### Activity on Arrow Diagrams

When we teach students about the Critical Path Method in our PMP^{®} exam preparation courses, we use the Activity on Node (AoN) technique exclusively to illustrate the concepts. However, there is an Activity on Arrow (AoA) representation that might be of interest to advanced schedulers.

An important point to note at the outset is that AoA is no longer referenced in the Guide to the Project Management Body of Knowledge (PMBOK^{®} Guide), so if you are currently studying for the PMP^{®} exam, this will be of no relevance.

AoA is very similar to AoN – the forward and backward passes are calculated in exactly the same way, but the layout of the diagram is different and can be difficult to interpret.

If you understand the AoN format, then you should have no difficulty coming to grips with AoA. Remember the AoN node format from the Project Schedule Management knowledge area in your PMP^{®} course notes?

Coming into the exercise, we knew the Activity Names and their durations. The names came from the Define Activities process while the durations were estimated during Estimate Activity Durations. To determine the critical path, we have to use a forward pass to determine the Early Start and the Early Finish. Then the backward pass gives us the Late Finish and Late Start, allowing us to calculate the float by subtracting the Late Finish from the Early Finish.

While the mechanics of AoN and AoA are the same, the information is presented differently.

In AoA diagrams, the Activity Name and the Activity Duration are specified on the arrow between two nodes. Then each node contains three numerical values – the Start Time, the Finish Time and the Float. This layout has consequences for the forward and the backward passes. In the forward pass, we calculate the Start Times of all the nodes.

In the example above, we take the initial Start Time of “0” and add the Duration for the first activity (“A1”) to get the Start Time for the second node. Then we proceed by adding the Duration for Activity “A2” to the Start Time in the second node to get the Start Time for the third node.

Just like in AoN representations, if we arrive at a point in the diagram where a node has several predecessors, then the largest Start Time from among the predecessors is added to the activity’s duration.

With the backward pass, the end node’s Start Time becomes its Finish Time and the backward pass proceeds by subtracting the durations of the activities leading to the end node from the end node’s Finish Time to arrive at the Finish Times for the preceding nodes. As with the AoN approach, when a node has several successors, the smallest of the successors’ Finish Times is used to base the node’s Finish Time on. The following diagram shows a completed AoA diagram.

In this example, the float has also been calculated. This is more straightforward than in the AoN equivalent because it simply involves subtracting each node’s Start Time from its Finish Time.

If you find the AoA method more appealing than the AoN, you might well ask why is AoN given more prominence in the scheduling world? Given that the techniques are broadly similar, why not use one or the other as you prefer? One reason is because AoA does not illustrate parallel activities as clearly as AoN. Suppose we have three activities – A1, A2 and A3 – that may be carried out in parallel. A1 is expected to take 3 days, A2 2 days and A3 5days. Using AoN, we may represent this as:

This clearly shows the critical path – activity A3 needs to be managed carefully. However, look at the equivalent AoA representation:

This is not so clear. Okay, we have highlighted the critical path, but it is not so obvious that the A1 and A2 are not on the critical path. AoA gets around this problem by introducing “dummy activities” as shown:

It is not surprising that AoN won out – it is a lot easier to use in these circumstances.

Velopi only covers the AoN approach to Critical Path Analysis in its PMP^{®} exam preparation courses, as well as everything else you need to obtain PMP^{®} accreditation. Please visit our training page or contact us directly for more details.