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## Decision Theory

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### … Decision Theory

Classification of problems (in decision theory may be divided into five categories):

1.Decision under certainty issues are those in which each alternative action results in one and only one outcome and where that outcome is sure to occur.

2.Decision under probabilistic uncertainty issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with known probabilities. There are outcome uncertainties, and the probabilities associated with these are known precisely.

3.Decision under probabilistic imprecision issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with unknown or imprecisely specified probabilities. There are outcome uncertainties, and the probabilities associated with the uncertainty parameters are not all known precisely.

4.Decision under information imperfection issues are those in which one of several outcomes can result from a given action depending on the state of nature, and these states occur with imperfectly specified probabilities. There are outcome uncertainties, and the probabilities associated with these are not all known precisely. Imperfections in knowledge of the utility of the various event outcomes may exist as well.

5.Decision under conflict and cooperation issues are those in which there is more than a single decision maker, and where the objectives and activities of one decision maker are not necessarily known to all decision makers. Also, the objectives of the decision makers

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### … Decision Theory

Bases of normative decision theory. The general concepts of axiomatic or normative decision theory formalize and rationalize the decision-making process. Normative decision theory depends on the following assumptions:

1.Past preferences are valid indicators of present and future preferences.

2.People correctly perceive the values of the uncertainties that are associated with the outcomes of decision alternatives.

3.People are able to assess decision situations correctly, and the resulting decision situation structural model is well formed and complete.

4.People make decisions that accurately reflect their true preferences over the alternative courses of action, each of which may have uncertain outcomes.

5.People are able to process decision information correctly.

6.Real decision situations provide people with decision alternatives that allow them to express their true preferences.

7.People accept the axioms that are assumed to develop the various normative theories.

8.People make decisions without being so overwhelmed by the complexity of actual decision situations that they would necessarily use suboptimal decision strategies.

Given these necessary assumptions, there will exist departures between normative and descriptive decision theories. A principal task of those aiding others in decision making is to retain those features from the descriptive approach which enable an acceptable transition from normative approaches to prescriptive approaches. The prescriptive features should eliminate potentially undesirable features of descriptive approaches, such as flawed

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### Determination of utility

When choosing among alternatives, the decision maker must be able to indicate preferences among decisions that may result in diverse outcomes. In simple situations when only money is involved, an expected-value approach might be suggested, in which a larger expected amount of money is preferred to a smaller amount. However, in many situations the utility associated with money is not a linear function of the amount of money involved.

According to expected utility theory, the decision maker should seek to choose the alternative aiwhich makes the resulting expected utility the largest possible. The utility uij, of choosing decision ai and obtaining outcome event ej, will also depend upon the particular value of the probabilistically uncertain random variable ej as conditioned on the decision path that is selected. So, the best that the decision maker can do here is to maximize some function, such as the expected value or utility (EU), as shown below, where the maximization is carried out over all alternative decisions, and P(ej | ai) is the probability that the state of nature is ej given that alternative ai is implemented. The notation EU{ai} is often used to mean the expected utility of taking action ai. Generally, this is also called the subjective expected utility (SEU). “Subjective” denotes the fact that the probabilities may be based on subjective beliefs and the utilities may reflect personal consequences.

n

i

ij

i

i

i i j=1

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### … Decision Theory

Example 1: Assume the following probability distribution of daily demand for strawberries:

Also assume that unit cost = $3, selling price =$5 (i.e., profit on sold unit = $2), and salvage value on unsold units =$2 (i.e., loss on unsold unit = $1). We can stock either 0, 1, 2, or 3 units. The question is: How many units should be stocked each day? Assume that units from one day cannot be sold the next day. Then the payoff table can be constructed as follows: *Profit for (stock 2, demand 1) equals (no. Of units sold) (profit per unit) - (no. Of units unsold)(loss per unit) = (1)($5 - 3) - (1)($3 - 2) =$1

**Expected value for (stock 2) is: -2(.2) + 1(.3) + 4(.3) + 4(.2) = \$1.90. The optimal stock action is the one with the highest Expected Monetary Value, i.e., stock 2 units.

Suppose the decision maker can obtain a perfect prediction of which event (state of nature) will occur. The Expected Value With Perfect Information would be the total expected value of actions selected on the assumption of a perfect forecast. Expected value

Daily Demand 0 1 2 3

Probability 0.2 0.3 0.3 0.2

\ Demand Stock \ Probability

State of Nature Expected

Value

0 1 2 3

0.2 0.3 0.3 0.2

Actions

0 0 0 0 0 0

1 -1 2 2 2 1.40

2 -2 1* 4 4 1.90**

3 -3 0 3 6 1.50

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### … Decision Theory

The p-value is the probability under the assumption of null hypothesis.

