J. Vicente Riera

Aggregation of subjective evaluations based on discrete fuzzy numbers

In this paper aggregation functions defined on the set of all discrete fuzzy numbers whose support is a subset of consecutive natural numbers are introduced and the particular cases of uninorms and nullnorms are studied in detail. These aggregation functions are constructed from discrete aggregation functions (defined on a finite chain) and they are applied to the aggregation of subjective evaluations.

Discrete Fuzzy Numbers Defined on a Subset of Natural Numbers

We introduce an alternative method to approach the addition of discrete fuzzy numbers when the application of the Zadeh’s extension principle does not obtain a convex membership function.

Residual implications on the set of discrete fuzzy numbers

In this paper residual implications defined on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers are studied. A specific construction of these implications is given and some examples are presented showing in particular that such a construction generalizes the case of interval-valued residual implications. The most usual properties for these operations are investigated leading to a residuated lattice structure on the set of discrete fuzzy numbers, that in general is not an MTL-algebra.

Weighted Means of Subjective Evaluations

In this article, we recall different student evaluation methods based on fuzzy set theory. The problem arises is the aggregation of this fuzzy information when it is presented as a fuzzy number. Such aggregation problem is becoming present in an increasing number of areas: mathematics, physic, engineering, economy, social sciences, etc. In the previously quoted methods the fuzzy numbers awarded by each evaluator are not directly aggregated. They are previously defuzzycated and then a weighted mean or other type of aggregation function is often applied. Our aim is to aggregate directly the fuzzy awards (expressed as discrete fuzzy numbers) and to get like a fuzzy set (a discrete fuzzy number) resulting from such aggregation, because we think that in the defuzzification process a large amount of information and characteristics are lost. Hence, we propose a theoretical method to build n-dimensional aggregation functions on the set of discrete fuzzy number. Moreover, we propose a method to obtain the group consensus opinion based on discrete fuzzy weighted normed operators.

Aggregation functions on the set of discrete fuzzy numbers defined from a pair of discrete aggregations

Abstract In this paper we propose a method to construct aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers from a couple of discrete aggregation functions. The interest on these discrete fuzzy numbers lies on the fact that they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms. Finally, a linguistic decision making model based on a pair of aggregation functions defined on discrete fuzzy numbers is given.

On the distributivity property for uninorms

Abstract Distributivity between two operations is a property that was already posed many years ago and that is especially interesting in the framework of logical connectives. For this reason, the distributivity property has been extensively studied for several families of operations like triangular norms and conorms, some kinds of uninorms and nullnorms (also called t-operators) and even for some generalizations of them. In this paper we investigate the distributivity equation involving two uninorms lying in any one of the most studied classes of uninorms, leading to many new solutions.

Using discrete fuzzy numbers in the aggregation of incomplete qualitative information

In this article discrete fuzzy numbers are used to model complete and incomplete qualitative information and some methods to aggregate this kind of information are proposed. When the support of discrete fuzzy numbers is a closed interval of the chain L n = { 0 , 1 , ? , n } , they can be interpreted as linguistic expert valuations that increase the flexibility of the elicitation of qualitative information based on linguistic terms. On the other hand, when the support is not an interval of L n , the corresponding discrete fuzzy number can be interpreted as an incomplete linguistic expert valuation. In order to aggregate this kind of information, we propose two different methods. One of them deals with the construction of aggregation functions on the set of discrete fuzzy numbers from discrete aggregation functions defined on L n . The other one presents several procedures for estimating the missing information based on the so-called discrete associations. Finally, the proposed aggregation methods are used in a multi-expert decision making problem and a concrete example is given.

S-implications in the set of discrete fuzzy numbers

This paper is devoted to build S-implications on the bounded lattice A^L<inf>1</inf> of discrete fuzzy numbers whose support is a subset of consecutive natural numbers of the finite chain L = {0, 1, …, m}. Moreover, we propose a method to compare them.

Triangular Norms and Conorms on the Set of Discrete Fuzzy Numbers

In this paper, we study different methods to construct triangular operations (t-norms and t-conorms) on the bounded distributive lattice \(\mathcal{A}_1^{L}\), of discrete fuzzy numbers whose support is a subset of consecutive natural numbers on a finite chain L of consecutive natural numbers. Moreover, we propose a method to compare two t-norms (t-conorms) defined on \(\mathcal{A}_1^{L}\).

Aggregation of bounded fuzzy natural number-valued multisets

Multisets (also called bags) are like-structures where an element can appear more than once. Recently, several generalizations of this concept have been studied. In this article we deal with a new extension of this concept, the bounded fuzzy natural number-valued multisets. On this kind of bags, a bounded distributive lattice structure is presented and a partial order is defined. Moreover, we study operations of aggregations (t-norms and t-conorms) and we provide two methods for their construction.