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

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

3. 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

4. 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

5. 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

6. 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. 7-7 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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.

17. 7-17 The Shortest Route Problem Computer Solution with QM for Windows (1 of 2) Exhibit 7.1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

18. 7-18 The Shortest Route Problem Computer Solution with QM for Windows (2 of 2) Exhibit 7.2 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

19. 7-19 Formulation as a 0 - 1 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 12 + 9x 13 + 35x 14 + 12x 24 + 25x 25 + 15x 34 + 22x 36 + 14x 45 + 17x 46 + 19x 47 + 8x 57 + 14x 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

20. 7-20 Exhibit 7.3 The Shortest Route Problem Computer Solution with Excel (2 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

21. 7-21 Exhibit 7.4 The Shortest Route Problem Computer Solution with Excel (3 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

22. 7-22 Exhibit 7.5 The Shortest Route Problem Computer Solution with Excel (4 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

23. 7-23 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

24. 7-24 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

25. 7-25 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

26. 7-26 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

27. 7-27 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.

28. 7-28 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.

29. 7-29 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

30. 7-30 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

31. 7-31 The Minimal Spanning Tree Problem Computer Solution with QM for Windows Exhibit 7.6 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

32. 7-32 Figure 7.18 Network of Railway System The Maximal Flow Problem Definition and Example Problem Data Problem: Maximize the amount of flow of items from an origin to a destination. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

33. 7-33 Figure 7.19 Maximal Flow for Path 1-2-5-6 The Maximal Flow Problem Solution Approach (1 of 5) Step 1: Arbitrarily choose any path through the network from origin to destination and ship as much as possible. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

34. 7-34 Figure 7.20 Maximal Flow for Path 1-4-6 The Maximal Flow Problem Solution Approach (2 of 5) Step 2: Re-compute branch flow in both directions Step 3: Select other feasible paths arbitrarily and determine maximum flow along the paths until flow is no longer possible. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

35. 7-35 Figure 7.21 Maximal Flow for Path 1-3-6 The Maximal Flow Problem Solution Approach (3 of 5) Continue 0 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

36. 7-36 Figure 7.22 Maximal Flow for Path 1-3-4-6 The Maximal Flow Problem Solution Approach (4 of 5) Continue Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

37. 7-37 Figure 7.23 Maximal Flow for Railway Network The Maximal Flow Problem Solution Approach (5 of 5) Optimal Solution Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

38. 7-38 The Maximal Flow Problem Solution Method Summary 1.Arbitrarily select any path in the network from origin to destination. 2.Adjust the capacities at each node by subtracting the maximal flow for the path selected in step 1. 3.Add the maximal flow along the path to the flow in the opposite direction at each node. 4.Repeat steps 1, 2, and 3 until there are no more paths with available flow capacity. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

39. 7-39 The Maximal Flow Problem Computer Solution with QM for Windows Exhibit 7.7 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

40. 7-40 x ij = flow along branch i-j and integer Maximize Z = x 61 subject to: x 61 - x 12 - x 13 - x 14 = 0 x 12 - x 24 - x 25 = 0 x 13 - x 34 - x 36 = 0 x 14 + x 24 + x 34 - x 46 = 0 x 25 - x 56 = 0 x 36 + x 46 + x 56 - x 61 = 0 x 12  6 x 24  3 x 34  2 x 13  7 x 25  8 x 36  6 x 14  4 x 46  5x 56  4 x 61  17 x ij  0and integer The Maximal Flow Problem Computer Solution with Excel (1 of 4) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

41. 7-41 The Maximal Flow Problem Computer Solution with Excel (2 of 4) Exhibit 7.8 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

42. 7-42 The Maximal Flow Problem Computer Solution with Excel (3 of 4) Exhibit 7.9 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

43. 7-43 The Maximal Flow Problem Computer Solution with Excel (4 of 4) Exhibit 7.10 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

44. 7-44 The Maximal Flow Problem Example Problem Statement and Data 1.Determine the shortest route from Atlanta (node 1) to each of the other five nodes (branches show travel time between nodes). 2.Assume branches show distance (instead of travel time) between nodes, develop a minimal spanning tree. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

45. 7-45 Step 1 (part A): Determine the Shortest Route Solution 1. Permanent Set Branch Time {1} 1-2[5] 1-35 1-4 7 2. {1,2} 1-3[5] 1-47 2-511 3. {1,2,3} 1-4[7] 2-511 3-47 4. {1,2,3,4} 4-510 4-6[9] 5. {1,2,3,4,6} 4-5[10] 6-513 6. {1,2,3,4,5,6} The Maximal Flow Problem Example Problem, Shortest Route Solution (1 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

46. 7-46 The Maximal Flow Problem Example Problem, Shortest Route Solution (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

47. 7-47 The Maximal Flow Problem Example Problem, Minimal Spanning Tree (1 of 2) 1.The closest unconnected node to node 1 is node 2. 2.The closest to 1 and 2 is node 3. 3.The closest to 1, 2, and 3 is node 4. 4.The closest to 1, 2, 3, and 4 is node 6. 5.The closest to 1, 2, 3, 4 and 6 is 5. 6.The shortest total distance is 17 miles. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

48. 7-48 The Maximal Flow Problem Example Problem, Minimal Spanning Tree (2 of 2) Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

49. 7-49 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall