# Solving Problems with Matrices, Graphs and Diagrams

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```							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
   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
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
   a task table is a table that lists all tasks associated
with a project...

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.

   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

```
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