Solving Problems with Matrices, Graphs and Diagrams

							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)
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)
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
    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
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.
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.
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.
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
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