Ls And Lf In Cpm Homework

If you have experience in project management, you might have heard about the critical path method (CPM) — a project modeling technique developed by Morgan R. Walker and James E. Kelly in late 1950s (Wikipedia).

The critical path method (CPM) is used extensively by project planners worldwide for developing the project schedule in all types of projects including IT, research, and construction.

This method is a basis of the project schedule and is discussed very broadly in the PMBOK Guide. You can expect to see two to three questions, or more, in your PMP exam from this topic.

In this blog post, I will discuss the critical path with a real world example, identify the critical path in a network diagram, and calculate the float for each path. I will then list some of the benefits and limitations of the critical path method. 

Once you become familiar with it, I will walk you through every step required for calculating Early – Start, Early – Finish with forward pass, and then calculate Late – Start, Late – Finish with backward pass.

Make sure you understand each step described, otherwise you might face some difficulties when working with these calculations. If you feel you need any clarification, feel free to reach out to me at any time.

Critical Path

If you look at any network diagram, you will notice many paths originating from one point and ending at another point. Every path will have some duration. The path with the longest duration is known as the critical path.

The critical path can be defined in many ways including:

  • The longest path in the network diagram, or
  • The shortest duration in which the project can be completed.

Now, are these two definitions are similar, or are they opposite to each other?

In fact, both definitions are trying to convey the same message. They might seem opposite to you because the first definition is talking about the longest path and the second definition is talking about the shortest duration.

However, they both are the same.

For example, let’s say you received a project to build three buildings in one location. The first building is the largest building, the second building is a medium-sized building, and the third building is the smallest building.

You develop the network diagram which consists of three paths; each path resembles each building.

You calculate the duration for each path. For the first building, the duration is 31 months, the second building will take 18 months, and the third building will require 13 months.

The first path represents the largest building; the second path represents the medium-sized building, and the third path, the smallest building.

 


Now, review the above diagram.

Did you notice that the path for the first building is the longest duration of all three? It is thirteen months longer than the second path, and 18 months longer than the third path. This means that if you start working on the first building, you can wait 13 months before working on the second building because you can complete second building in 18 months.

Likewise, you could wait 18 months to start working on the third building because it will take only 13 months to complete. This means that even if you start working on the third building after 18 months from the project start date, you can complete it on time.

This waiting period is known as the float or slack.

So, which is the critical path in this network diagram of three paths?

Of course, it is the longest path on the network diagram, because you cannot complete your project before constructing the first building. Although you can complete the other two buildings quickly, until you complete the first building, your project is not considered complete.

Hence, the critical path is the longest path on the network diagram.

Now, what is the shortest duration in which you can complete the project?

Sure enough, it is 31 months, because you cannot complete your project before 31 months, and this is the duration of the critical path.

Hence, the critical path is the shortest duration in which you can complete the project.

So, you see, both definitions are the same.

We can conclude that the critical path is the sequence of activities from start to end, and it has the longest duration among all paths in a network diagram.

In ideal conditions, a network diagram, and therefore, the project, should have only one critical path. However, if the network diagram has more than one critical path, you will be in a difficult situation. In this case, you will have to manage more than one path in parallel.

As we know, the critical path has the longest duration, and its duration is known as the duration of the project. Since activities on a critical path have no float or slack, no activity should be delayed. If this happens, the project will be delayed. However, if there are delays, you can use a schedule compression tool, such as fast tracking or schedule crashing, to bring the project on track.

Visit: Fast Tracking and Crashing

Note: You must always update the network diagram if there is any change to it so that you can have a better understanding of activities and predict the float, project completion dates, etc.

Procedure for Finding the Critical Path in a Network Diagram

The following is the procedure to find the critical path on a network diagram:

  • Draw the network diagram.
  • Identify all paths in the network diagram.
  • Find the duration of each path.
  • The path with the largest duration is the critical path.

Let’s look at the above procedure in action.

Example

Based on the below network diagram, identify the total paths, critical path, and float for each path.


The above network diagram has five paths; the paths and their duration are as follows:

  1. Start -> A -> B -> C-> End, duration: 31 days.
  2. Start ->D -> E ->F -> End, duration: 18 days.
  3. Start -> D -> B -> C -> End, duration: 26 days.
  4. Start -> G ->H ->I -> End, duration: 13 days.
  5. Start -> G -> E ->F -> End, duration: 16 days.

