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Unformatted text preview: Lessons in Business Statistics
Prepared By
P.K. Viswanathan Chapter 11: Decision Analysis Introduction
Managers must be capable of
making decisions under conditions
of uncertainty. They must also be
capable of using information, which
may be inadequate for decisionmaking. The question that must be
asked before making the final
decision is, "is it worth gathering
additional information?"
This
Chapter provides a framework for
decision making with minimum risk
under uncertain environment. 1) Systematic Problem SolvingA
Conceptual Framework Many decisions will have to be made very
often with a little bit of incomplete data in
an environment of uncertainty. It is in this context that the role of
systematic problem solving looms large. Schematic DiagramProblem Solving 2) How to Structure a Decision
Problem
A decision problem involves a scientific and
systematic problem solving approach in which
a quantitative analysis using monetary value
becomes a basis for decisionmaking. 2) How to Structure a Decision
ProblemContinues A decision problem is characterized by a set of
alternatives states of nature, and the consequences
expressed in monetary value. The alternatives pave the way for possible strategies
the decision maker can implement.
The states of nature refer to events that are not
under the control of the decision maker. States of
nature should be so defined that they become
mutually exclusive and collectively exhaustive. For each alternative and state of nature, there is a
resulting monetary value. These are displayed in
matrix form called a payoff table. Example
A small fruit merchant has got a problem on hand. He has to decide
how many dozens of a particular type of fruit to stock on a given
day. Total demand per day is uncertain. He has analyzed the past
data and found the following pattern of demand distribution based
on 360 days.
Total demand
Number of
Probability of
per day
days each
demand (based
(In dozens)
demand level
on relative
was recorded
frequency)
25
30
35
40 72
90
108
90 0.20
0.25
0.30
0.25 Fruits not sold on any day perish and have to be thrown out. Selling
price of the fruit per dozen is $ 30. Cost of procurement and other
incidentals add to $ 20 per dozen. How many dozens per day should
the merchant stock? Pay Off Matrix($ Value)
Demand
(States of
Nature)
25(0.20)
30(0.25)
35(0.30)
40(0.25) If you decide to stock
(Decision Alternatives)
25
30
35 40
250 150
50 50
250 300 200 100
250 300 350 250
250 300 350 400 3) Expected Monetary value If probabilities regarding the states of nature are
available, one can use the expected monetary value
(EMV) criterion to select the best alternative. EMV for each decision is computed by summing the
products of the pay off of each state of nature and
the probability of the respective state of nature
taking place.
The alternative giving the highest expected
monetary value (EMV) is selected as the best
decision. EMV for the Example
Demand
(States of
Nature)
25(0.20)
30(0.25)
35(0.30)
40(0.25)
EMV If you decide to stock
(Decision Alternatives)
30
35 40
25
250 150
50 50
250 300 200 100
250 300 350 250
250 300 350 400
250 270 252.5 190 EMV Optimum Expected Value of Perfect
Information(EVPI) Very often in a decision problem, any manager of
an enterprise has to answer the question “whether
to take action now, or gather additional
information and act later. Is the cost of additional information justified in
terms of the additional profit you may get? This
issue can be resolved using EVPI. EVPI is calculated as the difference between two
factors expected profit with perfect information
(EPPI) and EMV optimum. Pay Off Matrix with Perfect
Information for the Example
Demand
(States of
Nature)
25(0.20)
30(0.25)
35(0.30)
40(0.25) (Best Decision Alternatives)
Stock
30
35
25 40 250
300
350
400 EVPI for the Example
Now use probabilities of state of nature as weights and
compute the expected value as before. You get EPPI.
EPPI = (250)(0.20)+(300)(0.25)+(350)(0.30)+(400)(0.25)
=330.
Hence EVPI = EPPIEMV optimum = 330270 = $60.
EVPI acts as an upper bound measuring worth of gathering
additional information. Opportunity Loss Table
If the pay off matrix is expressed in terms of
opportunity loss for every consequence, it is called a
loss table. Opportunity loss is based on the concept
of opportunity cost that measures the cost of
sacrificing one alternative against another. Opportunity Loss Table for the
Example(figs in $)
Demand
(States of
Nature)
25(0.20)
30(0.25)
35(0.30)
40(0.25)
EOL If you decide to stock
(Decision alternatives)
25
30
35 40 0
50
100
150 100
0
50
100 200
100
0
50 300
200
100
0 $80 $60 $77.5 $140 Please note that the optimum decision is to stock 30 dozens
because it gives the minimum EOL($60). Also note that
Minimum EOL =EVPI! 4) Decision Tree A decision tree is a graphical representation of a
decision problem. Each decision tree has two types of nodes; circles
represent the states of nature and squares represent
the decision alternatives.
The branches emanating from each circle denote
different states of nature. The branches emanating from
squares denote the different decision alternatives. At the end of each extremity of a tree, the payoffs from
all the branches are displayed. Decision Tree for the Example Explanation on the Decision Tree for the
Example Branches emanating from the square represent decision
alternatives of stocking 25, 30, 35, and 40 dozens. For each decision alternative, there are four states of nature
(demand pattern) with associated probabilities. The branches that emanate from the circle represent the states of
nature of the uncertain demand that could be 25, 30, 35, or 40
dozens. The figures with in brackets for each demand represent
the probability of demand occurring. The pay offs for each decision alternative is given at the
extremity of the tree for every branch of states of nature. Explanation on the Decision Tree for the
Example For every decision option, working backwards, expected
monetary value (EMV) is calculated as the sum product of
two terms namely state of nature (demand) and probability of
state of nature (demand). The pay offs for each decision alternative is given at the
extremity of the tree for every branch of states of nature. EMV optimum = $270 because this is the highest you can get.
