Namaskar and welcome to this series of lectures
on Principles of Construction Management and
in this lecture today, we will talk about
uncertainties in duration of activities using
the PERT in scheduling.
So far what we have said is that the time
of completion of an activity is firm, that
is, it is a given number and then we have
constructed networks to determine project
completion times based on interdependencies.
You would recall that in a first example,
when we said there was 500 man days involved
then we divided this by 10 which was a number
of people and came up with 50 days of activity
time.
And then we have also constructed in the last
class, a slightly different and more elaborate
network and illustrated the concept of critical
path, that is that path which needs to be
monitored more regularly, more continuously
than the others and no slippage can be allowed.
But in any case, when the interdependencies
were taken into account, the time associated
was taken to be firm.
An activity ij, it was known that this will
be completed in a time of tij.
Now, today, what we will do is we will take
a slightly different look at the activity
duration, that is, this tij with the emphasis
of incorporating the element of uncertainty.
How do you incorporate a situation when the
tij is not a firm number but more like a probability
distribution and activity may be completed
in 10 days, but it might spill over to 12
days, and if something good happens, and we
are lucky, then it might be completed in just
5 to 6 days.
So, this idea that the time associated with
each activity is not necessarily a fixed number
- that is something which we will focus on
today in our discussion.
So, how do we actually go about estimating
the times for the different activities?
One simple approach is to keep records that
will give us the average durations.
So, a construction company keeps a record
for erection process and so on and finds out
that okay, in different projects, what is
the kind of time that it takes and from there
we try to find out the productivities and
then try to map that into a new project and
try to estimate the time involved for that
project.
We have talked about in the initial part of
this course that each construction project
is really unique.
So, the bridge built at a particular place
is surely different from maybe a similar bridge,
but built elsewhere.
Also, the productivity of labour during the
course of the project changes and that could
also affect the duration of activities.
So, if there is an activity, you put a gang
of labour or a certain set of workers on day
one, they may not be able to produce the maximum
output, but as they learn the activity, and
as they learn their role in the whole process,
their productivity improves.
So, how do we keep track of these kind of
things?
Going back to the basics, usually, the duration
Dij of an activity is determined using this
formula which is - Aij / (Pij*Nij, where ij
is the activity that we are talking about.
Aij is the required amount of work, Pij is
the productivity of the person which could
possibly take care of the learning curve and
so on, and Nij is the number of persons working
on that job independently and together.
So, with this, we can find out the Dij, but
each of these parameters has their own variations,
for example, as far as Aij is concerned, the
only estimate that we can make from is the
‘good for construction’ drawings.
It might happen, that there will be some changes
in the total amount of work at site, whether
it is paid for or not, it does not matter,
but the fact is that if additional work is
carried out, or less work is carried out,
then it will affect the time duration.
So, coming to the next part, which is labour
productivity, this could also be an important
aspect of determining the total time involved
and that is where the learning curve becomes
important.
What is the learning curve?
A general understanding of the productivity
rates can be obtained from the concept of
learning curve and the base of the learning
curve lies in the fact that as crew becomes
familiar with the activity and the work habits,
their productivity tends to improve.
Of course, there is a limit beyond which it
does not improve, i.e there is a limit to
the capacity, or productivity, of each worker.
This capacity may be different for different
workers, but sure enough there is a starting
point and it goes through a learning phase
and a study phase for each worker.
In other words, productivity increases as
the crew gains experience, which is possible
if sufficient time is available and most construction
projects, that is indeed typically the case.
So now, another thing is the number of workers.
This equation here assumes that the duration
of an activity is inversely proportional to
the number of persons in the crew or the number
of workers in the team.
But is this assumption absolutely true?
Is it true in the absolute sense?
If we keep increasing this Nij, is there no
limit for the reduction in the duration that
we get?
There are two reasons why this assumption
is not really true - one is the crew may not
be able to work independently.
In a certain construction site, there is a
limit to the number of workers which can be
accommodated without interfering with each
other’s productivity.
The second thing is coordination between individual
tasks has to be ensured, and that also becomes
difficult as the number increases.
We all know the saying that “too many cooks
spoil the broth”.
So, if there are too many people trying to
do an activity, that also has an adverse effect
on the time duration.
Apart from the workers working independently,
the coordination between individual tasks
and the individual crew members has to be
ensured.
More persons in the crew may result in the
delay in the following situations.
An example is, design tasks are often divided
between architects and engineers in which
greater coordination is required among themselves.
In such situations, more the number of people
in the crew will obviously affect the coordination,
and therefore, it affects the time.
