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 its a given number and then we have constructed networks to determine project
completion times based on interdependencies so you would recall that in a first example
when we said there was five hundred mandays involved then we divided this by ten which
was a number of people and came up with fifty days of activity time and then we have also
constructed in the last class slightly different network 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 that 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 ten days but it might bellower to twelve
days if something could happens if we are lucky then it might be completed in just five
to six 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 of process and so on and finds out that ok 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 may be 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 has they learn the activity 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 upon pij multiplied by
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 doesnt 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 we 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 so 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 there is a limit to the capacity 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 in 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 who can be accommodated without interfering with each others productivity
so thats one part of it the second thing is coordination between individual task has to
be ensured and that also becomes difficult we all know the saying that too many cooks
spoil the broth so if there are too many people trying to an activity that also has an adverse
effect on the time duration apart from the workers working independent the coordination
between individual task 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 task 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 affect the time ensuring a smooth flow
of material or information of any nature may become increasing the difficult if there is
more number of people in the crew further in projects which are being executed for the
first time it may not be possible for planers to finalize the exact estimates of the parameters
like productivity rates optimum number of people in the crew other considerations may
include practical difficulties like the weather contractors 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 a 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 night 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 in its not only the
project but also it has the affect obviously on individual activities and those individual
activities intern affect the project this underlines the need to understand that
there exists an uncertainty in estimating activity duration and therefore considering
to use continuous probability distribution functions becomes relevant in this case so
what we are saying in this sentence is instead of assigning a particular time t for any activity
can we now talk in terms of a probability distribution that is yes this activity will
take some time but that time could be may be little less may be 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 a statistic of the whole issue lets 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 compare 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 so the problem with our 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 or 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 time where we will
definitely be able to complete the project under most adverse conditions then there exist
only one most likely time which would be free top move between these two extreme conditions
so what we are saying is that there is an optimistic time t naught and there is a pessimistic
time tp between this t naught 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 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 comes over here in this case we will probability will 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 this is the kind of thought process
and 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 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 or random variables satisfying a beta distribution
function so this is what we have been talking about and now lets try to see how we actually
implemented on ground as far as activities are concerned so we have been talking off
an activity ij all the time so here also we are saying that there is an activity i and
j and we are talking of three time associated with it optimistic time t o which is the shortest
possible time of completing the activity under ideal conditions the most likely time t m
which is the best guess of the time required to complete the activity and the pessimistic
time t p which is the maximum time required to complete this activity so usually the notation
followed is given here for an activity ij we give t o t p and t m now from here the
expected time for completion and the standard deviation in the activity we know the t o
t p and t m then how do we calculate the t expected of the activity or for the activity
ij now this is calculated using this equation that is t o plus t p plus four times t m divided
by six so i am not getting into the derivation of this what it really says is the this on
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 fifty percent time of completion this time does not mean
that the activity will be completed fifty 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 t e does not confirm to a fifty percent probability of completion of that
activity continuing the variance that is s t square
associated in estimating the expected time of that activity evaluated as t p minus t
naught divided by six squared and then we have must retreat the t e is taken as a random
variable moving forward lets try to implement this thought process in a example lets talk
of four activities a b c and d and if these were the times that we used in a c p m calculation
that is the firm times now this table here or this part of the table here gives you the
t o t p and the t m for these activities and the way this t o t p and t m have been chosen
is to ensure that the time in cpm and the expected time of the activity are taken to
be the same so activity has ten here it is t e here it is eight here and eight here fifteen
here fifteen here and twelve and twelve so this effectively ensures that the t e 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 t naught is a little bit of an outlier these two times are fairly close to each other
but t naught 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 try 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 this skew to the left and skew to the right has a different meaning when its come
to interpreting the time duration associated with an activity so now lets actually consider
some numbers lets 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 t o t m and t p for activity a which is one two and t o t m and t p for
activity b are six fourteen and sixteen which is two three 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 the respective optimistic dates pessimistic dates
and the most likely dates and then of course we can discuss most results a little bit so
working out the t e and the standard deviations we find that for activities a and b we convert
this information to this table here the using these formulae we calculate the expected times
of completion of the activities and the variances so the total project duration considering
the expected time of the activities is twenty three that is ten and thirteen 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
access and try to say that a is completed in four days then b can immediately follow
from here and in ten days the project will be over however if they started the pessimistic
dates the project will be over only in twenty eight days that is this will take twelve days
and then this will gone for sixteen days as far as the most likely dates are concerned
then we are talking of eleven and fourteen which is giving us a number of twenty five
so the project duration obtained by considering the most likely times of activities is twenty
five is very close to the results obtained by the expected times that is twenty three
that is this number here the time to complete the project if activities started on their
pessimistic date that is twenty eight days is also pretty close to this result so this
is twenty eight and this is also close to this result of twenty five 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 ten days what we understand that the probability
of being able to complete this project in that time is indeed very low so in the most
simple 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
times 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 become 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 too find out what are the probability of completing the project
in that time and so on so expected length or duration of the project capital t e 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 be 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 of finding the 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 capital t e is the sum of small tes it is indeed a random variable and follows
normal distribution according to the central limit theorem of statistics the fact that
capital te follows normal distribution can now be used to do these two things find the
probability of completing the project within a target duration if there are two activities
we can ask the question that well there are some time associated with these two activities
what what is the probability that this project will be completed within a certain time whatever
that number is that we will see example later on but also 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 t d minus t e upon s t 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 say that there is a network which is given here which is a part of a larger
network so there is 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 t o t m and t p for all these activities
so what we are trying to find out is the total project duration and the probability of completing
this project within seventeen weeks now seventeen weeks please see is if we do a very simple
calculation four four and half and six will give us fourteen point five 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 fourteen and half so what we are
talking about is that what is the probability associated with completing this project not
an fourteen and half or fifteen weeks but an seventeen weeks so what the probability
should be what we need to calculate so moving forward we first calculate the tes and the
st squares using this formulae and we found out the capital t which is the total project
duration by summing up these three small tes and we find fifteen
similarly the variance which is associated with the project is the sum of the variance
of the activities which is one in this case so the standard deviation is one 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 fifteen here we have seventeen and here we have one seventeen is our target we
want to find out what is the probability associated with completing the project in seventeen days
which is higher than fifteen which is the most likely time of completing this project
so this number here which has been arrived at using the expected times is fifteen and
we are trying to find out what is the probability associated with completing the project in
seventeen and this is the standard deviation so the z value is two and with this if we
calculate the required probability we will find that the answer is ninety seven point
seven percent so what we can see is that the probability of completing this project in
seventeen weeks is ninety seven point seven percent here we can say that yes the capital
te represents that time which has a fifty 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 seventeen we were talking of thirteen if
we go back to the data given here the t naught for the activities is three four and four
so the idea is that instead of fifteen which represent the fifty percent probability of
completion we are talking of going minus two here at thirteen so what will be the probability
of completing this project in thirteen days of course there is a finite possibility because
the optimistic times associated with these three activities is indeed less than thirteen
this is something which i am leaving to you as a 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