Since the duration of the first path is the longest, it is the critical path. The float on the critical path is zero.

The float for the second path “Start ->D -> E ->F -> End” = duration of the critical path – duration of the path “Start ->D -> E ->F -> End”

= 31 – 18 = 13

Hence, the float for the second path is 13 days.

Using the same process, we can calculate the float for other paths as well.

Float for the third path = 31 – 26 = 5 days.
Float for the fourth path = 31 – 13 = 18 days.
Float for the fifth path = 31 – 16 = 15 days.

Calculate Early Start (ES), Early Finish (EF), Late Start (LS), and Late Finish (LF)

We have identified the critical path, and the duration of the other paths, it’s time to move on to more advanced calculations, Early Start, Early Finish, Late Start and Late Finish.

Calculating Early Start (ES) and Early Finish (EF)

To calculate the Early Start and Early Finish dates, we use forward pass; we will start from the beginning and proceed to the end.

Early Start (ES) for the first activity on any path will be 1, because no activity can be started before the first day. The start point for any activity or step along the path is the end point of the predecessor activity on the path plus one.

The formula used for calculating Early Start and Early Finish dates.

  • Early Start of the activity = Early Finish of predecessor activity + 1
  • Early Finish of the activity = Activity duration + Early Start of activity – 1

Early Start and Early Finish Dates for the path Start -> A -> B -> C -> End


Early Start of activity A = 1 (Since this is the first activity of the path)

Early Finish of activity A = ES of activity A + activity duration – 1
= 1 + 10 – 1 = 10

Early Start of activity B = EF of predecessor activity + 1
= 10 +1 = 11

Early Finish of activity B = ES of activity B + activity duration – 1
= 11 + 12 – 1 = 22

Early Start of activity C = EF of predecessor activity + 1
= 22 +1 = 23

Early Finish of activity C = ES of activity C + activity duration – 1
= 23 + 9 – 1 = 31

Early Start and Early Finish Dates for the path Start -> D -> E -> F -> End


Early Start of activity D = 1 (Since this is the first activity of the path)

Early Finish of activity D = 1 + 5 – 1 = 5

Early Start of activity E = EF of predecessor activity + 1

Since the Activity E has two predecessor activities, which one will you select? You will select the activity with the greater Early Finish date. Early Finish of activity D is 5, and Early Finish of activity G is 3 (we will calculate it later).

Therefore, we will select the Early Finish of activity D to find the Early Start of activity E.

Early Start of activity E = EF of predecessor activity + 1
= 5 + 1 = 6

Early Finish of activity E = 6 + 7 – 1 = 12

Early Start of activity F = 12 + 1 = 13

Early Finish of activity F = 13 + 6 -1 = 18

Early Start and Early Finish Dates for the path Start -> G -> H -> I -> End


Early Start of activity G = 1 (Since this is the first activity of the path)
Early Finish of activity G = 1 + 3 – 1 = 3

Early Start of activity H = 3 + 1 = 4
Early Finish of activity H = 4 + 4 – 1 = 7

Early Start of activity I = 7 +1 = 8
Early Finish of activity I = 8 + 6 – 1 = 13

Calculating Late Start (LS) and Late Finish (LF)

We have calculated Early Start and Early Finish dates of all activities. Now it is time to calculate the Late Start and Late Finish dates.

Late Finish of the last activity in any path will be the same as the Last Finish of the last activity on the critical path, because you cannot continue any activity once the project is completed.

The fula used for Late Start and Late Finish dates:

  • Late Start of Activity = Late Finish of activity – activity duration + 1
  • Late Finish of Activity = Late Start of successor activity – 1

To calculate the Late Start and Late Finish, we use backward pass; i.e. we will start from the last activity and move back towards the first activity.

Late Start and Late Finish Dates for the path Start -> A -> B -> C -> End

On a critical path, Early Start, and Early Finish dates will be the same as Late Start and Late Finish dates.

