statistical decision theory-english statistical decisio

8/13/2019 Statistical Decision Theory-English Statistical Decisio....

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Statistical Decision Theory

Every individual, big or small, rich or poor, educated or

uneducated has to take decision almost every day.

Some of these decisions are of a routine type which do not involve

high states and are consequently trivial in nature.

Example

A student may decide whether to put on a white or a red shirt

while going to the college on a particular day or a house

wife may decide to serve lemon uice or pineapple uice to aguest or business e!ecutive may decide whether to go bytrain or car to meet a potential customer in a nearby town.

"owever, in contract to these situations, we have quite often to

make decisions which we consider to be important from manypoints of view and which entails a lot of reasoning and thinking.

Example

#ecision to buy or not to buy shares of particular company,

to accept or not to accept a new recruitment and promotionspolicy are, by any standard, significant and importantdecisions and consequently would not be made in a haste orwithout a detailed analysis of the various pros and consinvolved in each situation.

Some of the decisions, like buying or not buying the shares of a

company or accepting a new ob affect the decision maker only. "ealone has to suffer the consequences of his decision or his familymembers may also be affected by it. Such decisions are personaldecisions.

As against these some people have to make decisions which affect

other people like consumers of the products, shareholders of thebusiness unit and employees of the organi$ation.


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Such decisions which affect other people in a society require very

careful and objective analyses of the different group of persons

aredifferently affected by these and their interests often clashes.

Decision Making Science or an Art

%he very first doubt that arises in the mind of a common person is

that whether decision making could ever be science obeyingdefinite laws which may give high of precisions to theconsequences of a particular decision.

%he ustification for this doubt is because in decision making we

make inferences about unknown rather than known.

%here is much that we do not know about all the problems that

beset us and that is why most of the time we find ourselvesguessing and making subective decisions.

&ery often it is felt that the choice of the decision is related to the

personality of the decision maker and his subective assessment ofthe situation that an unemotional and abstract scientific analysis isin appropriate in this area.

't is also argued that successful men in business have made right

decisions without the aid of scientific tools and techniques and thatthe choice of a decision is largely a matter of intuition based one!perience.

(n a closer e!amination it would become obvious that this feeling

is not right and its e!ists because people enoy guessing. (therwisewe cannot e!plain the popularity of many games of chance which

make guessing synonymous with entertainment.

%here can be no shortage of quantitative techniques which can be

appropriately and profitably used in such situations and yet are notmade rise of our shortage is only of more effective techniques andalso our willingness to apply the techniques we already have.

Even if it is accepted that there is always a strong human element

in decision making in the final stage, process which are amenable

to scientific analysis and treatment.


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Efforts are made to evolve a method by which these components

could be woven to help the person to make a coherent andconsistent decision.

't can, therefore, be stated that decision making in the field ofbusiness need not be e!clusive subective in character. %here aremany areas where it is possible to apply statistical tools andtechniques and thereby make decision making more obective innature.

#ecision making would, therefore, remain both an art as well as

science.

Elements in the Decision Making

)or all decisions whether routine or comple! there are some

common elements.

A decision situation arises only when the decision maker has

more than one course of action open to him.

'f there is only one alternative, there is nothing to decide.

%he first step, therefore, is any decision situation is to find out and

list all possible alternatives available.

%he list of alternative choices should be as far as possible

e!haustive and the list so drawn should provide courses of actionwhich are e!clusive of each other. 't means that out of variouscourses of action which are e!clusive of each other. 't means thatout of various courses of action, if any one is chosen, the others

have to be reected. (ne and only one decision can be choose as itis the best.

Another common element in most of decision problems is

uncertainty. %his uncertainty is referred to as State of Nature(N)or the State of the World.

%he strategy which a decision maker has to choose would depend

on the level of uncertainty of the state of nature.


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Statisticians make an effort to reduce the element of uncertainty by

trying to assign probabilities to various states of nature on the basisof past records about the problem under study.

Measuring Conseuences of !arious Decision

'n order to select a strategy from amongst the various available

strategies one has to know the consequences of selecting differentstrategies.

'n other words one should know the e!tent to which a particular

strategy would achieve the obective which the decision maker hasin his mind.

%he problem before the decision maker would be a problem of

measurement of the e!tent to which an obective is being reali$ed.

