Case Study: PERT / CPM  Calculating Floats
 Case Study: Using CPM or PERT in time planning & control
This hub is an extension to my previous hub CPM/PERT. It would be followed by another hub. The three have interlinks. These have been broken into three parts for ease of reading.
NET WORK # 1
This is in continuation of my previous Case Study on CPM & PERT. Here I would describe ways and means to calculate Floats or Slacks. A little change has been made in Sketch # 3 in the previous article. Description has been reduced to one cell. Since there is not enough space for full description, identifying letters have been inserted in the respective Cells.
Two more Cells have been introduced for accommodating Free Float (FF) and Total Float (TF). All other figures are the same as shown in Net Work #1.
WHAT IS A FLOAT?
A float shows time available for delaying an activity without delaying Finish Date of the Project.
In other words, delay in some activity would not increase the project duration. Please note that Critical Activities cannot be delayed. However, NonCritical Activities can start late or finished late within the given limitation. Other names for float are: slack, cushion, margin, excess time or flexibility.
For calculating floats, there are different Methods depending upon format of the Network used. There are two types of Net Works: ActivityOnNode (AON) and ActivityonArrow (AOA). Results are same in both cases.
Net Work # 2
 Arrow Diagram  ASQ
The arrow diagram shows the required order of tasks in a project or process, the best schedule for the entire project, and potential scheduling and resource problems and their solutions.
 Difference between AOA & AON
Method for constructing a unique AOA net with a node for each precedence constraint of its corresponding AON network (yielding small number of dummy arcs).
TWO TYPES OF PRESENTATIONS
For better understanding, two versions of the same net work are given as per Network #1 and #2 and briefly described below:
AON ( see Net Work #1)

A new format used by project management softwares.

Better at showing different types of dependencies.

Easy to understand. Can be constructed through placing cards each with name of an activity. Good for group discussion.
AOA (see Net Work # 2)
 Nodes or circles are the starting and ending points of activities.
 Activities are represented by arrows showing relationships between activities.
 Sometimes, dummy activities (dotted lines) are used for linking two activities.
 Can only show finishtostart dependencies.
THREE TYPES OF FLOATS
INDEPENDENT FLOAT (IF)
 Shows the time available even if an activity has a Late Start and Early Finish.

It is most adverse type of float and often results in a negative figure.
FREE FLOAT (FF)

Also called Normal Float, it shows time for which an activity can be delayed without delaying the early start of successor activity/activities.

If there are more than one succeeding activities, minimum of the floats would be taken as free float.

FF will always be less than or equal to TF and never more.

In all critical activities, FF is always zero.
TOTAL FLOAT (TF)
 It shows time for which an activity can be delayed from its ES without delay in project completion. (In FF, when one activity was delayed, the succeeding activity was started on time as per its ES and not delayed.)
 As in FF, in case of more succeeding activities, minimum to be taken.
 in TF, however, if one activity is delayed, the succeeding activity or activities would be affected or rescheduled.
 On Critical Activities, TF is always zero ( as well as FF)
SUBSET OF AON
CALCULATIONS OF FLOATS UNDER AON
 INDEPENDENT FLOAT (IF) : EF – LS = 10 4 =6
 FREE FLOAT (FF) : G(ES) –D(EF)= 10 – 4 =6
 TOTAL FLOAT (TF): D (LF) – D(EF)= 13 – 4 = 9
In case of free float, the formula is: Early Start of Successor Activities (G) minus Early Finish of Existing Activity (D). Naturally, if we can finish an activity early but the next will start sometimes later, we have a free float to delay our activity.
CALCULATION OF FLOAT UNDER AOA
 INDEPENDENT FLOAT (IF) : EF – LS = 10 4 =6
 FREE FLOAT (FF) : G(ES) –D(EF)= 10 – 4 =6
 TOTAL FLOAT (TF): D (LF) – D(EF)= 13 – 4 = 9
formulae are:

IF(i,j)=EF(j) ES(i)D(i,j)
 FF(i,j) = EF(j)  ES(i)  D(i,j), (If there are more than one activities, minimum thereof)

