WEBVTT
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and welcome to this series of lectures on
principles of construction management and
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in this lecture today we will talk about uncertainties
in duration of activities using the pert in
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scheduling so far what we have said is that
the time of completion of an activity is firm
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that is its a given number and then we have
constructed networks to determine project
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completion times based on interdependencies
so you would recall that in a first example
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when we said there was five hundred mandays
involved then we divided this by ten which
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was a number of people and came up with fifty
days of activity time and then we have also
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constructed in the last class slightly different
network more elaborate network and illustrated
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the concept of critical path that is that
path which needs to be monitored more regularly
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more continuously than the others and no slippage
can be allowed
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but in any case when the interdependencies
were taken into account that time associated
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was taken to be firm an activity ij it was
known that this will be completed in a time
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of tij now today what we will do is we will
take a slightly different look at the activity
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duration that is this tij with the emphasis
of incorporating the element of uncertainty
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how do you incorporate a situation when the
tij is not a firm number but more like a probability
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distribution and activity may be completed
in ten days but it might bellower to twelve
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days if something could happens if we are
lucky then it might be completed in just five
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to six days so this idea that the time associated
with each activity is not necessarily a fixed
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number that is something which we will focus
on today in our discussion
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so how do we actually go about estimating
the times for the different activities one
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simple approach is to keep records that will
give us the average durations so a construction
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company keeps a record for erection of process
and so on and finds out that ok in different
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projects what is the kind of time that it
takes and from there we try to find out the
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productivities and then try to map that into
a new project and try to estimate the time
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involved for that project we have talked about
in the initial part of this course that each
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construction project is really unique so the
bridge built at a particular place is surely
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different from may be a similar bridge but
built elsewhere also the productivity of labour
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during the course of the project changes and
that could also affect the duration of activities
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so if there is an activity you put a gang
of labour or a certain set of workers on day
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one they may not be able to produce the maximum
output but has they learn the activity as
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they learn their role in the whole process
their productivity improves
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so how do we keep track of these kind of things
going back to the basics usually the duration
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dij of an activity is determined using this
formula which is aij upon pij multiplied by
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nij where ij is the activity that we are talking
about aij is the required amount of work pij
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is the productivity of the person which could
possibly take care of the learning curve and
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so on and nij is the number of persons working
on that job independently and together so
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with this we can find out the dij but each
of these parameters has their own variations
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for example as far as aij is concerned the
only estimate that we can make from is the
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good for construction drawings it might happen
that there will be some changes in the total
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amount of work at site whether it is paid
for or not it doesnt matter but the fact is
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that if additional work is carried out or
less work is carried out then it will affect
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the time duration so we coming to the next
part which is labour productivity this could
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also be an important aspect of determining
the total time involved and that is where
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the learning curve becomes important so what
is the learning curve a general understanding
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of the productivity rates can be obtained
from the concept of learning curve and the
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base of the learning curve lies in the fact
that as crew becomes familiar with the activity
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and the work habits their productivity tends
to improve of course there is a limit beyond
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which it does not improve there is a limit
to the capacity of each worker this capacity
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may be different for different workers but
sure enough there is a starting point and
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it goes through a learning phase and a study
phase for each worker in other words productivity
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increases as the crew gains experience which
is possible if sufficient time is available
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and most construction projects that is in
typically the case
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so now another thing is the number of workers
this equation here assumes that the duration
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of an activity is inversely proportional to
the number of persons in the crew or the number
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of workers in the team but is this assumption
absolutely true is it true in the absolute
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sense if we keep increasing this nij is there
no limit for the reduction in the duration
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that we get there are two reasons why this
assumption is not really true one is the crew
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may not be able to work independently in a
certain construction site there is a limit
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to the number of workers who can be accommodated
without interfering with each others productivity
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so thats one part of it the second thing is
coordination between individual task has to
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be ensured and that also becomes difficult
we all know the saying that too many cooks
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spoil the broth so if there are too many people
trying to an activity that also has an adverse
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effect on the time duration apart from the
workers working independent the coordination
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between individual task and the individual
crew members has to be ensured
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more persons in the crew may result in the
