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Similarity measures of Pythagorean fuzzy soft sets and clustering analysis

Soft Computing volume 27, pages 3007–3022 (2023)Cite this article

Abstract

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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References

  • Aktaş H, Çağman N (2007) Soft sets and soft groups. Inf Sci 177(13):2726–2735

    Article  MathSciNet  MATH  Google Scholar 

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    Article  MathSciNet  MATH  Google Scholar 

  • Athira TM, John SJ, Garg H (2019a) Entropy and distance measures of Pythagorean fuzzy soft sets and their applications. J Intell Fuzzy Syst 1–14 (Preprint)

  • Athira TM, John SJ, Garg H (2019b) A novel entropy measure of Pythagorean fuzzy soft sets. AIMS Mathematics 5(2):1050–1061

  • Athira T, John SJ, Kumar PR (2020) Incomplete Pythagorean fuzzy soft sets. In: AIP conference proceedings, vol 2261, p 030143. AIP Publishing LLC (2020)

  • Babitha K, Sunil J (2010) Soft set relations and functions. Comput Math Appl 60(7):1840–1849

    Article  MathSciNet  MATH  Google Scholar 

  • Cagman N, Enginoglu S, Citak F (2011) Fuzzy soft set theory and its applications. Iran J Fuzzy Syst 8(3):137–147

    MathSciNet  MATH  Google Scholar 

  • Çağman N, Deli I (2013) Similarity measures of intuitionistic fuzzy soft sets and their decision making. arXiv preprint arXiv:1301.0456

  • Garg H (2016) A new generalized Pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. Int J Intell Syst 31(9):886–920

    Article  Google Scholar 

  • Garg H (2017) Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process. Comput Math Organ Theory 23(4):546–571

    Article  Google Scholar 

  • Garg H, Rani D (2019) Some results on information measures for complex intuitionistic fuzzy sets. Int J Intell Syst 34(10):2319–2363

    Article  Google Scholar 

  • Guleria A, Bajaj RK (2019) On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis. Soft Comput 23(17):7889–7900

    Article  MATH  Google Scholar 

  • Hameed MS, Mukhtar S, Khan HN, Ali S, Mateen MH, Gulzar M (2021) Pythagorean fuzzy n-soft groups. Int J Electr Comput Eng (IJECE) 21(2):1030–1038

    Google Scholar 

  • Hong DH, Hwang SY (1994) A note on the value similarity of fuzzy systems variables. Fuzzy Sets Syst 66(3):383–386

    Article  MathSciNet  MATH  Google Scholar 

  • Hu J, Pan L, Yang Y, Chen H (2019) A group medical diagnosis model based on intuitionistic fuzzy soft sets. Appl Soft Comput 77:453–466

    Article  Google Scholar 

  • John SJ (2021) Soft sets. In: Soft sets, pp 3–36. Springer, Berlin

  • Kharal A (2010) Distance and similarity measures for soft sets. New Math Natural Comput 6(03):321–334

    Article  MathSciNet  MATH  Google Scholar 

  • Liang Z, Shi P (2003) Similarity measures on intuitionistic fuzzy sets. Pattern Recogn Lett 24(15):2687–2693

    Article  MATH  Google Scholar 

  • Maji PK, Biswas R, Roy AR (2001) Intuitionistic fuzzy soft sets. J Fuzzy Math 9(3):677–692

    MathSciNet  MATH  Google Scholar 

  • Maji P, Biswas R, Roy A (2003) Soft set theory. Comput Math Appl 45(4–5):555–562

    Article  MathSciNet  MATH  Google Scholar 

  • Majumdar P, Samanta SK (2008) Similarity measure of soft sets. New Math Natural Comput 4(01):1–12

  • Majumdar P, Samanta S (2011) On similarity measures of fuzzy soft sets. Int J Adv Soft Comput Appl 3(2):1–8

    MATH  Google Scholar 

  • Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37(4–5):19–31

    Article  MathSciNet  MATH  Google Scholar 

  • Mukherjee A, Sarkar S (2015) Distance based similarity measures for interval-valued intuitionistic fuzzy soft sets and its application. New Trends Math Sci 3(4):34–42

    MathSciNet  Google Scholar 

  • Muthukumar P, Krishnan GSS (2016) A similarity measure of intuitionistic fuzzy soft sets and its application in medical diagnosis. Appl Soft Comput 41:148–156

    Article  Google Scholar 

  • Öztürk TY, Yolcu A (2020) Some structures on Pythagorean fuzzy soft topological spaces. In: 3rd International conference on mathematical and related sciences: current trends and developments proceedings book, p 154

  • Peng X, Garg H (2019) Multiparametric similarity measures on Pythagorean fuzzy sets with applications to pattern recognition. Appl Intell 1–39

  • Peng X, Selvachandran G (2017) Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev 1–55

  • Peng X, Yang Y, Song J, Jiang Y (2015) Pythagorean fuzzy soft set and its application. Comput Eng 41(7):224–229

    Google Scholar 

  • Riaz M, Naeem K, Afzal D (2020) A similarity measure under Pythagorean fuzzy soft environment with applications. Comput Appl Math 39(4):1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Sarala N, Suganya B (2016) An application of similarity measure of intuitionistic fuzzy soft set based on distance in medical diagnosis. Int J Sci Res 5(3):559–563

    Google Scholar 

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539

    MATH  Google Scholar 

  • Wang P (1982) Theory of fuzzy sets and their applications. Shanghai Science and Technology Publishing House, Shanghai

    Google Scholar 

  • Xu Z, Chen J, Wu J (2008) Clustering algorithm for intuitionistic fuzzy sets. Inf Sci 178(19):3775–3790

    Article  MathSciNet  MATH  Google Scholar 

  • Xuecheng L (1992) Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy Sets Syst 52(3):305–318

    Article  MathSciNet  MATH  Google Scholar 

  • Yager RR, Abbasov AM (2013) Pythagorean membership grades, complex numbers, and decision making. Int J Intell Syst 28(5):436–452

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  • Zeng W, Li D, Yin Q (2018) Distance and similarity measures of Pythagorean fuzzy sets and their applications to multiple criteria group decision making. Int J Intell Syst 33(11):2236–2254

    Article  Google Scholar 

Download references

Acknowledgements

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.

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This research received no external funding.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Calicut, 673601, Kerala, India

    T. M. Athira & Sunil Jacob John

  2. School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, 147004, Panjab, India

    Harish Garg

Authors
  1. T. M. Athira
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  2. Sunil Jacob John
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  3. Harish Garg
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Contributions

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

Corresponding author

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). https://doi.org/10.1007/s00500-022-07463-4

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Keywords

  • Pythagorean fuzzy soft sets
  • Similarity measures
  • Clustering algorithms