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QTBD
I MBA I SEM
QISET COLLEGE
K.V.RAMESH BABU
ASSISTANT PROFESSOR IN STATISTICS
2.
Decision theory
“Decision making” is an „integral part‟ of
1.
2.
3.
4.
Management planning
Organizing
Controlling
Motivation process
The “Decision maker” selects one strategy (course of action) over others
depending on some criteria like utility, sales, cost or rate of return
The specific combination of „goals‟ is not entirely depending on the
decision-maker i.e. the value of system is usually modified to other
interested groups, like stock holders, employers, unions, creditors,
government etc
Definition:“Decision theory” provides a method for rational „decision- making‟ when
the consequences are not fully “deterministic” Decision maker has to apply
various methods to decision problems.
The „Decision theory‟ identifies the best alternative or course of action for
specific „activity‟
The Decision theory provides a frame work for better understanding of the
decision situation and for evaluating alternatives, when the alternatives criteria
are not defined
The relationship between „decision theory‟ and „decision making‟ can be
represented in the following diagram
Define the
Define the problems
Search for Alternatives
Abstraction
Decision Theory
Evaluate Alternatives
Asses Consequences
Select an Alternative
Apply Decision Criteria
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3.
Decision:If a “decision” is to be made on the basis of one of the “decision criteria”
then the selection of an alternative course of action is a direct effect of “decision
theory” But, if alternative course of actions is compared with the
considerations other than so called “pure criteria” then the “decision theory”
helps us in the evaluation of alternatives.
Types of decisions:In general decisions can be classified into 3 categories
1. Strategic decisions
2. Administrative decision
3. Operating decision
1. Strategic decisions:These decisions are concerned with external environment of the
organization
Example:- Decision of selection of product-mix which a firm will produce and
the markets to which it with sell are under strategic decisions
2. Administrative decisions:This is concerned with structuring and accusation of the organization
resources, so as to optimize the performance of the organization
Example: - 1. Selection of distribution 2. Location of facilities
3. Operating Decision :This is the primary concerned with „day-to-day‟ operations of the
organization
Examples: - 1. Pricing 2.Production scheduling 3. Inventory levels etc
Components of Decision Making:Various components of the problem which a problem person should
know are briefly discussed as follows
1. The decision maker
2. Objectives
3. The system or environment
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4.
4. Alternative course of action
5. Choices
1. Decision Maker (or) Policy-Maker (or) Executive:Most obvious component is the fact that someone belonging „some group‟
must have „some problem‟ this individual or group is dissatisfied with some
aspect of the state of affairs and consequently wants to make a decision with
regarding to altering it, which is known as „decision maker‟ if the „decision
maker‟ controls the operations of an organized system of men or machines then
he is referred to as “policy maker” or “executive”.
2. Objectives
In order to have a problem the “decision maker” he must know
something other than what he has i.e. he must have some objectives which he
has not obtained to the degree he desires
3. System (or) environment:The „decision maker‟ has the problem in an environment or setting that
constraints or lacks various resources
“Environment” is an “organized system” usually embracing machines as
well as man
4. Alternative Course of Action:A problem cannot exist unless the „decision maker‟ has a choice from
among at least 2 alternative courses of „action or policies‟ A problem always
involved a question what to do this question becomes a problem only when
alternative courses of action are available
Decision Models
One of the primary functions of management is to make decisions
that determine the future course of action for the organization involving short
term or long-term consequences
Decision models are classified into various categories, depending upon
their nature and complexity, such as allocation so as to optimize the given
objective function subject to certain restrictions
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5.
I.
Goals to be achieved:The objective which the decision maker wants to achieve by his
action
II.
The Decision-Maker:The decision-maker refers to an individual or a group of
individuals responsible for making a choice of an appropriate course of
action amongst the available course of action
III.
Course of Action (or) Decision- Alternatives:-
For a specified problem all possible course of action should be included the number of
possible courses of action may be large or small but these are under control of
decision-maker
Example: - A course of action has a numerical description such as stocking of 150
units of a particular item or non-numerical description.
IV.
