2. decision theory

7/30/2019 2. Decision Theory

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Decision Theory


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Introduction

Decision theory is concerned with how to

assist people in decision making. The

theory provides a meaningful framework

for improved decision making.


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Decision Analysis

Define the

problem

Search for

alternatives

Evaluate the

alternative

Select an

alternative

Apply decision

criteria

Assess

Consequences

Decision theory

Abstraction

Design

Choice

Intelligence


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Decision Theory

Decision theory steps:

1. Decision making environment

Deterministic Situation (Certainty)

Stochastic situation (risk)

Situation of uncertainty

2. Objective of decision maker

3. Alternative plans of action or strategies

4. Decision payoff (indication of effectiveness of

strategies)


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Decision Theory

Deterministic Situation (Certainty)-Action leads onlyone outcome. Important techniques: linear

programming, transportation and assignment

problems.

Stochastic situation (risk)- many state of and decisionmaker knows the probability of occurrence. Important

techniques: Decision matrix, decision tree, PERT,

Markov Chain, Bayesion analysis

Situation of uncertainty- Probability associated with

state of nature are unknown . Important techniques:

Matrix games, minimax actions and strategies.


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Decision Theory

Decision process calls for:

1. Identification ofstates of nature, or events

2. Identification ofcourses of action, or acts

3. Determination of thepay-offs, depicting outcomes

of various combinations of acts and events (pay-

offs resulting from various act-event combinations

can be either gains/losses or costs)

4. Choosing, on the basis of some criterion, from

among different alternatives


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Introduction to DecisionAnalysis

Steps in Decision Theory Approach

Step 1: Determine the various alternative courses of action (or

choices or strategies) from which the final decision is to be

made. Step 2: Identify the possible outcomes, called the states

of nature or events for the decision problem. The events arebeyond the control of the decision-maker.

Step 3: Construct a pay off table

Step 4: The decision-maker will choose the criterion which

results in largest pay off. The criterion may be economic,

quantitative or qualitative (for example, market share, profit,fragrance of a perfume and so on)



Decision theory question

A book store sells a particular book of Tax Law for Rs

100. it purchases book for Rs. 80 per copy. Unsold

copies becomes outdated by the end of year and can

be disposed off for Rs. 30. According to past

experience, the annual demand for this book isbetween18 and 23 copies. Assuming the order can be

placed only once in an year , the problem before sales

manager is to decide on how many number of copies

should be purchased for the next year.



Payoff Table

Let Q is Quantity, D is demand, P is Profit

If D>Q Then P= 100Q-80Q or

P = 20 Q.

If D< Q then P= 100D +30(Q-D)-80Q or

P = 70D-50Q



Payoff Table

.Event EiDemanded

copies

Act Aj (Course of action)

A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 360 310 260 210 160 110

E2: 19 360 380 330 280 230 180

E3: 20 360 380 400 350 300 250

E4: 21 360 380 400 420 370 320

E5: 22 360 380 400 420 440 390

E6: 23 360 380 400 420 440 460



Opportunity Loss or Regret Table

.Event EiDemanded

copies


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 0 50 100 150 200 250

E2: 19 20 0 50 100 150 200

E3: 20 40 20 0 50 100 150

E4: 21 60 40 20 0 50 100

E5: 22 80 60 40 20 0 50

E6: 23 100 80 60 40 20 0



Decision under uncertainity

Decisions under Uncertainty

Laplace Principle: assumes equal likelihood of various

states of nature

Maximin (or Minimax in case of pay-offs as cost)

Principle: pessimists criterion considers best of the

worst

Maximax (or Minimin in case of pay-offs as cost)

Principle: optimists criterion considers best of the

best

Hurwicz Principle: uses decision-makers degree of

optimism

Savage Principle: uses minimum of the maximum regret

values of various acts



Laplace Criterion or EquallyLikely Decision Criterion

Step 1: Assign equal probabilities 1/(number of states of

nature) to each pay off of a strategy.

Step 2: Determine the expected pay off value for eachalternative.

Step 3: Select that alternative which corresponds to the

maximum (and minimum for cost) of the above expected pay

offs.


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Decision under uncertainity

Laplace Principle: highest mean value is adopted.

Stock Mean (Expected) Pay-off

A1 (18) (360+360+360+360+360+360)/ 6= Rs 360.0

A2 (19) (310+380+380+380+380+380)/6 = Rs 368.3

A3 (20) (260+330+400+400+400+400)/6 = Rs 365.0

A4 (21) (210+280+350+420+420+420)/6 = Rs 350.0

A5 (22) (160+230+300+370+440+440)/6 = Rs 323.3

A6 (23) (110+180+250+320+390+460)/6 = Rs 285.0


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Criterion of Pessimism(Wald decision)(Minimax or Maximin)

Step 1: Find the minimum assured pay off for each

alternative (course of action).

