Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering

Yang, Miin-Shen; Hussain, Zahid

doi:10.1007/s00500-018-3248-0

Cited by 36 publications

(26 citation statements)

References 40 publications

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“…The following are the results of the other proposed measures Equations (24) to (27) Obviously, the sample Q is similar to the pattern P 1 according to the principle of minimum degree of dissimilarity, but there are some changes in numerical values. Further, we observe that the following results are analogous to each other: To further check the validity and applicability of the proposed similarity measures of Equations (30) to (32) in pattern recognition, we next exhibit the following numerical examples.…”

Section: Numerical Examples and Comparisonsmentioning

confidence: 53%

“…Suppose that there are three patterns denoted by PFSs in the universe of discourse X x x x = { , , }. From the above results using the proposed similarity measures Equations (30) to (32), it is obvious that P Q =˜. 1 Furthermore, we can see that the sample Q is more similar to the pattern P 2 than the pattern P .…”

Section: Numerical Examples and Comparisonsmentioning

confidence: 85%

“…2 Therefore, it is evident that the sample Q is similar to the pattern P , 1 according to the principle of the maximum degree of similarity between PFSs. According to the above results using the similarity measures of Equations (30) to (32) between PFSs, we have that the sample Q is closed to the pattern P 3 as compared to the patterns P 1 and P . 2 Thus, according to the principle of the maximum degree of similarity between PFSs, we have that the sample Q is similar to the pattern P .…”

Section: Numerical Examples and Comparisonsmentioning

confidence: 91%

“…Distance and similarity measures play a vital role to differentiate two sets or objects. Distance and similarity have seen a lot of applications . Therefore, we give novel distance and similarity measures between two PFSs based on the Hausdorff metric in the next section.…”

Section: Preliminariesmentioning

confidence: 99%

“…2 Thus, according to the principle of the maximum degree of similarity between PFSs, we have that the sample Q is similar to the pattern P . 3 We next give an example pertaining to characterization of the similarities between linguistic variables by using the proposed similarity measures of Equations (30) to (32). In the literature, the usage of fuzzy sets for querying a regular database was suggested by Tahani 33 and database queries with fuzzy linguistic quantifiers were introduced by Kacprzyk and Ziolkowski 34 and Petry.…”

Section: Numerical Examples and Comparisonsmentioning

confidence: 99%

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Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

2019

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Pythagorean fuzzy sets (PFSs) were proposed by Yager in 2013 to treat imprecise and vague information in daily life more rigorously and efficiently with higher precision than intuitionistic fuzzy sets. In this paper, we construct new distance and similarity measures of PFSs based on the Hausdorff metric. We first develop a method to calculate a distance between PFSs based on the Hasudorff metric, along with proving several properties and theorems. We then consider a generalization of other distance measures, such as the Hamming distance, the Euclidean distance, and their normalized versions. On the basis of the proposed distances for PFSs, we give new similarity measures to compute the similarity degree of PFSs. Some examples related to pattern recognition and linguistic variables are used to validate the proposed distance and similarity measures. Finally, we apply the proposed methods to multicriteria decision‐making by constructing a Pythagorean fuzzy Technique for Order Preference by Similarity to an Ideal Solution and then present a practical example to address an important issue related to social sector. Numerical results indicate that the proposed methods are reasonable and applicable and also that they are well suited in pattern recognition, linguistic variables, and multicriteria decision‐making with PFSs.

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Section: Numerical Examples and Comparisonsmentioning

confidence: 53%

Section: Numerical Examples and Comparisonsmentioning

confidence: 85%

Section: Numerical Examples and Comparisonsmentioning

confidence: 91%

Section: Preliminariesmentioning

confidence: 99%

Section: Numerical Examples and Comparisonsmentioning

confidence: 99%

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Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

2019

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Hesitant Mahalanobis distance with applications to estimating the optimal number of clusters

Chao

Zhao

Zhang

2021

Int J of Intelligent Sys

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Distance measure is an essential tool to characterize the difference between two samples. Recently, lots of distance measures have been proposed for hesitant fuzzy sets (HFSs). In this paper, we shall propose some novel distance formulas to measure the deviation between two HFSs. First, we define some new concepts including the hesitant fuzzy variance, covariance, and correlation coefficient. Based on these concepts and the idea of the traditional Mahalanobis distance, the hesitant Mahalanobis distance between two HFSs is developed. Then we discuss the properties of the new distance measure and uncover the significant characteristic of the introduced distance measure that it can give the attributes an adaptive weight and can eliminate the influence of the correlation between the attributes under hesitant fuzzy environment. And then, some extensions of this new distance measure are also developed. Second, to show the validity and applicability of the proposed distance measures, we compare them with the existing ones in decision making and cluster analysis with some numerical examples. Third, using the proposed distance measures, we develop two algorithms to estimate the optimal number of clusters, which is a new application area of the hesitant fuzzy distance measures. Finally, the two algorithms are applied in a numerical example to illustrate their applicability and efficiency.

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How to Make Decisions with Uncertainty Using Hesitant Fuzzy Sets?

Kizielewicz

Shekhovtsov

Sałabun

2022

Lecture Notes in Networks and Systems

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Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering

Cited by 36 publications

References 40 publications

Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

Distance and similarity measures of Pythagorean fuzzy sets based on the Hausdorff metric with application to fuzzy TOPSIS

Hesitant Mahalanobis distance with applications to estimating the optimal number of clusters

How to Make Decisions with Uncertainty Using Hesitant Fuzzy Sets?

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