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Ja en on para la toma de decisiones multicriterio Funciones de agregaci c Torra 1 Vicen Octubre, 2014 1 Institut dInvestigaci o en Intel lig` encia Artificial (IIIA-CSIC); Universidad de Sk ovde (HiS, Suecia) - a partir de

  1. MCDM • Representaci´ on de preferencias – Funciones de utilidad. ◦ Ford T: U precio = 0 . 2 , U calidad = 0 . 8 , U confort = 0 . 3 ◦ Peugeot308: U precio = 0 . 7 , U calidad = 0 . 7 , U confort = 0 . 8 – Relaciones de preferencia (comparaci´ on entre varias alterntivas) ◦ R precio : R precio ( P 308 , FordT ) , ¬ R precio ( FordT, P 308) ◦ R calidad : ¬ R calidad ( P 308 , FordT ) , R calidad ( FordT, P 308) ◦ R confort : R confort ( P 308 , FordT ) , ¬ R confort ( FordT, P 308) Vicen¸ c Torra; Modeling decisions Ja´ en 17 / 97

  2. MCDM • Representaci´ on de preferencias – Ejemplo. Relaciones de preferencia. N´ umero Seguridad Precio Confort Maletero asientos Ford T + ++ + ++ + Seat 600 +++ + +++++ + +++ Simca 1000 +++++ +++ ++++ ++++ ++++ VW escarabajo ++++ +++++ ++ +++++ +++++ Citro¨ en Acadiane ++ ++++ +++ +++ ++ Vicen¸ c Torra; Modeling decisions Ja´ en 18 / 97

  3. MCDM • Representaci´ on de preferencias – Ejemplo. Funciones de utilidad. N´ umero Seguridad Precio Confort Maletero asientos Ford T 0 20 0 20 0 Seat 600 60 0 100 0 50 Simca 1000 100 30 100 50 70 VW escarabajo 80 50 30 70 100 Citro¨ en Acadiane 20 40 60 40 0 Vicen¸ c Torra; Modeling decisions Ja´ en 19 / 97

  4. MCDM • Representaci´ on de preferencias: Relaciones de preferencia – Formalizaci´ on: Conjunto de referencia X Propiedades (para todo x, y, z ) ∗ Relaci´ on binaria: I.e., subconjunto de R ⊆ X × X ∗ Denotamos x ≥ y sii ( x, y ) ∈ R ∗ Relaci´ on total o completa: x ≥ y o y ≥ x ∗ Relaci´ on transitiva: x ≥ y , y ≥ z entonces x ≥ z ∗ Relaci´ on reflexiva: x ≥ x Vicen¸ c Torra; Modeling decisions Ja´ en 20 / 97

  5. MCDM • Representaci´ on de preferencias: Relaciones de preferencia – Formalizaci´ on: Conjunto de referencia X Propiedades (para todo x, y, z ) ∗ Relaci´ on binaria: I.e., subconjunto de R ⊆ X × X ∗ Denotamos x ≥ y sii ( x, y ) ∈ R ∗ Relaci´ on total o completa: x ≥ y o y ≥ x ∗ Relaci´ on transitiva: x ≥ y , y ≥ z entonces x ≥ z ∗ Relaci´ on reflexiva: x ≥ x – Definici´ on: (en toma de decisiones) Una relaci´ on es una relaci´ on de preferencia racional si total, transitiva i reflexiva. – en matem´ aticas: preorden total Vicen¸ c Torra; Modeling decisions Ja´ en 20 / 97

  6. MCDM • Representaci´ on de preferencias – Ejemplo. Relaciones de preferencia racional Satisfacen completitud, transitividad, reflexividad N´ umero Seguridad Precio Confort Maletero asientos Ford T + ++ + ++ + Seat 600 +++ + +++++ + +++ Simca 1000 +++++ +++ ++++ ++++ ++++ VW escarabajo ++++ +++++ ++ +++++ +++++ Citro¨ en Acadiane ++ ++++ +++ +++ ++ Vicen¸ c Torra; Modeling decisions Ja´ en 21 / 97

  7. MCDM • Representaci´ on de preferencias: Funciones de utilidad – Formalizaci´ on: Conjunto de referencia X ◦ U : X → D para un cierto dominio D on: Una utilidad u representa una preferencia ≥ para – Representaci´ todo x, y ∈ X cuando x ≥ y si y solo si u ( x ) ≥ u ( y ) . Vicen¸ c Torra; Modeling decisions Ja´ en 22 / 97

