Critical Path Analysis – Solved Example

How To Calculate Critical Path
Schedule Network Analysis Example

Project scheduling tools like MS Project generate critical path with utmost ease. However, it is necessary to know the calculations in critical path analysis. Understanding basic concepts of critical path analysis not only help project managers but also assist pmp certification aspirants. Therefore, solving a critical path method problem improves knowledge of this technique. Hence, this post demonstrates steps in critical path analysis with a solved example.

Critical Path Analysis – An Overview

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

In order to establish a critical path it is necessary to draw a schedule network diagram. Therefore, critical path analysis also specifies link between project activities. 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.

Network Diagram for Critical Path Analysis

Techniques used to construct a project schedule network diagram are Arrow Diagram Method (ADM) and Precedence Diagram Method (PDM).

In order to construct the schedule network diagram, ADM uses Activity-on-Arrow (AOA) technique where as PDM uses Activity-on-Node (AON).

Program Evaluation And Review Technique (PERT) uses Activity-on-Arrow (AOA) where as Critical Path Method (CPM) uses Activity-on-Node (AON).

Most of the scheduling programs use PDM technique to construct schedule network diagram. Hence Critical Path Method (CPM) is the most popular schedule network analysis technique. Therefore, this article will demonstrate steps in critical path analysis with a solved example using CPM.

Critical Path Method (CPM) Example

Activity-on-Node (AON) method of constructing project schedule network diagram represents project activities on nodes. PMBOK uses the following nomenclature.

Precedence Diagraming Method Nomenclature

Critical Path Analysis Steps

To understand the steps involved in critical path analysis, consider the following schedule network diagram.

Schedule Network Diagram
Project Schedule Network Diagram

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

  • First path is Start (S) – A – D – E – End (E’) the duration of which is equal to 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 which is equal to 22 weeks
  • Fourth path is S – B – F – G – E’ the duration of which is equal to 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 above network diagram.

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

Schedule Network Diagram With Critical Path
Project Schedule Network Diagram With Critical Path

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.

Calculate ES and EF of activities on Critical Path.

First Node

The convention used here is that the project starts on day one. Another convention states that the project starts on day zero. We will stick to the convention indicated in PMBOK, which is, 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.

CPM Calculation Forward Pass on Critical Path
Calculate Early Start & Early Finish of Activities on Critical Path

Step – 4 : Perform Backward Pass on Critical Path

Perform backward pass to calculate Late Start and Late Finish.

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. Therefore, no backward pass calculation for activities on critical path. On completing the backward pass network diagram will look like this.

CPM-Calculation-Total Float Critical Path
Calculate Late Start & Late Finish of Activities on Critical Path

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 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

The following formula represents total float of an activity.

  • 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

Analysed Schedule Network Diagram
Analysed Project Schedule Network Diagram

Step – 8 : Calculate Free Float

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


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.

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

Visit the following page to know more about the formulas in critical path analysis.

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