So, stock 30 dozens is the optimal solution to the problem. 5) Value of Sample Information
This is calculated using expected value of sample
information (EVSI). EVSI intuitively and logically
must be = Expected profit with sample information
(EPSI) EMV optimum. In order to get EPSI, we
have to revise the probabilities of the states of nature,
based on sample information we hope to get. Value of Sample Information Example
The Marketing manager of a company producing consumer
durable wants to decide whether to launch a new product or
not. The sales quantity of the new product has two states of
nature namely high sales or low sales. The pay off table is as
follows (Figures in $ Million). States of Probability Launch
Nature
High Sales 0.4
5 Do not
Launch
0 Low Sales 0.6 0 4 Value of Sample Information Example
A market research agency is willing to do a study that will
cost $0.3million. There are two possibilities. The agency may
predict high sales for the new product or it may predict low
sales for the new product. Past record of the agency in terms
of predicting ability is tabulated below: States of
Nature Prior
Agency Predicts Agency Predicts
Probability High Sales given Low Sales given
States of Nature States of Nature High Sales 0.4
Low Sales 0.6 0.6
0.3 0.4
0.7 Value of Sample Information Example
Questions:
a) Should the product be launched based on EMV
criterion?
b) Should the marketing manager take the help of the
agency or not? Solution to the Example Problem
a) The product should not be launched. EMV for
launching = (5)(0.4)+(4)(0.6) = $0.4ml. Since this
is less than $ 0(EMV optimum) that you get if
product is not launched, the product should not be
launched. Solution to the Example Problem
b) In order to solve this problem, we need to work out the
revised probabilities of the states of nature from prior
probabilities of the states nature. This means we want
to obtain posterior probabilities of high sales and low
sales given agency prediction. This statisticians call as
inverse probability approach based on Bayes’ theorem.
You can easily get this by constructing joint
probability table given in the next slide Solution to the Example Problem
Joint Probability Table
States of
Agency
Nature
Predicts
High Sales Agency
Predicts
Low Sales High Sales 0.24 0.16 Marginal
Probability
of States of
Nature
0.40 Low Sales 0.18 0.42 0.60 Marginal
Probability
of Agency
Prediction 0.42 0.58 1.00 Solution to the Example Problem
Now the calculation of revised probabilities is very simple.
Probability of high sales given agency predicts high sales =
0.24/0.42 = 0.57. Probability of low sales given agency predicts
high sales =0.18/0.42 =0.43. Extending this logic leads to the
following revised probability table. Revised Posterior Probability Table Posterior
Probability of Given Agency Predicts
High Sales Low Sales High Sales 0.57 0.28 Low Sales 0.43 0.72 Decision Tree for the Example Explanation on the Tree The manager has two options in the first stage of decision namely
“go for agency” or “no agency”. This appears in the first square
node. If you don’t go for agency, then you have two decision optionslaunch the new product or do not launch the new product. These two
options are shown in the next square node. If you launch, states of
nature (demand) may be high sales or low sales with prior
probabilities 0.4 and 0.6 respectively. Monetary values
corresponding to these two states of nature are given as 5 and –4 at
the extreme end of the branch. These are in $million. If you don’t
launch, you get $0 million. EMV is calculated in the usual way for
these two options. They are –0.4 and 0 respectively. So, EMV
optimum = 0. This appears in the square node corresponding to no
agency. Explanation on the Tree If you decide to go for agency, there are two states of nature.
Agency predicts high sales with a probability of 0.42 and low sales
with a probability of 0.58. These probabilities are from the revised
posterior probability table. If agency predicts high sales, you have again got twodecision
optionslaunch or no launch. Under each of these options, the states
of nature could be high sales or low sales. However, the
probabilities of these states of nature are posterior probabilities.
These are given in brackets. Please refer again revised posterior
probability. Explanation on the Tree EMV values are worked out as before by multiplying probabilities
of states of nature with corresponding pay offs and then summing
them. EMVs are displayed in circle and square nodes in the
diagram. Please also note that while working out EMV values, cost of doing
the study by agency ($0.3million) is taken out to represent the
correct net monetary value. Under the scenario of taking agency
help, if you choose not to launch, you still incur cost of
$0.3million(0.3 EMV) When you finally fold back the tree, you find going for agency
produces a positive EMV of $0.17 million. This is better than
getting $0 million of no agency. Hence go for agency. Final Conclusions for this Example based
on the Decision tree EVSI = EPSIEMV optimum = 0.170 =$0.17 million.
This is the gain you have got by using sample
information. Take the decision of going to the agency. If agency predicts high sales, then launch the new
product (EMV $0.83 million) If agency predicts low sales, then do not launch the
new product (EMV $0.3 million) Expected value of sample information (EVSI) is $0.17
million. ...
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 Spring '12
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 Business

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