SUOWA operators: Constructing semi-uninorms and analyzing specific cases

Llamazares, Bonifacio

doi:10.1016/j.fss.2015.02.017

Cited by 19 publications

(27 citation statements)

References 32 publications

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“…Corollary Let

bold-italicw

be a weighting vector such that

w_{1} \leq w_{2} \leq \dots \leq w_{n}

. Then, for any weighting vector

bold-italicp

υ_{bold-italicp, bold-italicw}^{U_{P}}

is a normalized capacity on

N

.Proof The proof is immediate taking into account that if

w_{1} \leq w_{2} \leq \dots \leq w_{n}

, then

bold-italicw = bold-italic η

\sum_{i = 1}^{j} w_{i} < j ∕ n

for all

j \in {1, \dots, n - 1}

(see Lemma 1 in Llamazares). So, in both cases,

\frac{1}{j} \sum_{i = 1}^{j} w_{i} \leq \min (w_{j + 1}, 1 ∕ n)

for all

j \in {1, \dots, n - 1}

.…”

Section: Some New Resultsmentioning

confidence: 93%

“…In addition to the previous ones, continuous semiuninorms can be obtained through a procedure introduced in Llamazares

\begin{matrix} right U_{T_{L}} (x, y) & center = & left \{\begin{matrix} left \max (x, y) & left if (x, y) \in {[1 ∕ n, 1]}^{2}, \\ left \max (x + y - 1 ∕ n, 0) & left otherwise . \end{matrix} \\ right U_{\overset{P}{true}} (x, y) & center = & left \{\begin{matrix} left \max (x, y) & left if (x, y) \in {[1 ∕ n, 1]}^{2}, \\ left n x y & left otherwise . \end{matrix} \\ right U_{T_{M}} (x, y) & center = & left \{\begin{matrix} left \max (x, y) & left if (x, y) \in {[1 ∕ n, 1]}^{2}, \\ left \min (x, y) & left if (x, y) \in [0, 1 ∕ n {())}^{2}, \\ left x + y - 1 ∕ n & left otherwise . \end{matrix} \\ right U_{P} (x, y) & <...> \end{matrix}

…”

Section: Suowa Operatorsmentioning

confidence: 99%

See 1 more Smart Citation

SUOWA operators: A review of the state of the art

Llamazares

2018

Int J Intell Syst

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SUOWA operators are a particular case of Choquet integral that simultaneously generalize weighted means and OWA operators. Because they are constructed by using normalized capacities, they possess properties such as continuity, monotonicity, idempotency, compensativeness, and homogeneity of degree 1. Besides these ones, some articles published in recent years have shown that SUOWA operators also exhibit other interesting properties. So, we think that the time has come to summarize existing knowledge of these operators. The aim of this paper is to collect the main results obtained so far on SUOWA operators. Moreover, we also introduce some new results and illustrate the usefulness of SUOWA operators by using an example given by Beliakov (2018).

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“…Corollary Let

bold-italicw

be a weighting vector such that

w_{1} \leq w_{2} \leq \dots \leq w_{n}

. Then, for any weighting vector

bold-italicp

υ_{bold-italicp, bold-italicw}^{U_{P}}

is a normalized capacity on

N

.Proof The proof is immediate taking into account that if

w_{1} \leq w_{2} \leq \dots \leq w_{n}

, then

bold-italicw = bold-italic η

\sum_{i = 1}^{j} w_{i} < j ∕ n

for all

j \in {1, \dots, n - 1}

(see Lemma 1 in Llamazares). So, in both cases,

\frac{1}{j} \sum_{i = 1}^{j} w_{i} \leq \min (w_{j + 1}, 1 ∕ n)

for all

j \in {1, \dots, n - 1}

.…”

