Correlation is an important index in decision-making. In weighted extended hesitant fuzzy sets (WEHFSs) environment, researchers have only defined a class of correlation coefficients between WEHFSs with values in the unit interval [ 0 , 1 ] . This is not ideal because it does not extend the classical correlation coefficient in the case of classical sets. In fact, the negative values of the interval [ − 1 , 1 ] are ignored, and such neglectfulness leads to unreasonable results in decision-making. In other words, the existing definitions are unconvincing and lack consistency, which hinder their application potentials. This article addresses this issue by introducing a new class of weighted correlation coefficients of WEHFSs with values in the interval [ − 1 , 1 ] . Three decision-making methodologies based on the weighted correlation coefficients of WEHFSs are compared with the existing methodologies based on their respective correlation coefficients in the unit interval [ 0 , 1 ] . The comparative analysis shows both the efficiency and effectiveness of the new correlation index.

One worthwhile way of expressing imprecise information is the q-rung orthopair fuzzy sets (q-ROFSs), which extend intuitionistic fuzzy sets and Pythagorean fuzzy sets. The main goal of this contribution is to further extend the concept of similarity measure for q-ROFSs, which not only endows the similarity framework with more ability to create new ones but also inherits all essential properties of a logical similarity measure. This contribution proposes a class of novel similarity measures for q-ROFSs by drawing a general framework of existing q-ROFS similarity and q-ROFS distance measures. These q-ROFS similarity measures enable us to overcome the theoretical drawbacks of the existing measures in the case where they are used individually. In the application part of the contribution, a pattern recognition problem on classification of building materials with a number of known building materials is reconsidered. The study of this particular case shows that the proposed family of similarity measures consistently classify the unknown building material pattern with the same known building material pattern. Then, an experimental case study regarding a problem of classroom teaching quality is re-examined for the comparison of the performance of proposed similarity measures against the existing ones. The salient features of the proposed similarity measures in comparison to the existing qROFS similarity measures, are as follows: (i) a number of existing q-ROFS similarity measures are inherently correlation coefficients, and they satisfy only a limited number of essential properties of a comprehensive similarity measure; (ii) several existing q-ROFS similarity measures lead sometimes to nonlogical results, more specifically, to the same maximum similarity value for different q-ROFSs; (iii) a variety of existing q-ROFS similarity measures depend on subjective parameters, which either hinder their application in practice or increase their computational cost. In brief, following this direction of research, we will prove the superiority of the developed similarity measures over the existing ones from both theoretical and experimental viewpoints.

This contribution reviews critically the existing entropy measures for probabilistic hesitant fuzzy sets (PHFSs), and demonstrates that these entropy measures fail to effectively distinguish a variety of different PHFSs in some cases. In the sequel, we develop a new axiomatic framework of entropy measures for probabilistic hesitant fuzzy elements (PHFEs) by considering two facets of uncertainty associated with PHFEs which are known as fuzziness and nonspecificity. Respect to each kind of uncertainty, a number of formulae are derived to permit flexible selection of PHFE entropy measures. Moreover, based on the proposed PHFE entropy measures, we introduce some entropy-based distance measures which are used in the portion of comparative analysis. Eventually, the proposed PHFE entropy measures and PHFE entropy-based distance measures are applied to decision making in the strategy initiatives where their reliability and effectiveness are verified.

Ever since fuzzy set has been introduced, several extensions have been established, such as interval-valued fuzzy sets, Atanassov's intuitionistic fuzzy sets, interval-valued Atanassov's intuitionistic fuzzy sets, fuzzy multisets, hesitant fuzzy sets, interval-valued hesitant fuzzy sets, and dual hesitant fuzzy sets. In this contribution, we propose dual interval-valued hesitant fuzzy sets, which encompass fuzzy sets and its aforementioned extensions as special cases. Because of the importance of correlation measure in data analysis, we propose an approach for deriving the correlation coefficient of dual hesitant fuzzy sets, and then extend the approach to the dual interval-valued hesitant fuzzy set theory. We also put forward some formulas to create new correlation coefficients for fuzzy sets and its extensions in a general way. In addition, we give a practical example to illustrate the application of correlation coefficient for dual hesitant fuzzy sets in medical diagnosis.

Despite of several generalizations of fuzzy set theory, the notion of hesitant fuzzy set (HFS), which permits the membership having a set of possible values, is interesting and very useful in modeling real-life problems with anonymity. In this article, we introduce a new score function for ranking hesitant fuzzy elements (HFEs), which are the fundamental units of HFSs. Comparison with the existing score function shows that the proposed method meets all the well-known properties of a ranking measure and has no counterintuitive examples. On the basis of the relationships between the aggregation operators for HFEs, we derive a series of interesting properties of the new score function. Finally, we apply the proposed score function to solve the hesitant fuzzy multiattribute decision-making problems.