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Decisi o: agregaci o i consens Vicen c Torra Universitat de Sk ovde (HiS, Su` ecia) Novembre, 2015 Bibliografia (i/o spam) Bibliografia Gilboa, I., Theory of decision under uncertainty, Cambridge University Press, 2009.

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SLIDE 1

Decisi´

  • : agregaci´
  • i consens

Vicen¸ c Torra

Universitat de Sk¨

  • vde (HiS, Su`

ecia) Novembre, 2015

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SLIDE 2

Bibliografia (i/o spam)

  • Bibliografia

– Gilboa, I., Theory of decision under uncertainty, Cambridge University Press, 2009. – Rapoport, A., Decision Theory and Decision Behaviour, Dordrecht, Kluwer Academic Publishers, 1989. – Webb, J.N, Game Theory: Decisions, Interaction and Evolution, Berl´ ın, Springer, 2007.

  • Bibliografia pr`
  • pia

– Torra, V., Narukawa, Y. (2007) Modeling decisions: Information fusion and aggregation operators, Springer – Torra, V., Narukawa, Y. (2007) Modelitzaci´

  • de decisions: Fusi´
  • d’informaci´
  • i
  • peradors d’agegaci´
  • , Edicions UAB. (traducci´
  • de l’anterior)

– Torra, V. (2015) Cuando las matem´ aticas van a las urnas. Los procesos de decisi´

  • n,

RBA.

UdG 2015 1 / 102

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SLIDE 3

´ Index

  • Presa de decisions

– Presa de decisions multicriteri

  • Funcions d’agregaci´
  • : una introducci´
  • Agregaci´
  • de funcions d’utilitat (num`

eriques) – De la mitjana ponderada a les integrals difuses – Models jer` arquics

  • Agregaci´
  • de relacions de prefer`

encia

  • Resum de temes relacionats

Vicen¸ c Torra; Modeling decisions UdG 2015 2 / 102

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SLIDE 4

Introducci´

  • Presa de decisions

– Triar entre diverses alternatives

  • Un exemple: (un exemple cl`

assic) – Volem comprar un cotxe i hi ha diversos models – Alternatives: {Peugeot308, FordT., . . . }

  • Altres exemples: el problema del presoner, escollir un moviment en els

jocs, etc.

Vicen¸ c Torra; Modeling decisions UdG 2015 3 / 102

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SLIDE 5

Introducci´

  • Marc general de la presa de decisions

– Caracter´ ıstiques del problema

  • Diverses alternatives
  • {Peugeot308, FordT., . . . }

Vicen¸ c Torra; Modeling decisions UdG 2015 4 / 102

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SLIDE 6

Introducci´

  • Marc general de la presa de decisions

– Dificultats del problema

  • Criteris en contradicci´
  • Incertesa i risc
  • Adversari

Vicen¸ c Torra; Modeling decisions UdG 2015 5 / 102

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SLIDE 7

Introducci´

  • Marc general en la presa de decisions

– Dificultat: Criteris en contradicci´

  • No es possible trobar una alternativa que satisfagi tots els criteris
  • Un cotxe barat, assequible per`
  • potser no tan confortable
  • Preu vs. seguretat i confort

Vicen¸ c Torra; Modeling decisions UdG 2015 6 / 102

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SLIDE 8

Introducci´

  • Marc general en la presa de decisions

– Dificultat: Incertesa i risc

  • Coneixem o no l’efecte de la nostra acci´
  • Quan escollum un cotxe sabem el seu preu i la capacitat del

portaequipatges

  • Quan comprem un bitllet de loteria, no sabem si guanyarem
  • Quan el metge proposa un tractament, no est`

a segur del seu efecte

Vicen¸ c Torra; Modeling decisions UdG 2015 7 / 102

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SLIDE 9

Introducci´

  • Marc general en la presa de decisions

– Dificultat: Decisions amb adversari

  • La nostra decisi´
  • cal confrontar-se amb la dels oponents
  • Jocs amb adversari:

darrera el nostre moviment hi ha el de l’adversari

Vicen¸ c Torra; Modeling decisions UdG 2015 8 / 102

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SLIDE 10

Introducci´

  • Algunes notes sobre el marc general de la presa de decisions

– Incertesa vs. risc: conceptes diferents

Vicen¸ c Torra; Modeling decisions UdG 2015 9 / 102

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SLIDE 11

Introducci´

  • Algunes notes sobre el marc general de la presa de decisions

– Incertesa vs. risc: conceptes diferents

  • Decisi´
  • sota risc:

∗ Cada acci´

  • porta a diversos estats amb probabilitats conegudes

· Cas de la loteria · Cas dels jocs (amb daus)

Vicen¸ c Torra; Modeling decisions UdG 2015 9 / 102

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SLIDE 12

Introducci´

  • Algunes notes sobre el marc general de la presa de decisions

– Incertesa vs. risc: conceptes diferents

  • Decisi´
  • sota risc:

∗ Cada acci´

  • porta a diversos estats amb probabilitats conegudes

· Cas de la loteria · Cas dels jocs (amb daus)

  • Decisi´
  • sota incertesa:

∗ Les probabilitats s´

  • n desconegudes o no s´
  • n comparables

· Cas del metge ∗ No ´ unicament probabilitats, tamb´ e informaci´

  • vaga o imprecisa

· Una mica de febre: al voltant de 38?

Vicen¸ c Torra; Modeling decisions UdG 2015 9 / 102

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SLIDE 13

Introducci´

  • Marc general en la presa de decisions: classificaci´
  • (0)

– Presa de decisions amb certesa – Presa de decisions amb incertesa i risc – Presa de decisions amb adversari

Vicen¸ c Torra; Modeling decisions UdG 2015 10 / 102

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SLIDE 14

Introducci´

  • Marc general en la presa de decisions: classificaci´
  • (Ia)

– Presa de decisions amb certesa ∗ Presa de decisions: · Diverses alternatives, cadascuna d’elles avaluada d’acord amb diversos criteris. Efectes de la decisi´

  • sense incertesa.

· Exemple. Alternatives (cotxes) i criteris (preu, confort, etc.)

Vicen¸ c Torra; Modeling decisions UdG 2015 11 / 102

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SLIDE 15

Introducci´

  • Marc general en la presa de decisions: classificaci´
  • (Ia)

– Presa de decisions amb certesa ∗ Presa de decisions: · Diverses alternatives, cadascuna d’elles avaluada d’acord amb diversos criteris. Efectes de la decisi´

  • sense incertesa.

· Exemple. Alternatives (cotxes) i criteris (preu, confort, etc.) · Nombre finit d’alternatives: presa de decisions multicriteri · Nombre infinit d’alternatives: presa de decisions multiobjectiu

Vicen¸ c Torra; Modeling decisions UdG 2015 11 / 102

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SLIDE 16

Introducci´

  • Marc general en la presa de decisions: classificaci´
  • (Ia)

– Presa de decisions amb certesa ∗ Presa de decisions: · Diverses alternatives, cadascuna d’elles avaluada d’acord amb diversos criteris. Efectes de la decisi´

  • sense incertesa.

· Exemple. Alternatives (cotxes) i criteris (preu, confort, etc.) · Nombre finit d’alternatives: presa de decisions multicriteri · Nombre infinit d’alternatives: presa de decisions multiobjectiu – MCDA: Eines per capturar, entendre, analitzar les difer` encies (punt de vista constructivista) – MCDM: Eines per a descriure el proc´ es de decisi´

  • . Se suposa que es

pot formalitzar. (punt de vista descriptiu)

Vicen¸ c Torra; Modeling decisions UdG 2015 11 / 102

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SLIDE 17

Introducci´

  • Marc general en la presa de decisions: classificaci´
  • (Ib)

– Presa de decisions amb certesa ∗ Multicriteria Decision Aid (MCDA): finit / punt de vista descriptiu / modelitzaci´

  • ∗ Multicriteria Decision Making (MCDM):

finit / punt de vista constructivista ∗ Multiobjective Decision Making (MODM): infinit

Vicen¸ c Torra; Modeling decisions UdG 2015 12 / 102

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SLIDE 18

Introducci´

  • Exemple. Multiobjective decision making:

nombre infinit d’alternatives – Selecci´

  • de les quantitats de carb´
  • (entre dos tipus de carb´
  • ) per a

la generaci´

  • d’electricitat (Dallenbach, 1994, p.314).

