7-1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Network Flow Models Chapter 7

7-2 Chapter Topics ■The Shortest Route Problem ■The Minimal Spanning Tree Problem Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-3 Network Components ■A network is an arrangement of paths (branches) connected at various points (nodes) 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 representation and enhanced understanding. ■A large number of real-life systems can be modeled as networks which are relatively easy to conceive and construct. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-4 ■Network diagrams consist of nodes and branches. ■Nodes (circles), represent junction points, or locations. ■Branches (lines), connect nodes and represent flow. Network Components (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-5 Figure 7.1 Network of Railroad Routes ■Four nodes, four branches in figure. ■“Atlanta”, node 1, termed origin, any of others destination. ■Branches identified by beginning and ending node numbers. ■Value assigned to each branch (distance, time, cost, etc.). Network Components (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-6 Problem: Determine the shortest routes from the origin to all destinations. Figure 7.2 The Shortest Route Problem Definition and Example Problem Data (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-7 Figure 7.3 Network Representation The Shortest Route Problem Definition and Example Problem Data (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-8 Figure 7.4 Network with Node 1 in the Permanent Set The Shortest Route Problem Solution Approach (1 of 8) Determine the initial shortest route from the origin (node 1) to the closest node (3). Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-9 Figure 7.5 Network with Nodes 1 and 3 in the Permanent Set The Shortest Route Problem Solution Approach (2 of 8) Determine all nodes directly connected to the permanent set. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-10 Figure 7.6 Network with Nodes 1, 2, and 3 in the Permanent Set Redefine the permanent set. The Shortest Route Problem Solution Approach (3 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-11 Figure 7.7 Network with Nodes 1, 2, 3, and 4 in the Permanent Set The Shortest Route Problem Solution Approach (4 of 8) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-12 The Shortest Route Problem Solution Approach (5 of 8) Figure 7.8 Network with Nodes 1, 2, 3, 4, & 6 in the Permanent Set Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-13 The Shortest Route Problem Solution Approach (6 of 8) Figure 7.9 Network with Nodes 1, 2, 3, 4, 5 & 6 in the Permanent Set Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-14 The Shortest Route Problem Solution Approach (7 of 8) Figure 7.10 Network with Optimal Routes Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-15 The Shortest Route Problem Solution Approach (8 of 8) Table 7.1 Shortest Travel Time from Origin to Each Destination Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-16 The Shortest Route Problem Solution Method Summary Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall 1.Select the node with the shortest direct route from the origin. 2.Establish a permanent set with the origin node and the node that was selected in step 1. 3.Determine all nodes directly connected to the permanent set of nodes. 4.Select the node with the shortest route from the group of nodes directly connected to the permanent set of nodes. 5.Repeat steps 3 & 4 until all nodes have joined the permanent set.

7-17 Formulation as a integer linear programming problem. x ij = 0 if branch i-j is not selected as part of the shortest route and 1 if it is selected. Minimize Z = 16x x x x x x x x x x x x 67 subject to: x 12 + x 13 + x 14 = 1 x 12 - x 24 - x 25 = 0 x 13 - x 34 - x 36 = 0 x 14 + x 24 + x 34 - x 45 - x 46 - x 47 = 0 x 25 + x 45 - x 57 = 0 x 36 + x 46 - x 67 = 0 x 47 + x 57 + x 67 = 1 x ij = 0 or 1 The Shortest Route Problem Computer Solution with Excel (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-18 Exhibit 7.3 The Shortest Route Problem Computer Solution with Excel (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-19 Exhibit 7.4 The Shortest Route Problem Computer Solution with Excel (3 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-20 Exhibit 7.5 The Shortest Route Problem Computer Solution with Excel (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-21 Figure 7.11 Network of Possible Cable TV Paths The Minimal Spanning Tree Problem Definition and Example Problem Data Problem: Connect all nodes in a network so that the total of the branch lengths are minimized. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-22 The Minimal Spanning Tree Problem Solution Approach (1 of 6) Figure 7.12 Spanning Tree with Nodes 1 and 3 Start with any node in the network and select the closest node to join the spanning tree. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-23 The Minimal Spanning Tree Problem Solution Approach (2 of 6) Figure 7.13 Spanning Tree with Nodes 1, 3, and 4 Select the closest node not presently in the spanning area. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-24 The Minimal Spanning Tree Problem Solution Approach (3 of 6) Figure 7.14 Spanning Tree with Nodes 1, 2, 3, and 4 Continue to select the closest node not presently in the spanning area. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-25 The Minimal Spanning Tree Problem Solution Approach (4 of 6) Figure 7.15 Spanning Tree with Nodes 1, 2, 3, 4, and 5 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Continue to select the closest node not presently in the spanning area.

7-26 The Minimal Spanning Tree Problem Solution Approach (5 of 6) Figure 7.16 Spanning Tree with Nodes 1, 2, 3, 4, 5, and 7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Continue to select the closest node not presently in the spanning area.

7-27 The Minimal Spanning Tree Problem Solution Approach (6 of 6) Figure 7.17 Minimal Spanning Tree for Cable TV Network Optimal Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-28 The Minimal Spanning Tree Problem Solution Method Summary 1.Select any starting node (conventionally, node 1). 2.Select the node closest to the starting node to join the spanning tree. 3.Select the closest node not presently in the spanning tree. 4.Repeat step 3 until all nodes have joined the spanning tree. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

7-29 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall