A dual hesitant fuzzy set (DHFS) describes the uncertainty in the real world by using the membership degree and nonmembership degree. It can collect fuzzy information comprehensively and apply them into decision-making tasks efficiently. In this article, we extract some characteristics, such as the average function, variance function, hesitancy degree to describe a dual hesitant fuzzy element, and develop novel distance measures of DHFSs based on these characteristics. Further, we investigate their properties and prove the triangle inequality of distance measure. Finally, we apply it in practical medical diagnosis to illustrate the validity of our proposed distance measures.

In recent decades, image segmentation has aroused great interest of many researchers, and has become an important part of machine learning, pattern recognition, and computer vision. Among many methods of image segmentation, fuzzy C-means (FCM) algorithm is undoubtedly a milestone in unsupervised method. With the further study of FCM, various different kinds of FCM algorithms are put forward to deal with the specific problems in image segmentation. Because there exist uncertainties in different regions of the image and similarity in the same region, reducing the uncertainty is still the main problem in image segmentation. Considering that Pythagorean fuzzy set (PFS) is a powerful tool to deal with uncertainty, in this paper, we use PFS to describe the uncertainty of image segmentation, including introducing fuzzification and defuzzification process and Pythagorean fuzzy element to describe the membership degree of pixel, combine the neighborhood information with weights and Pythagorean fuzzy distance, and propose Pythagorean fuzzy C-means (PFCM) algorithm. Finally, we apply PFCM algorithm in image segmentation, such as different size images and Berkeley Segmentation Data Set to illustrate the effectiveness and applicability of our proposed algorithm. Meanwhile, we do comparison analysis between PFCM, fully convolution network and Deep-image-Prior networks, these results show that our proposed PFCM has good intuition and effectiveness.

The main feature of Pythagorean fuzzy sets is that it is characterized by four parameters, namely membership degree, nonmembership degree, strength of commitment about membership, and direction of commitment. In this paper, we propose a variety of distance measures for Pythagorean fuzzy sets and Pythagorean fuzzy numbers, which take into account the four parameters of Pythagorean fuzzy sets. Finally, a numerical example is provided to illustrate the validity and applicability of the presented distance measures.

The ordered weighted averaging (OWA) operator introduced by Yager is one of the most popular aggregation technique. In this paper, we develop two kinds of argument-dependent OWA (DOWA) operators including the pessimistic-dependent OWA (PE-DOWA) operator and optimistic-dependent OWA (OP-DOWA) operator, that point out that the PE-DOWA operator is decreasing and the OP-DOWA operator is increasing, and investigate some properties of our proposed monotonic DOWA operators in detail. Furthermore, we introduce the concept of original function in which a gradient vector generates the weights of the PE-DOWA and OP-DOWA operators. Meanwhile, we propose two classes of original functions including summing-type original function and multiplying-type original function and investigate the sufficient monotonic conditions for the DOWA operators generated by the original functions. Finally, we discuss the characteristics and properties of our proposed DOWA operators in detail and use a numerical example to illustrate the flexibility of our proposed operators.

The main feature of Pythagorean fuzzy sets is that it is characterized by five parameters, namely membership degree, nonmembership degree, hesitancy degree, strength of commitment about membership, and direction of commitment. In this paper, we first investigate four existing comparison methods for ranking Pythagorean fuzzy sets and point out by examples that the method proposed by Yager, which considers the influence fully of the five parameters, is more efficient than the other ones. Later, we propose a variety of distance measures for Pythagorean fuzzy sets and Pythagorean fuzzy numbers, which take into account the five parameters of Pythagorean fuzzy sets. Based on the proposed distance measures, we present some similarity measures of Pythagorean fuzzy sets. Furthermore, a multiple criteria Pythagorean fuzzy group decision-making approach is proposed. Finally, a numerical example is provided to illustrate the validity and applicability of the presented group decision-making method.

The Bonferroni mean has been extensively applied in multicriteria decision-making and support system and developed intuitionistic fuzzy set theory. Based on the second interpretation of the Bonferroni mean, in this paper, we introduce the geometric Bonferroni mean, which is a generalization of the Bonferroni mean and geometric mean and generalized geometric Bonferroni mean, and investigate their properties. To describe the uncertainty and fuzziness more objectively, we further develop the intuitionistic fuzzy geometric Bonferroni mean and the generalized intuitionistic fuzzy geometric Bonferroni mean, which describe the relationship between arguments, and the weighted intuitionistic fuzzy geometric Bonferroni mean and the generalized weighted intuitionistic fuzzy geometric Bonferroni mean, which consider the importance of each argument. Finally, we investigate their properties in detail.

The inclusion measure, the similarity measure, and the fuzziness of fuzzy sets are three important measures in fuzzy set theory. In this article, we investigate the relations among inclusion measures, similarity measures, and the fuzziness of fuzzy sets, prove eight theorems that inclusion measures, similarity measures, and the fuzziness of fuzzy sets can be transformed by each other based on their axiomatic definitions, and propose some new formulas to calculate inclusion measures, similarity measures, and the fuzziness of fuzzy sets. These results can be applied in many fields, such as pattern recognition, image processing, fuzzy neural networks, fuzzy reasoning, and fuzzy control. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 639–653, 2006.