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Manual methods – part 3e – task flow charts

Activity-on-Arrow networks

Look at the diagram below for the tasks identified earlier under the Activity-on-Node network.
There are some fundamental differences.

Firstly, the tasks lie on the ‘arrows’ and not the ‘bubble’. Hence, each of the arrows is labelled according to the task it represents.

That is, the arrows represent activity only. Below each arrow is the task duration (green boxes).

AOA network arrow diagramAOA network arrow diagram large

The circles here represent ‘events’ (light grey boxes) and NOT task activities. This is very important to remember later when looking at the information attached to each of the ‘event’ circles.

In these diagrams the ‘start’ and ‘end’ are also events. An activity arrow will always begin at an event and end at an event.
The event circles appear in position of occurrence with time going to the right, as for the Activity-on-Node network. Notice that the event circle at the end of task #5 (event ‘D’) is a little to the right of the event circle at the end of task #1 (event ‘A’).

In the Activity-on-Node network we had 6 tasks and only two events, the ‘start’ and the ‘end’ of the project.
For the Activity-on-Arrow network we still have 6 tasks (represented by the arrows) but we now have a total of 6 events as well.
This is a common trait for Activity-on-Arrow networks it puts the focus on events and not tasks.

Earliest Time (ET)

Note right from the start that the information is attached to the ‘event’ AND NOT the task.
An event is a moment in time that indicates when a task is complete and as such does not have any activity associated with it.
Hence, it CANNOT have a ‘start time’ or a ‘finish time’. It can only have a point in time within the project that contains it.

Thus, the Earliest Time (ET) for an event is the first project day it can exist.
The Earliest Time (ET) for the ‘start’ will always be ZERO, that is project day 0. The duration of task #1 is 2 days so the earliest event ‘A’ can occur is project day 2, its Earliest Time (ET).

Likewise, for event ‘B’ and ‘C’ they will have Earliest Times (ET) of project day 6 and project day 12.
Also, event ‘D’ will have an Earliest Time (ET) of project day 4.
The ‘end’ event will have an Earliest Time (ET) of project day 15. This is how long the project will last and agrees with the Activity-on-Node network.

Latest Time (LT)

In a similar fashion to the Activity-on-Node network we get these by working backward from the ‘end’ event.
These are the blue boxes.
For task #4 the event circle has a Latest Time (LT) of project day 15, which is the project ‘end’.
The event circle after the completion of task #3 becomes 12, that is the Latest Time (LT) of task #4 minus the duration of task #4.
In similar fashion, the completion events associated with tasks #2 and #1 have Latest Times (LT) of 6 and 2 project days.
The ‘start’ is of course at zero project day.

If we now look at the event for the completion of task #6 we see this is also the ‘end’ event.
Moving backwards to task #5 completion event the Latest Time (LT) will be 15 minus the duration of task #6, that is 2 days.
Hence, the Latest Time (LT) will be project day 13.

Note that the slack for this event ‘D’ is 13 – 4 = 9 (the orange box). This happens to be the same as the float for task #5 and task #6. The reason that these are the same is because event ‘D’ only has one task going into it.
If more than one Activity Arrow were going into the event ‘D’ then the situation could be different.
This is shown better next.

Critical path

When the slack for both events at each end of a task are zero it will be on the critical path.


We know that float applies to the task and not to the event from the completion of the task.
The total float for a task can be found from the Activity-on-Arrow network.

For task #5
Total Float = [Latest Time (event ‘D’) – Earliest Time (event ‘A’)] – duration of task #5
Total Float = [13 – 2] – 2 = 11 – 2 = 9 days.

Remember, this is the number of days that the task can be delayed before it becomes part of the critical path.

We can also calculate the Free Float.

Free Float = [Earliest Time (event ‘D’) - Earliest Time (event ‘A’)] - duration of task #5
Free Float = [4 -2] – 2 = 0

We can see from this that task#5 has zero Free Float, meaning that any delay will affect the start of task #6.

In addition, Total Float – Free float = 9 – 0 = 9 days
Which is also,

[Latest Time (event ‘D’) – Earliest Time (event ‘A’) – duration of task #5] - [Earliest Time (event ‘D’) - Earliest Time (event ‘A’) - duration of task #5]

Hence, we have
[Latest Time (event ‘D’) – Earliest Time (event ‘A’) – duration of task #5 - Earliest Time (event ‘D’) + Earliest Time (event ‘A’) + duration of task #5]

Which becomes,

[Latest Time (event ‘D’) - Earliest Time (event ‘D’)] = 13 – 4 = 9 days

Notice in this case that the ‘slack’ for event ‘D’ = Latest Time – Earliest Time = 13 – 4 = 9 days.
This happens to be the same as the Total Float for task #5.
This will be the case when only one task completes at an event.
In this case, it is saying that event ‘D’ can finish any time between project day 4 and project day 13, a ‘slack’ of 9 days.

Comparison of Activity-on-Node networks and Activity-on-Arrow networks

Both types of flow chart will give you the correct schedule. Both will give you the same overall time scale for the project.
The Earliest Time of an event will be the Earliest Start Time of the task that follows it.

The arrow-on-arrow technique is not specifically referenced in PRINCE2® 2009 but the arrow-on-node technique is described under PRINCE2 2009.
PRINCE2 2005 does not specifically refer to either technique.

An activity-on-node diagram (sometimes called an arrow diagram) can be used to schedule dependent activities within a plan.
It helps a Project Manager to work out the most efficient sequence of events needed to complete any plan and enables the creation of a realistic schedule.
[see Plans - The PRINCE2 approach - Prepare the schedule - Define activity sequence]

Under PRINCE2 2009 [see ‘The Complete Project Management plus PRINCE2’] planning is covered by the Plans theme.
The purpose of the Plans theme is to facilitate communication and control by defining the means of delivering the products (the where and how, by whom, and estimating the when and how much).
[see Plans - Purpose]

Identifying the plan activities and dependencies for a schedule are also covered within this theme.
[see Plans - The PRINCE2 approach - Identify activities and dependencies]

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