A Multicommodity Flow Approach for a Routing Problem with Time Synchronization Constraint Alain QUILLIOT

Loïc YON

{alain.quilliot, loic.yon}@isima.fr LIMOS, UMR 6158 - CNRS Université Blaise Pascal Clermont-Ferrand - France

CO 2004, 28-30 March

Introduction Related Problems MODT Problem Conclusion

Introduction

Modeling of Transportation Problems Configuration of a transportation offer for a bus company in an urban network I

cost for the bus company

I

customer satisfaction (Quality of Service)

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Introduction Related Problems MODT Problem Conclusion

Contents Introduction Related Problems Vehicle Routing Problem Network Design

MODT Problem Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Conclusion Numerical Experiments Alain QUILLIOT, Loïc YON

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Introduction Related Problems MODT Problem Conclusion

Public transportation

Quality of Service :

Routes : I

shape, duration, labor constraints

I

heterogeneous fleet : capacity, autonomy

I

areas : political, commercial

I

frequencies

Alain QUILLIOT, Loïc YON

I

demand

I

riding/travel time

I

frequency

I

price

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Introduction Related Problems MODT Problem Conclusion

Vehicle Routing Problem Network Design

School Bus Routing Problem "Providing public transportation service for students from their home to their school" I

equity

I

efficiency

I

lowest costs

A Multiobjective Optimization Approach to Urban School Routing: Formulation and Solution Method, BOWERMAN, HALL, CALAMAI, Transportation Research, 1995 A Computerized Approach to the New-York City School Bus Routing Problem, BRACA, BRAMEL, POSNER, SIMCHI-LEVI, IIE Transactions, 1997 Decision-aid Methodology for the School Bus Routing and Scheduling Problem, SPADA, BIERLAIRE, LIEBLING, 3rd STRC, 2003

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Vehicle Routing Problem Network Design

Dial-a-Ride Problem "Designing routes and schedules for vehicles to pick up or drop off people and/or goods" (Ex: disabled or elderly people transportation) I

best possible service

I

lowest costs

(PDVRP/VRPTW extension)

The Dial-a-Ride Problem: Variants, Modeling Issues and Algorithms, CORDEAU, LAPORTE, 4-OR, 2003

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Introduction Related Problems MODT Problem Conclusion

Vehicle Routing Problem Network Design

Capacity & Flow Assignment Problem "Designing topology for packet-switch networks with QoS constraints" I

capacity requirements

I

lowest costs

trade-off between investment costs and congestion Minimum Cost Capacity Installation for Multicommodity Network Flows, BIENSTOCK, CHOPRA, GÜNLÜCK, TSAI, Mathematical Programming, 1998 Capacity and Flow assignment of data networks by generalized Benders decomposition, MAHEY, BENCHAKROUN, BOYER, Journal of Global Optimization, 2001

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Passengers behavior pattern

SP-mode

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Realistic mode

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Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Notations G = (V , E ) directed graph E = A ∪ A where A is the set of "fast" arcs F integer flow f fractional multiflow P Sum(f ) = e∈E fe (ce ), (pe ) costs `(e) length of arc e k k (MAXe ), (Cmin ),(Cmax ) capacities e e o k , d k , D k for each commodity bvk for flow conservation constraints

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Multi-Origin/Destination Transportation Problem minimize c.F + p.Sum(f ) subject to

X

F ≤ MAX

(1)

Cmin ≤ f ≤ Cmax X Fe − Fe = 0

(2)

e∈ω − (v )

X e∈ω − (v )

∀v ∈ V

(3)

∀v ∈ V , ∀k ∈ K

(4)

∀e ∈ A

(5)

e∈ω + (v )

fek −

X

fek = bvk

e∈ω + (v )

dSum(f )e e ≤ Fe F ∈N

(6)

+

(7)

f ∈R Alain QUILLIOT, Loïc YON

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Application (1) K commodities (o k , d k , D k , tfk , Lkf ) I

D k : demand

I

tfk : maximal time for arrival at destination

I

Lkf : maximal length of the travel

1 depot : each tour begins and ends at this depot homogenous fleet of vehicles design and schedule a route system for a shuttle fleet I

minimal cost : c.F + p.f

I

demands

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Application (2) : Dynamic Graph

temporal discrete dynamic graph: 0 −→ N.δ each node is duplicate to correspond to an instant i ∈ {0, . . . , N} special fictitious nodes D and U arc [x, y ] ∈ E becomes [xr , yr +`∗ (e) ], ∀0 ≤ r ≤ N − `∗ (e) `(e) e where `∗ (e) = d δ Drawback : this graph can be really big.