"ere the decision maker has to face a new type of problem. %here

are some obectives which are either achieved or not achieved, fore!ample, whether a new product is developed or not developedwhether a particular quality is achieved or not achieved.

(n the other hand, there are many obectives which provide a vastrange of attainments. Such obectives are profits, cost, sales,

production, employee motivation or market goodwill.

Some of these obectives offer natural way of measuring the degree

of achievement in term of say rupees or numbers or some otherunits. *hereas, there are some other obectives like goodwill oremployee morals or motivation which cannot be measured in thismanner.

"owever, when we use+quantitative techniques in decision making

these qualitative terms have to be studied in the light of relatedcharacteristics. )or e!ample, labour satisfaction may be studiedthrough labour turn over rate or labour productivity etc.

't should be remembered that when our obectives can be directly

defined in terms of a natural unit, the inferences have to becarefully drawn. rofit in terms of rupees is a natural way of

measuring profit but does it really reflects the e!tent of


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achievement of the obectives of higher profits. %he value of rupeediffers from unit to unit and individual to individual.

%hus, the same rupees have different measures to different persons.

't is the utility which is more relevant measure of achievementof an obective rather than the natural unit like a rupee or any otherunit of measurement.

'f the obectives to be achieved are multiple and are of types which

cannot be measured on same scale, the problem becomes morecomple!. E!ample, if the obectives of the firm are, higher profit,higher productivity and a high quality standard of the product, thenthere is no single scale of measurement of these obectives.

"owever, in such comple! situations also, efforts are made toconvert these obectives into a single utility measure.

"ay #ff

*hen the value of a consequence is e!pressed directly in terms of

gains e!pressed in money, it is called a $pay off.

%he consequences of various decisions are given monetary values

and when the conditional outcome of the various strategies topossible states of nature is put in the shape of a table, it is called"ay off Table or "ay off Matrix.

A pay of matri! takes into account two things -

a Alternative strategies /or alternatives which may be denotedby S0, S1, 2S

b &arious states of nature denoted by 30, 31,23k.

(bviously, the total number of cell in a ay off 4atri! would

depend on the number of strategies which are available and thevarious states of nature which are identified.

'f there are 5 strategies and there are 6 states of nature, the total

number of cells in the pay off matri! would be 5 ! 6 or 17.

A pay off matri! generally assumes the following shape.


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Strategy

NatureofState %& %' %( %) *++ %k

S0 00 01 08 05 2.. 0kS1 10 11 18 15 2.. 1k

S8 80 81 88 85 2.. 8kS5 50 51 58 55 2.. 5k

2..2.. 2.. 2.. 2.. 2.. 2.. 2..

S 0 1 8 5 2.. k

%here is no rigidity about the rules that state of nature be shown in

column and the strategies in rows. %he state of nature can be shownin rows and strategies or actions in columns.

'n pay off matri! 00in the pay off the strategy S0when the state ofnature is 30. Similarly, 05is the pay off of strategy S0when thestate of nature is 35. 9ikewise, 0kwill be the pay off strategy S0when the state of nature is 3k.

Example

A carpenter is offered 6 tables for :s. 877 /:s. ;7 per table. "e

thinks that he may be able to sell each table for :s. 067 thismaking a profit of :s.


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'f the decision is not to buy, the pay off matri! would show 7 in all statesthe nature.

"ay #ff Matrix

Strategy

NatureofState %, %& %' %( %) %-

S0= >uy +877 +067 7 067 877 567

S1 = #o not>uy

7 7 7 7 7 7

)rom the above table it is clear that if the carpenter takes a decision

to buy the tables, he will earn of profit of :s. 567 only if he can

sell 6 tables at :s. 067 each and he will have no profit /if he sellsonly 1 tables.

'f he sells less than two tables then he suffers also.

Some times instead of gain we may measure less or cost and in

such cases the table is known as 9oss or ?ost %able. 9oss or?ost is a negative gain.

'n pay off table we select a strategy which gives us ma!imum payoff or minimum cost or loss.

Savage has suggested a different measure to assess the

consequences of a strategy. According to him if we have selected astrategy and we know our pay off. 3ow we should compare this

pay off with the ma!imum pay off which we might have got hadwe selected another in the state of nature which has occurred.

