Project Management

Prof. Raghunandan Sengupta

Module No # 3
Lecture No # 12
Decision Tree Analysis

In this lecture we will consider that how decision analysis can be used and what are the salient
points of the decision tree analysis the key steps in decision analysis
(Refer Slide Time: 01:29)

Delineate the decision tree that means draw it very clearly where the probabilities are given the
actual values are given specifies the probabilities and the monetary outcome. So you have to
make combination of the monetary values and outcome of the probabilities. Evaluate the various
decision alternatives which are there and take a decisions accordingly based on whatever
different criteria it is whether it is want to maximize the return minimize the risk other different
combinations.Remember the decision points are also called the decision folks so in the initial
when we are discussion we said decision gates alternative gates. So they are something where
you are on same line where you make a decision accordingly they are denoted by D and the
alternative action available for experimentations and actions are possible at this point. So at a
decision point you either take the right fork the left fork or if they are three outcomes you can
take alternate one r decisions one or two or three.Well the chance points are called the chance
folks where there are chances of one alternative being two or the decision being true they are
generated by C and these are the points with outcomes are dependent on a chance occurrence and
the likely outcomes at this points are a conglomeration of the decision which is already been
taken. Now you are trying to proceed for the decision remember that he would not basically go
from start to the finish but backwards why it is that I will come back to that within few minutes.
(Refer Slide Time: 03:10)

So we now discuss a problem before going to the theory so we will discuss the problem because
this will give you an idea that how decision and expected values and later on when we consider
the different concepts of utility it will give you good picture and how it can be utilized. So we
will just consider the problems in an holistic framework with probabilities and values are given.
(Refer Slide Time: 03:36)

Now I have shifted from PPT to word document because drawing the diagram a little bit
complicated with all the decision modes and all these things was much easier involved even
though there are other software which you can use. So consider this first set of points which is
there we are already discussed so I will just bang on with the problem. The scientist in a
spectrum has come up with an electric moped the firm is ready for pilot production and test
marketing this will cost about rupees twenty million and take six months.Management believes
that there is a seventy percent chance at the pilot production and test marketing will be successful
in the case of success. Spectrum can build a plan costing about one fifty million rupees the plan
will generate at annual cash flow of thirty millions for twenty years that means for a twenty year
period do you expect that. If the demand is high on an annual basis a cash flow of twenty
millions then the demand is low.There are two such occurrences it can be very high medium and
low also in that case there must be three such occurrences but in this examples we will take two.
High demand has the probability of point six low demand as a probability of point four. So if
there are three different type of demand probability the remember the sum should always be one
like here in this case point six point four is one in that case if there are three different folks
means this one I am trying to basically consider there let me go back to the slide it would be
easier for me to make you understand. Say for example we have this is one and there are three
alternatives so this probability which is given as P1, P2, P3 the sum is 1 in this case it is true
because point 6, point 4, is 1. So again let me switch back to the decision tree problem. So now if
you read the problem if you note down on a piece of paper based on that we will slowly proceed
and draw the diagram.
(Refer Slide Time: 06:00)

The diagram is like this stages of evaluation if you remember are the green vertical lines so this
say for example stage 1, stage 2, stage 3 and the decision is like this either you go or do not go
that means do not do anything and this twenty which is read in color mean basically means that
you are investing that means some amount of money is going out of your pocket and this blue
one which is twenty million per year actually means the overall amount of money which is
coming in your pocket on a yearly basis quarterly basis whatever it is.
If I consider the problem from the mode extreme because what I will try to do is that try to
basically analyze the problem from end result and slowly go into the actual source where the
decision would be made. for the extreme right values thirty million means there is a probability
was sixty percent of high demand and you would get on an annual basis thirty millions as a
payback for a twenty year period in case if the demand was on the lower side which are
probability of point four that means there are probability of arms then the value would to twenty
million per year for again the same duration.
This twenty year being same for both of them need not be so it can be different time frames.
Now this is what I called a C2 2 that means there are two now alternative accuracy there is a
chance. Now this one fifty means that you are investing if you read the problem let me go back
to the problem again spectrum can build a plant considering one fifty. So this one fifty is the
actual investment and if I consider this two hundred actually this two hundred means it will cost
and take the test from the pilot production to happen so this is basically the well the concept the
firm is ready to invest so this is the twenty million.
This time frame is six months so once it is done eight you go and go and make a decision
accordingly. Now let us go one by one at C2 what you are trying to do you are trying to find out
the expected value of that decision at C2. So at C2 how many folks are there are two so
probabilities are 0.6, 0.4. What are the values? Thirty million, twenty million per annum now if
you see the whole column inside the bracket it basically means the time value of money as of
now. If I consider an interest rate of twelve percent which is constant if you remember the
problem in the instances where we are discussing different type of financial concepts which can
be utilized to make a decision I did mentioned that the interest rate can either be fluctuate hence
you have to take an expected value here we are considering two the expected value to two to
twelve percent for twenty year time period on an annual basis which is very simplistic it can
change. If I consider these for a twenty year time period for the thirty million which will be paid
back to me for a very high demand one it means that after the first year if I get the money what is
the money what is the present value of that. After the second year if I get back the money after
two years gap what is the value of that so that means thirty multiplied by this term is after one
year present value thirty million by the second term is the present value for the second term.

