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

Absorbent tuples of aggregation operators

We generalize the notion of an absorbent element of aggregation operators. Our construction involves tuples of values that decide the result of aggregation. Absorbent tuples are useful to model situations in which certain decision makers may decide the outcome irrespective of the opinion of the others. We examine the most important classes of aggregation operators in respect to their absorbent tuples, and also construct new aggregation operators with predefined sets of absorbent tuples.<br /

Some characterizations of T-power based implications

Recently, the so-called family of T-power based implications was introduced. These operators involve the use of Zadeh’s quantifiers based on powers of t-norms in its definition. Due to the fact that Zadeh’s quantifiers constitute the usual method to modify fuzzy propositions, this family of fuzzy implication functions satisfies an important property in approximate reasoning such as the invariance of the truth value of the fuzzy conditional when both the antecedent and the consequent are modified using the same quantifier. In this paper, an in-depth analysis of this property is performed by characterizing all binary functions satisfying it. From this general result, a fully characterization of the family of T-power based implications is presented. Furthermore, a second characterization is also proved in which surprisingly the invariance property is not explicitly used.Peer ReviewedPostprint (author's final draft

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 primer on triangle functions II

In [32] we presented an overview of concepts, facts and results on triangle functions based on the notions of t-norm, copula, (generalized) convolution, semicopula, quasi-copula. Here, we continue our presentation. In particular, we treat the concept of duality and study a few important cases of functional equations and inequalities for triangle functions like, e.g., convolution, Cauchy's equation, dominance, and Jensen convexity

Noncommutative Logic Systems with Applications in Management and Engineering

Zadeh's (min-max, standard) fuzzy logic and various other logics are commutative, but natural language has nuances suggesting the premises are not equal, with premises contributing to the conclusion according to their prominency. Therefore, we suggest variants of salience-based, noncommutative and non-associative fuzzy logic (prominence logic) that may better model natural language and reasoning when using linguistic variables. Noncommutative fuzzy logics have several theoretical and applicative motivations to be used as models for human inference and decision making processes. Among others, asymmetric relations in economy and management, such as buyer-seller, provider-user, and employer-employee are noncommutative relations and induce noncommutative logic operations between premises or conclusions. A class of noncommutative fuzzy logic operators is introduced and fuzzy logic systems based on the corresponding noncommutative logics are described and analyzed. The prominence of the operators in the noncommutative operations is conventionally assumed to be determined by their precedence. Specific versions of noncommutative logics in the class of the salience-based, noncommutative logics are discussed. We show how fuzzy logic systems may be built based on these types of logics. Compared with classic fuzzy systems, the noncommutative fuzzy logic systems have improved performances in modeling problems, including the modeling of economic and social processes, and offer more flexibility in approximation and control. Applications discussed include management and engineering problems and issues in the field of firms’ ethics or ethics of AI algorithms

The ⊛-composition of fuzzy implications: Closures with respect to properties, powers and families

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications from a given pair of fuzzy implications. Viewing this as a binary operation ⊛ on the set II of fuzzy implications they obtained, for the first time, a monoid structure (I,⊛)(I,⊛) on the set II. Some algebraic aspects of (I,⊛)(I,⊛) had already been explored and hitherto unknown representation results for the Yager's families of fuzzy implications were obtained in [53] (N.R. Vemuri and B. Jayaram, Representations through a monoid on the set of fuzzy implications, fuzzy sets and systems, 247 (2014) 51–67). However, the properties of fuzzy implications generated or obtained using the ⊛-composition have not been explored. In this work, the preservation of the basic properties like neutrality, ordering and exchange principles , the functional equations that the obtained fuzzy implications satisfy, the powers w.r.t. ⊛ and their convergence, and the closures of some families of fuzzy implications w.r.t. the operation ⊛, specifically the families of (S,N)(S,N)-, R-, f- and g-implications, are studied. This study shows that the ⊛-composition carries over many of the desirable properties of the original fuzzy implications to the generated fuzzy implications and further, due to the associativity of the ⊛-composition one can obtain, often, infinitely many new fuzzy implications from a single fuzzy implication through self-composition w.r.t. the ⊛-composition

On the Sheffer stroke operation in fuzzy logic

From the beginnings of fuzzy logic, the Sheffer stroke operation has been overlooked and the efforts of the researchers have beendevoted to other logical connectives. In this paper, the Sheffer stroke operation is introduced in fuzzy logic generalizing the classical operation when the truth values are restricted to {0, 1}2. Similar to what happens in Boolean logic, the fuzzy Sheffer stroke is functionally complete and it can be used to generate any other fuzzy logical connective by combinations of itself. Two construction methods are presented and the close connection of this operation with a pair of fuzzy conjunction and negation is analysed

T-generable indistinguishability operators and their use for feature selection and classification.

Peer ReviewedPostprint (author's final draft