Decision Theory

Decision Theory

C_ 4 / 22.10.2019

A broad range of concepts which have been developed to both describe and prescribe the process of decision making, where a choice is made from a finite set of possible alternatives. Normative decision theory describes how decisions should be made in order to accommodate a set of axioms believed to be desirable; descriptive decision theory deals with how people actually make decisions;

and prescriptive decision theory formulates how decisions should be

made in realistic settings. Thus, this field of study involves people

from various disciplines: behavioral and social scientists and

psychologists who generally attempt to discover elaborate

descriptive models of the decision process of real humans in real

settings; mathematicians and economists who are concerned with

the axiomatic or normative theory of decisions; and engineers and

managers who may be concerned with sophisticated prescriptive

… 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

… 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

… Decision Theory

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 a_iwhich makes the resulting expected utility the largest possible. The utility u_ij, of choosing decision a_i and obtaining outcome event e_j, will also depend upon the particular value of the probabilistically uncertain random variable e_j 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(e_j | a_i) is the probability that the state of nature is e_j given that alternative a_i is implemented. The notation EU{a_i} is often used to mean the expected utility of taking action a_i. 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

Max EU{a

} = Max Σ ^u

^P(e

^{| a}

⁾

i i j=1

… Decision Theory

Systematic approach to making decisions especially under uncertainty.

Although statistics such as Expected Value and Standard Deviation are essential

for choosing the best course of action, the decision problem can best be

approached, using what is referred to as a payoff table (or decision matrix), which

is characterized by: (1) the row representing a set of alternative Courses of Action

available to the decision maker; (2) the column representing the State of Nature or

conditions that are likely to occur and over which the decision maker has no

control; and (3) the entries in the body of the table representing the outcome of the

decision, known as payoffs, which may be in the form of costs, revenues, profits,

or cash flows. By computing expected value of each action, we will be able to pick

the best one.

… 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

… 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

of about 10%, might be:

• 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

… 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

Decision Tree Analysis

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

Decision Trees are useful tools for helping you to choose between several courses of action:

• They provide a highly effective structure within which you can explore options, and investigate the possible outcomes of choosing those options.

• They also help you to form a balanced picture of the risks and rewards associated with each possible course of action.

• This makes them particularly useful for choosing between

different strategies, projects or investment opportunities,

particularly when your resources are limited

… Decision Tree Analysis

The Decision Tree start with the decision that you need to make, drawing a square to represent this on the left hand side.

From this box draw out lines towards the right for each possible solution, and write a short description of the solution along the line.

At the end of each line, consider the results. If the result of taking that decision is uncertain, draw a circle. If the result is another decision that you need to make, draw another square (squares represent decisions, and circles represent uncertain outcomes). Write the decision or factor above the square or circle.

Starting from the new decision squares, draw out lines representing

the options that you could select, and so on.

… Decision Tree Analysis

Example Decision Tree :

Should we develop a new

product or consolidate?

… Decision Tree Analysis

Evaluating the Decision Tree :

• Start by assigning a cash value or score to each possible outcome.

• Estimate the probability of each outcome :

•percentages

total = 100 % ,

•fractions

total = 1 ,

at each circle.

… Decision Tree Analysis

Calculating Tree Values:

Start on the right side of the decision tree, and work back towards the left.

As you complete a set of calculations on a node (decision square or uncertainty circle), then record the result.

You can ignore all the calculations that lead to that result from then on.

0.4 1,000,000 400,000

0.4 50,000 20,000

0.2 2,000 400

1 + ^420,400

… 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.

… Decision Tree Analysis

The best option is to develop a new product. It is worth much more to us to take our time and get the product right, than to rush the product to market. And it's better just to improve our existing products than to botch a new product, even though it costs us less.

Decision trees provide an effective method of decision making because they:

• Clearly lay out the problem so that all options can be challenged.

• Allow us to analyze the possible consequences of a decision fully.

• Provide a framework to quantify the values of outcomes and the probabilities of achieving them.

• Help us to make the best decisions on the basis of existing information and best guesses.

As with all decision making methods, decision tree analysis should be

used in conjunction with common sense - decision trees are just one

Teoria Deciziilor

Managerul trebuie sa ia decizii eficiente!