  8. MCDM • Representaci´ on de preferencias: Funciones de utilidad – Formalizaci´ on: Conjunto de referencia X ◦ U : X → D para un cierto dominio D on: Una utilidad u representa una preferencia ≥ para – Representaci´ todo x, y ∈ X cuando x ≥ y si y solo si u ( x ) ≥ u ( y ) . Ejemplo. En el precio, la utilidad no representa la relaci´ on Es cierto u precio ( Simca 1000) ≥ u precio ( Seat 600) pero es falso Simca 1000 ≥ Seat 600 Vicen¸ c Torra; Modeling decisions Ja´ en 22 / 97

  9. MCDM • Representaci´ on de preferencias: Funciones de utilidad – Formalizaci´ on: Conjunto de referencia X ◦ U : X → D para un cierto dominio D on: Una utilidad u representa una preferencia ≥ para – Representaci´ todo x, y ∈ X cuando x ≥ y si y solo si u ( x ) ≥ u ( y ) . Ejemplo. En el precio, la utilidad no representa la relaci´ on Es cierto u precio ( Simca 1000) ≥ u precio ( Seat 600) pero es falso Simca 1000 ≥ Seat 600 – Relaci´ on: Podemos establecer una relaci´ on entre las utilidades y las relaciones de preferencia Vicen¸ c Torra; Modeling decisions Ja´ en 22 / 97

  10. MCDM • Representaci´ on de preferencias: Funciones de utilidad – Formalizaci´ on: Conjunto de referencia X ◦ U : X → D para un cierto dominio D – Representaci´ on: Una utilidad u representa una preferencia ≥ para todo x, y ∈ X cuando x ≥ y si y solo si u ( x ) ≥ u ( y ) . Ejemplo. En el precio, la utilidad no representa la relaci´ on Es cierto u precio ( Simca 1000) ≥ u precio ( Seat 600) pero es falso Simca 1000 ≥ Seat 600 – Relaci´ on: Podemos establecer una relaci´ on entre las utilidades y las relaciones de preferencia ◦ Teorema. Dado un conjunto de alternativas, existe una funci´ on de utilidad que representa a la relaci´ on de preferencia si y s´ olo si la relaci´ on de preferencia es racional. Vicen¸ c Torra; Modeling decisions Ja´ en 22 / 97

  11. MCDM • Representaci´ on de preferencias: Funciones de utilidad – Ejemplo: definici´ on para precio ◦ Presupuesto maximo de 10000 euros. ◦ Menor que 1000 es perfecto. ◦ Funcion lineal entre 1000 y 10000  x ≤ 1000 100 if   (10000 − x ) / 90 x ∈ (1000 , 10000) u p ( x ) = if x ≥ 10000 0  if  Vicen¸ c Torra; Modeling decisions Ja´ en 23 / 97

  12. MCDM • Representaci´ on de preferencias: Funciones de utilidad – Ejemplo: definici´ on para capacidad del maletero No siempre hay una relaci´ on mon´ otona entre los valores de un criterio y la utilidad optimo es de 1 m 3 . ◦ El maletero ´ ◦ Ni demasiado peque˜ no, ni demasiado grande  x ≤ 0 . 8 0 if   u m ( x ) = 100 − 500 | x − 1 | x ∈ (0 . 8 , 1 . 2) if 0 x ≥ 1 . 2  if  Vicen¸ c Torra; Modeling decisions Ja´ en 24 / 97

  13. MCDM • Decisi´ on – Modelizaci´ on del problema: representaci´ on de los criterios – Agregaci´ on – Selecci´ on de las alternativas Vicen¸ c Torra; Modeling decisions Ja´ en 25 / 97

  14. MCDM • Agregaci´ on, seg´ un la representaci´ on de las preferencias – Funciones de utilidad ◦ Ford T: U precio = 0 . 2 , U calidad = 0 . 8 , U confort = 0 . 3 ∗ Dadas unas utilidades, tenemos que agregarlas – Relaciones de preferencia (comparaci´ on entre varias alterntivas) ◦ R precio : R precio ( P 308 , FordT ) , ¬ R precio ( FordT, P 308) ◦ R calidad : ¬ R calidad ( P 308 , FordT ) , R calidad ( FordT, P 308) ∗ Dadas unas relaciones de preferencia, tenemos que agregarlas Vicen¸ c Torra; Modeling decisions Ja´ en 26 / 97

  15. MCDM • Decisi´ on con relaciones de preferencia Modelizaci´ on, agregaci´ on, selecci´ on N´ umero Seguridad Precio Confort Maletero Preferencia asientos agregada Ford T + ++ + ++ + + Seat 600 +++ + +++++ + +++ ++ Simca 1000 +++++ +++ ++++ ++++ ++++ ++++ VW esc. ++++ +++++ ++ +++++ +++++ +++++ Citr. Acadiane ++ ++++ +++ +++ ++ +++ Vicen¸ c Torra; Modeling decisions Ja´ en 27 / 97

  16. MCDM • Decisi´ on con funciones de utilidad Modelizaci´ on, agregaci´ on = AM, selecci´ on N´ umero Seguridad Precio Confort Maletero Preferencia asientos agregada Ford T 0 20 0 20 0 8 Seat 600 60 0 100 0 50 42 Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66 Citr. Acadiane 20 40 60 40 0 32 Vicen¸ c Torra; Modeling decisions Ja´ en 28 / 97

  17. Aggregation functions: an introduction Vicen¸ c Torra; Modeling decisions Ja´ en 29 / 97

  18. 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 Ja´ en 30 / 97

  19. 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 a i /N (AM arithmetic mean) – � N i =1 p i · a i (WM weighted mean) Vicen¸ c Torra; Modeling decisions Ja´ en 30 / 97

  20. 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 a i /N (AM arithmetic mean) – � N i =1 p i · a i (WM weighted mean) • Different functions, lead to different results – In our case, different orderings, different selections! Vicen¸ c Torra; Modeling decisions Ja´ en 30 / 97

  21. 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 Ja´ en 31 / 97

  22. Aggregation functions • Information fusion studies . . . . . . all aspects related to combining information: • Goals of data aggregation ( goals of the area ): Vicen¸ c Torra; Modeling decisions Ja´ en 32 / 97

  23. 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 Ja´ en 32 / 97

  24. 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 Ja´ en 32 / 97

  25. Aggregation functions • Terms: – Information integration – Information fusion: concrete functions / techniques concrete process to combine several data into a single datum. – Aggregation functions: C : D N → D ( C from C onsensus) → i C with parameters (background knowledge): C P Vicen¸ c Torra; Modeling decisions Ja´ en 33 / 97

  26. Aggregation functions • Terms: – Information integration – Information fusion: concrete functions / techniques concrete process to combine several data into a single datum. – Aggregation functions: C : D N → D ( C from C onsensus) → i C with parameters (background knowledge): C P • Aggregation functions: basic properties Vicen¸ c Torra; Modeling decisions Ja´ en 33 / 97

  27. Aggregation functions • Terms: – Information integration – Information fusion: concrete functions / techniques concrete process to combine several data into a single datum. – Aggregation functions: C : D N → D ( C from C onsensus) → i C with parameters (background knowledge): C P • Aggregation functions: basic properties – Unanimity and idempotency: C ( a, . . . , a ) = a for all a Vicen¸ c Torra; Modeling decisions Ja´ en 33 / 97

  28. Aggregation functions • Terms: – Information integration – Information fusion: concrete functions / techniques concrete process to combine several data into a single datum. – Aggregation functions: C : D N → D ( C from C onsensus) → i C with parameters (background knowledge): C P • Aggregation functions: basic properties – Unanimity and idempotency: C ( a, . . . , a ) = a for all a – Monotonicity: C ( a 1 , . . . , a N ) ≥ C ( a ′ 1 , . . . , a ′ N ) , if a i ≥ a ′ i Vicen¸ c Torra; Modeling decisions Ja´ en 33 / 97

  29. Aggregation functions • Terms: – Information integration – Information fusion: concrete functions / techniques concrete process to combine several data into a single datum. – Aggregation functions: C : D N → D ( C from C onsensus) → i C with parameters (background knowledge): C P • Aggregation functions: basic properties – Unanimity and idempotency: C ( a, . . . , a ) = a for all a – Monotonicity: C ( a 1 , . . . , a N ) ≥ C ( a ′ 1 , . . . , a ′ N ) , if a i ≥ a ′ i – Symmetry: For all permutation π over { 1 , . . . , N } C ( a 1 , . . . , a N ) = C ( a π (1) , . . . , a π ( N ) ) Vicen¸ c Torra; Modeling decisions Ja´ en 33 / 97

