multi depot vehicle routing

7/27/2019 Multi Depot Vehicle Routing

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MultipleDepot Vehicle

RoutingProblem


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Contents

Where the Problem Comes From

Introduction

VRP Description

MDVPR Description

Motivation

Abstraction (Problem Formulation)

NP Proof

NP-C Proof

Strong or Weak?


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Where the Problem

Comes From Each day at Sears Home Appliance Repair, a

fleet of technicians must have routes made forthem for the next day in order to service

customers


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Introduction

Vehicle Routing Problem (VRP)

Originally formulated as The Truck

Dispatching Problem by Dantzigand R.H. Ramser, 1959

Routes must be made for multiplevehicles to drop off goods orservices at multiple destinations,constrained on total distance (butcould be some other cost).


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VRP

Depot

LegendService Destination


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VRP

Depot

Route1

Route3

Route2


Route (Path)


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MDVRP

Multiple Depot Vehicle Routing Problem (MDVRP)

Variant of VRP

Same as VRP but with more than one depot

Depot

Depot

Depot

Depot



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MDVRP

A Solution might look something like this

Notice that solutions allows revisiting depots

Depot

Depot

Depot

Depot


Route1

Route2

Route3 Route

4


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Motivation

Real-world applicable:transportation, distribution, and

logistics [1]Appliance Repair

Parcel Delivery

Good routes save money More competitive businesses

Savings passed down to the buyer

Morally, we should save resources


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Abstraction

MDVRP problem can be modeled in terms of aGraph with weighted edges

Vertices are service destinations and depots

Edges connect any two vertices and has someweight

There is one vehicle per depot


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MDVRP Problem

Formulation Given

Directed Graph G=(V,E)

S = { all service destinations }

D = { all depots } V = S D

E = { weighted positive cost between any two distinct v V }

W(e), is the weight for edge e E

Question

Does there exist a set of closed walks C, such that,

s S implies s c, for some c C,

AND sum{ W(c) }, c C, is less than or equal to somek?


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NP Proof

MDVTP can be answered by yes OR nomaking it a decision problem

A witness can be provided (the set containing

closed walks C) which we can verify in polynomialtime with respect to k to have the followingproperties:

s S implies s c, for some c C,

sum{ cost(c) },

c

C, is less than or equal to somek

Simple iteration through S and C will suffice


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NP-Complete Proof

Show that MSVRP is NP (last slide)

Show that a polynomial transformation from someknown NP-C problem to MSVRP exists

Traveling Salesman Problem (TSP) will be used


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TSP

Given

A undirected graph G=(V,E)

V = { all cities }

E = { weighted postive cost between any two distinctv V }

W(e), is the weight for edge e E

Question

Is there a Hamiltonian Cycle C with sum { W(c) },c C, less than or equal to some k?


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Construction

Construct an instance of MDVRP for eachinstance of TSP such that

MDVRP answers yes iff TSP answers yes

MDVRP answers no iff TSP answers no


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Transformation

For an instance of TSP: G=(V,E) and k

v V create vin Vin and vout Vout

Vin Vout = V

e with endpoints vi and vj create a directed edgefrom the corresponding vi out to vj in and a directededge from vj out to vi in such that |E| = 2|E|

Now create |V| edges with weight k going fromeach vin to its corresponding vout so the new |E| =2|E| + |V|

k = k(|V| + 1)

Randomly select one element of V to be D so that |D|= 1 and all other elements of V are in the set S so S D= V


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Polynomial Sized

Reduction The G(V,E) and k are created from G(V,E) and k

|V| = 2|V| so vertex creation is polynomial withrespect to V

|E| = 2|E| + |V| and since the maximumnumber of edges in a TSP is limited by |V|2, |E|= 2|V|2 + |V| so edge creation is polynomialwith respect to V

k is created in linear time so the reduction ispolynomial with respect to V


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Yes Instances

If a TSP returns yes a Hamiltonian Circuit was foundwith weight less than k

The MDVRP is always capable of following the

same graph as the TSP because the edges areidentical whether the graph is Euclidian or not.

5 6

7

Euclidian

i = 19

5 6

12

non-Euclidian

i = 24

7

7

6

6

5

19

19

19

Euclidian

k= 76

12

12

6

6

5

24

24

24

non-

Euclidian

k= 96


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No Instances

The TSP yields a no if the instance requires a vertexto be visited more than once or if it cannotcomplete with a weight less than or equal to k

In case a non-Hamiltonian cycle is required theMDVRP reduction will also fail because a v in to voutedge will be traversed more than once causing kto be exceeded.

5 5

i = 21

5 555

21 21 21

k=

84


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References

[1] G. B. Dantzig and R.H. Ramser."The Truck Dispatching Problem".

Management Science 6, 8091.1959

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