Solving Problems with Matrices, Graphs and Diagrams

1
Solving Problems with Matrices, Graphs and
Diagrams

• Chapter 6.5 Solving Problems with Matrices,
  Graphs and Diagrams
• Mathematics of Data Management (Nelson)
• MDM 4U
• Authors Gary Greer and James Gauthier
• (with K. Myers)

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Graph Theory

• In mathematics and computer science, a graph is a
  generalization of the simple concept of a set of
  dots, called vertices, connected by links, edges.
  Depending on the applications, edges may or may
  not have a direction vertices and/or edges may
  be assigned labels. A numeric label is often
  called a weight.
• Graph theory is the branch of mathematics that
  examines the properties of graphs.
• Informally, a graph is a set of objects called
  vertices connected by links called edges
• Wikipedia (2004)

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Terminology

• sequential tasks are dependent on each other and
  must be performed in order
• a project graph is a digraph on which each task
  to be completed is represented by a vertex
• in a project graph, the weight of an edge is the
  duration (to completion) of a preceding task

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Terminology (continued)

• parallel tasks are independent and can be
  performed at the same time
• critical path analysis is the process of finding
  the optimal way to complete a complex task
• earliest start time the earliest that a task can
  begin if all tasks on which it depends begin as
  early as possible

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Another Term Task Table

• a task table is a table that lists all tasks
  associated with a project...
• ... including the duration of each task and the
  tasks on which each depends.

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What is OR?

• Known in the United Kingdom as Operational
  Research and as Operations Research in North
  America, OR is the use of mathematical models,
  statistics and algorithms to aid in
  decision-making.
• OR is most often used to analyze complex
  real-world systems, typically with the goal of
  improving or optimizing performance.
• It is one form of applied mathematics.
• The Royal Air Force (RAF) and the Royal Navy
  began to study operations in the 1930s as a way
  to make the armed forces more efficient.

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OR (continued)

• After World War II, OR methods spread to other
  nations. Government, military and industrial
  sectors now regularly employ OR.
• OR is a bit of a misnomer... it is no longer
  concerned only with operations, nor does its
  application involve any research in the
  traditional sense (although OR research is still
  carried out to find new and better techniques).
• The growth of OR is, to a large extent, the
  result of the increasing power and widespread
  availability of computers.

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OR Links

• Operations Everything (Article from the Boston
  Globe)
• http//www.orie.cornell.edu/
• http//www.coe.ubc.ca/
• http//www.brunel.ac.uk/depts/ma/research
  /jeb/or/intro.html
• http//en.wikipedia.org/wiki/Operations_research

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Exercises / Homework

• Read pp. 374 - 378
• most of our 6.5 problems will look like example 1
  or example 2
• Homework
• page 379, 1, 3, 5
• page 380, 6
• page 381, 7