Network Models Minimum spanning tree

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Network ModelsMinimum spanning tree

• Operational Research
• By
• TMJA Cooray

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Network definitionsEssential terms

• Graph
• Nodes
• Arcs (branches)
• Directed networks
• Path
• Loop
• Cycle

• Directed loop (circuit)
• Connected network
• Tree
• Spanning tree

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What is a network?

• A graph in which every edge (or vertex) is
  assigned a weight is called a network.
• The weight of an edge might represent the time or
  cost ,length or capacity depending on the
  situation considered.
• Networks and directed networks are use in OR to
  solve variety of problems ,not only in the areas
  of transportation and communication, but also
  problems arising in the planning of production
  and construction projects.

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Applications

• There are many operational research situations
  which can be solved as network models.
• Some such examples are
• The design of an off shore natural gas pipe line
  network connecting the well heads in the gulf of
  Mexico to an inshore delivery point.
• (The objective of the model is to minimize the
  cost of constructing the pipeline.)

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• 2. The determination of the shortest route
  between two cities in an existing network of
  roads.
• 3. The determination of the maximum capacity (in
  tons per year) of a coal slurry pipe line network
  joining the coal mines in one city with the power
  plants in another city.
• 4. The determination of the minimum cost flow
  schedule from oil fields to refineries through a
  pipe line network.

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• 5. The determination of the time schedule for the
  activities of a construction project.
• The solutions of these situations and other
  similar situations (communication and
  transportation and power) can be accomplished
  through a variety of network optimization
  algorithms.
• For situation 1 -- Minimal spanning tree
• For situation 2 -- Shortest route algorithm.
• For situation 3 --Maximal flow algorithm
• For situation 4 Minimum cost capacitated
  network algorithm
• For situation 5 Critical path algorithm

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• A common practical problem that arises, in the
  design of communication and transportation
  networks is that of connecting a number of sites
  at minimum cost

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Example

• The vertices in the network N represent eight
  offshore oil wells. The edges represent
  possible connections that can be made between the
  wells. The weight on each edge is proportional to
  the cost of constructing that link. Design a
  network that will connect the wells at minimum
  cost

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f
b
2 2 4 5 3
a
d
h
4 6
3 3 7 3 6
g
c
4 1
e
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• Minimum Spanning Tree
• Prims algorithm is used to find the minimum
  spanning tree.
• Three different approaches
• Using set notations for disconnected trees and
  vertices until all are connected
• Using a sketch of the network for disconnected
  trees and vertices until all are connected
• 3. Using a table of weights of the network as
  shown in the next slide.

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Prims algorithm for finding a minimum weight
spanning tree.

• Given a connected network N, we build a sub graph
  T of N as follows
• 1.(Initialization step) select arbitrarily any
  vertex of N and put this vertex into T.
• 2.(Recursive step) Scan the edges joining a
  vertex of T to a vertex not in T, select one such
  edge with minimum weight. Add this edge and its
  other end vertex to T .
• (Stop condition) Stop when all the vertices of N
  are in T.

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• Prims algorithm (set theory approach)
• Iteration 1 Select any vertex arbitrarily (say
  a) and put this into the tree T1. Let T1a,
  b,c,d,e,f,g,h,E1?
• Scan the edges incident with the vertices of T1
  ,and select the one with minimum weight .the
  minimum weight on the edges incident with a is 2
  and theres just one edge with weight 2,the edge
  ab.
• Add the vertex b and the edge ab to the
  tree .
• Iteration 2. Let T2a,b, c,d,e,f,g,h
• E2 ab and the length is 2

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• Iteration 3. Similarly ,select d, since bd has
  the minimum weight Let T3a,b,d
  c,e,f,g,h
• E3 ab,bd and the length is 224
• Iteration 4. There is more than one edge which
  has same weight. Edges ac, dc and dg each have
  minimum weight 3. All ties can be decided
  arbitrarily . Suppose we select ac.
• Let T4a,b,d,c, e,f,g,h
• E4 ab,bd,ac and the length is 437
• It can be E4 ab,bd,dc and the length is
  223 or
• E4 ab,bd,dg and the length is 223 (hence
  the spanning tree will not be unique.

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• Iteration 5. Let T5a,b,d,c,g, e,f,h
• E5 ab, bd, ac,dg and the length is 7310
• Iteration 6. Let T6a,b,d,c,g,e, f,h
• E6 ab, bd, ac,dg,ge and the length is 10111
• Iteration 7. Let T7a,b,d,c,g,e,f h
• E7 ab, bd, ac,dg,ge,df with length is11415
  Iteration 8. Let T8a,b,d,c,g,e,f,h ?
• E8 ab, bd, ac,dg,ge,df,fh with length is
  15318

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Minimum spanning tree
f
b
2
4 3
2
d
h
a
3
g
3
1
c
(e
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Put a into the tree. Highlight column aand
delete row a
new edge is ab. It gives the min distance from a

• 
  

a b c d e f g
h
a 2 3 4
b 2 2 6
c 3 3 4
d 4 2 3 7 4
3 6
e 4 7
1
f 6 4
5 3
g 3 1
5 6
h 6
3 6
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Delete row band highlight column b
. new edge is bd.

• 
  

a b c d e f g
h
b 2 2 6
c 3 3 4
2
d 4 3 7 4
3 6
e 4 7
1
f 6 4
5 3
g 3 1
5 6
h 6
3 6
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Min distance from the a,b,d is3. . new edge
is dg.
Delete row d and highlight column d

• 
  

a b c d e f
g h
c 3 3 4
d 4 2 3 7 4
3 6
e 4 7
1
f 6 4
5 3
g 1
5 6
3
h 6
3 6
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Min distance from the a,b,d ,g is1. . new
edge is ge.
Delete row g and highlight column g

• 
  

a b c d e f g
h
c 3 3 4
e 4 7
1
f 6 4
5 3
g 1
5 6
3
h 6
3 6
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Min distance from the a,b,d ,g,e is3. . new
edge is dc.
Delete row e and highlight column e

• 
  

a b c d e f g
h
c 3 3 4
e 4 7
1
f 6 4
5 3
3
h 6
3 6
20

Min distance from the a,b,d ,g,e is3. . new
edge is dc.
Delete row e and highlight column e

• 
  

a b c d e f g
h
c 3 3 4
f 6 4
5 3
3
h 6
3 6
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Min distance from the a,b,d ,g,e,c,f is4. .
new edge is ac or df.
Delete row f and highlight column f

• 
  

a b c d e f g
h
f 6 4
5 3
h 6
3 6
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Min distance from the a,b,d ,g,e,c,f,h
is3. . new edge is ac or fh.
Delete row h and highlight column h

• 
  

a b c d e f g
h
h 6
3 6
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Min distance from the a,b,d ,g,e,c,f,h
is3. . new edge is ac or fh.
Delete row h and highlight column h

• 
  

a b c d e f g
h
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