Similarly for the last value to be thirty multiplied by last term whether term as the denominator
1+ 0.12 which is the interest rate to the power twenty that means for a twenty year time period.
So if you add up all of them I am able to add up thirty is a common factor I take it out I find out
multiplying by sixty percentage of 0.60 is what is the value of that overall thirty million which
would be paid back to the firm considering is a high demand. If i find it for the forty percent
which is on the load demand so I multiply them by another expected value now here just note as
I mentioned again I am mentioning twelve percent may change for both the cases timeframe
twenty twenty may change for both the cases. So the expected value is one ninety four now if
you consider the one ninety four here it means that if I conglomerate all the values here the
overall net worth is one twenty four. So what I will try to consider is that whether one ninety four
is greater than one fifty if it is yes I will invest if it is no obviously I would not go into that.
let us go into the next stage at B2 where you will take a decision that means in D2 you would
either stop or else you will ahead that means go ahead and invest one fifty to build up that that
path such that production would be there with the probability of high demand sixty percent more
demand forty percent. So add D2 you find out the expected value at D21 at this point is forty
four point 2 hence as this decision with respect to the other one. Here what you will do is there
you will stop production if it is negative and you would not do anything so what you do is that
we compare D2 and D3 this is positive means you will definitely take that which is shown here.
So as D21 is greater than D22 hence D2 is same 21 is selected so these suffix is based on from
which point to which point we are going the nodes and the arcs. Even though it does not have
any direct implication with the PERT and CPM just I wanted to mention that. Now similarly at
the point of C1 so what I will do at C1 is like this this forty four point two is already the value
which is positive here. What I will try to find out is that whether the probability multiplied by
forty four point two and the value where I am not taking a decision whether that is positive or
negative in the overall expected value sets. Similarly what I will do is that seventy percent of
point seven multiplied by point four I am just skipping going up and down the slide but please
bear with me. So it is point seven into forty four point two plus thirty percent into zero because Y
the zero because there is no cost involved in case there was a cost involved if it was negative.
obviously it can be true that seventy percent into forty four point two multiplied plus thirty
percent into negative value if that was negative obviously you will stop that means even even if
this value is giving you a positive one it will definitely mean that here there is not at all any logic
in trying to proceed in the later stage are the right hand side of the file where the cursor is now
hovering about. This means I am trying to go from left to the right and take a decision and each
and every step so at C1 the value is thirty point nine now at D1 what I do is there this value I
have already found is expected value is forty four into point seven plus zero into two point three
and the trying to compare whether twenty which is the amount money which is going out of my
pocket is more or less so again I find out this value is thirty point nine. Thirty point nine is
greater than twenty which is ten point cents it would actually means I will definitely invest into
that projects.
In case say for example these value is considered it was forty so in and in that case would have
been thirty point nine – forty P which would be almost in the tune of -10 hence we would not go
into the project. So this was the very simple example let us consider it and extend this concept
for a second example.