Late Start and Late Finish Dates for the path Start -> D -> E -> F -> End


Late Finish of activity F = 31 (because you cannot allow any activity to cross the project completion date)

Late Start of activity F = LF of activity F – activity duration + 1
= 31 – 6 +1 = 26

Late Finish of activity E = LS of successor activity – 1
= LS of activity F – 1
= 26 – 1 = 25

Late Start of Activity E = LF of activity E – activity duration + 1
= 25 – 7 + 1 = 19

Late Finish of activity D = LS of successor activity – 1

If you look at the network diagram, you will notice that activity D has two successor activities, B and E. So, which activity will you select?

You will select the activity with the earlier(least) Late Start date. Here, Late Start of activity B is 11, and Late Start of activity E is 19.

Therefore, you will select activity B which has the earlier Late Start date.

Hence,

Late Finish of activity D = LS of activity B – 1
= 11 – 1 = 10

Late Start of Activity D = LF of activity D – activity duration + 1
= 10 – 5 + 1 = 6

Late Start and Late Finish Dates for the path Start -> G -> H -> I -> End


Late Finish of activity I = 31 (because you cannot allow any activity to cross the project completion date)
Late Start of activity I = 31 – 6 + 1 = 26

Late Finish of activity H = 26 – 1 = 25
Late Start of activity H = 25 – 4 + 1 = 22

Late Finish of Activity G = 19 – 1= 18 (we will choose the late start of activity E, not activity H, because the Late Start of activity E is earlier than the Late Start of activity H)

Late Start of activity G = 18 – 3 + 1
= 16

Calculate the Free Float

I strongly recommend you read my blog post on total float and free float to get a better understanding before proceeding further.

Visit: Total Float and Free Float

The formula for the Free Float is:

  • Free Float = ES of next activity – EF of current activity – 1

Benefits of the Critical Path Method

The following are a few benefits of the critical path method:

  • It shows the graphical view of the project.
  • It discovers and makes dependencies visible.
  • It helps in project planning, scheduling, and controlling.
  • It helps in contingency planning.
  • It shows the critical path, and identifies critical activities requiring special attention.
  • It helps you assign the float to activities and flexibility to float activities.
  • It shows you where you need to take action to bring project back on track.

Although the critical path is very useful tool in project planning, it also has some limitations and drawbacks.

Limitations and drawbacks of the Critical Path Method

  • Because the critical path method is an optimal planning tool, it always assumes that all resources are available for the project at all times.
  • It does not consider resource dependencies.
  • There are chances of misusing float or slack.
  • Less attention on non-critical activities, though sometimes they may also become critical activities.
  • Projects based on the critical path often fail to be completed within the approved time duration.

To overcome these shortcomings of the critical path, the critical chain method was developed. In the critical chain method resource constraints are also taken into consideration while developing the network diagram.

Visit: Critical Chain Method in Project Management

Summary

The critical path method has helped many project managers develop and manage their schedule. In the critical path method, you will draw a network diagram with multiple paths. The path with the longest duration is known as the critical path. During your project execution your main emphasis will be on this path, because this is the longest duration path and the duration of this path will be duration of the project.

As a project manager you have to keep an eye on your network diagram and take prompt corrective action whenever necessary.

If you have any comments or questions, you can do so in the comments section below.

If you are interested in learning all the mathematical formulas for the PMP exam, you can try my PMP Formula Guide. You can also try my PMP Question Bank and PMP Mock Test to practice PMP exam sample questions.

Gantt chart this will help calculate the early start, late start, and slack time. The calculations for early start (ES), early finish ( EF ), late start ( LS ), late finish (LF) and slack time ( SL ) are as follows ( SHENOY, S. 2013): EF = ES + Duration -1 ES = 1 (The earliest start on any activity is always 1) SL = LS – ES LS = LF – Duration + 1 LF = (Late finish of last activity on the critical path is same as its early finish.) SL = LF – EF CPM can help a project manager figure out whether you should try to speed up the project and if so, what the least costly way to speed the project up will be by putting in the information regarding the cost of each activity and how much it cost to speed up each activity ( Baker, S. (2013). After listing all the activities for the project list the predecessors and any dependencies for the activities. Once the activities, predecessors, and dependencies are written out and you

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