# Critical Path Analysis Solved Example

Project scheduling tools like MS Project generate critical path with utmost ease. However, it is necessary to understand the critical path analysis calculations. Understanding basic concepts of critical path analysis not only help project managers but also assist pmp certification aspirants. Moreover, solving a critical path analysis example problem also improves knowledge of CPM technique.

Today, Critical Path Method (CPM) is the most popular schedule network analysis technique. Therefore, this post demonstrates steps in critical path analysis with a solved example using critical path method technique.

## Critical Path Analysis Definition

Critical path analysis is the process of identifying the longest path in a schedule network diagram. The analysis not only indicates completeness of project schedule but also ascertains degree of scheduling flexibility. Moreover, critical path analysis also reveals the minimum duration required to complete a project.

Critical path analysis also specifies link between project activities. It is important to realize that not all activities can start and finish on time. Hence critical path analysis specifies permissible delays to activities that are not on critical path. Thus schedule flexibility is the permissible delay that does not affect project completion date.

## Critical Path Analysis Schedule Network Diagramming Techniques

The first activity of critical path analysis is to draw a schedule network diagram. The critical path analysis uses Precedence Diagram Method (PDM) to construct the schedule network. Further, PDM method uses Activity-on-Node (AON) diagramming technique to represent the schedule logic. In AON the nodes represent the schedule activities. Most of the scheduling programs use PDM technique to construct project network diagram. Therefore, critical path analysis example problem uses activity-on-node method to represent the schedule network logic.

The following figure represents PMBOK nomenclature for activity nodes in precedence diagram method.

## Critical Path Method Example With Solution

To understand the critical path analysis calculation steps consider the following schedule network diagram.

## Step – 1 : Calculate the total number of paths and their duration.

The first and the most crucial critical path analysis step is to identify the critical path. To achieve this first identify all the paths in the network. The schedule network diagram shown above has four paths. The path with longest duration is the critical path. The following indicates description of all the paths.

• First path is Start (S) – A – D – E – End (E’) the duration of this path is 16 weeks
• The second path is  S – A – E – G – E’ the duration of which is also equal to 16 weeks
• The third path is S – B – C – E – G – E’ the duration of this path is 22 weeks
• Fourth path is S – B – F – G – E’ the duration of this path is equal to 20 weeks

The longest path in the network above is S-B-C-E-G-E’ with a duration of 22 weeks. Hence, path S-B-C-E-G-E’ is the critical path of the above schedule network diagram.

## Step – 2 : Indicate the Critical Path

Indicate the critical path on the network diagram with a bold line. The network diagram with critical path will look as follows

## Step – 3 : Perform Forward Pass on Critical Path

The next step is to calculate early start and early finish of each activity. We need to start with activities on critical path.

Step – 3.1 : Calculate Early Start and Early Finish of activities on Critical Path.

First Node

Here, it is essential to point out that critical path analysis has two conventions for calculation. This critical path method (cpm) example problem uses the convention that project starts on day one.

Another convention for cpm analysis states that the project starts on day zero.

However, we will stick to the convention indicated in PMBOK, which states that, the project starts on day 1. Hence ES of first activity B on critical path is 1.

• EF = ES + Activity Duration – 1
• EF of Activity B = 1 + 6 – 1 = 6

Node B

• ES = EF of first node + 1 = 6 + 1 = 7
• EF = ES + Activity Duration -1 = 7 + 4 – 1 = 10

Repeat the above step till you reach the last node

Once the forward pass is complete the network diagram will look as follows. The following diagram looks different because of spreadsheet calculations. However, the schedule network logic has not changed.

## Step – 4 : Perform Backward Pass on Critical Path

Perform backward pass to calculate Late Start and Late Finish.

Step – 4.1 : Calculate Late Start and Late Finish of activities on critical path

The total float of activities on critical path is zero. Hence on critical path LS = ES and LF = EF. Therefore, no backward pass calculation for activities on critical path. Once backward pass calculation is complete the network diagram will look as follows.

## Step – 5 : Perform Forward Pass on Activities Not On Critical Path

Node A

• ES = 1 and EF = 1+4-1 =4

Node D

• ES = 4 and EF = 16

For activities with more than one preceding activity ES is latest of the earliest finish times of the preceding activities

When we have ES and EF of a particular node we can calculate the Total Float using the formula

Total Float = LS – ES or LF – EF

## Step – 6 : Perform Backward Pass on Activities Not On Critical Path.

Node F

• LF = LS of Previous node -1 = 18 – 1 = 17
• LS = LF – Activity Duration + 1 = 17 – 9 + 1 = 9

Node D

• LF = 22 since this is the last activity not on critical path it can finish on week 22
• LS = 22 – 12 + 1 = 11

Node A

This node connects to two different activities D and E. In such conditions LF of node A is the earliest of the latest start times of the preceding activity. In this case it is same hence

• LF of node A is 11 – 1 = 10 and
• LS = 10 – 4 + 1 = 7

## Step – 7 : Calculate Total Float

The following formula represents total float of an activity.

• Total Float = LS – ES or LF – EF

Consider the first node, Node A.

Here LS = 7 and ES =1, therefore total float of Node A = 7 – 1 = 6.

On critical path LS = ES and LF = EF hence total float is equal to zero.

Once backward pass and total float calculation is complete the resultant network diagram will look as follows.

## Step – 8 : Calculate Free Float

For formula to calculate free float please refer to the following post

For tools and techniques to establish logical relationships between project activities, please read the following post.

Also read the following post for a comprehensive list of project management formulas for pmp examination.

Visit the following page to know more about the formulas in Critical Path Method (CPM).

Take the Critical Path Method Quiz

## Conclusion

To summarize, critical path analysis reveals network information such as critical path, total float, and activity float. This data forms the basis for further project execution. Moreover, other advanced schedule analysis techniques use output of critical path analysis.

## 19 thoughts on “Critical Path Analysis Solved Example”

1. Denis says:

your blog is quite illustrative and easiest to comprehend, that I found on the net.
Thank you.

2. Atul Gaur says:

Reblogged this on Atul Gaur.

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