Precedency Networks  Critical Path Analysis
Activities often depend on other activities' completion before they can be started. A precedence/dependency table shows us the information we need.
Above, we can see a precedence table. The figures in brackets represent the duration of the activity, i.e. the time required for its completion (in units of time).
The vertices are each individually numbered. The start vertex has a number 1 and the final vertex, known as the terminal vertex, has the largest number.
The direction of the arrows shows the order in which the activities must be completed. Each arrow must travel from a lowernumber node than its destination node.
Each pair of events in a network should only be linked by one activity.
A dummy activity has zero duration. They are represented by dotted lines and are needed to show the dependency of one activity on another.
Example:
Given a precedency table:
The vertices are each individually numbered. The start vertex has a number 1 and the final vertex, known as the terminal vertex, has the largest number.
The direction of the arrows shows the order in which the activities must be completed. Each arrow must travel from a lowernumber node than its destination node.
Each pair of events in a network should only be linked by one activity.
A dummy activity has zero duration. They are represented by dotted lines and are needed to show the dependency of one activity on another.
Example:
Given a precedency table:
We draw our basic precedency network:
Three dummies are needed in this network. This is because C depends on A but D depends on both A and B. Similarly, E and F both depend on C, but G depends on C and D. And finally, since activities may not have identical start and end nodes and H depends on both E and F, we could have a dummy going either from E’s end node to F’s end node – in which case activity H would protrude from F’s end node instead, or vice versa, as above (where H protrudes from E’s end node.
Then we add the numbers to the nodes so that each activity travels to a node of a higher value:
Then we add the numbers to the nodes so that each activity travels to a node of a higher value:


Now to add the earliest and latest times to the events, we draw our boxes at the nodes such that:
Earliest event time (EET)
The earliest event time for node i is denoted by eᵢ and represents the earliest time of arrival at event i with all dependent activities completed. EETs are calculated using a forwards pass from the start node to the terminal node.
Latest event time (LET)
The latest event time for vertex i is denoted by ☆ᵢ and represents the latest time that event i may be left without extending the time for the project. These times are calculated using a backwards pass from the terminal node back to the start node.
The critical path is the longest path through the network. The activities on this path are the critical activities. If any critical activity is delayed, then this will increase the time needed to complete the project. The events on the critical path are the critical events which each have zero float – for each of these eᵢ = ☆ᵢ. Here, the critical path is ACEH, time = 15. To see out how to find the critical activities, check out the ‘scheduling’ sections.
The earliest event time for node i is denoted by eᵢ and represents the earliest time of arrival at event i with all dependent activities completed. EETs are calculated using a forwards pass from the start node to the terminal node.
Latest event time (LET)
The latest event time for vertex i is denoted by ☆ᵢ and represents the latest time that event i may be left without extending the time for the project. These times are calculated using a backwards pass from the terminal node back to the start node.
The critical path is the longest path through the network. The activities on this path are the critical activities. If any critical activity is delayed, then this will increase the time needed to complete the project. The events on the critical path are the critical events which each have zero float – for each of these eᵢ = ☆ᵢ. Here, the critical path is ACEH, time = 15. To see out how to find the critical activities, check out the ‘scheduling’ sections.
See the ‘scheduling’ sections for important details on how to use this precedence network.