This preview shows pages 1–7. Sign up to view the full content.
1
Minimum Cost Flows
CONTENTS
Introduction to Minimum Cost Flows (Section 9.1)
Applications of Minimum Cost Flows (Section 9.2)
Structure of the Basis (Section 11.11)
Optimality Conditions (Section 9.3)
Obtaining Primal and Dual Solutions (Section 11.4)
Network Simplex Algorithm (Section 11.5)
Strongly Feasible Basis (Section 11.6)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 2
Minimum Cost Flow Problem
Determine a least cost shipment of a commodity through a
network in order to satisfy demands at certain nodes from
available supplies at other nodes. Arcs have capacities and
cost associated with them.
Distribution of products
Flow of items in a production line
Routing of cars through street networks
Routing of telephone calls
1
2
3
4
5
6
7
10
10
5
15
5
3
6
4
3
1
2
2
4
6
5
3
Minimum Cost Flow Problem (contd.)
Mathematical Formulation
:
Minimize
c x
subject to
b i
for each node i
N
x
u for each arc i j
N
ij
ij
i j
A
ij
ij
( , )
( )
( , )
∈
∈
∈
∑
∑
∑

=
∈
≤
≤
∈
x
x
ij
ji
{j:(j,i)
A}
{j:(i,j)
A}
0
where
Supply nodes (b(i) > 0); Demand nodes (b(i) < 0),
Transhipment nodes (b(i) = 0)
Mass balance constraints (flow in  flow out = supply/demand)
Flow bound constraints (lower and upper bound constraints)
b i
i N
( )
.
=
∈
∑
0
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 4
Assumptions
All data (cost, supply/demand, capacity) are integral.
The network is directed.
The sum of supplies equals the sum of demands, and the
problem admits a feasible flow.
The network contains an uncapacitated directed path
between every pair of nodes.
All arc costs are nonnegative.
5
Distribution Problems
A car manufacturer has several manufacturing plants and
produces several car models at each plant that is shipped to
various retailers.
The manufacturer must determine the
production plan for each model and shipping pattern that
minimizes the overall cost of production and transportation.
r
1
r
2
p
1
p
2
p
1
/m
1
p
1
/m
2
p
2
/m
1
p
2
/m
2
p
2
/m
3
r
1
/m
1
r
1
/m
2
r
1
/m
3
r
2
/m
1
r
2
/m
2
Plant
nodes
Plant/model
nodes
Retailer/model
nodes
Retailer
nodes
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 6
Airplane Hopping Problem
A
hopping flight visits the cities 1, 2, 3, … , n, in a fixed
sequence.
The plane can pickup passengers at any node and drop
them off at any other node.
Let b
ij
denote the number of passengers available at node i
who want to go to node j and let f
ij
denote the fare for such
passengers.
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 08/27/2011 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.
 Spring '07
 SIRIPHONGLAWPHONGPANICH

Click to edit the document details