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 features or give a contribution to its theoretieal examine asses its growing to be value within the box of Fuzzy Logie and in others the place uncertainty and imprecision play a proper function. when you consider that those papers are pub­ lished in lots of journals, few books and a number of other complaints of meetings, books on aggregation are partieularly welcome. To my wisdom, "Agrega­ tion Operators. New tendencies and functions" is the 1st ebook aiming at generality , and that i take it as a honour to write down this Foreword according to the light call for of its editors, Radko Mesiar, Tomasa Calvo and Gaspar Mayor. My excitement additionally derives from the truth that two decades aga i used to be one of many first Spaniards drawn to the learn of aggregation services, and this publication contains paintings via a number of Spanish authors. The e-book comprises great and proper unique papers, authored via probably the most impressive 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 worthy 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 important in gaining perception at the destiny developments.

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0) . A = A for all considered u, are appropriate operators for evaluating information with possible collapse of some of the inputs, for instance if the i-th sensor is broken (with unknown i). 5 , (85) 42 and Ko. 5 = M. , the dual operator to 11, and they are constrained invariant for all u E [0,1] up to or 1, respectively. Observe that for each a E [0,1], a = K (O,l) = K (l,O), that is, the parameter a corresponds to the aggregation output when aggregating the contradictory inputs and 1. ° Q ° Q Augmenting and decreasing There are several reasons for augmenting (or decreasing) the output given by chosen aggregation operators.

Define two opemtors A (w) = « ai, bi , Ai > li = 1, ... , k)w : = U[0, 1t --+ [0,1] nEN and A(S) =« ai,bi,A i > li = 1, ... ,k)s: U [O,lt --+ [0,1] nEJIl by where ak+l = 1 and a = min(xl, ... and ° ,Xn ), where bo = and b = max(xl, ... ,xn ). Then A(w) and A(s) are the weakest and the strongest aggregation opemtors, respectively, such that the corresponding restrictions to the inputs from [ai, bi ] 36 coincide with aggregation operators Ai, i = 1, ... , k. The operator A (w) is called the lower ordinal sum (of aggregation operators Ai), while A(s) is calted the upper ordinal sum (oi aggregation operators Ai).

2, (81) where (x~, ... , x~) is a non-decreasing permutation of (Xl, ••• , X n ). (ii) Similarly to (i), D k = Mk'in is an OWA operator related to a decreasing quantifier q(x) = (1- x)k, n Dk(X1, ... )k - (I_t~ . I)k Ix~, I-~ = M? 3. (iv) The operator G~, k > 1, is a new operator not belonging to any of classes discussed in next sections. For example, for n = 2 we obtain while In general, 41 Proposition 6 describes two-step aggregation. First, the inputs are aggregated by means of operators B1.

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