[PPT] - Aggregation functions and information fusion. Modeling decisions PowerPoint Presentation

SLIDE 1

Aggregation functions and information fusion. Modeling decisions

Vicen¸ c Torra

Universitat de Sk¨

vde (HiS, Sweden)

July, 2017

SLIDE 2

Summary

Aggregation functions
There is life beyond the (weighted) mean
Important concepts: the Pareto front (what is relevant to study)
Fuzzy integrals to express non-independence
Indices and methods to select functions and their parameters

2017 1 / 91

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Index (I)

I. An introduction
1. Defining the problem
2. The goals of the field (aggregation functions)
II. On the definition of aggregation functions
1. Definition from properties
2. Definition heuristically
3. Definition from examples

2017 2 / 91

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Index (II)

III. From the weighted mean to fuzzy integrals
1. An example
2. WM, OWA; and WOWA operators
3. Choquet integral
4. Weighted minimum and maximum
5. Sugeno integral
IV. Fuzzy measures
V. Preference relations
VI. Related topics
VII. Summary

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Aggregation:

I. an introduction

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Aggregation:

I. an introduction
1. defining the problem

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Aggregation functions

Aggregation and information fusion
How to combine information
Focus here: information about criteria to make decisions
In general,
it is a broad area, with different types of applications

fusion for robotics (sensors), computer vision, running times, economy (GDP), biology (DNA sequences), education, . . .

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Aggregation functions

Aggregation functions C(a1, . . . , aN) : DN → D:

(aggregate, combine, fuse information)

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Aggregation functions

Aggregation functions C(a1, . . . , aN) : DN → D:

(aggregate, combine, fuse information)

Well known examples for numerical data:

⋆ N

i=1 ai/N (AM arithmetic mean)

⋆ N

i=1 pi · ai (WM weighted mean, p weights)

Vicen¸ c Torra; Modeling decisions 2017 7 / 91

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Aggregation functions

Aggregation functions C(a1, . . . , aN) : DN → D:

(aggregate, combine, fuse information)

Well known examples for numerical data:

⋆ N

i=1 ai/N (AM arithmetic mean)

⋆ N

i=1 pi · ai (WM weighted mean, p weights)

Why study them, why different functions?
Different functions, lead to different results

In decision, different orderings, different selections!

Some functions lead to inconsistent results

Expected properties on the aggregated data!

Vicen¸ c Torra; Modeling decisions 2017 7 / 91

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Aggregation functions

Aggregation functions:
N

i=1 pi · ai (WM weighted mean, p weights)

Different functions/parameters, lead to different results

In decision, different orderings, different selections!

Vicen¸ c Torra; Modeling decisions 2017 8 / 91

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Aggregation functions

Aggregation functions:
N

i=1 pi · ai (WM weighted mean, p weights)

Different functions/parameters, lead to different results

In decision, different orderings, different selections!

Example (book II). 0 (min) – 100 (max)

Number of Security Price Confort trunk 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˜ A<<n Acadiane 20 40 60 40

WM with p#Seats = 0.5, pprice = 0.5: Select Simca 1000
WM with pSecurity = 0.5, pConfort = 0.5: Select VW Beetle

Vicen¸ c Torra; Modeling decisions 2017 8 / 91

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Aggregation functions

Aggregation functions: Other examples (p. 92, book)
log(( eai)/N) (EM, exponential mean)
(logc((1/N) c1/ai))−1 (Radical mean)
Some functions lead to inconsistent results: Expected properties
Example 4.21 (book). Assessing performance of Java runtime systems.
7 benchmark programs. How aggregation should be done?

Can we use our nice means: exponential mean ?, radical mean?

NO!!: If we want to have consistent results when time is changed

from seconds to miliseconds, or to minutes!

227- 202- 201- 209- 222- 228- 213- Runtime system mtr jess compress db mpegaudio jack javac GM Sun JDK 1.5.0 Client VM 325 221 204 43.4 251 192 96.1 162.13 Sun JDK 1.4.2 Client VM 318 186 199 43.6 249 181 90.9 154.51 Kaffe 32 21.3 191 24.8 101 32.9 21.3 41.95 Vicen¸ c Torra; Modeling decisions 2017 9 / 91

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Aggregation functions

Aggregation functions and information fusion

(in general, not restricted to decision): (Book, Ch. 1)

used to produce a comprehensive and specific datum about an entity,
datum produced from information supplied by different information

sources (or the same source over time).

