Pythagorean fuzzy sets (PFSs), characterized by membership degrees and non-membership degrees, are a more effective and flexible way than intuitionistic fuzzy sets (IFSs) to capture indeterminacy. In this paper, some new diverse types of similarity measures, overcoming the blemishes of the existing similarity measures, for PFSs with multiple parameters are studied, along with their detailed proofs. The various desirable properties among the developed similarity measures and distance measures have also been derived. A comparison between the proposed and the existing similarity measures has been performed in terms of the division by zero problem, unsatisfied similarity axiom conditions, and counter-intuitive cases for showing their effectiveness and feasibility. The initiated similarity measures have been illustrated with case studies of pattern recognition, along with the effect of the different parameters on the ordering and classification of the patterns.
This is a preview of subscription content, log in to check access.
The authors are very appreciative to the reviewers for their precious comments which enormously ameliorated the quality of this paper. Our work is sponsored by the National Natural Science Foundation of China (No. 61462019), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 18YJCZH054), Natural Science Foundation of Guangdong Province (No. 2018A030307033, 2018A0303130274), Social Science Foundation of Guangdong Province (No. GD18CFX06).
Compliance with Ethical Standards
Conflict of interests
The authors declare no conflict of interests regarding the publication for the paper.
Chen SM, Chang CH (2015) A novel similarity measure between Atanssov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114CrossRefGoogle Scholar
Hung WL, Yang MS (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recognit Lett 25(14):1603–1611CrossRefGoogle Scholar
Peng XD, Yang Y (2016) Fundamental properties of interval-valued Pythagorean fuzzy aggregation operators. Int J Intell Syst 31(5):444–487CrossRefGoogle Scholar
Zhang C, Li D, Ren R (2016) Pythagorean fuzzy multigranulation rough set over two universes and its applications in merger and acquisition. Int J Intell Syst 31(9):921–943CrossRefGoogle Scholar
Peng XD, Yang Y (2016) Multiple attribute group decision making methods based on Pythagorean fuzzy linguistic set. Comput Eng Appl 52(23):50–54Google Scholar
Liu ZM, Liu PD, Liu WL, Pang JY (2017) Pythagorean uncertain linguistic partitioned bonferroni mean operators and their application in multi-attribute decision making. J Intell Fuzzy Syst 32(3):2779–2790MathSciNetCrossRefGoogle Scholar
Liang DC, Xu ZS (2017) The new extension of TOPSIS method for multiple criteria decision making with hesitant Pythagorean fuzzy sets. Appl Soft Comput 60:167–179CrossRefGoogle Scholar
Peng XD, Yang Y, Song J, Jiang Y (2015) Pythagorean fuzzy soft set and its application. Comput Eng 41(7):224–229Google Scholar
Peng HG, Wang JQ (2018) A multicriteria group decision-making method based on the normal cloud model with Zadeh’s Z-numbers. IEEE Trans Fuzzy Syst 26:3246–3260CrossRefGoogle Scholar
Chen TY (2019) Multiple criteria decision analysis under complex uncertainty: a pearson-like correlation-based Pythagorean fuzzy compromise approach. Int J Intell Syst 34(1):114–151CrossRefGoogle Scholar
Yang W, Pang Y (2018) New Pythagorean fuzzy interaction Maclaurin symmetric mean operators and their application in multiple attribute decision making. IEEE Access 6:39241–39260CrossRefGoogle Scholar
Peng XD, Li WQ (2019) Algorithms for interval-valued Pythagorean fuzzy sets in emergency decision making based on multiparametric similarity measures and WDBA. IEEE Access 7:7419–7441CrossRefGoogle Scholar
Yang Y, Ding H, Chen ZS, Li YL (2016) A note on extension of TOPSIS to multiple criteria decision making with Pythagorean fuzzy sets. Int J Intell Syst 31(1):68–72CrossRefGoogle Scholar
Wei G, Wei Y (2018) Similarity measures of Pythagorean fuzzy sets based on the cosine function and their applications. Int J Intell Syst 33(3):634–652CrossRefGoogle Scholar
Li DQ, Zeng WY, Qian Y (2017) Distance measures of Pythagorean fuzzy sets and their applications in multiattribute decision making. Control Decis 32(10):1817–1823zbMATHGoogle Scholar
Zhang X (2016) A novel approach based on similarity measure for Pythagorean fuzzy multiple criteria group decision making. Int J Intell Syst 31(6):593–611CrossRefGoogle 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):2236–2254CrossRefGoogle Scholar
Peng X, Yuan H, Yang Y (2017) Pythagorean fuzzy information measures and their applications. Int J Intell Syst 32(10):991–1029CrossRefGoogle Scholar
Huang HH, Liang Y (2018) Hybrid L1/2 + 2 method for gene selection in the Cox proportional hazards model. Comput Meth Prog Bio 164:65–73CrossRefGoogle Scholar