Critical Path Analysis (Edexcel A Level Further Maths: Decision Maths 1)

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

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 leading to an event
  • (or, look at each of the arcs ending at a node) 
  • for each activity's starting node, work out the earliest time it can start
    • this would be the earliest event time from its start node add its duration
  • the earliest event time is then the maximum of these times
    • this is the earliest time an event ('stepping stone') can be moved on from

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

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 leading back to an event
  • (or, look at each of the arcs starting at a node) 
  • for each activity's starting node, work out the latest time it can start
    •  this would be the latest event time from its end node subtract its duration
  • the latest event time is then the minimum of these times
    • this is the latest time an event ('stepping stone') 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)

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 by 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 by considering 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!

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 such that the activities along the path have a total float of zero
  • Each activity in a critical path will be critical
• Where events/nodes are labelled with 0, 1, 2, etc the critical path would be given in terms of the events the path passes though
  • e.g.  a critical path starting at source node 0, passing through events 2, 4, 5 and finishing at sink node 8 would be written 0 - 2 - 4 - 5 - 8
• Where events/nodes are not labelled, 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