Vicen¸ c Torra; Modeling decisions UdG 2015 13 / 102

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SLIDE 19

Introducci´

  • Exemple. Multiobjective decision making:

nombre infinit d’alternatives – Selecci´

  • de les quantitats de carb´
  • (entre dos tipus de carb´
  • ) per a

la generaci´

  • d’electricitat (Dallenbach, 1994, p.314).
  • Vapor m`

axim (producci´

  • )? Benefici m`

axim?

  • Hem de tenir en compte les restriccions
  • Cada carb´
  • t´

e els seus inconvenients (emissions diferents)

  • No poden generar-se massa emissions (exc´

es) ∗ Formulaci´

  • /resoluci´
  • mitjan¸

cant optimitzaci´

  • (e.g., Simplex)

Vicen¸ c Torra; Modeling decisions UdG 2015 13 / 102

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SLIDE 20

Introducci´

  • Exemple. Multicriteria decision making:

nombre finit d’alternatives – Volem comprar un cotxe

  • Alternatives: {Peugeot308, FordT., . . . }
  • Punts de vista/criteris: Preu, qualitat, confort

→ representem les nostres prefer` encies sobre les alternatives

Vicen¸ c Torra; Modeling decisions UdG 2015 14 / 102

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SLIDE 21

Introducci´

  • n
  • Marc general de la presa de decisions: classificaci´
  • (III)

– Presa de decisions amb adversari

  • Jocs est`

atics: els jugadors actuen a la vegada Teoria de jocs (game theory), jocs no cooperatius, jocs cooperatius

  • Jocs din`

amics: els jugadors actuen sequencialment Algorismes de jocs (minimax, poda α-β – I.A.)

Vicen¸ c Torra; Modeling decisions UdG 2015 15 / 102

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SLIDE 22

MCDM: Multicriteria decision making

Vicen¸ c Torra; Modeling decisions UdG 2015 16 / 102

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SLIDE 23

MCDM

  • Preference representation

– Utility functions.

  • A function for each criterion
  • The function is applied to each alternative
  • The value of the function is larger, if the satistaction is larger

(the larger the satisfaction, the larger the value)

Vicen¸ c Torra; Modeling decisions UdG 2015 17 / 102

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SLIDE 24

MCDM

  • Preference representation

– Utility functions.

  • A function for each criterion
  • The function is applied to each alternative
  • The value of the function is larger, if the satistaction is larger

(the larger the satisfaction, the larger the value) – Preference relations (compare two altenatives)

  • Binary relation for each criterion
  • Each relation orders the alternatives

Vicen¸ c Torra; Modeling decisions UdG 2015 17 / 102

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SLIDE 25

MCDM

  • Preference representation

– Utility functions.

  • A function for each criterion
  • The function is applied to each alternative
  • The value of the function is larger, if the satistaction is larger

(the larger the satisfaction, the larger the value) – Preference relations (compare two altenatives)

  • Binary relation for each criterion
  • Each relation orders the alternatives
  • Utility functions are (or can be seen as) a mathematical description of

preference relations

Vicen¸ c Torra; Modeling decisions UdG 2015 17 / 102

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SLIDE 26

MCDM

  • Representation of preferences

– Utility functions.

  • Ford T: Uprecio = 0.2, Ucalidad = 0.8, Uconfort = 0.3
  • Peugeot308: Uprecio = 0.7, Ucalidad = 0.7, Uconfort = 0.8

– Preference relations (comparison between pairs of alternatives)

  • Rprecio: Rprecio(P308, FordT), ¬Rprecio(FordT, P308)
  • Rcalidad: ¬Rcalidad(P308, FordT), Rcalidad(FordT, P308)
  • Rconfort: Rconfort(P308, FordT), ¬Rconfort(FordT, P308)

Vicen¸ c Torra; Modeling decisions UdG 2015 18 / 102

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SLIDE 27

MCDM

  • Preference representation

– Example. Preference relations.

Number of Security Price Confort trunk1 seats Ford T + ++ + ++ + Seat 600 +++ + +++++ + +++ Simca 1000 +++++ +++ ++++ ++++ ++++ VW Beetle ++++ +++++ ++ +++++ +++++ Citro¨ en Acadiane ++ ++++ +++ +++ ++

Vicen¸ c Torra; Modeling decisions UdG 2015 19 / 102

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SLIDE 28

MCDM

  • Preference representation

– Example. Utility functions.

Number of Security Price Confort trunk2 seats Ford T 20 20 Seat 600 60 100 50 Simca 1000 100 30 100 50 70 VW Beetle 80 50 30 70 100 Citro¨ en Acadiane 20 40 60 40

Vicen¸ c Torra; Modeling decisions UdG 2015 20 / 102

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SLIDE 29

MCDM

  • Preference representation: Preference relations

– Formalization: Reference set X Properties (for all x, y, z) ∗ Binary relation: I.e., a subset R ⊆ X × X ∗ We denote by x ≥ y if and only if (x, y) ∈ R ∗ Total or complet relation: x ≥ y o y ≥ x ∗ Transitive relation: x ≥ y, y ≥ z entonces x ≥ z ∗ Reflexive relation: x ≥ x

Vicen¸ c Torra; Modeling decisions UdG 2015 21 / 102

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SLIDE 30

MCDM

  • Preference representation: Preference relations

– Formalization: Reference set X Properties (for all x, y, z) ∗ Binary relation: I.e., a subset R ⊆ X × X ∗ We denote by x ≥ y if and only if (x, y) ∈ R ∗ Total or complet relation: x ≥ y o y ≥ x ∗ Transitive relation: x ≥ y, y ≥ z entonces x ≥ z ∗ Reflexive relation: x ≥ x – Definition: (in decision making) A relation is a rational preference relation if it is total, transitive and reflexive. – in mathematics: a total preorder

Vicen¸ c Torra; Modeling decisions UdG 2015 21 / 102

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SLIDE 31

MCDM

  • Preference representation

– Example. Preference relation.

Number of Security Price Confort trunk3 seats Ford T + ++ + ++ + Seat 600 +++ + +++++ + +++ Simca 1000 +++++ +++ ++++ ++++ ++++ VW Beetle ++++ +++++ ++ +++++ +++++ Citro¨ en Acadiane ++ ++++ +++ +++ ++

Vicen¸ c Torra; Modeling decisions UdG 2015 22 / 102

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SLIDE 32

MCDM

  • Preference representation: Utility functions

– Formalization: Reference set X

  • U : X → D for a given domain D

– Representation: A utility u represents a preference ≥ when for all x, y ∈ X when x ≥ y if and only if u(x) ≥ u(y).

Vicen¸ c Torra; Modeling decisions UdG 2015 23 / 102

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SLIDE 33

MCDM

  • Preference representation: Utility functions

– Formalization: Reference set X

  • U : X → D for a given domain D

– Representation: A utility u represents a preference ≥ when for all x, y ∈ X when x ≥ y if and only if u(x) ≥ u(y).

  • Example. For the price, the utility does not represent the relation

It is true uprecio(Simca1000) ≥ uprecio(Seat600) but it is false Simca 1000 ≥ Seat 600

Vicen¸ c Torra; Modeling decisions UdG 2015 23 / 102

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SLIDE 34

MCDM

  • Preference representation: Utility functions

– Formalization: Reference set X

  • U : X → D for a given domain D

– Representation: A utility u represents a preference ≥ when for all x, y ∈ X when x ≥ y if and only if u(x) ≥ u(y).

