International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems

Previous work has shown that the Choquet integral w.r.t. a symmetric k-order additive fuzzy measure is characterized as a weighted sum, with possibly negative weights, of k specific OWA operators, and that the Sugeno integral w.r.t. a symmetric k-order maxitive fuzzy measure is characterized as a weighted maximum of k order statistics. In this paper, these characterizations are extended to the general, non-symmetric case, thereby providing additional insight in the symmetric case. More specifically, a new general decomposition of the Choquet integral w.r.t. a k-order additive fuzzy measure and of the Sugeno integral w.r.t. a k-order maxitive fuzzy measure are established.

The intepretation of aggregation functions in multicriteria decision making is often based on indices such as importance indices that measure the importance of one criterion. For instance, for the Choquet integral, the importance index is the so-called Shapley value. The use of an index must always be limited to the specific context it has been designed for. Here, we are interested in determining on which criteria acts should be improved if we want their global evaluation to increase as much as possible. The Shapley value is not suited for describing this. So, we introduce a new index of importance which represents the mean worth for acts to reach higher scores in a set of criteria. This index is defined for general aggregation functions with the help of several axioms. This importance index is then applied to the Choquet integral. In particular, we computer the worth to reach higher levels in one attribute, and in a couple of attributes. Interestingly, this leads to quantities that are closely related to the Shapley and interaction indices.

We shall consider non-associative and non-commutative pseudo-addition and pseudo-multiplication, i.e., generalized pseudo-operations. More precise, we shall consider special class of generalized pseudo-operations that have the following form: x⊕ y=k^-1(ε k(x)+k(y)), x⊙ y=k^-1(k(x)^ε k(y)), where ε is arbitrary fixed positive real number and k is a positive strictly monotone function. Using previous pseudo-operations, corresponding pseudo-measure and pseudo-integral will be introduced. Pseudo-convolution based on pseudo-measure and pseudo-integral will be constructed.

In the paper the possibility of the construction of relevancy transformation operators introduced in ⁹ by means of triangular norms and copulas is studied. Several examples are included.

Generalizing the construction of weighted quasi–arithmetic means from generated t–conorms, we propose a new class of weighted means related to continuous t–conorms. By duality, t–norm based weighted means can be obtained, too. Our method fits also in the case of finite ordinal scales. Several examples are given.

The natural properties of the aggregation operators and the most elementary ones are the idempotence, the monotonicity and the continuity from below. We assume only these properties for the aggregation operators with infinitely many inputs, defined by functionals on the family of measurable functions. A family of fuzzy measures is associated with each aggregation operator. The properties of horizontal or vertical pseudo-additivity are recognizable by means of this family of fuzzy measures.

Generated aggregation operators on continuous and discrete scales derived by means of a generating system are introduced. Linear generating systems are shown to be equivalent with the class of weighted means. The following classes are characterized: symmetric generated aggregation operators, idempotent generated aggregation operators, generated aggregation operators with a neutral element. Several examples are included.

Collapsed input-based aggregation leading to constrained aggregation operators is discussed. Constrained aggregation operators eliminating a non–distinguishing evaluation are studied. Moreover, collapse invariant aggregation operators are defined and a complete characterization of symmetric collapse invariant aggregation operators, which are independent of the position of the collapsed input, is given. Several examples are included.

We deal with the system of functional equations I(x,T(y,z))=T(I(x,y), I(x,z)), I(x,I(y,z))=I(T(x,y),z), where function T:[0,1]²→ [0,1] is a strict t-norm and I:[0,1]²→ [0,1] is an unknown function. Under some assumptions imposed on function I we obtain that I is almost conjugate with the Yager fuzzy implication.

Transformation of the Minimum t-norm T_M by means of a non-decreasing transformation φ of the unit interval [0, 1] yielding triangular norm is studied. Full characterization of such φ is given. In the same spirit, transformations of the Drastic product T_D are studied.

Some aspects of the relationship between Goodman and Nguyen's one-point coverage interpretation of a fuzzy set and Zadeh's possibilistic interpretation are discussed. As a result of this, we derive a new interpretation of the strong α-cut of a normalized fuzzy set, namely that of being the most precise set we are sure to contain an unknown parameter with probability greater than or equal to 1-α.

Bell-type inequalities, used in mathematical physics as a criterion to check whether a physical situation allows description in terms of classical (Kolmogorovian) or quantum probability calculus are applied to various fuzzy probability models. It occurs that the standard set of Bell-type inequalities does not allow to distinguish Kolmogorovian probabilities from fuzzy probabilities based on the most frequently used Zadeh intersection or probabilistic intersection, but it allows to distinguish all these models from fuzzy probability models based on Giles (Łukasiewicz) intersection. It is proved that if we use fuzzy set intersections pointwisely generated by Frank's fundamental triangular norms T_s(x,y), then the borderline between fuzzy probability models that can be distinguished from Kolmogorovian ones and these fuzzy probability models that cannot be distinguished is for .

This section is maintained by Vladik Kreinovich. Please send you abstract (or copies of papers that you want to see reviewed here) to vladik@cs.utep.edu, or by regular mail to: Vladik Kreinovich, Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968, USA.