(Refer Slide Time: 14:12)

An oil company while evaluating the oil base in is considering few alternatives now you will see
there are three alternatives is not two. So it is either drill conduct such missed test which is cost
of twenty thousand and find the nature of underlying oil basing or do nothing. So doing nothing
is also an alternative so if the company drills the oil basing can be of three types depending on
the geophysical characteristics it can be try wet or soaking. If I dry well yields nothing that
means it would not give you any oil wet well will provide you moderate quality oil and soaking
will definitely give you the maximum amount of oil. So now what will depend that underlying
oil basin structure may can also be of different qualities or or variety one can be no structure one
can be open structure and one can be closed structure these are just characteristic basic where
you are trying to drill.
If no structure is found then the prospect of finding oil is very week almost zero if an open
structure is found then finding out the probability of oil is fairly high and for if the structure is
closed then the prospect of finding oil is very high that means probability is very high. So these
are the qualities feel of the problem which is there now with that I just slowly tried to give you
on the probability sense how this or situation makes sense that you have all the values and in the
quantitative sense.
(Refer Slide Time: 15:46)

Here are the values probability P means probability state is died is fifty percent probability is
weight is twenty five percent probability is soaking is twenty five percent.
if you add up all them then it is one which as should be as it is then I go to the probability of
geophysical structure geological states no structure forty percent open structure thirty percent
closed structure thirty percent again the sum is one as it should be also we have a joint
probability a conditional probability based on different type of oil bearing states and juridical
structure that means I am trying to how compare and find out the condition probabilities
depending on one as occurred other may occur or vice versa.
(Refer Slide Time: 16:31)
So this is where slowly complications will come but it will give you a very good idea how the
problem can be solved so the pink one which you see are the oil bearing states though they are a
drive at soaking as I have already mentioned and if we remember dry was fifty percent see the
value on the right most column. Wet was twenty five percent soaking so these add up 2 1 if I
consider the underlying geographical structure a geological structure which is in the blue color
there is no structure open structure close structure or close structure has a probability of thirty
percent open structure thirty percent no structure forty percent again probabilities 1.
The value which you have are long or top most the bottom most row and the right most column
are the marginal probability of the state considering that the geological and he oil bearing states
are very nicely shown. Now if I want to find out these values which is given inside they are
provided it means that provided is not structure and dry the probability which is give is the
conditional value point three two or say for example with soaking and close it is point one six or
sixteen percent. You have the different values which are given finding the oil company also has
the following set of information which means the net present value of the dice dry state is this is
minus value which is negative net present value for the positive set is positive point eight and for
the soaking it is two point four. So this is minus zero point five plus zero point eight plus two
point four. So if I go try to basically find out the overall problem the problem in diagrammatic
sense in the decision analysis since looks like this.
the first I will show you the diagram which I will try to basically reduce size of the word
document and try to basically show you at one go rather than going up and down in word
document.
(Refer Slide Time: 18:38)