  • Example. For the price, the utility does not represent the relation

It is true uprecio(Simca1000) ≥ uprecio(Seat600) but it is false Simca 1000 ≥ Seat 600

– Relation: We can establish a relationship between utilities and preference relations

Vicen¸ c Torra; Modeling decisions UdG 2015 23 / 102

slide-35
SLIDE 35

MCDM

  • Preference representation: Utility functions

– Formalization: Reference set X

  • U : X → D for a given domain D

– Representation: A utility u represents a preference ≥ when for all x, y ∈ X when x ≥ y if and only if u(x) ≥ u(y).

  • Example. For the price, the utility does not represent the relation

It is true uprecio(Simca1000) ≥ uprecio(Seat600) but it is false Simca 1000 ≥ Seat 600

– Relation: We can establish a relationship between utilities and preference relations

  • Theorem. Given a set of alternatives, there exist a utility function

that represents the preference relation if and only if the preference relation is rational.

Vicen¸ c Torra; Modeling decisions UdG 2015 23 / 102

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SLIDE 36

MCDM

  • Preference representation: Utility functions

– Example: definition for price

  • Maximum budget 10000 euros.
  • Less that 1000 is perfect
  • Lineal function between 1000 and 10000

up(x) =      100 if x ≤ 1000 (10000 − x)/90 if x ∈ (1000, 10000) if x ≥ 10000

Vicen¸ c Torra; Modeling decisions UdG 2015 24 / 102

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SLIDE 37

MCDM

  • Preference representation: Utility functions

– Example: definition for trunk capacity Not always is a monotonic relationship

  • f utility with respect to the values of a criterion
  • The trunk is optimal for 1 m3.
  • Neither too small, nor too large

um(x) =      if x ≤ 0.8 100 − 500|x − 1| if x ∈ (0.8, 1.2) if x ≥ 1.2

Vicen¸ c Torra; Modeling decisions UdG 2015 25 / 102

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SLIDE 38

MCDM

  • Decision

– Modeling the problem: representation of the criteria – Aggregation – Selection of alternatives

Vicen¸ c Torra; Modeling decisions UdG 2015 26 / 102

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SLIDE 39

MCDM

  • Aggregation, it depends on the representation for preferences

– Utility functions

  • Ford T: Uprecio = 0.2, Ucalidad = 0.8, Uconfort = 0.3

∗ Given utilities, we aggregate them – Preference relationships (comparison between pairs of alternatives)

  • Rprecio: Rprecio(P308, FordT), ¬Rprecio(FordT, P308)
  • Rcalidad: ¬Rcalidad(P308, FordT), Rcalidad(FordT, P308)

∗ Given preference relations, we aggregate them

Vicen¸ c Torra; Modeling decisions UdG 2015 27 / 102

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SLIDE 40

MCDM

  • Decision for preference relations

Modelling, aggregation, selection

Number of Security Price Confort trunk4 Aggregated seats Preference Ford T + ++ + ++ + + Seat 600 +++ + +++++ + +++ ++ Simca 1000 +++++ +++ ++++ ++++ ++++ ++++ VW Beetle ++++ +++++ ++ +++++ +++++ +++++

  • Citr. Acadiane

++ ++++ +++ +++ ++ +++ Vicen¸ c Torra; Modeling decisions UdG 2015 28 / 102

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SLIDE 41

MCDM

  • Decision for utility functions

Modelling, aggregation = AM, selection

Number of Security Price Confort trunk5 Aggregated seats Utility Ford T 20 20 8 Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66

  • Citr. Acadiane

20 40 60 40 32 Vicen¸ c Torra; Modeling decisions UdG 2015 29 / 102

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SLIDE 42

Aggregation functions

Vicen¸ c Torra; Modeling decisions UdG 2015 30 / 102

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SLIDE 43

Aggregation functions

  • Outline

– Introduction – Aggregation for (numerical) utility functions: basics – A tour on (numerical) aggregation: from WM to Fuzzy integrals – Aggregation for preference relations

Vicen¸ c Torra; Modeling decisions UdG 2015 31 / 102

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SLIDE 44

Aggregation functions: an introduction

Vicen¸ c Torra; Modeling decisions UdG 2015 32 / 102

slide-45
SLIDE 45

Aggregation functions

  • Aggregation and information fusion

– In our case, how to combine information about criteria

  • In general,

– it is a broad area, with different types of applications

Vicen¸ c Torra; Modeling decisions UdG 2015 33 / 102

slide-46
SLIDE 46

Aggregation functions

  • Aggregation and information fusion

– In our case, how to combine information about criteria

  • In general,

– it is a broad area, with different types of applications

  • Examples of aggregation functions:

– N

i=1 ai/N (AM arithmetic mean)

– N

i=1 pi · ai (WM weighted mean)

Vicen¸ c Torra; Modeling decisions UdG 2015 33 / 102

slide-47
SLIDE 47

Aggregation functions

  • Aggregation and information fusion

– In our case, how to combine information about criteria

  • In general,

– it is a broad area, with different types of applications

  • Examples of aggregation functions:

– N

i=1 ai/N (AM arithmetic mean)

– N

i=1 pi · ai (WM weighted mean)

  • Different functions, lead to different results

– In our case, different orderings, different selections!

Vicen¸ c Torra; Modeling decisions UdG 2015 33 / 102

slide-48
SLIDE 48

Aggregation functions

  • Goal of aggregation functions (in general, not restricted to MCDM):

– To produce a specific datum, and exhaustive, on an entity – Datum produced from information supplied by different information sources (or the same source over time) – Techniques to reduce noise, increase precision, summarize information, extract information, make decisions, etc.

Vicen¸ c Torra; Modeling decisions UdG 2015 34 / 102

slide-49
SLIDE 49

Aggregation functions

  • Information fusion studies . . .

. . . all aspects related to combining information:

  • Goals of data aggregation (goals of the area):

Vicen¸ c Torra; Modeling decisions UdG 2015 35 / 102

slide-50
SLIDE 50

Aggregation functions

  • Information fusion studies . . .

. . . all aspects related to combining information:

  • Goals of data aggregation (goals of the area):

– Formalization of the aggregation process

  • Definition of new functions
  • Selection of functions

(methods to decide which is the most appropriate function in a given context)

  • Parameter determination

Vicen¸ c Torra; Modeling decisions UdG 2015 35 / 102

slide-51
SLIDE 51

Aggregation functions

  • Information fusion studies . . .

. . . all aspects related to combining information:

  • Goals of data aggregation (goals of the area):

– Formalization of the aggregation process

  • Definition of new functions
  • Selection of functions

(methods to decide which is the most appropriate function in a given context)

  • Parameter determination

– Study of existing methods:

  • Caracterization of functions
  • Determination of the modeling capabilities of the functions
  • Relation between operators and parameters

(how parameters influence the result: can be achieve dictatorship?, sensitivity to data → index).

Vicen¸ c Torra; Modeling decisions UdG 2015 35 / 102

slide-52
SLIDE 52

Aggregation functions

  • Terms:

– Information integration – Information fusion: concrete functions / techniques

concrete process to combine several data into a single datum.

– Aggregation functions: C : DN → D (C from Consensus)

→ i C with parameters (background knowledge): CP

Vicen¸ c Torra; Modeling decisions UdG 2015 36 / 102

slide-53
SLIDE 53

Aggregation functions

  • Terms:

– Information integration – Information fusion: concrete functions / techniques

concrete process to combine several data into a single datum.

– Aggregation functions: C : DN → D (C from Consensus)

→ i C with parameters (background knowledge): CP

  • Aggregation functions: basic properties

Vicen¸ c Torra; Modeling decisions UdG 2015 36 / 102

slide-54
SLIDE 54

Aggregation functions

  • Terms:

– Information integration – Information fusion: concrete functions / techniques

concrete process to combine several data into a single datum.