Funcţiile manageriale - planificarea, organizarea, leadership-ul şi controlul - implică luarea unor decizii eficiente.

• La decizii strategice se preferă să se recurgă la experienţă şi intuiţie (deciziile executive nu se pretează la abordări cantitative deoarece ele sunt caracterizate de aspecte calitative).

• La decizii tactice (operative) deciziile se pot programa, cuantificarea fiind posibilă / utilă.

Modelarea matematică permite folosirea analizei deciziilor în

procesul decizional - permite rezolvarea de probleme complexe, în

care factorii de incertitudine şi risc sunt luaţi în considerare.

Elementele Teoriei Deciziilor

Decizia

acţiunea sau procesul de alegere sau selectare a unei alternative din mai multe posibile.

Rolul deciziei

rezolvarea unei probleme

, prin alegerea

o unei soluţii dintre mai multe variante posibile

.

Sistemul decizional este constituit din ansamblul deciziilor de conducere elaborate, adoptate şi aplicate în cadrul instituţii.

Activitatea managerilor este o înlănţuire de decizii interdependente.

Componentele unei decizii: decidentul, obiectivele, mulţimea

alternativelor decizionale

, mulţimea

… Elementele Teoriei Deciziilor

Variantele decizionale

alternativele posibile.

Criteriile stabilesc obiectivele de atins (profitul - maxim , costul -

minim , cheltuielile - minime , … Clasificări ale deciziilor:

• strategice, tactice şi curente - timpul pentru care se adoptă;

• în condiţii de certitudine, de risc şi de incertitudine - gradul de cunoaştere a datelor de intrare şi consecinţelor ;

• individuale sau de grup – numărul de persoane care participă la luarea deciziei;

• structurate, semistructurate şi nestructurate - structura problemei de decizie;

• unicriteriale şi multicriteriale - numărul criteriilor de decizie .

… Elementele Teoriei Deciziilor

Etapele procesului decizional tradiţional:

• identificarea şi definirea problemei de rezolvat (deciziei) - Ce trebuie făcut pentru a rezolva problema? ;

• stabilirea obiectivelor şi criteriilor decizionale - Obiectivul sau scopul unui proces decizional este analizat în funcţie de criteriul ales ;

• culegerea informaţiilor - are ca scop certificarea faptelor relevante şi se poate reduce la o problemă de căutare - informaţiile trebuie să fie exacte, operative şi prezentate sugestiv ;

• construirea variantelor (soluţiilor) posibile - generarea de alternative realiste posibile ;

• evaluarea variantelor şi alegerea variantei optime – se compară avantajele şi dezavantajele fiecărei alternative, rezultând variante posibile dintre care se alege varianta optima - cea care satisface cel mai bine criteriile alese ;

• comunicarea şi aplicarea (implementarea) deciziei;

• controlul (urmărirea, monitorizarea) aplicării deciziei şi evaluarea

Analiza deciziilor

Analiza deciziilor

o abordare raţională a procesului decizional, foloseşte un model formal pentru reprezentarea alternativelor şi criteriilor decizionale în scopul luării unei decizii optime - când riscul este semnificativ.

Analiza deciziilor permite decidentului să abordeze probleme de decizie caracterizate de incertitudine. Ea construieşte un model normativ pentru reprezentarea problemei de decizie, care uşurează analiza ulterioară a sa şi produce o decizie bazată pe considerente de ordin obiectiv. Modelul formal obţinut este capabil să genereze strategii optimale pentru probleme de decizie în mai multe etape.

Analiza deciziilor se bazează pe separarea elementelor

controlabile de cele necontrolabile - face distincţie între acţiunile pe

care decidentul le poate lua şi circumstanţele care sunt în afara

controlului acestuia.

… Analiza deciziilor

Etapele analizei deciziilor:

• Recunoaşterea problemei - Problemă aparentă

sau problemă reală ? Problemă pozitivă sau negativă?

