multi depot vehicle routing
TRANSCRIPT
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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
LegendService Destination
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
LegendService Destination
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MDVRP
A Solution might look something like this
Notice that solutions allows revisiting depots
Depot
Depot
Depot
Depot
LegendService Destination
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