– Aggregation functions: C : DN → D (C from Consensus)

→ i C with parameters (background knowledge): CP

  • Aggregation functions: basic properties

– Unanimity and idempotency: C(a, . . . , a) = a for all a

Vicen¸ c Torra; Modeling decisions UdG 2015 36 / 102

slide-55
SLIDE 55

Aggregation functions

  • Terms:

– Information integration – Information fusion: concrete functions / techniques

concrete process to combine several data into a single datum.

– Aggregation functions: C : DN → D (C from Consensus)

→ i C with parameters (background knowledge): CP

  • Aggregation functions: basic properties

– Unanimity and idempotency: C(a, . . . , a) = a for all a – Monotonicity: C(a1, . . . , aN) ≥ C(a′

1, . . . , a′ N), if ai ≥ a′ i

Vicen¸ c Torra; Modeling decisions UdG 2015 36 / 102

slide-56
SLIDE 56

Aggregation functions

  • Terms:

– Information integration – Information fusion: concrete functions / techniques

concrete process to combine several data into a single datum.

– Aggregation functions: C : DN → D (C from Consensus)

→ i C with parameters (background knowledge): CP

  • Aggregation functions: basic properties

– Unanimity and idempotency: C(a, . . . , a) = a for all a – Monotonicity: C(a1, . . . , aN) ≥ C(a′

1, . . . , a′ N), if ai ≥ a′ i

– Symmetry: For all permutation π over {1, . . . , N} C(a1, . . . , aN) = C(aπ(1), . . . , aπ(N))

Vicen¸ c Torra; Modeling decisions UdG 2015 36 / 102

slide-57
SLIDE 57

Aggregation functions

  • Terms:

– Information integration – Information fusion: concrete functions / techniques

concrete process to combine several data into a single datum.

– Aggregation functions: C : DN → D (C from Consensus)

→ i C with parameters (background knowledge): CP

  • Aggregation functions: basic properties

– Unanimity and idempotency: C(a, . . . , a) = a for all a – Monotonicity: C(a1, . . . , aN) ≥ C(a′

1, . . . , a′ N), if ai ≥ a′ i

– Symmetry: For all permutation π over {1, . . . , N} C(a1, . . . , aN) = C(aπ(1), . . . , aπ(N)) – Unanimity + monotonicity → internality: mini ai ≤ C(a1, . . . , aN) ≤ maxi ai

Vicen¸ c Torra; Modeling decisions UdG 2015 36 / 102

slide-58
SLIDE 58

Aggregation functions

Definition of aggregation functions:

  • Definition from properties

properties − → function

  • Heuristic definition

properties ← − function

  • Definition from examples

examples − → function

Vicen¸ c Torra; Modeling decisions UdG 2015 37 / 102

slide-59
SLIDE 59

Aggregation functions

  • Definition from properties

properties − → function

Vicen¸ c Torra; Modeling decisions UdG 2015 38 / 102

slide-60
SLIDE 60

Aggregation functions

  • Definition from properties

properties − → function

  • Some ways

a) Using functional equations

Vicen¸ c Torra; Modeling decisions UdG 2015 38 / 102

slide-61
SLIDE 61

Aggregation functions

  • Definition from properties

properties − → function

  • Some ways

a) Using functional equations b) Aggregation of a1, a2, . . . , aN ∈ D, as the datum c which is at a minimum distance from ai: C(a1, a2, . . . , aN) = arg min

c {

  • ai

d(c, ai)}, d is a distance over D.

Vicen¸ c Torra; Modeling decisions UdG 2015 38 / 102

slide-62
SLIDE 62

Aggregation functions

  • Example (case (a)): Functional equations

– Cauchy equation φ(x + y) = φ(x) + φ(y) – find φ !

Vicen¸ c Torra; Modeling decisions UdG 2015 39 / 102

slide-63
SLIDE 63

Aggregation functions

  • Example (case (a)): Functional equations

– Cauchy equation φ(x + y) = φ(x) + φ(y) – find φ ! – φ(x) = αx for an arbitrary value for α

Vicen¸ c Torra; Modeling decisions UdG 2015 39 / 102

slide-64
SLIDE 64

Aggregation functions

  • Example (case (a)): Functional equations

– distribute s euros among m projects according to the opinion of N experts

Proj 1 Proj 2 · · · Proj j · · · Proj m E1 x1

1

x1

2

· · · x1

j

· · · x1

m

E2 x2

1

x2

2

· · · x2

j

· · · x2

m

. . . . . . . . . . . . Ei xi

1

xi

2

· · · xi

j

· · · xi

m

. . . . . . . . . . . . EN xN

1

xN

2

· · · xN

j

· · · xN

m

DM f1(x1) f2(x2) · · · fj(xj) · · · fm(xm)

Vicen¸ c Torra; Modeling decisions UdG 2015 40 / 102

slide-65
SLIDE 65

Aggregation functions

  • The general solution of the system (Proposition 3.11) for a given m > 2

fj : [0, s]N → R+ for j = {1, · · · , m} (1)

m

  • j=1

xj = s implies that

m

  • j=1

fj(xj) = s (2) fj(0) = 0 for j = 1, · · · , m (3) is given by

Vicen¸ c Torra; Modeling decisions UdG 2015 41 / 102

slide-66
SLIDE 66

Aggregation functions

  • The general solution of the system (Proposition 3.11) for a given m > 2

fj : [0, s]N → R+ for j = {1, · · · , m} (1)

m

  • j=1

xj = s implies that

m

  • j=1

fj(xj) = s (2) fj(0) = 0 for j = 1, · · · , m (3) is given by f1(x) = f2(x) = · · · = fm(x) = f((x1, x2, . . . , xN)) =

N

  • i=1

αixi, (4) where α1, · · · , αN are nonnegative constants satisfying N

i=1 αi = 1,

but are otherwise arbitrary.

Vicen¸ c Torra; Modeling decisions UdG 2015 41 / 102

slide-67
SLIDE 67

Aggregation functions

  • Example (case (b)): Consider the following expression

C(a1, a2, . . . , aN) = arg min

c {

  • ai

d(c, ai)}, where ai are numbers from R and d is a distance on D. Then,

Vicen¸ c Torra; Modeling decisions UdG 2015 42 / 102

slide-68
SLIDE 68

Aggregation functions

  • Example (case (b)): Consider the following expression

C(a1, a2, . . . , aN) = arg min

c {

  • ai

d(c, ai)}, where ai are numbers from R and d is a distance on D. Then,

  • 1. When d(a, b) = (a − b)2, C is the arithmetic mean

I.e., C(a1, a2, . . . , aN) = N

i=1 ai/N.

  • 2. When d(a, b) = |a − b|, C is the median

I.e., the median of a1, a2, . . . , aN is the element which occupies the central position when we order ai.

  • 3. When d(a, b) = 1 iff a = b, C is the plurality rule (mode or voting).

I.e., C(a1, a2, . . . , aN) selects the element of R with a largest frequency among elements in (a1, a2, . . . , aN).

Vicen¸ c Torra; Modeling decisions UdG 2015 42 / 102

slide-69
SLIDE 69

Aggregation for (numerical) utility functions

Vicen¸ c Torra; Modeling decisions UdG 2015 43 / 102

slide-70
SLIDE 70

Aggregation for (numerical) utility functions

  • Decision for utility functions

Modelling, aggregation = C, selection

Seats Security Price Comfort trunk C = AM Ford T 20 20 8 Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66

  • Citr. Acadiane

20 40 60 40 32

Vicen¸ c Torra; Modeling decisions UdG 2015 44 / 102

slide-71
SLIDE 71

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria

Vicen¸ c Torra; Modeling decisions UdG 2015 45 / 102

slide-72
SLIDE 72

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria
  • But there are occasions in which ordering is clear

when ai ≤ bi it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70

Vicen¸ c Torra; Modeling decisions UdG 2015 45 / 102

slide-73
SLIDE 73

Aggregation for (numerical) utility functions

  • MCDM: Aggregation to deal with contradictory criteria
  • But there are occasions in which ordering is clear

when ai ≤ bi it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 100 50 42 Simca 1000 100 30 100 50 70 70

  • Pareto dominance:

Given two vectors a = (a1, . . . , an) and b = (b1, . . . , bn), we say that b dominates a when ai ≤ bi for all i and there is at least one k such that ak < bk.

