what is critical path method in operation research

What Is the Meaning of the Critical Path Method in Operations Research?

The critical path method, a product of operations research, helps you set timetables for complicated projects made up of multiple interdependent tasks. Getting these kinds of projects done in the most efficient manner means determining which tasks to complete in which order -- and identifying which tasks must be done on schedule and which have some wiggle room.

Operations Research

Operations research involves applying mathematical concepts to decision making. When you have a project or proposal you're considering, you set it up much like a math problem: You identify all your variables, such as time requirements and costs, and define your desired outcome. You then "solve the equation" -- or, more likely, you have a computer do it by running simulations until it finds the most efficient solution. Operations research has generated an array of tools now common in business, including the critical path method.

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Interlocking Tasks

Some projects are straightforward, with one task leading to the next until the project is done, much like an assembly line. Other projects, though, may be made up of dozens, hundreds, even thousands of separate tasks, many of them interdependent. For example: You must finish Task A before you can start Task B; Task C can start anytime, but you can't start Task D until both C and A are finished; Task E depends on completion of B, and so on. Construction projects are often like this, but smaller-scale projects, such as switching to a new computer system or remodeling a store, can get complicated, too.

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Key milestones in projects, how to draw a schedule for a large construction project, critical path planning and scheduling, why scheduling issues affect project management, examples of time constraints in project planning, on the critical path.

In the critical path method, you lay out all the tasks involved in the project, determine how long it will take to complete each task and identify how each task is dependent upon other tasks. With that information, you can identify a target date for completing the whole project. For each task within the project, you identify the date by which the task should start and the date by which it should be finished. Some tasks will have "slack" or "float," meaning they can start late or finish late without upsetting the overall timetable. Other tasks, however, must be completed on time or the entire project will fall behind. Those tasks are said to be on the "critical path." A day's delay in a critical path task sets the whole project back a day. Project managers commonly use computers to help them identify the critical path within complicated projects.

When the Path Veers

The critical path in a project can take detours. Say you've got a peripheral task with plenty of slack, something that can take place anytime within a three-month span. That project may not be on the critical path when the project starts, but if it doesn't get done, it may well force its way onto the critical path. Installing the doors in a house under construction, for example, may not be identified as a critical path task. But you can't put the doorknobs and locks on until the doors are up, and you can't install a security system until that's done, and you can't install window treatments until the security system is in. So if the doors just never get put on, the whole project is delayed. Project managers must be focused on the critical path, but it can't be their only focus.

Stanford University: Critical Path Method

Cam Merritt is a writer and editor specializing in business, personal finance and home design. He has contributed to USA Today, The Des Moines Register and Better Homes and Gardens"publications. Merritt has a journalism degree from Drake University and is pursuing an MBA from the University of Iowa.

Operations Research

Lesson 20. CRITICAL PATH METHOD (CPM)

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Critical Path Method for Project management

Critical Path Method (CPM) is a method used in project planning, generally for project scheduling for the on-time completion of the project. It helps in the determination of the earliest time by which the whole project can be completed. There are two main concepts in this method namely critical task and critical path.

Table of Content

What is a Critical task in project management?

What is the critical path in project management, benefits of using the critical path method in project management:.

How to find the critical path in a project:

Rules for Designing the Activity-on-Node network diagram:

Node representation:, activity-on-node diagram:, forward pass in critical path in project management:, backward pass in critical path in project management:.

It is the task/activity that can’t be delayed otherwise the completion of the entire project will be delayed. It must be completed on time before starting the other dependent tasks.

It is a sequence of critical tasks/activities and is the largest path in the project network. It gives us the minimum time which is required to complete the entire project. The activities in the critical path are known as critical activities and if these activities are delayed then the completion of the entire project is also delayed.

Show the project schedule visually.
Highlight important tasks with CPM.
Use CPM to find and handle risks.
CPM helps the project team communicate better.

How to find the critical path in a project :

Step 1: Identify all tasks required to complete the project
Step 2: Determine the sequence of tasks
Step 3: Estimate the duration of each task
Step 4: Draw a network diagram
Step 5: Identify the critical path
Step 6: Calculate the float
Step 7: Monitor the critical path

The table given below contains the activity label, its respective duration (in weeks), and its precedents. We will use the critical path method to find the critical path and activities of this project.

A project network should have only one start node
A project network should have only one end node
A node has a duration
Links normally have no duration
“Precedents” are the immediate preceding activities
Time moves from left to right in the project network
A network should not contain loops
A network should not contain dangles

what is critical path method in operation research

Activity label is the name of the activity represented by that node.
Earliest Start is the date or time at which the activity can be started at the earliest.
Earliest Finish is the date or time at which the activity can be completed at the earliest.
Latest Start is the date or time at which the activity can be started at the latest.
The latest Finish is the date or time at which the activity can be finished at the latest.
Float is equal to the difference between the earliest start and latest start or earliest finish and latest finish.

The forward pass is carried out to calculate the earliest dates on which each activity may be started and completed.

Activity A may start immediately. Hence, the earliest date for its start is zero i.e. ES(A) = 0. It takes 6 weeks to complete its execution. Hence, earliest it can finish is week 6 i.e. EF(A) = 6.
Activity B may start immediately. Hence, the earliest date for its start is zero i.e. ES(B) = 0. It takes 4 weeks to complete its execution. Hence, the earliest it can finish is week 4 i.e. EF(B) = 4.
Activity F may start immediately. Hence, the earliest date for its start is zero i.e. ES(F) = 0. It takes 10 weeks to complete its execution. Hence, the earliest it can finish is week 10 i.e. EF(F) = 10.
Activity C starts as soon as Activity A completes its execution. Hence, the earliest week it can start its execution is week 6 i.e. ES(C) = 6. It takes 3 weeks to complete its execution. Hence, the earliest it can finish is week 9 i.e. EF(C) = 9.
Activity D starts as soon as Activity B completes its execution. Hence, the earliest week it can start its execution is week 4 i.e. ES(D) = 4. It takes 4 weeks to complete its execution. Hence, the earliest it can finish is week 8 i.e. EF(D) = 8.
Activity E starts as soon as Activity B completes its execution. Hence, the earliest week it can start its execution is week 4 i.e. ES(E) = 4. It takes 3 weeks to complete its execution. Hence, the earliest it can finish is week 7 i.e. EF(E) = 7.
Activity G starts as soon as activity E and activity F completes their execution. Since the activity requires the completion of both for starting its execution, we would consider the MAX(ES(E), ES(F)). Hence, the earliest week it can start its execution is week 10 i.e. ES(G) = 10. It takes 3 weeks to complete its execution. Hence, the earliest it can finish is week 13 i.e. EF(G) = 13.
Activity H starts as soon as activity C and activity D completes their execution. Since the activity requires the completion of both for starting its execution, we would consider the MAX(ES(C), ES(D)). Hence, the earliest week it can start its execution is week 9 i.e. ES(H) = 9. It takes 2 weeks to complete its execution. Hence, the earliest it can finish is week 11 i.e. EF(H) = 11.

The backward pass is carried out to calculate the latest dates on which each activity may be started and finished without delaying the end date of the project. Assumption: Latest finish date = Earliest Finish date (of project).

Activity G’s latest finish date is equal to the earliest finish date of the precedent activity of finish according to the assumption i.e. LF(G) = 13. It takes 3 weeks to complete its execution. Hence, the latest it can start is week 10 i.e. LS(G) = 10.
Activity H’s latest finish date is equal to the earliest finish date of the precedent activity of finish according to the assumption i.e. LF(H) = 13. It takes 2 weeks to complete its execution. Hence, the latest it can start is week 11 i.e. LS(H) = 11.
The latest end date for activity C would be the latest start date of H i.e. LF(C) = 11. It takes 3 weeks to complete its execution. Hence, the latest it can start is week 8 i.e. LS(C) = 8.
The latest end date for activity D would be the latest start date of H i.e. LF(D) = 11. It takes 4 weeks to complete its execution. Hence, the latest it can start is week 7 i.e. LS(D) = 7.
The latest end date for activity E would be the latest start date of G i.e. LF(G) = 10. It takes 3 weeks to complete its execution. Hence, the latest it can start is week 7 i.e. LS(E) = 7.
The latest end date for activity F would be the latest start date of G i.e. LF(G) = 10. It takes 10 weeks to complete its execution. Hence, the latest it can start is week 0 i.e. LS(F) = 0.
The latest end date for activity A would be the latest start date of C i.e. LF(A) = 8. It takes 6 weeks to complete its execution. Hence, the latest it can start is week 2 i.e. LS(A) = 2.
The latest end date for activity B would be the earliest of the latest start date of D and E i.e. LF(B) = 7. It takes 4 weeks to complete its execution. Hence, the latest it can start is week 3 i.e. LS(B) = 3.

Backward Pass in Critical path in project management

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Everything You Need To Know About CPM: The Critical Path Method With Examples

Critical Path Method is an algorithm or a tool to ease complex project scheduling. It finds the sequence of critical tasks that determines the minimum completion time of a project, thus enabling the manager to optimize resources, predict delays, and improve efficiency. Though CPM brings some benefits in the form of greater control and visualization, it could very well miss out on some factors of a large-scale project. It applies key CPM terms such as early start, late start, slack, and float. Other features that can be added to enhance CPM include automated scheduling and resource management. Even though it is usually compared to PERT or Gantt, the choice of method to use should be based on the needs of the project. Steps in applying CPM are given to ensure readiness for its application.

