To represent a set whose members are known partially, the graded ill-known set is proposed. In this paper, we investigate calculations of function values of graded ill-known sets. Because a graded ill-known set is characterized by a possibility distribution in the power set, the calculations of function values of graded ill-known sets are based on the extension principle but generally complex. To reduce the complexity, lower and upper approximations of a given graded ill-known set are used at the...

We consider the internal rate of return (IRR) decision rule in capital budgeting problems with fuzzy cash flows. The possibility distribution of the IRR at any r ≥ 0, is defined to be the degree of possibility that the (fuzzy) net present value of the project with discount factor r equals to zero. Generalizing our earlier results on fuzzy capital budegeting problems [Car99] we show that the possibility distribution of the {IRR} is a highly nonlinear function which is getting more and more unbalanced...

Commuting is an important property in any two-step information merging procedure where the results should not depend on the order in which the single steps are performed. In the case of bisymmetric aggregation operators with the neutral elements, Saminger, Mesiar and Dubois, already reduced characterization of commuting $n$-ary operators to resolving the unary distributive functional equations. And then the full characterizations of these equations are obtained under the assumption that the unary...

In the present paper, fuzzy order relations on a real vector space are characterized by fuzzy cones. It is well-known that there is one-to-one correspondence between order relations, that a real vector space with the order relation is an ordered vector space, and pointed convex cones. We show that there is one-to-one correspondence between fuzzy order relations with some properties, which are fuzzification of the order relations, and fuzzy pointed convex cones, which are fuzzification of the pointed...

Distortion of fuzzy measures is discussed. A special attention is paid to the preservation of submodularity and supermodularity, belief and plausibility. Full characterization of distortion functions preserving the mentioned properties of fuzzy measures is given.

A characterization of regular lattices of fuzzy sets and their isomorphisms is given in Part I. A characterization of involutions on regular lattices of fuzzy sets and the isomorphisms of De Morgan algebras of fuzzy sets is given in Part II. Finally all classes of De Morgan algebras of fuzzy sets with respect to isomorphisms are completely described.

The paper deals with the concept of coalitional preferences in the group decision-making situations in which the agents and coalitions have only vague idea about the comparative acceptability of particular outcomes. The coalitional games with vague utilities (see, e. g., [6]) can serve for a good example when some types of the game solutions (e. g., the von Neumann– Morgenstern one) are to be extended to the fuzzy game case. In this paper, we consider the fuzzy analogies of coalitional preferences...

Using techniques for modeling indices by means of functional equations and resources from fuzzy set theory, the classical Balthazard index used in order to combine several degrees of impairment is characterized in two natural ways and its use is criticized. In addition some hints are given on how to study better solutions than Balthazard's one for the problem of combining impairment degrees.

A subobjects structure of the category $\Omega$- of $\Omega$-fuzzy sets over a complete $MV$-algebra $\Omega =(L,\wedge ,\vee ,\otimes ,\to )$ is investigated, where an $\Omega$-fuzzy set is a pair $𝐀=(A,\delta )$ such that $A$ is a set and $\delta A\times A\to \Omega$ is a special map. Special subobjects (called complete) of an $\Omega$-fuzzy set $𝐀$ which can be identified with some characteristic morphisms $𝐀\to {\Omega }^{*}=(L\times L,\mu )$ are then investigated. It is proved that some truth-valued morphisms ${\neg }_{\Omega }{\Omega }^{*}\to {\Omega }^{*},{\cap }_{\Omega }$, ${\cup }_{\Omega }{\Omega }^{*}\times {\Omega }^{*}\to {\Omega }^{*}$ are characteristic morphisms of complete subobjects.

In this paper, the ordinal sum construction methods of implications on bounded lattices are studied. Necessary and sufficient conditions of an ordinal sum for obtaining an implication are presented. New ordinal sum construction methods on bounded lattices which generate implications are discussed. Some basic properties of ordinal sum implications are studied.

In this paper, two construction methods have been proposed for uni-nullnorms on any bounded lattices. The difference between these two construction methods and the difference from the existing construction methods have been demonstrated and supported by an example. Moreover, the relationship between our construction methods and the existing construction methods for uninorms and nullnorms on bounded lattices are investigated. The charactertics of null-uninorms on bounded lattice $L$ are given and a...

In this paper, we propose the general methods, yielding uninorms on the bounded lattice $(L,\le ,0,1)$, with some additional constraints on $e\in L\setminus \{0,1\}$ for a fixed neutral element $e\in L\setminus \{0,1\}$ based on underlying an arbitrary triangular norm $T_e$ on $[0,e]$ and an arbitrary triangular conorm $S_e$ on $[e,1]$. And, some illustrative examples are added for clarity.