Download e-book for kindle: Aggregation Operators: New Trends and Applications by Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko

By Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko Mesiar (auth.), Professor Tomasa Calvo, Professor Gaspar Mayor, Professor Radko Mesiar (eds.)

ISBN-10: 3662003198

ISBN-13: 9783662003190

ISBN-10: 3790817872

ISBN-13: 9783790817874

1. The expanding variety of learn papers seemed within the final years that both utilize aggregation services or give a contribution to its theoretieal research asses its starting to be significance within the box of Fuzzy Logie and in others the place uncertainty and imprecision play a proper position. on account that those papers are pub­ lished in lots of journals, few books and several other lawsuits of meetings, books on aggregation are partieularly welcome. To my wisdom, "Agrega­ tion Operators. New developments and functions" is the 1st booklet aiming at generality , and that i take it as a honour to write down this Foreword based on the mild call for of its editors, Radko Mesiar, Tomasa Calvo and Gaspar Mayor. My excitement additionally derives from the truth that 20 years aga i used to be one of many first Spaniards attracted to the research of aggregation capabilities, and this booklet contains paintings through numerous Spanish authors. The publication comprises great and appropriate unique papers, authored through essentially the most notable researchers within the box, and because it might serve, because the editors indicate within the Preface, as a small guide on aggregation, the ebook is particularly helpful for these getting into the topic for the 1st time. The ebook additionally includes aside facing power parts of software, so it may be priceless in gaining perception at the destiny developments.

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U [0, l]n ~ [0,1] is a symmetric sum if and only if there is an aggregation operator B: U [0, l]n ~ [0,1] Proposition 3. An aggregation operator A: nEN nEN such that A = Bü where B Ü( Xl, ... with convention ,Xn )_ - B ( B(xl, ... ,xn ) Xl, ... , X n ) + B (1 - Xl, ... 5. Observe that symmetrie sums ean be characterized either by the equality A d or by the equality A = AU. However, the ~-operator introdueed in (67) gives a hint how to construct symmetrie sums. Moreover, we ean introduee an equivalence relation", on the class of all aggregation operators, namely, A '" B if and only if Aü = Bü.

1: m; ;=1 However, applying (68) to the max operator, we get operator also discussed in [13], maxm(xl. , x n ) = max{xi; i E {I, ... , n}, mi > O}. Hence, if all weights mi are positive, maxm = max. Because of the rieh variety of aggregation operators, there is no general approach leading to a satisfactory incorporation of weights into a symmetrie aggregation. Therefore we sketch only some of the most important approaches. First note that if the weights are successfully incorporated into an aggregation operator A, then we can do the same in the case of any transformed aggregation operator AI(>' Based on (68), we can define the weighted arithmetic mean Mv for any non-zero weighting vector v = (Vb ...

Because of the boundary condition (9), any aggregation operator fulfilling at least one of the properties introduced in Definition 13 is idempotent. Each shift-invariant aggregation operator can be constructed (and characterized) as folIows. Take any aggregation operator B: U [O,I]n -t [0,1] and define the operator SB: U [0, l]n -t [0,1] nEI\I by nEI\I (53) where a = min(xl,"" x n ). Evidently, SB is a shift-invariant operator. Further, an aggregation operator A: U [O,I]n -t [0,1] is shift-invariant if and nEI\I = SA.

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Aggregation Operators: New Trends and Applications by Tomasa Calvo, Anna Kolesárová, Magda Komorníková, Radko Mesiar (auth.), Professor Tomasa Calvo, Professor Gaspar Mayor, Professor Radko Mesiar (eds.)


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