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As the theory of picture fuzzy sets has been developed, more information in life can be expressed in mathematical terms. Similarity measure is a special tool for quantifying the similarity between two sets, so studying similarity measure on picture fuzzy sets has become a trending topic. This new research direction has drawn a great deal of attention from experts and has led to a number of important results which have led to significant results in a number of practical applications. By examining these new findings, we discovered that there are many studies on similarity measure of picture fuzzy sets, some of them are deficient in solving certain problems, and such similarity measures can lead to the calculation of unreasonable data in practical applications, affecting the final results. Secondly, there is still room for research similarity measures on exponential functions. Considering these two aspects, we propose two new similarity measures based on exponential function, which not only satisfy the axiomatic definition of similarity measures, but also show reasonable computational results in practical applications. [ABSTRACT FROM AUTHOR]

As a direct extension of intuitionistic fuzzy sets, picture fuzzy sets have a powerful ability to deal with vague, uncertain and inconsistent information. The distance between picture fuzzy sets is an effective mathematical tool to distinguish the difference between objects. Although various distances have been proposed, there are some distances that do not meet the axiomatic definition of distance or some distances that induce counter-intuitive results. To address the above defects, this paper gives a new distance between picture fuzzy sets by aggregating three-dimensional divergence, due to the capability of divergence to distinguish information. This new distance not only satisfies the axiomatic definition, but also overcomes the counter-intuitive defects. Furthermore, the proposed distance is applied to solve the multi-attribute decision-making problem. Through comparison and analysis of decision results, this distance not only provides reasonable decision results, but also has high degree of confidence. [ABSTRACT FROM AUTHOR]

Similarity measure is an important tool to evaluate the degree of similarity between different objects. Therefore, a large number of similarity measures on picture fuzzy sets have been studied. However, some existing similarity measures have not been unified into a framework. The aim of this paper is to study a construction method of similarity measure on picture fuzzy sets, based on this, we can unify some existing similarity measures into a framework, and then propose some new similarity measures. Firstly, the concept of picture fuzzy equivalence is introduced. Some construction methods for picture fuzzy equivalence are given. Secondly, a construction method of similarity measure on picture fuzzy sets is proposed. Some new similarity measures on picture fuzzy sets based on picture fuzzy equivalences are given and some existing similarity measures are unified into a framework. Finally, the proposed similarity measures are applied to pattern recognition. By comparing the degrees of confidence, we find that our degrees of confidence are higher than some similarity measures. The experimental results show that the proposed similarity measures can effectively distinguish some similar picture fuzzy sets. [ABSTRACT FROM AUTHOR]

Two important basic inference models of fuzzy reasoning are Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT). In order to solve FMP and FMT problems, the full implication triple I algorithm, the reverse triple I algorithm and the Subsethood Inference Subsethood (SIS for short) algorithm are proposed, respectively. Furthermore, the existing reasoning algorithms are extended to intuitionistic fuzzy sets and interval-valued fuzzy sets according to different needs. The purpose of this paper is to study the relationship between intuitionistic fuzzy reasoning algorithms and interval-valued fuzzy reasoning algorithms. It is proven that there is a bijection between the solutions of intuitionistic fuzzy triple I algorithm and the interval-valued fuzzy triple I algorithm. Then, there is a bijection between the solutions of intuitionistic fuzzy reverse triple I algorithm and the interval-valued fuzzy reverse triple I algorithm. At the same time, it is shown that there is also a bijection between the solutions of intuitionistic fuzzy SIS algorithm and interval-valued fuzzy SIS algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we focus on parametric interval-valued full implication algorithms. The interval-valued full implication algorithms based on a class of interval-valued t-norms with parameters are proposed. Meanwhile, the [ a , b ] -type interval-valued full implication algorithms based on a class of interval-valued t-norms with parameters are introduced. Furthermore, the solutions of these algorithms are given. Finally, the proposed algorithms are applied to pattern recognition. [ABSTRACT FROM AUTHOR]

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule's antecedent and input. Both algorithms did not consider the similarity of rule's antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods. [ABSTRACT FROM AUTHOR]

In previous studies, based on picture fuzzy sets, full implication algorithm and quintuple implication principle have been studied. The problem is, full implication algorithm did not consider the relationship between the input and antecedent of rule, while quintuple implication principle only considered the inclusion relationship between the rule's antecedent and input. Both algorithms did not consider the similarity of rule's antecedent and input. This leads to unreasonable calculation results in some cases. Therefore, we study fuzzy reasoning methods based on picture similarity measure for generalized modus ponens and generalized modus tollens in this paper. First of all, based on picture biresiduation, a new picture similarity measure between picture fuzzy sets is constructed. Then, representations of solutions of fuzzy reasoning methods based on picture similarity measure are given. Moreover, reducibility and robustness of fuzzy reasoning methods based on picture similarity measure are studied. Finally, two issues related to medical diagnosis are addressed by the proposed methods, demonstrate the rationality and effectiveness of our suggested methods. [ABSTRACT FROM AUTHOR]

In this paper, the concept of α (x, y)-interval-valued pointwise sustaining degree based on the left-continuous t-representable t-norms is put forward. And then, as a general extension based on the interval-valued pointwise sustaining degree, the interval-valued α (x, y)-full implication triple I method model, the interval-valued α (x, y)-quintuple implication principle models and the interval-valued α (x, y)-similarity measure method models are given. Moreover, the interval-valued R -type α (x, y)-fuzzy reasoning solutions with triple I method, quintuple implication principle and similarity measure method are given. Some existing results are special cases of the main conclusions in this paper. [ABSTRACT FROM AUTHOR]

As an extension of intuitionistic fuzzy sets, picture fuzzy sets can deal with vague, uncertain, incomplete and inconsistent information. The similarity measure is an important technique to distinguish two objects. In this study, a similarity measure between picture fuzzy sets based on relationship matrix is proposed. The new similarity measure satisfies the axiomatic definition of similarity measure. It can be testified from a numerical experiment that the new similarity measure is more effective. Finally, we apply the proposed similarity measure to multiple-attribute decision making. [ABSTRACT FROM AUTHOR]