CPM, or the Critical Path Method, is an algorithm used in project management to schedule project activities. The critical path refers to the longest stretch of the activities and a measure of them from start to finish. The primary goals of CPM are to determine the critical path, estimate the minimum project duration, and highlight the tasks that cannot be delayed without affecting the overall timeline.

What is the Critical Path Method (CPM)?

The critical path method (CPM) is a strategy for surveying plan adaptability and distinguishing tasks fundamental for project completion. In the project, the critical path is the longest succession of tasks that must be done on time for the project to be completed. For instance, in a construction project, pouring the foundation might be a critical task. If such tasks are delayed, the project will also be delayed.

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Why Use the Critical Path Method?

Critical path method (CPM) empowers project managers to set priorities, distribute resources, and schedule projects with confidence. There are various reasons to use this method, including the ones listed below:

Improves Future Planning

Improves future planning by utilizing critical path method (CPM) to compare expectations with actual progress. Future undertaking thoughts can be affected by the information accumulated from progressing projects.

Facilitates More Effective Resource Management

It enables project managers to efficiently prioritize tasks, giving them a clear understanding of how and where to deploy resources, thereby enhancing productivity.

Helps Avoid Bottlenecks

Project bottlenecks can be a source of stress and time loss. By outlining project dependencies using a network diagram, you can more accurately decide which tasks can and cannot be finished in parallel, saving time and reducing stress.

With the help of critical path method (CPM), we’ll be able to create a model that enables you to determine the following:

Tasks required to complete the project
Dependencies between tasks
The duration required to complete an activity

Before we can get started with CPM or the Critical Path Method, we’ll have to understand two major concepts: events and Activities. To help us understand them better, let’s look at the process's network diagram (which is also the output).

This output represents some of the most important parts of the process: Events and Activities.

Events are represented by a circle and will occur at the start and end of an activity. Event 1 is the tail event and Event 2 is the head event. In the case of our example, the events are 1, 2,3,4, 5, and 6. Taking into consideration, nodes 1 and 2, and the connection between them, 1 will be referred to as the tail event, and 2 will be referred to as the head event.

Similarly, for 2 and 3, 2 is the tail event, and 3 is the head event.

Activities represent action and consumption of resources like time, money, and energy required to complete the project. In the case of our example, A, B, C, D, E, and F represent the activities taking place between their respective events.

Dummy Activity

A dummy activity represents a relationship between two events. In the case of the example below us, the dotted line represents a relationship between nodes 4 and 3. The activity between these nodes will not have any value.

Other rules to consider

The network should have a unique starting and ending node. In the case of our example, event 1 represents a unique starting point, and 6 represents the unique completion node.
No activity can be represented by more than a single arc (the line with an arrow connecting the events) in the network.
No two activities can have the same starting and ending node.

Now, let’s talk about the process of the Critical Path Method with an example.

The Critical Path Method

The objective of the question below is to determine the critical path, based on the information available, like activity, immediate predecessor, and duration (which in this case, we’ll take as months)

First, let’s analyze the activities and their immediate predecessors.

Activities A, B, and C don’t have any immediate predecessors. This means that each of them will have individual arcs connecting to them. First, we’ll draw nodes 1 (which is the starting point) and 2. We’ll add the activity on the arc, along with the duration.

We’ll have to also keep in mind that A acts as the immediate predecessor for both nodes E and F. Similarly, let’s draw the arcs for nodes B and C.

Before we can draw the nodes for activity D, a quick look at the table will tell us that it is preceded by activity B and that a combination of activities C and D act as immediate predecessors for activities H and J. This means that both activities, C and D, have to connect at some point. That’s why we’ll draw an arc from events 3 and 4.

So now, we’ve completed activities A, B, C, and D of the critical path method. Next, let’s take a look at activity E.

Activity E is preceded by activity A and acts as the immediate predecessor for activity J. Since this is an independent activity, we’ll be able to draw an arc like this.

If we have a look at activity F, it’s preceded by activity A, and a combination of F, G, and H act as immediate predecessors for activities K and L. So, let’s wait before we take it up. Instead, let’s shift our attention to activity G. It’s preceded by B. So, we’ll draw it like so.

Now, let’s take a look at activity H. It is preceded by both C and D and will act as the immediate predecessor for K and L, along with F and G. So, we can connect node 4 to 6.

Now that we’ve done that, let’s go back to activity F. Now that we know where activities G and H connect to, we can combine nodes 2 and 6, fulfilling the conditions required for activities K and L.

Following this, we have an activity I. The activity I is preceded by activities C and D. It also acts as an immediate predecessor to activity M. Since it’s an independent activity, we can draw it like so.

Next, let’s take a look at activity J. Activity J is preceded by activity E. We can also see that a combination of J and K will act as an immediate predecessor for activity N. We can then draw an arc like this.

Let’s go on to activity K. Here we can see that K is preceded by F, G, and H. It also acts as an immediate predecessor to activity N. So, we’ll connect nodes 6 to 8.

Next, let’s continue with activity L. The table now shows that L, M, and N don’t act as immediate predecessors for any other activity. Hence it can be assumed that it’ll connect to the final node.

L is preceded by activities by F, G, and H. The arc can be drawn like so.

We’ll now go to activity M. This activity is preceded by activity I. Similarly, we can connect an arc from node 8 to 9 for activity N.

Now, the network is complete!

Now, to find the critical path. For this, we’ll need to find two values, Earliest Start Time (Es) and Latest Completion Time (Lc).

The process of determining the Es for all events is called a forward pass.

The process of determining the Lc for all events is called a backward pass.

Let’s get into the forward pass. For this, first, we’ll need to create boxes at all nodes. These are then divided into two. The lower half of the box represents the earliest start time of the node, while the upper half represents the latest completion time.

Your network diagram should look something like this.

For this, we’ll be using the formula, Esj = max (Esi + Dij)

Which when simplified, the earliest start time for the second node (head node), is the maximum of the combination of the earliest start time of the tail node and the duration between the two nodes.

So, for node 1, the earliest start time is always zero.

For node 2, it would be, Es2 = 0 (earliest start time for node 1) + 3 (duration between 1 and 2) = 3

For node 3,

it would be, Es3 = 0(Es1) + 4(D1 to 3) = 4

For node 4, we can see that two arcs connect to it. This means that we’ll need to choose among the largest of the two options available to us.

Es4 = 0(Es0) + 6 = 6 or

Es4 = 4(Es3) + 3 = 7

We’ll choose 7 since it’s larger.

Similarly, we have three options to choose from when it comes to node 6. Since three arcs connect to it.

Es6 = 3(Es2) + 1(D2-3) = 4

Es6 = 4(Es3) + 4(D3-6) = 8

Es6 = 7(Es4) + 5(D4-6) = 12

Hence we’ll select the last option since it’s the largest among the three.

Now, for node 5. Since it’s directly connected to node 2, we can directly apply the formula.

Es5 = 3(Es2) + 9(D2-5) = 12

Let’s take node 8.

Es8 = 12(Es5) + 3(D5-8) = 15 or

Es8 = 12 (Es6) + 6(D6-8) = 18

We’ll choose Es8 as 18 since it’s the larger of the two.

Now for node 7. We can directly apply the formula to these nodes.

Es7 = 7(Es4) + 4(D4-7) = 11

Finally, we’ve got node 9.

It has 3 nodes connecting towards it. We’ll have to choose the maximum of the three.

Es9 = 18(Es8) + 9(D8-9) = 27

Es9 = 12(Es6) + 3(D6-9) = 15

Es9 = 11(Es7) + 6(D7-9) = 17

We’ll choose the arc from node 8 since it’s got the highest value.

And like that, the forward pass is complete. Now, for the second part of the critical path method. Let’s take up the backward pass. For that’ we will be using the following formula.

Lci = min(Lcj - Dij)

This, when put simply, means the latest completion time of the tail node is equal to the latest completion time of the head node minus the distance between the two.

Let’s start from the final node, number 9.

The Lc for this node will always be equal to its Es. So, Lc9 = 27.

Next, let’s have a look at the latest completion time for the 8th node. Since it’s directly connected only to the 9th node, we can directly apply the formula mentioned earlier.

Lc8 = 27(Lc9) - 9(D9-8) = 18

Now, let’s have a look at the latest completion time for node 7. Since there’s a direct connection between nodes 9 and 7.

Lc7 = 27(Lc9) - 6(D9-7) = 21

Let’s move on to node 6. As we can see in the diagram, there are two points extending to nodes 8 and 9 from node 6. So we have two options to choose from.

Lc6 = 18(Ls8) - 6(D6-8) = 12 or

Lc6 = 27(Ls9) - 3(D6-9) = 24

We’ll choose the Lc of node 6 as 12.

We’ll now go to node 5. Since it’s directly connected to the 8th node, we can directly apply the equation.

Lc5 = 18(Lc8) - 3(D5-8) = 15

Next up, let’s find the latest completion time for node 4.

Since there are two connections extending from the node, to nodes 6 and 7 respectively, we’ll need to select the minimum between the two.

Lc4 = 21(Lc7) - 4(D4-7) = 17

Lc4 = 12(Lc6) - 5(D4-6) = 7

We’ll choose 7 as the latest completion time for node 4.

Now for node 3.

Since there are two nodes connecting from node 3 to nodes 4 and 6. So, we’ll need to choose between the 2.

Lc3 = 12(Lc6) - 4(D3-6) = 8 or

Lc3 = 7(Lc4) - 3(D3-4) = 4

We’ll choose 4 as the latest completion time for node 3.

Let’s now go to node 2. Again, since there are two connections made from 2 to node 5 and 6, we’ll need to choose the minimum among the two.