States of Action:-
Before applying decision theory we must develop an exhaustive list of
possible future events. Forever decision maker has no direct control over the
occurrence of particular event such future events are referred to as “states of nature”
and it is assumed but these are mutually exclusive collectively exhaustive
Examples:- the state of nature can be a numerical description such as demand of
some units of a given item or a non- numerical description like employees strike.
V.
Preference (or) Value System
This refers to the criteria that the decision maker uses in making a choice of the best
course of action. It could include maximization of income utility profit etc.
VI.
Pay of (or)Profit (or)Conditional Value
It is the effectiveness associated with specified with combination of a course of action
and state of nature. These are also known as profit or conditional value.
Example: -The conditional profit can also be of Rs 15 associated with the action of
stocking 20units of an item when the outcome is a demand of 17 units of that item.
costs can be considered as negative profits.
VII.
Opportunity Loss Table :-Opportunity loss is incurred due to failure of not
adopting most favorable course of action or strategy the opportunity los values
are determined separately for each state of nature or outcome by 1 stfinding the
most favorable course of action for that state for nature or outcome. And then
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6.
taking the difference between pay off value for given course of action and the
payoff value for the best possible course of action which would be chosen.
VIII.
Pay off Table
For a problem a payoff table exists the states of nature which are
mutually exclusive and collectively exhaustive and as set of given courses of
action (strategies). For each combination of state of nature and course of action
the payoff is determined
Course of Action
…………
…………
States of
Nature
:
:
:
:
:
:
:
:
:
:
:
:
…………
…………
…………
…………
…………
…………
:
:
:
:
…………
…………
…………
…………
…………
:
:
…………
The weighted profit associated with the given combination of state of nature and
course of action is obtained by multiplying the payoff the state of nature and
course of action by the probability of occurrence of the specified state of nature
The table shown in the above is one such type of pay off table in this
table „m‟ states of nature and denoted by O, O2 … Om with respect to „n‟ course of
action S1 S2 …. Sn for a specified combination of state of nature and course of
action the corresponding payoff is represented by ai j.
Types of Environment
Decision theory helps the decision maker in selecting the best course of
action from the available course of action. The decision models are classified such that
the type of information which is given about the occurrence of the various state of
nature as well as depending upon the decision environment basically there are 4
different states of decision environment
i.
ii.
iii.
iv.
Decision
Decision
Decision
Decision
making
making
making
making
under
under
under
under
certainty
un-certainty
risk
conflict
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7.
I.
Decision Making Under Certainty
This is all easiest form of „decision making‟ the outcome resulting from
the selecting of a particular course of action given with certainty. There is just one
state of nature for each course of action and has probability „I‟ we are given
complete and accurate knowledge of the consequence of each choice. Since the
decision maker has perfect knowledge of the future and the outcome, he simply
selects that course of action for which the payoff is optimum
Example
The analysis of cost profit and volume is a “decision problem under
certainty”
Where the information regarding costs and profits is given with respect to volume of
sales. Similarly in L.P.P the amount of resources required and the corresponding unit
profit (cost) is given with certainty the other techniques used for solving problems
under certainty are
I.
II.
III.
IV.
V.
Input –output analysis
Break- even analysis
Goal programming
Transportation and assignment methods
Inventory models
1. Decision Making Under Uncertainty:In the absence of knowledge about the probability of any state of nature
occurring, the decision maker must arrive at a decision only on the actual conditional
pay off values, together with a policy. There are several different criteria of decisionmaking in this situations
Types of criteria
1.
2.
3.
4.
5.
Optimism criterion or maximum criterion or minimum criterion
Pessimism criterion or maximum criterion or minimum criterion
Laplace criterion or equal probabilities criterion
Hurwitz criterion or coefficient of optimism
Regret criterion or salvage criterion
1. Optimism Criterion:In this criterion the decision-maker ensures that he should not miss
the opportunity to achieve the largest possible profit (maximum) or the lowest possible
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8.
cost (minimum). Thus, he selects the alternative that represents the maximum of the
maximum payoff. The working method is summarized as follows.
