Generation of Weighting Triangles Associated with Aggregation Functions

Calvo, Tomasa; Mayor, Gaspar; Torrens, Joan; Suñer, Jaume; Mas, Margarita; Carbonell, M.

doi:10.1016/s0218-4885(00)00029-0

Cited by 19 publications

(14 citation statements)

References 8 publications

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“…Then, in particular, if aggregation operators F and G of last theorem are taken with the same weighting triangle, cases (2)-(4) are equivalent. Note that, in the case where OWA operators F and G of previous theorem coincide, cases (3) and (4) are equal, that is, the characterization is the same for the two mixed orders, 6 ÿ and 6 ÿ : a double aggregation operator A F;F;H , where F is an OWA operator, is 6 ÿ -monotone (6 ÿ -monotone) if and only if the OWA's weighting triangle is, according to [4], a left regular triangle.…”

Section: Monotonicitymentioning

confidence: 94%

“…Weighting triangles that satisfy the property given in case (2) are called left descending triangles [4]. In addition, remark that any triangle that is left descending also veriÿes the condition given in case (1).…”

Section: Monotonicitymentioning

confidence: 99%

“…This characterization is interesting since, along with already known results on the 6 i -monotonicity (i ∈ { ; ÿ}) of some ordinary aggregation operators [4], it allows to describe some classes of 6 ij -monotone double aggregation operators. With that purpose, ÿrst we recall the deÿnition of two important ordinary aggregation operators: quasi-linear weighted means and ordered weighted averagings (OWA) (see e.g.…”

Section: Monotonicitymentioning

confidence: 99%

“…[2]). Both of them are weighted operators deÿned by means of a weighting triangle, i.e., a set of numbers = (w i;n ) n∈N such that w i;n ∈ [0; 1] and n i=1 w i;n = 1 for all n ∈ N (for more details on weighting triangles see [4]). …”

Section: Monotonicitymentioning

confidence: 99%

“…In both cases, the proof is obtained applying Theorem 1 and the characterizations for 6 i -monotone weighted operators given in [4].…”

Section: Monotonicitymentioning

confidence: 99%

See 4 more Smart Citations

Double aggregation operators

Calvo

Pradera

2004

Fuzzy Sets and Systems

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Section: Monotonicitymentioning

confidence: 94%

Section: Monotonicitymentioning

confidence: 99%

Section: Monotonicitymentioning

confidence: 99%

Section: Monotonicitymentioning

confidence: 99%

“…In both cases, the proof is obtained applying Theorem 1 and the characterizations for 6 i -monotone weighted operators given in [4].…”

Section: Monotonicitymentioning

confidence: 99%

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Double aggregation operators

Calvo

Pradera

2004

Fuzzy Sets and Systems

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Domination of ordered weighted averaging operators over t-norms

Mesiar¹,

Saminger²

2003

Soft Computing

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The fusion of transitive fuzzy relations preserving the transitivity is linked to the domination of the involved aggregation operator. The aim of this contribution is to investigate the domination of OWA operators over t-norms whereas the main emphasis is on the domination over the Łukasiewicz t-norm. The domination of OWA operators and related operators over continuous Archimedean t-norms will also be discussed.Keywords Domination, OWA operators MotivationIn several applications of fuzzy logics and fuzzy systems the processing of data based on the strongest t-norm, i.e., the minimum T M suffers from the increase of uncertainty.Recall only the addition of fuzzy numbers where the final sum spreads equal to the sum of all incoming spreads. In order to avoid this undesirable effect, alternative approaches have to be taken into account. One of the most promising t-norms for reducing the enormous growth of uncertainty is the Łukasiewicz t-norm T L . The addition of triangular (trapezoidal) fuzzy numbers based on T L leads to output spreads equal to the maximal (left and right) incoming spreads -a property which is often required in models dealing with uncertainty. Moreover, the Łukas-iewicz t-norm T L is often applied when fuzzy rule based systems are designed with the purpose to reduce redundancy, and especially in clustering algorithms (likeness relations of Bezdek and Harris [2]). Recall also the approximate solutions of fuzzy relational equations [6] or several relations possessing T L -transitivity as a genuine counterpart of the classical triangle inequality of the Euclidean metric on R. Furthermore, when solving complex problems we sometimes need either to refine several fuzzy relations or to introduce their Cartesian product. However, we expect that the new fuzzy relation will be again T L -transitive if the original fuzzy relations have also been T L -transitive.As it has been shown in [11], the preservation of T-transitivity of fuzzy relations during an aggregation process is guaranteed if the involved aggregation operator dominates the corresponding t-norm T. Several special operators dominating the Łukasiewicz t-norm are already known, e.g. T L itself, the minimum and the arithmetic mean, see [11]. The later two are special cases of so called OWA operators, one of the most important family of aggregation operators applied in many domains (see [14,16]), which are used for summarizing singular data into a single output where inputs are ordered with respect to their value and are assigned certain weights before being aggregated.If we want to preserve the T L -transitivity of fuzzy relations the domination of an OWA operator O over T L should be checked before fusing the T L -transitive fuzzy relations by means of this OWA operator O. Therefore the characterization of OWA operators dominating the Łukasiewicz t-norm T L seemed to be important. The extension of the obtained results for some other t-norms and other dominating aggregation operators will be the final task of this contribution. Preliminaries 2.1

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Stability of weighted penalty-based aggregation functions

Beliakov

Liu

2013

Fuzzy Sets and Systems

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In many practical applications, the need arises to aggregate data of varying dimension. Following from the self-identity property, some recent studies have looked at the stability of aggregation operators in terms of their behavior as the dimensionality is increased from n−1 to n. We use the penalty-based representation of aggregation functions in order to investigate the conditions for weighting vectors associated with some important weighted families, extending on the results already established for quasi-arithmetic means. In particular, we obtain results for quasi-medians and functions that involve a reordering of the inputs such as the OWA and order statistics.

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Generation of Weighting Triangles Associated with Aggregation Functions

Cited by 19 publications

References 8 publications

Double aggregation operators

Double aggregation operators

Domination of ordered weighted averaging operators over t-norms

Stability of weighted penalty-based aggregation functions

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