Lc2 = 15(Lc5) - 9(D2-5) = 6

Lc2 = 12(Lc6) - 1(D2-6) = 11

We’ll choose the latest completion time of 2, as 6.

And finally, we have node 1.

Since there are connections to 2, 3, and 4 from 1, we’ll need to choose from the three.

Lc1 = 6(Lc2) - 3(D1-2) = 3

Lc1 = 4(Lc3) - 4(D1-3) = 0

Lc1 = 7(Lc4) - 6(D1-4) = 1

We’ll choose 0 as the latest completion time for the node.

And there we go! The backward pass is complete.

Now, for the final step of the critical path method. To determine the critical path, there are three major criteria that need to be satisfied.

Esj - Esi = Lcj - Lci = Dij

From the diagram, we can see that nodes that satisfy the requirements are: 1, 3, 4, 6, 8, and 9.

Hence the activities on the critical path are B - D - H - K - N.

Hence the critical path is B + D + H + K + N = 4 + 3 + 5 + 6 + 9 = 27.

And there we go! We’ve found the critical path!

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Advantages of Using CPM in Project Management

The advantages of using critical path method (CPM) are as follows:

Effective Communication: When creating critical path method schedules, all phases of a project's life span must be considered. The program's structure becomes more achievable and firm when the skills shared by various team members are integrated. Therefore, effective communication is the key.
Easier to Prioritize Tasks: Project managers can prioritize tasks effectively and estimate their float by determining the critical path. Float indicates the amount of time a task may be put off before it affects its completion. A lower float indicates a greater priority.
Accurate Scheduling: Critical path method (CPM) is a popular and dependable methodology for enhancing the precision of project schedules. Several project managers utilize CPM with the Programme Evaluation and Review Technique (PERT), which supports teams in estimating overall project length.
Better Visualisation: Gantt charts and CPM network diagrams, which show critical path timelines, can help project managers understand a project's timeline and progress more quickly. These visual tools allow them to understand a project's direction more intuitively than a less eye-catching alternative.

Disadvantages of Using CPM in Project Management

Some of the disadvantages of using the critical path method are as follows:

Multiple Complexities: The critical path method (CPM) involves several moving elements and detailed computations. Even with the aid of software, the potential for human error in data entry is a significant concern, underscoring the need for caution and attention to detail.
Limited Applicability: Only some project types are suited to the critical path method. For instance, projects requiring creativity, like product design or research work, tend to come along in unforeseen forms and fail to lend themselves well to CPM. Similarly, projects with high uncertainty or those that involve a high degree of risk may not be suitable for critical path methods (CPM).

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Key Terminologies in CPM

Some of the fundamental terminologies that are important to understand in critical path method (CPM) for the effective management of project schedules are given below:

Critical Path: A series of activities in a project which, when delayed, will delay the project completion time. In other words, the longest chain of activities must be completed on schedule if the project must be completed by its due date. Any delay in the chain will directly impact the completion time for the entire project.
Early Start (ES): The earliest time an activity may start if all activities preceding it begin as early as possible.
Late Start(LS): The latest time at which an activity can start without causing the project's end date to be delayed.
EF(EF): Early Finish: The earliest time an activity may finish, calculated from its ES and duration.
Late Finish (LF): It is the latest date a task can be finished without impacting the overall project timeline.
Slack (Float): The time, usually measured in days, that an activity can be delayed without impacting the overall completion of a project. Activities on the critical path have zero slack.
Total Float: The total time an activity may be delayed without impacting the end date of a project. This is referred to as Total float .
Free Float: The time that an activity can be delayed without delaying the start of any subsequent activities.
Duration: The total time for an activity from its commencement to completion.

All these terminologies are basic but form the core for computing and understanding the critical path, which assists the project manager in effectively planning, monitoring, and controlling the project's timeline.

How to Calculate the Critical Path of a Project

Calculating the critical path of a project involves identifying the sequence of activities that will determine the minimum completion time for a project. Here's how you can calculate the critical path:

List all tasks: Identify all the things that need to be done to complete a project and list them. Every task is supposed to have a defined duration.
Next, determine dependencies: Understanding the tasks that must be completed before others can start is a strategic move. It puts you in control of the project's sequence and ensures a smooth flow of tasks.
Create Network Diagram: A network diagram or flowchart can be drawn showing each of the tasks and their dependencies. Every task or event gets represented on a node. Arrows depict relationships among these nodes.
Estimate Task Durations: State a duration for every task, usually measured in days or weeks. Improving time management skills can be beneficial.
Forward Pass: Compute ES, the earliest start, and EF, the earliest finish times for every task from the project start. The procedure for doing this is called the forward pass. ES will tell when a task can start as soon as possible, and EF will tell when it can finish as quickly as possible, considering its duration.
Perform Backward Pass: Work the latest start and finish times from the project's end date. This is called the backward pass. LS represents the latest time a task can start without delaying the project, and LF denotes the latest it can finish.
Identify the Critical Path: This is the set of activities with ES equalling LS. This means zero slack or float for those activities. Simply put, delays in these activities will directly impact the project completion date.

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How to Use the Critical Path Method

The critical path method (CPM) gives you an insight into the status of your project and enables you to keep track of activities and their turnaround times. These are some additional uses for CPM.

Compress Schedules

There are situations when project deadlines may be advanced, but this could be better. You can use fast-tracking or crashing as a schedule compression strategy in such circumstances, that would offer you an opportunity to meet your project goals more efficiently.

Fast-tracking : Analyze the critical path to identify tasks that can be completed concurrently. For instance, if Task A and Task B are not dependent on each other, they can be fast-tracked. By using parallel processes like these, the entire length will be shortened.

Increasing resources is a step in the process of 'crashing' operations. It is crucial to ensure the additional resources fit within the project's scope before acquiring them. Equally important is to inform the stakeholders of any modifications, demonstrating your responsibility and consideration for their involvement.

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Resolve Resource Shortages

Remember that the critical path method (CPM) does not account for resource availability. However, resource-leveling tactics can be a game-changer, helping you resolve resource deficits like an overbooked team member or a lack of equipment and ultimately leading to a more successful project.

Resource-leveling strategies play a crucial and reassuring role in managing project schedules. They help alleviate resource over-allocation problems and ensure that a project can be completed with available resources, instilling a sense of confidence in your project management.

Resource-leveling tactics are a tool of control for project managers. They are highly adaptable, allowing you to modify the critical route or apply this strategy to floated activities. This adaptability makes you feel flexible and agile, giving you the power to manage your project resources effectively and ensuring a sense of security in your project management.

Compile Data for Future Use

Since you're working with informed estimations for activity durations, the schedule generated by the critical path method (CPM) is liable to alter. Therefore, as the project progresses, you can contrast the original critical path with the current one. Future studies can use this information to predict work durations more precisely.

Critical Path Method Software

Programs or software specifically designed for project management that lets you create critical path schematics for a given project are called critical path software or critical path method (CPM) software. These tools make your daily activities more accessible by helping you analyze, schedule, and manage project tasks, reliance, and resources.

Here are some of the critical path method software:

Office Timeline
Zoho Projects
Liquid Planner
Project Manager
Lucid Chart

Features of Critical Path Software

The general features of critical path software are as follows:

Complete process visibility using Gantt charts and Kanban boards
Set a task, an overview of the task, assignees, and to-do lists
Interact on discussions or challenges to projects
Make dependencies between tasks
Set both the actual and projected dates
Control spending and produce a financial summary
Identify challenges and risks, eventually delegate them
Integrations by third parties
Track your tasks

Key Critical Path Terms for the PMP Exam

Some of the critical path terms that can be important for the PMP exam, which includes the question of what critical path method (CPM) is, are as follows: 1. Critical Path Method: This sequential project management approach for process development is not just a theoretical concept but a practical tool that distinguishes between essential and minor duties, thereby avoiding delays and workflow constraints. 2. Critical Path DRAG: The total time of an essential action adds to the project's duration. However, reducing the length of one basic activity to a minimum could significantly shorten the time needed to complete the project, underscoring the impact of your decisions. 3. Criticality Index: It is a crucial tool in risk analysis, displaying how often a specific activity has been on the critical path throughout the study. High Criticality Index activities are more inclined to be placed on the critical path, which increases the urgency for you to master this concept to avoid project delays 4. Total Float: The amount of time that can be added to an activity's early start date yet to prevent the project as a whole from being pushed back. 5. Free Float: A task's duration can only be postponed by advancing the early start time of a succeeding task. 6. Forward Pass: This is the strategy for determining the critical path method's early start or finish times for tasks. 7. Backward Pass: The strategy to determine when an activity in the critical path method will have a late start or finish. 8. Network Diagram: A network diagram shows the connections between project activities. It is typically created from left to right to symbolize the project's sequence. 9. Network Analysis: This involves deconstructing a complex project into its various components (tasks, timelines, etc.) and graphing those parts to show how they relate. This method can help project managers identify the critical path and allocate resources effectively. It also aids in visualizing the project's timeline and dependencies, which is crucial for project planning and execution.

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Example of Critical Path Method in Action

Here is an example of how the Critical Path Method (CPM) is applied in a real-world project: organizing a large corporate conference. This project involves several tasks, such as booking the venue, arranging catering, sending out invitations, setting up the stage, and preparing presentations.

