The Mathematics of Networks

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Untitled The Mathematics of Networks (NAME ) Network Models There is a multitude of operation research situations that can be modeled

and solved as networks (nodes connected by branch . Some of the surveys said that almost 70 of the real-world mathematical programming problems can be represented by network-related models . Some of the possible applications of network models on real-world situations are : Design of an offshore natural gas pipeline network connecting wellheads in the Gulf of Mexico to an inshore delivery point . The objective of the model is to minimize the cost of constructing the pipeline . Another is the determination of the shortest route between two cities in a network of roads , and determination of the maximum capacity of a coal slurry pipeline network joining the coal mines in Wyoming with the power plants in Houston (Taha , 2003 ) The solution of these situations , and others like it , is accomplished through a variety of network optimization algorithms . Some of these algorithms are the minimal spanning tree algorithm and the Kruskal`s Algorithm . The situations for which these algorithms apply can also be formulated and solved as explicit linear programs . However , the proposed network-based algorithms are more efficient than the simplex method . This will show the analysis on network models using the minimal spanning tree algorithm and the Kruskal`s Algorithm . Before going to the analysis , some definitions about network models must be first establish . Properties of Trees A network consists of nodes that are linked by arcs or branches . The notation for describing a network is (N ,A , where N is the set of nodes , and A is the set of arcs . Associated with each network is some type of flow (eg , oil products flow in a pipeline and automobile traffic on highways . In general , the flow in a network is limited by the capacity of its arcs , which may be finite or infinite . An arc is to be directed or oriented if it allows positive flow in one direction and zero flow in the opposite direction . A directed network has all directed arcs "Managerial Decision Modeling with Spreadsheets " 2007 ) Another definition is path . Path is a sequence of distinct arcs that join two nodes through other nodes regardless of the direction of flow in each arc . A path can form a cycle if it connects a node to itself through other nodes . And a cycle is said to be directed if it consists of a directed path . A connected network is such that every two distinct nodes are linked by at least one path . A tree is a connected network that may involve only a subset of all the...