Critical Path Method

by Buck Jefferson

The Critical Path Method is one big, bad mutha.  Using it, you can find the critical activities that cannot be delayed. 

Similarly, you can also find the non-critical activities that can be delayed without delaying the project. 

For those non-critical activities, the Critical Path Method computes how much "slack" you have in getting them done. 

Finally, the Critical Path Method computes the start and finish dates for all your activities.  What a creation!

Finding your critical activities helps you allocate your resources where they are needed most.  The "slack" or "float" of your non-critical activities is simply the "extra" time you can use to finish the activity without delaying the project completion date.

You'll no doubt hear about lags and leads.  For instance, a lag means 10 days must pass AFTER A before B can begin.  A lead means B can start 10 days BEFORE A finishes.  Don't worry at this point if you're confused.

Critical Path Method: An Example

Let's refer again to the get-ready-for-work activities identified in the example discussed elsewhere:

A) Wake up

B) Wash face/Brush teeth

C) Style hair

D) Get dressed

E) Drive to work

F) Watch TV

G) Eat breakfast

H) Park

The first thing we need to do is estimate how long each activity will take.  Doing that, we come up with:

A) Wake up = 5 minutes (I'm slow and groggy)

B) Wash face/Brush teeth = 5 minutes (the water needs time to get warm)

C) Style hair = 2 minutes (unless it's a bad hair day)

D) Get dressed = 5 minutes (I suppose I should pick out my outfit the night before)

E) Drive to work = 20 minutes (this is a lie, but it makes the example easier)

F) Watch TV = 10 minutes (how many times can you watch the traffic/weather report?)

G) Eat breakfast = 10 minutes (I'm an all-day snacker)

H) Park = 3 minutes (the lot is always full)

Now, for Critical Path Method computations, we use the AON diagram.  If you remember, our AON looks like this:

Figure 1: Activity-on-Node (AON) Diagram

 

Since we'll be making a lot of calculations, we need room to write in some numbers.  Each node will have little boxes for that purpose.  Let's magnify just the "A" node to really understand this:

 

Figure 2: Node A

• Duration is the amount of time our activity takes to complete.
   

• The Early Start Time (EST) is the EARLIEST possible date you can START the activity.
   

• The Early Finish Time (EFT) is the EARLIEST possible date you can FINISH the activity.
   

• The Late Start Time (LST) is the LATEST possible date you can START the activity.
   

• The Late Finish Time (LFT) is the LATEST possible date you can FINISH the activity.
   

• Float is the amount of time an activity can be delayed from its early start time without delaying the project's end date.

Our goal is to find the float times for all the activities, which will enable us to determine the critical path.

In order to compute the floats, we need to compute the other numbers (EST, EFT, LST, and LFT) first.

Now, we must run through each activity and fill in what we know.  The first time we do this is called a forward pass.  The main formula you must use here is:

EFT = EST + Duration

Using activity A, we get:

Figure 3:  Node A

 

Since A is the first activity, our initial EST is 0.  The duration of A is 5 minutes.  And using the formula above, we get:

EFT = (EST + Duration) = (0 + 5) = 5

Let's continue along our network diagram.  Remember, during our forward pass we're just computing the EST, EFT, and duration for each activity.  We'll get the other numbers later, during our backward pass.

According to our diagram, both F and B are our next activities.  Let's do B first (it doesn't matter which).  B also has a duration of 5 minutes.

Figure 4: Nodes A and B

OK, since A comes right before B, you'll notice that the EFT for A (which is 5) becomes the EST for B.  So using our formula, for B we get:

EFT = (EST + Duration) = (5 + 5) = 10

That's B.  Now let's do F, since F is also immediately after A in our AON diagram.  F has a duration of 10 minutes.

Figure 5: Nodes A and B

Just like for B, the EFT for A becomes the EST for F, since F comes right after A.  With an EST of 5, and a duration of 10, we get:

EFT = (EST + Duration) = (5 + 10) = 15

With me so far?  We've finished A, B, and F.  The next one we do now is C, since it comes right after B on our AON diagram.  We can't do D before C, since in the diagram, C has to finish before D begins.  C is easy, so let's do it.

Figure 6: Nodes A, B, C, and D

The EFT for B becomes the EST for C, since C comes right after B.  With an EST of 10, and a duration of 2, we get:

EFT = (EST + Duration) = (10 + 2) = 12

The next activity is D, which follows both F and C.  Let's add D to our diagram:

Figure 7: Nodes F, C, and D

Uh-oh.  What's wrong here?

We want to write in D's EST, but there are two arrows pointing to D.  What do we do now? 

For the previous activities, it was easy to compute the EST.  We simply used the preceding activity's EFT.  But now… we have two preceding activities.  Do we use the EFT from F, or the EFT from C?

Good question, because you will encounter this a lot using Critical Path Method.  The answer is this:

When faced with multiple preceding activities, choose the one with the GREATEST EFT.

In this case, F's EFT is 15.  C's EFT is 12.  We choose F since 15 > 12.

Why do we choose the greater value?  Because D cannot start until both F and C finish.  Since C finishes at 12 minutes, we cannot start D yet because F isn't finished.  F won't be finished until 15 minutes. 

D must wait 15 minutes until beginning.  Therefore, D's EST is 15.  Since D's duration is 5, we now have:

EFT = (EST + Duration) = (15 + 5) = 20

Updating our diagram, we get:

Figure 8: Nodes A, B, C, D, and F

Our next two activities are E and G, which both directly follow D.  Since D points to both of them, the EFT for D (which is 20) becomes the EST for both E and G.  E's duration is 20, and G's duration is 10.  Therefore,

for E: EFT = (EST + Duration) = (20 + 20) = 40

for G: EFT = (EST + Duration) = (20 + 10) = 30

Our new graph is now:
 

Figure 9: Nodes A, B, C, D, E, F, and G

All right, there's just one activity left, and it's H.  Once again, we are faced with two preceding activities.  Now that you know how to deal with those, finishing up is a snap.

