# Critical Path Analysis – Solved Example.

Arrow Diagram Method (ADM) and Precedence Diagram Method (PDM) are the two methodologies used to construct a project schedule network diagram. ADM uses Activity-on-Arrow (AOA) technique to construct the schedule network diagram where as PDM uses Activity-on-Node (AON) technique. The schedule network analysis methodologies associated with these techniques respectively are Program Evaluation And Review Technique (PERT) and Critical Path Method (CPM). Since most of the scheduling programs use PDM technique to construct the schedule network diagram the most popular network analysis technique these days happens to be the Critical Path Method (CPM). Hence this article will demosntrate steps involved in constructing the project schedule network diagram, identifying the critical path of the schedule, and conducting a complete project schedule network analysis using the Critical Path Method (CPM).

## Critical Path Method Analysis – An Overview

The process of drawing the schedule network diagram begins with identifying the activities for the project, sequencing the activities, estimating the activity resources, applying the resources to the activities, estimating the activity duration and then finally developing the project schedule. A project schedule can be considered complete only when a critical path for the project is established and the degree of flexibility in the schedule established. A project critical path indicates the longest path and the minimum project duration required to complete the project. Since the critical path can be established only after drawing the network diagram, we also learn how each activity in the project is related to other activities and what is the degree of flexibility in the project schedule. Not all activities can start and finish on time, hence network analysis specifies the extent to which activities that are not on critical path are permitted to be delayed without affecting the project completion date.

## Critical Path Method (CPM) Example

In Activity-on-Node (AON) methodology of construting project schedule network diagram the project schedule activities are represented on nodes. The nomenclature used by PMBOK is appended below.

Consider the following schedule network diagram

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

1. Start (S) – A – D – E – End (E’) = 16 weeks
2. S – A – E – G – E’ = 16 weeks
3. S – B – C – E – G – E’ = 22 weeks
4. S – B – F – G – E’ = 20 weeks

The longest path is S-B-C-E-G-E’ with a duration of 22 weeks hence this is the critical path of the project.

## Step – 2 : Indicate the Critical Path

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

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

Once we have established the critical path we need to calculate the early start and early finish times of each activity in the schedule. We need to start with the activities on critical path.

### Calculate ES and EF of activities on Critical Path.

#### The First Node

The convention used is that the project starts on day 1. There is other conventions also which states that the project starts on day 0. We will stick to the convention indicated in PMBOK and consider 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
##### The Second Node
• 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 indicated in the figure below. The network diagram may look different as the calculations have been performed on a spreadsheet however, the schedule network logic has not changed.

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

On reaching the last node and finishing with EF it is time to perform backward pass to calculate LS and LF

### Calculate LS and LF of activities on critical path

The total float of activities on critical path is zero, hence on critical path LS = ES and LF = EF.

The backward pass of activities on critical path requires no calculation and can be completed in no time, the network diagram will look something like this.

## 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 which is not on critical path it can finish on week 22
• LS = 22 – 12 + 1 = 11

### Node A

This node has two activities connected to it i.e 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 & Free Float

Once we have all the variable we can calculate the total float of any activity is represented as

• Total Float = LS – ES or LF – EF

Upon completion of backward pass and calculation of total float the resultant network diagram will look as follows

For formula to calculate free float and definition of various terms used in Critical Path Analysis please refer to my post

## Conclusion

A schedule network diagram with all the relevant data like critical path, total float, activity float forms the basis for further project execution, probabilistic determination of schedule completion dates, what-if analysis of the schedule model, resource optimization, crashing of project schedule, and risk analysis.

For tools and techniques used to establish logical relationships between project activities, please read my post

For the complete list of formulas used for the schedule network analysis visit

Hope this was useful, if there are any queries you may use the contact form available on Contact page.

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