Aggregation

to reduce noise, increase precision, summarize information, extract information, make decisions, etc.

Vicen¸ c Torra; Modeling decisions 2017 10 / 91

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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)

→ C with parameters (background knowledge): CP

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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)

→ C with parameters (background knowledge): CP

Aggregation functions: basic properties

Vicen¸ c Torra; Modeling decisions 2017 11 / 91

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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)

→ C with parameters (background knowledge): CP

Aggregation functions: basic properties
Unanimity and idempotency: C(a, . . . , a) = a for all a (some?)

Vicen¸ c Torra; Modeling decisions 2017 11 / 91

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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)

→ C with parameters (background knowledge): CP

Aggregation functions: basic properties
Unanimity and idempotency: C(a, . . . , a) = a for all a (some?)
Monotonicity: C(a1, . . . , aN) ≥ C(a′

1, . . . , a′ N), if ai ≥ a′ i

Vicen¸ c Torra; Modeling decisions 2017 11 / 91

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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)

→ C with parameters (background knowledge): CP

Aggregation functions: basic properties
Unanimity and idempotency: C(a, . . . , a) = a for all a (some?)
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 2017 11 / 91

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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)

→ C with parameters (background knowledge): CP

Aggregation functions: basic properties
Unanimity and idempotency: C(a, . . . , a) = a for all a (some?)
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 2017 11 / 91

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Aggregation functions

Unanimity and idempotency:

C(a, . . . , a) = a for a = 0 and a = 1.

t-norms: associative, symmetric, monotonic,

and has a neutral element 1

t-conorms: associative, symmetric, monotonic

and has a neutral element 0

uninorms: associative, symmetric, monotonic

and has a neutral element e in (0, 1)

Vicen¸ c Torra; Modeling decisions 2017 12 / 91

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Aggregation functions

Unanimity and idempotency:

C(a, . . . , a) = a for a = 0 and a = 1.

t-norms: associative, symmetric, monotonic,

and has a neutral element 1

t-conorms: associative, symmetric, monotonic

and has a neutral element 0

uninorms: associative, symmetric, monotonic

and has a neutral element e in (0, 1)

t−norms t−conorms means min max

Vicen¸ c Torra; Modeling decisions 2017 12 / 91

SLIDE 23

Aggregation:

I. an introduction
2. the goals of the field

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SLIDE 24

Aggregation functions

Goals of data aggregation (goals of the area):

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Aggregation functions

Goals of data aggregation (goals of the area):
Formalization of the aggregation process
Definition of new functions (and impossibility results!)
Selection of functions

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

Parameter determination

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Aggregation functions

Goals of data aggregation (goals of the area):
Formalization of the aggregation process
Definition of new functions (and impossibility results!)
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: dictatorship?, sensitivity to data → index).

Vicen¸ c Torra; Modeling decisions 2017 14 / 91

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Aggregation:

II. on the definition of aggregation functions

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Aggregation functions

Definition of aggregation functions:

1. Definition from properties:

E.g., results consistent with changes of scale: C(ra1, . . . , ran) = rC(a1, . . . , an)

2. Heuristic definition: from functions to properties

E.g., after some testing we decide to use xxxxx: properties?

3. Definition from examples

E.g., find C that approximates the examples

properties function examples

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Aggregation functions

1. Definition from properties

properties − → function

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Aggregation functions

1. Definition from properties

properties − → function

Some alternatives

(1.a) Expressing properties as equations: functional equations

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Aggregation functions

1. Definition from properties

properties − → function

Some alternatives

(1.a) Expressing properties as equations: functional equations (1.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 2017 17 / 91

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Aggregation functions

Functional equations (case 1.a).
What is a functional equation ?
Equations where the unknown are functions
Example. Cauchy equation (a well known functional equation):

φ(x + y) = φ(x) + φ(y)

find φ !

