Figure 7.3 Network Representation The Shortest Route Problem Definition and Example Problem Data 2 of 2 Copyright © 2010 Pearson Education, Inc.. Figure 7.7 Network with Nodes 1, 2, 3,
Trang 2Chapter Topics
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Trang 3Network Components
connected at various points (nodes) through which one or more items move from one point to another.
of the system thus enabling visual representation and enhanced understanding.
networks which are relatively easy to conceive and
construct.
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Trang 4■ Nodes (circles), represent junction points , or
Trang 5Figure 7.1 Network of
Railroad Routes
■ “Atlanta”, node 1, termed origin , any of others
Trang 6Problem: 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)
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Trang 7Figure 7.3 Network
Representation
The Shortest Route Problem
Definition and Example Problem
Data (2 of 2)
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Trang 8Determine the initial shortest route from the origin
(node 1) to the closest node (3).
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Trang 10Figure 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)
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Trang 11Figure 7.7 Network with Nodes 1, 2, 3, and 4 in
the Permanent Set
The Shortest Route Problem
Solution Approach (4 of 8)
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Trang 12The Shortest Route Problem
Solution Approach (5 of 8)
Figure 7.8 Network with Nodes 1, 2, 3, 4, & 6 in
the Permanent Set
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Trang 16The Shortest Route Problem
Solution Method Summary
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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.
Trang 17The Shortest Route Problem
Computer Solution with QM for
Windows (1 of 2)
Exhibit 7.1
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Trang 18The Shortest Route Problem
Computer Solution with QM for
Windows (2 of 2)
Exhibit 7.2
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Trang 19Formulation 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
The Shortest Route Problem
Computer Solution with Excel (1 of
4)
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Trang 20Exhibit 7.3
The Shortest Route Problem
Computer Solution with Excel (2 of
4)
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Trang 21Exhibit 7.4
The Shortest Route Problem
Computer Solution with Excel (3 of
4)
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Trang 22Exhibit 7.5
The Shortest Route Problem
Computer Solution with Excel (4 of
4)
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Trang 23total of the branch lengths are minimized.
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Trang 24Start with any node in the network and select the
closest node to join the spanning tree.
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Trang 302 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.
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Trang 31Copyright © 2010 Pearson Education, Inc Publishing as
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Trang 32Figure 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.
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Trang 33Step 1: Arbitrarily choose any path through the
network from origin to destination and ship as
much as possible.
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Trang 34Figure 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.
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Trang 36Figure 7.22 Maximal Flow for Path 1-3-4-6
The Maximal Flow Problem
Solution Approach (4 of 5)
Continue
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Trang 38The 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.
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Trang 39The Maximal Flow Problem
Computer Solution with QM for
Windows
Exhibit 7.7
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Trang 40The Maximal Flow Problem
Computer Solution with Excel (1 of
4)
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Trang 41The Maximal Flow Problem
Computer Solution with Excel (2 of
4)
Exhibit 7.8
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Trang 42The Maximal Flow Problem
Computer Solution with Excel (3 of
4)
Exhibit 7.9
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Trang 43The Maximal Flow Problem
Computer Solution with Excel (4 of
4)
Exhibit 7.10
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Trang 44The 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.
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Trang 45Step 1 (part A): Determine the Shortest Route Solution
1 Permanent Set Branch Time
The Maximal Flow Problem
Example Problem, Shortest Route
Solution (1 of 2)
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Trang 46The Maximal Flow Problem
Example Problem, Shortest Route
Solution (2 of 2)
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Trang 47The 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.
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Trang 48The Maximal Flow Problem
Example Problem, Minimal Spanning Tree (2 of 2)
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Trang 49Copyright © 2010 Pearson Education, Inc Publishing as
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