Vicen¸ c Torra; Modeling decisions UdG 2015 45 / 102

slide-74
SLIDE 74

Aggregation for (numerical) utility functions

  • Pareto set, Pareto frontier, or non dominance set:

Seats Security Price Comfort trunk C = AM Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66

  • Citr. Acadiane

20 40 60 40 32

  • Each one wins at least in one criteria to another one

Vicen¸ c Torra; Modeling decisions UdG 2015 46 / 102

slide-75
SLIDE 75

Aggregation for (numerical) utility functions

  • Pareto set, Pareto frontier, or non dominance set:

Given a set of alternatives U represented by vectors u = (u1, . . . , un), the Pareto frontier is the set u ∈ U such that there is no other v ∈ U such that v dominates u. PF = {u|there is no v s.t. v dominates u}

  • Pareto optimal: an element u of the Pareto set

x1 f1(x2) f1(x1) f1 f2 f2(x2) f2(x1) x2

Vicen¸ c Torra; Modeling decisions UdG 2015 47 / 102

slide-76
SLIDE 76

Aggregation for (numerical) utility functions

  • MCDM: we aggregate utility, and order according to utility
  • The function of aggregation functions
  • Different aggregations lead to different orders
  • Aggregation establishes which points are equivalent
  • Different aggregations, establish different curves of points (level

curves)

Ranking alt alt Consensus alt Criteria Satisfaction on: Price Quality Comfort FordT 206 0.2 0.8 0.3 0.7 0.7 0.8 FordT 206 FordT 206 0.35 0.72 0.72 0.35 ... ... ... ... ... ... x1 f1(x2) f1(x1) f1 f2 f2(x2) f2(x1) x2

Vicen¸ c Torra; Modeling decisions UdG 2015 48 / 102

slide-77
SLIDE 77

Aggregation for (numerical) utility functions

  • Why alternatives to the arithmetic mean?

– Not all criteria are equally important (security and comfort) – There are mandatory requirements (price below a threshold) – Compensation among criteria – Interactions among criteria

Vicen¸ c Torra; Modeling decisions UdG 2015 49 / 102

slide-78
SLIDE 78

Aggregation: from the weighted mean to fuzzy integrals

Vicen¸ c Torra; Modeling decisions UdG 2015 50 / 102

slide-79
SLIDE 79

Aggregation: from the weighted mean to fuzzy integrals An example

Vicen¸ c Torra; Modeling decisions UdG 2015 51 / 102

slide-80
SLIDE 80

Aggregation: example

  • Example. A and B teaching a tutorial+training course w/ constraints
  • The total number of sessions is six.
  • Professor A will give the tutorial, which should consist of about three

sessions; three is the optimal number of sessions; a difference in the number of sessions greater than two is unacceptable.

  • Professor B will give the training part,

consisting of about two sessions.

  • Both professors should give more or less the same number of sessions.

A difference of one or two is half acceptable; a difference of three is unacceptable.

Vicen¸ c Torra; Modeling decisions UdG 2015 52 / 102

slide-81
SLIDE 81

Aggregation: example

  • Example. Formalization
  • Variables

– xA: Number of sessions taught by Professor A – xB: Number of sessions taught by Professor B

  • Constraints

– the constraints are translated into ∗ C1: xA + xB should be about 6 ∗ C2: xA should be about 3 ∗ C3: xB should be about 2 ∗ C4: |xA − xB| should be about 0 – using fuzzy sets, the constraints are described ...

Vicen¸ c Torra; Modeling decisions UdG 2015 53 / 102

slide-82
SLIDE 82

Aggregation: example

  • Example. Formalization
  • Constraints

– if fuzzy set µ6 expresses “about 6,” then, we evaluate “xA + xB should be about 6” by µ6(xA + xB). → given µ6, µ3, µ2, µ0, – Then, given a solution pair (xA, xB), the degrees of satisfaction: ∗ µ6(xA + xB) ∗ µ3(xA) ∗ µ2(xB) ∗ µ0(|xA − xB|)

Vicen¸ c Torra; Modeling decisions UdG 2015 54 / 102

slide-83
SLIDE 83

Aggregation: example

  • Example. Formalization
  • Membership functions for constraints

1 2 3 4 5 6 7 µ0 µ2 µ3 µ6

Vicen¸ c Torra; Modeling decisions UdG 2015 55 / 102

slide-84
SLIDE 84

Aggregation: example

  • Example. Application

alternative Satisfaction degrees Satisfaction degrees (xA, xB) (µ6(xA + xB), µ3(xA), C1 C2 C3 C4 µ2(xB), µ0(|xA − xB|)) (2, 2) (µ6(4), µ3(2), µ2(2), µ0(0)) 0.5 1 1 (2, 3) (µ6(5), µ3(2), µ2(3), µ0(1)) 0.5 0.5 0.5 0.5 (2, 4) (µ6(6), µ3(2), µ2(4), µ0(2)) 1 0.5 0.5 (3.5, 2.5) (µ6(6), µ3(3.5), µ2(2.5), µ0(1)) 1 0.5 0.5 0.5 (3, 2) (µ6(5), µ3(3), µ2(2), µ0(1)) 0.5 1 1 0.5 (3, 3) (µ6(6), µ3(3), µ2(3), µ0(0)) 1 1 0.5 1

Vicen¸ c Torra; Modeling decisions UdG 2015 56 / 102

slide-85
SLIDE 85

Aggregation: from the weighted mean to fuzzy integrals WM, OWA, and WOWA operators

Vicen¸ c Torra; Modeling decisions UdG 2015 57 / 102

slide-86
SLIDE 86

Aggregation: WM, OWA, and WOWA operators

  • Operators

– Weighting vector (dimension N): v = (v1...vN) iff vi ∈ [0, 1] and

i vi = 1

– Arithmetic mean (AM :RN → R): AM(a1, ..., aN) = (1/N) N

i=1 ai

– Weighted mean (WM: RN → R): WMp(a1, ..., aN) = N

i=1 piai

(p a weighting vector of dimension N) – Ordered Weighting Averaging operator (OWA: RN → R): OWAw(a1, ..., aN) =

N

  • i=1

wiaσ(i), where {σ(1), ..., σ(N)} is a permutation of {1, ..., N} s. t. aσ(i−1) ≥ aσ(i), and w a weighting vector.

Vicen¸ c Torra; Modeling decisions UdG 2015 58 / 102

slide-87
SLIDE 87

Aggregation: WM, OWA, and WOWA operators

  • Example. Application
  • Let us consider the following situation:

– Professor A is more important than Professor B – The number of sessions equal to six is the most important constraint (not a crisp requirement) – The difference in the number of sessions taught by the two professors is the least important constraint WM with p = (p1, p2, p3, p4) = (0.5, 0.3, 0.15, 0.05).

