2.3 Solving Network Problems with Matrices

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2.3 Solving Network Problems with Matrices
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Planning travel between different cities can
become very complicated. If the number of cities
and the number of alternative routes are small,
the problem is relatively easy and can be handled
with little more than paper and pencil. If the
number of cities is large (even just 5 cities) or
if there are many alternative routes, the human
mind has great difficulty organizing and
considering all the alternatives.A matrix is a
mathematical tool that is useful for organizing
and dealing with large amounts of data. Matrices
(plural for matrix) can be used to summarize the
routes between cities and to even calculate the
different number of routes. Airlines, train and
bus companies, and truck dispatchers are some of
the organizations that use matrices as they make
plans for moving people or goods to and from
various locations.To see how matrices are used
in transportation planning, lets start with
something simple a problem involving three
cities and then well extend what weve learned
to a more realistic problem involving six cities.
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Example 1 Three cities

• A small airline serves three cities, Atlanta
  (ATL), Boston (BOS), and Charlotte (CLT), using a
  limited number of airplanes. The flight service
  planner needs a convenient method for keeping
  track of all possible trips connecting the three
  cities. She is concerned with direct, one stop
  over and two stop over trips. A one stop over
  trip means that you start at one city, say
  Atlanta, and make one stop, say Boston, then
  continue on to a final destination city, either
  Atlanta or Charlotte.

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Network Diagram

• The following diagram illustrates the routes
  between the three cities. An arrow illustrates a
  route in that direction.

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Network Matrix

• We can create a matrix that represents the routes
  between our three cities. Such a matrix will need
  three rows and three columns, each row and column
  represent each of the three cities.
• For each cell in the matrix, we can place a 1 to
  indicate that there is one route between the city
  in the row and column for that cell. For example,
  there is one route between Atlanta and Boston, so
  there should be a 1 in the cell on the first row,
  second column.

Complete the matrix
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Network Matrix

• The following is the completed network matrix for
  the direct routes between Boston, Atlanta and
  Charlotte.

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Network Matrix Calculations

• The power of mathematics is derived from what
  information one can gains from its use. To see
  this power, first produce all the one stop over
  routes between the three cities. To get you
  started, is there a one stop over route between
  Atlanta and Atlanta? Yes, you fly Atlanta to
  Boston then back to Atlanta. Any others? No. So 1
  goes in the first cell. Is there a one stop over
  route from Atlanta to Boston? No, you can fly
  from Atlanta to Boston, but your next hop will
  take you away from Boston, so 0 will go in the
  next cell. Is there a one stop over route from
  Atlanta to Charlotte? Yes, you fly Atlanta to
  Boston then Boston to Charlotte. So 1 goes in
  the third cell.

Complete the table
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Network Matrix Calculations

• Here is the completed table

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Network Matrix Calculations

• Take the original network matrix and enter it
  into A in your calculator.
• Then calculate A2.

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Network Matrix Calculations

• Compare A2 to the table you created for one
  stop over.

What do you notice???
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Network Matrix Calculations

• So, if you want to write the matrix that
  represents how many routes there are with one
  stop over you calculate A2.
• What would you do to write the matrix that
  represents the number of routes with 2 stop
  overs?
• How would you calculate the matrix that
  represents at most 2 stop overs?

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Network Matrix Calculations

• A2 1 stop over
• A3 2 stopovers
• A A2 A3 at most 2 stopovers

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Example 2 Six cities

• Consider six cities Los Angeles (LAX), Atlanta
  (ATL), Pittsburgh (PIT), Miami (MIA), Charlotte
  (CLT), and Boston (BOS). Suppose the following
  network diagram represents train routes between
  the different cities.

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Example 2 Six cities

• Complete the following table that represents the
  direct routes between cities.

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Example 2 Six cities

• Here is the table representing direct routes
  between different cities

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Example 2 Six cities

• Determine the matrix that represents all of the
  routes with one stop over.
• 2. Determine the matrix that represents at most
  one stop over.

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Example 2 Six cities
Question 1
Question 2
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Example 3 Spy Network

• A network of spies has been created according to
  these specifications
• - Alison is able to talk to Greg and Kari
  directly
• Greg is able to contact Steve and Kari directly
• Kari can contact Alison, Greg and Toni directly
• Steve can contact Greg directly
• Toni can contact everyone directly
• - Each spy can contact him or herself

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Example 3 Spy Network

• Draw a network diagram based on the description
  and using the following layout

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Example 3 Spy Network