Step-1 locate the maximum (minimum) payoff values corresponding to each
alternative
Step-2 Select an alternative with best anticipated payoff value (maximum for profit)
and minimum for cost)
Since in this criterion the decision-maker selects an alternative with largest (or
lowest) possible payoff value, It is also called an “optimistic decision criterion”
2. Pessimism Criterion
In this criterion the decision-maker ensures that he would earn no less
than some specified amount. Thus, he selects the alternative that represents the
maximum of the minimum payoff in the case of profits. The working method is
summarized as follows
Step-1 Locate the minimum (or maximum in the case of profit). Payoff value in the
case of loss data corresponding to each alternative.
Step-2 Select an alternative with the best anticipated pay off value (maximum for
profit and minimum for loss)
Since in this criterion the “decision –maker” is conservative about the
future and always anticipates the “worst possible” outcome. Then it is called a
“pessimistic decision criterion”. This criterion is also known as “wald‟s criterion”
3. Laplace Criterion
Since the probabilities of states of nature are not known, it is assumed that all states
of a nature will occur with equal probability .i.e. each state of nature is assigned an
equal probability. As states of nature are mutually exclusive and collectively
exhaustive, so the probability of each of these must be
the
working method is summarized as follows.
Step-1 Assign equal probability value to each state of nature by using formula
Step -2 Compute the expected (average) pay off for each alternative by adding all the
payoffs and dividing by the number of possible states of nature by applying the
formula
⟹ ( Probability of state of nature ) X
(Payoff value for the combinations of alternative „i‟ and state of nature „j‟)
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9.
Step-3 Select the best expected pay off value (maximum for profit and minimum for
cost)
This criterion is also known as the criterion of “insufficient reason” this is because
except in a few cases, some information of the likelihood of occurrence of states of
nature is available
4. Hurwitz Criterion:
This criterion suggest that a rational decision- maker should be neither
completely optimistic nor pessimistic and must display a mixture of both Hurwitz also
suggested this criterion, introduced the idea of a coefficient of optimism which is
denoted to measure the decision maker‟s degree of optimism. This coefficient lies
between o & 1.
Where 0= A complete pessimistic attitude about the future
1= A complete optimistic attitude about the future
= Coefficient of optimism attitude about the future
(1-
)= Coefficient of pessimism
The Hurwitz approach suggests that the decision-maker must select alternative that
maximizes
H (Criterion of realism) = (maximum in column) + (1working method is summarized as follows.
)(minimum in column the
Step-1 Decide the coefficient of optimism. And the coefficient of pessimism (1- )
Step-2 For each alternative select the largest and lowest pay off value and multiply
these with and (1- ) values respectively. Then calculate the weighted average H by
using above formula
Step-3 Select the course of action with the smallest anticipated opportunity loss value
Step-3 Select an alternative with best anticipated weighted average payoff value.
5. Regret Criterion
This criterion is also known as opportunity loss decision” criterion or
“Minimax Regret decision” criterion. This is because decision- making regrets the fact
that he adopted a wrong course of action resulting in an opportunity loss of payoff.
Thus, he always intends to minimize this regret the working method is summarized as
follows
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10.
Step-1 From the given payoff matrix, develop an opportunity loss matrix as follows
1. Find the best pay off corresponding to each sate of nature
2. Subtract all other entries in that row form this value
Step-2 For each course of action identify the worst or maximum regret value record
this number in a new row
Step -3 Select the course of action with the smallest anticipated opportunity loss
value
Problems on – Decision Making Under Uncertainty
Problem -1
A food products company is contemplating the introducing of a revolutionary new
product with new packaging or replacing the existing product at much higher price
(s1). It may even make a moderate change in the composition of the existing product,
with a new packaging at a small increase in price (s2) or may be a small change in the
composition of the existing product, backing it with the word “new” and a negligible
increase in price (S3) the 3 possible states of nature or events are high increase in
sales (N1), No change in sales (N2), Decrease in sales (N3)
The marketing department of the company worked out of payoffs in terms of yearly net
profits for each of the strategies of „3‟ events this is presented in the following table
States of nature
strategies
N1
N2
N3
S1
7,00,000
3,00,000
1,50,000
S2
5,00,000
4,50,000
0
S3
3,00,000
3,00,000
3,00,000
Which strategy should the concerned executive choose on the basis of
I.
