Tomasa Calvo

Aggregation operators: properties, classes and construction methods

Aggregation (fusion) of several input values into a single output value is an indispensable tool not only of mathematics or physics, but of majority of engineering, economical, social and other sciences. The problems of aggregation are very broad and heterogeneous, in general. Therefore we restrict ourselves in this contribution to the specific topic of the aggregation of finite number of real inputs only. Closely related topics of aggregating infinitely many real inputs [23,109,64,52,43,42,44,99], of aggregating inputs from some ordinal scales [41,50], of aggregating complex inputs (such as probability distributions [107,114], fuzzy sets [143]), etc., are treated, among others, in the quoted papers, and we will not deal with them. In this spirit, if the number of input values is fixed, say n, an aggregation operator is a real function of n variables. This is still a too general topic. Therefore we restrict our considerations regarding inputs as well as outputs to some fixed interval (scale) I = [a, b] ⊑ [-∞, ∞]. It is a matter of rescaling to fix I = [0,1].

The functional equations of Frank and Alsina for uninorms and nullnorms

The aim of this work is to study the functional equations of Frank and Alsina for two classes of commutative, associative and increasing binary operators. The first one is the class of uninorms introduced by Yager and Rybalov. The second one is the class of nullnorms arising from our study of the Frank equation for uninorms. Both classes contain t-norms and t-conorms as special cases. Moreover, the structure of the other uninorms and nullnorms is closely related to t-norms and t-conorms. These observations are the motivation for studying some generalizations of the Frank and Alsina equations. However, it is shown that all considerations lead back to the already known t-norm and t-conorm solutions. Important consequences in fuzzy preference modelling are pointed out.

Quantitative weights and aggregation

Based on the strong idempotency of aggregation operators, quantitative weights are incorporated into the fusion process. Our general method is compared with some previous specific cases. More details about weighted aggregation based on some penalty function is given. Further, weighted integral based aggregation linked to quantifier-based fuzzy measures is investigated, especially weighted OWA operators. Several examples are included.

Aggregation functions based on penalties

This article studies a large class of averaging aggregation functions based on minimizing a distance from the vector of inputs, or equivalently, minimizing a penalty imposed for deviations of individual inputs from the aggregated value. We provide a systematization of various types of penalty based aggregation functions, and show how many special cases arise as the result. We show how new aggregation functions can be constructed either analytically or numerically and provide many examples. We establish connection with the maximum likelihood principle, and present tools for averaging experimental noisy data with distinct noise distributions.

A class of fuzzy multisets with a fixed number of memberships

The main aim of this work is to present a generalization of Atanassovs operators to higher dimensions. To do so, we use the concept of fuzzy set, which can be seen as a special kind of fuzzy multiset, to define a generalization of Atanassovs operators for n-dimensional fuzzy values (called n-dimensional intervals). We prove that our generalized Atanassovs operators also generalize OWA operators of any dimension by allowing negative weights. We apply our results to a decision making problem. We also extend the notions of aggregating functions, in particular t-norms, fuzzy negations and automorphism and related notions for n-dimensional framework.

A Review of Aggregation Functions

Several local and global properties of (extended) aggregation functions are discussed and their relationships are examined. Some special classes of averaging, conjunctive and disjunctive aggregation functions are reviewed. A special attention is paid to the weighted aggregation functions, including some construction methods

Consensus in multi-expert decision making problems using penalty functions defined over a Cartesian product of lattices

In this paper we introduce an algorithm to aggregate the preference relations provided by experts in multi-expert decision making problems. Instead of using a single aggregation function for the whole process, we start from a set of aggregation functions and select, by means of consensus done through penalty functions, the most suitable aggregation function in order to aggregate the individual preferences for each of the elements. An advantage of the method that we propose is that it allows us to recover the classical methods, just by using a single aggregation function. We also present a generalization of the concepts of restricted dissimilarity function and distance between sets for the case where we are working with a Cartesian product of lattices and use such concepts to build penalty functions. Finally, we propose an algorithm that allows us to choose the best combination of aggregation functions for a multi-expert decision making problem.

Generation of weighting triangles associated with aggregation functions

In this work, we present several ways to obtain different types of weighting triangles, due to these types characterize some interesting properties of Extended Ordered Weighted Averaging operators, EOWA, and Extended Quasi-linear Weighted Mean, EQLWM, as well as of their reverse functions. We show that any quantifier determines an EOWA operator which is also an Extended Aggregation Function, EAF, i.e., the weighting triangle generated by a quantifier is always regular. Moreover, we present different results about generation of weighting triangles by means of sequences and fractal structures. Finally, we introduce a degree of orness of a weighting triangle associated with an EOWA operator. After that, we mention some results on each class of triangle, considering each one of these triangles as triangles associated with their corresponding EOWA operator, and we calculate the ornessof some interesting examples.

Aggregation for Atanassov’s Intuitionistic and Interval Valued Fuzzy Sets: The Median Operator

Atanassovs intuitionistic fuzzy sets (AIFS) and interval valued fuzzy sets (IVFS) are two generalizations of a fuzzy set, which are equivalent mathematically although different semantically. We analyze the median aggregation operator for AIFS and IVFS. Different mathematical theories have lead to different definitions of the median operator. We look at the median from various perspectives: as an instance of the intuitionistic ordered weighted averaging operator, as a Fermat point in a plane, as a minimizer of input disagreement, and as an operation on distributive lattices. We underline several connections between these approaches and summarize essential properties of the median in different representations.

On some solutions of the distributivity equation

Abstract The aim of this paper is to present some interesting results on the distributivity equation, for which several pairs of binary operations on [0, 1] are considered. Moreover, different methods to solve distributivity equations are used in this work.