Alexandra Šipošová

On some Transformations of Fuzzy Measures

Abstract In this paper, some transformations of fuzzy measures are reviewed. Then, based on them, two new transformations are introduced and their properties are investigated. Also, some examples are provided to illustrate these notions and invariantness under them.

A note on the superadditive and the subadditive transformations of aggregation functions

We expand the theoretical background of the recently introduced superadditive and subadditive transformations of aggregation functions. Necessary and sufficient conditions ensuring that a transformation of a proper aggregation function is again proper are deeply studied and exemplified. Relationships between these transformations are also studied.

Aggregation functions with given super-additive and sub-additive transformations

Aggregation functions and their transformations have found numerous applications in various kinds of systems as well as in economics and social science. Every aggregation function is known to be bounded above and below by its super-additive and sub-additive transformations. We are interested in the “inverse” problem of whether or not every pair consisting of a super-additive function dominating a sub-additive function comes from some aggregation function in the above sense. Our main results provide a negative answer under mild extra conditions on the super- and sub-additive pair. We also show that our results are, in a sense, best possible.

On the existence of aggregation functions with given super-additive and sub-additive transformations

Abstract In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, A ↦ A ⁎ and A ↦ A ⁎ , of an aggregation function A. We prove that if A ⁎ has a slightly stronger property of being strictly directionally convex, then A = A ⁎ and A ⁎ is linear; dually, if A ⁎ is strictly directionally concave, then A = A ⁎ and A ⁎ is linear. This implies, for example, the existence of pairs of functions f ≤ g sub-additive and super-additive on [ 0 , ∞ [ n , respectively, with zero value at the origin and satisfying relatively mild extra conditions, for which there exists no aggregation function A on [ 0 , ∞ [ n such that A ⁎ = f and A ⁎ = g .

Super- and subadditive constructions of aggregation functions

Transformation of partially given aggregation function.Generalization of decomposition integral.Subadditive and superadditive fusion of information. Two construction methods for aggregation functions based on a restricted a priori known decomposition set and decomposition weighing function are introduced and studied. The outgoing aggregation functions are either superadditive or subadditive. Several examples, including illustrative figures, show the potential of the introduced construction methods. Our approach generalizes several known constructions and optimization methods, including decomposition and superdecomposition integrals. We present also an economic applications of the introduced concepts.

Event-based transformations of capacities and invariantness

We discuss and study several kinds of capacity transformations based on a given event B and a fixed implication function I. In particular, local, global and total invariantness of these transformations are examined and described.

Continuous functions with given super-additive and sub-additive transformations

ABSTRACT Results of F. Kouchakinejad and the authors [Internat. J. General Systems 46 (2017), 225–234] imply that if are continuous functions with zero value at the origin, such that , f is strictly super-additive and g is sub-additive but not linear, then there is no continuous aggregation function such that f and g are super-additive and sub-additive transformations of h, respectively. In this paper we extend this result to all continuous functions , as well as to a corresponding dual statement obtained by swapping the roes of super- and sub-additivity. We also show that the continuity assumption here cannot be dropped even at a single point.

On fuzzy solution of a linear optimization problem with max-aggregation function relation inequality constraints

Crisp constraints in the case of linear optimization problem linked to max-aggregation function composition as rule may result in unsatisfactory solution. We propose and discuss a method of getting more feasible solutions, relaxing crisp inequalities/extremes by means of fuzzy sets.