• Definirea problemei - elemente generale

, elemente specifice

(

)

• Construirea modelului – prototip (matricea de decizie / arborele de decizie)

• Culegerea datelor necesare - constante, parametri şi variabile

• Execuţia modelului – efectuare calcule – reguli de decizie

• Analiza rezultatelor obţinute - stabilirea deciziei (acţiunii prescrise), analiza sensibilităţii (de senzitivitate)

• Interpretarea rezultatelor - determină maxime sau minime bazate pe

structura modelului şi ipotezele de lucru

… Analiza deciziilor -

Elementele specifice problemei

structurarea modelului :

• Alternativele decizionale ( variantele de acţiune

exclusive şi exhaustive) : A = {A

, A

, ..., A

}.

• Stările naturii (situaţii în funcţie de care se analizează fiecare alternativă,

şi ex.

) :

S = {S

, S

, ..., S

}.

• Consecinţele - măsuri cantitative

ale alegerii unei alternative A

combinată cu apariţia unei stări S

:

R = {r

, 1  i  m; 1  j  n}.

unde: r

reprezintă câştigul net (r

> 0) sau pierderea netă (r

< 0).

• Probabilităţile - asociate stărilor S

(1  j  n) caracterizează incertitudinea apariţiei acestora:

P = {p , p , ..., p }.

… Analiza deciziilor – Construirea

Construirea modelului

Matricea de decizie :

• modalitate tabelară de reprezentare a elementelor problemei de decizie.

Liniile sale reprezintă alternativele, iar coloanele stările:

D = {A, S, R, P}.

Matricea de decizie Alternativele

decizionale

Stările naturii

p

p

... p

S

S

... S

A

r

r

... r

A

r

r

... r

... ... ... ... ...

… Analiza deciziilor – Construirea

Construirea modelului

Arborele de decizie :

• modalitate grafică de reprezentare a elementelor problemei de decizie.

Nodurile pot fi de decizie ( □ ) sau de stare

○

:

D = {A, S, R, P}.

p_n p₁ p₂ S₁

r₁₁ S₂

r₁₂ ...

S_n

r_1n

p_n p₁ p₂ S₁

r₂₁ S₂

r₂₂ ...

S_n

r_2n A₁

A₂ ...

p₁ p₂ S₁

r_m1 S₂

r_m2 ...

A_m

Metode monocriteriale de analiză a deciziilor

1. Metode elementare ( fără probabilităţi ) 2. Metode bazate pe valoarea medie

Alternative

Stări – profituri

S

S

S

A

15 3 -6

A

9 4 -2

A 3 2 1

O alternativă A

se numeşte dominantă pentru A

dacă r

r

pentru toate coloanele j (1 j n) - consecinţele pentru alternativa A

sunt întotdeauna mai bune decât cele pentru A

,

Reciproc, spunem că alternativa A

este dominată de alternativa A

.

Alternative

Stări – profituri

S

S

S

A

15 3 -6

A

9 4 -2

A 3 2 1

Alternativele

dominate

trebuie

eliminate

… Metode elementare (nu folosesc probabilităţile P )

Criteriul optimist - descrie comportamentul decizional al unui optimist atras de câştigurile mai mari, dispus să rişte oricât pentru a le obţine.

Se poate modela cu regula:

• MAXIMAX (consecinţele referă ceva pozitiv, iar scopul este maximizarea),

MINIMIN (

negativ,

minimizarea).

Alternative

Stări – profituri maxiMAX

S₁ S₂ S₃

A₁ 15 3 -6 15 15 [-6] A₁

A 9 4 -2 9

Regula MAXIMAX :

Determină consecinţa maximă maxiMAX

pentru fiecare alternativă A

: maxiMAX

= max {r

, r

, ..., r

} (1  i  m).

Dintre consecinţele maxiMAX={maxiMAX

, maxiMAX

, ..., maxiMAX

} se selectează cea mai mare:

MAXIMAX

= max {maxiMAX

, maxiMAX

, ..., maxiMAX

}.

… Metode elementare ⁽ nu folosesc probabilităţile P )

Criteriul pesimist - descrie comportamentul decizional al unui pesimist speriat de pierderile mari, care ignoră câştigurile atractive cu riscuri mari.

Se poate modela cu regula:

• MAXIMIN (consecinţele referă ceva pozitiv),

• MINIMAX (consecinţele reprezintă ceva negativ).

