Aggregation for Atanassov's intuitionistic and interval valued fuzzy sets: the median operator

Atanassov's 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.

(S,N)-implications on bounded lattices

Since the birth of the fuzzy sets theory several extensions have been proposed. For these extensions, different sets of membership functions were considered. Since fuzzy connectives, such as conjunctions, negations and implications, play an important role in the theory and applications of fuzzy logics, these connectives have also been extended. An extension of fuzzy logic, which generalizes the ones considered up to the present, was proposed by Joseph Goguen in 1967. In this extension, the membership values are drawn from arbitrary bounded lattices. The simplest and best studied class of fuzzy implications is the class of (S,N)-implications, and in this chapter we provide an extension of (S,N)-implications in the context of bounded lattice valued fuzzy logic, and we show that several properties of this class are preserved in this more general framework.

Representing images by means of interval-valued fuzzy sets. Application to stereo matching

Stereo matching tries to find correspondences between locations in a pair of displaced images of the same scene in order to extract the underlying depth information. Pixel correspondence estimation suffers from occlusions, noise or bias. In this work, we introduce a novel approach to represent images by means of interval-valued fuzzy sets to overcome the uncertainty due to the above mentioned problems. Our aim is to take advantage of this representation in the stereo matching algorithm. The image interval-valued fuzzification process that we propose is based on image segmentation in a different way to the common use of segmentation in stereo vision. We introduce interval-valued fuzzy similarities to compare windows whose pixels are represented by intervals. In the experimental analysis we show the goodness of this representation in the stereo matching problem. The new representation together with the new similarity measure that we introduce shows a better overall behavior with respect to other very well-known methods.

A construction method of interval-valued fuzzy sets for image processing

In this work we present a new construction method of IVFSs from Fuzzy Sets. We use these IVFSs for image processing. Concretely, in this contribution we introduce a new image magnification algorithm using IVFSs. This algorithm is based on block expansion and it is characterized by its simplicity.

Color image magnification with interval-valued fuzzy sets

In this work we present a simple magnification algorithm for color images. It uses Interval-Valued Fuzzy Sets in such a way that every pixel has an interval membership constructed from its original intensity and its neighbourhood's one. Based on that interval membership, a block is created for each pixel, so this is a block expansion method.

A new method for deriving priority weights by extracting consistent numerical-valued matrices from interval-valued fuzzy judgement matrix

It is important to derive priority weights from interval-valued fuzzy preferences when a pairwise comparative mechanism is used. By focusing on the significance of consistency in the pairwise comparison matrix, two numerical-valued consistent comparison matrices are extracted from an interval fuzzy judgement matrix. Both consistent matrices are derived by solving the linear or nonlinear programming models with the aid of assessments from Decision Makers (DMs). An interval priority weight vector from the extracted consistent matrices is generated. In order to retain more information hidden in the intervals, a new probability-based method for comparison of the interval priority weights is introduced. An algorithm for deriving the final priority interval weights for both consistent and inconsistent interval matrices is proposed. The algorithm is also generalized to handle the pairwise comparison matrix with fuzzy numbers. The comparative results from the five examples reveal that the proposed method, as compared with eight existing methods, exhibits a smaller degree of uncertainty pertaining to the priority weights, and is also more reliable based on the similarity measure. © 2014 Elsevier Inc. All rights reserved.

On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts

Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous extensions of the weighted arithmetic mean and ordered weighted averaging operator also have the absorbent element 〈1,0〉, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the operations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit analogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.

Negations generated by bounded lattices t-norms

From the birth of fuzzy sets theory, several extensions have been proposed changing the possible membership values. Since fuzzy connectives such as t-norms and negations have an important role in theoretical as well as applied fuzzy logics, these connectives have been adapted for these generalized frameworks. Perhaps, an extension of fuzzy logic which generalizes the remaining extensions, proposed by Joseph Goguen in 1967, is to consider arbitrary bounded lattices for the values of the membership degrees. In this paper we extend the usual way of constructing fuzzy negations from t-norms for the bounded lattice t-norms and prove some properties of this construction.

A penalty-based aggregation operator for non-convex intervals

In the case of real-valued inputs, averaging aggregation functions have been studied extensively with results arising in fields including probability and statistics, fuzzy decision-making, and various sciences. Although much of the behavior of aggregation functions when combining standard fuzzy membership values is well established, extensions to interval-valued fuzzy sets, hesitant fuzzy sets, and other new domains pose a number of difficulties. The aggregation of non-convex or discontinuous intervals is usually approached in line with the extension principle, i.e. by aggregating all real-valued input vectors lying within the interval boundaries and taking the union as the final output. Although this is consistent with the aggregation of convex interval inputs, in the non-convex case such operators are not idempotent and may result in outputs which do not faithfully summarize or represent the set of inputs. After giving an overview of the treatment of non-convex intervals and their associated interpretations, we propose a novel extension of the arithmetic mean based on penalty functions that provides a representative output and satisfies idempotency.

Averaging aggregation functions for preferences expressed as Pythagorean membership grades and fuzzy orthopairs

Rather than denoting fuzzy membership with a single value, orthopairs such as Atanassov's intuitionistic membership and non-membership pairs allow the incorporation of uncertainty, as well as positive and negative aspects when providing evaluations in fuzzy decision making problems. Such representations, along with interval-valued fuzzy values and the recently introduced Pythagorean membership grades, present particular challenges when it comes to defining orders and constructing aggregation functions that behave consistently when summarizing evaluations over multiple criteria or experts. In this paper we consider the aggregation of pairwise preferences denoted by membership and non-membership pairs. We look at how mappings from the space of Atanassov orthopairs to more general classes of fuzzy orthopairs can be used to help define averaging aggregation functions in these new settings. In particular, we focus on how the notion of 'averaging' should be treated in the case of Yager's Pythagorean membership grades and how to ensure that such functions produce outputs consistent with the case of ordinary fuzzy membership degrees.

A characterization theorem for t-representable n-dimensional triangular norms

n-dimensional fuzzy sets are an extension of fuzzy sets that includes interval-valued fuzzy sets and interval-valued Atanassov intuitionistic fuzzy sets. The membership values of n-dimensional fuzzy sets are n-tuples of real numbers in the unit interval [0,1], called n-dimensional intervals, ordered in increasing order. The main idea in n-dimensional fuzzy sets is to consider several uncertainty levels in the memberships degrees. Triangular norms have played an important role in fuzzy sets theory, in the narrow as in the broad sense. So it is reasonable to extend this fundamental notion for n-dimensional intervals. In interval-valued fuzzy theory, interval-valued t-norms are related with t-norms via the notion of t-representability. A characterization of t-representable interval-valued t-norms is given in term of inclusion monotonicity. In this paper we generalize the notion of t-representability for n-dimensional t-norms and provide a characterization theorem for that class of n-dimensional t-norms. © 2011 Springer-Verlag Berlin Heidelberg.

Vector valued similarity measures for Atanassov's intuitionistic fuzzy sets

We present a new approach for defining similarity measures for Atanassov's intuitionistic fuzzy sets (AIFS), in which a similarity measure has two components indicating the similarity and hesitancy aspects. We justify that there are at least two facets of uncertainty of an AIFS, one of which is related to fuzziness while other is related to lack of knowledge or non-specificity. We propose a set of axioms and build families of similarity measures that avoid counterintuitive examples that are used to justify one similarity measure over another. We also investigate a relation to entropies of AIFS, and outline possible application of our method in decision making and image segmentation. © 2014 Elsevier Inc. All rights reserved.