Skip to main content
SpringerLink
Log in

Novel distance measures for Pythagorean fuzzy sets with applications to pattern recognition problems

  • Original Paper
  • Published:
Granular Computing Aims and scope Submit manuscript

Abstract

Pythagorean fuzzy set (PFS) is a concept that generalizes intuitionistic fuzzy sets. The notion of PFSs is very much applicable in decision science because of its unique nature of indeterminacy. The main feature of PFSs is that it is characterized by membership degree, non-membership degree, and indeterminate degree in such a way that the sum of the square of each of the parameters is one. In this paper, we propose some novel distance measures for PFSs by incorporating the conventional parameters that describe PFSs. We provide a numerical example to illustrate the validity and applicability of the distance measures for PFSs. While analyzing the reliability of the proposed distance measures in comparison with similar distance measures for PFSs in the literature, we discover that the proposed distance measures, especially, \(d_5\) yields the most reasonable measure. Finally, some applications of \(d_5\) to pattern recognition problems are explicated. These novel distance measures for Pythagorean fuzzy sets could be applied in decision making of real-life problems embedded with uncertainty.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Atanassov KT (1983) Intuitionistic fuzzy sets. VII ITKR’s Session, Sofia

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Atanassov KT (1989) Geometrical interpretation of the elements of the intuitionistic fuzzy objects. Preprint IM-MFAIS-1-89, Sofia

  • Chen SM, Chang CH (2015) A novel similarity measure between atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114

    Article  Google Scholar 

  • Chen SM, Chang YC (2011) Weighted fuzzy rule interpolation based on ga-based weight-learning techniques. IEEE Trans Fuzzy Syst 19(4):729–744

    Article  Google Scholar 

  • Chen SM, Cheng SH, Chiou CH (2016a) Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology. Inf Fusion 27:215–227

    Article  Google Scholar 

  • Chen SM, Cheng SH, Lan TC (2016b) Multicriteria decision making based on the topsis method and similarity measures between intuitionistic fuzzy values. Inf Sci 367–368(1):279–295

    Article  Google Scholar 

  • Chen SM, Huang CM (2003) Generating weighted fuzzy rules from relational database systems for estimating null values using genetic algorithms. IEEE Trans Fuzzy Syst 11(4):495–506

    Article  Google Scholar 

  • Chen SM, Lee SH, Lee CH (2001) A new method for generating fuzzy rules from numerical data for handling classification problems. Appl Artif Intell 15(7):645–664

    Article  Google Scholar 

  • Chen SM, Munif A, Chen GS, Liu HC, Kuo BC (2012) Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights. Expert Syst Appl 39(7):6320–6334

    Article  Google Scholar 

  • Chen SM, Tanuwijaya K (2011) Fuzzy forecasting based on high-order fuzzy logical relationships and automatic clustering techniques. Expert Syst Appl 38(12):15425–15437

    Article  Google Scholar 

  • Chen SM, Wang JY (1995) Document retrieval using knowledge-based fuzzy information retrieval techniques. IEEE Trans Syst Man Cybern 25(5):793–803

    Article  Google Scholar 

  • Cheng SH, Chen SM, Jian WS (2016) Fuzzy time series forecasting based on fuzzy logical relationships and similarity measures. Inf Sci 327:272–287

    Article  MathSciNet  MATH  Google Scholar 

  • Davvaz B, Sadrabadi EH (2016) An application of intuitionistic fuzzy sets in medicine. Int J Biomath 9(3):1650037

    Article  MathSciNet  MATH  Google Scholar 

  • De SK, Biswas R, Roy AR (2001) An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Set Syst 117(2):209–213

    Article  MATH  Google Scholar 

  • Dick S, Yager RR, Yazdanbakhsh O (2016) On Pythagorean and complex fuzzy set operations. IEEE Trans Fuzzy Syst 24(5):1009–1021