Example 2: For two sets (A and B)

An informal interpretation of a p-value, based on a significance level

p≤0.01: very strong presumption against null hypothesis

• 0.01<p≤0.05 : strong presumption against null hypothesis

• 0.05<p≤0.1 : low presumption against null hypothesis

p>0.1 : no presumption

against the null hypothesis

Day 1 2 3 4 5 =T.TEST(B3:G3,B4:G4,2,1)

A: 33 35 36 38 39

B: 22 23 22 24 23 0.000104521

A: B:

Mean 36.2 22.8

Variance 5.7 0.7

Observations 5 5

Pearson Correlation 0.650814027 Hypothesized Mean Difference 0

df 4

t Stat 15.37085417

P(T<=t) one-tail 0.000052260 t Critical one-tail 2.131846786 P(T<=t) two-tail 0.000104521 t-Test: Paired Two Sample for Means

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… Decision Theory Values of the t-distribution (two-tailed):

DF

A 0.8 0.9 0.95 0.98 0.99 0.995 0.998 0.999 P 0.2 0.1 0.05 0.02 0.01 0.005 0.002 0.001 1 3.078 6.314 12.706 31.82 63.657 127.32

1

318.30 9

636.61 9 2 1.886 2.92 4.303 6.965 9.925 14.089 22.327 31.599 3 1.638 2.353 3.182 4.541 5.841 7.453 10.215 12.924 4 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.61 5 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6.869 6 1.44 1.943 2.447 3.143 3.707 4.317 5.208 5.959 7 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.408 8 1.397 1.86 2.306 2.897 3.355 3.833 4.501 5.041 9 1.383 1.833 2.262 2.821 3.25 3.69 4.297 4.781 10 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.587

t-Test: Paired Two Sample for Means

A: B:

Mean 36.2 22.8

Variance 5.7 0.7

Observations 5 5

Pearson Correlation 0.65081403 Hypothesized Mean Difference 0

df 4

t Stat 15.3708542

P(T<=t) one-tail 5.226E-05 t Critical one-tail 2.13184679 P(T<=t) two-tail 0.00010452 t Critical two-tail 2.77644511

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### Decision Tree Analysis

(http://www.mindtools.com/dectree.html)

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(11)

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### at each circle.

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… Decision Tree Analysis

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### … Decision Tree Analysis

The benefit calculated for new product, thorough development was 420,400. We estimate the future cost of this approach as 150,000.

This gives a net benefit of 270,400. The net benefit of new product, rapid development was 31,400. We choose the most valuable option and allocate this value to the decision node.

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… Decision Tree Analysis

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::=

::=

### rezolvarea unei probleme

(de conducere, coordonare, reglare, control sau previziune a activităţilor din aria de competenţă a managerului)

### , prin alegerea

(de catre manager)

### o unei soluţii dintre mai multe variante posibile

(de rezolvare a problemei date)

### alternativelor decizionale

(variantele sau soluţiile posibile)

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::=

(19)

(20)

::=

(21)

### • Recunoaşterea problemei - Problemă aparentă

(manifestată prin simptome)

### • Definirea problemei - elemente generale

(decidentul, scopul deciziei, restricţiile)

*

### (

alternativele decizionale, stările naturii, consecinţele, probabilităţile

*

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### … Analiza deciziilor -

Elementele specifice

:

1

2

m

ex.

şi ex.

1

2

n

(numerice)

i

j

ij

ij

ij

ij

j

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### … Analiza deciziilor – Construirea

modelului - Matricea de decizie

1

2

n

1

2

n

1

11

12

1n

2

21

22

2n

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### … Analiza deciziilor – Construirea

modelului - Arborele de decizie

(

)

pn p1 p2 S1

r11 S2

r12 ...

Sn

r1n

pn p1 p2 S1

r21 S2

r22 ...

Sn

r2n A1

A2 ...

p1 p2 S1

rm1 S2

rm2 ...