E's EFT is 40, and G's EFT is 30.  We choose the higher value, so H's EST becomes 40.  With a duration of 3, we get:

EFT = (EST + Duration) = (40 + 3) = 43

We're done with our forward pass.  We're 33% done with our Critical Path Method example.  So far, it looks like this:
 

Figure 10: Critical Path Method (Forward Pass)

Now it's time for the backward pass.  It's just like the forward pass, except we go in the opposite direction.  We started with A last time, and now that we're going backwards, we start with our last activity: H.
 

Figure 11: Node H

The purpose of the backward pass is to compute the Late Finish Time (LFT) and Late Start Time (LST) for each activity.  Our new formula to help us out is:

LST = LFT – Duration

We need an initial LFT to get us going.  Where do we get it?  Well, by definition:

LFT of the last activity = EFT of the last activity

Our last activity is H, and its EFT is 43.  Voila, our initial LFT is 43.  Using the above formula, we get for H:

LST = (LFT – Duration) = (43 – 3) = 40

Working backwards, our diagram becomes:

 

Figure 12: Node H

One activity down, seven to go.  Working backwards, E and G are next.  You guessed it, the LST of H (which is 40) becomes the LFT of both E and G.  First, let's do E (again, it doesn't matter which):

LST = (LFT – Duration) = (40 – 20) = 20

 

Figure 13: Node E

And for G:

LST = (LFT – Duration) = (40 – 10) = 30

 

Figure 14: Node G

And our diagram:

 

Figure 15: Nodes E, G, and H

We're ready to compute activity D.  Remember, we take the LST from the succeeding activity to get the LFT of the current activity. 

Once again, uh-oh.

D precedes both E and G.  We have two LST's to choose from, so which do we pick?

For the forward pass, we picked the greater value.  But for the backward pass, it's the opposite:

When faced with multiple succeeding activities, choose the one with the lowest LST.

In this case, E's LST is 20.  G's LST is 30.  We will choose E since 20 < 30.

Therefore, D's LFT is 20.  Since D's duration is 5, we now have:

LST = (LFT - Duration) = (20 - 5) = 15

Updating our diagram, we get:
 

Figure 16: Nodes D, E, G, and H

Let's keep going.  C and F are next, and their computations are straight-forward.  They both precede D, so both C and F have an LFT of 15. 

For C: LST = (LFT - Duration) = (15 - 2) = 13

For F: LST = (LFT - Duration) = (15 - 10) = 5

And our diagram is now:

 

Figure 17: Nodes C, D, E, F, G, and H

Quickly taking care of B, which comes right before C:

LST = (LFT - Duration) = (13 - 5) = 8

 

Figure 18: Nodes B, C, D, E, G, and H

Only one activity left, and that's our first one: A.  Much like D, activity A has two succeeding activities: B and F.  Remember, in this instance we take the lower LST of them.  B's LST is 8, and F's LST is 5, so we use 5.

LST = (LFT - Duration) = (5 - 5) = 0

It makes sense for our first activity to have an LST of 0, so we did this correctly.  We're now 66% done, and our updated diagram becomes:
 

Figure 19: Critical Path Method (Backward Pass)

Looks like a beauty. 

The hard stuff is done.  You'll notice that only one box remains, and that's the box containing the value of the float.  Remember, the float is the amount of time an activity can be delayed from its early start time without delaying the project's end date.

Luckily, computing the float is a breeze.  We use a nifty little formula that says:

Float = LFT – EFT

Using that for each activity, we can simply calculate the following float times:

For activity A: Float = (LFT – EFT) = (5 – 5) = 0

For activity B: Float = (LFT – EFT) = (13 – 10) = 3

For activity C: Float = (LFT – EFT) = (15 – 12) = 3

For activity D: Float = (LFT – EFT) = (20 – 20) = 0

For activity E: Float = (LFT – EFT) = (40 – 40) = 0

For activity F: Float = (LFT – EFT) = (15 – 15) = 0

For activity G: Float = (LFT – EFT) = (40 – 30) = 10

For activity H: Float = (LFT – EFT) = (43 – 43) = 0

We now have all our values.  The diagram looks more complete now:

 

Figure 20: Critical Path Method (with float)

We're 99% done.

All those numbers are nice, but what we're after is the CRITICAL PATH.  Fortunately, that's easy now, because we just did all the dirty work. 

Figuring your critical path is as simple as selecting those activities that have the lowest float values.  9.9 times out of 10, that value will be 0.  And whaddya know, it's 0 here.

Based on our figures, these activities have zero float: A, F, D, E, and H.

And there you go, we have our critical path!  Let's bold and highlight it for all the world to see:

 

Figure 21: Critical Path Method (Final)

Now that you know your critical path, you know which activities are more important than others to finish.  Now you know that if you're running late, brushing your teeth and styling your hair can be delayed a little, but that watching TV is critical.  Isn't it always. 

There are some projects with unimaginable lists of activities, and doing the critical path methods on them would be suicide.  Thankfully, that's what software programs are for.  Any tool like Microsoft Project can easily whip out Critical Path Method's and network diagrams for you. 

Quite honestly, it's quicker to do it by hand for smaller projects.  But if you're building the Sears Tower, use a computer.

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