Step 1: List All Tasks

First, we identify and list all the necessary tasks for the conference:

Task A: Book Venue
Task B: Arrange Catering
Task C: Send Invitations
Task D: Set Up Stage and Audio-Visual Equipment
Task E: Prepare Presentations

Step 2: Determine Dependencies

Next, we identify which tasks depend on the completion of others:

Task B (Arrange Catering) can only start after Task A (Book Venue) is completed.
Task C (Send Invitations) also depends on Task A (Book Venue).
Task D (Set Up Stage) depends on Task A (Book Venue).
Task E (Prepare Presentations) is independent and can start at any time.

Step 3: Estimate Durations

We then assign a duration to each task, estimating how long each will take:

Task A: Book Venue - 5 days
Task B: Arrange Catering - 3 days
Task C: Send Invitations - 2 days
Task D: Set Up Stage - 1 day
Task E: Prepare Presentations - 4 days

Step 4: Create a Network Diagram

A network diagram is created to visualize the tasks and their dependencies. This helps in seeing the flow of tasks:

Task A is the starting point.
Tasks B, C, and D branch off from Task A.
Task E is independent.

Step 5: Perform a Forward Pass

We calculate the earliest start (ES) and earliest finish (EF) times for each task:

Task A: ES = 0, EF = 5 (since it’s the first task)
Task B: ES = 5, EF = 8 (starts after Task A finishes)
Task C: ES = 5, EF = 7 (starts after Task A finishes)
Task D: ES = 5, EF = 6 (starts after Task A finishes)
Task E: ES = 0, EF = 4 (independent, can start at any time)

Step 6: Perform a Backward Pass

Now, we calculate the latest start (LS) and latest finish (LF) times for each task to determine flexibility:

Task A: LS = 0, LF = 5
Task B: LS = 5, LF = 8
Task C: LS = 6, LF = 8
Task D: LS = 7, LF = 8
Task E: LS = 0, LF = 4

Step 7: Identify the Critical Path

The critical path is the sequence of tasks with no slack (float), meaning any delay in these tasks will delay the entire project. In this case:

The critical path is Task A → Task B, with a total duration of 8 days. This path has zero slack, so any delay here directly affects the conference date.

Step 8: Monitor and Adjust

As the project progresses, the project manager will closely monitor Task A (Book Venue) and Task B (Arrange Catering) to ensure they stay on schedule, as these tasks would delay the entire conference. Other tasks, like Task C and Task D, have some flexibility but still need attention to avoid knock-on effects.

This example shows how CPM helps project managers focus on the most critical tasks, optimize resources, and ensure that the project is completed on time.

Critical Path Method vs PERT

While CPM stands for Critical Path Method and PERT for Program Evaluation and Review Technique , the former applies to known and fixed project durations of tasks. In contrast, the latter deals with uncertain or variable task durations. It finds out the critical path that enables one to minimize project time. This method is deterministic and thus quite ideal for repetitive projects like construction work. However, PERT is used on projects which have uncertainty about task time ranges. It uses three time estimates, optimistic, pessimistic, and most likely, to evaluate expected durations. PERT is probabilistic and very useful in research and development projects where time estimation is uncertain.

Critical Path Method vs Gantt Chart

Horizontal bar charts, called Gantt charts, layout project activities that may be monitored within a predetermined time frame. The dependencies between tasks are displayed using both critical path methods (CPM) and Gantt charts.

Here are some distinctions between the two tools:

CPM (Critical Path Method)

Project duration is calculated, and critical and non-critical pathways are visualized.
Shown as a network diagram with connected boxes.
Does not indicate the resources needed
Plot activity without a time frame on a network diagram

Gantt Diagram

Visualizes the development of project activity
Presented as a horizontal bar graph .
Demonstrates the resources needed for each action
Creates a timetable of activities

Gantt charts and CPM can be used in conjunction to monitor critical pathways over time and keep your project on schedule.

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1. What is the critical path formula?

It's an independent sequence of activities that determines the minimum duration required for the completion of a project. When locating the critical path, it identifies all activities that have zero slack or zero float: the ES equates with LS and EF equals LF. These important steps include a forward pass to determine the earliest start and finish times and a backward pass for the latest start and finish times.

2. What is a CPM schedule?

A critical path method (CPM) schedule is one of the project management tools used in outlining the sequence of tasks that must be completed to finish a project. It identifies the critical path, thereby enabling efficient scheduling and resource allocation. It also gives the project manager a better view of the whole timeline of the project, which would help monitor the progress of completion of tasks in the project and adjust the plan to focus on the critical activities to meet the deadlines.

3. What are the four main steps involved within the CPM?

The four main steps involved in the Critical Path Method (CPM) are:

List All Tasks: Identify and list all tasks required to complete the project, including their durations and dependencies.
Create a Network Diagram: Develop a visual representation of the tasks and their dependencies, usually in the form of a flowchart.
Perform Forward and Backward Passes: Calculate the earliest and latest start and finish times for each task to identify the critical path.
Identify and Monitor the Critical Path: Focus on the tasks in the critical path, ensuring they are completed on time to avoid delays in the overall project.

4. Can the critical path change during the course of a project?

Yes a project's critical path may change in the course of a project execution. It might be changed at any time because of changes to the length of tasks, the introduction of new tasks, or delays in non-critical tasks that actually turn out to be critical. In this, continuous monitoring and updating of the CPM schedule are important to account for changes and assure that the project stays on track.

5. Can CPM be used for all types of projects?

While the Critical Path Method is amazingly proficient in most project works, especially those with well-defined tasks and dependencies, it does not quite fit some projects. It applies best in projects with well-defined workflows and durations that are fairly predictable. For projects with high uncertainty, artistic/creative processes, or tasks that are extremely interdependent and iterative, more flexible methods of project management, such as Agile, are applied.

Find our PMP® Certification Training Online Classroom training classes in top cities:

About the author.

Rahul is a Senior Research Analyst at Simplilearn. Blockchain, Cloud Computing, and Machine Learning are some of his favorite topics of discussion. Rahul can be found listening to music, doodling, and gaming.

Recommended Resources

An Introduction to Project Management: A Beginner’s Guide

PERT and CPM: Vital Gears of Contemporary Project Management

How to Pass Your PMP Exam in One Attempt

Project Management Interview Guide

Acknowledgement
PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, OPM3 and the PMI ATP seal are the registered marks of the Project Management Institute, Inc.

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39 Project Planning: Critical Path Method (CPM)

Vikas Singla

39.1 OBJECTIVES

This chapter would help students to understand:

Importance of project planning
Rules of network construction
Importance of CPM in project planning
Application of CPM in estimating trade-off between time and cost of project

39.2 INTRODUCTION

Building of a house requires carrying out numerous activities sequentially as well as simultaneously. It is necessary to identify type of activities and resources required to carry out such activities. Some of such activities might be creating foundation, construction, wood work, electric work, interior designing etc. Each activity can be further broken down into number of activities. Thus, a project in this example construction of house involves combination of number of activities arranged in such a sequence so as to achieve completion of project in significant period of time. Definition of project involves following important aspects:

Identification of activities: which activities need to be performed for completion of project?
Sequence of activities: which activities are to be performed in what sequence and can some of them be performed simultaneously or can some of them be combined?
Resources required: each activity requires certain resources in terms of man, machine, material, time and effort. How many individuals are required to perform a particular activity? Can they be used for
performance of other activities? How much time does each activity take? What is the cost required to perform each activity?
Critical activities: which are the most important activities and what is the time required to complete those activities? Can these activities be delayed and if yes what would be the effect on project completion time? Which are non-critical activities or activities which can be delayed without delaying the project?

Thus, a project involves number of aspects and its successful completion requires a scientific and systematic approach. PERT and CPM are two of the such approaches which answers above mentioned questions. These two methods are network oriented techniques used primarily to determine project completion time. This chapter discusses CPM in detail with illustrations. CPM is used to find out critical and non-critical activities. The identification of such activities helps a manager to evaluate which activities can be delayed and which cannot with an effect on project completion time. Also, chapter discusses time-cost model of project which provides the effect of change in time of activity on total cost of completion of project.

39.3 PROJECT MANAGEMENT: IMPORTANCE

Project management primarily involves following three managerial functions:

· Planning: involves identification and breaking down of activities into viable and feasible tasks that must be performed for completion of project. It also involves determination and allocation of resources to each activity. It requires estimation of number of man hours, amount of material required, cost that would be incurred etc. for each activity.

· Scheduling: involves arranging activities in a logical sequence that would help in converting allocated resources to desired outputs in most productive manner. It is important to note that identified activities can be arranged in different ways to obtain certain outputs. But it is important to find out the most productive way of arranging activities. Such approach would also require identification of critical activities and estimation of available idle time for which activities can be delayed.

· Control: function involves review of project progress. This function determines whether each activity is performed in estimated duration, uses allocated resources or is deviating from planned schedule. This function allows controlling any difference between planned and actual schedule. The reasons for such a difference are analyzed and remedial measures adopted.

PERT and CPM methods of project management helps management to achieve these functions effectively as these methods identifies and estimates time for performance of activities, arranges them in most efficient sequence and provides data regarding any deviation from planned schedule.

Application of these methods requires understanding of construction of network diagrams representing arrangement of project activities. This has been discussed in following section.

39.1 NETWORK CONSTRUCTION: SOME DEFINITIONS

Network is a web of identified activities showing their arrangement or sequence in a diagrammatic fashion. Construction of network diagram requires understanding of following terms:

Activity: An activity is a task that requires certain resource such as time, money and labor for its completion. In a network diagram it is represented by an arrow. An activity occurs over a period of time.

Event: An event indicates start or finish of an activity. It is represented by a circle called as node. An event does not consume any resource and it occurs at a point in time.