  30. Aggregation functions • Terms: – Information integration – Information fusion: concrete functions / techniques concrete process to combine several data into a single datum. – Aggregation functions: C : D N → D ( C from C onsensus) → i C with parameters (background knowledge): C P • Aggregation functions: basic properties – Unanimity and idempotency: C ( a, . . . , a ) = a for all a – Monotonicity: C ( a 1 , . . . , a N ) ≥ C ( a ′ 1 , . . . , a ′ N ) , if a i ≥ a ′ i – Symmetry: For all permutation π over { 1 , . . . , N } C ( a 1 , . . . , a N ) = C ( a π (1) , . . . , a π ( N ) ) – Unanimity + monotonicity → internality: min i a i ≤ C ( a 1 , . . . , a N ) ≤ max i a i Vicen¸ c Torra; Modeling decisions Ja´ en 33 / 97

  31. 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 Ja´ en 34 / 97

  32. Aggregation functions • Definition from properties properties − → function Vicen¸ c Torra; Modeling decisions Ja´ en 35 / 97

  33. Aggregation functions • Definition from properties properties − → function • Some ways a) Using functional equations Vicen¸ c Torra; Modeling decisions Ja´ en 35 / 97

  34. Aggregation functions • Definition from properties properties − → function • Some ways a) Using functional equations b) Aggregation of a 1 , a 2 , . . . , a N ∈ D , as the datum c which is at a minimum distance from a i : � c { d ( c, a i ) } , C ( a 1 , a 2 , . . . , a N ) = arg min a i d is a distance over D . Vicen¸ c Torra; Modeling decisions Ja´ en 35 / 97

  35. Aggregation functions • Example (case (a)): Functional equations – Cauchy equation φ ( x + y ) = φ ( x ) + φ ( y ) – find φ ! Vicen¸ c Torra; Modeling decisions Ja´ en 36 / 97

  36. 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 Ja´ en 36 / 97

  37. 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 x 1 x 1 x 1 x 1 · · · · · · E 1 1 2 m j x 2 x 2 x 2 x 2 E 2 · · · · · · 1 2 j m . . . . . . . . . . . . x i x i x i x i E i · · · · · · 1 2 j m . . . . . . . . . . . . x N x N x N x N · · · · · · E N 1 2 j m DM f 1 ( x 1 ) f 2 ( x 2 ) · · · f j ( x j ) · · · f m ( x m ) Vicen¸ c Torra; Modeling decisions Ja´ en 37 / 97

  38. Aggregation functions • The general solution of the system (Proposition 3.11) for a given m > 2 f j : [0 , s ] N → R + for j = { 1 , · · · , m } (1) m m � � x j = s implies that f j ( x j ) = s (2) j =1 j =1 f j ( 0 ) = 0 for j = 1 , · · · , m (3) is given by Vicen¸ c Torra; Modeling decisions Ja´ en 38 / 97

  39. Aggregation functions • The general solution of the system (Proposition 3.11) for a given m > 2 f j : [0 , s ] N → R + for j = { 1 , · · · , m } (1) m m � � x j = s implies that f j ( x j ) = s (2) j =1 j =1 f j ( 0 ) = 0 for j = 1 , · · · , m (3) is given by N � f 1 ( x ) = f 2 ( x ) = · · · = f m ( x ) = f (( x 1 , x 2 , . . . , x N )) = α i x i , (4) i =1 where α 1 , · · · , α N are nonnegative constants satisfying � N i =1 α i = 1 , but are otherwise arbitrary. Vicen¸ c Torra; Modeling decisions Ja´ en 38 / 97

  40. Aggregation functions • Example (case (b)): Consider the following expression � c { d ( c, a i ) } , C ( a 1 , a 2 , . . . , a N ) = arg min a i where a i are numbers from R and d is a distance on D . Then, Vicen¸ c Torra; Modeling decisions Ja´ en 39 / 97

