problem)

Graph Theoretic Model Integer Programming Model

Linear Programming Relaxation

5 4 0

34

33

76 7

8 10

7 8

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}

+ ++ + ++ + +

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

≥∈

KK

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

9

Ralf Borndörfer

16

"Shadow Prices"

Ralf Borndörfer

17


The „Successive Shortest Path“ Algorithm

0 0 033

443

3

33 7

7

66

77 8

81010

0 0 0

Ralf Borndörfer

18


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 = 0

Partial sol.cost = 0

Buses

Trips

Ralf Borndörfer

19



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

Ralf Borndörfer

20



0 0 030

40-3

0

30 7

4

62

74 8

5106

3 3 4

00 0 0

000 Buses

Trips

Boundcost = 10Partial sol.cost = 3

Ralf Borndörfer

21



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


Ralf Borndörfer

22



0 0 0-30

403

0

-30 7

4

62

74 8

5106

3 3 4

00 0 0

000 Buses

Trips


Ralf Borndörfer

23



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

Ralf Borndörfer

24



5 4 030

-403

1

-30 7

3

61

70 -8

0101

7 8 9

00 0 0

000 Buses

Trips


Ralf Borndörfer

25

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

Ralf Borndörfer

26

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

Ralf Borndörfer

27

SPEC

Ralf Borndörfer

28

Outline

Introduction






Trip Scheduling

Ralf Borndörfer

29

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.

Ralf Borndörfer

30

Graph Theoretic Model

Ralf Borndörfer

31

Example: Regensburg

Ralf Borndörfer

32

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

Ralf Borndörfer

33

Timelines

d

d

Ralf Borndörfer

34

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

Ralf Borndörfer

35

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.

Ralf Borndörfer

36

Outline

Introduction






Trip Scheduling

Ralf Borndörfer

37


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)

Ralf Borndörfer

38

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

Ralf Borndörfer

39

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− → →∞%

Ralf Borndörfer

40

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)

⇔

⇔

Ralf Borndörfer

41

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

Ralf Borndörfer

42

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

43

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

44

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

45

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

Ralf Borndörfer

46

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

Ralf Borndörfer

47

Lagrangean Relaxation Algorithm

Ralf Borndörfer

48

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

Ralf Borndörfer

49

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

Ralf Borndörfer

50

Outline

Introduction






Trip Scheduling

Ralf Borndörfer

51

Trip Shifting

Ralf Borndörfer

52

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

Ralf Borndörfer

53

Outline

Introduction






Trip Scheduling

Ralf Borndörfer

54

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

Ralf Borndörfer

55

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

Ralf Borndörfer

56

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

Ralf Borndörfer

57


Dual Problem: Find multipliers λ maximizing f

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

Note: Integrality relaxed

∈∈

=

=

−− +

=

6 4 4 4 4 4 4 4 44 7 4 4 4 4 4 4 4 4 48

1 4 4 44 2 4 4 4 43 1 4 4 44 2 4 4 4 43

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

Ralf Borndörfer

58

Outline

Introduction






Trip Scheduling

Ralf Borndörfer

59

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

Ralf Borndörfer

60

Graph Theoretic Model

6:306:31

7:30

8:30

9:30

Ralf Borndörfer

61

Integer Programming Model

00

0

min

0 , Vehicle flow

1 Timetabled trips

Frequencies

Pax volume

Depot capacities


∈

∈

− = ∀

≤ ∀

≥ ∀

≥ ∀

≤ ∀

∈ ∀

∑∑

∑ ∑

∑∑

∑∑

∑∑

∑

l

l

l

l

l

l

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

62

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

63

Thank you for your attention.

Documents

Vehicle Scheduling Problems - co-at-work.zib.deco-at-work.zib.de/berlin/download/CD/Talks/09M1-VSP-051012.pdfRalf Borndörfer DFG Research Center MATHEON “Mathematics for key technologies”