II.
III.
IV.
Maximum criterion
Maximax criterion
Minimax criterion
Laplace criterion
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11.
Solution
The payoff matrix is rewritten as follows
a) Maximin criterion or (pessimistic criterion)
States of nature
strategies
S1
S2
S3
N1
7,00,000
5,00,000
3,00,000
N2
3,00,000
4,50,000
3,00,000
N3
1,50,000
0
3,00,000
Column Minimum
1,50,000
0
3,00,000
The maximum of column minima is 3,00,000. Hence, the company should adopt
strategy S3
b) Maximax criterion or (optimistic criterion)
States of nature
Strategies
S1
S2
S3
N
7,00,000
5,00,000
3,00,000
N
3,00,000
4,50,000
3,00,000
N
1,50,000
0
3,00,000
Column maximum
7,00,000
5,00,000
3,00,000
The maximum of column maxima is 7, 00,000. Hence the company should adopt
strategy S1
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12.
C) Minimax Regret Criterion or (Regret Criterion :
States of
nature
Strategies
S1
S2
N1
7,00,000-7,00,000=0
7,00,000-5,00,000= 2,00,000
7,00,0003,00,000=4,00,000
N2
4,50,0003,00,000=1,50,000
4,50,000-4,50,000=0
4,50,0003,00,000=1,50,000
N3
3,00,0001,50,000=1,50,000
3,00,000-0=3,00,000
3,00,0003,00,000=4,00,000
Column
maximum
1,50,000
S3
3,00,000
4,00,000
Hence the company should adopt minimum opportunity loss strategy of „S1‟
D) Laplace Criterion or (Equal Probability Criterion):
Since we do not know the probabilities of states of nature, assume that they
are equal. For example we should assume that state of nature has probability
of
occurrence. Thus
Strategy
Expected return (Rs)
S1
(7,00,000+3,00,000+1,50,000)/3 = 3,83,333.33
S2
(5,00,000+4,50,000+0)/3
S3
(3,00,000+3,00,000+3,00,000)/3 = 3,00,000
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= 3,16,666.66
Page 12
13.
Since the largest expected return is from strategy S1 then the executive most select
strategy S1
Solution:1.
2.
3.
4.
Maximin criterion is strategy S3 ( pessimistic criterion)
Maximax criterion is strategy S1 ( optimistic criterion )
Minimax criterion is strategy S1 (regret criterion)
Laplace criterion is strategy S1 ( equal probability criterion)
Problem-2
A manufacturer manufactures a product, of which the principal ingredient is a
chemical „X‟. At the moment the manufacturer spends Rs 1,000/- per year on supply
of „X‟ but there is a possibility that the price may soon increase to „4‟ time it‟s another
chemical „Y‟ which the manufacturer could use in conjunction with a 3rd chemical Z,
in order to give the same effect as chemical „X‟ chemicals Y and Z would together cost
the manufacturer Rs 3,000 for year, but their prices are unlikely to rise. What action
should the manufacturer taken? Apply the maximin and minimax criteria for decisionmaking and give „2‟ sets of solution. If the coefficient of optimism is 0.4, then find the
course of action that minimizes the cost
Solution :The data of the problem is summarized in the following table
States of nature
Courses of action
S1 ( use Y and Z)
S2 ( use X)
N1 ( price of X increases)
-3,000
-4,000
N2 ( price of X does not
increase )
-3,000
-1,000
1. Maximin Criterion or ( Pessimistic Criterion):
States of nature
Courses of action
S1
S2
N1
-3,000
-4,000
N2
-3,000
-1,000
Column minimum
-3,000
-4,000
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14.
The maximum of column minimum is -3,000. Hence the manufacture should adopt
the action „S1‟
2. Minmax Criterion or ( Pessimistic Criterion )
States of nature
Course of action
S1
S2
N1
-3,000-(-3,000)=0
-3,000-(-4,000) =1,000
N2
-1,000-(-3,000)= 2,000
-1,000-(-1,000)=0
2,000
1,000
Maximum opportunity
Hence, manufacturer should adopt minimum opportunity less course of action
3. Hurnicz criterion or ( Coefficient of Optimistic)
Given that coefficient of optimism = = 0.4
Coefficient of pessimism = (1-
) = 1-0.4=0.6
Then according to Hurwitz, select course of action that optimizes (maximum for
profit and minimum for loss) the payoff value.