Ensuring a smooth flow of material or information
of any nature may become increasingly difficult
if there is a higher number of people in the
crew.
Further, in projects which are being executed
for the first time, it may not be possible
for planners to finalize the exact estimates
of the parameters like productivity rates,
optimum number of people in the crew, etc.
Other considerations may include practical
difficulties like the weather, contractor’s
failure to deliver materials, machinery breakdowns
and so on.
Also, most of the activities that are outside
the control of the client are certainly uncertain.
One simple example could be the time required
to get approvals from outside agencies and
that could affect the project in different
ways.
What this competent authority means is, for
example, if there is a project which requires
an approval from a Pollution Control Board
and such regulatory authorities, that might
be not within the control or within the powers
of the client to be able to implement and
that introduces an element of uncertainty
as far as the completion of the project is
concerned.
And it’s not only the project, but also
it has the affect, obviously, on individual
activities and those individual activities
in turn affect the project.
This underlines the need to understand that
there exists an uncertainty in estimating
activity duration, and therefore, considering
use of (continuous) probability distribution
functions becomes relevant.
So, what we are saying in this sentence is,
instead of assigning a particular time duration,
t, for any activity, can we not talk in terms
of a probability distribution, that is, yes,
this activity will take some time, but that
time could be maybe a little less, maybe a
little more.
So, whether this distribution should be normal
or not is the next question that we need to
answer.
So, without going into the statistics of the
whole issue, let us try to understand the
basic story of the normal distribution.
The normal distribution with values widely
spread away from the average is the situation
where their standard deviation is high.
Compared to this picture, this picture here
is a situation where the peak is sharper which
means that the area in this part is closer
to the mean.
If you want to go to the same area here, we
will go more standard deviations away from
the mean.
However, in both cases, the normal distribution
is symmetric.
The problem with construction activities is
that they are not necessarily symmetrically
distributed.
So now, what are the requirements that we
need to impose in determining or finding the
kind of distribution that helps us say that
yes, this is what really meets our requirements.
The probability of reaching the most optimistic
time, which is early completion, should be
very less.
So, this is the situation where we become
lucky and we are able to achieve the target
very quickly.
The probability of reaching the most pessimistic
time should be very less, that is, there should
be a time where we will definitely be able
to complete the project under most adverse
conditions.
Then there exists only one most likely time
which would be free to move between these
two extreme conditions.
So, what we are saying is that there is an
optimistic time, to, and there is a pessimistic
time, tp - between this to and tp we want
to define a distribution which is not normal.
So, either this most likely time will move
towards the left, in which case the pessimistic
time becomes an outlier, or it will move towards
the right in which case the optimistic time
becomes an outlier.
So, this amount of uncertainty should also
be measurable.
So, once we understand these requirements,
we find that the beta distribution that is
used in statistics satisfies these requirements
for us.
So, we can use the properties of the beta
distribution and move forward as far as determining
the expected times of the activities and so
on is concerned.
So, now what is a beta distribution?
We assume that this is the most likely time,
this is the pessimistic time and this is the
optimistic time and this distribution is skewed
to the right.
In this case however, on the figure on the
right, for the most likely time, the distribution
is skewed to the left.
So, this is a situation where most of the
times we will be able to complete the project
here, except if we become very lucky we might
hit the jackpot and come somewhere here.
In this case, we will probably be able to
complete the project somewhere here and only
if we are not running in luck at all, we will
be able to complete the project within the
pessimistic time.
So, with this kind of thought process we move
forward and try to see how the uncertainty
in estimating activity times can now be incorporated
in a scheduling process like PERT.
PERT is the Program Evaluation and Review
Technique, and the uncertainty in estimating
the duration of an activity is considered
in this technique as against the Critical
Path Method that we saw in the last class,
where the activities had a firm time of completion.
It is more suitable to control jobs that have
not been done before and there is more uncertainty
in the activities.
PERT assumes the three estimates of time are
random variables satisfying a beta distribution
function.
So, this is what we have been talking about
and now let us try to see how we actually
implemented on ground as far as activities
are concerned.
So, we have been talking of an activity ij
all the time.
So, here also we are saying that there is
an activity between nodes ind j, and we are
talking of three times associated with it
- optimistic time, to, which is the shortest
possible time of completing the activity under
ideal conditions, the most likely time tm,
which is the best guess of the time required
to complete the activity and the pessimistic
time tp, which is the maximum time required
to complete this activity.
So, usually the notation followed is given
here, for an activity ij, we give to, tp and
tm.
