OR 215
Network Flows
Spring 1999
M. Hartmann
THE SIMPLEX ALGORITHM FOR
GENERALIZED NETWORK FLOWS
Generalized Networks
Generalized Min-Cost Flow Problem
Sending Flows in Paths and Cycles
Augmented Trees (and Forests)
Calculating Simplex Multipliers
Pivoting in Generalized Networks
Data Structures
GENERALIZED NETWORKS
ij
i
j
ij > 0 is called the flow multiplier; for each unit of flow
leaving i along arc (i,j), ij units of flow arrive at node j.
EXAMPLE 1. Financial networks.
 node i : Bank i (in the US)
 node j : Bank j (in Great Britain)
ij = conversion factor from dollars to pounds
EXAMPLE 2. Energy networks.
node i : a power plant
 node j : the end of a high power transmission line
 ij = .90 to reflect a 10% loss in energy shipped
along the line.
EXAMPLE 3. Communication networks.
 node i : a communication node (sender)
node j: a communication node (receiver)
 ij < 1 reflects that communication may be lost in
transmission.
THE GENERALIZED MIN-COST FLOW PROBLEM
uij = capacity of arc (i,j).
cij = unit cost of shipping flow from node i to node j on (i,j).
xij = amount shipped on arc (i,j) (decision variables)
b(i) = supply/demand at node i (> 0 for a supply node)
ij = the multiplier of arc (i,j)
minimize
cijxij
subject to
 j xij

-
 k kixki
0  xij  uij

= b(i) for all i  N
for all (i,j)  A
 An arc (i,j) is a gainy arc
if
ij > 1.
 An arc (i,j) is a lossy arc
if
ij < 1.
 An arc (i,j) is a gainfree arc
if
ij = 1.
Note: The constraint matrices of generalized network
flow problems have at most two non-zero elements per
column.
GENERALIZED ASSIGMENT PROBLEM
EXAMPLE 4. Job shop scheduling.
 node Ji: job i
node Mj: machine j
 pij is the processing time of job i on machine j
p11
J1
M1
xij =
assigned
p12
to machine j
p13
J2
1 if job i is
Tj
=
processing time
alloted for machine j
M2
minimize
J3
subject to
cx
 j xij = 1 for all i
 i pijxij  Tj for all


j J
4
M3
x  0, integer
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2
2
1
1
2
3
4
3
(1,2) (1,3) (2,3) (3,4) (4,1)
1
1
1
0
0
-2
2
-1
0
1
0
0
3
0
-2
-1
1
0
4
0
0
0
-3
1
node-arc incidence matrix
Assumption: The rank of the node-arc incidence matrix
is n. (The generalized network is connected and is not
equivalent to an ordinary min cost network flow problem.)
PATHS AND CYCLES
Sending Flow along Paths: Suppose that P is a path
from node i to node j. If we send one unit of flow from i
and flow is conserved at all non-terminal nodes of P,
how much flow arrives at j?
Example 1: A directed path.
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1/4
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The multiplier of a directed path P is (P) = (i,j)P ij.
Example 2: A directed cycle.
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The multiplier of a directed cycle C is (C) = (i,j)C ij.
The multiplier of a directed cycle C is (C) = (i,j)C ij.
 A cycle is a gainy cycle
if
(C) > 1.
 A cycle is a lossy cycle
if
 (C) < 1.
 A cycle is a gainfree cycle
if
 (C) = 1.
A gainy cycle is also called a flow generating cycle.
A lossy cycle is also called a flow absorbing cycle.
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For a non-directed cycle, the multiplier of the cycle is the
product of the multipliers of the forward arcs divided by
the product of the multipliers of the reverse arcs.
Theorem. Let G = (N,A) be a generalized network, and
let C be a gainy or lossy directed circuit of G. Then there
is a unique way of sending flow in C so as to create one
unit of supply at node i of C.
AUGMENTED TREES
An augmented tree is a tree T plus an arc (v,w).
Suppose that the cycle C in an augmented tree is gainy
or lossy. Then T+(v,w) is a basis. (Note: the number of
nodes in an augmented tree equals the number of arcs.)
Proof: It suffices to show that the following procedure
uniquely determines the flow (WLOG node v is the root):
Step 1. Calculate the unique solution x such that xvw = 0
and the flows in T satisfy demands b(i) for i  v.
Step 2. Send flow from v to v around C (or its reversal) so
as to satisfy the suppy/demand constraint at v.
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Corollary. A 2-tree is dependent (a tree plus 2 arcs)
BASES FOR GENERALIZED NETWORKS
An augmented forest is a subgraph of n arcs such that
each component is an augmented tree.
Theorem. A basis an augmented forest with the additional property that no circuit of the augmented forest is
gainfree.
Proof: We leave as an exercise that there is some
augmented tree in G with a gainy or lossy cycle. Thus
some basis has n arcs, and thus all bases have n arcs.
Suppose that H is a basis. If there is some component H'
with more nodes than arcs, then H' is dependent. Thus
each component has the same number of arcs as nodes,
and thus each component has a unique cycle.
If some such cycle was gainfree, the arcs on the cycle
would be dependent, contrary to assumption.
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CALCULATING SIMPLEX MULTIPLIERS
minimize
cijxij
subject to
 j xij

-
 k kixki
0  xij  uij

= b(i) for all i  N
for all (i,j)  A
i is the dual variable for the supply constraint at node i.
c ij cijiijj.
Rule: Calculate simplex multipliers so that c ij = 0 for all
arcs (i,j) in the augmented forest solution.
Let j be the cost of getting rid of one unit of supply from
node j in the augmented forest.
Claim: i = cij ijj for each arc (i,j) in the augmented
forest, and thus  is the vector of simplex multipliers.
CALCULATING THE SIMPLEX MULTIPLIERS
OF AN AUGMENTED TREE T+(v,w)
Assume without loss of generality that node v is the root.
Step 1. Calculate the simplex multiplier for node v. (It is
the cost of getting rid of one unit of supply from node v.)
Step 2. Calculate the simplex multiplier for every other
node of the tree T in the order of the thread of T.
root
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$10,1
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$5, 2
3
$6, 3
$8, 3
$10, 2
4
5
Arc labels are costs cij and multipliers ij.
PIVOTING
Case 1. The entering arc spans two components.
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1. Send  units of flow along the entering arc.
2. Compute the flows in all other arcs as a function of .
3. Determine the leaving arc.
4. Determine the new component(s) of the basis, and
recompute the data structures, including the new
prices and flows.
Case 2. The entering arc is within one component.
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A bicycle is a 2-tree with no nodes of degree 1. In a
non-degenerate pivot, we either send flow on each arc of
a gainfree cycle, or we send flow on each arc of a 2-tree.
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DATA STRUCTURES
Pred, Depth and Thread will suffice. Other data structures have proved to be slightly faster, but not by much.
Time to perform the pivot with entering arc (i,j):
0[ (# of arcs scanned in pricing out) +
(# of of arcs in the components of the
augmented forest containing i and j) ]
Remark: Strong feasibility extends to generalized
networks when each multiplier is positive (i.e., if a column
of the constraint matrix has 2 non-zero elements, then
one is positive and one is negative.)
Remark: Computation times are 2 to 4 times greater
for the simplex algorithm in generalized networks than
in pure networks.