H= (best pay off) + (1=
) (Worst payoff)
(maximum in column) + (1-
) (minimum in column)
States of nature
Courses of action
S1
S2
N1
-3,000
-4,000
N
-3,000
-1,000
Column maximum
-3,000
-1,000
Column minimum
-3,000
-4,000
2
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15.
Course of action
Best payoff
Worst pay off
H= 0.4 (best payoff) +0.6 (worst
payoff)
S1
-3,000
-3,000
= 0.4 (-3,000) +0.6 (-3,000) = (1200)
–(1800)= - (3,000)
S2
-1,000
-4,000
= 0.4 (-1000) +0.6 (-4,000) = - 4002400= - (2800)
Since course of action S2 has the least cost (maximum profit) = Rs 2,800. The
manufacturer should adopt strategy „S2‟
I.
Decision Making – Under Certainty
Decision Tree Analysis Problems
Problem-1
You are given the following estimates concerning a and development programmer
Decision Di
Probability of
decision (Di)
given (R) P
(Di/R)
Outcome
Probability of
given research
(R) P(Xi/Di)
Pay off value of
outcome Xi (Rs‟
000)
1
0.5
QIS COLLEGE OF ENGINEERING & TECHNOLOGY
0.3
-100
0.1
0
0.0
600
2
0.0
-100
3
Do not develop
2
1
0.5
600
3
Develop
0.6
1.0
0
Page 15
16.
P (xi/di)
3
=0.6
P(x2/di)
D1
1
4
Develop
0.3
D(x3/Di)
5
=0.1
P(xi/D2)
6
Do not
develop
=0
2
D(x2/D2)
D2
7
=0
P(x3/D2)
8
=1.0
Construct and evaluate the decision tree diagram for the above data. Show working for
evaluation
Solution The “decision tree” of the given problem along with necessary calculations is
shown in the following diagram
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17.
Probability
Pay off in (Rs‟ 000)
Expected pay off (in Rs‟ 000)
0.5 x 0.6 =0.3
600
=0.3 x 600= 180
0.5 x 0.3= 0.15
-100
=0.15 x -100=-15
0.5 x0.1= 0.05
0
=0.05 x 0=0
Total
165
0.5 x 0= 0
600
=0 x 600 =0
0.5 x 0 = 0
-100
= 0 x (-100) =0
0
=(0.05) x 0 = 0
0.5 x 1.0= 0.05
Total
0
Problem -2
A businessman has 2 independent investment portfolios A and B available to him,
but he lacks the capital to undertake both of them simultaneously. He can either
choose a first and then stop, or if A is not successful, then take. B or vice versa the
probability of success of A is 0.6 white for B it is 0.4. Both investment schemes require
are initial capital outlay of Rs 10,000 and both return nothing if the venture process to
unsuccessful. Successful completion of A will return Rs 20,000 and Successful
completion of B will return Rs 24,000 Draw a decision tree in order to determine the
best strategy.
Solution: - The decision tree corresponding to the given information is as followed
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18.
DECISION POINT
PROBABILITY
CONDITIONAL
VALUE (RS/-)
EXCEPTED VALUE
SUCCESS/
1.ACCPT A
OUT COME
0.6
23,000/-
12,000/-
FAILURE
0.4
-10,000/-
-4,000
8,000
2.STOP
0
SUCCESS/
1.ACCEPT B
FAILURE
0.4
24,000/-
9,600/-
0.6
-10,000/-
-6,000/-
3,6000/2.STOP
0
SUCCESS/
1.ACCPT A
0.6
FAILURE
0.4
20,000+3,600 =
23600 (0.6)
14,160
-10,000
-4,000
10,160
2.ACCEPT B
SUCCESS/
FAILURE
0.4
0.6
24000+8000 =
32000
(0.4)
-10,000 (0.6)
12800
-6,000
6,800
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19.