TF(i,j) = LF(j)  ES(i)  D(i,j) (If there are more than one activities, minimum thereof)
DISPLAY OF CALCULATED FLOATS UNDER BOTH FORMATS
Network # 3 and Table #1 show calculated floats in their respective places. AON is verstile format and can accommodate all type of floats. ( I have given only two types TF & FF but it can be expanded.) In case of AOA, the floats are to be shown separately which makes it rather inconvenient to follow.
NET WORK # 3
TABLE 1
TABLE #2
 CASE STUDY  Probability in PERT
Next article on PERT has been added.
CONFLICTS IN CALCULATION & COMMON SENSE
Sometimes, floats given by formulae do not appeal to the common sense. It happens in both cases whether AON or AOA. The floats are, however, result of a natural flow and a visionary project manager would easily spot a flaw in calculation.
For example, FF in case of activity 23 in AOA is different from the one shown under AON. The reason is that in AOA, dummies are used which distort the straight calculation.
Here we can resort to Gantt Chart to reconcile the difference as shown in Table #2.
Interference Float
Interference float is the difference between total float and independent float.
Supposing there is a conference of 5,000 participants in an Island. The conference would be for 5 days starting from 17th June and ending on 21st June. To transfer a large number of people, the flights to island would start from 1st June to 16th June and the return would be from 22nd June to 30th June. Suppose further that the Island has a lot of sightseeing opportunities on the side.
Now if some big boss is interested only in conference, he or she can leave on 16th June and return by 22nd June. In this way, one day would still be available for shopping or sightseeing calculated as follows: (22 – 16  5). This one day is called Independent Float and is available in any case.
On other hand, total float (or total time for sightseeing) would be for 24 days when someone starts on 1st June and returns on 30th June.
So one cannot devote more than 24 days for sightseeing while one has to spend one day because of flight schedule. It is evident that one can interfere or modify the schedule for sightseeing for 23 days ( 24 1).
WHY FLOAT IS IMPORTANT?
If every activity had a fixed start date and a fixed end date, it would be very difficult to complete the project in time.
Problems do occur such as shortage of funds or workers or machines. If there are floats, the manager can decide which activity to give priority and which one to reschedule. In this way, it is possible not to let the project suffer because of shortages. Some activities can be slipped or splitted for a good cause.
In some other article, I would explain how these floats are used for a sound management.
A case study on floats is given at another hub Floats in a Network
Popular
Comments
I am an Engineering student at University, and I have a Project Management module. All the exam papers ask what the dependent float is. I have searched and searched, but I cant find anything on how to calculate dependent float. Can you be of any help?
thank u....its very help full.
I had no idea hw CPM worke until I read this case study, you made it look easy. It helped me a lot. Many many many thanks.
FLOAT FOR CRITICAL ACTIVITY IN PERT
wonderful article ...thnks a lot
the info is precise and that makes it awesome.
a very informative and a very well define article of Floats.
sir what is the end float?
what is the dependent float?
maybe the total float can be in minus?
(TotalFloat)ij = (LateFinish)j  (EarlyFinish)i  Tij
which of the floats can be under zero? what does it mean? why the other's can't be under zero?
VERY USEFUL
sir,wat if there are more than one activities who don't have a predecessor,how do we represent that in an activity on node diagram
Thank you very much, now i know how i will tackle my assignment
Formal education will make you a living; selfeducation will make you a fortune. and for me you are very fortune .sir we love you may Allah gives you a very very long life ameen .
Dear Sir,
Thanks for very clear presentation. Please keep up the good work.
Manjunath, India.
Thank you Sir for your reply.
The link about risk management is very usefull.
Dear Sir,
I just discover your hub and would like to thank you for your contribution on sharing your knowledge in project management.
I went thru many readings about project Networks seeking for dangers and problems commonly faced by Project manger and the associated risks. Also i never understand in what circumstances these dangers or problem can be advantageous.
For example, it is known that:
 WBS is not easy to do
 Duration estimates are most of the time subjective
 near critical paths can become critical and affect project completion on time and schedule.