%he ma!imum pay off minus the pay off which we have received isthe .egretof the decision makers.

)or e!ample, if the pay off of a decision maker under the

strategy chosen with the state of nature that has occurred is :s.677 and if an alternative strategy under this state of naturewould have given him :s. @77, then his regret is :s. @77 = :s.677 or :s. 877.

Savage argues that the decision maker should try to minimi$ethis regret.


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't is called minimi$ing the regret or loss of opportunity.

%he regret or the opportunity loss is simply the difference between

the pay off reali$ed and the ma!imum pay off which could havebeen reali$ed if another strategy was chosen.

.egret Tablefrom the above e!ample would be of the following

form

Sate of

%ature %, %& %' %( %) %-Strategy

S0>uy 877 067 7 7 7 7S1#ont >uy 7 7 7 067 877 567

)rom the above it is clear that if the carpenter decides to buy the

tables his ma!imum regret if the State of 3ature is 3 7is :s. 877B+and in case of 30it is :s. 067B+. After this he has no regret.

'n case he selects strategy 1 /dont buy his regret would start when

the state of nature would be 38, 35or 36. 'f the state of 3ature is

37, 30, or 31, he has no regret.

Example

A fruit dealer buys oranges at the rate of :s. 8 per do$en and

sells them at the rate of :s. 6 per do$en. (ranges not soldduring the day are treated as stale and thrown away. %hedaily sale of oranges in the past has not been less than


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%he e!pected pay off /E, under various strategies with

different states of nature will be-

00C pay of strategy, is S0and state of nature is 30

00C D


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31/


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'n such cases the pay off matri! is of great help and an optimal

decision can be arrived at by assigning probabilities to variousstates of nature.

/iii2 Decision making under uncertainty

A decision process is said to be under condition of uncertainty

when in states of nature are unknown and no obective informationis available about their probabilities of occurrence.

'n such cases there is no historical data or no relative frequency

which could indicate the probability of occurrence of a particularstate of nature.

Such situation arises when a new product is introduced in the

market or a new plant is set up.

(f course even in such cases some market surveys are conducted

and relevant information gathered but generally it is not sufficientto indicate a probability figure for the occurrence of a particularstate of nature.

/iv2 Decision making under partial information

%his is a situation some where between the condition of risk and

the condition of uncertainty.

'n the case of condition of risk the probability of the occurrence of

various states of nature is known on the basis of past e!perienceand in condition of uncertainty, there is no such data available."owever, there might be many situations where thee is partial

information available of the data. 'f is so, decision making is saidto done on the basis of partial information.

/v2 Decision making under conflicts

A condition of conflict is supposed to be e!it when instead of state

of nature we are dealing with rational opponent. "ere the decision


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maker has to choice a strategy which takes into account the actionand counter action of his opponents.

4arket place, brand competition military weapons etc. are

problems which come under this category.

%he choice of strategy in such situationBconditions is done on the

basis of game theory 3here a decision maker anticipates the

action of his opponent and then determines his o3n strategy.

't is like paying a game of chess.

Choice of a Decision Criteria

'n order to select a strategy from amongst the many in different

types of decision situations, it is necessary that the selected strategyis one which is most appropriate for achieving the obective inmind of the decision maker.

%he nature of decision criteria would depend on the types of the

decision situation.

4nder condition of certainty

Fnder the condition of certainty there is a pay off for each strategy.

%he pay off measured as utility, i.e., profit represents the degree ofachievement of the obective, hence the largest pay off is chosenand the corresponding strategy is selected.

'f, however, the measure is the cost, then the strategy with the

lowest cost is picked up.

4nder condition of risk

Fnder the condition of risk there would be more than one state of

nature but the probabilities of their occurrence are known on thebasis of their past e!perience.

'n this situation each strategy will have as many pay off as the

states of nature. %o pick up the correct strategy we will have to

transform all possible pay offs of a strategy into a single figure, on


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the basis of their probabilities of the states of nature and thee!pected pay off.

%he strategy which gives ma!imum pay off is selected in such

cases.

4nder condition of uncertainty

Fnder the condition of uncertainty, since we do not know the

probability of the occurrence of various states of nature, theproblem becomes more comple! and the personality of the decisionmaker plays an important role in the selection of the strategy, thusthe decision taken under uncertainty are necessarily subective.