Step 2: Choose that alternative which corresponds to the

maximum of the above minimum pay off.

FOR COSTS

Step 1: Determine the maximum possible cost for each

alternative.


minimum of the above costs.


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Criterion of Pessimism(Maximin or Minimax)

Event Ei

Demanded

copies


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 360 310 260 210 160 110

E2: 19 360 380 330 280 230 180

E3: 20 360 380 400 350 300 250

E4: 21 360 380 400 420 370 320

E5: 22 360 380 400 420 440 390

E6: 23 360 380 400 420 440 460


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Criterion of Optimism(Maximax or Minimin)

Step 1: Determine the maximum possible pay off for each

alternative.

Step 2: Select that alternative which corresponds to the

maximum of the above maximum pay offs.

In decision problems dealing with costs, the minimum for

each alternative is considered and then the alternative which

minimises the above minimum cost is selected. This is

termed as minimin principle.


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Criterion of Optimism(Maximax or Minimin)

Event Ei

Demanded

copies


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 360 310 260 210 160 110

E2: 19 360 380 330 280 230 180

E3: 20 360 380 400 350 300 250

E4: 21 360 380 400 420 370 320

E5: 22 360 380 400 420 440 390

E6: 23 360 380 400 420 440 460


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Criterion of Realism(Hurwicz Criterion)

1. Decide the coefficient of optimism and then the coefficient

of pessimism 1 - .

2. Determine the maximum as well as minimum pay off for each

alternative and obtain the quantities.h = maximum for each alternative + (1 - ) minimum for

each alternative.

3. Select an alternative with maximum value.


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Criterion of Realism(Hurwicz Criterion)

Act Max Min Criterion Value=(Max Value) +(1-)

A1 360 360 0.6 X 360 + 0.4 X 360 360

A2 380 310 0.6 X 380 + 0.4 X 310 352

A3 400 260 0.6 X 400 + 0.4 X 260 344

A4 420 210 0.6 X 420 + 0.4 X 210 336

A5 440 160 0.6 X 440 + 0.4 X 160 328

A6 460 110 0.6 X 460 + 0.4 X 110 320


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Criterion of Regret(savage Criterion)or Minimax Regret Criterion

Step 1: From the given pay off matrix, develop an

opportunity-loss (or regret) matrix.

(i) Find the best pay off corresponding to each state of nature

(maximum for profit and minimum for cost)

(ii)ith regret =(maximum pay off-ith pay off) for the jth event ifthe pay offs represent profits (minimum pay off-ith pay off) for

the jth event if the pay offs represent costs.

Step 2: Determine the maximum regret amount for each

alternative.


minimum regrets.


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Criterion of Regret(savage Criterion)or Minimax Regret Criterion

Event Ei

Demanded

copies


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 0 50 100 150 200 250

E2: 19 20 0 50 100 150 200

E3: 20 40 20 0 50 100 150

E4: 21 60 40 20 0 50 100

E5: 22 80 60 40 20 0 50

E6: 23 100 80 60 40 20 0


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Decision Under Certainty

Since under this environment, only one state of nature exists,

the decision-maker simply picks up the best pay off in that

one column and chooses the associated alternative. Under

conditions of certainty, the particular state of nature is

associated with probability Though the state of nature is only one, possible alternatives

could be numerous. Linear programming, transportation and

assignment techniques, input out analysis, activity analysis and

economic order quantity models are used for such situations.


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Decision-making Under Risk

The decision situations where the decision maker chooses toconsider several possible outcomes and the probabilities of

occurrence can be stated under risk. Probability of

occurrence can be determined from past records.

Suppose the book seller observes from the past sales thatfor the number of copies sold 18,19,20,21,22,23 copies the

the probability 0.05, 0.10, 0.30, 0.40, 0.10, 0.05


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Under condition of risk, there are two criteria: Maximum likelihood principle: for event with highest

probability, choose the act with best pay-off

Expectation principle: choose the act with best

expected pay-off (highest in case of gains and lowest incase of costs)


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Maximum likelihood principle

Event Ei

Demandedcopies

Probability

Pi


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 0.05 360 310 260 210 160 110

E2: 19 0.10 360 380 330 280 230 180

E3: 20 0.30 360 380 400 350 300 250

E4: 21 0.40 360 380 400 420 370 320

E5: 22 0.10 360 380 400 420 440 390

E6: 23 0.05 360 380 400 420 440 460

Expected pay-off 360 376.5 386 374.5 335 288.5


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Expectation principle:

Expected Monetary Value (EMV) criterion

Expected Opportunity Loss (EOL) criterion

Expected Value of Perfect Information (EVPI)

E t d M t V l (EMV)


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Expected Monetary Value (EMV)Criterion

Step 1: List conditional profit for each act-event

combinations, along with the corresponding event

probabilities.

