Introduction to management science 10e by bernard taylor chapter 07

Network Flow
Models
Chapter 7

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Chapter Topics
■ The Shortest Route Problem
■ The Minimal Spanning Tree Problem
■ The Maximal Flow Problem

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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.

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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.

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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

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Railroad
Routes
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7-5


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

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Figure
7.2

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The Shortest Route Problem
Definition and Example Problem
Data (2 of 2)

Figure 7.3 Network
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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
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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

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SetInc. Publishing as
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The Shortest Route Problem
Solution Approach (3 of 8)
Redefine the permanent set.

Figure 7.6 Network with Nodes 1, 2, and 3 in
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Permanent Set

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The Shortest Route Problem
Solution Approach (4 of 8)

Figure 7.7 Network with Nodes 1, 2, 3, and 4 in
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Permanent Set
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The Shortest Route Problem
Solution Approach (5 of 8)

Figure 7.8

Network with Nodes 1, 2, 3, 4, & 6 in

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The Shortest Route Problem
Solution Approach (6 of 8)

Figure
7.9 Network with Nodes 1, 2, 3, 4, 5 & 6 in
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The Shortest Route Problem
Solution Approach (7 of 8)

Figure 7.10 Network with
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Optimal
Routes

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The Shortest Route Problem
Solution Approach (8 of 8)

Table 7.1 Shortest Travel Time from Origin to
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Each
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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
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

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

Exhibit
7.1Education, Inc. Publishing as
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The Shortest Route Problem
Computer Solution with QM for
Windows (2 of 2)

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Exhibit

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The Shortest Route Problem
Computer Solution with Excel (1 of
4)Formulation as a 0 - 1 integer linear programming
problem.
xij = 0 if branch i-j is not selected as part of the
shortest route and 1 if it is selected.
Minimize Z = 16x12 + 9x13 + 35x14 + 12x24 + 25x25 +
15x34 +
22x36 + 14x45 + 17x46 + 19x47 +
8x57 + 14x67
subject to:

x12 + x13 + x14= 1
x12 - x24 - x25 = 0
x13 - x34 - x36 = 0
x14 + x24 + x34 - x45 - x46 - x47 = 0
x25 + x45 - x57 = 0
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x +x -x =0

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

Exhibit
7.3
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The Shortest Route Problem
Computer Solution with Excel (3 of
4)

Exhibit
7.4
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The Shortest Route Problem
Computer Solution with Excel (4 of
4)

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Exhibit 7.5
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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
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Paths
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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
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Nodes
1
and
3
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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,
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3,
and
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Hall 4

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