STOP
Rs/- 3600
RS/- 0
(0.6)
EMV=3600
SUCCESS
D2
SUCCES
EMV =10,160
0.6
FAILURE
RS/-20,000
1
ACCEPT A
0.4 24,000=9600
8
0.4
4
ACCEPT B
S
3
0.6
-10,000 =-6,000
0.6
9
20,000 =12,000
RS/-10,000
FAILUR
E
FAILURE
RS/10,000
D1
ACCEPT B
5
SUCCESS (0.6)
6
2
EMV=6,800
RS/-20000
RS/-24000
ACCEPT A
SUCCESS
(0.4)
7
EMV=8000
D3
10
-19000
11
0.4
-10,000 =
FAILURE (0.4)
RS/-0
RS/-8000
12
4. Decision – Making with Utilities
The previous problems were analyzed with probabilistic states of nature,
where the selection of an optimal course of action was on the criterion of expected
profit (loss) expressed in monetary terms
However, in many situations such criterion that involves expected
monetary payoff may not be appropriate. This is because of the fact that different
individuals attach different utility to money, under different conditions.
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-4000
20.
The term utility is the “measure of preference” for various alternatives in terms
money. The utility of a given alternative is unique to the individual decision- maker
and unlike a simple monetary amount, can incorporate intangible factors or
subjective standards form their own value system.
Example
Mr. Ran has won Rs 1,000 in a quiz programme. In the last round he is asked
either to complete or quit now he has alternatives.
Quit and take his winnings
Take a least chance in which he has 50.50 chance of winning Rs 4,000 or nothing
The question now is: what should he do? On EMV basis he has
EMV (b) =0.50 (4,000) + 0.50 (0) = Rs 2,000
The amount is twice what he was already won. But would he really give up Rs
1,000 for 50-50 chance or Rs 4,000 or nothing? Many individuals would not
because they would think of all the alternatives they could do with Rs 1,000 and
how they would regret it if they end up with nothing hence a new payoff measure
utility reflecting the decision- makers attitude and preference has to be introduced.
The basis axioms of utility may be stated as follows.
1. If outcome A is proffered to outcome B, they the utility U(A) of outcome A is
greater than the utility u(B) of outcome B vice versa
If both are equally proffered then u(A) = u(B)
2. If the decision-maker is different between the „2‟ alternatives and outcome „A‟ is
received with probability P1 and outcome C with probability (1-P) then
U (B) = P [U (A)] + (1-P) [U/C)]
Under this alternative criterion, it is assumed that a rational decision-maker
will choose that alternative which optimize the “expected utility” rather than
expected monetary value. Once we know that individual‟s utility function, along
with the probability assigned to outcome in a particular situation then the total
expected utility for each course of action can be obtained by multiplying the
utility values with their probabilities the strategy that corresponds to the
optimum utility function is called that equal strategy”
Utility function
“Utility function” is a formula or method that is used to describe the
relative preference value that individuals have for a given criterion such as money,
goods etc.
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Once derive, a “utility function” can be used to convert a decision criteria value into
utilities so that a decision can be made on the bass of maximizing the expected “utility
value” (EUV) rather than, say the “EMV”
Example
Preference are often determined by proposing a situation where by decision- maker
must choose between receiving a given amount, say Rs 20,000 for a certain thing
versus a 50-50 chance of gambling a larger amount or nothing say Rs 60,000 or Zero.
The gamble amount of Rs 60,000 is then adjusted upward or down ward until the
individual is indifferent to whether decision – maker receives the certain amount of Rs
20,000 or the gamble
Utility Curve
A utility curve that relates utility values to rupee value is construction such a curve
is usually obtained by placing the decision maker in various hypothetical decision
situations and plotting the decision makers pattern of choices in terms of risk and
utilities.
