Skip to main content
SpringerLink
Log in

Distance and similarity measures for Pythagorean fuzzy sets

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

Abstract

The concept of Pythagorean fuzzy sets is very much applicable in decision science because of its unique nature of indeterminacy. The main feature of Pythagorean fuzzy sets is that it is characterized by three parameters, namely, 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 present axiomatic definitions of distance and similarity measures for Pythagorean fuzzy sets, taking into account the three parameters that describe the sets. Some distance and similarity measures in intuitionistic fuzzy sets, viz, Hamming, Euclidean, normalized Hamming, and normalized Euclidean distances, and similarities are extended to Pythagorean fuzzy set setting. However, it is discovered that Hamming and Euclidean distances and similarities fail the metric conditions in Pythagorean fuzzy set setting whenever the elements of the two Pythagorean fuzzy sets, whose distance and similarity are to be measured, are not equal. Finally, numerical examples are provided to illustrate the validity and applicability of the measures. These measures are suggestible to be resourceful in multicriteria decision-making problems (MCDMP) and multiattribute decision-making problems (MADMP), respectively.

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

    MATH  Google Scholar 

  • Atanassov KT (1999) Intuitionistic fuzzy sets: theory and applications. Physica-Verlag, Heidelberg

    MATH  Google Scholar 

  • Atanassov KT (2012) On intuitionistic fuzzy sets theory. Springer, Berlin

    MATH  Google Scholar 

  • Beliakov G, James S (2014) Averaging aggregation functions for preferences expressed as Pythagorean membership grades and fuzzy orthopairs. In: Proc IEEE Int Conf Fuzzy Syst (FUZZ-IEEE), pp 298–305

  • 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

    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

    MathSciNet  Google Scholar 

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

    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

    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):15425–1543

    Google Scholar 

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

    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

    MATH  Google Scholar 

  • Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets theory and applications. World Scientific, Singapore

    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

    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

    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 Math Decis Sci 15(6):1–6

    Google Scholar 

  • Ejegwa PA, Akubo AJ, Joshua OM (2014) Intuitionistic fuzzy 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 

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

    Google Scholar 

  • Garg H (2016b) A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem. J Intell Fuzzy Syst 31(1):529–540

    MATH  Google Scholar 

  • Garg H (2016c) A novel correlation coefficients between Pythagorean fuzzy sets and its applications to decision making processes. Int J Intell Syst 31(12):1234–1252

    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

    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

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  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

    Google Scholar 

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

    MATH  Google Scholar 

  • Kacprzyk J (1997) Multistage fuzzy control. Wiley, Chichester

    MATH  Google Scholar 

  • Khan M S. A, Abdullah S, Ali A, Amin F (2018a) An extension of VIKOR method for multiattribute decision making under Pythagorean hesitant fuzzy setting. Granul Comput. https://doi.org/10.1007/s41066-018-0102-9

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  • Liang D, Xu Z (2017) The new extension of topsis method for multiple criteria decision making with hesitant Pythagorean fuzzy sets. Appl Soft Comput 60:167–179

    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

    Google Scholar 

  • Mohagheghi V, Mousavi SM, Vahdani B (2017) Enhancing decision-making flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and Pythagorean fuzzy sets. Appl Soft Comput 61:527–535

    Google Scholar 

  • Mohd WRW, Abdullah L (2018) Similarity measures of Pythagorean fuzzy sets based on combination of cosine similarity measure and Euclidean distance measure. AIP Conf Proc 1974:030017–1–030017-7. https://doi.org/10.1063/1.5041661

    Article  Google Scholar 

  • Peng X, Selvachandran G (2017) Pythagorean fuzzy set: state of the art and future directions. Artif Intell Rev. https://doi.org/10.1007/s10462-017-9596-9

    Google Scholar 

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

    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

    MATH  Google Scholar 

  • Rahman K, Abdullah S (2018) Generalized interval-valued Pythagorean fuzzy aggregation operators and their application to group decision making. Granul Comput. https://doi.org/10.1007/s41066-018-0082-9

    Google Scholar 

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

    Google Scholar 

  • Rahman K, Abdullah S, Shakeel M, Khan M. S. A, Ullah M (2017) Interval-valued Pythagorean fuzzy geometric aggregation operators and their application to group decision making problem. Cogent Math. https://doi.org/10.1080/23311835.2017.1338638

  • Rahman K, Ali A, Abdullah S, Amin F (2018b) Approaches to multi-attribute group decision making based on induced interval-valued Pythagorean fuzzy einstein aggregation operator. New Math Nat Comput 14(3):343–361

    MathSciNet  MATH  Google Scholar 

  • Szmidt E (2014) Distances and similarities in intuitionistic fuzzy sets. Springer, Switzerland

    MATH  Google Scholar 

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

    MathSciNet  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

    Google Scholar 

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

    Google Scholar 

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

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

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

    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

    Google Scholar 

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

    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

    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

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is thankful to the Editors-in-Chief: Professor Witold Pedrycz and Professor Shyi-Ming Chen for their technical comments, and to the anonymous reviewers for their suggestions, which have improved the quality of this paper.

Author information

Authors and Affiliations

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

    Paul Augustine Ejegwa

Authors
  1. Paul Augustine Ejegwa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Augustine Ejegwa.

Ethics declarations

Conflict of interest

The author declares 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. Distance and similarity measures for Pythagorean fuzzy sets. Granul. Comput. 5, 225–238 (2020). https://doi.org/10.1007/s41066-018-00149-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41066-018-00149-z

Keywords

Search

Navigation