    Article  Google Scholar 

  • Du Y, Hou F, Zafar W, Yu Q, Zhai Y (2017) A novel method for multiattribute decision making with interval-valued Pythagorean fuzzy linguistic information. Int J Intell Syst 32(10):1085–1112

    Article  Google Scholar 

  • Ejegwa PA (2015) Intuitionistic fuzzy sets approach in appointment of positions in an organization via max-min-max rule. Glob J Sci Front Res F Math Decis Sci 15(6):1–6

    Google Scholar 

  • Ejegwa PA (2018) Distance and similarity measures for Pythagorean fuzzy sets. Granul Comput. https://doi.org/10.1007/s41066-018-00149-z

    Article  Google Scholar 

  • Ejegwa PA (2019a) Improved composite relation for Pythagorean fuzzy sets and its application to medical diagnosis. Granul Comput. https://doi.org/10.1007/s41066-019-00156-8

    Article  Google Scholar 

  • Ejegwa PA (2019b) Pythagorean fuzzy set and its application in career placements based on academic performance using max-min-max composition. Complex Intell Syst. https://doi.org/10.1007/s40747-019-0091-6

    Article  Google Scholar 

  • Ejegwa PA, Akubo AJ, Joshua OM (2014) Intuitionistic fuzzzy sets in career determination. J Inf Comput Sci 9(4):285–288

    Google Scholar 

  • Ejegwa PA, Modom ES (2015) Diagnosis of viral hepatitis using new distance measure of intuitionistic fuzzy sets. Intern J Fuzzy Math Arch 8(1):1–7

    Google Scholar 

  • Ejegwa PA, Onasanya BO (2019) Improved intuitionistic fuzzy composite relation and its application to medical diagnostic process. Note IFS 25(1):43–58

    Google Scholar 

  • Gao H, Wei GW (2018) Multiple attribute decision making based on interval-valued Pythagorean fuzzy uncertain linguistic aggregation operators. Int J Knowl Based Intell Eng Syst 22:59–81

    Google Scholar 

  • Garg H (2017) Generalized Pythagorean fuzzy geometric aggregation operators using einstein t-norm and t-conorm fo multicriteria decision making process. Int J Intell Syst 32(6):597–630

    Article  Google Scholar 

  • Garg H (2018) Linguistic Pythagorean fuzzy sets and its applications in multiattribute decision making process. Int J Intell Syst 33(6):1234–1263

    Article  Google Scholar 

  • Gou XJ, Xu ZS, Ren PJ (2016) The properties of continuous pythagorean fuzzy information. Int J Intell Syst 31(5):401–424

    Article  Google Scholar 

  • Hadi-Venchen A, Mirjaberi M (2014) Fuzzy inferior ratio method for multiple attribue decision making problems. Inf Sci 277:263–272

    Article  MATH  Google Scholar 

  • Hatzimichailidis AG, Papakostas AG, Kaburlasos VG (2012) A novel distance measure of intuitionistic fuzzy sets and its application to pattern recognition problems. Int J Intell Syst 27:396–409

    Article  Google Scholar 

  • He X, Du Y, Liu W (2016) Pythagorean fuzzy power average operators. Fuzzy Syst Math 30(6):116–124

    MATH  Google Scholar 

  • Khan MSA, Abdullah S, Ali A, Amin F (2018) Pythagorean fuzzy prioritized aggregation operators and their application to multiattribute group decision making. Granul Comput. https://doi.org/10.1007/s41066-018-0093-6

    Article  Google Scholar 

  • Lee LW, Chen SM (2008) Fuzzy risk analysis based on fuzzy numbers with different shapes and different deviations. Expert Syst Appl 34(4):2763–2771

    Article  Google Scholar 

  • Li DQ, Zeng WY (2018) Distance measure of Pythagorean fuzzy sets. Int J Intell Syst 33:348–361

    Article  Google Scholar 

  • Liu P, Chen SM (2017) Group decision making based on Heronian aggregation operators of intuitionistic fuzzy numbers. IEEE Trans Cybern 47(9):2514–2530

