Generalized qualitative Sugeno integrals

Abstract

Sugeno integrals are aggregation operations involving a criterion weighting scheme based on the use of set functions called capacities or fuzzy measures. In this paper, we define generalized versions of Sugeno integrals on totally ordered bounded chains, by extending the operation that combines the value of the capacity on each subset of criteria and the value of the utility function over elements of the subset. We show that the generalized concept of Sugeno integral splits into two functionals, one based on a general multiplevalued conjunction (we call integral) and one based on a general multiple-valued implication (we call cointegral). These fuzzy conjunction and implication connectives are related via a so-called semiduality property, involving an involutive negation. Sugeno integrals correspond to the case when the fuzzy conjunction is the minimum and the fuzzy implication is Kleene-Dienes implication, in which case integrals and cointegrals coincide. In this paper, we consider a very general class of fuzzy conjunction operations on a finite setting, that reduce to Boolean conjunctions on extreme values of the bounded chain, and are nondecreasing in each place, and the corresponding general class of implications (their semiduals). The merit of these new aggregation operators is to go beyond pure lattice polynomials, thus enhancing the expressive power of qualitative aggregation functions, especially as to the way an importance weight can affect a local rating of an object to be chosen.

General form of Chebyshev type inequality for generalized Sugeno integral

“…The most important examples of fuzzy conjunction are: the Gödel conjunction ⊗ G (see Example 2.4) and the contrapositive Gödel conjunction a ⊗ GC b = a½ {a>1−b} (a, b). Dubois et al [8] introduced and studied the q-integral defined as…”

“…This definition is motivated by alternative ways of using weights of qualitative criteria in min-and max-based aggregations, that make intuitive sense as tolerance thresholds. Note that the research on Chebyshev type inequality for q-integral has been initiated by Kaluszka et al [13] even before its formal definition by Dubois [8], see for example [13, ϕ i (1) for i = 1, 2, 3. Then the following statements are equivalent:…”

New Directions in Ordinal Evaluation: Sugeno Integrals and Beyond

“…If the fuzzy conjunction ⊗ satisfies λ ⊗ 1 = λ, then under the assumptions of Theorem 6, the functional I is of the form I(f ) = ∫ ⊗ µ f where µ(A) = I(1 A ). There is a specific result in [57] when the functional I is fully maxitive, to characterize possibilistic q-integrals of the form I(f ) = ∫ ⊗ Π f . Since the fuzzy conjunction ⊗ is not supposed to be commutative, there is a companion q-integral ∫ ⋆ µ f with λ ⋆ λ ′ = λ ′ ⊗ λ and a similar characterization result.…”

“…Note that the homogeneity condition I(λ ∨ 1 A ) = ρ c (A) → λ for q-cointegrals is better understood if we express the latter expression the symmetric contrapositive of a residual fuzzy implication induced by a conjunction having two-sided identity 1, such as ∧, for which we have λ → CG 0 = 1 − λ). There is a specific result in [57] when the functional I is fully minitive, to characterize possibilistic q-integrals of the form I(f ) = ∫ → N f . In the same paper, representation results for the companion q-cointegral defined from a q-cointegral by contrapositive symmetry are proposed.…”

“…[57] Let I : L C → L be a mapping. There are a capacity µ and a fuzzy conjunction ⊗ such thatI(f ) = ∫ ⊗ µ f for every f ∈ L C ifand only if (i) I(f ∨ g) = I(f ) ∨ I(g), for any comonotone f, g ∈ L C .…”