Date post: | 02-Jun-2018 |
Category: |
Documents |
Upload: | moeshfieq-williams |
View: | 217 times |
Download: | 0 times |
of 29
8/11/2019 RSH_10_Ch_12
1/29
2008 Prentice-Hall, Inc.
Chapter 12
To accompanyQuant i tat ive Analysis for Management, Tenth Edit io n,
by Render, Stair, and HannaPower Point slides created by Jeff Heyl
Network Models
2009 Prentice-Hall, Inc.
8/11/2019 RSH_10_Ch_12
2/29
2009 Prentice-Hall, Inc. 122
Learn ing Ob ject ives
1. Connect all points of a network whileminimizing total distance using the minimal-
spanning tree technique2. Determine the maximum flow through a
network using the maximal-flow technique
3. Find the shortest path through a network
using the shortest-route technique4. Understand the important role of software insolving network problems
After completing this chapter, students will be able to:
8/11/2019 RSH_10_Ch_12
3/29
2009 Prentice-Hall, Inc. 123
Chapter Outl ine
12.1 Introduction
12.2 Minimal-Spanning Tree Technique
12.3 Maximal-Flow Technique
12.4 Shortest-Route Technique
8/11/2019 RSH_10_Ch_12
4/29
2009 Prentice-Hall, Inc. 124
This chapter covers three network models thatcan be used to solve a variety of problems
The minimal-spann ing tree techniquedeterminesa path through a network that connects all the
points while minimizing the total distance
The maximal-f low techniquefinds the maximumflow of any quantity or substance through anetwork
The shortest-route techniquecan find theshortest path through a network
In t roduct ion
8/11/2019 RSH_10_Ch_12
5/29
2009 Prentice-Hall, Inc. 125
Large scale problems may require hundreds orthousands of iterations making efficient computerprograms a necessity
All types of networks use a common terminology
The points on a network are called nodesandmay be represented as circles of squares
The lines connecting the nodes are called arcs
In t roduct ion
8/11/2019 RSH_10_Ch_12
6/29
2009 Prentice-Hall, Inc. 126
Minimal-Spann ing Tree Technique
The minimal-spanning tree technique involvesconnecting all the points of a network togetherwhile minimizing the distance between them
The Lauderdale Construction Company is
developing a housing project They want to determine the least expensive way
to provide water and power to each house
There are eight houses in the project and the
distance between them is shown in Figure 12.1
8/11/2019 RSH_10_Ch_12
7/29 2009 Prentice-Hall, Inc. 127
Minimal-Spann ing Tree Technique
Steps for the minimal-spanning treetechnique
1. Select any node in the network
2. Connect this node to the nearest node thatminimizes the total distance
3. Considering all the nodes that are nowconnected, find and connect the nearest nodethat is not connected. If there is a tie, select
one arbitrarily. A tie suggests there may bemore than one optimal solution.
4. Repeat the third step until all nodes areconnected
8/11/2019 RSH_10_Ch_12
8/29 2009 Prentice-Hall, Inc. 128
Minimal-Spann ing Tree Technique
Network for Lauderdale Construction
3
3
2
3
2
4
2
5
6
7
1
5
1
2
3
4
5
6
7
8
3
GulfFigure 12.1
8/11/2019 RSH_10_Ch_12
9/29 2009 Prentice-Hall, Inc. 129
Minimal-Spann ing Tree Technique
Start by arbitrarily selecting node 1
The nearest node is node 3 at a distance of 2 (200feet) and we connect those nodes
Considering nodes 1 and 3, we look for the nextnearest node
This is node 4, the closest to node 3
We connect those nodes
We now look for the nearest unconnected node to
nodes 1, 3, and 4 This is either node 2 or node 6
We pick node 2 and connect it to node 3
8/11/2019 RSH_10_Ch_12
10/29 2009 Prentice-Hall, Inc. 1210
Minimal-Spann ing Tree Technique
Following this same process we connect fromnode 2 to node 5
We then connect node 3 to node 6
Node 6 will connect to node 8
The last connection to be made is node 8 to node7
The total distance is found by adding up thedistances in the arcs used in the spanning tree
2 + 2 + 3 + 3 + 3 + 1 + 2 = 16 (or 1,600 feet)
8/11/2019 RSH_10_Ch_12
11/29 2009 Prentice-Hall, Inc. 1211
Minimal-Spann ing Tree Technique
All iterations for Lauderdale Construction
Figures 12.212.5
3
3
2
3
2
4
2
5
6
7
1
5
1
2
3
4
5
6
7
8
3
Gulf
8/11/2019 RSH_10_Ch_12
12/29 2009 Prentice-Hall, Inc. 1212
Maximal-Flow Techn ique
The maximal-flow technique allows us todetermine the maximum amount of a material thatcan flow through a network
Waukesha Wisconsin is in the process of
developing a road system for the downtown area They want to determine the maximum number of
cars that can flow through the town from west toeast
The road network is shown in Figure 12.7 The numbers by the nodes indicate the number of
cars that can flow f romthe node
8/11/2019 RSH_10_Ch_12
13/29 2009 Prentice-Hall, Inc. 1213
Maximal-Flow Techn ique
Four steps of the Maximal-Flow Technique1. Pick any path from the start (source) to the
finish (sink) with some flow. If no path withflow exists, then the optimal solution has
been found.2. 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.
