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1
15.082 and 6.855J
The Network Simplex Algorithm
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Calculating A Spanning Tree Flow
A tree with
supplies and
demands.
(Assume that all
other arcs have a
flow of 0)
1
3
6
4
5
2
7
1
6
4
1
3
3
What is the flow
in arc (4,3)?
2
See the animation.
3
What would happen if the flows in non
tree arcs were not 0?
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3
6
4
5
2
7
1
6
4
1
3
Suppose that non
tree arcs had a non
zero flow.
How
would this change
the computations?
3
1
3
2
2
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What would happen if the flows
in nontree arcs were not 0?
1
3
6
4
5
2
7
1
3
6
4
1
3
Adjust the
supplies/demands.
They will be
interpreted as
excesses and
deficits
.
2
2
1
3
4
2
6
The compute flows
as in the previous
method; e.g., what is
the flow in (4,3)?
0
5
What would happen if the
flow were negative?
If the direction of (4,3)
were reversed, the flow
in (3,4) would be
negative.
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3
6
4
5
2
7
1
3
6
4
1
3
2
3
64
43
A spanning tree flow
is
guaranteed
to satisfy the
supply/demand
constraints.
It may
violate an upper or lower
bound.
2
A spanning tree flow is
called
feasible
if it
satisfies its upper and
lower bound.
Otherwise,
it is
infeasible
.
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Basic Flows
A
basis structure
consists of a spanning tree T, a set L of
arcs, and a set U of arcs, such that T
∪
L
∪
U = A.
For each (i,j)
∈
L, x
ij
= 0.
For each (i,j)
∈
U, x
ij
= u
ij
.
The arc flows in T are selected so that each node satisfies its
supply/demand constraint.
The basis structure is
feasible
if the arc flows also satisfy the
upper and lower bounds.
It is possible for a basis structure to be infeasible.
In fact, this
is normally the case in the dual simplex algorithm.
7
Another way of calculating flows in arcs
Case 1.
If (i,j) is not in the tree, then x
ij
= 0.
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This note was uploaded on 03/15/2010 for the course IE 505 taught by Professor Yok during the Spring '10 term at Galatasaray Üniversitesi.
 Spring '10
 yok

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