If in a network diagram an activity is represented by arrow and an event by node then such a diagram is called as activity on arrow (A-o-A) network diagram. Whereas if in a network diagram an activity is represented by a node and an event by an arrow then such a diagram is called as activity on node (A-o-N) network diagram.

Predecessor: An activity which has to be completed before the start of another activity is called as predecessor to start of new activity.

Successor: An activity which starts after or succeeds the finish of another activity is called successor or succeeding activity.

Concurrent activity: An activity which can be performed simultaneously with some other activity is called as concurrent or parallel activities.

These activities are illustrated in figure 39.3.1. This is an activity on arrow network diagram. A is preceding activity to C which is preceding to E. Similarly, B is preceding to D which is preceding to F. Activities can also be explained in successor terms. E activity will start after finish of C i.e. it succeeds C which succeed .Similarly, F succeeds D which succeeds B. Events are depicted by circles called as nodes. They indicate the start and finish of an activity. It is important to note that A and B start from same node at same time making them concurrent or parallel activities.

39.5 CRITICALPATHMETHOD (CPM)

CPM is a deterministic approach of network technique where duration of each activity involved in completion of project is known with certainty. Certain activity times would be known in cases of projects which are repetitive or are not new for an organization. For example an IT firm specializing in creating software for different banks would apply CPM for a new banking project. As company has already created software for other banks so creating software for new bank would involve similar tasks for which completion time is known with high certainty.

The purpose of CPM technique is to:

Find scheduled project completion time.
Scheduled start and finish times for each activity.
Identification of most important or critical activities which should be completed within scheduled time without delaying the project.
Identification of non-critical activities which can be delayed by specific time without delaying the completion of project from its scheduled time.

To achieve objectives of CPM certain terms have to be understood:

Earliest Start Time (EST) of an activity indicates the time at which an activity can start as soon as possible. Earliest Finish Time (EFT) of an activity indicates the earliest possible time at which an activity can be completed. It is the sum of EST and duration of that activity.

Calculation of EST is governed by forward path rule. This rule states that for calculating EST we should move forward from first event to the last event in a network diagram. As an activity cannot start until all preceding activities have been completed so EST of an activity is equal to the largest EFT of immediate preceding activity. Latest Finish Time (LFT) of an activity indicates finish of an activity as late as possible without having an effect on scheduled finish of entire project.

Latest Start Time (LST) of an activity indicates start of an activity as late as possible without having an effect on scheduled finish of entire project. LST is the difference of LFT and duration of particular activity.

Calculation of LFT is governed by backward path rule. This rule states that for calculating LFT we should move backwards from last event to the first event in a network diagram. As purpose is to finish an activity as early as possible so LFT for an activity is the smallest of latest start time of immediate succeeding activities.

Slack is determined after finding out all earliest and latest times by moving forward and backward in a network diagram. Slack indicates idle time of an activity with which it can be delayed without increasing the project completion time. It can be computed as: LFT – EFT or LST – EST.

LFT is calculated by moving backwards from last event in the network diagram. E6 indicates finish of all the paths. Now E6 can be reached by moving along A—C—F, A—D—G and B—E—G. Total time for each of the path would be 19 days, 22 days and 20 days respectively. As project cannot be finished till all activities are completed so LFT of E6 would be 22 days. Moving backwards C finishes at E4. Its LFT would be 22-5 = 17 days and so on. Interestingly, finish of E2 can be achieved by moving backwards from F to C or from G to D. For F to C LFT would be 22-5-5 = 12 days and for G to D LFT would be 22-5-8 = 9 days. As an activity should be finished as early as possible so by moving backwards minimum LFT should be considered. So finish of E2 or LFT of activity A would be 9 days. All other times are calculated and shown in table.

Step 3: Calculation of slack:

Zero slack of an activity would indicate that such an activity cannot be delayed or it has to be completed within scheduled time. Such activities are most important activities in a project and constitute critical path. In this case A—D—G constitute critical path which would take 22 days. This would also be the project completion time i.e. project comprising of above mentioned activities would take minimum of 22 days. Activities with some slack are called as non-critical activities. For instance, activity B with a slack of two days implies that even if start of this activity is delayed by two days then also project would be completed in 22 days

39.6 CRASHING

The major application of CPM is to estimate the impact on total cost associated with project if project completion time is reduced. The knowledge of scheduled start and finish times of each activity, slack of each activity and identification of critical and non-critical activities obtained from application of CPM is used in estimating cost of project with change in project completion time. In network techniques of PERT and CPM the resource used with each activity is the duration of that activity. But performance of each activity also uses other resources such as labor, materials etc. which incur cost. Crashing method uses this information of cost associated with performance of each activity to achieve the objective of estimating project time. It is important to understand that with reduction in activity time more resources would be required for its completion thus, increasing its cost. That is why there would always be a trade-off between time and cost.

So, question that a project manager faces:

· Which activities to be crashed i.e. duration time of which activities should be reduced so that project time gets reduced?

What is the increase in cost of project completion with reduction in time period of identified activities?
For how much time period an activity can be crashed?

Crashing should not be done indefinitely. A stage will come when further crashing of activities will not reduce total project completion time but it would increase cost significantly. As illustration discussed below would show that with decrease in time of an activity it increases direct cost such as cost of machinery, labor etc. and reduces indirect cost such as cost of lighting, rent, security etc. At first total cost would reduce with reduction in project time because proportion of increase in direct cost is less than decrease in indirect cost. But after certain reductions in time period proportion of increase in direct cost would outweigh reduction in indirect cost thus increasing total cost. This is an indication that further crashing would not result in optimal results. Thus objective of crashing gets defeated. So, a project manager should stop crashing when all the paths in network diagram become critical or total cost starts increasing.

Following are certain terms that must be understood in application of crashing:

Normal time: It is the expected time of completion of an activity.
Crash time: It is the reduced time of completion of an activity.
Normal cost: is the cost of performance of an activity when performed under normal conditions.

Crash cost: is the cost of performance of an activity when performed under reduced time.

Following are the steps to be used in crashing of a project:

Step 1: Draw network diagram

Step 2: Find cost slope of each activity. Cost slope is the increase in direct cost of an activity when its duration is reduced by one day. Cost slope = change in cost / change in time

Step 3: Find critical path. As discussed critical path is found by calculating EST, EFT, LST and LFT. By definition critical path is the longest path in the network diagram. In example shown under critical path is estimated by finding the longest path in diagram.

Step 4: In the critical path find an activity with least cost slope because aim is to reduce project time with least increase in cost.

Step 5: Estimate the time for which selected activity can be crashed. For this a general rule that is followed is that activity should be crashed for so much time that it does not result in change of critical path.

Step 6: Keep on repeating the process till further reduction in time period leads to increase in cost or when all paths in network diagram have become critical.

Example: For the following project estimate total project cost under normal times and crash times. Indirect cost is Rs.100 per day.

39.7 SUMMARY

Project is a combination of activities arranged in a sequence to achieve objective of completion of project. Planning of project becomes increasingly difficult if it involves numerous activities. Thus, need of managing project arises which involves arranging activities in a sequence which would help in completing the project most productively. Management of project can be done by using two time oriented network techniques namely PERT and CPM. These techniques are similar to each other in fulfilling objectives of finding project completion time and identifying critical and non-critical activities. They differentiate from each other in a manner that CPM is a deterministic approach used for projects which have been carried out earlier also. Whereas PERT is a probabilistic technique used for finding out probability of completion of project within scheduled time of those projects which are new and project manager has little experience for their conduct. Crashing is a time-cost trade-off technique used to find minimum possible time in which project can be completed with minimal impact on cost.

39.1 GLOSSARY

Project: is a non-repetitive combination of activities arranged in a logical sequence.
Critical Path Method (CPM): is a deterministic time oriented network technique used to determine critical and non-critical activities.
Forward pass: is a procedure that involved moving forward in network diagram for determining earliest start and finish times.
Backward pass: is a procedure that involved moving backwards in network diagram for determining latest start and finish times.
Crashing: is a method of shortening of activity times by adding resources leading to increase in costs.

39.1 REFERENCES/ SUGGESTED READINGS

Chase, B.R., Shankar, R., Jacobs, F.R. and Aquilano, N.J., Operations & Supply Chain Management, 12thEdition, McGraw Hill.
Stevenson, W.J., Operations Management, 9thEdition, Tata McGraw Hill.
Lee J. Krajewski, Operations Management, Prentice-Hall of India, New Delhi, 8th Edition.

How to calculate the Critical Path with Examples 🥇

The critical path method is one of the most important concepts in project management; however, many students have difficulties in its application. In this post, you will learn how to calculate the critical path of a project step by step by following practical examples.

Before you start we recommend you to review our post how to create a PERT CPM chart step by step with examples ; where we explain how to graphically represent the relationships between the activities of a project. Creating the chart of project is the starting point to calculate our critical path.

You can also check our online critical path method calculator that performs the steps to calculate the critical path automatically.

What is the Critical Path Method?

The Guide for Project Management (better known as PMBOK Guide), defines the critical path as:

“The sequence of scheduled activities that determines the duration of the project”.

To complete a project, we must perform all the activities that compose it. Some activities can be done simultaneously, but others must be done in a certain order. For example, if I am going to build a house, I must first build the walls before painting them. Of all the sequences of activities that make up the project, the sequence that has the longest duration will be the critical path and determines the total project time . As a consequence, the critical path method specifies the algorithm for finding it.

Why is Critical Path Important?

By identifying the critical path, teams recognize the most important tasks to be performed in a project.

Further, any delay in a critical path activity can cause the entire project to be delayed, so managers take extra care to ensure those activities take place as planned.