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Several approaches

NP-hard I

Lagrangian relaxation

I

Hierarchical decomposition

I

Metaheuristics (GRASP, Tabu)

⇒ all but metaheuristics lead to an auxiliary problem

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Lagrangian Relaxations

I

integrity constraints

I

flow constraints (*)

I

coupling constraints (*)

if MAX = ∞, all optimal values are equal

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Master Slave Decomposition Heuristic Scheme Initializing F et f , feasible for MODT Stop ← false while not (Stop) do Solving MODTf Extracting dual vector λ (coupling constraints) Evaluating ∆ = c.F + p.Sum(f ) Solving|Improving P-aux(λ) if f is unchanged OR ∆ not improved during a given number of iterations then Stop ← true end if end while

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

P-aux(λ,f) (1)

minimize λ. dSum(f )e + p.Sum(f ) subject to

X e∈ω − (v )

Cmin ≤ f ≤ Cmax X fek − fek = bvk

(8) ∀v ∈ V , ∀k ∈ K

(9)

e∈ω + (v )

Sum(f )e ≤ Fe f ∈R

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∀e ∈ A

+

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

P-aux(λ,f) (2)

NP-hard (location problem) Special case of P-aux : f is a single flow (or all components of f can be treated separately) solved by finding iteratively a cycle improving the objective function in the residual graph : negative cycle canceling (CYGEN)

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Special Case : Aggregated flow

transportation application : commodity set really big D k small compared to total demand Assumption : each component f is routed on a single path only possible only if G is strong connected f → Sum(f ) ???

f ← Sum(f )

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Introduction Related Problems MODT Problem Conclusion

Problem definition Resolution Schemes Hierarchical decomposition Auxiliary problem

Metric inequalities for a flow g Given Z ⊂ V , OD(Z ) = {k ∈ K /ok ∈ Z et dk ∈ / Z} ∀Z ⊂ V ,

X e∈ω + (Z )

X

ge ≥

Dk

k∈OD(Z )

I

necessary conditions for Sum(f ) → f

I

cuts (Benders) injected in CYGEN and MSD scheme

On feasibility conditions of multicommodity flows in networks, ONAGA, KALUSHO, IEEE Transactions on Circuit Theory, 1971

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Introduction Related Problems MODT Problem Conclusion

Numerical Experiments

Numerical Experiments 20 instances G = (V , E ) 10 ≤ |V | ≤ 20 2|V | ≤ |E | ≤ 4|V | same results for MSD and relaxations on 12 instances I

Decomposition: 5%

I

Coupling constraints relaxation: 3%

I

Flow constraint relaxation: 3%

CYGEN : 4%

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Introduction Related Problems MODT Problem Conclusion

Numerical Experiments

Conclusion & Further work

I

instances are too small

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→ bigger instances with metaheuristics

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more tests on CYGEN with the aggregation process

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integration of travel patterns, elastic demand

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Introduction Related Problems MODT Problem Conclusion

Numerical Experiments

Clermont-Ferrand city 80

81

47 83

82

6 46

77 48 76

42 45

41

16

49 75 5

17

43 78

18

73

50

74

40

19

39

15

44

51

92 91 20

37

38 14

4

89

52

87

88

21 79

13 36 53

57

54 55

58 90

56

22

72

12

35

3 71

59

61

60 23

62

34

70

11

65

66

63

69 67

64

2

68

24

1

33 10 25

84

9 85 86 32 8

7

26

31

27

30 28 29

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Nodes Arcs Commodities

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