Now, from here, the expected time for completion
and the standard deviation in the activity,
we know the to, tp and tm, then how do we
calculate the texpected (te) of the activity
or for the activity ij.
Now, this is calculated using this equation,
that is, (to + tp + 4*tm)/6.
So, I am not getting into the derivation of
this, what it really says is that these are
the pessimistic and optimistic times and we
are giving a higher weightage to the most
likely time and we are getting an estimate
of the time at which, or during which, this
activity will be completed.
Please remember that this expected time does
not mean or does not give you an idea of the
50 percent probability for the time of completion.
In other words, this time does not mean that
the activity will be completed 50 percent
of the time - that would happen if the time
of completion of that activity was normally
distributed.
Given that it is not normally distributed,
te does not conform to a 50 percent probability
of completion of that activity.
Continuing, the variance, that is, St2, associated
in estimating the expected time of that activity
evaluated as ((tp–to)/6)2 and then we must
reiterate that te is taken as a random variable.
Moving forward, let us try to implement this
thought process in an example.
Let us talk of four activities A, B, C, and
D and these were the times that we used in
a CPM calculation, that is, the firm times.
Now, this table here, or this part of the
table here, gives you the to, tp and the tm
for each of these activities and the way these
times have been chosen is to ensure that the
time in CPM, and the expected time of the
activity can be taken to be the same.
So, activity A has 10 here, it is te here,
it is 8 here and 8 here, 15 here 15 here,
and, 12 and 12.
So, this effectively ensures that the te is
the same.
Similarly using this equation here, we have
calculated the St associated with each of
these activities.
So, if we examine this table closely, we find
that for activity A, this to is a little bit
of an outlier.
These two times are fairly close to each other,
but to is the optimistic time.
In the case of B, this number here, that is,
the pessimistic time is the outlier and for
activity C again, this is the outlier and
this is the outlier here.
So, what we have tried to do through this
arrangement of numbers is to covey to you
that these activities are not normally distributed
and sometimes they are skewed to the left
or they are skewed to the right.
And we have already talk before that skewing
to the left and skewing to the right have
a different meaning when it is comes to interpreting
the time duration associated with an activity.
So, now let us actually consider some numbers
- let us consider a network that has two activities
A and B, as shown here.
The three estimates for the duration of these
activities and days is given on the arrow.
So, here is the to, tm and tp for activity
A which is 1-2, and to, tm and tp for activity
B are 6, 14 and 16 which is 2-3, that is,
the activity B. So, what we are required to
do is to compute the total project duration
using the concept of expected times of activities
as we have defined earlier and compute the
project durations in the following cases.
If the activities are started on their respective
optimistic dates, pessimistic dates, and the
most likely dates and then, of course, we
can discuss those results a little bit.
So, working out the te and the standard deviations,
we find that for activities A and B, we convert
this information to this table here.
So, using these formulae, we calculate the
expected times of completion of the activities
and the variances.
So, the total project duration considering
the sum of the expected time of the activities
is 23, that is sum of 10 and 13.
Now, coming to the second part of the problem
which was to say that well, if activities
are completed at the optimistic times, then
of course, if we go back to our representation
on the time axis and try to say that A is
completed in 4 days, then B can immediately
follow from here, and in 10 days, the project
will be over.
However, if they started the pessimistic dates
the project will be over only in 28 days,
that is, this will take 12 days and then this
will go on for 16 days.
As far as the most likely dates are concerned,
then we are talking of 11 and 14, which is
giving us a number of 25.
So, the project duration obtained by considering
the most likely times of activities is 25
is very close to the results obtained by the
expected times, i.e., 23, that is this number
here.
The time to complete the project if activities
started on their pessimistic date, that is
28 days, is also pretty close to this result.
So, this is 28 and this is also close to this
result of 25.
However, the project being completed in these
two dates - the fact that both of these were
outliers - only shows that the project can
be completed, yes, in 10 days, what we understand
that the probability of being able to complete
this project in that time is indeed very low.
So, in the simplest terms, what we are saying
is that we become lucky not only once, but
twice; not only the activity A is completed
in the optimistic time, but activity B is
also completed in optimistic time.
So, we all know the basic probability kind
of questions that A and B has to happen in
this case.
We have to have A being completed in its optimistic
time and we have to have B also being completed
in its optimistic time.
Of course, there are other combinations that
A is completed in its optimistic time and
B in its pessimistic time, and so on.
So, that is the kind of range in the time
estimation that is important or that becomes
relevant as far as construction management
is concerned.