delay in the following situations an example
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is design task are often divided between architects
and engineers in which greater coordination
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is required among themselves in such situations
more the number of people in the crew will
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obviously affect the coordination and therefore
it affect the time ensuring a smooth flow
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of material or information of any nature may
become increasing the difficult if there is
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more number of people in the crew further
in projects which are being executed for the
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first time it may not be possible for planers
to finalize the exact estimates of the parameters
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like productivity rates optimum number of
people in the crew other considerations may
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include practical difficulties like the weather
contractors failure to deliver materials machinery
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breakdowns and so on also most of the activities
that are outside the control of the client
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are certainly uncertain one simple example
could be the time required to get a approvals
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from outside agencies and that could affect
the project in different ways what this competent
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authority means is for example if there is
a project which requires an approval from
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a pollution control board and such regulatory
authorities that night be not within the control
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or within the powers of the client to be able
to implement and that introduces an element
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of uncertainty as far as the completion of
the project is concerned in its not only the
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project but also it has the affect obviously
on individual activities and those individual
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activities intern affect the project
this underlines the need to understand that
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there exists an uncertainty in estimating
activity duration and therefore considering
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to use continuous probability distribution
functions becomes relevant in this case so
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what we are saying in this sentence is instead
of assigning a particular time t for any activity
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can we now talk in terms of a probability
distribution that is yes this activity will
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take some time but that time could be may
be little less may be a little more so whether
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this distribution should be normal or not
is the next question that we need to answer
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so without going into the a statistic of the
whole issue lets try to understand the basic
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story of the normal distribution the normal
distribution with values widely spread away
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from the average is the situation where their
standard deviation is high compare to this
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picture this picture here is a situation where
the peak is sharper which means that the area
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in this part is closer to the mean if you
want to go to the same area here we will go
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more standard deviations away from the mean
however in both cases the normal distribution
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is symmetric
so the problem with our construction activities
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is that they are not necessarily symmetrically
distributed so now what are the requirements
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that we need to impose in determining or finding
the kind of distribution that helps or say
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that yes this is what really meets our requirements
the probability of reaching the most optimistic
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time which is early completion should be very
less so this is the situation where we become
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lucky and we are able to achieve the target
very quickly the probability of reaching the
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most pessimistic time should be very less
that is there should be time where we will
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definitely be able to complete the project
under most adverse conditions then there exist
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only one most likely time which would be free
top move between these two extreme conditions
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so what we are saying is that there is an
optimistic time t naught and there is a pessimistic
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time tp between this t naught and tp we want
to define a distribution which is not normal
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so either this most likely time will move
towards the left in which case the pessimistic
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time becomes an outlier or it will move towards
the right in which case the optimistic time
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becomes an outlier so this amount of uncertainty
should also be measurable so once we understand
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these requirements we find that the beta distribution
that is used in statistics satisfies these
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requirements for us
so we can use the properties of the beta distribution
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and move forward as far as determining the
expected times of the activities and so on
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is concerned so now what is a beta distribution
we assume that this is the most likely time
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this is the pessimistic time and this is the
optimistic time and this distribution is is
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skewed to the right in this case however on
the figure on the right for the most likely
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time the distribution is skewed to the left
so this is a situation where most of the times
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we will be able to complete the project here
except if we become very lucky we might hit
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the jackpot and comes over here in this case
we will probability will able to complete
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the project somewhere here and only if we
are not running in luck at all we will be
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able to complete the project within the pessimistic
time so this is the kind of thought process
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and we move forward and try to see how the
uncertainty in estimating activity times can
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now be incorporated in a scheduling process
like pert pert is the program evaluation and
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review technique and the uncertainty in estimating
the duration of an activity is considered
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in this technique as against the critical
path method that we saw in the last class
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where the activities had a firm time of completion
is more suitable to control jobs that have
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not been done before and there is more uncertainty
in the activities
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pert assumes the three estimates of time or
random variables satisfying a beta distribution
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function so this is what we have been talking
about and now lets try to see how we actually
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implemented on ground as far as activities
are concerned so we have been talking off
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an activity ij all the time so here also we