Suppose the relationship between monetary gains, losses and utilities for gains and
for small negative losses is established. The following diagram shows that if the curve
is bent down non- linearly then we assign, to large losses a disproportionately large
negative utility. It is important not to make the curve bend down too steeply or to start
the bending too quickly since this could lead individuals into a situation where they
attach such a heavy to the possibility of loss they never take any risk and tell us never
masse any gains
Once the +ve side of the curve, it is usual for the curve to eventually
bend away from the straight line. This indicates that increasing units of money are
resulting in smaller additional gains in utility
RISK AVERSION (AVOIDERS)
UTILITY
RISK INDIFFERENCE (NEUTRALITY)
RISK AFFINITY (SEEKERS)
QIS COLLEGE OF ENGINEERING & TECHNOLOGY
MONEY
RS/- MAX
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Problem 1
A manager must choose between 2 investments A&B that are calculated to yield net
profit of Rs 1,200 and Rs 1,600 respectively. With probabilities subjectively estimated
at 0.75 and 0.60. Assume the manager‟s utility function reveals that utilities for Rs
1,200 and Rs 1,600 are 45 and 50 units respectively what is the best choice on the
basis of the expected utility value (EUV)?
Solution
The except utility value (EUV) is expressed as
where
EUV =
EUV (A) = (
EUV (B) = (
= utility value of state of nature i
= probability value of state of nature i
( ) = (0.75) (45) =33.75 = EUV (A) =33.75 utilities
( ) = (0.60) (50) =30.00 = EUV (B) =30 utilities
⟹ 30< 33.75
Since EUV (A) > EUV (B)
The best choice is investment “A”
Problem 2
Mr. X has an after –tax annual of Rs 90,000 and is considering to buy accident
insurance for his car. The probability of accident during the year is 0% (Assume that
at most one accident will occur) in which case the damage to the car will be Rs 11,600
which a utility function U(x)= , what is the insurance premium he will be willing to
pay?
Solution:
Let A = Venture when Mr. X does not buy the accident insurance for his
car. Then in that case of accident he would spend Rs. 11,600 on damages and will be
left with Rs. 78,400. In the case for no accident he retains Rs. 90,000. Then we have
=(
X 0.1) + (
U (x) =
=
=
X 0.9)
1
2
= 280 utilities
=
= 300 utilities
= (280 X 0.1) + (300 X 0.9) = 298 utilities
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So the amount Rs. X which will give the same utility of the venture
A=
= Rs. 88,804
[∵ U (x) =
⟹X = [
]
Thus Mr. X will be indifferent to an amount of Rs.88, 804 with certainty and the
venture A. The amount he is willing to pay as car premium would be
90,000 – 88,804 = Rs. 1,196
3) Decision Making Under Risk:
Decision making under risk is a probabilistic decision situation in which more
than one state of nature exists. And the decision maker has sufficient information to
assign probability values to the likely occurrence of each of these states. Knowing the
probability distribution of these states of nature, the best decision is to select that
course of action which has the largest expected payoff value.
The expected (average) payoff of an alternative is the sum of all possible payoffs of
that alternative, weight by the probabilities of the occurrence of those payoffs.
The most widely used criterion for evaluating various course of action (alternatives)
under risk is the “expected monetary value (EMV)” or “expected utility”
Expected Monetary Value (EMV)
The expected monetary value (EMV) for a given course of action is the
weighted sum of possible payoffs for each alternative. The expected value is the longrun average value that would result if the decision were repeated a large number of
times. Mathematically EMV is stated as follows.
EMV [ Course of avtion, Sj ] =
Where m = number of possible states of nature
Pi = probability of occurrence of nature, N:
Pij= payoff associated with state of nature Ni and course of action, Sj
Steps for calculating EMV
The various steps involved in the calculation of EMV are as follows
1. Construct of payoff matrix listing all possible courses of action and states of
nature. Enter the conditional payoff values associated with each possible
combination of courses of action and state of nature along with the probabilities
of the occurrence of each state of nature
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2. Calculate the EMV for each course of action by multiplying the conditional
payoff by the associated probabilities and adding these weighted values for each
of action
3. Select the course of action that yields the optimal EMV
Expected Profit with Perfect Information (EPPI )
The expected profit with perfect information (EPPI) is the maximum
attainable expected money value (EMV) based on perfect information about the state of
nature that will occur. The expected profit with perfect information may be defined as
the sum of the product of best state of nature corresponding to each optimal course of
action and its probability
Expected Value of PerfectInformation ( EVPI)
The expected value of perfect information (EVPI) may now be defined as the maximum
amount one would be willing to pay to obtain perfect information about the state of
nature that would be willing to pay to obtain perfect information about the state of
nature that would occur. EMV Represents the maximum attainable expected monetary
value given only the prior outcome probabilities with no information as to which state
of nature will actually occur. Therefore perfect information would increase profit from
EMV up to the value of EPPI. This increased amount is termed as EVPI.