Important: Projects may have multiple critical paths.

What methods are used to calculate the Critical Path?

The methods used to calculate the critical path are the Project Evaluation and Review Technique (PERT) and the Critical Path Method (CPM).

The PERT and CPM methods began to be developed in the 1950s to assist managers in scheduling, monitoring and controlling large, complex projects. The CPM method (1957) was implemented as a procedure to assist in the construction and maintenance of chemical plants. The PERT technique was developed independently in 1958 for the U.S. Navy's Polaris project.

Although both methods are similar, they used different techniques to estimate task durations. The PERT method used three different time estimates for the duration of each task, plus it calculated the probability of completing the project at a given time. The CPM method used a single duration for each task; however, it also analyzed the additional costs that would be incurred if the project is accelerated.

The differences between the two techniques are now considered to be minimal; therefore, we can refer to both as an overall method called PERT CPM .

How to calculate the critical path?

To calculate the critical path we will follow the following steps:

Step 1: Obtain the project data. Make a list of all the activities of the project along with their dependencies and their specific times.
Step 2: Elaborate the network diagram. We have written a post that explains how to elaborate the project network diagram step by step.
Step 3: Calculate the Early Start and Late Start Times. Determine the Early Start and Late Start Times for each activity.
Step 4: Calculate the Early Finish and Late Finish Times. Determine the Early Finish and Late Finish Times for each activity.
Step 5: Calculate the slack time. The critical path must be determined by finding out the slack for each activity of the process. The activities where there is no slack are the ones making up the critical path.

How to Calculate Early Start, Late Start, Early Finish, Late Finish and Slack (Float)

To visualize the calculation of these values, we will use the following graphic representation of the activity:

Duration (t):

Indicates the time it takes to complete the activity.

Early Start Time (ES):

This is the earliest time that an activity can be started assuming all previous activities have been completed beforehand. For activities that have more than one precedent, the ES is the greatest of the late start time of their precedents.

Early finish Time (EF):

This is the earliest time that an activity can finish. It is equal to the early start time plus its estimated duration (t):

EF = ES + t

Late Finish Time (LF):

This is the latest time at which an activity can be completed without delaying the entire project. It is obtained by equaling the late start time of the activity that immediately follows. If activities have more than one task immediately following them, the LF will be the least of the late start time of those activities.

Late Start Time (LS):

It is the latest time an activity can begin without delaying the whole project. It is equal to the Late Finish Time minus the expected duration of that activity (t):

LS = LF – t

Slack – Float (S):

A period of time when an activity can be delayed without causing the entire project to be delayed. All activities contained in the critical path have zero slack.

Here is how it is calculated mathematically:

S = LS – ES = LF – EF

We will illustrate these concepts with an example:

Imagine that your professor asks you to write an essay to be submitted in 15 days.

According to the length of the topic, it will take you approximately 1 week to prepare it. If you are a very responsible person, you will probably start the essay as soon as possible. The earliest start time (ES=0) , represents the first moment when you can start your essay:

If you start your project at point 0; you will have it ready at point 7. That final value represents the early finish time (EF=7) .

Now let's suppose that you like to do your activities at the last minute; most likely you want to finish your essay right on the day of the presentation. That day would be point 15 and represents the late finish time (LF=15) . You cannot exceed that point because you will no longer submit the paper on time.

In order for you to finish your paper on the 15th day, you have to start 7 days earlier (the time it takes to do the rehearsal); therefore you must start on the 8th day. This point represents the late start time (LS=8) . If you start after this date you will not be able to submit your work on time.

The 8 free days you have between the time you start work and the date of submission is the slack (float) in your activity.

Slack = LS – ES = LF – EF

Slack = 8 – 0 = 15 – 7 = 8

If we change the scenario and our teacher gives us only the one-week deadline to submit the essay. We will no longer have any free time (slack) so I must start working on the essay from the very beginning to get it done on time. Since this activity has no chance of being delayed, we call it a critical activity .

In project management, it is required to calculate these values for each activity which is achieved through a two-path process: a forward path to calculate ES and EF; and a backward path to calculate LS, LF and slack (float).

This procedure will be detailed by means of solved examples:

Draw the activity-on-node (AON) project network associated with the following activities for Dave Carhart’s consulting company project.

How long should it take Dave and his team to complete this project?
What are the critical path activities?

Solution 1:

The network diagram would look as follows:

The development of this diagram was explained in detail in our article on creating PERT CPM diagrams . The number at each node represents the duration of each activity.

Forward path

We will now perform the forward path to calculate the ES and EF with an analysis of each activity.

Start node:

This dummy node has all values equal to zero.

Activity A:

Since it is the first activity, its ES will be equal to the EF of the starting node (zero); the EF is calculated as follows:

EF = ES + activity time

EF A = 0 + 3 = 3

Activity B:

It has as precedent only activity A; therefore its ES will be equal to the EF of activity A. In the same way as the previous node, the EF of activity B is calculated by adding its ES + the corresponding time:

EF B = 3 + 4 = 7

Activity C:

It has as precedent only activity A; therefore its ES will be equal to the EF of activity A. The EF of activity C is calculated by adding its ES + the corresponding time:

EF C = 3 + 6 = 9

Activity D:

It has as a precedent only activity B; therefore its ES will be equal to the EF of activity B. The EF of activity D is calculated by adding its ES + the corresponding time:

EF D = 7 + 6 = 13

Activity E:

It has as a precedent only activity B; therefore its ES will be equal to the EF of activity B. The EF of activity E is calculated by adding its ES + the corresponding time:

EF E = 7 + 4 = 11

Activity F:

It has as precedent only activity C; therefore its ES will be equal to the EF of activity C. The EF of activity F is calculated by adding its ES + the corresponding time:

EF F = 9 + 4 = 13

Activity G:

It has as precedent only activity D; therefore its ES will be equal to the EF of activity D. The EF of activity G is calculated by adding its ES + the corresponding time:

EF G = 13 + 6 = 19

Activity H:

This activity has two precedents: E and F; therefore its ES will be equal to the highest EF of both activities. In this case, activity F has the highest value with 13. The EF of activity H is calculated by adding its ES + the corresponding time:

EF H = 13 + 8 = 21

The end fictitious node is joined with the last activities G and H; and the highest value of the EF of both activities is placed as ES: 21. This value represents the total duration of the project. As this node has zero duration (because it is fictitious) its EF will be equal to 21 + 0 = 21.

Backward path

To finalize the critical path calculation we will perform the backward traversal to calculate the LF and LS, starting from the final node; placing the values at the bottom of the node as follows:

For the end node the LF value is equal to the project duration (21). The LS is calculated by subtracting the LF minus the duration (zero).

LS End = 21 – 0 = 21

Since the final node is the only successor to activity H, its LF will be equal to the LS of the final node (21). The LS of activity H is calculated by subtracting its LF minus its duration:

LS H = 21 – 8 = 13

Since the end node is the only successor of activity G, its LF will be equal to the LS of the end node (21). The LS of activity G is calculated by subtracting its LF minus its duration:

LS G = 21 – 6 = 15

Since activity H is the only successor of activity F, its LF will be equal to the LS of activity H (13). The LS of activity F is calculated by subtracting its LF minus its duration:

LS F = 13 – 4 = 9

Since activity H is the only successor of activity E, its LF will be equal to the LS of activity H (13). The LS of activity E is calculated by subtracting its LF minus its duration:

LS E = 13 – 4 = 9

Since activity G is the only successor of activity D, its LF will be equal to the LS of activity G (15). The LS of activity D is calculated by subtracting its LF minus its duration:

LS D = 15 – 6 = 9

Since activity F is the only successor of activity C, its LF will be equal to the LS of activity F (9). The LS of activity C is calculated by subtracting its LF minus its duration:

LS C = 9 – 6 = 3

How activity B has as successors activities D and E, its LF will be equal to the smaller value of the LS of both. In this case, since both have a value of 9; that value will be the LF of activity B. The LS of activity B is calculated by subtracting its LF minus its duration:

LS B = 9 – 4 = 5

How activity A has as successors activities B and C, its LF will be equal to the smaller value of the LS of both. In this case, the lowest value is that of activity C (3); therefore, that value will be the LF of activity A. The LS of activity A is calculated by subtracting its LF minus its duration:

LS A = 3 – 3 = 0

Initial Node:

Using the same analysis as above we complete the values of the initial node with zero.

Finally we calculate the slack for each node with the following formula:

The final graph would look like this:

The activities with zero float (marked in red) are the critical activities. In this example, these activities follow a single path that will be our critical path: A – C – F – H . Total project time is 21 days.

Shirley Hopkins is developing a program in leadership training for middle-level managers. Shirley has listed a number of activities that must be completed before a training program of this nature can be conducted. The activities, immediate predecessors, and times appear in the accompanying table.

Provide the following:

AON diagram for these precedencies
What is the critical path?
What is the total project completion date?
What is the slack time for each individual activity?

Solution 2:

We will solve this example by showing the results and analysis of our critical path method calculator :

Each activity is broken down by its precedents and descendants:

This table serves as a guide to build our network diagram . Two dummy activities were added to show the start and end. The activities with 0 (zero) slack are the ones that make up the critical path .

CPM Network Diagram

CPM Network Diagram with Calculations of ES, EF, LS, LF and Float

The following table presents the results to determine the critical path:

Early Start (ES): It is equal to the Early Finish to the activity's precedent. If it has more than one precedent, the highest value is taken.
Early Finish (EF): It is equal to the Early Start of the activity plus its duration (t). EF = ES + t.
Late Start (LS): It is equal to the Late Finish minus its duration (t). LS = LF – t.
Late Finish (LF): It is equal to the late start of the activity that follows. If it has more than one successor, the lowest value is taken.
Slack (S): It can be calculated in two ways. S = LS – ES = LF – EF. Activities with zero clearance make up the critical path.

