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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.)

1. The expanding variety of examine papers seemed within the final years that both utilize aggregation capabilities or give a contribution to its theoretieal research asses its becoming value within the box of Fuzzy Logie and in others the place uncertainty and imprecision play a suitable position. considering those papers are pub­ lished in lots of journals, few books and several other court cases of meetings, books on aggregation are partieularly welcome. To my wisdom, "Agrega­ tion Operators. New tendencies and purposes" is the 1st e-book aiming at generality , and that i take it as a honour to jot 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 features, and this e-book contains paintings by way of numerous Spanish authors. The publication comprises great and suitable unique papers, authored via one of the most awesome researchers within the box, and because it will probably serve, because the editors indicate within the Preface, as a small guide on aggregation, the booklet is particularly helpful for these getting into the topic for the 1st time. The e-book additionally comprises aside facing strength parts of program, so it may be invaluable in gaining perception at the destiny developments.

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Example text

N i=l while for I(x) = log x we have A (log)( Xl,···,Xn ) -_ rr k (i) (i) Ai(xl , ... , X n b· i=l ) . • Remarkably, if all operators Ai are t-norms (t-conorms, copulas, dual copulas) then for any I, A (f) is the standard ordinal sum of t-norms (t-conorms, copulas, dual copulas). Observe that (77) is a special case of the D-based ordinal sum construction [95] where D: U [O,l]n --+ [0,1] is asymmetrie continuous idempotent nEN aggregation operator which is strietly monotone (cancellative) on Then the D-ordinal sum A D Ai: : U [ai-I.

Bß is just the Sugeno A- addition introduced in [119] for A = ß - 1. 1. For more examples and applications of these aggregation operators we refer the reader to [138,25]. An alternative approach was proposed in [69]. , A,r(Xl, ... ,xn) = A (tin) (xt), ... ,J~n) (x n)) , where :F = (ti(n) In E N, i E {I, ... ,n}) is a system of [0,1] decreasing mappings with fixed points and 1. ° --? [0,1] non- 43 Example 7 Put fi(n) : [0,1]-+ [0,1], fi(n)(X) = x l / n , n E N, i E {I, ... ,n}. As f~n) (x) ;::: x, we augment the input values.

Similarly, starting from the product operator II acting either on [0,1] or [0,00], by the described procedure we get the geometrie mean G. Further, the idempotization of the operator Q given in (30) by n Q(X1,'''' xn ) = I1 xL i=l leads to the weighted geometrie mean GLl : U[0, 1t -+ [0,1], n GLl(XI, ... , x n ) = nEN with weights Win = I1 xY'in , i=l n(~~l)' i = 1, ... , n, n E N, see also (112). , max = I max and min = Imin. , the Einstein sum E[l] from (41), whose binary form is X+y E[l](X,y) = - - .

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