Are these examples limitations of the project networks or really problems?
I have a question.
I know it is not possible for Float to be negative.
For example,if a project expected to finish after 138 days. and it finished in 130. there will be differences in my project how can I avoid the negative float?
Thank you sir brilliant it helped me a lot. thanks a lot..
very helpful article thnx a lot for such a good article .
I consider PERT/CPM a secret weapon of the allies in WWII because it enabled the building of a ship in 56 days instead of over 300 days before WWII.
Thank you Sir, article is informative.
VERY well written! Struggled with this quite a bit could not find a helpful approach  excellent!
Good Morning Sir,
I think, now i have understood, i have to carefully consider succeeding activity for both free float and total float. Sorry for disturbing you
Without developing AON, how one can calculate total float in Gantt Chart
Thank you very much. I had a problem with the independent float calculations and you cleared it up!
The article is really very good. Floats are always very important in determining the Critical Time of the Project, because uncertainties are always there. A very comprehensive study is present in this hub. Thanks Sir.
thanks for refering this article. this help me to clear my concepts about float.
thnx for providing such a detailed knowledge
a very well define article of Floats
Dear Sir,
Thanks for the explanation.
But independent float formula mentioned earlier is Late Start (LS)  Early Finish (EF) which gives answer as (133)4=6
But later in example you have mentioned formula for Independent float as Early Finish(=4)  Late Start(=10)  Duration(=3) gives answer as 9.
So the answer is different.
Kindly explain this.
thanks brief and clear
thanks a lot your material has been very helpful in my course of study.please keep it coming.
Hi, I came across this link following my query to explain Floats and i found much more than this..Thank you
Salem,
Montreal,Canada
Thank you
sir can you tell about the ineterference float
this article is presented in a simple but very informative way...thanks, it made me understand our lesson well.
now its so easy to prepare for exams in barely 1 hr. thanks a tonne
thanks.. very helpful
Your suggested formula for calculating free float for an activity on arrow network (The original Du Pont CPM algorithm) I  J is incorrect and cannot be applied in all scenarios. For example when you have two or more preceeding activities converging into the same finish event, which becomes the start event of a succeeding actvity, the ESD of that succeeding activity is governed by the maximum EFD value of one of the preceeding activities as a result of the forward pass calculation. Only this value will be recorded as the ESD event of the succeeding activity. Therefore the formula for calculation of free float for an activity I to J is flawed. Instead Free float calculations need to be undertaken for each individual link to the non critical I activity i.e. j1, j2 etc. Then the smallest of of the individual free float values will become a free float of activity I, i.e. MIN (ESD j1, j2...  EFDi).
ESD being the Earliest Start Date / EFD being the Earliest Finish Date of an activity i to j.
Dear Mr. Hafeez,
Thank you very much, your way of explanation makes one to understand with ease.
A.A
THANKS FOR ADDING UP IN OUR KNOWLEDGE, AND ALWAYS BE EAGER TO RECEIVE SUCH VALUABLE INFORMATION.
ASLAMOALAIKUM
THIS ARTICLE IS REALLY IMPORTANT TO READ, BECAUSE IT ENHANCE OUR KNOWLEDGE IN THE FIELD OF PROJECT MANAGEMENT.
REGARDS
ASSALAM O ALAIKUM
you done a great job but these tables are little complex to understand...but they will help me in many cases
thanks
regards
SAAD RAIS
this article is very informative i learn many thing through by this aricle
This case study helped me a lot, the graphical representation was very useful.
Regards,
M. Fahad Khan
ASSALAM U ALAKUM
VERY INFORMATIVE. CLEARING MY CONCEPTS MUCH MORE.
THANK YOU
very much informative article :)
Thanks you are providing such a great information about flot it is really informative
thanks a lot
Thats really informativeI think that these pictorial views give me more understanding to have in your articles.
Regards,
Mubin Ashraf.
IF (fec), FF(fdc) , TF(gdc) is this given or we can identify ourselves the subtraction of activities??? if it is given so uptill now it is clear to me and if we have to identify ourselves so how we can identify the subtraction of activities among each other??
i can't understand the table 1, the IF(fec), FF(fdc), TF(gdc).. how these activities are subtracting with each other for a float???
The importance of Floats is obvious from the article.......
Thank you v. much..
good article, very informative.
Good Article
Thanks!
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