This is the overall diagram which I have when I am just moving my cursor so it means that these
values which are there dry wet and soaking are the structure which I have discussed so this is dry
so quit soaking dry wet soaking and so and hence so forth and this minus and plus probabilities
have been very nicely shown in color in order for the students to understand. I did mention
negative point six so this is zero point six in the red color then we mentioned positive point eight
which is point zero eight point eight in the blue color and then I also mentioned for the soaking
part the values returned was plus two point four which is again in the blue color. So these are
given for all the states now if it is given for the all the states now let us go through all the arts in
little bit detail it will give you a very good picture how the problem has been found out.
If you drill you have to invest which is the upper most part if you do not drill do not do nothing
this is the which is there which is D13 do nothing. Now if you drill without considering any test
so immediately you have either dry, wet and soaking. Now if you conduct a test for which you
remember there was twenty thousand cost overlay for that because that is the cost which
you have to incur for doing the test the doing the test would give you the structure has not
structure which is the some open structure which is C22 arm and this value basically give you
the closed structure.
Closed, open and no structure would always lead to the fact which it is dry, wet or soaking
depending on the first part consider no drill or not test and this whole half this whole three set of
decision which are there depending on whether you drill or do not drill considered that you have
already conducted or test and found out the properties. So these are the values which are very
sure clone clearly this green one are the decision making stage and as already discussed the
circular part and deep part basically gives you the decision and the chance for folks.
Now again let me go back to the problem and solve it one by one so if you remember note down
these values on a piece of paper these are the probabilities. Note down the overall problem which
is given with the probabilities and let us slowly solve the problem. So at seat one between that is
the value for which is there for C1 is this part the probabilities would be now just I did skip but
let me again come back to this diagrammatic and discuss the probabilities if you note these
probabilities fifty percent twenty five percent twenty five percent then four by eight which is
eighty percent one tenth one tenth.
These are the values which are already there so either you utilize this matrix which is there
considering the marginal value or conditional values based on that you have basically seen on all
the probabilities values which are there. So if you consider probability C2 and four by ten if you
consider the probability C2 3 is 3 by 10. So all these values are considering the fact that you have
conducted a test for this one conducted the test and then you have found out that there was
basically closed structure here which is 3 by 10. Considering you conducted the test then his
open structure that the corresponding probabilities are given. So all these values are coming from
the table which is already provided so if you go one step at a time at C1 there would be three
values negative point six positive point four positive two point four but they would be multiplied
by the corresponding other probabilities what are the probabilities the probabilities are fifty
percent twenty five percent twenty five percent.You find a value of point five which is positive
now at C3 again the values of minus point six plus eight two point four remain the same as this
blue and red values show but the corresponding probabilities are changing depending on the
conditional values. So now if you have C3 so let me go through C3 is this portion it will be
minus point six into four by fifth plus point eight into one by ten plus two point four into one by
ten which is give here.
You find the value of minus point one six which means the value over net value which you have
at this point is negative. So negative with no drill definitely would mean that you would not take
any action technically means if there is conducted a suspect test there is no structure which
means you would never invest in that one that means because if it is negative if you have drilled
the value would be negative.Similarly let us combine to this set of arc’s and this set of arc’s one
at a time at C4 again minus point six point eight two point four. So point A and two point four
are positive and what are the probabilities half, one third, one six. So let us go so these are the
probabilities half, one third, one six again I am repeating these values which are there the
probabilities can be found out from the matrix.
Similarly if I go into those last so called node it is minus point six into one by ten here where the
cursor is then again plus point eight into eleven by thirty and this value of two point four into
sixteen by thirty. So now sixteen by thirty actually means that what how you are getting is
basically you have conducted the test then you found out that there was basically closed structure
and based on that when you had drilled there was a soaking based on which you find out the
maximum probability of getting the oil. So this is one point five three one three. So hence the
decision you have to make is that at this point when you have basically this was negative.So
obviously you would not do anything now if I consider this value with do not drill this value with
now drill obviously it will mean that in all the instances they would be corresponding no action
some action some action means the drill concept you can say. So at D2 it is negative hence do
nothing at B3 positive drill D4 positive drill what are the values at D2 it was negative which is so
hence compare with zero at zero and D3 it was positive zero point three six seven.Hence the
value is zero point three six seven and D4 it is one point five one three hence it is one point four
five one three now if I want to find out the values at this point what do I have this whole value is
zero. Zero multiplied by the probability four by ten here it is four by ten this arm had a positive
value what was the positive value zero point three six seven here it is again point three six seven
multiplied the probability thirty percent and one point five one three multiplied by point six five
point five six four.
Hence again I compare zero five six four and considering this so this is zero point five six four
you had a value already found out here and this was zero. So you will basically try to find
compare them and then find out the values accordingly.
(Refer Slide Time: 26:45)

Based on the above evaluation the alternatives can be found out there is set of decision strategies
are as follows. So this basically give you a probabilities and the values in the negative and the
positive sense are given in the red and the blue color. So the expected value if I find out so if you
follow up any arm D 12 C21 D22 the probability are given from the tables we can find out and
the net present values if which are already there. Once the net present values have all been found
out considering the calculations which you have already done the expected value can be found
out as five four four triple zero that means positive and in case if you want to find out the
variance you can use the simple concept of the variance to try out the value as given in the
formula. We can make our own decisions accordingly to the where there that we want to
maximize the return as highlighted minimize the risk or make a balance between them where you
take basically a ratio of trying to maximize return to risk or minimize the risk return or different
type of multi-criteria decisions can be made in order to make the decision.
These two problems which I discussed one was there was no conditional probabilities in the
second case there was conditional probabilities which is simple values and the concept of present
value was also there. So in assignments we will also try to tackle such simple problems and there
and the assignment solutions would already be which will also be provided to the students to
understand both the concept of decision trees which is the second set of problems we did it as
well as in the concept of AHP in order to make you understand how these two different
techniques are used in a big way.