  41. Aggregation functions • Example (case (b)): Consider the following expression � C ( a 1 , a 2 , . . . , a N ) = arg min c { d ( c, a i ) } , a i where a i 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 ( a 1 , a 2 , . . . , a N ) = � N i =1 a i /N . 2. When d ( a, b ) = | a − b | , C is the median I.e., the median of a 1 , a 2 , . . . , a N is the element which occupies the central position when we order a i . 3. When d ( a, b ) = 1 iff a = b , C is the plurality rule (mode or voting). I.e., C ( a 1 , a 2 , . . . , a N ) selects the element of R with a largest frequency among elements in ( a 1 , a 2 , . . . , a N ) . Vicen¸ c Torra; Modeling decisions Ja´ en 39 / 97

  42. Aggregation for (numerical) utility functions Vicen¸ c Torra; Modeling decisions Ja´ en 40 / 97

  43. Aggregation for (numerical) utility functions • Decisi´ on con funciones de utilidad Modelizaci´ on, agregaci´ on = C , selecci´ on Seats Security Price Comfort trunk C = AM Ford T 0 20 0 20 0 8 Seat 600 60 0 100 0 50 42 Simca 1000 100 30 100 50 70 70 VW 80 50 30 70 100 66 Citr. Acadiane 20 40 60 40 0 32 Vicen¸ c Torra; Modeling decisions Ja´ en 41 / 97

  44. Aggregation for (numerical) utility functions • MCDM: Aggregation to deal with contradictory criteria Vicen¸ c Torra; Modeling decisions Ja´ en 42 / 97

  45. Aggregation for (numerical) utility functions • MCDM: Aggregation to deal with contradictory criteria • But there are occasions in which ordering is clear when a i ≤ b i it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 0 100 0 50 42 Simca 1000 100 30 100 50 70 70 Vicen¸ c Torra; Modeling decisions Ja´ en 42 / 97

  46. Aggregation for (numerical) utility functions • MCDM: Aggregation to deal with contradictory criteria • But there are occasions in which ordering is clear when a i ≤ b i it is clear that a ≤ b E.g., Seats Security Price Comfort trunk C = AM Seat 600 60 0 100 0 50 42 Simca 1000 100 30 100 50 70 70 • Pareto dominance: Given two vectors a = ( a 1 , . . . , a n ) and b = ( b 1 , . . . , b n ) , we say that b dominates a when a i ≤ b i for all i and there is at least one k such that a k < b k . Vicen¸ c Torra; Modeling decisions Ja´ en 42 / 97

  47. 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 0 32 • Each one wins at least in one criteria Vicen¸ c Torra; Modeling decisions Ja´ en 43 / 97

  48. Aggregation for (numerical) utility functions • Pareto set, Pareto frontier, or non dominance set: Given a set of alternatives U represented by vectors u = ( u 1 , . . . , u n ) , 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 f 2 f 2 ( x 2 ) x 2 f 2 ( x 1 ) x 1 f 1 f 1 ( x 2 ) f 1 ( x 1 ) Vicen¸ c Torra; Modeling decisions Ja´ en 44 / 97

  49. 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) Criteria f 2 Satisfaction on: alt Price Quality Comfort alt Consensus alt Ranking FordT 0.2 0.8 0.3 FordT 0.35 206 0.72 f 2 ( x 2 ) x 2 FordT 0.35 206 0.7 0.7 0.8 206 0.72 ... ... ... ... ... ... f 2 ( x 1 ) x 1 f 1 f 1 ( x 2 ) f 1 ( x 1 ) Vicen¸ c Torra; Modeling decisions Ja´ en 45 / 97

  50. Aggregation for (numerical) utility functions • Why alternatives de 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 Ja´ en 46 / 97

  51. Aggregation: from the weighted mean to fuzzy integrals Vicen¸ c Torra; Modeling decisions Ja´ en 47 / 97

  52. Aggregation: from the weighted mean to fuzzy integrals An example Vicen¸ c Torra; Modeling decisions Ja´ en 48 / 97

  53. 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 Ja´ en 49 / 97

  54. Aggregation: example Example. Formalization • Variables – x A : Number of sessions taught by Professor A – x B : Number of sessions taught by Professor B • Constraints – the constraints are translated into ∗ C 1 : x A + x B should be about 6 ∗ C 2 : x A should be about 3 ∗ C 3 : x B should be about 2 ∗ C 4 : | x A − x B | should be about 0 – using fuzzy sets, the constraints are described ... Vicen¸ c Torra; Modeling decisions Ja´ en 50 / 97

