Vehicle routing problems with alternative paths Dominique Feillet - - PowerPoint PPT Presentation

vehicle routing problems with alternative paths
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Vehicle routing problems with alternative paths Dominique Feillet - - PowerPoint PPT Presentation

Vehicle routing problems with alternative paths Dominique Feillet University of Avignon ( moving soon to Ecole des Mines de Saint-Etienne ) Co-authors: T. Garaix, C. Artigues, D. Josselin 1 Outline Data modeling in vehicle routing problems


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Vehicle routing problems with alternative paths

Dominique Feillet University of Avignon (moving soon to Ecole des Mines de Saint-Etienne)

Co-authors: T. Garaix, C. Artigues, D. Josselin

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Outline

  • Data modeling in vehicle routing problems
  • Methodological impact of the introduction of

alternative paths

  • Practical impact on computing times and

quality of results

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Data collection from a Geographical Information System

  • Too many pieces of information delivered by

the GIS

  • every consistent portion of road is described
  • Usual approach for vehicle routing:
  • introduce a vertex for every important location

(depot, customer location…)

  • consider the best route between every pair of

vertices

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Limit

  • How to compute the best route between two

vertices when arcs are described with more than one attribute?

  • multicriteria shortest path problem
  • the solution is a set of Pareto efficient solutions
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Illustration: VRPTW

From St Olavs Gate To Grefsen-Kjelsas Route A: 7 kilometers (cost) 15 minutes Route B: 5 kilometers (cost) 20 minutes

A B

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Illustration: VRPTW

[0,20] [0,30]

(7,15)

[0,20] [0,30]

(5,20)

Route Olavs – Grefsen - Olavs Feasible, cost 14 kms Unfeasible

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Illustration: VRPTW

[0,20] [0,40]

(7,15)

[0,20] [0,40]

(5,20)

Route Olavs – Grefsen - Olavs Feasible, cost 14 kms Feasible, cost 10 kms

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Multigraph modeling

[0,20] [0,40]

(7,15) (5,20)

Multigraph G=(V,A) V = set of important locations An arc (i,j)k exists in A for every pair of vertices (i,j) and every Pareto efficient path Pk between i and j

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Methodological impact

  • Construction of the graph
  • difficult to find the set of Pareto efficient paths
  • multicriteria shortest path problem
  • the set might be of very large size
  • in practice
  • one can expect a limited number of attributes on arcs
  • one can expect a set of limited size due to correlations
  • one can consider a subset of the efficient set (possibly

user-defined)

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Methodological impact

  • Finding the sequence of arcs when the

sequence of vertices is known is NP-hard

Called Fixed Sequence Arc Selection Problem (FSASP) = Multidimensional Multiple Choice Knapsack Problem Addressed as a shortest path problem with resource constraints (in an acyclic graph)

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Methodological impact

  • Three complex decisions when solving

vehicles routing on multigraphs

  • assign customers to vehicles
  • sequence customers
  • select arcs (FSASP)
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Methodological impact

  • Heuristic / Metaheuristic
  • Any algorithm can be applied, accepting that

evaluating the cost and the feasibility of a solution involves the solution of a FSASP

  • Integer programming
  • Classical models can be adapted, with new

decision variables (new flow variables for new arcs…)

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Computational impact

  • Set of random Euclidean instances for a Dial-

A-Ride problem

* : generated from a simple graph with 10% of additional arcs about 30% cheaper and slower ; results in a graph with up to 10 times more arcs Simple graph Best insertion Exact Multigraph* Best insertion Exact 0%

  • 17%

12%

  • 8%

VALUE

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Computational impact

  • Set of realistic instances (computed from a

GIS) for a Dial-A-Ride problem in a rural zone

* : up to 10 times more arcs, with arcs up to 50% cheaper or slower Simple graph Best insertion Exact Multigraph* Best insertion Exact 0%

  • 8%

12% 2% VALUE

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Some improvements

  • Best insertion
  • Find the best neighbor with the solution of a single

Shortest Path Problem with Resource Constraints

  • add a vertex for every possible insertion location
  • add a binary resource that imply to visit exactly one of

these vertices possible insertion locations: initial sequence:

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Some improvements

  • Exact method
  • Branch and Price
  • The multigraph only impacts:
  • the subproblem: extend labels with every outgoing arcs
  • the branching rule: select or forbid a successor (i.e., a

set of parallel arcs)

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Computational impact

  • Set of random Euclidean instances for a Dial-

A-Ride problem

* : generated from a simple graph with 10% of additional arcs about 30% cheaper and slower ; results in a graph with up to 10 times more arcs Simple graph Best insertion Exact Multigraph* Best insertion Exact 0%

  • 17%

12%

  • 8%

VALUE 100s 1000s 0s 10s TIME

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Computational impact

  • Set of realistic instances (computed from a

GIS) for a Dial-A-Ride problem in a rural zone

* : up to 10 times more arcs, with arcs up to 50% cheaper or slower Simple graph Best insertion Exact Multigraph* Best insertion Exact 0%

  • 8%

12% 2% VALUE 100s 1000s 0s 2s TIME

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Other improvements

  • Exact method
  • When solving the subproblem, replace set of arcs

(i,j)k with a single idealized arc (i,j)

  • c(i,j) = min {c((i,j)k)
  • t(i,j) = min {t((i,j)k)
  • A set of promising vertex-sequences are obtained
  • Solve the FSASP on these sequences and only

keep the feasible routes of negative reduced cost

  • If no route is obtained, solve the original subproblem
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Conclusion

  • Improving the completeness of the data is a

real issue

  • A very simple heuristic can beat an exact method

with the multigraph representation

  • An « automatic » adaptation of the algorithms

looks simple most of the times (once the FSASP tool is developed)

  • Some possibilities exist to really consider the

multigraph issue in the algorithms

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Perspectives

  • Evaluate this modeling in other contexts
  • Multimodal transportation (time, cost)
  • Transportation with congestion (time, cost)
  • Tourist tours (time, scenic interest)
  • Implement more efficient algorithms