Network Models 7-1

Learning Objectives Students will be able to: Find the shortest path through a network using the shortest-route technique. Connect all points of a network while minimizing total distance using the minimal-spanning tree technique. Determine the maximum flow through a network using the maximal-flow technique. Understand the important role of software in solving network problems. 7-2

Chapter Outline 7.1 Introduction 7.2 Shortest-Route Technique 7.3 Maximal-Flow Technique 7.4 Minimal-Spanning Tree Technique 7-3

Introduction Graph/ Network Nodes / Vertices / Nod Edges / Arcs / Arka Consists of a set of points and set of lines connecting certain pairs of the point. Nodes / Vertices / Nod The point It is refer to sources or destination. Edges / Arcs / Arka A connection between two vertices or nodes. Consist an arrow to show the path. 7-4

Introduction Example Nodes A graph with 4 nodes 14 Johor Nodes A graph with 4 nodes A graph with arcs and nodes 7-5

Material-handling routes Introduction Components of typical network Nodes Arcs Flow Intersections Roads Vehicles Airports Air lanes Aircraft Switching points Wires, Channels Messages Pumping stations Pipes Fluids Work Centers Material-handling routes Jobs 7-6

Introduction The presentation will cover three network models that can be used to solve a variety of problems: the shortest-route technique, the maximal-flow technique, the minimal-spanning tree technique 7-7

Shortest Path Technique Definition: Shortest path technique can find the shortest path through a network. For example: This technique can find the shortest route from one city to another through a network of roads. Technique: Floyd-Warshall 7-8

The Maximum Flow Technique Definition: The maximal-flow technique finds the maximum flow of any quantity or substance through a network. For example: This technique can determine the maximum number of vehicles (cars, trucks, etc.) that can go through a network of roads from one location to another. Technique : Ford-furkurson 7-9

Minimal-Spanning Tree Technique Definition: The minimal-spanning tree technique determines the path through the network that connects all the points while minimizing total distance. For example: If the points represent houses in a subdivision, the minimal spanning tree technique can be used to determine the best way to connect all of the houses to electrical power, water systems, etc. in a way that minimizes the total distance or length of power lines or water pipes. Technique : Prim 7-10

The Shortest-Path Technique The shortest-path technique minimizes the distance through a network. The shortest-path technique finds how a person or item can travel from one location to another while minimizing the total distance traveled. The shortest-path technique finds the shortest path to a series of destinations. 7-11

Example 2 7 4 1 3 4 1 4 5 3 4 5 3 7-12

Example 1 2 1 2 1 3 2 4 1 3 2 4 from 1-5 = 8km 1 5 3 1st iteration 1 1-2 =   1-3 4 2nd iteration 1,2 2-3 2-5 8 2-4 5 3rd iteration 1,2,3 3-4 7 3-5 9 4th iteration 1,2,3,4 4-5 Result 1,2,3,4,5 1 1 2 3 1 2 4 3 1 2 5 4 3 1 2 from 1-5 = 8km 7-13

Maximal-Flow Technique The maximal-flow technique allows the maximum amount of a material that can flow through a network to be determined. For example: It has been used to find the maximum number of automobiles that can flow through a state highway system. An example: Kuala Terengganu is in the process of developing a road system for downtown. City planners would like to determine the maximum number of cars that can flow through the town from west to east. The road network is shown in Figure 12.6 (next slide). 7-14

Road Network for KT Traffic can flow in both directions. 7-15

Maximal-Flow Technique (continued) The Four Maximal-Flow Technique Steps: Pick any path from the start (source) to the finish (sink) with some flow. If no path with flow exists, then the optimal solution has been found. Find the arc on this path with the smallest flow capacity available. Call this capacity C. This represents the maximum additional capacity that can be allocated to this route. For each node on this path, decrease the flow capacity in the direction of flow by the amount C. For each node on this path, increase the flow capacity in the reverse direction by the amount C. Repeat these steps until an increase in flow is no longer possible. 7-16

Solving the KT Example Start by arbitrarily picking the path 1–2–6, at the top of the network. What is the maximum flow from west to east? It is 2 because only 2 units (200 cars) can flow from node 2 to node 6. Now we adjust the flow capacities (Figure 12.7). As you can see, we subtracted the maximum flow of 2 along the path 1–2–6 in the direction of the flow (west to east) and added 2 to the path in the direction against the flow (east to west). The result is the new path in Figure 12.7 (next slide). 7-17

1 2 3 4 5 6 10 Initial (C) 7-18

1 2 3 4 5 6 -3 10 +1 -2 +2 1st iteration Rantai = {1-2-6} Θ = min{3,2} +2 1st iteration Rantai = {1-2-6} Θ = min{3,2} = 2 7-19

