Critical Paths (Edexcel International AS Maths) : Revision Note

Introduction to Critical Path Analysis

What is critical path analysis?

• Critical path analysis (CPA) is the use of an activity network for a project that determines

  • The minimum duration of the entire project

  • The critical activities

    • These are the activities that must start and finish at the earliest possible time so that the project can be completed in its minimum duration

  • The critical path

    • The path from the source node to the sink node that encompasses all the critical activities

    • In harder problems there may be more than one critical path

  • Any possible delay to starting non-critical activities

    • The total float of an activity is the time it can be delayed by as to not affect the minimum project duration

Earliest Event Times

What are earliest (or early) event times?

• The earliest event time is the earliest time at which an activity can commence from its starting event

  • This will determine the minimum project duration

• On an activity network, each event (node) will have two boxes that require completing

  • Sometimes the boxes will replace the circled event (node) number

  • Sometimes the boxes will be drawn next to the circled event (node) number

  • The top box will be the earliest event time

  • The bottom box is for the latest event time (see the next section of this revision note)

    • The earliest event times must be completed before the latest event times

How do I work out early event times?

• Starting at the source node, this will be 0

  • This is the starting point of the project, i.e. time zero

• Working forwards from the source node, look at each of the activities (arcs) leading to an event (node)

  • For each preceding activity of an event (node) 

    • Add the early event time from that activity's start node to its duration

  • The earliest event time is then the maximum of these times

    • This is the earliest time an event can be moved on from

• Completing the earliest event times is sometimes referred to as a forwards pass (through the activity network)

Latest Event Times

What are latest (or late) event times?

• The latest event time is the latest time at which an activity can commence from its starting event

  • without affecting the minimum project duration

• On an activity network, each event/node will have two boxes that require completing

  • Sometimes the boxes will replace the circled event/node number

  • Sometimes the boxes will be drawn near to the circled event/node number

  • The bottom box will be the latest event time

• The top box is for the earliest event time (see the previous section of this revision note)

  • The earliest event times must have been completed before the latest event times

How do I work out late event times?

• Starting at the sink node, this will be the same as the early event time

  • This is the minimum duration of the project

    • I.e. The shortest time in which all activities can be completed

• Working backwards from the sink node, look at each of the activities (arcs) leading back to an event (node)
  

  • For each activity after an event (node)

    • Subtract that activity's duration from the latest event time at its end node

  • The latest event time is then the minimum of these times

    • This is the latest time an event can be moved on from such that all activities can be completed in the minimum duration of the project

• Completing the latest event times is sometimes referred to as a backwards pass (through the activity network)

Total Float of an Activity

What is the total float of an activity?

• The total float of an activity is the time the start of an activity can be delayed

  • without affecting the minimum project duration

• Critical activities will have zero float

How do I work out the total float of an activity?

• For non-critical activities the total float can be calculated

  • Consider the earliest time it can commence and the latest time it can finish

• In words

• Formally, you may see this written as

  • The total float, , of activity , is defined to be

    • where  is the earliest time for event  and  is the latest time for event 

• Do not be put off by the word 'total'

  • The total float of an activity is one value for one activity!

Critical Path & Critical Activities

What are the critical activities?

• Critical activities
  

  • have a total float of zero

  • must commence at their earliest event time in order for the minimum project duration to be achieved

• A critical event is an event where the earliest start time is equal to the latest finish time

• To find the critical activities look for the critical events

  • If an activity is critical then the start and finish events must be critical

  • However the converse is not true

    • It is possible for a non-critical activity to be between two critical events 

• Critical activities should be stated as a list

  • E.g.  the critical activities are A, C, D and G

What is a critical path?

• A critical path through an activity network runs from the source node to the sink node

  • The activities along the path have a total float of zero

  • Each activity in a critical path will be critical

• The critical path would be given in terms of the activities/arcs along the path

  • E.g. a critical path encompassing activities A, C, E, F and G would be written A - C - E - F - G

• It is possible that there is more than one critical path through an activity network

  • E.g. A-C-D-G and A-C-E-F-G could both be critical paths

    • In this case the critical activities would be A, C, D, E, F, G

How do I find the minimum project duration?

• The minimum project duration is the quickest time in which all activities in a project can be completed

• It is found by completing the early event times for each node in an activity network

  • The early event time for the sink node is the minimum project duration