Step 2: For each act, determine the expected conditionalprofits.

Step 3: Determine EMV for each act where

EMV (Aj) = pij Pi

pij - pay off; Pi- probability of occcurrence

Step 4: Choose the act which corresponds to the optimal

EMV.

Expected Payoff or Monetary


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Expected Payoff or Monetaryvalue(EMV/ EPV)

EMV A3 = 0.05X 260 + 0.10X 330 + 0.30 X 400 + 0.40 X 400 + 0.10 X

400 +0.05 X 400 = Rs 386Event Ei

Demande

d copies

Probabilit

y Pi


A1 :18 A2: 19 A3: 20 A4: 21 A5:22A6:

23

E1 :18 0.05 360 310 260 210 160 110

E2: 19 0.10 360 380 330 280 230 180

E3: 20 0.30 360 380 400 350 300 250

E4: 21 0.40 360 380 400 420 370 320

E5: 22 0.10 360 380 400 420 440 390

E6: 23 0.05 360 380 400 420 440 460

Expected pay-off 360 376.5 386 374.5 335 288.5

E t d O t it L


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Expected Oppurtunity Loss(EOL) Criterion

Step 1: Construct the conditional profit table for each act-

event combination, along with the corresponding event

probabilities.

Step 2: For each state of nature calculate the Conditional

Opportunity Loss (COL) values by subtracting each pay offfrom the maximum pay off for that event. Step 3: For each act, determine the expected COL values

and add these values to get the Expected Opportunity Loss

(EOL) for that act where

EOL(Ni) = lij pilij - opppurtunity loss ; pi - probability

Step 4: Choose that act which corresponds to the minimum

EOL value.


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Expected opportunity loss (EOL)

EOL A3 = 0.05X 100 + 0.10X 50 + 0.30 X 0 + 0.40 X 20 + 0.10 X

40 +0.05 X 60 = Rs 25

Event Ei

Demande

d copies

Probability

Pi


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 0.05 0 50 100 150 200 250

E2: 19 0.10 20 0 50 100 150 200

E3: 20 0.30 40 20 0 50 100 150

E4: 21 0.40

60 40 20 0 50 100E5: 22 0.10 80 60 40 20 0 50

E6: 23 0.05 100 80 60 40 20 0

Expected pay-off51 34.5 25 36.5 76 122.5

E t d P ff f P f t


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Expected Pay-off of PerfectInformation (EPPI)

When the bookstore manager knows that next year demand

is 18 copies to obtain profit of 360 with probability 0.05 then

Expected profit = 0.05X 360 =Rs. 18

With demand 19 on 2

nd

year with probability 0.10 thenExpected profit = 0.10X 360 =Rs 38

Similarly for the each level of demand each year;

E t d P ff f P f t


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Expected Pay-off of PerfectInformation (EPPI)

EPPI = 0.05X 360 + 0.10X 360 + 0.30 X 400 + 0.40 X 420 + 0.10

X 440 +0.05 X 460 = Rs 411

Event Ei

Demanded

copies

Probability

Pi


A1 :18 A2: 19 A3: 20 A4: 21 A5:22 A6: 23

E1 :18 0.05360 310 260 210 160 110

E2: 19 0.10 360 380 330 280 230 180

E3: 20 0.30 360 380 400 350 300 250

E4: 21 .040360 380 400 420 370 320

E5: 22 0.10 360 380 400 420 440 390

E6: 23 0.05 360 380 400 420 440 460

E t d V l f P f t


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Expected Value of PerfectInformation (EVPI)

Expected profit with perfect information (EPPI) represents

the expected profit, in the long run, if we have perfect

information before a decision is made.

EVPI = EPPI - Expected profit without information=EPPI -EMV *

where EMV* - maximum expected monetary value (386 in

above case).

EVPI = 411 386 = Rs. 25


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Question

A news paper Boy has following probabilities of selling

magazine:

No. of Copies sold. Probability

10 0.10

11 0.15

12 0.20

13 0.25

14 0.30

Cost of a copy is 30 paise and selling price is 50 paise. Hecan not return the unsold copies? How much should he

order? Find EVPI.


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Decision Trees

Decision tree is a schematic representation of a sequential

and multi-dimensional decision problems.