  55. Aggregation: example Example. Formalization • Constraints – if fuzzy set µ 6 expresses “about 6,” then, we evaluate “ x A + x B should be about 6” by µ 6 ( x A + x B ) . → given µ 6 , µ 3 , µ 2 , µ 0 , – Then, given a solution pair ( x A , x B ) , the degrees of satisfaction: ∗ µ 6 ( x A + x B ) ∗ µ 3 ( x A ) ∗ µ 2 ( x B ) ∗ µ 0 ( | x A − x B | ) Vicen¸ c Torra; Modeling decisions Ja´ en 51 / 97

  56. Aggregation: example Example. Formalization • Membership functions for constraints µ 2 µ 0 µ 3 µ 6 1 2 3 4 5 6 7 Vicen¸ c Torra; Modeling decisions Ja´ en 52 / 97

  57. Aggregation: example Example. Application alternative Satisfaction degrees Satisfaction degrees ( x A , x B ) ( µ 6 ( x A + x B ) , µ 3 ( x A ) , C 1 C 2 C 3 C 4 µ 2 ( x B ) , µ 0 ( | x A − x B | ) ) (2 , 2) ( µ 6 (4) , µ 3 (2) , µ 2 (2) , µ 0 (0) ) 0 0 . 5 1 1 (2 , 3) ( µ 6 (5) , µ 3 (2) , µ 2 (3) , µ 0 (1) ) 0 . 5 0 . 5 0 . 5 0 . 5 (2 , 4) ( µ 6 (6) , µ 3 (2) , µ 2 (4) , µ 0 (2) ) 1 0 . 5 0 0 . 5 (3 . 5 , 2 . 5) ( µ 6 (6) , µ 3 (3 . 5) , µ 2 (2 . 5) , µ 0 (1) ) 1 0 . 5 0 . 5 0 . 5 (3 , 2) ( µ 6 (5) , µ 3 (3) , µ 2 (2) , µ 0 (1) ) 0 . 5 1 1 0 . 5 (3 , 3) ( µ 6 (6) , µ 3 (3) , µ 2 (3) , µ 0 (0) ) 1 1 0 . 5 1 Vicen¸ c Torra; Modeling decisions Ja´ en 53 / 97

  58. Aggregation: from the weighted mean to fuzzy integrals WM, OWA, and WOWA operators Vicen¸ c Torra; Modeling decisions Ja´ en 54 / 97

  59. Aggregation: WM, OWA, and WOWA operators • Operators – Weighting vector (dimension N ): v = ( v 1 ...v N ) iff v i ∈ [0 , 1] and � i v i = 1 – Arithmetic mean (AM : R N → R ): AM ( a 1 , ..., a N ) = (1 /N ) � N i =1 a i – Weighted mean (WM: R N → R ): WM p ( a 1 , ..., a N ) = � N i =1 p i a i ( p a weighting vector of dimension N ) – Ordered Weighting Averaging operator (OWA: R N → R ): N � OWA w ( a 1 , ..., a N ) = w i a σ ( i ) , i =1 where { σ (1) , ..., σ ( N ) } is a permutation of { 1 , ..., N } s. t. a σ ( i − 1) ≥ a σ ( i ) , and w a weighting vector. Vicen¸ c Torra; Modeling decisions Ja´ en 55 / 97

  60. 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 = ( p 1 , p 2 , p 3 , p 4 ) = (0 . 5 , 0 . 3 , 0 . 15 , 0 . 05) . Vicen¸ c Torra; Modeling decisions Ja´ en 56 / 97

  61. Aggregation: WM, OWA, and WOWA operators Example. Application • WM with p = ( p 1 , p 2 , p 3 , p 4 ) = (0 . 5 , 0 . 3 , 0 . 15 , 0 . 05) . alternative Aggregation of the Satisfaction degrees WM ( x A , x B ) WM p ( C 1 , C 2 , C 3 , C 4 ) (2 , 2) WM p (0 , 0 . 5 , 1 , 1) 0.35 (2 , 3) WM p (0 . 5 , 0 . 5 , 0 . 5 , 0 . 5) 0.5 (2 , 4) WM p (1 , 0 . 5 , 0 , 0 . 5) 0.675 (3 . 5 , 2 . 5) WM p (1 , 0 . 5 , 0 . 5 , 0 . 5) 0.75 (3 , 2) WM p (0 . 5 , 1 , 1 , 0 . 5) 0.725 (3 , 3) WM p (1 , 1 , 0 . 5 , 1) 0.925 Vicen¸ c Torra; Modeling decisions Ja´ en 57 / 97