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Aggregation functions

Functional equations (case 1.a).
What is a functional equation ?
Equations where the unknown are functions
Example. Cauchy equation (a well known functional equation):

φ(x + y) = φ(x) + φ(y)

find φ !
Solution: For continuous φ,

φ(x) = αx for an arbitrary value for α

Vicen¸ c Torra; Modeling decisions 2017 18 / 91

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Aggregation functions

Functional equations (case 1.a). Example I in aggregation.
Distribute s euros among m projects according to the opinion of N

j

· · · xN

m

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

Vicen¸ c Torra; Modeling decisions 2017 19 / 91

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Aggregation functions

Functional equations (case 1.a). Example I in aggregation.
The general solution of the system (Prop. 3.11) for m > 2

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

m

j=1

xj = s implies that

m

j=1

fj(xj) = s fj(0) = 0 for j = 1, · · · , m

Vicen¸ c Torra; Modeling decisions 2017 20 / 91

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Aggregation functions

Functional equations (case 1.a). Example I in aggregation.
The general solution of the system (Prop. 3.11) for m > 2

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

m

j=1

xj = s implies that

m

j=1

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

N

i=1

αixi, where α1, · · · , αN are nonnegative constants satisfying N

i=1 αi = 1, but are

therwise arbitrary.

Vicen¸ c Torra; Modeling decisions 2017 20 / 91

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Aggregation functions

Functional equations (case 1.a). Example II in aggregation.
(Prop.

4.17) An operator C is separable in terms of a unique monotone increasing g (that is, C(a1, . . . , aN) = g(a1) ◦ · · · ◦ g(aN)) (with ◦ continuous, associative, and cancellative)

Vicen¸ c Torra; Modeling decisions 2017 21 / 91

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Aggregation functions

Functional equations (case 1.a). Example II in aggregation.
(Prop.

4.17) An operator C is separable in terms of a unique monotone increasing g (that is, C(a1, . . . , aN) = g(a1) ◦ · · · ◦ g(aN)) (with ◦ continuous, associative, and cancellative) C(a1, . . . , aN) = φ−1(

N

i=1

φ(g(ai))) and satisfies unanimity

C(a, . . . , a) = a

if and only if it is of the form (quasi-arithmetic mean) C(a1, . . . , aN) = φ−1( 1 N

N

i=1

φ(ai)).

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Aggregation functions

Functional equations (case 1.a). Quasi-arithmetic means

Name Generator function C(a1, . . . , aN) Arithmetic mean φ(x) = x

N

i=1 xi

N

Geometric mean φ(x) = log x

N

i=1 xi

N

Harmonic mean φ(x) = 1/x

N N

i=1 1 xi

Root-mean-square φ(x) = x2 N

i=1 x2 i

N

Root-mean-power φ(x) = xα

α

N

i=1 xα i

N

Exponential mean φ(x) = ex log N

i=1 exi

N

Radical mean

φ(x) = c1/x

logc

N

i=1 c1/xi

N

−1 Basis-exponential mean φ(x) = xx m s.t. mm =

N

i=1 xxi i

N

Basis-radical mean φ(x) = x1/x m s.t. m1/m =

N

i=1 x1/xi i

N

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Aggregation functions

Functional equations (case 1.a). Example III in aggregation.
(Prop.

4.20) An operator C is separable in terms of a unique monotone increasing g (that is, C(a1, . . . , aN) = g(a1) ◦ · · · ◦ g(aN)) (with ◦ continuous, associative, and cancellative) C(a1, . . . , aN) = φ−1(

N

i=1

φ(g(ai))) and satisfies unanimity

C(a, . . . , a) = a

and positive homogeneity C(ra1, . . . , raN) = rC(a1, . . . , aN)

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Aggregation functions

Functional equations (case 1.a). Example III in aggregation (cont.)
if and only if C is either the root-mean-power

C(a1, . . . , aN) = ( 1 N

N

i=1

aα

i )1/α

with parameter α = 0 (RMPα) or the geometric mean C(a1, . . . , aN) = (

N

i=1

ai)1/N.

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Aggregation functions

Functional equations (case 1.a). Example III in aggregation (cont.)
if and only if C is either the root-mean-power

C(a1, . . . , aN) = ( 1 N

N

i=1

aα

i )1/α

with parameter α = 0 (RMPα) or the geometric mean C(a1, . . . , aN) = (

N

i=1

ai)1/N.