are saying that there is an activity i and
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j and we are talking of three time associated
with it optimistic time t o which is the shortest
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possible time of completing the activity under
ideal conditions the most likely time t m
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which is the best guess of the time required
to complete the activity and the pessimistic
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time t p which is the maximum time required
to complete this activity so usually the notation
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followed is given here for an activity ij
we give t o t p and t m now from here the
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expected time for completion and the standard
deviation in the activity we know the t o
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t p and t m then how do we calculate the t
expected of the activity or for the activity
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ij now this is calculated using this equation
that is t o plus t p plus four times t m divided
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by six so i am not getting into the derivation
of this what it really says is the this on
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the pessimistic and optimistic times and we
are giving a higher weightage to the most
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likely time and we are getting an estimate
of the time at which or during which this
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activity will be completed please remember
that this expected time does not mean or does
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not give you an idea of the fifty percent
time of completion this time does not mean
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that the activity will be completed fifty
percent of the time that would happen if the
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time of completion of that activity was normally
distributed given that it is not normally
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distributed t e does not confirm to a fifty
percent probability of completion of that
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activity
continuing the variance that is s t square
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associated in estimating the expected time
of that activity evaluated as t p minus t
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naught divided by six squared and then we
have must retreat the t e is taken as a random
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variable moving forward lets try to implement
this thought process in a example lets talk
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of four activities a b c and d and if these
were the times that we used in a c p m calculation
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that is the firm times now this table here
or this part of the table here gives you the
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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
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is to ensure that the time in cpm and the
expected time of the activity are taken to
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be the same so activity has ten here it is
t e here it is eight here and eight here fifteen
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here fifteen here and twelve and twelve so
this effectively ensures that the t e is the
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same similarly using this equation here we
have calculated the st associated with each
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of these activities so if we examine this
table closely we find that for activity a
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this t naught is a little bit of an outlier
these two times are fairly close to each other
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but t naught is the optimistic time in the
case of b this number here that is the pessimistic
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time is the outlier and for activity c again
this is the outlier and this is the outlier
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here
so what we have try to do through this arrangement
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of numbers is to covey to you that these activities
are not normally distributed and sometimes
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they are skewed to the left or they are skewed
to the right and we have already talk before
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that this skew to the left and skew to the
right has a different meaning when its come
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to interpreting the time duration associated
with an activity so now lets actually consider
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some numbers lets consider a network that
has two activities a and b as shown here the
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three estimates for the duration of these
activities and days is given on the arrow
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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
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activity b are six fourteen and sixteen which
is two three that is the activity b so what
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we are required to do is to compute the total
project duration using the concept of expected
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times of activities as we have defined earlier
and compute the project durations in the following
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cases if the activities are started on the
respective optimistic dates pessimistic dates
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and the most likely dates and then of course
we can discuss most results a little bit so
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working out the t e and the standard deviations
we find that for activities a and b we convert
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this information to this table here the using
these formulae we calculate the expected times
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of completion of the activities and the variances
so the total project duration considering
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the expected time of the activities is twenty
three that is ten and thirteen now coming
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to the second part of the problem which was
to say that well if activities are completed
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at the optimistic times then of course if
we go back to our representation on the time
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access and try to say that a is completed
in four days then b can immediately follow
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from here and in ten days the project will
be over however if they started the pessimistic
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dates the project will be over only in twenty
eight days that is this will take twelve days
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and then this will gone for sixteen days as
far as the most likely dates are concerned
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then we are talking of eleven and fourteen
which is giving us a number of twenty five
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so the project duration obtained by considering
the most likely times of activities is twenty
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five is very close to the results obtained
by the expected times that is twenty three
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that is this number here the time to complete
the project if activities started on their
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pessimistic date that is twenty eight days
is also pretty close to this result so this
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is twenty eight and this is also close to
this result of twenty five however the project
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being completed in these two dates the fact
that both of these were outliers only shows
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that the project can be completed yes in ten
days what we understand that the probability
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of being able to complete this project in
that time is indeed very low so in the most
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simple terms what we are saying is that we
become lucky not only once but twice not only
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the activity a is completed in the optimistic