  62. 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 ( x A , x B ) OWA w ( C 1 , C 2 , C 3 , C 4 ) (2 , 2) OWA w (0 , 0 . 5 , 1 , 1) 0.8333 (2 , 3) OWA w (0 . 5 , 0 . 5 , 0 . 5 , 0 . 5) 0.5 (2 , 4) OWA w (1 , 0 . 5 , 0 , 0 . 5) 0.6666 (3 . 5 , 2 . 5) OWA w (1 , 0 . 5 , 0 . 5 , 0 . 5) 0.6666 (3 , 2) OWA w (0 . 5 , 1 , 1 , 0 . 5) 0.8333 (3 , 3) OWA w (1 , 1 , 0 . 5 , 1) 1.0 Vicen¸ c Torra; Modeling decisions Ja´ en 58 / 97

  63. Aggregation: WM, OWA, and WOWA operators • Weighted Ordered Weighted Averaging WOWA operator (WOWA : R N → R ): WOWA p , w ( a 1 , ..., a N ) = � N i =1 ω i a σ ( i ) where ω i = w ∗ ( � j ≤ i p σ ( j ) ) − w ∗ ( � j<i p σ ( j ) ) , with σ a permutation of { 1 , ..., N } s. t. a σ ( i − 1) ≥ a σ ( i ) , and w ∗ a nondecreasing function that interpolates the points { ( i/N, � j ≤ i w j ) } 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 Ja´ en 59 / 97

  64. Aggregation: WM, OWA, and WOWA operators • Construction of the w ∗ quantifier (a) (b) ( ) w w N N w w 2 2 ! 1 w 1 w 1 ! 1 � 0 � p � (1) p � (1) 1 = N 1 = N ::: 1 = N 0 0 0 0 p p p p p � (1) � (1) � (2) � ( N ) � (1) • Rationale for new weights ( ω i , for each value a i ) in terms of p and w . – If a i is small, and small values have more importance than larger ones, increase p i for a i (i.e., ω i ≥ p σ ( i ) ). (the same holds if the value a i is large and importance is given to large values) – If a i is small, and importance is for large values, ω i < p σ ( i ) (the same holds if a i is large and importance is given to small values). Vicen¸ c Torra; Modeling decisions Ja´ en 60 / 97

  65. 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 Ja´ en 61 / 97

  66. 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 ( x A , x B ) WOWA p , w ( C 1 , C 2 , C 3 , C 4 ) (2 , 2) WOWA p , w (0 , 0 . 5 , 1 , 1) 0.4666 (2 , 3) WOWA p , w (0 . 5 , 0 . 5 , 0 . 5 , 0 . 5) 0.5 (2 , 4) WOWA p , w (1 , 0 . 5 , 0 , 0 . 5) 0.8333 (3 . 5 , 2 . 5) WOWA p , w (1 , 0 . 5 , 0 . 5 , 0 . 5) 0.8333 (3 , 2) WOWA p , w (0 . 5 , 1 , 1 , 0 . 5) 0.8 (3 , 3) WOWA p , w (1 , 1 , 0 . 5 , 1) 1.0 Vicen¸ c Torra; Modeling decisions Ja´ en 62 / 97

  67. Aggregation: WM, OWA, and WOWA operators • Properties – The WOWA operator generalizes the WM and the OWA operator. ◦ When p = (1 /N . . . 1 /N ) , OWA WOWA p , w ( a 1 , ..., a N ) = OWA w ( a 1 , ..., a N ) for all w and a i . ◦ When w = (1 /N ... 1 /N ) , WM WOWA p , w ( a 1 , ..., a N ) = WM p ( a 1 , ..., a N ) for all p and a i . ◦ When w = p = (1 /N ... 1 /N ) , AM WOWA p , w ( a 1 , ..., a N ) = AM ( a 1 , ..., a N ) Vicen¸ c Torra; Modeling decisions Ja´ en 63 / 97