Remember Ex. 4.21. Assessing performance of Java runtime systems.

Only RMP and GM are acceptable !! (Note: limα→0 RMPα = GM)

Root-mean-powers, known as rth power mean, generalized mean.

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Aggregation functions

Functional equations (case 1.a). Example IV in aggregation.
(Prop. 4.24) When we add the equation of reciprocity

C(1/a1, . . . , 1/aN) = 1/C(a1, . . . , aN) the only operator C satisfying all conditions is the geometric mean C(a1, . . . , aN) = (

N

i=1

ai)1/N.

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Aggregation functions

Functional equations (case 1.a). Root-mean-powers

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,

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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).

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Aggregation functions

2. Heuristic definition: from functions to properties

function − → properties

We can use functional equations for this purpose:

characterizations of functions

We have propositions with “if and only if”

eq1, eq2, eq3 if and only if aggr. function so, given a function we know the fundamental properties of the function. E.g., geometric mean: a fundamental property is reciprocity

Note: characterizations are not unique

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Aggregation:

III. from the weighted mean to fuzzy integrals

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Aggregation:

III. from the weighted mean to fuzzy integrals
1. An example

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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.

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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 ...

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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|)

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Aggregation: example

Example. Formalization
Membership functions for constraints

1 2 3 4 5 6 7 µ0 µ2 µ3 µ6

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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

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Aggregation:

III. from the weighted mean to fuzzy integrals
2. WM, OWA, and WOWA operators

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SLIDE 56

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.

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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).

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SLIDE 58

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

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SLIDE 59

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

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SLIDE 60

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.

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SLIDE 61

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).

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SLIDE 62

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)

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SLIDE 63

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

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SLIDE 64

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)

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SLIDE 65

Aggregation:

III. from the weighted mean to fuzzy integrals
3. Choquet integral

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SLIDE 66

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???

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SLIDE 67

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)

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SLIDE 68

Choquet integrals

Example.
We need to evaluate students (who is best?) using marks in three

subjects X={Mathematics, Physics, Literature} (M,P,L)

pM = 0.4, pP = 0.4, pL = 0.2.

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???

p(Mathematics, Physics) ?

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SLIDE 69

Choquet integrals

Example.
We need to evaluate students (who is best?) using marks in three

subjects X={Mathematics, Physics, Literature} (M,P,L)

pM = 0.4, pP = 0.4, pL = 0.2.

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???

p(Mathematics, Physics) ?

. . . fuzzy measures µ(A)

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SLIDE 70

Choquet integrals

fuzzy measures µ(A): X ⊂ X = {M, P, L}

M P L

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SLIDE 71

Choquet integrals

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)

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SLIDE 72

Choquet integrals

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)

Similar to probability or standard (additive) measures,

but additivity condition is removed replaced by monotonicity

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SLIDE 73

Choquet integrals

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)

Similar to probability or standard (additive) measures,

but additivity condition is removed replaced by monotonicity

Why? to represent redundancy and support (for A ∩ B = ∅)

µ(A ∪ B) < µ(A) + µ(B) µ(A ∪ B) > µ(A) + µ(B)

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SLIDE 74

Choquet integrals

Fuzzy measures µ(A): Example with X = {M, P, L}
1. Boundary conditions:

µ(∅) = 0, µ({M, P, L}) = 1

2. Relative importance of scientific versus literary subjects:

µ({M}) = µ({P}) = 0.45, µ({L}) = 0.3

3. Redudancy between mathematics and physics:

µ({M, P}) = 0.5 < µ({M}) + µ({P})

4. Support between literature and scientific subjects:

µ({M, L}) = µ({P, L}) = 0.9 > µ({P}) + µ({L}) = 0.45 + 0.3 = 0.75 µ({M, L}) = µ({P, L}) = 0.9 > µ({M}) + µ({L}) = 0.45 + 0.3 = 0.75

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SLIDE 75

Choquet integrals

Now, we have a fuzzy measure µ(A)

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

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SLIDE 76

Choquet integrals

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

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

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SLIDE 77

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))

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SLIDE 78

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) = ∅

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SLIDE 79

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))],

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SLIDE 80

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.