i.e. EVPI= EPP1- EMV
Expected Opportunity Loss ( EOL):
Another useful way of maximizing monetary value is to minimize the expected loss
or expected value of regret. The conditional opportunity los (COL) or regret function for
a particular course of action is determined by taking the difference between payoff
values of the most favorable course of action. And some other course of action .Which
may be considered as loss due to loosing the opportunity of choosing the most
favorable course of action, thus opportunity loss can be obtained separately for each
course of action by 1St obtaining the best state of nature for the prescribed course of
action and then taking the difference between that best outcome and each outcome for
those courses of action. The opportunity loss for each course of action is known as the
conditional opportunity loss
After calculating the opportunity loss value for each course of action,
the E0L for ith course of action Si is then computed by
EOL ( ,) =
, ). P ( )
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Where ( , )= Conditional Opportunity loss associated with the course of action
And state of nature
P(
.
) = Probability of occurrence of state of nature
.
In other words EOL denotes the expected difference between the payoff of right
decision and the payoff of actual decision.
Problem-1
A modern home appliances dealer finds that the cost of holding a mini cooking
range is stock for a month is Rs 200 (Insurance, minor deterioration, interest on
borrowed capital, etc) customer who cannot obtain a working range immediately tends
to go to other dealer and he estimates that for every customer who cannot get
immediate delivery, he loses an average of Rs 500 the probabilities of a demand of
0,1,2,3,4,5 mini cooking ranges in a month are 0.05, 0.10, 0.20, 0.30, 0.20, 0.15
respectively determine the optimum stock level of consuming rangers. Also find EVPI
Solution:
The cost function =Rs 500(D-S); If D>S
= Rs 200 (S-D); If D < S
Where S= The number of units purchased and
QIS COLLEGE OF ENGINEERING & TECHNOLOGY
D= the number of units demanded.
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Event
Probability
Conditional Cost (Rs/-)
(S)
Expected Cost (Rs/-)
Demanded
0
(D)
1
2
3
4
5
0
1
2
3
4
5
0
0.05
0
200
400
600
800
1000
0
10
20
30
40
50
1
0.10
500
0
200
400
600
800
50
0
20
40
60
80
2
0.20
1000
500
0
200
400
600
200
100
0
40
80
120
3
0.30
1500
1000
500
0
200
400
450
300
200
100
0
40
4
0.20
2000
1500
1000
500
0
200
400
300
200
100
0
40
5
0.15
2500
2000
1500
1000
500
0
375
300
225
150
7
50
1475
1010
615
360
315
410
Expected cost
Since the expected cost is minimum if 4 cooking ranges are stocked each month.
The optimum act is to stock 4 looking ranges.
Problem -2
Under an employment promotion programming, it is proposed to allow sale of
newspapers on the buses during of peak hours. The vendor can purchase the
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newspaper at a special concessional rate of 25 paisa per copy against the selling price
of 40 paise. Unsold copies are however a dead loss. A vendor has estimated the
following probability distribution for the number of copies demanded
No of copies demanded
15
16
17
18
19
20
Probability
0.04
0.19
0.33
0.26
0.11
0.07
How many copies should he order so that his expected profit will be maximum?
Solution:The vendor does not purchase less than 15 copies or more than 20 copies
Let n = The number of copies of newspaper demanded
The vendor would loss 25 paise on each copy in case of demand is less than „n‟
otherwise , if the demand is more than or equal to „n‟ then he would gain 15 paise on
each newspaper copy .
The incremental profit =( Excepted profit –Expected loss), for each value on „n‟ is
given in the following table
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