8/11/2019 RSH_10_Ch_12
14/29
2009 Prentice-Hall, Inc. 1214
Maximal-Flow Techn ique
Four steps of the Maximal-Flow Technique3. For each node on this path, decrease the flow
capacity in the direction of flow by theamount C. For each node on the path,
increase the flow capacity in the reversedirection by the amount C.
4. Repeat these steps until an increase in flow isno longer possible
8/11/2019 RSH_10_Ch_12
15/29
8/11/2019 RSH_10_Ch_12
16/29
2009 Prentice-Hall, Inc. 1216
Maximal-Flow Techn ique
We start by arbitrarily picking the path 126which is at the top of the network
The maximum flow is 2 units from node 2 tonode 6
The path capacity is adjusted by adding 2 to thewestbound flows and subtracting 2 from theeastbound flows
The result is the new path in Figure 12.7 which
shows the new relative capacity of the path atthis stage
8/11/2019 RSH_10_Ch_12
17/29
2009 Prentice-Hall, Inc. 1217
Maximal-Flow Techn ique
Capacity adjustment for path 126 iteration 1
Figure 12.7
22
1
3
1
2
6
403
1
1
2
6
Old Path
New Path
Add 2
Subtract 2
8/11/2019 RSH_10_Ch_12
18/29
2009 Prentice-Hall, Inc. 1218
Maximal-Flow Techn ique
We repeat this process by picking the path 1246
The maximum capacity along this path is 1
The path capacity is adjusted by adding 1 to the
westbound flows and subtracting 1 from theeastbound flows
The result is the new path in Figure 12.8
We repeat this process by picking the path 13
56 The maximum capacity along this path is 2
Figure 12.9 shows this adjusted path
8/11/2019 RSH_10_Ch_12
19/29
2009 Prentice-Hall, Inc. 1219
Maximal-Flow Techn ique
Second iteration for Waukesha road system
Figure 12.8
10
0 2
13
1
2
0
40
4
0
6
02
0 2
0
1
2
3
4
5
6
1
1
3
1
1
1
1
2
4
6
Old Path
New Network
Add 1
Subtract 1
8/11/2019 RSH_10_Ch_12
20/29
2009 Prentice-Hall, Inc. 1220
Maximal-Flow Techn ique
Third and final iteration for Waukesha roadsystem
Figure 12.9
8
2 0
33
1
2
0
40
4
0
4
22
0 2
0
1
2
3
4
5
6
8/11/2019 RSH_10_Ch_12
21/29
2009 Prentice-Hall, Inc. 1221
Maximal-Flow Techn ique
There are no more paths from nodes 1 to 6 withunused capacity so this represents a finaliteration
The maximum flow through this network is 500
cars
PATH FLOW (CARS PER HOUR)
126 200
1246 100
1356 200
Total 500
8/11/2019 RSH_10_Ch_12
22/29
2009 Prentice-Hall, Inc. 1222
Shortest-Rou te Techn ique
The shortest-route techniquefinds how a personor item can travel from one location to anotherwhile minimizing the total distance traveled
It finds the shortest route to a series of
destinations Ray Design, Inc. transports beds, chairs, and
other furniture from the factory to the warehouse
They would like to find the route with the shortest
distance The road network is shown in Figure 12.10
8/11/2019 RSH_10_Ch_12
23/29
8/11/2019 RSH_10_Ch_12
24/29
2009 Prentice-Hall, Inc. 1224
Shortest-Rou te Techn ique
Steps of the shortest-route technique1. Find the nearest node to the origin (plant). Put
the distance in a box by the node.
2. Find the next-nearest node to the origin andput the distance in a box by the node. Severalpaths may have to be checked to find thenearest node.
3. Repeat this process until you have gone
through the entire network. The last distanceat the ending node will be the distance of theshortest route.
8/11/2019 RSH_10_Ch_12
25/29
2009 Prentice-Hall, Inc. 1225
Shortest-Rou te Techn ique
We can see that the nearest node to the plant isnode 2
We connect these two nodes
After investigation, we find node 3 is the next
nearest node but there are two possible paths The shortest path is 123 with a distance of 150
We repeat the process and find the next node isnode 5 by going through node 3
The next nearest node is either 4 or 6 and 6 turnsout to be closer
The shortest path is 12356 with a distance of290 miles
8/11/2019 RSH_10_Ch_12
26/29
2009 Prentice-Hall, Inc. 1226
Shortest-Rou te Techn ique
First iteration for Ray Design
Plant
Warehouse
50
40
200
1501
2
3
4
5
6
Figure 12.11
100
8/11/2019 RSH_10_Ch_12
27/29
2009 Prentice-Hall, Inc. 1227
Shortest-Rou te Techn ique
Second iteration for Ray Design
Figure 12.12
Plant
Warehouse
50
40
200
1501
2
3
4
5
6
100
150
8/11/2019 RSH_10_Ch_12
28/29
2009 Prentice-Hall, Inc. 1228
Shortest-Rou te Techn ique
Third iteration for Ray Design
Figure 12.13
Plant
Warehouse
50
40
200
1501
2
3
4
5
6
100
150 190
8/11/2019 RSH_10_Ch_12
29/29
Shortest-Rou te Techn ique
Fourth and final iteration for Ray Design
Figure 12.14
Plant
Warehouse
50
40
200
1501
2
3
4
5
6
100
150 190
290