Vicen¸ c Torra; Modeling decisions UdG 2015 59 / 102

slide-88
SLIDE 88

Aggregation: WM, OWA, and WOWA operators

  • Example. Application
  • WM with p = (p1, p2, p3, p4) = (0.5, 0.3, 0.15, 0.05).

alternative Aggregation of the Satisfaction degrees WM (xA, xB) WMp(C1, C2, C3, C4) (2, 2) WMp(0, 0.5, 1, 1) 0.35 (2, 3) WMp(0.5, 0.5, 0.5, 0.5) 0.5 (2, 4) WMp(1, 0.5, 0, 0.5) 0.675 (3.5, 2.5) WMp(1, 0.5, 0.5, 0.5) 0.75 (3, 2) WMp(0.5, 1, 1, 0.5) 0.725 (3, 3) WMp(1, 1, 0.5, 1) 0.925

Vicen¸ c Torra; Modeling decisions UdG 2015 60 / 102

slide-89
SLIDE 89

Aggregation: WM, OWA, and WOWA operators

  • Example. Application
  • Compensation: how many values can have a bad evaluation
  • One bad value does not matter: OWA with w = (1/3, 1/3, 1/3, 0)

(lowest value discarded) alternative Aggregation of the Satisfaction degrees OWA (xA, xB) OWAw(C1, C2, C3, C4) (2, 2) OWAw(0, 0.5, 1, 1) 0.8333 (2, 3) OWAw(0.5, 0.5, 0.5, 0.5) 0.5 (2, 4) OWAw(1, 0.5, 0, 0.5) 0.6666 (3.5, 2.5) OWAw(1, 0.5, 0.5, 0.5) 0.6666 (3, 2) OWAw(0.5, 1, 1, 0.5) 0.8333 (3, 3) OWAw(1, 1, 0.5, 1) 1.0

Vicen¸ c Torra; Modeling decisions UdG 2015 61 / 102

slide-90
SLIDE 90

Aggregation: WM, OWA, and WOWA operators

  • Weighted Ordered Weighted Averaging WOWA operator

(WOWA :RN → R):

WOWAp,w(a1, ..., aN) = N

i=1 ωiaσ(i)

where ωi = w∗(

j≤i pσ(j)) − w∗( j<i pσ(j)),

with σ a permutation of {1, ..., N} s. t. aσ(i−1) ≥ aσ(i), and w∗ a nondecreasing function that interpolates the points {(i/N,

j≤i wj)}i=1,...,N ∪ {(0, 0)}.

w∗ is required to be a straight line when the points can be interpolated in this way.

Vicen¸ c Torra; Modeling decisions UdG 2015 62 / 102

slide-91
SLIDE 91

Aggregation: WM, OWA, and WOWA operators

  • Construction of the w∗ quantifier
1= N 1= N ::: 1= N p
  • (1)
p
  • (2)
p
  • (N
) w 2 w N w 2 w N w 1 w 1 ! 1 p
  • (1)
p
  • (1)
p
  • (1)
p
  • (1)
  • !
1 (a) (b) ( )
  • Rationale for new weights (ωi, for each value ai) in terms of p and w.

– If ai is small, and small values have more importance than larger

  • nes, increase pi for ai (i.e., ωi ≥ pσ(i)).

(the same holds if the value ai is large and importance is given to large values)

– If ai is small, and importance is for large values, ωi < pσ(i)

(the same holds if ai is large and importance is given to small values).

Vicen¸ c Torra; Modeling decisions UdG 2015 63 / 102

slide-92
SLIDE 92

Aggregation: WM, OWA, and WOWA operators

  • The shape of the function w∗ gives importance

– (a) to large values – (b) to medium values – (c) to small values – (d) equal importance to all values

(a) (b) (c) (d)

Vicen¸ c Torra; Modeling decisions UdG 2015 64 / 102

slide-93
SLIDE 93

Aggregation: WM, OWA, and WOWA operators

  • Example. Application
  • Importance for constraints as given above: p = (0.5, 0.3, 0.15, 0.05)
  • Compensation as given above: w = (1/3, 1/3, 1/3, 0) (lowest value

discarded) → WOWA with p and w. alternative Aggregation of the Satisfaction degrees WOWA (xA, xB) WOWAp,w(C1, C2, C3, C4) (2, 2) WOWAp,w(0, 0.5, 1, 1) 0.4666 (2, 3) WOWAp,w(0.5, 0.5, 0.5, 0.5) 0.5 (2, 4) WOWAp,w(1, 0.5, 0, 0.5) 0.8333 (3.5, 2.5) WOWAp,w(1, 0.5, 0.5, 0.5) 0.8333 (3, 2) WOWAp,w(0.5, 1, 1, 0.5) 0.8 (3, 3) WOWAp,w(1, 1, 0.5, 1) 1.0

Vicen¸ c Torra; Modeling decisions UdG 2015 65 / 102

slide-94
SLIDE 94

Aggregation: WM, OWA, and WOWA operators

  • Properties

– The WOWA operator generalizes the WM and the OWA operator.

  • When p = (1/N . . . 1/N), OWA

WOWAp,w(a1, ..., aN) = OWAw(a1, ..., aN) for all w and ai.

  • When w = (1/N ... 1/N), WM

WOWAp,w(a1, ..., aN) = WMp(a1, ..., aN) for all p and ai.

  • When w = p = (1/N ... 1/N), AM

WOWAp,w(a1, ..., aN) = AM(a1, ..., aN)

Vicen¸ c Torra; Modeling decisions UdG 2015 66 / 102

slide-95
SLIDE 95

Aggregation: from the weighted mean to fuzzy integrals Choquet integral

Vicen¸ c Torra; Modeling decisions UdG 2015 67 / 102

slide-96
SLIDE 96

Choquet integrals

  • In WM, we combine ai w.r.t. weights pi.

→ ai is the value supplied by information source xi. Formally

Vicen¸ c Torra; Modeling decisions UdG 2015 68 / 102

slide-97
SLIDE 97

Choquet integrals

  • In WM, we combine ai w.r.t. weights pi.

→ ai is the value supplied by information source xi. Formally – X = {x1, . . . , xN} is the set of information sources – f : X → R+ the values supplied by the sources → then ai = f(xi) Thus, WMp(a1, ..., aN) =

N

  • i=1

piai =

N

  • i=1

pif(xi) = WMp(f(x1), ..., f(xN))

Vicen¸ c Torra; Modeling decisions UdG 2015 68 / 102

slide-98
SLIDE 98

Choquet integrals

  • In the WM, a single weight is used for each element

I.e., pi = p(xi) (where, xi is the information source that supplies ai) → when we consider a set A ⊂ X, weight ofA???

Vicen¸ c Torra; Modeling decisions UdG 2015 69 / 102

slide-99
SLIDE 99

Choquet integrals

  • In the WM, a single weight is used for each element

I.e., pi = p(xi) (where, xi is the information source that supplies ai) → when we consider a set A ⊂ X, weight ofA??? . . . fuzzy measures µ(A) Formally, – Fuzzy measure (µ : ℘(X) → [0, 1]), a set function satisfying (i) µ(∅) = 0, µ(X) = 1 (boundary conditions) (ii) A ⊆ B implies µ(A) ≤ µ(B) (monotonicity)

Vicen¸ c Torra; Modeling decisions UdG 2015 69 / 102

slide-100
SLIDE 100

Choquet integrals

  • Now, we have a fuzzy measure µ(A)

then, how aggregation proceeds? ⇒ fuzzy integrals as the Choquet integral

Vicen¸ c Torra; Modeling decisions UdG 2015 70 / 102

slide-101
SLIDE 101

Choquet integrals

  • Choquet integral of f w.r.t. µ (alternative notation, CIµ(a1, . . . , aN)/CIµ(f))

(C)

  • fdµ =

N

  • i=1

[f(xs(i)) − f(xs(i−1))]µ(As(i)), where s in f(xs(i)) is a permutation so that f(xs(i−1)) ≤ f(xs(i)) for i ≥ 1, f(xs(0)) = 0, and As(k) = {xs(j)|j ≥ k} and As(N+1) = ∅.

