Quasi-arithmetic means and OWA functions in interval-valued and Atanassov's intuitionistic fuzzy set theory

In this paper we propose an extension of the well-known OWA functions introduced by Yager to interval-valued (IVFS) and Atanassov’s intuitionistic (AIFS) fuzzy set theory. We first extend the arithmetic and the quasi-arithmetic mean using the arithmetic operators in IVFS and AIFS theory and investigate under which conditions these means are idempotent. Since on the unit interval the construction of the OWA function involves reordering the input values, we propose a way of transforming the input values in IVFS and AIFS theory to a new list of input values which are now ordered

Implication functions in interval-valued fuzzy set theory

Interval-valued fuzzy set theory is an extension of fuzzy set theory in which the real, but unknown, membership degree is approximated by a closed interval of possible membership degrees. Since implications on the unit interval play an important role in fuzzy set theory, several authors have extended this notion to interval-valued fuzzy set theory. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory. In particular, we describe several possibilities to represent such implications using implications on the unit interval, we give a characterization of the implications in interval-valued fuzzy set theory which satisfy the Smets-Magrez axioms, we discuss the solutions of a particular distributivity equation involving strict t-norms, we extend monoidal logic to the interval-valued fuzzy case and we give a soundness and completeness theorem which is similar to the one existing for monoidal logic, and finally we discuss some other constructions of implications in interval-valued fuzzy set theory

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.<br /

A Decision Method for Online Purchases Considering Dynamic Information Preference Based on Sentiment Orientation Classification and Discrete DIFWA Operators

© 2013 IEEE. Online reviews are crucial for evaluating product features and supporting consumers' purchase decisions. However, as a result of online buying behaviors, consumer habits, and discrete dynamic distribution characteristics of online reviews, and consumers typically randomly choose a limited number of reviews from discrete time frames among all reviews and give more weight to recent review information and less weight to earlier information to support their online purchase decisions; moreover, existing studies have ignored the discrete random dynamic characteristics and dynamic information preferences of consumers. To address this issue, this paper proposes a method based on sentiment orientation classification and discrete DIFWA (DDIFWA) operators for online purchase decisions considering dynamic information preferences. In this method, we transformed review texts from original discrete time slices to discrete random features, extracted product features based on the constructed feature and sentiment dictionaries, and matched pairs of features and sentiment phrases in the dictionaries. We subsequently employed three types of semantic orientation by defining semantic rules to extract the product features of each review. Of note, the semantic orientations were transformed from discrete time to semantic intuitionistic fuzzy numbers and semantic intuitionistic fuzzy information matrixes. Furthermore, we proposed two DDIFWA operators to aggregate the dynamic semantic intuitionistic fuzzy information. Specifically, we obtained the rankings of alternative products and their features to support consumers' purchase decisions using the intuitionistic fuzzy scoring function and the 'vertical projection distance' method. Finally, comparisons and experiments are provided to demonstrate the plausibility of our methods

Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms

This work studies the aggregation operators on the set of all possible membership degrees of typical hesitant fuzzy sets, which we refer to as H, as well as the action of H-automorphisms which are defined over the set of all finite non-empty subsets of the unitary interval. In order to do so, the partial order ≤H, based on α-normalization, is introduced, leading to a comparison based on selecting the greatest membership degrees of the related fuzzy sets. Additionally, the idea of interval representation is extended to the context of typical hesitant aggregation functions named as the H-representation. As main contribution, we consider the class of finite hesitant triangular norms, studying their properties and analyzing the H-conjugate functions over such operators. © 2013 Elsevier Inc. All rights reserved.Peer Reviewe

Power of Continuous Triangular Norms with Application to Intuitionistic Fuzzy Information Aggregation

The paper aims to investigate the power operation of continuous triangular norms (t-norms) and develop some intuitionistic fuzzy information aggregation methods. It is proved that a continuous t-norm is power stable if and only if every point is a power stable point, and if and only if it is the minimum t-norm, or it is strict, or it is an ordinal sum of strict t-norms. Moreover, the representation theorem of continuous t-norms is used to obtain the computational formula for the power of continuous t-norms. Based on the power operation of t-norms, four fundamental operations induced by a continuous t-norm for the intuitionistic fuzzy (IF) sets are introduced. Furthermore, various aggregation operators, namely the IF weighted average (IFWA), IF weighted geometric (IFWG), and IF mean weighted average and geometric (IFMWAG) operators, are defined, and their properties are analyzed. Finally, a new decision-making algorithm is designed based on the IFMWAG operator, which can remove the hindrance of indiscernibility on the boundaries of some classical aggregation operators. The practical applicability, comparative analysis, and advantages of the study with other decision-making methods are furnished to ascertain the efficacy of the designed method

A novel method for interval-value intuitionistic fuzzy multicriteria decision-making problems with immediate probabilities based on OWA distance operators

The goal of this work is to develop a novel decision-making method which can solve some complex decision problems that include the following three-aspect information: (1) information represented in the form of interval-valued intuitionistic fuzzy values (IVIFVs) not only intuitionistic fuzzy values (IFVs), (2) the probability information and the weighted information, and (3) the importance degree of each concept in the process of decision-making. Firstly, by integrating OWA operator, probabilistic weight (PW), and individual distance of two IVIFNs in the same formulation, we introduce two new distance operators named PIVIFOWAD operator and IPIVIFOWAD operator, respectively. Secondly, satisfaction degree of an alternative is proposed based on the positive ideal IVIFS and the negative ideal IVIFS and applied to MCDM. Finally, we use an illustrative example to show the feasibility and validity of the new method by comparing with the other existing methods