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Operations Research II Course,, September 20131 Part 2: Network Flow Operations Research II Dr. Aref Rashad.

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Presentation on theme: "Operations Research II Course,, September 20131 Part 2: Network Flow Operations Research II Dr. Aref Rashad."— Presentation transcript:

1 Operations Research II Course,, September 20131 Part 2: Network Flow Operations Research II Dr. Aref Rashad

2 Topics The Shortest Route Problem The Minimal Spanning Tree Problem Operations Research II Course,, September 2013 2

3 3 Overview A network is an arrangement of paths connected at various points through which one or more items move from one point to another. The network is drawn as a diagram providing a picture of the system thus enabling visual interpretation and enhanced understanding. A large number of real-life systems can be modeled as networks which are relatively easy to conceive and construct. Operations Research II Course,, September 20133

4 Network diagrams consist of nodes and branches. Nodes (circles), represent junction points, or locations. Branches (lines), connect nodes and represent flow. Network Components Operations Research II Course,, September 20134 distance, time, cost, etc. Origin

5 Operations Research II Course,, September 20135 GraphNodesEdges Transportation networks street intersectionshighways communicationcomputersfiber optic cables World Wide Webweb pagesHyperlinks socialpeopleRelationships software systemsfunctionsPrecedence SchedulingtasksConstraints circuitsgatesWires Network Applications

6 Operations Research II Course,, September 20136 Connected Graphs Undirected Graphs Directed Graphs Bipartite Graph Complete Graph Network Types

7 Operations Research II Course,, September 20137 Cycles Trees Def. An undirected graph is a tree if it is connected and does not contain a cycle. Network Types

8 Operations Research II Course,, September 20138 Shortest-Route Algorithm Used for determining the shortest time, distance, or cost from an origin to a destination through a network. Minimum Spanning Tree Algorithm Used in determining the minimum distance (cost, time) needed to connect a set of locations into a single system. Maximal Flow Algorithm Used for determining the greatest amount of flow that can be transmitted through a system in which various branches, or connections, have specified flow capacity limitations. Network Models

9 Problem: Determine the shortest routes from the origin to all destinations. Shipping Routes from Los Angeles The Shortest Route Problem Definition and Example Problem Data (1 of 2) Operations Research II Course,, September 20139

10 Network of Shipping Routes The Shortest Route Problem Definition and Example Problem Data (2 of 2) Operations Research II Course,, September 2013 10

11 Network with Node 1 in the Permanent Set The Shortest Route Problem Solution Approach Determine the initial shortest route from the origin (node 1) to the closest node (3). Operations Research II Course,, September 2013 11

12 Network with Nodes 1 and 3 in the Permanent Set The Shortest Route Problem Solution Approach Determine all nodes directly connected to the permanent set. Operations Research II Course,, September 2013 12

13 Network with Nodes 1, 2, and 3 in the Permanent Set Redefine the permanent set. The Shortest Route Problem Solution Approach Operations Research II Course,, September 2013 13

14 Network with Nodes 1, 2, 3, and 4 in the Permanent Set The Shortest Route Problem Solution Approach Operations Research II Course,, September 2013 14

15 The Shortest Route Problem Solution Approach Network with Nodes 1, 2, 3, 4, and 6 in the Permanent Set Operations Research II Course,, September 201315

16 The Shortest Route Problem Solution Approach Network with Nodes 1, 2, 3, 4, 5, and 6 in the Permanent Set Continue Operations Research II Course,, September 201316

17 The Shortest Route Problem Solution Approach Network with Optimal Routes from Los Angeles to All Destinations Optimal Solution Operations Research II Course,, September 201317

18 The Shortest Route Problem Solution Method Summary Select the node with the shortest direct route from the origin. Establish a permanent set with the origin node and the node that was selected in step 1. Determine all nodes directly connected to the permanent set nodes. Select the node with the shortest route (branch) from the group of nodes directly connected to the permanent set nodes. Repeat steps 3 and 4 until all nodes have joined the permanent set. Operations Research II Course,, September 201318

19 Operations Research II Course,, September 201319 A tree is a set of connected arcs that does not form a cycle. A spanning tree is a tree that connects all nodes of a network. The minimal spanning tree problem seeks to determine the minimum sum of arc lengths necessary to connect all nodes in a network. The criterion to be minimized in the minimal spanning tree problem is not limited to distance. Other criteria include time and cost. The Minimal Spanning Tree Problem

20 Required: Connect all nodes at minimum cost. 1 3 2 The Minimal Spanning Tree Problem

21 Required: Connect all nodes at minimum cost. Cost is sum of edge weights 1 2 3 2 1 3 The Minimal Spanning Tree Problem

22 Required: Connect all nodes at minimum cost. Cost is sum of edge weights 1 2 3 2 1 3 S.T. = 3S.T. = 5S.T. = 4 The Minimal Spanning Tree Problem

23 Required: Connect all nodes at minimum cost. Cost is sum of edge weights 1 3 2 1 3 2 1 3 2 M.S.T. = 3S.T. = 5S.T. = 4 The Minimal Spanning Tree Problem

24 Network of Possible Cable TV Paths The Minimal Spanning Tree Problem Example Problem Problem: Connect all nodes in a network so that the total branch lengths are minimized. Operations Research II Course,, September 201324

25 The Minimal Spanning Tree Problem Solution Approach (1 of 6) Spanning Tree with Nodes 1 and 3 Start with any node in the network and select the closest node to join the spanning tree. Operations Research II Course,, September 201325

26 The Minimal Spanning Tree Problem Solution Approach (2 of 6) Spanning Tree with Nodes 1, 3, and 4 Select the closest node not presently in the spanning area (not to create a cycle). Operations Research II Course,, September 201326

27 The Minimal Spanning Tree Problem Solution Approach (3 of 6) Spanning Tree with Nodes 1, 2, 3, and 4 Continue Operations Research II Course,, September 201327

28 The Minimal Spanning Tree Problem Solution Approach (4 of 6) Spanning Tree with Nodes 1, 2, 3, 4, and 5 Continue Operations Research II Course,, September 201328

29 The Minimal Spanning Tree Problem Solution Approach (5 of 6) Spanning Tree with Nodes 1, 2, 3, 4, 5, and 7 Continue Operations Research II Course,, September 201329

30 The Minimal Spanning Tree Problem Solution Approach (6 of 6) Minimal Spanning Tree for Cable TV Network Optimal Solution Operations Research II Course,, September 201330

31 The Minimal Spanning Tree Problem Solution Method Summary Select any starting node (conventionally, node 1). Select the node closest to the starting node to join the spanning tree (not to create a cycle). Select the closest node not presently in the spanning tree. Repeat step 3 until all nodes have joined the spanning tree. Operations Research II Course,, September 201331

32 4 2 7 5 18 6 9 3 15 14 19 8 10 12 20 9 17 11 13 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201332

33 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 Start 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201333

34 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201334

35 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201335

36 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201336

37 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201337

38 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201338

39 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201339

40 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 NO! The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201340

41 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 LOOP The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201341

42 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201342

43 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201343

44 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201344

45 4 2 7 5 18 6 9 3 15 14 19 8 10 12 9 17 11 13 20 21 24 The Minimal Spanning Tree Problem-2 Operations Research II Course,, September 201345

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