The critical path is: B → D → E → G

The total project time is: 26 hours

Final Reflection

The critical path shows us the main activities that will determine the total time of the project; that is why Project Managers must be aware of the fulfillment of each one of them. This does not mean that the other activities should be neglected; on the contrary, if they are delayed longer than their slack, the project will have problems to finish on time.

With the help of our entry, we are confident that you will gain more understanding about the CPM technique and how it can be used to calculate critical path.

Finally, we invite you to follow us on our social networks and join our Facebook group where we can exchange more information with the participants.

Operations > CPM

CPM - Critical Path Method

In 1957, DuPont developed a project management method designed to address the challenge of shutting down chemical plants for maintenance and then restarting the plants once the maintenance had been completed. Given the complexity of the process, they developed the Critical Path Method (CPM) for managing such projects.

CPM provides the following benefits:

Provides a graphical view of the project.
Predicts the time required to complete the project.
Shows which activities are critical to maintaining the schedule and which are not.

CPM models the activities and events of a project as a network. Activities are depicted as nodes on the network and events that signify the beginning or ending of activities are depicted as arcs or lines between the nodes. The following is an example of a CPM network diagram:

CPM Diagram

Steps in cpm project planning.

Specify the individual activities.
Determine the sequence of those activities.
Draw a network diagram.
Estimate the completion time for each activity.
Identify the critical path (longest path through the network)
Update the CPM diagram as the project progresses.

1. Specify the Individual Activities

From the work breakdown structure, a listing can be made of all the activities in the project. This listing can be used as the basis for adding sequence and duration information in later steps.

2. Determine the Sequence of the Activities

Some activities are dependent on the completion of others. A listing of the immediate predecessors of each activity is useful for constructing the CPM network diagram.

3. Draw the Network Diagram

Once the activities and their sequencing have been defined, the CPM diagram can be drawn. CPM originally was developed as an activity on node (AON) network, but some project planners prefer to specify the activities on the arcs.

4. Estimate Activity Completion Time

The time required to complete each activity can be estimated using past experience or the estimates of knowledgeable persons. CPM is a deterministic model that does not take into account variation in the completion time, so only one number is used for an activity's time estimate.

5. Identify the Critical Path

The critical path is the longest-duration path through the network. The significance of the critical path is that the activities that lie on it cannot be delayed without delaying the project. Because of its impact on the entire project, critical path analysis is an important aspect of project planning.

The critical path can be identified by determining the following four parameters for each activity:

ES - earliest start time: the earliest time at which the activity can start given that its precedent activities must be completed first.

EF - earliest finish time, equal to the earliest start time for the activity plus the time required to complete the activity.

LF - latest finish time: the latest time at which the activity can be completed without delaying the project.

LS - latest start time, equal to the latest finish time minus the time required to complete the activity.

The slack time for an activity is the time between its earliest and latest start time, or between its earliest and latest finish time. Slack is the amount of time that an activity can be delayed past its earliest start or earliest finish without delaying the project.

The critical path is the path through the project network in which none of the activities have slack, that is, the path for which ES=LS and EF=LF for all activities in the path. A delay in the critical path delays the project. Similarly, to accelerate the project it is necessary to reduce the total time required for the activities in the critical path.

6. Update CPM Diagram

As the project progresses, the actual task completion times will be known and the network diagram can be updated to include this information. A new critical path may emerge, and structural changes may be made in the network if project requirements change.

CPM Limitations

CPM was developed for complex but fairly routine projects with minimal uncertainty in the project completion times. For less routine projects there is more uncertainty in the completion times, and this uncertainty limits the usefulness of the deterministic CPM model. An alternative to CPM is the PERT project planning model, which allows a range of durations to be specified for each activity.

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Critical Path Method

Learn how to find the critical path of a project and effectively implement the critical path method. See its importance and advantages and disadvantages.

The critical path method was developed in the late 1950s by James E. Kelley and Morgan R. Walker. Critical Path Method (CPM) is used to estimate the minimum project duration and determine the amount of schedule flexibility on the logical network paths within the schedule model. The critical path method is a widely recognized project management technique included in the Project Management Body of Knowledge (PMBOK). PMBOK provides a standardized framework for project management practices and methodologies. It emphasizes the importance of critical path analysis in project planning, scheduling, and control.

What is the Critical Path Method in Project Management?

The critical path method involves several key steps, including defining project activities, identifying task dependencies, estimating durations, constructing the network diagram, determining the critical path, and analyzing the project schedule. The process ensures a systematic approach to project planning and enables project managers to make informed decisions as the project progresses.

Successful project delivery requires essential project management methods in all types of organisations. For example, a project in any corporation would essentially have a range of activities that need to be completed in sequence or in parallel to achieve the project goal. For timely project completion, it is vital to understand the activities involved and how they relate to each other. The critical path method is particularly valuable for managing complex projects that involve numerous interdependent tasks and a significant number of resources. In such projects, it is crucial to have a clear understanding of the critical tasks that directly impact the project's overall timeline. Critical path tasks depend on their immediate predecessors, which are tasks that must be completed before the critical path tasks can begin. Understanding the immediate predecessors is vital for accurate critical path analysis. By identifying and sequencing these tasks correctly, project managers can establish the logical flow of work and determine the project's critical path. Accurate identification of immediate predecessors is essential for creating a reliable project schedule.By identifying these critical tasks, project managers can allocate resources strategically, closely monitor their progress, and ensure timely completion of the project.

Critical path analysis involves a thorough examination of the critical path and its associated activities. By conducting this analysis, project managers gain valuable insights into the sequence and duration of critical path activities. They can identify potential bottlenecks or areas of concern, allocate additional resources if needed, and implement contingency plans to mitigate risks.to proactively address issues that could potentially delay the project completion.

For instance, consider a House Construction Project. This project would have several tasks in sequence, such as compacting the ground, forming solid bases, footing reinforcement, shuttering, footing concrete, column casting, construction of walls, masonry work to keep openings for windows, roofing, plastering work, fixing of doors and windows. Each of these tasks will require a different amount of time and resources.

Given the above example,

(1) The project consists of a defined set of activities that mark the end of the project when completed.

(2) The activities may be started and stopped independently within a given sequence.

(3) The tasks are performed in order; for example, the walls must be constructed before starting the roofing.

With the use of CPM analysis, builders can estimate the duration (hours, days, or weeks it will take to complete the task, including elapsed time), the effort (hours, days, or weeks that take to perform the actual work), and based on that they can estimate the cost or budget each task.

Accurate task identification is essential for effective critical path analysis. Project managers need to break down the project into discrete tasks and determine their dependencies. By clearly defining and sequencing the tasks, project managers can identify the critical path tasks that have the most significant impact on the project timeline. Thorough task identification lays the foundation for accurate critical path analysis and project scheduling.

Importance of Critical Path Method (CPM)

Optimum utilisation of time and resources

In projects, all tasks take time, but some are more time-consuming and labour-intensive than others. Therefore, it is vital to identify a precise day-to-day plan of what activities to prioritise. A critical path method is a helpful tool that facilitates task prioritisation, giving organisations a better understanding of how and where to utilise resources. In addition, it provides the team with an assessment of actual time versus planned time.

The critical path method relies on the critical path algorithm to calculate the critical path and determine the project's duration. The algorithm takes into account task dependencies, activity durations, and constraints to identify the critical path activities. Through mathematical calculations and analysis, the algorithm identifies the sequence of tasks that must be completed without delay to ensure the project's timely completion.

The tool enables organisations to understand which tasks are taking longer than expected, which are ahead and scheduled, and which activities are right on track. The use of the tool will be helpful in terms of resource management .

Improved planning

There is a risk of losing valuable time in the project process due to unanticipated bottlenecks. Certain activities must be completed in order for other activities to begin. If one person cannot start a particular activity because they're waiting on a required item to be completed, it will eventually slow down the progress. This can also create a chain of events that can put the project behind schedule. The critical path method plays a significant role in mapping out parallel tasks or activities in sequence and managing project risks that impact critical path tasks have the potential to delay the entire project. By proactively addressing and mitigating risks associated with critical path tasks, project managers can minimize the likelihood of project delays and ensure successful project completion. A thorough assessment and analysis using critical path scheduling techniques enable better time estimates.

How to find the Critical Path of Project

The following are the steps to find the critical path of the project:

1. Divide the activities of the projects

Make a list of project activities and assign each activity with a name or a shortcode.

2. Create a dependency chart.

Put the activities in a logical line-up and identify dependencies.

3. Create a network diagram

Once all of the necessary activities are divided, make a visual activity line-up visual. The network diagrams are beneficial in connecting activities in the chart.

4. Estimate activity duration

It is crucial when planning any project to calculate how long each section will take. First, clearly define the beginning and end date for each task. Then set each task's estimated duration considering the order and dependencies of the activities. Then, calculate the project completion date.

5. Determine the Critical Path

The Critical Path is the sequence of activities that represents the longest path through a project, which determines the shortest possible duration and any delay in completing activities on the critical path will delay the overall project completion. Therefore, project managers must focus their attention on managing and monitoring critical path activities to ensure the project stays on track. The work breakdown structure is a hierarchical representation of the project's deliverables, sub-deliverables, and tasks. It provides a framework for organizing and decomposing the project into manageable components. The critical path method relies on the work breakdown structure to identify the sequence of tasks and their dependencies, enabling project managers to determine the critical path and schedule the project accordingly.

