TM 631 Optimization Fall 2006 Dr' Frank Joseph Matejcik

1
TM 631 Optimization Fall 2006Dr. Frank Joseph
Matejcik
10th Session Ch. 9 Network Optimization
Models 11/13/06
2
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

3
Tentative Schedule
Chapters Assigned 8/28/2006 1,
2 ________ 9/04/2006 Holiday 9/11/2006 3
3.1-8,3.2-4,3.6-3 9/18/2006 4 4.3-6, 4.4-6,
4.7-6 9/25/2006 6 6.3-1, 6.3-5, and
6.8-3(abce) 10/02/2006 Exam 1 10/09/2006 Holiday 1
0/16/2006 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
Chapters Assigned 11/13/2006 9
11/20/2006 9 cont. 11/27/2006 11 12/04/2006 11
or 13 12/11/2005 Final
4
Web Resources

• Class Web site on the HPCnet system
• http//sdmines.sdsmt.edu/sdsmt/directory/courses/2
  006fa/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.c
  om/ 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

5
9.0 Network Optimization Models

• Exciting development
• Common in applications
• Many are special types of LPs
• Distribution Unlimited (3.4) min cost flow
• Citgo Petro (3.5) min cost flow (18.5M/yr)
• CPM tradeoffs in project management

6
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

7
9.1 Prototype Example
8
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)

9
9.2 Network Terminology
10
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

11
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)

12
9.2 Network Terminology
13
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

14
9.3 Algorithm S-P
15
9.3 S-P for Park Ex.
16
9.3 S-P for Park Ex.
17
9.3 Excel for S-P
18
9.3 Excel for S-P
19
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", "gt32", or "apples".
• Sum_range are the actual cells to add if their
  corresponding cells in range match criteria.

20
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

21
9.4 Minimum Spanning Tree
22
9.4 Minimum Spanning Tree
23
9.4 Applications

• Telecommunications networks (fiber-optic,
  computer, telephone, cable TV)
• Lightly used transportation networks
• High voltage power transmission lines
• Wiring on electrical equipment
• Design of pipelines to connect locations

24
9.4 Algorithm Min Span Tree

• Its a simple greedy algorithm
• Just pick the smallest link and attach to the tree

25
9.4 Algorithm Min Span Tree
26
9.4 Span Tree for Park Ex.
27
9.4 Span Tree for Park Ex.
28
9.4 Span Tree for Park Ex.
29
9.4 Span Tree for Park Ex.
30
9.4 Span Tree for Park Ex.
31
9.4 Span Tree for Park Ex.
32
9.5 Maximum Flow
33
9.5 Applications

• Maximize the flow through
• A companys distribution network
• A companys supply from its vendors to its
  factories
• Oil through a pipeline system
• Vehicles through a transportation system
• Extended to multiple sinks and sources

34
9.5 Algorithm

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

35
9.5 Algorithm
36
9.5 Algorithm
37
9.5 Maxflow for Park Ex.
38
9.5 Maxflow for Park Ex.
39
9.5 Maxflow for Park Ex.
40
9.5 Maxflow for Park Ex.
41
9.5 Maxflow for Park Ex.
42
9.5 Maxflow for Park Ex.
43
9.5 Maxflow for Park Ex.
44
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)

45
9.5 Find Augmenting Path
46
9.5 Excel for Maxflow
47
9.5 Excel for Maxflow
48
Enough for today

• Next week we will finish the chapter