Section: Some New Resultsmentioning

confidence: 93%

“…In addition to the previous ones, continuous semiuninorms can be obtained through a procedure introduced in Llamazares

\begin{matrix} right U_{T_{L}} (x, y) & center = & left \{\begin{matrix} left \max (x, y) & left if (x, y) \in {[1 ∕ n, 1]}^{2}, \\ left \max (x + y - 1 ∕ n, 0) & left otherwise . \end{matrix} \\ right U_{\overset{P}{true}} (x, y) & center = & left \{\begin{matrix} left \max (x, y) & left if (x, y) \in {[1 ∕ n, 1]}^{2}, \\ left n x y & left otherwise . \end{matrix} \\ right U_{T_{M}} (x, y) & center = & left \{\begin{matrix} left \max (x, y) & left if (x, y) \in {[1 ∕ n, 1]}^{2}, \\ left \min (x, y) & left if (x, y) \in [0, 1 ∕ n {())}^{2}, \\ left x + y - 1 ∕ n & left otherwise . \end{matrix} \\ right U_{P} (x, y) & <...> \end{matrix}

…”

Section: Suowa Operatorsmentioning

confidence: 99%

SUOWA operators: A review of the state of the art

Llamazares

2018

Int J Intell Syst

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“…There are some studies that are working on the construction of weighted uninorm operators, such as the work of

\overset{S}{true}

abo et al., Mas et al., Grabisch et al., and Llamazares . However, our methods in this paper have significant advantages over their methods.…”

Section: Introductionmentioning

confidence: 92%

“…More specifically, in this section we shall give two methods of constructing uninorm weighting operators: the first one is to construct a uninorm weighting operator from t‐norms and t‐conorms, and the second is to do it from an existing uninorm weighting operator. A benefit of doing so is that a number of researchers have already worked out some methods of constructing uninorm weighting operators; actually, their basic ideas can be traced back to the work of Silvert, and similar work has been proposed by

\overset{S}{true}

abo et al., Mas et al., Grabisch et al., and Llamazares . Therefore, we can obtain many of the various uninorm weighting operators on the same line.…”

Section: Construction Of Uninorm Weighting Operatorsmentioning

confidence: 99%

A Spectrum of Weighted Compromise Aggregation Operators: A Generalization of Weighted Uninorm Operator

Luo

Zhong

Leung

2015

Int. J. Intell. Syst.

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(2-3): 161-184, 2007. Luo and Jennings identify and analyze the complete spectrum of compromise aggregation operators that can be used to model the various attitudes that decision-making agents can have toward risk in aggregation. In this paper, we extend these operators to deal with aggregation when the ratings have different degrees of importance. Specifically, we generalize the method of weighted uninorms to handle this issue. We choose this approach because uninorm compromise operators are a kind of common ones, and their weighted counterparts, which are widely accepted, can cover other common operators, such as weighted t-norms and t-conorms, as special cases. As per the analysis of weighted uninorms, we identify common properties that the weighting operators of the various compromise operators should satisfy, and in so doing, we introduce the concept of a general weighting operator for compromise operators and reveal the different properties that a specific type of weighting operator should obey. This, in turn, defines the concepts of the various weighting operators of the various compromise operators. We then go onto discuss the construction issue of weighting operators associated with the various compromise operators. C 2015 Wiley Periodicals, Inc.

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“…In addition to the previous ones, several procedures to construct semiuninorms have been introduced by Llamazares . One of them, which is based on ordinal sums of aggregation operators, allows us to get continuous semiuninorms.…”

Section: Preliminariesmentioning

confidence: 99%

A Behavioral Analysis of WOWA and SUOWA Operators

Llamazares

2016

Int. J. Intell. Syst.

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Weighted ordered weighted averaging (WOWA) and semiuninorm-based ordered weighted averaging (SUOWA) operators are two families of aggregation functions that simultaneously generalize weighted means and OWA operators. Both families can be obtained by using the Choquet integral with respect to normalized capacities. Therefore, they are continuous, monotonic, idempotent, compensative and homogeneous of degree 1 functions. Although both families fulfill good properties, there are situations where their behavior is quite different. The aim of this paper is to analyze both families of functions regarding some simple cases of weighting vectors, the capacities from which they are building, the weights affecting the components of each vector, and the values they return.

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SUOWA operators: Constructing semi-uninorms and analyzing specific cases

Cited by 19 publications

References 32 publications

SUOWA operators: A review of the state of the art

SUOWA operators: A review of the state of the art

A Spectrum of Weighted Compromise Aggregation Operators: A Generalization of Weighted Uninorm Operator

A Behavioral Analysis of WOWA and SUOWA Operators

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