5o3ever6 the analyst has devised some decision rules to impartsome objectivity to the subjective decisions. %he followingdecisions choices reflect the attitude of the decision maker.

/i2 7ald8s Maxim in Decision criterion

*alds ma!im in decision criterion tells us that the

decision makers should specify first the worst possibleoutcome of each strategy and accept a strategy that

gives the best out of the worst outcomes.

%o illustrate the e!ample let us consider a hypothetical

pay off matri! as

"ay off Matrix

States of %ature

Strategy %& %' %( %)

S0lant G 4achinery 17 01 ; 6S1Equity 06 0; 5 +1

S8Hovernment >onds 0; @ ; +0

S5:eal Estate 6 01 8 1

't is assumed that an investor conceives of four

strategic investment proects S0, S1, S8 and S5/investment in plant and machinery, equity, real estateand government bonds respectively under four

different states of nature of the economy, 30, 31, 38


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and 35 /high growth, low growth, stagnation andrecession respectively.

%o apply ma!imin criterion, the decision maker needs

to find out the worst /minimum outcome of eachstrategy. %his can be done by reading the pay off tablerow wise.

%he ma!imin column presents the worst outcome of

each strategy. %he best or the highest outcome of theworst outcome is 6 of strategy.

Hoing by the ma!imin criterion, the decision maker

would accept strategy S0.

'f the ma!imin rules are closely looked, it implies a

pessimistic approach to investment decision making.9t gives a conservative decision rule for risk

avoidance.

/ii2 Maximax Criteria

According to this criterion, if the decision maker is anoptimist by nature he would always think that the stateof nature would be the best from his point of view.

"e would find out the e!pected pay off of all the

strategies and will pick up the strategy which gives thema!imum pay off out of ma!imum pay offs of all thestrategies. "e always thinks that the states of naturewould be favourable to him and his eyes are on the

ma!imum possible pay off of all the strategies.

/iii2 Minimax .egret Criteria

4inima! regret criterion is another decision rule

under uncertainty.

%his criteria suggests that the decision maker should

select a strategy that minimi$e the maximum regret

of a wrong decision.


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Example

Suppose an investor has three strategies for

investment, S0, S1 and S8, giving returns of :s.

07,777B+ :s. @,777B+ and :s. ;,777B+ respectively. 'fthe investor opts for strategy S0, he gets the ma!imum

possible return, he has no regret. >ut, if he opts for S1,by way of incorrect decision, then his regret onopportunity cost would be :s. 07,777B+ = :s. @,777B+ C:s. 1,777B+. 9ikewise if he opts for S8, his regretequals :s. 07,777B+ + :s. ;,777B+ C :s. 5,777B+.

Hoing by the minima! regret criterion, the investor

should opt for strategy S1 because it minimi$e theregrets.

Suppose we have the following pay off table, we can

construct the regret table. %he method is simple.Select a column /the state of nature, find thema!imum pay off and subtract it from the pay off ofall strategies. %his process gives the pay off column.

StrategyStates of %ature .egret Matrix Max:Min

.egret%& %' %( %) %& %' %( %)S0 17 01 ; 6 7 7 7 7 7

S1 06 07 5 +1 6 1 1 8 6

S8 0; @ ; +0 5 5 7 5 5

S5 6 01 8 1 06 7 8 8 06

)or e!ample, under column 30, strategy S0 has the

ma!imum pay off /17. 't means that if S 0 is chosenunder the states of nature 30, the regret is $ero. 'n thene!t strategy S1, the pay off is 06 and the regret isequal to 6.

>y repeating this process for all strategies and all the

states of nature we get regret matri!. )rom the regret


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matri! we can find Ima!imin regret by listing thema!imum regret for each strategy.

%he regret table shows that the ma!imum regret is

minimum /06 in case of strategy S5. %herefore,strategy S5strategy should be selected for investment.

/iv2 ;aplace Decision criterion

%he laplace criterion was >ayesian rule to calculate

the Iexpected value of each strategy.

According to the >ayesian rule, where meaningful

estimate of probabilities is not available, the outcomeof each strategy under each state of nature must beassigned the same probability and the sum of

probabilities of out come of each strategy must add upto one. )or this reason, the 9aplace criterion is alsocalled I>ayesian criterion.