1 2 3 4 5 6 -1 10 +3 +1 2nd iteration Rantai = {1-2-4-6} +1 2nd iteration Rantai = {1-2-4-6} Θ = min{1,1,1} = 1 7-20

1 2 3 4 5 6 -10 +0 -2 +1 -6 3rd iteration Rantai = {1-3-5-6} -10 +0 -2 +1 -6 3rd iteration Rantai = {1-3-5-6} Θ = min{10,2,6} = 2 7-21

1 2 3 4 5 6 8 4th iteration Rantai = {1-4-3-5-6} Θ = min{2,1,0,4} = 0 8 4th iteration Rantai = {1-4-3-5-6} Θ = min{2,1,0,4} = 0 7-22

1 2 3 4 5 6 8 5th iteration Rantai = {1-4-6} Θ = min{2,0} = 0 7-23

1 2 3 4 5 6 8 6th iteration Rantai = {1-3-4-6} Θ = min{8,3,0} = 0 7-24

1 2 3 4 5 6 8 7th iteration (C*) Rantai = {1-3-4-2-6} Θ = min{8,3,2,0} 8 7th iteration (C*) Rantai = {1-3-4-2-6} Θ = min{8,3,2,0} = 0 7-25

Result X = C-C* 1 2 3 4 5 6 # if C-C* = -ve, change to 0 7-26

1-2-6 200 (dapat dari min table) 1-2-4-6 100 1-3-5-6 200 Total =500 (Cars per Hour) PATH FLOW 1-2-6 200 (dapat dari min table) 1-2-4-6 100 1-3-5-6 200 Total =500 7-27

Summary Theme Park O C B T E D A 2 1 5 4 7 3 7-28

Summary Example Theme Park Car are not allowed Tram system Road for the park ranger 3 Problem Determine which route form the park entrance to station T has the small total distance for the operation of the tram. (Shortest path problem) Telephone line must be installed under the roads to established telephone communication among all the stations. The installation is expensive. Where is the line should be laid to accomplish this with a minimum total number of line installed. (minimum spinning tree problem) More people want to take the tram ride from the park entrance to station T than can be accommodated during the peak season. (maximum flow problem) 7-29

Minimal-Spanning Tree Steps Select any node in the network. Connect this node to the nearest node minimizing the total distance. Find and connect the nearest unconnected node to one of the connected nodes. If there is a tie for the nearest node, one can be selected arbitrarily. A tie suggests that there may be more than one optimal solution. Repeating the third step until all nodes are connected. 7-30

Figure 12.1: Network for Lauderdale Construction 7-31

C = {1} C = {2,3,4,5,6,7,8} 7-32

1st iteration C = {1, 3} C = {2,4,5,6,7,8} 7-33

2nd iteration C = {1, 3, 4} C = {2, 5, 6, 7, 8} 7-34

3rd iteration C = {1, 3, 4, 2} C = {5, 6, 7, 8} 7-35

4th iteration C = {1, 3, 4, 2, 6} C = {5, 7, 8} 7-36

5th iteration C = {1, 3, 4, 2, 6} C = {5, 7, 8} 7-37

6th iteration C = {1, 3, 4, 2, 6, 8} C = {5,7} 7-38

7th iteration C = {1, 3, 4, 2, 6,8, 7} C = {5} 7-39

8th iteration C = {1, 3, 4, 2, 6, 8, 7, 5} C = {} 7-40

Minimal-Spanning Tree Technique Solving the network for Melvin Lauderdale construction Start by arbitrarily selecting node 1. Since the nearest node is the third node at a distance of 2 (200 feet), connect node 1 to node 3. Shown in Figure 12.2 (2 slides hence) Considering nodes 1 and 3, look for the next-nearest node. This is node 4, which is the closest to node 3 with a distance of 2 (200 feet). Once again, connect these nodes (Figure 12.3a (3 slides hence). 7-41

Summarize: Minimal-Spanning Tree Technique Step 1: Select node 1 Step 2: Connect node 1 to node 3 Step 3: Connect the next nearest node Step 4: Repeat the process The total number of iterations to solve this example is 7. This final solution is shown in the following slide. 7-42

Final Solution to the Minimal-Spanning Tree Example Nodes 1, 2, 4, and 6 are all connected to node 3. Node 2 is connected to node 5. Node 6 is connected to node 8, and node 8 is connected to node 7. All of the nodes are now connected. The total distance is found by adding the distances for the arcs used in the spanning tree. In this example, the distance is: 2 + 2 + 3 + 3 + 3 + 1 + 2 = 16 (or 1,600 feet). 7-43