    Article  Google Scholar 

  • Liu P, Chen SM (2018) Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Inf Sci 430–431:599–619

    Article  MathSciNet  MATH  Google Scholar 

  • Liu P, Chen SM, Liu J (2017) Multiple attribute group decision making based on intuitionistic fuzzy interaction partitioned bonferroni mean operators. Inf Sci 411:98–121

    Article  MathSciNet  MATH  Google Scholar 

  • Peng X (2018) New similarity measure and distance measure for Pythagorean fuzzy set. Complex Intell Syst. https://doi.org/10.1007/s40747-018-0084-x

    Article  Google Scholar 

  • Peng X, Yang Y (2015) Some results for Pythagorean fuzzy sets. Int J Intell Syst 30:1133–1160

    Article  Google Scholar 

  • Perez-Dominguez L, Rodriguez-Picon LA, Alvarado-Iniesta A, Cruz DL, Xu Z (2018) Moora under Pythagorean fuzzy sets for multiple criteria decision making. Complex. https://doi.org/10.1155/2018/2602376

    Article  MATH  Google Scholar 

  • Rahman K, Abdullah S, Ali A (2018) Some induced aggregation operators based on interval-valued Pythagorean fuzzy numbers. Granul Comput. https://doi.org/10.1007/s41066-018-0091-8

    Article  Google Scholar 

  • Szmidt E (2014) Distances and similarities in intuitionistic fuzzy sets. Springer International Publishing, New York

    Book  MATH  Google Scholar 

  • Szmidt E, Kacprzyk J (2000) Distances between inuitionistic fuzzy sets. Fuzzy Set Syst 114(3):505–518

    Article  MATH  Google Scholar 

  • Szmidt E, Kacprzyk J (2001) Intuitionistic fuzzy sets in some medical applications. Note IFS 7(4):58–64

    MATH  Google Scholar 

  • Szmidt E, Kacprzyk J (2004) Medical diagnostic reasoning using a similarity measure for intuitionistic fuzzy sets. Note IFS 10(4):61–69

    MATH  Google Scholar 

  • Wang HY, Chen SM (2008) Evaluating students’ answerscripts using fuzzy numbers associated with degrees of confidence. IEEE Trans Fuzzy Syst 16(2):403–415

    Article  Google Scholar 

  • Wang W, Xin X (2005) Distance measure between intuitionistic fuzzy sets. Pattern Recognit Lett 26:2063–2069

    Article  Google Scholar 

  • Yager RR (2013a) Pythagorean fuzzy subsets. In: Proc joint IFSAWorld congress NAFIPS annual meeting, pp 57–61

  • Yager RR (2013b) Pythagorean membership grades in multicriteria decision making. Technical Report MII-3301 Machine Intelligence Institute, Iona College, New Rochelle, NY

  • Yager RR (2014) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965

    Article  Google Scholar 

  • Yager RR (2016) Properties and applications of Pythagoean fuzzy sets. Springer, Berlin

    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 8:338–353

    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. https://doi.org/10.1002/int.22027

    Article  Google Scholar 

  • Zhang XL, Xu ZS (2014) Extension of topsis to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078

    Article  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the Editors-in-chief, Professors Withold Pedrycz and Shyi-Ming Chen for their technical comments, and to the anonymous reviewers for their suggestions, which have improved the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373, Makurdi, Nigeria

    P. A. Ejegwa & J. A. Awolola

Authors
  1. P. A. Ejegwa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. Awolola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. A. Ejegwa.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest toward the publication of this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ejegwa, P.A., Awolola, J.A. Novel distance measures for Pythagorean fuzzy sets with applications to pattern recognition problems. Granul. Comput. 6, 181–189 (2021). https://doi.org/10.1007/s41066-019-00176-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41066-019-00176-4

Keywords

Search

Navigation