>y assuming eual probability for all events, the

environment of uncertainty is converted into an

environment of risk

%his decision rule avoids the problem that arises due

to subectivity in assuming a probability of pay off.

Example

A news paper vender buys a newly started local paper

for 6 n.p. and sells it at the rate of 07 n.p. %he unfold

paper do not have any value. %he vendor knows thathe cannot sell more than 17 papers in a day and theminimum sale would not be less than 05. "ow many

papers should he buy.

Solution

%he strategies open to the newspaper vender are to

buy 05, 06, 0;, 0J, 0@, 0< or 17 papers.


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%he states of nature are the number of newspaper he

can sell. %he states of nature would be 05, 06, 0;, 0J,0@, 0< or 17.

rofit of vendor wills as follows if he buys only 05profit 6 n.p. on each paper J7 n.p.

'f he sells 06 papers he earn a profit of J6 n.p.

9ikewise, the profit of other class can be calculated.

Strategy

S&) S&- S&< S&= S&1 S&0 S',States of

%ature305 J7 ;6 ;7 66 67 56 57

306 J7 J6 J7 ;6 ;7 66 67

30; J7 J6 @7 J6 J7 ;6 ;7

30J J7 J6 @7 @6 @7 J6 J7

30@ J7 J6 @7 @6


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0. 4a!imin #ecision - >uy 05 papers. "e will ma!imi$ethe minimum pay off. %he largest

minimum pay off is J7, so he willbuy 05 papers.

1. 4a!ima! #ecision - >uy 17 papers. %he highestma!imum pay off is 077. "e will

buy 17 papers as it ma!imi$es thepay off.

8. 4inimi$ing the 4a!imum :egret?riterion

- )or studying this it is necessaryto construct or prepare a regrettable.

.egret Table

Strategy

S&) S&- S&< S&= S&1 S&0 S',States of

%ature

305 7 6 07 06 17 16 87

306 6 7 6 07 06 17 16

30; 07 6 7 6 07 06 17

30J 06 07 6 7 6 07 06

30@ 17 06 07 6 7 6 07

30< 16 17 06 07 6 7 6

317 87 16 17 06 07 6 7

3ow if the state of nature is 305 and S05 the regret

would be 7 as he loses nothing but if the strategy isS05and state of nature is 306his regret would be 6,

because the ma!imum pay off with 306to J6 and he

would reali$e only J7.

9ikewise with S05and 30;his regret would be 07 as

the ma!imum pay off under 30;is @7.

4nder Minimax Criteria

"e will buy 0J news paper. %here he is minimi$ing

the ma!imum regret. %he ma!imum regret with S0JC

06. 'n all the other strategies the ma!imum regret ismore than 06.


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All the above decision have been taken under conditions of

uncertainty. %he decision maker did not have any probabilityvalues for various states of nature.

'f the decision maker on the basis of past e!perience could assign

some probabilities to the various states of nature.

State of %ature "robabilities

305 7.7@

306 7.1130; 7.1730J 7.0530@ 7.0530< 7.01317 7.07

0.77

*ith the above information the decision maker is in a better position to

take a rational decision.

%he e!pected pay off table would be.

"ay off Table

Strategy

" S&) S&- S&< S&= S&1 S&0 S',States of

%ature

305 77.7@

76.;7

76.17

75.7@

75.57

75.77

78.;7

78.17

306 77.11

06.05

0;.67

06.57

05.87

08.17

01.07

00.77

30; 77.17

05.77

06.77

0;.77

06.77

05.77

08.77

01.77

30J 77.05

7


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1 7 7 7 7 7 7

317 77.07

7J.77

7J.67

[email protected]

[email protected]

7


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'n 9aplace decision theory, the decision maker /in the absence of

any past experience2 3ould assign eual probabilities to all

states of nature.

Since in this case there are J states of nature, hence the probabilityof each state of nature would be 0BJ.

%he e!pected pay off under laplace decision would be obtained by

multiplying each pay off with 0BJ. %he pay off matri! would therebe as follows-

;aplace "ay #ff Table

Strategy" S&) S&- S&< S&= S&1 S&0 S',States of

%ature

305 0BJ 07.77

7


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%here are two specific problems in this kind of decision making.

