19.2 Critical Path - StudyNinja

HSC Standard Maths Resources

Browse: 1. Home  »  19. Critical Path Analysis  » 19.2 Critical Path

19.2 Critical Path

When it comes to activity charts, there are some paths that do not impact the overall timeline of the project even if delayed to a point.

However, there is a critical path in the activity chart that can delay the overall timeline of the whole project if delayed. This module is about identifying that critical path.

To identify a critical path, we will work through an example to understand all the processes, but to start-off, the following is a summary of the most important things that will make your life very easy if you memorize.

Don’t worry even if it doesn’t make sense now; it will make sense once we do the example.

Congrats you have now done the hardest part of this section! Now, it is only a matter of seeing all these in action in an example; it will all make sense.


Consider the following activity chart, identify EST, EFT, LST, LFT, Float for each node and use it to identify its critical path.


As the first step, it is a good idea to redraw the chart with the matrices along with activities and weights filled in for each node:

Next, we will work out EST and EFT. This would be done through a process called ‘Forward Scan’ where EST and EFT for each node would be calculated one-by-one from left to right.

EST and EFT for nodes A to D are filled below:

As an example, for node A: EST = 0 (since it is the starting node), EFT = EST + weight = 0 + 2 = 2.

For node B, EST = EFT of node A = 2, EFT = EST + weight = 2 + 2 = 4 and so on.

When it comes to node E, there is a problem in calculating EST, as two nodes (C and D) are coming into it. In this kind of situation, always choose the highest value as EST; it will be 7 in this case.

The following are the completed EST and EFT for the whole chart:

Next, let’s work out LST and LFT. It would be calculated through a process called ‘Backward Scan’ where LST and LFT for each node would be calculated one-by-one from right to left.

LST and LFT for nodes F to B are filled as below.

As an example, for node F: LFT = EFT = 12 (since it is the last node), LST = LFT – weight = 12 – 3 = 9.

For node E: LFT = LST of node F = 9, LST = LFT – weight = 9 – 2 = 7 and so on.

At node A, we are again stuck with a situation where we have to choose between LST values of node B or C as LFT value for node A.

In backward scan, always choose the lower of the LST values for all relevant nodes; so, in this case, LFT for node A = Lower of LST of node B or node C = 2.

The following is the completed chart:

Now, to update float values, simply use formula LST – EST. The following chart has float values updated:

The critical path is simply the path where all floats have value of 0; it is ABDEF in this case.

Before concluding this section, let’s take a moment to think about what some of these numbers are telling us. You can either look at starting times or finishing times.

For example, any activity on path ABDEF has same EST and LST indicating you cannot afford to delay any of these activities, and if you do so, your overall timeline of the project will be affected.

However, notice on node C EST is 2 and LST is 6, which means that the earliest you can start this activity is at time 2 (you cannot start it before activity A is finished which takes 2 days) and the latest you can start this activity is at time 6.

Why 6? This is because even if you start at day 6, this activity will take a day to finish and will be completed by day 7 when activities ABD will be finished at their earliest and activity E starts.

So, the latest you can start activity C is day 6. The same thing is indicated by EFT and LFT but from the perspective of finishing times instead of starting times.

Float is just the difference between LST and EST, and if it is not 0, it means that given activity has some time buffer and can be delayed, which means it is not critical to the overall project.

Get access to 20 Mock Exams with over 700 exam-style questions for HSC Standard Maths.

Click here to check them out!!

For the other sections:

– 19.1 Activity Chart

19.3 Max-Flow Min-Cut

The following is the type of questions you can expect in exam:

19.4 Exam Questions

Study notes of this section and other resources can be accessed here:

19.5 Study Notes & Resources

Browse: 1. Home  »  19. Critical Path Analysis  » 19.2 Critical Path

Check out all the hacks for this exam. Full playlist here