This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: MA3252 LINEAR AND NETWORK OPTIMIZATION Topic 6: Introduction to Network Optimization 1 / 49 Network flow problems are a special case of linear programs and are among the most frequently solved optimization problems. Everywhere around us networks are apparent. Examples include electrical and power networks, road networks, airline route maps, Internet network, printed circuit boards. In almost all these domains, we wish to move some commodity (electricity, cars, people, airlines, messages) between points in the underlying network as efficiently as possible. Network flow models provide the underlying mathematical framework to analyze such problems. 2 / 49 3 / 49 Graphs Network flow problems are defined on graphs. We introduce graphs formally in this section. A graph G = ( V , E ) consists of: • Vertex set V : Set of vertices or nodes • Edge set E : Set of edges or arcs that connect vertices A graph can be: • Directed with arcs specified as ordered pair of distinct nodes (eg: One way street) • Undirected with arcs specified as unordered pair of distinct nodes (eg: Two way street) 4 / 49 Undirected Graphs An undirected graph G = ( V , E ) consists of a set V of vertices and a set E of unordered pair of vertices. An edge { i , j } in an undirected graph is incident to node i and j . It can also be denoted as { j , i } . Typically, selfarcs like { i , i } will not be allowed. The degree of a node in an undirected graph is the number of edges incident to the node. The degree of an undirected graph is the maximum of the degrees of it’s nodes. A walk in an undirected graph is a finite sequence of nodes i 1 i 2 ... i t with { i k , i k +1 } ∈ E for k = 1 , 2 ,..., t 1. A path is a walk with no repeated nodes. 5 / 49 A cycle is a path i 1 i 2 ... i t along with the additional arc { i t , i 1 } . A cycle has the same starting and ending nodes. In addition, the number of distinct nodes must be at least 3 to exclude a walk i j i where the same arc is traversed back and forth. An undirected graph is connected if there is at least one path between every distinct pair of nodes; otherwise the graph is disconnected. The number of nodes is defined by  V  = n and number of edges by  E  = m . 6 / 49 Example 5.1 (a) Define the vertex set and edge set for the graph. (b) What is the degree of the graph? (c) Find an example of a walk, a path and a cycle in the graph. (d) Is the graph connected? 7 / 49 Directed Graphs A directed graph G = ( V , E ) consists of a set V of vertices and set E of ordered pair of vertices. An edge ( i , j ) in a directed graph is outgoing from node i and incoming to node j . The set of incoming and outgoing nodes for node i are defined as: I ( i ) = { j ∈ V  ( j , i ) ∈ E } O ( i ) = { j ∈ V  ( i , j ) ∈ E } The indegree and outdegree of a node is the number of incoming and outgoing edges for the node respectively. The degree of a node is the sum of its indegree and outdegree....
View
Full Document
 Spring '10
 TanBanPin
 Graph Theory, Shortest path problem, Flow network, cij xij

Click to edit the document details