<p>University of Southern California Daniel J. Epstein Department of Industrial and Systems Engineering ISE 330: Introduction to Operations Research Lecture 17 Outline</p><p>Read: H & L chapter 9.1-9.3</p><p>Many network optimization models are actually special types of LP problems.</p><p>Four important kinds of network problems:</p><p>[1] the shortest path problem</p><p>[2] the minimum spanning tree problem</p><p>[3] the maximum flow problem</p><p>[4] the minimum cost flow problem</p><p>Network terminology:</p><p>Nodes</p><p>Arcs: directed arcs, undirected arcs (links)</p><p>Networks: directed networks, undirected networks</p><p>Path: directed path, undirected path</p><p>Cycle</p><p>Connected:connected nodes, connected network</p><p>Spanning Tree: a connected network with no undirected cycles Corresponds to BFS for network simplex method.</p><p>Arc capacity</p><p>Supply node, demand node, transshipment node</p><p>1 ISE 330: Lecture 16 Notes http://www-classes.usc.edu/engr/ise/330</p><p>E.C. 2 ISE 330: Lecture 16 Notes http://www-classes.usc.edu/engr/ise/330</p><p>SEERVEDA PARK: Road system for trams and jeeps driven by park rangers.</p><p>A 7 2 2 5 T O 5 B 4 D</p><p>4 1 3 1 7</p><p>C E 4</p><p>O is the park entrance, and there is a scenic wonder at station T. The park management currently faces three problems:</p><p>[1] Determine which route from the park entrance to station T has the smallest total distance for the operation of the trams.</p><p>[2] Telephone lines must be installed under the roads to establish communication among all the stations. Where should the lines be laid to provide connection between every pair of stations while minimizing the total lines laid?</p><p>[3] To reduce tourist impact on the park, there is a ration on the number of tram trips that can be made on each of the roads per day. How should the tram routes be planned so as to maximize the number of trips that can be made per day without violating the limits?</p><p>E.C. 3 ISE 330: Lecture 16 Notes http://www-classes.usc.edu/engr/ise/330</p><p>The Shortest Path Problem</p><p>Consider an undirected and connected network with two special nodes called the origin and the destination. Associated with each of the links (undirected arcs) is a non-negative distance. The objective is to find the shortest path from the origin to the destination.</p><p>E.C. 4</p>