Flow in Network

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• Flow in Network

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Graph, oriented graph, network

• A graph G (V, E) is specified by a non empty set
  of nodes V and a set of edges E such that each
  edge a is identified by a pair of nodes (u, v).
• c
• a b
• d
• V 1, 2, 3
  E a, b, c, d
• a b (1, 2) c (1,
  3) d (2, 3)

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• A graph g is a sub graph of a graph G if all the
  nodes and all the edges of g are nodes and edges
  of G.
• c
  c
• a b
• d
• G
  g
  

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• A sub graph of a graph G including all the nodes
  of G is a partial graph of G.
• c
  c
• a b
• d
  d
• G
  g
  

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• A chain in a graph G is a sequence of distinct
  edges a1, a2, , ap
• such that there exist (p1) nodes u1, u2,
  , up1 where ai (ui, ui1).
• c
• a b
• d
• The sequence a, c is a chain.
  

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• A cycle in a graph G is a chain such that u1
  up1
• c
• a b
• d
• The sequence c, b, d is a cycle.
  

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• A graph G is connected if for all pair of nodes,
  there exists a chain linking them.
• c
• a b
• d
• This graph is connected.
  

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• A tree is a connected graph with no cycle
• Property A tree with n nodes includes
  exactly (n 1) edges

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• A partial tree ( spanning tree) of a connected
  graph G is a partial graph of G being a tree
• c
  c
• a b
• d
  d
• G
  partial tree
  

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• A fondamental cycle with respect to a partial
  tree is a cycle including an edge of the graph
  not included in the partial tree and edges of the
  tree.
• c
  c
• a b
• d
  d
• G
  partial tree

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• A fondamental cycle with respect to a spanning
  tree is a cycle including an edge of the graph
  not included in the partial tree and edges of the
  tree.
• c
  c
• a b
  a
• d
  d
• G
  fondamental cycle
  

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• An oriented graph G (V, E) is specified by a
  non empty set of nodes V and a set of arcs E such
  that each arc a is identified by an ordered pair
  of nodes (u, v).
• a
  b
• 
  c d
• e
  f
• V 1, 2, 3, 4 E a, b, c, d,
  e, f
• a (1, 2), b (2, 4), c (2, 3), d
  (3, 2), e (1, 3), f (3, 4)

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• A non oriented graph obtained from an oriented
  graph G by eliminating the direction of the arc
  is denoted the corresponding graph.
• The notions of chain, cycle, connectedness, tree,
  spanning tree, and fondamental cycle for an
  oriented graph refer to the corresponding graph.
• A path in an oriented graph is a sequence of
  distinct arcs a1, a2, , ap being a chain where
  all the arcs are oriented in the same direction.
• A directed graph is simple if the nodes
  identifying each arc are distinct, and if there
  is no pairs of arcs specified by the same pair of
  nodes.

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• A network is a connected oriented graph in which
  a flow can move over the arcs. Each arc (i, j)
  est characterized as follows
• a capacity dij corresponding to
  an upper bound on the flow in the
• arc
• a lower bound lij on the flow in
  the arc
• Moreover, 0 lij dij
• 
  0, 2
  2, 9
• 
  0, 6 3, 4
• 
  0, 4 1, 6
• At each arc (i, j) is associated a
  pair lij , dij.

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Minimum cost flow model

• Consider a network where the following attributs
  are associated with each arc (i, j)
• dij the capacity of the arc
• lij the lower bound on the flow in
  the arc
• cij flow unitary cost in the arc
• xij the variable including the
  value of the flow in the arc
• Furthermore, associate the following two sets of
  nodes with each node i

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• Exemple
• B1 F , B2 1, 3, B3
  1, 2, B4 2, 3
• P1 2, 3, P2 3, 4,
  P3 2, 4, P4 F
• 
  0, 2 2, 9
• 
  0, 6 3, 4
• 
  0, 4 1, 6

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• The minimum cost flow problem is to determine how
  to send a fixed quantity of flow v from a source
  to a destination along the arcs
  of the network satisfying the lower bound and the
  capacity constraints of the arcs in order to
  minimize the total cost.
• Constraints of the problem
• 1. lower bound and the capacity
  constraints of the arcs
• 2. flow conservation constraints
  (specific to network flow problems)
• For each node the total
  flow going in the node must be equal
• to the total flow going out of the
  node

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• For each node, the total flow going in the node
  must be equal to the total flow going out of the
  node.
• v

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• For each node, the total flow going in the node
  must be equal to the total flow going out of the
  node.
• 
  
• 
  v

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• For each node, the total flow going in the node
  must be equal to the total flow going out of the
  node.

