MS&E111
Lecture 10
Introduction to Optimization
May 15, 2006
Prof. Amin Saberi
From Prof. Van Roy’s Notes
1
Network Flows
A wide variety of engineering and management problems involve optimization of network
flows that is, how objects move through a network. Examples include coordination of trucks
in a transportation system, routing of packets in a communication network, and sequencing
of legs for air travel. Such problems often involve few indivisible objects, and this leads to a
finite set of feasible solutions. For example, consider the problem of finding a minimal cost
sequence of legs for air travel from Nukualofa to Reykjavik. Though there are many routes
that will get a traveller from one place to the other, the number is finite. This may appear
as a striking difference that distinguishes network flows problems from linear programs the
latter always involves a polyhedral set of feasible solutions. Surprisingly, as we will see in
this chapter, network flows problems can often be formulated and solved as linear programs.
1.1
Networks
A network is characterized by a collection of nodes and directed edges, called a directed
graph. Each edge points from one node to another. The figure below offers a visual rep
resentation of a directed graph with nodes labelled 1 through 8. We will denote an edge
pointing from a node
i
to a node
j
by (
i,j
). In this notation, the graph of the below figure
can be characterized in terms of a set of nodes
V
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
and a set of edges
E
=
{
(1
,
2)
,
(1
,
3)
,
(1
,
6)
,
(2
,
5)
,
(3
,
4)
,
(4
,
6)
,
(5
,
8)
,
(6
,
5)
,
(6
,
7)
,
(7
,
8)
}
. Graphs can be used to
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MS&E 111: Introduction to Optimization  Spring 06
model many real networked systems. For example, in modelling air travel, each node might
represent an airport, and each edge a route taken by some flight. Note that, to solve a specific
problem, one often requires more information than the topology captured by a graph. For
example, to minimize cost of air travel, one would need to know costs of tickets for various
routes.
1.2
MinCostFlow Problems
Consider a directed graph with a set
V
of nodes and a set
E
of edges. In a mincostflow
problem, each edge (
i,j
)
∈
E
is associated with a cost
c
ij
and a capacity constraint
u
ij
.
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 Spring '06
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 Operations Research, Linear Programming, Optimization, Shortest path problem, Flow network, Basic Feasible Solutions

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