Alternative

Stări – profituri maxiMIN

miniMAX

MAXIMIN MINIMAX

Decizia

S₁ S₂ S₃

A 15 3 -6 -6 [15]

Regula MAXIMIN :

Determină consecinţa minimă maxiMIN

pentru fiecare alternativă A

: maxiMIN

= min {r

, r

, ..., r

} (1  i  m).

Dintre consecinţele maxiMIN={maxiMIN

, maxiMIN

, ..., maxiMIN

} se selectează cea mai mare:

MAXIMIN

= max {maxiMIN

, maxiMIN

, ..., maxiMIN

}.

… Metode elementare ⁽

)

Criteriul lui Hurwicz - descrie un comportament aflat între optimist şi pesimist printr-o combinaţie ponderată a acestora. Pentru fiecare alternativă se se va calcula o combinaţie cu un coeficient 0a1 de realism (a=optimism, 1-a=pesimism) :

a ∙ maxiMax

+ (1-a) ∙ maxiMin

pentru fluxuri pozitive H(A

) =

a ∙ miniMin

+ (1-a) ∙ miniMax

pentru fluxuri negative

Regula Hurwicz :



Alege un coeficient de optimism a ;



Determină consecinţa ponderată H(A

) pentru fiecare alternativă A

;

Dintre consecinţele H={H(A

), H(A

), ..., H(A

)} se selectează cea mai bună astfel:

max (H) pentru fluxuri pozitive

min (H) pentru fluxuri negative

… Metode elementare

a=0.39

Alternative

Stări – profituri maxi-

MAX

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

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

0.33 A₃

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

… Metode elementare ⁽

)

Criteriul de regret MiniMax al lui (Leonard Jimmie) Savage –

minimizarea regretelor reultate dintr-o alegere nepotrivită. Regretul ol

se defineşte ca pierderea de avantaje suferită prin alegerea alternativei A

şi apariţia stării S

(diferenţa dintre cel mai bun câştig posibil din starea S

şi cel obţinut prin alegerea alternativei A

):

Max(S

) - r

pentru fluxuri pozitive ol

=

r

- Min(S

) pentru fluxuri negative

Regula Savage :



Se construieşte matricea regretelor OL din matricea consecinţelor R.



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

… Metode elementare ⁽

)

Regula Laplace :



Se atribuie probabilităţile p

= 1/n la stările naturii S

( 1  j  n ).



Se calculează valoarea aşteptată E(A

) pentru fiecare alternativă (

).



Se alege alternativa cu cea mai bună valoare aşteptată:

max / min { E(A

), E(A

), ..., E(A

) } pentru fluxuri pozitive / negative

Alternative Stări – profituri E(A_i) Decizia

1/3 1/3 1/3

A₁ 15 3 -6 4 A₁

A₂ 9 4 -2 3.67

Criteriul motivaţiei insuficiente - Laplace – Principiul motivaţiei insuficiente: dacă decidentul n-a atribuit stărilor probabilităţi de apariţie, atunci se consideră că toate stările naturii S

( 1  j  n ) sunt egal probabile, ceea ce înseamnă că p

= 1/n, ( 1  j  n ).

Criteriul lui Laplace foloseşte valoarea medie 







j

ij j

i

p r

A E

1

)

(

… Metode elementare ⁽

)

Regula verosimilităţii maximale (modală) :



Se selectează starea S

( 1  j  n ) cu şansa maximă.



Coloanele stărilor S

(1  k  n, k  j) se exclud din matricea de decizie.



Se alege alternativa cu cea mai bună consecinţă din coloana S

: max / min {r

, r

, ..., r

} pentru fluxuri pozitive / negative

Alterna-

tive

Stări – profituri

1/4

1/2

A₁ 15

3

A 9

4

Criteriul modal (al verosimilităţii maximale) – ia în considerare doar starea cu şansă maximă de realizare ( Verosimilitatea Maximală - modal

de la modul distribuţiei statistice).

Alternative

Stare

Decizia

1/2

A₁

3

A₂

4 A

• Metode bazate pe valoarea medie

(expected value)

…

• Metode multicriteriale de analiză a deciziilor Electre

…

… Next

^Dss_5

End of … 4.

Resources and Links:

• 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 ⁾

• Decision Theory Free Download - windows software (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 )