Network Flow Models. Chapter Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall

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1 Network Flow Models Chapter 7 7-1

2 Chapter Topics The Shortest Route Problem The Minimal Spanning Tree Problem The Maximal Flow Problem 7-2

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. 7-3

4 Network Components (1 of 2) Network diagrams consist of nodes and branches. Nodes (circles), represent junction points, or locations. Branches (lines), connect nodes and represent flow. 7-4

5 Network Components (2 of 2) 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.). Figure 7.1 Network of Railroad Routes 7-5

6 The Shortest Route Problem Definition and Example Problem Data (1 of 2) Problem: Determine the shortest routes from the origin to all destinations. Figure

7 The Shortest Route Problem Definition and Example Problem Data (2 of 2) Figure 7.3 Network Representation 7-7

8 The Shortest Route Problem Solution Approach (1 of 8) Determine the initial shortest route from the origin (node 1) to the closest node (3). Figure 7.4 Network with Node 1 in the Permanent Set 7-8

9 The Shortest Route Problem Solution Approach (2 of 8) Determine all nodes directly connected to the permanent set. Figure 7.5 Network with Nodes 1 and 3 in the Permanent Set 7-9

10 The Shortest Route Problem Solution Approach (3 of 8) Redefine the permanent set. Figure 7.6 Network with Nodes 1, 2, and 3 in the Permanent Set 7-10

11 The Shortest Route Problem Solution Approach (4 of 8) Figure 7.7 Network with Nodes 1, 2, 3, and 4 in the Permanent Set 7-11

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 7-12

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 7-13

14 The Shortest Route Problem Solution Approach (7 of 8) Figure 7.10 Network with Optimal Routes 7-14

15 The Shortest Route Problem Solution Approach (8 of 8) Table 7.1 Shortest Travel Time from Origin to Each Destination 7-15

16 The Shortest Route Problem Solution Method Summary 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 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. 7-16

17 The Shortest Route Problem Computer Solution with QM for Windows (1 of 2) Exhibit

18 The Shortest Route Problem Computer Solution with QM for Windows (2 of 2) Exhibit

19 The Shortest Route Problem Computer Solution with Excel (1 of 4) 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 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

20 The Shortest Route Problem Computer Solution with Excel (2 of 4) Exhibit

21 The Shortest Route Problem Computer Solution with Excel (3 of 4) Exhibit

22 The Shortest Route Problem Computer Solution with Excel (4 of 4) Exhibit

23 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. Figure 7.11 Network of Possible Cable TV Paths 7-23

24 The Minimal Spanning Tree Problem Solution Approach (1 of 6) Start with any node in the network and select the closest node to join the spanning tree. Figure 7.12 Spanning Tree with Nodes 1 and

25 The Minimal Spanning Tree Problem Solution Approach (2 of 6) Select the closest node not presently in the spanning area. Figure 7.13 Spanning Tree with Nodes 1, 3, and

26 The Minimal Spanning Tree Problem Solution Approach (3 of 6) Continue to select the closest node not presently in the spanning area. Figure 7.14 Spanning Tree with Nodes 1, 2, 3, and

27 The Minimal Spanning Tree Problem Solution Approach (4 of 6) Continue to select the closest node not presently in the spanning area. Figure 7.15 Spanning Tree with Nodes 1, 2, 3, 4, and

28 The Minimal Spanning Tree Problem Solution Approach (5 of 6) Continue to select the closest node not presently in the spanning area. Figure 7.16 Spanning Tree with Nodes 1, 2, 3, 4, 5, and

29 The Minimal Spanning Tree Problem Solution Approach (6 of 6) Optimal Solution Figure 7.17 Minimal Spanning Tree for Cable TV Network 7-29

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. 7-30

31 The Minimal Spanning Tree Problem Computer Solution with QM for Windows Exhibit

32 The Maximal Flow Problem Definition and Example Problem Data Problem: Maximize the amount of flow of items from an origin to a destination. Figure 7.18 Network of Railway System 7-32

33 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. Figure 7.19 Maximal Flow for Path

34 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. Figure 7.20 Maximal Flow for Path

35 The Maximal Flow Problem Solution Approach (3 of 5) Continue 0 Figure 7.21 Maximal Flow for Path

36 The Maximal Flow Problem Solution Approach (4 of 5) Continue Figure 7.22 Maximal Flow for Path

37 The Maximal Flow Problem Solution Approach (5 of 5) Optimal Solution Figure 7.23 Maximal Flow for Railway Network 7-37

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 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. 7-38

39 The Maximal Flow Problem Computer Solution with QM for Windows Exhibit

40 The Maximal Flow Problem Computer Solution with Excel (1 of 4) 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 5 x 56 4 x x ij 0 and integer 7-40

41 The Maximal Flow Problem Computer Solution with Excel (2 of 4) Exhibit

42 The Maximal Flow Problem Computer Solution with Excel (3 of 4) Exhibit

43 The Maximal Flow Problem Computer Solution with Excel (4 of 4) Exhibit

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. 7-44

45 The Maximal Flow Problem Example Problem, Shortest Route Solution (1 of 2) Step 1 (part A): Determine the Shortest Route Solution 1. Permanent Set Branch Time {1} 1-2 [5] {1,2} 1-3 [5] {1,2,3} 1-4 [7] {1,2,3,4} [9] 5. {1,2,3,4,6} 4-5 [10] {1,2,3,4,5,6} 7-45

46 The Maximal Flow Problem Example Problem, Shortest Route Solution (2 of 2) 7-46

47 The Maximal Flow Problem Example Problem, Minimal Spanning Tree (1 of 2) 1. The closest unconnected node to node 1 is node The closest to 1 and 2 is node The closest to 1, 2, and 3 is node The closest to 1, 2, 3, and 4 is node The closest to 1, 2, 3, 4 and 6 is The shortest total distance is 17 miles. 7-47

48 The Maximal Flow Problem Example Problem, Minimal Spanning Tree (2 of 2) 7-48

49 7-49

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