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Model for the minimum cost flow problem

• The minimum cost flow problem is to determine how
  to send a fixed quantity of flow v from a source
  to a destination along the arcs
  of the network satisfying the lower bound and the
  capacity constraints of the arcs in order to
  minimize the total cost.

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• The matrix associated with the flow conservation
  constraints are denoted
• the nodes-arcs incidence matrix
• row i ?
  node i
• column (i, j) ? arc
  (i, j)

colonne (i, j)
ligne i
ligne j
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• Exemple of an incidence matrix

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• The nodes-arcs incidence matrix has the
  unimodularity property indicating
• that the simplex algorithm generates an
  integer solution for problem (MCF) when lij, dij
  et v are integer.

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Properties of the incidence matrix

• Now we show that the columns of any basis
  correspond to arcs in a spanning tree and vice
  versa.
• Theorem The incidence matrix A of a connected
  simple oriented graph having m nodes and n arcs
  has a rank equal to (m1).
• Proof First we show that
  Each column of A includes only one
  component equal to 1, another one equal to -1,
  and all the other elements are equal to 0.
• Hence if we sum up the rows of A, then we
  obtain a row vector having all its elements equal
  to 0.
• Consequently the rows of A are linearly
  dependent.
• Thus

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• Exemple of incidence matrix

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• Exemple of incidence matrix

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• Theorem Consider an incidence matrix A
  associated with a simple connected directed graph
  G including m nodes and n arcs. A square
• (m 1)x (m 1) sub matrix of A is non
  singular if and only if the arcs associated to
  its columns are those specifying a spanning tree
  of G.

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• Proof Let T be a spanning tree of G illustrated
  in red.

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• Proof Let T be a spanning tree of G illustrated
  in red. Denote by A(T) the
• mx(m?1) sub matrix A being the incidence matrix
  of T.

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• Proof Let T be a spanning tree of G illustrated
  in red. Denote by A(T) the
• mx(m1) sub matrix A being the incidence matrix
  of T.
• Since T is a simple connected graph, it follows
  from the preceding theorem
• that rank(A(T)) (m1).

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• Proof Let T be a spanning tree of G illustrated
  in red. Denote by A(T) the
• mx(m1) sub matrix A being the incidence matrix
  of T.
• Since T is a simple connected graph, it follows
  from the preceding theorem
• that rank(A(T)) (m1).
• Then all (m 1)x (m 1) sub matrices obtained
  by eliminating a row of A(T)
• is non singular.
• But these sub matrices
• are also sub matrices
• of A.

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• Let B be a (m 1)x (m 1) square non
  singular sub matrix of A.

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• Let B be a (m 1)x (m 1) square non
  singular sub matrix of A.

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• Let B be a (m 1)x (m 1) square non
  singular sub matrix of A.
• B is obtained by eliminating a row from an
  incident matrix of a spanning tree g of G.

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• Let B be a (m 1)x (m 1) square non singular
  sub matrix of A.
• B is obtained by eliminating a row from an
  incident matrix of a spanning tree g of G.
• This sub graph g is simple, connected,
  including (m 1) arcs.
• Then g is a
• spanning tree
• of G.

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• Theorem Consider an incidence matrix A
  associated with a simple connected directed graph
  G including m nodes and n arcs. A square
• (m 1)x (m 1) sub matrix of A is non
  singular if and only if the arcs associated to
  its columns are those specifying a spanning tree
  of G.

• Theorem The incidence matrix A of a connected
  simple oriented graph having m nodes and n arcs
  has a rank equal to (m1).

Any base from the incidence matrix is such that
its columns correspond to variables associated
with the arcs of a spanning tree, and vice versa.
The basic variables of a basic solution of a
(MCF) problem correspond to the arcs of a
spanning tree and vice-versa.
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The simplex method forthe minimum cost flow
problem

• We use the variant of the simplex method when the
  variables are bounded to solve the minimum cost
  flow problem (MCF)

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• To analyse one iteration of the simplex method,
  suppose that we have a basic feasible solution x
  of the problem.
• Then the non basic variables xij are such
  that
• xij 0 ou dij.
• The basic variables xij correspond to the
  arcs of a spanning tree T of the network.
• 
  

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• Entering criterion
• determine the relative costs of the
  variables xij
• that can be written as
• Moreover, for the basic variables
  associated to the arcs
• of the spanning tree T

ligne i
ligne j
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• In a tree
• ( arcs) ( nodes) 1.
• The preceding system includes (m 1)
  équations (associated to the arcs
  ) et m unknown pi (associated to the nodes i of
  T).

