WOOD 492 MODELLING FOR DECISION SUPPORT

Lecture 22

Network Problems

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Example 13: shortest path problem

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HW

T3

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T1 T4 T7

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4.96.2

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Example 13

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Network Problems

• There are many types of network problems, we will focus on three types:

– Shortest path problems• Find the shortest route between the origin and the destination point

– Minimum spanning tree• Find the smallest network that has a path between each two points

– Critical path method (CPM) for time-cost trade-off• Find the optimal plan to expedite some activities within a project in

order to minimize the costs while meeting the project deadline

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Minimum Spanning Tree

• We have:– a set of nodes– a set of potential arcs and their lengths (undirected arcs)

• Objective:– Insert enough arcs so there is a path between every pair of nodes– Minimize the total length of the inserted arcs

• Note: for a network with n nodes, a minimum spanning tree can be found with only n-1 arcs

• A few applications– Design of telecom networks (fiber optic networks, cable TV networks)– Network of pipelines to connect a number of locations– Designing railway networks

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Minimum Spanning Tree – solution algorithm

• Being greedy works! It gives us the optimal solution• The greedy algorithm:

1. Select any node arbitrarily and connect to the nearest node

2. Identify the unconnected node that is closest to a connected node and then connect these two nodes*

3. Repeat until all nodes have been connected

* In case of ties, choose a node arbitrarily. Such ties usually (not always) mean there are multiple optimal solutions.

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Minimum Spanning Tree – Example 14

• Seervada Park example• The arcs now represent potential links• Problem: Find the minimum spanning tree for the park network

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11ParkEntrance

Backcountry Gate

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Example 14

1. We arbitrarily choose node O to start. Closest unconnected node is A

2. The closest unconnected node to either O or A is node B (closest to A)

3. Closest unconnected node to O , A or B is node C (closest to B)

4. Closest unconnected node to O , A , B or C is node E (closest to B)

5. Closest unconnected node to O , A , B ,C or E is node D (closest to E)

6. The only remaining node is T , and it’s closest to D

7. The resulting network is the minimum spanning tree with length of 14

Important: The selection of the first node will not impact the final solution

Oct 29, 2012

O

A

B D

T

EC

2

4

4

7

57

2

5

3

4

11ParkEntrance

Backcountry Gate

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Critical Path Method (CPM)

• A network used to represent a project is called a “project network”• Three types of information are needed before we can create a

project network:– Project activities: break down the project into individual tasks– Precedence relationships: Identify immediate predecessors of each

activity (which activities must be finished before each activity can start)– Time information: estimate activity durations

• To visualize the network: Activity-on-Node (AON) project networks are common.

– Each activity is represented with a node and arcs are used to depict precedence relationships

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Example 15 – Project network

• “Reliable Constructions Co.” has identified the activities within a plant construction project

• The deadline is in 40 weeks• The total of all estimated durations will be 79 weeks, but some

activities can be done in parallel• How long will the project take?• First, we need to visualize the project network

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Example 15