  68. Aggregation: from the weighted mean to fuzzy integrals Choquet integrals Vicen¸ c Torra; Modeling decisions Ja´ en 64 / 97

  69. Choquet integrals • In WM, we combine a i w.r.t. weights p i . → a i is the value supplied by information source x i . Formally Vicen¸ c Torra; Modeling decisions Ja´ en 65 / 97

  70. Choquet integrals • In WM, we combine a i w.r.t. weights p i . → a i is the value supplied by information source x i . Formally – X = { x 1 , . . . , x N } is the set of information sources – f : X → R + the values supplied by the sources → then a i = f ( x i ) Thus, N N � � WM p ( a 1 , ..., a N ) = p i a i = p i f ( x i ) = WM p ( f ( x 1 ) , ..., f ( x N )) i =1 i =1 Vicen¸ c Torra; Modeling decisions Ja´ en 65 / 97

  71. Choquet integrals • In the WM, a single weight is used for each element I.e., p i = p ( x i ) (where, x i is the information source that supplies a i ) → when we consider a set A ⊂ X , weight of A ??? Vicen¸ c Torra; Modeling decisions Ja´ en 66 / 97

  72. Choquet integrals • In the WM, a single weight is used for each element I.e., p i = p ( x i ) (where, x i is the information source that supplies a i ) → when we consider a set A ⊂ X , weight of A ??? . . . 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 Ja´ en 66 / 97

  73. 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 Ja´ en 67 / 97

  74. Choquet integrals • Choquet integral of f w.r.t. µ (alternative notation, CI µ ( a 1 , . . . , a N ) /CI µ ( f ) ) N � � ( C ) fdµ = [ f ( x s ( i ) ) − f ( x s ( i − 1) )] µ ( A s ( i ) ) , i =1 where s in f ( x s ( i ) ) is a permutation so that f ( x s ( i − 1) ) ≤ f ( x s ( i ) ) for i ≥ 1 , f ( x s (0) ) = 0 , and A s ( k ) = { x s ( j ) | j ≥ k } and A s ( N +1) = ∅ . • Alternative expressions (Proposition 6.18): N � � ( C ) fdµ = f ( x σ ( i ) )[ µ ( A σ ( i ) ) − µ ( A σ ( i − 1) )] , i =1 N � � ( C ) fdµ = f ( x s ( i ) )[ µ ( A s ( i ) ) − µ ( A s ( i +1) )] , i =1 where σ is a permutation of { 1 , . . . , N } s.t. f ( x σ ( i − 1) ) ≥ f ( x σ ( i ) ) , where A σ ( k ) = { x σ ( j ) | j ≤ k } for k ≥ 1 and A σ (0) = ∅ Vicen¸ c Torra; Modeling decisions Ja´ en 68 / 97

  75. Choquet integrals • Different equations point out different aspects of the CI fdµ = � N (6.1) ( C ) � i =1 [ f ( x s ( i ) ) − f ( x s ( i − 1) )] µ ( A s ( i ) ) , 0 a s (1) a s (2) a s (3) a s (4) a s (5) µ ( A s (2) ) µ ( A s (1) ) = { x s (1) , · · · , x s ( N ) } µ ( A s (4) ) = { x s (4) , · · · , x s ( N ) } fdµ = � N (6.2) ( C ) � i =1 f ( x σ ( i ) )[ µ ( A σ ( i ) ) − µ ( A σ ( i − 1) )] , Vicen¸ c Torra; Modeling decisions Ja´ en 69 / 97

  76. Choquet integrals � • fdµ = (for additive measures) (6.5) � x ∈ X f ( x ) µ ( { x } ) (6.6) � R i =1 b i µ ( { x | f ( x ) = b i } ) (6.7) � N i =1 ( a i − a i − 1 ) µ ( { x | f ( x ) ≥ a i } ) (6.8) � N � � i =1 ( a i − a i − 1 ) 1 − µ ( { x | f ( x ) ≤ a i − 1 } ) (a) (b) (c) b i b i a i b i − 1 b i − 1 a i − 1 x 1 x N x 1 x N x 1 { x | f ( x ) = b i } { x | f ( x ) ≥ a i } x • Among (6.5), (6.6) and (6.7), only (6.7) satisfies internality. Vicen¸ c Torra; Modeling decisions Ja´ en 70 / 97

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