time but activity b is also completed in optimistic
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times so we all know the basic probability
kind of questions that a and b has to happen
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in this case we have to have a being completed
in its optimistic time and we have to have
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b also being completed in its optimistic time
of course there are other combinations that
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a is completed in its optimistic time and
b in its pessimistic time and so on so that
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is the kind of range in the time estimation
that is important or that become relevant
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as far as construction management is concerned
so continuing our discussion further how do
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we determine the critical path in this method
the critical path is determined by carrying
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out the forward and backward passes of activities
with durations being taken as the expected
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time of completion so of course we take the
expected time of completion and then we use
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probability concepts too find out what are
the probability of completing the project
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in that time and so on so expected length
or duration of the project capital t e is
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calculated by summing up the expected durations
small tes of activities on the critical path
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so what we are saying is that if we say that
this is the critical path we are not denying
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that there are no other paths there can be
other paths but this is the critical path
21:26.029 --> 21:33.000
so for this critical path we take the activities
on this path and try to find out the expected
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time of the project using only the expected
times of the activities on the critical path
21:40.059 --> 21:47.600
these activities which are non critical will
have to be dealt be separately and the variance
21:47.600 --> 21:51.799
associated with the critical path is the sum
of the variances of the activities on the
21:51.799 --> 21:56.100
critical path and we can find out the standard
deviation of finding the the square root of
21:56.100 --> 22:02.320
this variance so this is the statistical concept
which we are not deriving in this course and
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i am leaving it to you to take a look at some
of the statistics text books and move forward
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now since the capital t e is the sum of small
tes it is indeed a random variable and follows
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normal distribution according to the central
limit theorem of statistics the fact that
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capital te follows normal distribution can
now be used to do these two things find the
22:26.149 --> 22:31.640
probability of completing the project within
a target duration if there are two activities
22:31.640 --> 22:37.559
we can ask the question that well there are
some time associated with these two activities
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what what is the probability that this project
will be completed within a certain time whatever
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that number is that we will see example later
on but also we can find the expected completion
22:51.600 --> 22:58.040
time with a given probability basically these
two questions are just two sides of the same
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coin correspondingly the standard normal deviate
z is evaluated as t d minus t e upon s t and
23:07.429 --> 23:11.220
this is something which comes in very handy
when we are trying to answer questions such
23:11.220 --> 23:17.100
as these
so now to use an illustrative example for
23:17.100 --> 23:22.830
pert let say that there is a network which
is given here which is a part of a larger
23:22.830 --> 23:28.100
network so there is a lot of activities here
and here but we are talking only of this part
23:28.100 --> 23:35.029
because that is what is our critical path
so for this critical path the durations are
23:35.029 --> 23:43.090
given here for that activities a b and c this
is the t o t m and t p for all these activities
23:43.090 --> 23:48.190
so what we are trying to find out is the total
project duration and the probability of completing
23:48.190 --> 23:55.500
this project within seventeen weeks now seventeen
weeks please see is if we do a very simple
23:55.500 --> 24:02.870
calculation four four and half and six will
give us fourteen point five so what we are
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saying is that if we just take the most likely
times we did that in an example just now that
24:09.690 --> 24:15.100
if we just take the most likely times it turns
out to be fourteen and half so what we are
24:15.100 --> 24:19.750
talking about is that what is the probability
associated with completing this project not
24:19.750 --> 24:25.029
an fourteen and half or fifteen weeks but
an seventeen weeks so what the probability
24:25.029 --> 24:32.179
should be what we need to calculate so moving
forward we first calculate the tes and the
24:32.179 --> 24:39.740
st squares using this formulae and we found
out the capital t which is the total project
24:39.740 --> 24:46.790
duration by summing up these three small tes
and we find fifteen
24:46.790 --> 24:50.800
similarly the variance which is associated
with the project is the sum of the variance
24:50.800 --> 24:56.279
of the activities which is one in this case
so the standard deviation is one so we have
24:56.279 --> 25:02.350
this calculation carried out now with this
calculation we go to the charts and tables
25:02.350 --> 25:10.720
relating to finding out how this area is related
to the number of standard deviations that
25:10.720 --> 25:16.230
you go away from the mean so if you do this
exercise for this particular project here
25:16.230 --> 25:22.369
we have fifteen here we have seventeen and
here we have one seventeen is our target we
25:22.369 --> 25:27.490
want to find out what is the probability associated
with completing the project in seventeen days
25:27.490 --> 25:33.440
which is higher than fifteen which is the
most likely time of completing this project
25:33.440 --> 25:40.039
so this number here which has been arrived
at using the expected times is fifteen and
25:40.039 --> 25:43.850
we are trying to find out what is the probability
associated with completing the project in
25:43.850 --> 25:49.480
seventeen and this is the standard deviation
so the z value is two and with this if we
25:49.480 --> 25:53.419
calculate the required probability we will
find that the answer is ninety seven point
25:53.419 --> 25:59.289
seven percent so what we can see is that the
probability of completing this project in
25:59.289 --> 26:05.010
seventeen weeks is ninety seven point seven
percent here we can say that yes the capital
26:05.010 --> 26:11.889
te represents that time which has a fifty
percent probability of completion so with
26:11.889 --> 26:18.260
this we have introduced the concept of pert
we have introduced the concept of uncertainty
26:18.260 --> 26:25.570
in the activity durations now suppose instead
of seventeen we were talking of thirteen if
26:25.570 --> 26:32.820
we go back to the data given here the t naught
for the activities is three four and four
26:32.820 --> 26:38.309
so the idea is that instead of fifteen which
represent the fifty percent probability of
26:38.309 --> 26:45.901
completion we are talking of going minus two
here at thirteen so what will be the probability
26:45.901 --> 26:50.850
of completing this project in thirteen days
of course there is a finite possibility because
26:50.850 --> 26:58.070
the optimistic times associated with these
three activities is indeed less than thirteen
26:58.070 --> 27:02.710
this is something which i am leaving to you
as a for thought you try to do this example
27:02.710 --> 27:06.880
on your own and i am sure you will get the
answer very quickly and i hope that you now
27:06.880 --> 27:12.830
understand how to handle these concepts and
be able to calculate or estimate project durations
27:12.830 --> 27:17.450
find out the probabilities associated with
completing a project in a given time and so
27:17.450 --> 27:23.090
on with this we will just give you the list
of references which i always do at the end
27:23.090 --> 27:26.139
of a lecture and look forward to seeing you
again
27:26.139 --> 27:26.580
thank you