So, continuing our discussion further - how
do we determine the critical path in this
method?
The critical path is determined by carrying
out the forward and backward passes of activities
with durations being taken as the expected
time of completion.
So, of course, we take the expected time of
completion and then we use probability concepts
to find out what is the probability of completing
the project in that time, and so on.
So, expected length, or duration of the project,
Te, is calculated by summing up the expected
durations small tes of activities on the critical
path.
So, what we are saying is that if we say that
this is the critical path, we are not denying
that there are no other paths.
There can be other paths, but this is the
critical path.
So, for this critical path, we take the activities
on this path and try to find out the expected
time of the project using only the expected
times of the activities on the critical path.
These activities which are non-critical, will
have to be dealt with separately.
And the variance associated with the critical
path is the sum of the variances of the activities
on the critical path and we can find out the
standard deviation by finding the square root
of this variance.
So, this is the statistical concept which
we are not deriving in this course, and I
am leaving it to you to take a look at some
of the statistics text books and move forward.
Now, since the Te is the sum of tes it is
indeed a random variable and follows normal
distribution according to the central limit
theorem of statistics.
The fact that Te follows normal distribution
can now be used to do these two things – the
first thing is to find the probability of
completing the project within a certain target
duration.
If there are two activities, we can ask the
question that well, there are some times associated
with these two activities, but what is the
probability that this project will be completed
within a certain time, whatever that number
is?
That we will see in an example later on.
But also, in the second variant, we can find
the expected completion time with a given
probability.
Basically, these two questions are just two
sides of the same coin.
Correspondingly, the standard normal deviate
Z is evaluated as (Td – Te) / ST and this
is something which comes in very handy when
we are trying to answer questions such as
these.
So, now to use an illustrative example for
PERT, let’s say that there is a network
which is given here, which is a part of a
larger network.
There are a lot of activities here and here,
but we are talking only of this part because
that is what is our critical path.
So, for this critical path, the durations
are given here for that activities A, B and
C, this is the to, tm and tp for all these
activities.
So, what we are trying to find out is (i)
the total project duration, and, (ii) the
probability of completing this project within
17 weeks.
Now, 17 weeks, please see, is if we do a very
simple calculation 4, 4.5 and 6, will give
us 14.5.
So, what we are saying is that if we just
take the most likely times, we did that in
an example just now, that if we just take
the most likely times, it turns out to be
14.5.
So, what we are talking about is that what
is the probability associated with completing
this project, not in 14.5 or 15 weeks, but
in 17 weeks.
So, what the probability should be is what
we need to calculate.
Moving forward, we first calculate the tes
and the St2 s using this formulae and we find
out the Te, which is the total project duration,
by summing up these three tes, and we find
15.
Similarly, the variance which is associated
with the project is the sum of the variance
of the activities, which is 1 in this case,
so the standard deviation is 1.
So, we have this calculation carried out.
Now, with this calculation we go to the charts
and tables relating to finding out how this
area is related to the number of standard
deviations that you go away from the mean.
So, if you do this exercise for this particular
project, here we have 15, here we have 17,
and here we have 1.
17 is our target.
We want to find out what is the probability
associated with completing the project in
17 days, which is higher than 15, which is
the most likely time of completing this project.
So, this number here, which has been arrived
at using the expected times, is 15 and we
are trying to find out what is the probability
associated with completing the project in
17 and this is the standard deviation.
So, the Z value is 2 and with this, if we
calculate the required probability, we will
find that the answer is 97.7 percent.
So, what we can see is that the probability
of completing this project in 17 weeks is
97.7 percent.
Here we can say that, yes, the Te represents
that time which has a 50 percent probability
of completion.
So, with this, we have introduced the concept
of PERT, we have introduced the concept of
uncertainty in the activity durations.
Now suppose instead of 17 we were talking
of 13, if we go back to the data given here,
the to for the three activities is 3, 4 and
4.
So, the idea is that instead of 15, which
represents the 50 percent probability of completion,
we are talking of going minus 2 here at 13,
so what will be the probability of completing
this project in 13 days?
Of course, there is a finite possibility because
the optimistic times associated with these
three activities is indeed less than 13.
This is something which I am leaving to you
as a food for thought.
You try to do this example on your own and
I am sure you will get the answer very quickly.
And I hope that you now understand how to
handle these concepts and be able to calculate
or estimate project durations, find out the
probabilities associated with completing a
project in a given time, and so on.
With this, we will just give you the list
of references which I always do at the end
of a lecture and look forward to seeing you
again.
Thank you.