?irst, the decision makers are required to make a choice /or series

of choices from the alternative investment avenues to them. %hey

are not supposed to leave the matter undecided.

Second, the decision makers know for sure that all the decisions

will yield a positive outcome, but they cannot tell in advance theexact outcome of a decision. %hey might be knowing that a

particular decision will yield a higher return than another but theydo not know for sure high or low the outcome will be.

%he question that decision+makers face under these conditions is

how to find the most profitable or gainful solution.

%he method that is used to find an acceptable solution under these

conditions is called a Tree:Decision.

A tree+decision is a graphical device to map all possible

managerial decisions in a sequence and their e!pected outcomesunder different states of economy.

Since all possible strategic decisions and their possible outcomesare arranged graphically in the form of branches of a tree, thetechnique is called decision:tree.

%he decision+tree presents the entire decision options and possible

outcomes in the form of a diagram and thereby guides the decisionmeters to a rational decision.

Decision Tree for 9nvestment Decision


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Fnder both these growth probabilities, the prospect for the product

demand again has three probabilities = high, medium and low =under both high and low growth of the economy.

't should also be noted that the probability distribution in respect ofdemand prospects = high, medium and low = under high growthadd up to 7.; and in case of low growth they add up to 7.5.

9et us now suppose that the investor has the information on the

present value of cash flow under each probability as presented incol. /6. 3ow when the present value of cash flow is multiplied bythe corresponding probability in col. /5, it gives the Ie!pectedvalue of the present value of cash flows.

?ol. /; gives the e!pected value of the two proects under all the

stipulated conditions. %he investor has now the full information fordecision making.

9nvestment decision

%he investor can easily find out the net e!pected value of each proectand decide in favour of the proect having a higher net e!pected value.

Crore80Rs.ValueexpectedNet

Crore500Rs.:CostProjectLess-Crore580Rs.:ValueExpectedotal

:Project !=

Crore"0Rs.ValueexpectedNet

Crore#00Rs.:CostProjectLess-

Crore#"0Rs.:ValueExpectedotal

:$Project=

According to this calculation, the net e!pected value of proect A

is higher than the proect >. %herefore, a rational investor woulddecide to invest in proect A and not in roect >.

Example

%he ?oca+?ola associates deals with instant soft drink. %hey have

two courses of action for selling their product in the market /a:egional distribution through distributors and /b #irect selling.


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%he prior probabilities of high penetration and low penetration ofregional distribution channel are 7.J and 7.8 respectively. %he

prior probabilities of high penetration and low penetration of directselling channel are 7.; and 7.5 respectively. %he pay off of high

and low penetration of regional distribution channel are :s. 67lakh and :s. 07 lakh respectively. %he pay off of high and low

penetration of regional distribution channel are :s. 67 lakh and :s.07 lakh respectively. %he pay off of high and low penetration ofdirect selling channel are :s. 87 lakh and :s. 6 lakh respectively.#raw the decision tree and determine the best selling channel i.e.strategy.

Solution

%he total e!pected monetary value by regional distribution

7.J K 67 L 7.87 K 07 C :s. 8@ lakh

%he e!pected monetary value by direct selling


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7.; K 87 L 7.5 K 6 C :s. 17 9akh

Since the e!pected monetary value is more i.e. regional

distribution in more than direct selling he is should opt for regionaldistribution.

;imitation of Distribution Theory

%he decision theories associated with conditions of risk and

conditions of uncertainty have many limitations.

%heories under conditions of risk and uncertainty e!ist because

thee are limitations in the relative frequency approach on the basisof which probabilities are assigned to various states of nature andalso because the e!pected pay off matri! does not always provideinfallible information for a decision situation.

Decision Tree Method


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Dra3ing a Decision Tree

Mou start a #ecision %ree with a decision that you needto make. #raw a small square to represent this towardsthe left of a large piece of paper.

)rom this bo! draw out lines towards the right for eachpossible solution, and write that solution along the line.Neep the lines apart as far as possible so that you cane!pand your thoughts.

