TM 631 Optimization Fall 2006 Dr. Frank Joseph Matejcik

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TM 631 Optimization Fall 2006Dr. Frank Joseph
Matejcik
11th Session Ch. 9 Network Optimization
Models 11/20/06
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Activities

• Review assignments and resources
• Hand back exams
• Assignment
• weird way of numbering problems
• Chapter 9 9.3-3, 9.4-1, 9.5-6, 9.6-1, 9.8-1
• Chapter 9 9.6, 9.8 H L

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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 9.3-3,
9.4-1, 9.5-6 11/20/2006 9 9.6-1,
9.8-1 11/27/2006 11 12/04/2006 11 or
13 12/11/2005 Final
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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

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9.6 Minimum Cost Flow Problem

• Central Position Network OR
• Can be solved efficiently
• Encompasses many problems including previous in
  the chapter
• Can be formulated as an LP
• Can be solved by the network simplex method
  (algorithm discussed in the 9.7)

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9.6 Minimum Cost Flow Problem

• Problem description
• Directed and connected network
• At least one node is a supply node
• At least one node is a demand
• Remaining nodes are transshipment nodes
• Flow thru arcs one way and with capacity
• Capacities large enough supply node flow
• Proportional cost of flow in arc
• Objective minimize total using supply for demand

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9.6 Some Applications
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9.6 Some Applications

• Sometimes many levels of transshipment nodes as
  in International Paper Co.

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9.6 Formulation of the Model
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9.6 Formulation of the Model
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9.6 Feasible Solutions Property9.6 Integer
Solution Property

• Feasible solutions property, necessary
  condition
• Integer solution propertyAll bs us are
  integers, then all BFS are.

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9.6 An Example
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9.6 An Example
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9.6 Using Excel
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9.6 Transportation Problem
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9.6 Assignment Problem

• Special type of Transportation Problem
• Additional restrictions
• Number of supply nodes Number of demand nodes
• bi 1 for each supply node
• bi -1 for each demand node

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9.6 Transshipment Problem

• All features of min cost flow except the us.
• If we removed the us from the previous example,
  it would be a transshipment
• Arise as generalizations of transportation
  problems with intermediate points
• More in chapter 23 on disk and online

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9.6 Shortest-Path Problem
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9.6 Maximum Flow Problem
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9.6 Final Comments

• Except for transshipment, we have seen all the
  special cases before
• Network simplex method works for a special cases,
  so .
• Network simplex method implementations are very
  and competitive to special case
• Min cost flow theory results are interesting and
  apply to all special cases.

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9.7 Network Simplex Method

• Skip this section
• Its difficult supplement in competitors text
• Algorithm is not explicitly given
• Difficult to put on an exam

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9.8alt. A Process for Scheduling

• 1. Think
• 2. List activities
• 3. Arrange activities considering precedence and
  relationships
• 4. Develop Gantt charts and PERT/CPM networks
• 5. Determine critical activities/critical path
• 6. Crash and adjust as necessary

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9.8alt. Gantt Charts

• Advantages
• Easy to understand
• Easy to show progress and status
• Easy to maintain
• Most popular view to communicate project status
  to client and/or senior management
• Disadvantages
• Can be superficial
• Not always easy to see precedence, relationships

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9.8alt. PERT/CPM Network Charts

• Advantages
• Allows visualization of task relationships
• Facilitates calculation of critical path
• Clarifies impact of decisions on downstream
  activities
• Disadvantages
• Complex, not easy to comprehend at a glance
• Charts dont readily depict durations, dates,
  progress

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9.8alt. Look at a Simple Network, for a Simple
Project
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9.8alt. A Simple Network (AON) (contd)
Calculate Critical Path Project Duration
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9.8alt. The Critical Path
ES 0 EF 14
ES 14 EF 17
ES 17 EF 21
ES 21 EF 31
A
14
C
3
Start
E
4
F
10
B
3
D
7
Finish
ES 0 EF 3
ES 3 EF 10
Critical Path
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9.8alt. Three Sequential Activities, AON Format
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9.8alt. Activity Network, AON Format
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9.8alt. Activity Network, AOA Format
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9.8alt. Sample of Network Construction,
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9.8alt. Sample of Network Construction,
AON
AOA
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9.8alt. Sample of Network Construction,
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9.8alt. Networking Concurrent Activities
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9.8alt. Activity c Not Required for e,
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9.8alt. Showing Precedents
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9.8alt. MSP Gantt Chart
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9.8alt. MSP AON Network
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9.8alt. An AON Network for a 10-Activity Project,
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9.8alt. Some Definitions

• Resource allocation permits efficient use of
  physical assets
• Within a project, or across multiple projects
• Drives both the identification of resources, and
  timing of their application
• There are generally two conditions
• Normal
• Crashed

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9.8alt. Normal and Crashing

• Normal Most likely task duration
• Crash Expedite an activity, by applying
  additional resources
• Specialized or additional equipment
• More people (e.g., borrowed staff, temps)
• More hours (e.g., overtime, weekends)

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9.8alt. No Free Lunch Crashing Creates a Ripple
Effect

• Crashing buys time, but nothing comes free
• Potential cost areas
• Additional equipment/material
• Extra labor
• Negative effects on other projects
• Reduced morale, from excessive hours/shifts
• Lower quality, from the pressure of time,
  inexperienced and tired staff
• If you want it bad, youll get it bad . . .

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9.8alt. Critical Path Method - Crashing a Project

• CPM includes a way of relating the project
  schedule to the level of physical resources
  allocated to the project
• This allows the project manager to trade time for
  cost, or vice versa
• In CPM, two activity times and two costs are
  specified, if appropriate for each activity

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9.8alt. Critical Path Method - Crashing a Project

• Careful planning is critical when attempting to
  expedite (crash) a project
• Expediting tends to create problems and the
  solution to one problem often creates several
  more problems that require solutions
• Some organizations have more than one level of
  crashing

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9.8alt. When Trying to Crash a Project . . .

• Two basic principles
• 1. Generally, focus on the critical path
• Usually not helpful to shorten non-critical
  activities
• Exception When a scarce resource is needed
  elsewhere, e.g., in another project
• 2. When shortening project duration, choose
  least expensive way to do it

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9.8alt. Compute Cost per Day of Crashing a Project

• Compute cost/time slope for each expeditable
  activity
• Slope crash cost normal cost
  crash time normal time

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9.8alt. An Example
Cost(normal, crash)
Days(normal, crash)
Predecessor
Activity
40, 80
3, 2
-
a
20, 80
2, 1
a
b
20, 20
2, 2
a
c
30, 120
4, 1
a
d
10, 80
3, 1
b
e
Partial crashing allowed Partial crashing
not allowed
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9.8alt. Example (contd) Cost per Day to Crash
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9.8alt. A CPM Example

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9.8alt. CPM Cost-Duration
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9.8 LP for Crashing

• Figure 9.31 summary in Excel
• Not included on the exam

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C9.1 Money in Motion

• International Currency Exchange uses network
  models

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C9.2 Aiding Allies

• Military Planning problem
• Uses Networks to supply problems involved in
  trying to stop a revolt.

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C9.3 Steps to Success