By Grabisch M., Marichal J.-L., Mesiar R., Pap E.

Aggregation is the method of mixing a number of numerical values right into a unmarried consultant price, and an aggregation functionality plays this operation. those features come up anywhere aggregating info is necessary: utilized and natural arithmetic (probability, facts, determination conception, practical equations), operations examine, computing device technological know-how, and lots of utilized fields (economics and finance, trend attractiveness and photograph processing, facts fusion, etc.). it is a finished, rigorous and self-contained exposition of aggregation services. periods of aggregation services coated comprise triangular norms and conorms, copulas, capacity and averages, and people in accordance with nonadditive integrals. The houses of every strategy, in addition to their interpretation and research, are studied extensive, including development tools and sensible id tools. certain cognizance is given to the character of scales on which values to be aggregated are outlined (ordinal, period, ratio, bipolar). it really is a great advent for graduate scholars and a different source for researchers

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X(n) ) (idempotency). 75. 74, we can readily see that, in the statement of the lemma, idempotency can be relaxed to the following condition: ran(F(n) ) ⊆ ran(F(kn) ) for all k, n ∈ N. 76. If F : ∪n∈N In → I is strongly decomposable and idempotent then, for any n, p ∈ N and any x(1) , . . , x(n) ∈ Ip , we have F(x(1) , . . , x(n) ) = F F(x(1) ), . . , F(x(n) ) . In particular, F is strongly idempotent. Proof. We simply have F(x(1) , . . , x(n) ) = F p · F(x(1) ), . . , p · F(x(n) ) = F F(x (1) ), .

Of course, we could as well consider only one candidate, n judges, and p criteria (assuming commensurateness of the scores along the criteria). In this latter setting, strong bisymmetry seems very natural as well. Obviously, for any strongly bisymmetric extended function F : ∪n∈N In → I, each n-ary function F(n) is bisymmetric. Moreover, we have the following result [284]. 84. If F : ∪n∈N In → I is strongly decomposable and idempotent then it is strongly bisymmetric. In particular F(2) is bisymmetric.

For more details, see for instance [180, 350, 371]. 16. 10). (ii) If f : [a, b] → R is absolutely continuous, it is of bounded variation on [a, b], so it is differentiable almost everywhere on [a, b], and its derivative is integrable over [a, b]. 17. Let f : [a, b] → R. Then the following are equivalent: (i) There is an integrable real-valued function g : [a, b] → R such that x f (x) = f (a) + g(t) dt a for every x ∈ [a, b]. x (ii) a f (t) dt exists and is equal to f (x) − f (a) for every x ∈ [a, b].