Am

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Alternative

1

2

3

1

2

i

k

ij

kj

i

k

### ,

indiferent de starea naturii Sj (1  j  n).

k

i

Alternative

1

2

3

1

2

(26)

### MINIMIN (

consecinţele reprezintă ceva

iar scopul este

### minimizarea).

Alternative

Stări – profituri maxiMAX

MAXIMAX Decizia

S1 S2 S3

A1 15 3 -6 15 15 [-6] A1

A 9 4 -2 9

i

i

i

i1

i2

in

1

2

m

k

1

2

m

(27)

### • MINIMAX (consecinţele reprezintă ceva negativ).

Alternative

Stări – profituri maxiMIN

miniMAX

MAXIMIN MINIMAX

Decizia

S1 S2 S3

A 15 3 -6 -6 [15]

i

i

i

i1

i2

in

1

2

m

k

1

2

m

(28)

### … Metode elementare (

nu folosesc probabilităţile P

i

i

i

i

i

i

i

1

2

m

(29)

### … Metode elementare

(Criteriul lui Hurwicz )

### a=0.39

Alternative

Stări – profituri maxi-

MAX

maxi-MIN H Decizia

S1 S2 S3

A1 15 3 -6 15 -6 2.19

A2 9 4 -2 9 -2 2.29 A2

A3 3 2 1 3 1 1.78

### a=0.33

Alternative

Stări – profituri maxi-

MAX

maxi-MIN H Decizia

S1 S2 S3

A1 15 3 -6 15 -6 0.93

A2 9 4 -2 9 -2 1.63

A3 3 2 1 3 1 1.66 A3

a=0.41

Alternative

Decizia

A1 A1

Analiza Senzitivitate (Sensibilitate)

### a

Decizia

0.33 A3

(30)

An. sens. : … Metode elementare (Criteriul lui Hurwicz )

1 2 3 4

Alt (1,2,3):

A:

α: 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46

-0.12 0.09 0.30 0.51 0.72 0.93 1.14 1.35 1.56 1.77 1.98 2.19 2.40 2.61 2.82 3.03 3.24 3.45 3.66 1.08 1.19 1.30 1.41 1.52 1.63 1.74 1.85 1.96 2.07 2.18 2.29 2.40 2.51 2.62 2.73 2.84 2.95 3.06

1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.56 1.58 1.60 1.62 1.64 1.66 1.74 1.85 1.96 2.07 2.18 2.29 2.40 2.61 2.82 3.03 3.24 3.45 3.66

A: 3 3 3 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 1

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### … Metode elementare (

nu folosesc probabilităţile P

ij

i

j

j

i

j

ij

ij

ij

j

### Se aplică regula MINIMAX la matricea OL.

Alterna- tive

Stări – profituri S1 S2 S3

A 15 3 -6

Alterna-

tive

Stări – profituri maxi- MAX

MAXI- MAX

Deci- S1 S2 S3 zia

A1 0 1 7 7

(32)

### … Metode elementare (

foloseşte probabilităţile P

j

j

i

1 i m

1

2

m

### ) } pentru fluxuri pozitive / negative

Alternative Stări – profituri E(Ai) Decizia

1/3 1/3 1/3

A1 15 3 -6 4 A1

A2 9 4 -2 3.67

j

j

, 1im.

n

j

ij j

i

1

(33)

### … Metode elementare (

foloseşte probabilităţile P

j

k

j

1j

2j

mj

### } pentru fluxuri pozitive / negative

Alterna-

tive

Stări – profituri

1/4

1/4

A1 15

-6

A 9

-2

Alternative

Stare

Decizia

A1

A2

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### End of … 4.

• DTREG Software For Predictive Modeling and Forecasting (http://www.dtreg.com/index.htm )

• Decision Tree Forests (http://www.dtreg.com/treeforest.htm?gclid=CIi7sdWI1Z0CFU1_3godpHJysA)

• Bayes Decision Theory: Discrete Features (http://www.cim.mcgill.ca/~friggi/bayes/ )

• Measurement Decision Theory (http://www.sciencecentral.com/site/494630 )

• Decision Theory (http://www.ierd.duth.gr/english/courses/syllabus_decision_makinglecture_e.htm )

• Decision Theory (http://www.ierd.duth.gr/english/courses/syllabus_decision_makinglecture_e.htm )

• Elementary Decision Theory (http://www.ebookee.com/Elementary-Decision-Theory_201022.html )

• Planning Algorithms, Steven M. LaValleCambridge University Press, , 2006 (http://planning.cs.uiuc.edu/ )

• Pdf & Doc book … decision theory pdf (http://pdfdatabase.com/index.php?q=decision+theory+pdf )

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