At the end of each line, consider the results. 'f the resultof taking that decision is uncertain, draw a small circle. 'fthe result is another decision that you need to make, drawanother square. Squares represent decisions, and circlesrepresent uncertain outcomes. *rite the decision orfactor above the square or circle. 'f you have completedthe solution at the end of the line, ust leave it blank.

Starting from the new decision squares on your diagram,draw out lines representing the options that you couldselect. )rom the circles draw lines representing possibleoutcomes. Again make a brief note on the line sayingwhat it means. Neep on doing this until you have drawnout as many of the possible outcomes and decisions asyou can see leading on from the original decisions.

An e!ample of the sort of thing you will end up with isshown in )igure 0-


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(nce you have done this, review your tree diagram.?hallenge each square and circle to see if there are anysolutions or outcomes you have not considered. 'f thereare, draw them in. 'f necessary, redraft your tree if partsof it are too congested or untidy. Mou should now have agood understanding of the range of possible outcomes ofyour decisions.

Evaluating @our Decision Tree

3ow you are ready to evaluate the decision tree. %his is

where you can work out which option has the greatestworth to you. Start by assigning a cash value or score to


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each possible outcome. Estimate how much you think itwould be worth to you if that outcome came about.

3e!t look at each circle /representing an uncertaintypoint and estimate the probability of each outcome. 'fyou use percentages, the total must come to 077O ateach circle. 'f you use fractions, these must add up to 0.'f you have data on past events you may be able to makerigorous estimates of the probabilities. (therwise writedown your best guess.

%his will give you a tree like the one shown in )igure 1 -


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Calculating Tree !alues

(nce you have worked out the value of the outcomes,and have assessed the probability of the outcomes ofuncertainty, it is time to start calculating the values thatwill help you make your decision.

Start on the right hand side of the decision tree, and workback towards the left. As you complete a set ofcalculations on a node /decision square or uncertaintycircle, all you need to do is to record the result. Mou can

ignore all the calculations that lead to that result fromthen on.

Calculating the !alue of 4ncertain #utcome %odes

*here you are calculating the value of uncertainoutcomes /circles on the diagram, do this by multiplyingthe value of the outcomes by their probability. %he total

for that node of the tree is the total of these values.

'n the e!ample in )igure 1, the value for Pnew product,thorough developmentP is-

7.5 /probability good outcome !Q0,777,777 /value C

Q577,777

7.5 /probability moderate outcome! Q67,777 /value C Q17,777

7.1 /probability poor outcome !Q1,777 /value C

Q577

L )',6),,

)igure 8 shows the calculation of uncertain outcomenodes-


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3ote that the values calculated for each node are shown in the bo!es.

Calculating the !alue of Decision %odes

*hen you are evaluating a decision node, write down thecost of each option along each decision line. %hensubtract the cost from the outcome value that you havealready calculated. %his will give you a value thatrepresents the benefit of that decision.

3ote that amounts already spent do not count for thisanalysis = these are Psunk costsP and /despite emotional


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counter+arguments should not be factored into thedecision.

*hen you have calculated these decision benefits,choose the option that has the largest benefit, and takethat as the decision made. %his is the value of thatdecision node.

)igure 5 shows this calculation of decision nodes in oure!ample-

'n this e!ample, the benefit we previously calculated forPnew product, thorough developmentP was Q517,577. *e


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estimate the future cost of this approach as Q067,777.%his gives a net benefit of Q1J7,577.

%he net benefit of Pnew product, rapid developmentP wasQ80,577. (n this branch we therefore choose the mostvaluable option, Pnew product, thorough developmentP,and allocate this valueto the decision node.

.esult

>y applying this technique we can see that the best

option is to develop a new product. 't is worth muchmore to us to take our time and get the product right,than to rush the product to market. 't is better ust toimprove our e!isting products than to botch a newproduct, even though it costs us less.

Bey "oints>

#ecision trees provide an effective method of #ecision4aking because they-

?learly lay out the problem so that all options canbe challenged.

Allow us to analy$e fully the possible consequencesof a decision.

rovide a framework to quantify the values ofoutcomes and the probabilities of achieving them.

"elp us to make the best decisions on the basis ofe!isting information and best guesses.

As with all #ecision 4aking methods, decision treeanalysis should be used in conunction with commonsense = decision trees are ust one important part of your

#ecision 4aking tool kit.

statistical decision theory-english statistical decisio

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