But since any sub matrix (m 1)x (m 1) of the
incidence matrix T is non singular, we can fix
the value of one multiplier and to evaluate the
others using the system.
The system of equations being triangular, it is
easy to show that the multipliers are evaluated
sequentially one by one.
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• In our exemple
• Let pt 0
• 
  

p11
pt0
ps5
1
p31
p24
3
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• Once the values of the multipliers are
  known, determine the relative costs
• of the non basic variables associated to the
  arcs that are not belonging to the spanning tree
  T.
• In our exemple
• 
  

pt0
p11
ps5
p31
p24
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• Once the values of the multipliers are
  known, determine the relative costs
• of the non basic variables associated to the
  arcs that are not belonging to the spanning tree
  T.

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• Once the values of the multipliers are
  known, determine the relative costs
• of the non basic variables associated to the
  arcs that are not belonging to the spanning tree
  T.
• In our exemple
• 
  

pt0
p11
ps5
x21 is admissible to increase
p31
p24
x3t is admissible to decrease
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Recall Step 1 Selecting the entering variable

• The criterion to select the entering
  variable must be modified to account for the non
  basic variables xj being equal to their upper
  bounds uj since these variables can be reduced.
• Hence, for an index
• if
  , it is interesting to increase xj
• if
  , it is interesting to
  decrease xj

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• Once the values of the multipliers are
  known, determine the relative costs
• of the non basic variables associated to the
  arcs that are not belonging to the spanning tree
  T.
• In our exemple
• 
  

pt0
p11
ps5
x21 is admissible to increase
p31
p24
x3t is admissible to decrease
Entering variable
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• Leaving criterion

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• 
  

0?
0?
4-?
4-?
4-?
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• 
  

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• Second iteration
• Entering criterion
• Determine the multipliers by
• solving the system

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• 
  
• Let pt 0

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• Determine the relavive costs of the
• non basic variables

p1 2
pt 0
ps 6
4
2
1
3
p2 5
p3 2
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4
2
1
3
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• Leaving criterion
• Then ? 2 since for this value,
• xs1 2 2 0
• x21 0 2 2 d21.

2-?
0?
2?
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• Leaving criterion
• Then ? 2 since for this value,
• xs1 2 2 0
• x21 0 2 2 d21.

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• Third iteration
• Entering criterion
• Determine the multipliers
• Determine the relative costs of
• the non basic variables

pt0
ps6
p13
2
2
1
3
p25
p32
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• The solution is optimal

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• The simplex version for the minimum cost flow
  problem is simplified because of the incidence
  matrix structure.
• There is a lot of degeneracy in flow problem.
  This may induce degenerate iteration (where the
  values of the variables dont change) sinply to
  modify the basis. There exists procedures to
  guide in chosing the basic solutions in order to
  reduce the number of degenerate iterations.

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Multiple sources and multiple destinations
s1
t1
Rest of network
s2
t2
sp
tq
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• Generate the network G1 (V1, E1 )

s1
t1
Rest of network
s2
t2
s
t
sp
tq
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• Generate the network G1 (V1, E1 )

s1
t1
Reste du réseau
s2
t2
s
t
sp
tq
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• Generate the network G1 (V1, E1 )

s1
t1
Rest of network
s2
t2
s
t
sp
tq
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• Generate the network G1 (V1, E1 )
• v
  
  v

s1
t1
Rest of network
s2
t2
s
t
sp
tq
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References

• M.S. Bazaraa, J.J. Jarvis, H.D. Sherali, Linear
  Programming and Network Flows, 3rd edition,
  Wiley-Interscience (2005), Chapter9
• F.S. Hillier, G.J. Lieberman, Introduction to
  Operations Research, Mc Graw Hill (2005),
  Section 9.7
• D. G. Luenberger,  Linear and Nonlinear
  Programming , 2nd edition, Addison-Wesley
  (1984), Chapter 5