  • Alternative expressions (Proposition 6.18):

(C)

  • fdµ =

N

  • i=1

f(xσ(i))[µ(Aσ(i)) − µ(Aσ(i−1))], (C)

  • fdµ =

N

  • i=1

f(xs(i))[µ(As(i)) − µ(As(i+1))],

where σ is a permutation of {1, . . . , N} s.t. f(xσ(i−1)) ≥ f(xσ(i)), where Aσ(k) = {xσ(j)|j ≤ k} for k ≥ 1 and Aσ(0) = ∅

Vicen¸ c Torra; Modeling decisions UdG 2015 71 / 102

slide-102
SLIDE 102

Choquet integrals

  • Different equations point out different aspects of the CI

(6.1) (C)

  • fdµ = N

i=1[f(xs(i)) − f(xs(i−1))]µ(As(i)),

µ(As(1)) = {xs(1), · · · , xs(N)} µ(As(4)) = {xs(4), · · · , xs(N)} µ(As(2)) as(1) as(2) as(3) as(4) as(5)

(6.2) (C)

  • fdµ = N

i=1 f(xσ(i))[µ(Aσ(i)) − µ(Aσ(i−1))],

Vicen¸ c Torra; Modeling decisions UdG 2015 72 / 102

slide-103
SLIDE 103

Choquet integrals

  • fdµ =

(for additive measures)

(6.5)

x∈X f(x)µ({x})

(6.6) R

i=1 biµ({x|f(x) = bi})

(6.7) N

i=1(ai − ai−1)µ({x|f(x) ≥ ai})

(6.8) N

i=1(ai − ai−1)

  • 1 − µ({x|f(x) ≤ ai−1})
  • bi

bi−1 ai ai−1 bi bi−1 x1 x1 x1 xN xN x {x|f(x) ≥ ai} {x|f(x) = bi} (a) (b) (c)

  • Among (6.5), (6.6) and (6.7), only (6.7) satisfies internality.

Vicen¸ c Torra; Modeling decisions UdG 2015 73 / 102

slide-104
SLIDE 104

Choquet integrals

  • Properties of CI

– Horizontal additive because CIµ(f) = CIµ(f ∧ c) + CIµ(f +

c )

(f = (f ∧ c) + f +

c is a horizontal additive decomposition of f)

where, f +

c is defined by (for c ∈ [0, 1])

f +

c =

  • if f(x) ≤ c

f(x) − c if f(x) > c.

f +

c

f ∧ c f c

Vicen¸ c Torra; Modeling decisions UdG 2015 74 / 102

slide-105
SLIDE 105

Choquet integrals

  • Definitions (X a reference set, f, g functions f, g : X → [0, 1])

– f < g when, for all xi, f(xi) < g(xi) – f and g are comonotonic if, for all xi, xj ∈ X, f(xi) < f(xj) imply that g(xi) ≤ g(xj) – C is comonotonic monotone if and only if, for comonotonic f and g, f ≤ g imply that C(f) ≤ C(g) – C is comonotonic additive if and only if, for comonotonic f and g, C(f + g) = C(f) + C(g)

  • Characterization. Let C satisfy the following properties

– C is comonotonic monotone – C is comonotonic additive – C(1, . . . , 1) = 1 Then, there exists µ s.t. C(f) is the CI of f w.r.t. µ.

Vicen¸ c Torra; Modeling decisions UdG 2015 75 / 102

slide-106
SLIDE 106

Choquet integrals

  • Properties

– WM, OWA and WOWA are particular cases of CI. ∗ WM with weighting vector p is a CI w.r.t. µp(B) =

xi∈B pi

∗ OWA with weighting vector w is a CI w.r.t. µw(B) = |B|

i=1 wi

∗ WOWA with w.v. p and w is a CI w.r.t. µp,w(B) = w∗(

xi∈B pi)

– Any symmetric CI is an OWA operator. – Any CI with a distorted probability is a WOWA operator. – Let A be a crisp subset of X; then, the Choquet integral of A with respect to µ is µ(A).

Here, the integral of A corresponds to the integral of its characteristic function,

  • r, in other words, to the integral of the function fA defined as fA(x) = 1 if and
  • nly if x ∈ A.

Vicen¸ c Torra; Modeling decisions UdG 2015 76 / 102

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SLIDE 107

Aggregation: from the weighted mean to fuzzy integrals Weighted minimum and maximum

Vicen¸ c Torra; Modeling decisions UdG 2015 77 / 102

slide-108
SLIDE 108

Weighted Minimum and Weighted Maximum

  • Possibilistic weighting vector (dimension N): v = (v1...vN) iff

vi ∈ [0, 1] and maxi vi = 1.

  • Weighted minimum (WMin: [0, 1]N → [0, 1]):

WMinu(a1, ..., aN) = mini max(neg(ui), ai)

(alternative definition can be given with v = (v1, . . . , vN) where vi = neg(ui))

  • Weighted maximum (WMax: [0, 1]N → [0, 1]):

WMaxu(a1, ..., aN) = maxi min(ui, ai)

Vicen¸ c Torra; Modeling decisions UdG 2015 78 / 102

slide-109
SLIDE 109

Weighted Minimum and Weighted Maximum

  • Exemple 6.34. Evaluation of the alternatives related to the course

– Weighting vector (possibilistic vector): u = (1, 0.5, 0.3, 0.1). – WMin:

∗ sat(2, 2) = WMinu(0, 0.5, 1, 1) = 0 ∗ sat(2, 3) = WMinu(0.5, 0.5, 0.5, 0.5) = 0.5 ∗ sat(2, 4) = WMinu(1, 0.5, 0, 0.5) = 0.5 ∗ sat(3.5, 2.5) = WMinu(1, 0.5, 0.5, 0.5) = 0.5 ∗ sat(3, 2) = WMinu(0.5, 1, 1, 0.5) = 0.5 ∗ sat(3, 3) = WMinu(1, 1, 0.5, 1) = 0.7.

– WMax: (with neg(u) = (0, 0.5, 0.7, 0.9), using neg(x) = 1 − x)

∗ sat(2, 2) = WMaxu(0, 0.5, 1, 1) = 0.5 ∗ sat(2, 3) = WMaxu(0.5, 0.5, 0.5, 0.5) = 0.5 ∗ sat(2, 4) = WMaxu(1, 0.5, 0, 0.5) = 1 ∗ sat(3.5, 2.5) = WMaxu(1, 0.5, 0.5, 0.5) = 1 ∗ sat(3, 2) = WMaxu(0.5, 1, 1, 0.5) = 0.5 ∗ sat(3, 3) = WMaxu(1, 1, 0.5, 1) = 1.

– weighted minimum, the best pair is (3, 3); with weighted maximum (3, 3), (2, 4) and (3, 5, 2, 5) indistinguishable

Vicen¸ c Torra; Modeling decisions UdG 2015 79 / 102

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SLIDE 110

Weighted Minimum and Weighted Maximum

  • Exemple 6.35. Fuzzy inference system

Ri: IF x is Ai THEN y is Bi.

– with disjunctive rules, the (fuzzy) output for a particular y0 is a WMax ˜ B(y0) = ∨N

i=1

  • Bi(y0) ∧ Ai(x0)
  • .

– with conjunctive rules, and Kleene-Dienes implication (I(x, y) = max(1 − x, y)) the (fuzzy) output of the system for a particular y0 is a WMin ˜ B(y0) = ∧N

i=1

  • I(Ai(x0), Bi(y0))
  • = ∧N

i=1 max(1 − Ai(x0), Bi(y0)).

that with u = (A1(x0), . . . , AN(x0)) ˜ B(y0) = WMinu(B1(y0), . . . , BN(y0)).

Vicen¸ c Torra; Modeling decisions UdG 2015 80 / 102

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SLIDE 111

Weighted Minimum and Weighted Maximum

  • Only operators in ordinal scales (max, min, neg) are used in WMax

and WMin.