6. Calculate the float

Float is the amount of time that a schedule activity can be delayed without delaying the early start date of any successor or violating a schedule constraint.

The float is determined by how long the activity takes to complete and how many days are available between the activity's start date and the one which follows directly afterward. For example, no float is available if a task takes four days to complete and only has four days available from the start to the following activity. Therefore it is a critical task.

However, float days would be available if a task only took one day to complete, but there are five days until the start of the following job.

Therefore, this task is not time critical and can be slightly delayed if necessary.

Critical Path Method Example

Let's consider a software development project as an example. The project consists of various activities such as requirement gathering, design, coding, testing, and deployment. By applying the critical path method, project managers can identify the critical tasks that directly impact the project's duration. For instance, if coding and testing activities are on the critical path, any delay in these tasks will extend the overall project completion time. By focusing resources and attention on these critical tasks, project managers can ensure timely delivery of the software.

How to use the Critical Path Method

Schedule compression.

CPM is applicable in every project that determines the estimated time of interdependent activities. The method establishes a trade-off between the overall cost of the Project and the total time for completion of desired activities.

It assesses whether the estimated time could be reduced through compromising with cost (i.e. allowing the escalation of cost) or the same is not sound from a financial perspective.

CPM provides project managers with key information that allows them to minimise the project duration within certain limits by deploying additional resources at an optimal cost.

For example, if the project has some activities with negative floats, extra resources are inducted to reduce the negative float value. The process of reducing the time involved by allowing the increase in cost is known as crashing.

Initially, the normal time estimates for each activity are taken to calculate the project duration. To reduce the project duration, we need to reduce the time of critical activities because the reduction in time of any noncritical activity will not help in minimising the project duration. Therefore, the cheapest activity is selected for the crashing, and the process continues until any critical activity remains.

Measure Project Progress

A critical path schedule is useful to provide a visual representation of the critical path and the associated tasks within a project. It highlights the critical path activities, their durations, and the interdependencies between them. The critical path schedule serves as a roadmap for project managers, allowing them to track progress, identify potential delays, and make necessary adjustments to keep the project on schedule. Critical path drag is a metric used in the CPM to measure the impact of delaying a particular task on the project's overall duration. It quantifies the amount of time that a non-critical task can be delayed without affecting the project completion date. Tasks with a high critical path drag value should be closely monitored and managed to prevent any potential delays to the project.

Resource levelling

CPM can also be used for resource-levelling. Resource levelling is a resource optimisation technique in which adjustments are made to the project schedule to optimise the allocation of resources and which may affect critical path.

Resource levelling balances the demand and supply of available resources. It allows adjustments in time deadlines to meet the constraints of resources.

Resource levelling is primarily done when the restriction is on the availability of the workforce.

Levelling assists in adjusting resources against the possible floats available in the activity.

It is helpful to reduce the burden on the limited resource and establish the trade-off between the overall duration of the project and the cost involved.

Critical Path Analysis is commonly used in large industrial houses where the time involved in completing an activity can be reasonably predicted, owing to previous experiences.

A few examples of industries where CPM has wide applications are construction, defence, installation of complex equipment, engineering, aerospace, maintenance and shifting of plants, launching space programmes, traffic flow patterns, and other large projects.

Critical Path Method and PERT

A critical path diagram, also known as a network diagram or a PERT chart, visually represents the activities and their dependencies in a project. In PERT, the focus is on the start and completion of events and not activities. The activities that occur between events cannot be specified. It is appropriate for projects where the required time to complete different activities is unknown. PERT is usually applicable for scheduling, organisation, and integration of various tasks within a project. By examining the critical path analysis chart, project managers can easily identify the critical tasks and understand the flow of work throughout the project.

On the other hand, CPM is used for projects where the time needed to complete the project is already known. It helps determine the approximate time during which a project can be completed. It provides a meaningful way to keep project costs and timelines checked and balanced effectively.

Critical Path Method and Gantt chart

The Critical Path Method shows how each activity is related to one another; however, a significant problem with the Gantt method is that the Gantt charts do not show how activities relate to one another.

The critical path method represents a network diagram that displays each project activity and connects them to show task dependencies. At the same time, the Gantt chart is a bar chart that lays out project activities and timelines for each activity.

Critical Path Method: Advantages and Disadvantages

Advantages of critical path method.

It has the following advantages:

1. The CPM method determines the activities that can run parallel to each other.

2. The CPM helps the project managers identify the project's most critical elements.

3. CPM offers a straightforward approach to communicating project plans, schedules, time, and cost performance.

4. The CPM considers the requirements well in advance to complete a project in the most efficient way possible.

5. With the help of the CPM, the project manager can determine the duration and estimate the time and budget of the project.

6. CPM enables project managers to compare the actual progress of tasks with the project schedule. By tracking the actual start and completion dates of critical tasks, project managers can assess whether the project is on track or if adjustments are necessary. If actual progress deviates from the project schedule, project managers can take corrective actions to bring the project back on schedule.

Disadvantages of Critical Path Method

It has the following disadvantages:

1. CPM could be time consuming. It is also challenging to estimate the activity completion time in a multidimensional project.

2. The CPM networks can be complicated for extensive projects.

3. It cannot effectively tackle sudden changes in implementing the plan on the ground. For example, redrawing the entire CPM chart is extremely difficult if the project's plan suddenly changes midway.

4. CPM cannot form and control the schedules of the persons involved in the project.

5. Resource constraints can have a significant impact on the critical path and overall project duration. Limited availability of key resources, such as skilled personnel or specialized equipment, can potentially create bottlenecks and extend the critical path. Project managers must carefully consider resource constraints when planning and scheduling tasks to optimize resource allocation and minimize the impact on critical activities.

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COMMENTS

What Is the Meaning of the Critical Path Method in Operations ...
The critical path method, a product of operations research, helps you set timetables for complicated projects made up of multiple interdependent tasks.
Operations Research: Lesson 20. CRITICAL PATH METHOD (CPM)
Module 9. Project planning and network analysis. Lesson 20. CRITICAL PATH METHOD (CPM) 20.1 Introduction. After the project network plan is constructed and activity times are known, the time analysis of the network becomes essential for planning various activities of the project as well as obtaining answers to questions like when the various activities are scheduled to be performed, how long ...
Critical Path Method for Project management - GeeksforGeeks
Apr 5, 2024 · Identifying Critical Path: The critical path is the path that gives us or helps us estimate the earliest time in which the whole project can be completed. Any delay to an activity on this critical path will lead to a delay in the completion of the entire project. To identify the critical path, we need to calculate the activity float for each ...
Critical path method - Wikipedia
The critical path method (CPM), or critical path analysis (CPA), is an algorithm for scheduling a set of project activities. [1] A critical path is determined by identifying the longest stretch of dependent activities and measuring the time [ 2 ] required to complete them from start to finish.
What Is Critical Path Method (CPM) - Simplilearn
Aug 20, 2024 · 6. Forward Pass: This is the strategy for determining the critical path method's early start or finish times for tasks. 7. Backward Pass: The strategy to determine when an activity in the critical path method will have a late start or finish. 8. Network Diagram: A network diagram shows the connections between project activities.
Project Planning: Critical Path Method (CPM) – Operations ...
In example shown under critical path is estimated by finding the longest path in diagram. Step 4: In the critical path find an activity with least cost slope because aim is to reduce project time with least increase in cost. Step 5: Estimate the time for which selected activity can be crashed. For this a general rule that is followed is that ...
How to calculate the Critical Path with Examples
The methods used to calculate the critical path are the Project Evaluation and Review Technique (PERT) and the Critical Path Method (CPM). The PERT and CPM methods began to be developed in the 1950s to assist managers in scheduling, monitoring and controlling large, complex projects.
Guide: Critical Path Method (CPM) - Learn Lean Sigma
The critical path method is a strategic tool used in project management that is important for effective planning, scheduling, and executing complex projects. The core focus of CPM is to identify the longest sequence of dependent tasks in a project and identify and calculate the minimum required time to complete them.
CPM - Critical Path Method - NetMBA
The critical path is the path through the project network in which none of the activities have slack, that is, the path for which ES=LS and EF=LF for all activities in the path. A delay in the critical path delays the project. Similarly, to accelerate the project it is necessary to reduce the total time required for the activities in the ...
What is the Critical Path Method and Why is it so Important ...
Aug 15, 2022 · The critical path method represents a network diagram that displays each project activity and connects them to show task dependencies. At the same time, the Gantt chart is a bar chart that lays out project activities and timelines for each activity. Critical Path Method: Advantages and Disadvantages Advantages of Critical Path Method. It has ...

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Step 1: List All Tasks

Step 2: Determine Dependencies

Step 3: Estimate Durations

Step 4: Create a Network Diagram

Step 5: Perform a Forward Pass

Step 6: Perform a Backward Pass

Step 7: Identify the Critical Path

Step 8: Monitor and Adjust

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What’s Next?

1. What is the critical path formula?

2. What is a CPM schedule?

3. What are the four main steps involved within the CPM?

4. Can the critical path change during the course of a project?

5. Can CPM be used for all types of projects?

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How to Calculate Early Start, Late Start, Early Finish, Late Finish and Slack (Float)

Duration (t):

Early Start Time (ES):

Early finish Time (EF):

Late Finish Time (LF):

Late Start Time (LS):

Slack – Float (S):

Solution 1:

Forward path

Start node:

Activity A:

Activity B:

Activity C:

Activity D:

Activity E: