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Similarity measures of Pythagorean fuzzy soft sets and clustering analysis


Pythagorean fuzzy set (PFS) is a broadening of intuitionistic fuzzy set that can represent the situations where the sum of membership and the non-membership values exceeds one. Adding parameterization to PFS, we obtain a structure named as Pythagorean fuzzy soft set (PFSS). It has a higher capacity to deal with vagueness as it captures both the structures of a PFS and a soft set. Several practical situations demand the measure of similarity between two structures, whose sum of membership value and non-membership value exceeds one. There are no existing tools to measure the similarity between PFSS and this paper put forward similarity measures for PFSS. An axiomatic definition for similarity measure is proposed for PFSS and certain expressions for similarity measure are introduced. Further, some theorems which express the properties of similarity measures are proved. A comparative study between proposed expressions for similarity measure is carried out. Also, a clustering algorithm based on PFSS is introduced by utilizing the proposed similarity measure.

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The first author would like to thank MHRD, Government of India, along with National Institute of Technology, Calicut, for the financial assistance and the DST, Government of India, for providing support to carry out this work under the scheme ‘FIST’ (No. SR/FST/MS-I/2019/40). The authors want to thank the editor and reviewers for their valuable and constructive comments to improve this paper.


This research received no external funding.

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ATM wrote the original manuscript. SJJ and HG contributed to writing, reviewing, and editing.

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Correspondence to Sunil Jacob John.

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Athira, T.M., John, S.J. & Garg, H. Similarity measures of Pythagorean fuzzy soft sets and clustering analysis. Soft Comput 27, 3007–3022 (2023).

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  • Pythagorean fuzzy soft sets
  • Similarity measures
  • Clustering algorithms