  • neg is completely determined in an ordinal scale

Proposition 6.36. Let L = {l0, . . . , lr} with l0 <L l1 <L · · · <L lr; then, there exists

  • nly one function, neg : L → L, satisfying

(N1) if x <L x′ then neg(x) >L neg(x′) for all x, x′ in L. (N2) neg(neg(x)) = x for all x in L. This function is defined by neg(xi) = xr−i for all xi in L

  • Properties. For u = (1, . . . , 1)

– WMINu = min – WMAXu = max

Vicen¸ c Torra; Modeling decisions UdG 2015 81 / 102

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SLIDE 112

Aggregation: from the weighted mean to fuzzy integrals Sugeno integral

Vicen¸ c Torra; Modeling decisions UdG 2015 82 / 102

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SLIDE 113

Sugeno integral

  • Sugeno integral of f w.r.t. µ (alternative notation, SIµ(a1, . . . , aN)/SIµ(f))

(S)

  • fdµ = max

i=1,N min(f(xs(i)), µ(As(i))),

where s in f(xs(i)) is a permutation so that f(xs(i−1)) ≤ f(xs(i)) for i ≥ 2, and As(k) = {xs(j)|j ≥ k}.

  • Alternative expression (Proposition 6.38):

max

i

min(f(xσ(i)), µ(Aσ(i))), where σ is a permutation of {1, . . . , N} s.t. f(xσ(i−1)) ≥ f(xσ(i)), where Aσ(k) = {xσ(j)|j ≤ k} for k ≥ 1

Vicen¸ c Torra; Modeling decisions UdG 2015 83 / 102

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SLIDE 114

Sugeno integral

  • Graphical interpretation of Sugeno integrals
f (x s(i) ) (A s(i) ) (S ) R f d (A s(i) ) f (x s(i) ) (A) f (x) f (x) (b) (a) ( )

Vicen¸ c Torra; Modeling decisions UdG 2015 84 / 102

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SLIDE 115

Sugeno integral

  • Properties

– WMin and WMax are particular cases of SI ∗ WMax with weighting vector u is a SI w.r.t. µwmax

u

(A) = maxai∈A ui. ∗ WMin with weighting vector u is a SI w.r.t. µwmin

u

(A) = 1 − maxai /

∈A ui.

Vicen¸ c Torra; Modeling decisions UdG 2015 85 / 102

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SLIDE 116

Sugeno integral

  • Example. Citation indices
  • Number of citations: CI
  • h-index: SI

In both cases,

  • X the set of papers
  • f(x) the number of citations of paper x
  • µ(A) ⊆ X the cardinality of the set

Vicen¸ c Torra; Modeling decisions UdG 2015 86 / 102

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SLIDE 117

Fuzzy integrals

  • Fuzzy integrals that generalize Choquet and Sugeno integrals

– The fuzzy t-conorm integral – The twofold integral

Vicen¸ c Torra; Modeling decisions UdG 2015 87 / 102

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SLIDE 118

Aggregation: Hierarchical models

Vicen¸ c Torra; Modeling decisions UdG 2015 88 / 102

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SLIDE 119

Hierarchical Models for Aggregation

  • Hierarchical model
  • Properties. The following conditions hold

(i) Every multistep Choquet integral is a monotone increasing, positively homogeneous, piecewise linear function. (ii) Every monotone increasing, positively homogeneous, piecewise linear function on a full-dimensional convex set in RN is representable as a two-step Choquet integral such that the fuzzy measures of the first step are additive and the fuzzy measure

  • f the second step is a 0-1 fuzzy measure.

Vicen¸ c Torra; Modeling decisions UdG 2015 89 / 102

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SLIDE 120

Aggregation for preference relations (MCDM: social choice)

Vicen¸ c Torra; Modeling decisions UdG 2015 90 / 102

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SLIDE 121

Aggregation for preference relations

  • MCDM (decision) and social choice

⇒ are two related areas

Vicen¸ c Torra; Modeling decisions UdG 2015 91 / 102

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SLIDE 122

Aggregation for preference relations

  • Social choice

– studies voting rules, and how the preferences of a set of people can be aggregated to obtain the preference of the set.

  • There is no formal difference between aggregation of opinions from

people and aggregation of criteria

Vicen¸ c Torra; Modeling decisions UdG 2015 92 / 102

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SLIDE 123

Aggregation for preference relations

  • Given preference relations, how aggregation is built?
  • Formalization of preferences with > an = (preference, indiference)
  • F(R1, R2, . . . , RN) to denote aggregated preference

Vicen¸ c Torra; Modeling decisions UdG 2015 93 / 102

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SLIDE 124

Aggregation for preference relations

  • Given preference relations, how aggregation is built?
  • Formalization of preferences with > an = (preference, indiference)
  • F(R1, R2, . . . , RN) to denote aggregated preference

– Problems (I): consider ∗ R1 : x > y > z ∗ R4 : y > z > x ∗ R5 : z > x > y → simple majority rule: u > v if most prefer u to v ∗ x > y, y > z, z > x (intransitive!!: x > y, y > z but not x > z) – Problems (II): → Arrow impossibility theorem

Vicen¸ c Torra; Modeling decisions UdG 2015 93 / 102

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SLIDE 125

Aggregation for preference relations

  • Given preference relations, how aggregation is built?
  • Axioms of Arrow impossibility theorem

C0 Finite number of voters and more than one Number of alternatives more or equal to three C1 Universality: Voters can select any total preorder C2 Transitivity: The result is a total preorder C3 Unanimity: If all agree on x better than y, then x better than y in the social preference C4 Independence of irrellevant alternatives: the social preference of x and y only depends on the preferences on x and y C5 No-dictatorship: No voter can be a dictatorship

  • There is no function F that satisfies all C0-C5 axioms

Vicen¸ c Torra; Modeling decisions UdG 2015 94 / 102

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SLIDE 126

Aggregation for preference relations

  • Given preference relations, how aggregation is built?
  • Circumventing Arrow’s theorem

– Ignore the condition of universality – Ignore the condition of independence of irrelevant alternatives

Vicen¸ c Torra; Modeling decisions UdG 2015 95 / 102

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SLIDE 127

Aggregation for preference relations

  • Given preference relations, how aggregation is built?

– Solutions failing the universality condition ∗ Simple peak, odd number of voters, Condorcet rule satisfies all other conditions

Vicen¸ c Torra; Modeling decisions UdG 2015 96 / 102

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SLIDE 128

Aggregation for preference relations

  • Given preference relations, how aggregation is built?

– Solutions failing the condition

  • f

independence

  • f

irrelevant alternatives ∗ Condorcet rule with Copeland6: ∗ Borda count7

6Defined by Ramon Llull s. xiii 7Defined by Nicolas de Cusa s. xv. Vicen¸ c Torra; Modeling decisions UdG 2015 97 / 102

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SLIDE 129

Related topics

Vicen¸ c Torra; Modeling decisions UdG 2015 98 / 102

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SLIDE 130

Related topics

  • Aggregation functions

– Functional equations (synthesis of judgements) – Fuzzy measures – Indices and evaluation methods – Model selection

  • Decision making

– Game theory (for decision making with adversary) – Decision under risk and uncertainty – Voting systems (social choice, aggregation of preferences)

Vicen¸ c Torra; Modeling decisions UdG 2015 99 / 102

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SLIDE 131

Summary

Vicen¸ c Torra; Modeling decisions UdG 2015 100 / 102

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SLIDE 132

Related topics

  • Funcions d’agregaci´
  • d’utilitats i prefer`

encies

  • Hi ha vida m´

es enll` a de la mitjana (ponderada)

  • Conceptes importants: la frontera de Pareto (all`
  • que val la pena mirar)

Vicen¸ c Torra; Modeling decisions UdG 2015 101 / 102

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SLIDE 133

Related topics

  • Funcions d’agregaci´
  • d’utilitats i prefer`

encies

  • Hi ha vida m´

es enll` a de la mitjana (ponderada)

  • Conceptes importants: la frontera de Pareto (all`
  • que val la pena mirar)
  • Integrals difuses quan volem expressar depend`

encies

  • ´

Indexs i m` etodes per triar les funcions i trobar els par` ametres

Vicen¸ c Torra; Modeling decisions UdG 2015 101 / 102

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SLIDE 134

Thank you

Vicen¸ c Torra; Modeling decisions UdG 2015 102 / 102