Decision Maths

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Decision Maths

• Lesson 4 Critical Path Analysis

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Problem

• Every morning you have three pieces of bread (A,B
  and C) under the grill.
• The grill will take two pieces of bread at a time
  and will take 30 seconds to toast one side of the
  bread.
• How long will it take to toast all three pieces
  of bread?

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Solution

• You could follow the schedule below.

• Or you could follow a more efficient plan.

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Critical Path Analysis

• This was a trivial example of how time can be
  saved by careful planning.
• We can apply similar thinking to larger problems
  involving construction and maintenance.
• Critical Path Analysis enables us to plan and
  monitor complex projects, so that they are
  approached and carried out as efficiently as
  possible.

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Critical Path Analysis

• Imagine you have a project to do which involves
  doing lots of different activities, some of which
  cannot be started until others have been
  completed.
• For example the stages involved in recording and
  promoting a compact disc are shown in the table
  below.

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Critical Path Analysis
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Critical Path Analysis

• Given the conditions previously shown, what is
  the shortest possible time the project can be
  completed in.
• We will assume that tasks can be carried out
  simultaneously whenever the conditions allow it.
• The first stage is construct a Precedence Network.

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Precedence Network

• Draw a start node
• From the start node
• add the activities that can be done immediately.

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Precedence Network

• Activity D needs A to have been done immediately
  before it.
• Add a node to A.
• Have D coming from that node.

C(3)
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Precedence Network

• We can do a similar thing with activity E

D(2)
2
A(10)
B(9)
1
C(3)
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Precedence Network

• Activities F and G have both of D and E as their
  immediately preceding activities.
• Bring D and E into a single node.
• F and G emerge from that node.

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Precedence Network

• Add nodes to the end of activities F and G.
• Activity H must follow F.
• Activity I must follow G.

D(2)
F(1)
2
A(10)
4
B(9)
E(2)
G(1)
1
3
C(3)
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Precedence Network

• J must follow C,H and I, so bring all of those in
  to one node.
• J can now be added and the network is complete.

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D(2)
F(1)
2
H(3)
A(10)
4
B(9)
E(2)
G(1)
1
3
I(2)
6
C(3)
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Complicated Example

• The example we have just looked through was a
  reasonably easy one.
• Consider the example to the right, it will give
  us more to think about.

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Complicated Example 1

• Draw a start node
• From the start node add the activities that can
  be done immediately.
• Connect nodes to the ends of these activities.

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Complicated Example 1

• Activity C follows A.
• D must follow both A and B.
• Where should D go?

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Complicated Example 1

• You must add in a dummy activity. This will have
  a duration of 0.
• Now to progress from node (3) you will have to
  have completed A and B.

• Why does the dummy go from node (2) to (3)?
• Activity C needs to follow A but has no need to
  follow B. If the dummy went the other way then it
  would be impossible to place activity C.

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Complicated Example 1

• Activity D can now be attached to node (3).
• Now place a node at the end of D.
• Activity E can be assigned.

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Complicated Example 2

• This starts in a similar way and we can assign A
  and B to node 1.
• Both C and D must follow A and B, Where does the
  dummy go?
• The dummy is set from (3) to (2) because E comes
  from B and does not follow A.

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Complicated Example 2

• C and D can now be added to node (2).
• E can be added to node (3).
• They all join the final node (4).

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Complicated Example 3

• Activities A and B are simple.
• C and E are also simple as they follow A and B
  respectively.
• Two dummies are required to a new node (4).
• D is attached and all meet at (5)

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Precedence Network

• Below is a precedence Network for a real life
  construction project. The times on the arc
  represent days. What is the quickest time that
  the project can be completed in?
• 170 days.

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Precedence Network

• Due to a resource problem activity H is delayed
  by 10 days. What effect will this have on the
  whole project?
• The entire project is delayed by 3 days.

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Precedence Network

• The supervisor in charge of activity I wants to
  use overtime to reduce the completion time. How
  would you respond to this request?
• No there is plenty of spare time for the
  activity to be completed.

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Precedence Network

• Money is available to reduce the time of activity
  B or Q. Where would you advise that the money is
  spent and why?
• Activity B It can reduce the overall completion
  time of the project.