Decision tree is made of

Nodes, Branches,

probability estimates and

Payoffs


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Decision Trees

Node- indicated by square and represents the place ofdecision making

Branches connects various nodes: has decision branches

and chance branches. Branch that indicate end of decision

tree is terminal branch

Associated Probabilitieslikelihood that the chance outcome

will assume the value assigned to the given branch. It is

represented alongside of respective chance branch

Payoff- positive (sales) or negative (cost) and associated

with decision or chance branches


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Steps In Decision Tree Analysis

Step 1: Identify the decision points and the alternative

courses of action at each decision point systematically. Step 2: At each decision point determine the probability and

the pay off associated with each course of action. Step 3: Commencing from the extreme right end, compute

the expected pay offs (EMV) for each course of action.

Step 4: Choose the course of action that yields the best payoff for each decisions. Step 5: Proceed backwards to the next stage of decision

points. Step 6: Repeat above steps till the first decision point is

reached. Step 7: Finally, identify the course of action to be adopted

from the beginning to the end under different possibleoutcomes for the situation as a whole.


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Decision Trees

Example:

Raman Industries ltd. Has a new product which they expect

a great potential. At the moment they have two course of

action S1= test market the product and S2= Drop theproduct.

Indicate the most preferred decisions.


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Decision Trees

1

2

3

A

B

D

E

F

G

H

C

S2

500,000

100,000

-100,000


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Decision Trees

Node E EMV(E) = 0.2x500,000+0.55x100,000+0.25x(-100,000)=Rs. 130,000

Selection of D = Rs 25,000 (certain payment)

Node 2 Max. payment = 130,000

Node 3 = Rs 25,0000 (Certain payment)Node A EMV (A) = 0.7 x 130,000 + 0.3 x 25000

= Rs. 98,000

Value of Branch (1- A) = 98000 -50,000 = Rs 48,000

Value of Branch (1-B) = 25,000 (Certain payment)

Optimal Decision: Test market the product , if the result is

positive , then market and if negative then drop.


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Decision Trees

Example:

A finance manager is considering drilling a well. In the past,only 70% of wells drilled were successful at 20 meters depth inthat area. Moreover, on finding, no water at 20 meters, somepersons in that area drilled it further up to 25 meters but only20% stuck water at that level. The prevailing cost of drilling isRs. 500 per meter. The finance manager estimated that in case

he does not get water in his own well, he will have to pay Rs.15000 to buy from outside for the same period of getting waterfrom the well.The following decisions are considered:1. Do not drill any well2. Drill up to 20 meters3. If no water is found at 20 meter, drill further up to 20

meters.Draw and appropriate decision tree and determine the optimalsolution

D i i T


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Decision Trees

Decision Tree for Example 13.19 (Vohra)

Buy Water

Drill 20 m

Rs 15,000

Water

0.7Rs 10,000

0.3

No Water

Stop

Rs 15,000 + Rs 500 X 20= Rs 25,000

Rs 25,000

Drill Upto 25 m

Water, 0.2

500X25=

Rs 12500

No Water

Rs 27,500

Decision Node 2

Decision Node 1

A l i T bl


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Operations Research

Analysis Table

Decision Node Options Expected Cost Decision

1 Drill to 25 m

0.8(15,000+1

2,500) +

0.212,500 =

Rs 24,500

Stop Rs 25,000

2 Drill to 20 m

0.324,500+0.7

10,000 = Rs

14,350

Do not Drill Rs 15,000

Optimal Decision: Drill up to 20 m,if no water, then drill up to 25 m

Advantages Of Decision Tree


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Advantages Of Decision TreeApproach

1. It structures the decision process and helps decision-making

in an orderly, systematic and sequential manner.2. It requires the decision-maker to examine all possible

outcomes, whether desirable or undesir- able.3. It communicates the decision-making process to others in an

easy and clear manner, illustrating each assumption about

the future.4. It displays the logical relationship between the parts of acomplex decision and identifies the time sequence in whichvarious actions and subsequent events would occur.

5. It is especially useful in situations wherein the initial decisionand its outcome affects the subsequent decisions.

6. It can be applied in various fields such as introduction of anew product, marketing, make or buy decisions, investmentdecisions and so on.

Limitations Of Decision Tree


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Limitations Of Decision TreeApproach

1. Decision tree diagrams become more complicated as the

number of decision alternative increases and more

variables are introduced.

2. It becomes highly complicated when inter-dependent

alternatives and dependent variables are present in theproblem.

3. It assumes that utility of money is linear with money.

4. It analyses the problem in terms of expected values and

thus yields an average valued solution.

5. There is often inconsistency in assigning probabilities fordifferent events.

2. decision theory

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