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SLIDE 81

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

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SLIDE 82

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.

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SLIDE 83

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 CI with a symmetric measure 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.

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SLIDE 84

Aggregation:

III. from the weighted mean to fuzzy integrals
4. Weighted minimum and maximum

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SLIDE 85

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)

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SLIDE 86

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

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SLIDE 87

Aggregation:

III. from the weighted mean to fuzzy integrals
5. Sugeno integral

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SLIDE 88

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

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SLIDE 89

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) ( )

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SLIDE 90

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.

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SLIDE 91

Fuzzy integrals

Fuzzy integrals that generalize Choquet and Sugeno integrals
The fuzzy t-conorm integral
The twofold integral

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SLIDE 92

Aggregation:

IV. fuzzy measures

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SLIDE 93

Fuzzy measures

Definition:

(i) µ(∅) = 0, µ(X) = 1 (boundary conditions) (ii) A ⊆ B implies µ(A) ≤ µ(B) (monotonicity)

Difficulty:
2|X| − 2 values (because µ(∅) = 0, µ(X) = 1)
Solution: Families of fuzzy measures (to reduce complexity)

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SLIDE 94

Fuzzy measures

Examples of families:
⊥-Decomposable Fuzzy Measures (⊥ a t-conorm)

µ(A ∪ B) = µ(A)⊥µ(B). Therefore, for a given A: ⊥xi∈Av(xi)

Sugeno λ-measures (for λ > −1)

µ(A ∪ B) = µ(A) + µ(B) + λµ(A)µ(B) ⊥-decomposable for ⊥(x, y) = min(1, x + y + λxy).

Extensively used in computer vision applications

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SLIDE 95

Fuzzy measures

Examples of families:
Distorted probabilities (P: probability, f: increasing function)

µ(A) = f(P(A))

Well known in economics (decision)
m-dimensional distorted probability X1, . . . , Xm, Pi, f

µ(A) = f(P1(A ∩ X1), P2(A ∩ X2), · · · , Pm(A ∩ Xm)).

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SLIDE 96

Aggregation:

V. preference relations

(MCDM: social choice)

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SLIDE 97

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.

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SLIDE 98

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

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SLIDE 99

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

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SLIDE 100

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 preorder1 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

1either x y or y x Vicen¸ c Torra; Modeling decisions 2017 75 / 91

SLIDE 101

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

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SLIDE 102

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

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SLIDE 103

Aggregation for preference relations

Given preference relations, how aggregation is built?
Solutions

failing the condition

f

independence

f

irrelevant alternatives ⋆ Condorcet rule with Copeland2: ⋆ Borda count3

2Defined by Ramon Llull s. xiii 3Defined by Nicolas de Cusa s. xv. Vicen¸ c Torra; Modeling decisions 2017 78 / 91

SLIDE 104

Aggregation:

VI. Related topics

MCDM, utility functions, selection, Pareto

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SLIDE 105

Aggregation

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

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SLIDE 106

Aggregation

MCDM: Aggregation to deal with contradictory criteria

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SLIDE 107

Aggregation

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

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SLIDE 108

Aggregation

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.

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SLIDE 109

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

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SLIDE 110

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

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SLIDE 111

Aggregation for (numerical) utility functions

MCDM: we aggregate utility, and order according to utility
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

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SLIDE 112

Aggregation:

VI. Related topics

Hierarchical models

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SLIDE 113

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.

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SLIDE 114

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SLIDE 119

References

Torra, V., Narukawa, Y. (2007) Modeling decisions: Information fusion

and aggregation operators, Springer

Torra, V. (2014) Cuando las matem´

aticas van a las urnas. Los procesos de decisi´

n, RBA.

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SLIDE 120

References

Torra, V., Narukawa, Y. (2007) Modeling decisions: Information fusion

and aggregation operators, Springer

Torra, V. (2014) Cuando las matem´

aticas van a las urnas. Los procesos de decisi´

n, RBA.
Beliakov, G., Bustince, H., Calvo, T. (2016) A Practical Guide to

Averaging Functions, Springer.

Grabisch, M., Marichal, J.-L., Mesiar, R., Pap, E. (2009) Aggregation

Functions, Cambridge University Press.

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