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					            TM 631 Optimization
                  Fall 2006
          Dr. Frank Joseph Matejcik
10th Session: Ch. 9:
Network Optimization Models
11/13/06




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                       Activities
• Review assignments and resources
• Hand back exams
• Assignment
   – weird way of numbering problems
   Chapter 9 thru 9.5

• Chapter 9 thru 9.5 H & L


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             Tentative Schedule
          Chapters Assigned               Chapters Assigned
8/28/2006   1, 2 ________ 11/13/2006 9
9/04/2006   Holiday             11/20/2006 9 cont.
9/11/2006 3 3.1-8,3.2-4,3.6-3 11/27/2006 11
9/18/2006 4 4.3-6, 4.4-6, 4.7-6 12/04/2006 11 or 13
9/25/2006 6 6.3-1, 6.3-5,       12/11/2005 Final
            and 6.8-3(abce)
10/02/2006 Exam 1
10/09/2006 Holiday
10/16/2006 8 8.1-5, 8.1-6, 8.2-6, 8.2-7(ab), 8.2-8
10/23/2006 8 8.4 Answers in Slides & HPCNET
10/30/2006 21 No problems
11/06/2006 Exam 2
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             Web Resources
• Class Web site on the HPCnet system
• http://sdmines.sdsmt.edu/sdsmt/directory/course
  s/2006fa/tm631021
• Streaming video http://its.sdsmt.edu/Distance/
• The same class session that is on the DVD is on
  the stream in lower quality.
  http://www.flashget.com/ will allow you to capture
  the stream more readily and review the lecture,
  anywhere you can get your computer to run.
• Answers have been posted through chapter 8

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    9.0 Network Optimization Models
•   Exciting development
•   Common in applications
•   Many are special types of LP’s
•   Distribution Unlimited (3.4) min cost flow
•   Citgo Petro (3.5) min cost flow ($18.5M/yr)
•   CPM tradeoffs in project management


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         9.1 Prototype Example
• Seervada Park
• Trams route smallest total distance
• Phone lines minimum spanning tree
• Tram limitations for environmental
  concerns limitations on each road
  Maximum Flow problem
• O is origin, T is endpoint


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         9.1 Prototype Example




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       9.2 Network Terminology
•   Network has points with lines connecting
•   Nodes (vertices)
•   Arcs (links, edges, or branches)
•   Directed arc: flow in one direction
    – AB or A → B not BA
• Undirected arc: flow in two directions (link)
• Directed Network (only directed arcs)
• Undirected Network (only undirected arcs)
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       9.2 Network Terminology




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       9.2 Network Terminology
• A network with a mixture of undirected arcs
  and directed arcs can be converted to a
  directed network by adding two directed
  arcs to replace each undirected arc.
• Path OB-BD-DT (O → B → D → T)
• Directed path
• Undirected path either way
• Connected nodes undirected path between
• Unconnected nodes not
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       9.2 Network Terminology
•   Cycle
•   Tree figure 9.3
•   Spanning Tree figure 9.3
•   Arc capacity
•   Supply node
•   Demand node
•   Transshipment node (conservation of flow)

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       9.2 Network Terminology




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              9.3 Shortest-Path
• Focus on simple version
• Undirected connected network with an
  origin and a destination
• Nonnegative distances on links
• Fans out from origin identifying shortest
  path of all points until destination is
  reached


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              9.3 Algorithm S-P




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            9.3 S-P for Park Ex.




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            9.3 S-P for Park Ex.




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               9.3 Excel for S-P




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               9.3 Excel for S-P




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9.3 Excel for S-P SUMIF 8th ed.
• SUMIF(Range,Criteria,Sum_range)
• Range is the range of cells that you want
  evaluated by criteria.
• Criteria is the criteria in the form of a
  number, expression, or text that defines
  which cells will be added. Criteria can be
  expressed as 32, "32", ">32", or "apples".
• Sum_range are the actual cells to add if
  their corresponding cells in range match
  criteria.
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    9.3 Other Shortest-Path Uses
•   Minimize Total distance
•   Minimize Total cost of travel
•   Minimize Total time of travel
•   Used in directed networks
•   To every node methods exist
•   To and from every node exist, too


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    9.4 Minimum Spanning Tree




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    9.4 Minimum Spanning Tree




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                9.4 Applications
1. Telecommunications networks (fiber-
   optic, computer, telephone, cable TV)
2. Lightly used transportation networks
3. High voltage power transmission lines
4. Wiring on electrical equipment
5. Design of pipelines to connect locations



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   9.4 Algorithm Min Span Tree
• It’s a simple greedy algorithm
• Just pick the smallest link and attach to
  the tree




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   9.4 Algorithm Min Span Tree




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     9.4 Span Tree for Park Ex.




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     9.4 Span Tree for Park Ex.




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     9.4 Span Tree for Park Ex.




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     9.4 Span Tree for Park Ex.




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     9.4 Span Tree for Park Ex.




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     9.4 Span Tree for Park Ex.




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             9.5 Maximum Flow




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                9.5 Applications
•   Maximize the flow through …
    1. A company’s distribution network
    2. A company’s supply from its vendors to its
       factories
    3. Oil through a pipeline system
    4. Vehicles through a transportation system


– Extended to multiple sinks and sources

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                  9.5 Algorithm
• Can be set up as an LP, so …
• Residual network (residual capacities)
• Augmenting path




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                  9.5 Algorithm




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                  9.5 Algorithm




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       9.5 Maxflow for Park Ex.




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       9.5 Maxflow for Park Ex.




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       9.5 Maxflow for Park Ex.




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       9.5 Maxflow for Park Ex.




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       9.5 Maxflow for Park Ex.




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       9.5 Maxflow for Park Ex.




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       9.5 Maxflow for Park Ex.




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      9.5 Find Augmenting Path
• Fanning procedure for finding path
  Iteration 7 is the illustration Cut (think of it
  as slicing through the plot)
• Cut value (sum of flows cut)
• Max-flow min-cut theorem
   – smallest cut value equals max-flow value
   – Can prove the max flow value without solving
     the problem (so know when done)

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      9.5 Find Augmenting Path




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          9.5 Excel for Maxflow




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          9.5 Excel for Maxflow




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               Enough for today
• Next week we will finish the chapter




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