Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process



Pythagorean fuzzy set, an extension of the intuitionistic fuzzy set which relax the condition of sum of their membership function to square sum of its membership functions is less than one. Under these environment and by incorporating the idea of the confidence levels of each Pythagorean fuzzy number, the present study investigated a new averaging and geometric operators namely confidence Pythagorean fuzzy weighted and ordered weighted operators along with their some desired properties. Based on its, a multi criteria decision-making method has been proposed and illustrated with an example for showing the validity and effectiveness of it. A computed results are compared with the aid of existing results.


Pythagorean fuzzy set MCDM Confidence levels Aggregation operators Decision making 


  1. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefGoogle Scholar
  2. Dalman H (2016) Uncertain programming model for multi-item solid transportation problem. Int J Mach Learn Cybern 1–9. doi: 10.1007/s13042-016-0538-7
  3. Dalman H, Güzel N, Sivri M (2016) A fuzzy set-based approach to multi-objective multi-item solid transportation problem under uncertainty. Int J Fuzzy Syst 18(4):716–729CrossRefGoogle Scholar
  4. Garg H (2016a) Generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein t-norm and t-conorm and their application to decision making. Comput Ind Eng 101:53–69CrossRefGoogle Scholar
  5. Garg H (2016b) Generalized intuitionistic fuzzy multiplicative interactive geometric operators and their application to multiple criteria decision making. Int J Mach Learn Cybern 7(6):1075–1092CrossRefGoogle Scholar
  6. Garg H (2016c) Generalized pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multicriteria decision-making process. Int J Intell Syst. doi: 10.1002/int.21860
  7. Garg H (2016d) A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Appl Soft Comput 38:988–999CrossRefGoogle Scholar
  8. Garg H (2016e) A new generalized pythagorean fuzzy information aggregation using Einstein operations and its application to decision making. Int J Intell Syst 31(9):886–920CrossRefGoogle Scholar
  9. Garg H (2016f) A novel accuracy function under interval-valued pythagorean fuzzy environment for solving multicriteria decision making problem. J Intell Fuzzy Syst 31(1):529–540CrossRefGoogle Scholar
  10. Garg H (2016g) A novel correlation coefficients between pythagorean fuzzy sets and its applications to decision-making processes. Int J Intell Syst 31(12):1234–1253CrossRefGoogle Scholar
  11. Garg H (2016h) Some series of intuitionistic fuzzy interactive averaging aggregation operators. SpringerPlus 5(1):999. doi: 10.1186/s40064-016-2591-9 CrossRefGoogle Scholar
  12. Garg H, Agarwal N, Tripathi A (2015) Entropy based multi-criteria decision making method under fuzzy environment and unknown attribute weights. Glob J Technol Optim 6:13–20Google Scholar
  13. Kumar K, Garg H (2016) TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment. Comput Appl Math 1–11. doi: 10.1007/s40314-016-0402-0
  14. Nancy Garg H (2016a) An improved score function for ranking neutrosophic sets and its application to decision-making process. Int J Uncertain Quantif 6(5):377–385CrossRefGoogle Scholar
  15. Nancy Garg H (2016b) Novel single-valued neutrosophic decision making operators under frank norm operations and its application. Int J Uncertain Quantif 6(4):361–375CrossRefGoogle Scholar
  16. Peng X, Yang Y (2015) Some results for pythagorean fuzzy sets. Int J Intell Syst 30(11):1133–1160CrossRefGoogle Scholar
  17. Wang W, Liu X (2012) Intuitionistic fuzzy information aggregation using Einstein operations. IEEE Trans Fuzzy Syst 20(5):923–938CrossRefGoogle Scholar
  18. Xu Y, Wang H, Merigo JM (2014) Intuitionistic fuzzy einstein choquet intergral operators for multiple attribute decision making. Technol Econ Dev Econ 20(2):227–253CrossRefGoogle Scholar
  19. Xu Z, Zhao N (2016) Information fusion for intuitionistic fuzzy decision making: an overview. Inf Fusion 28:10–23CrossRefGoogle Scholar
  20. Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187CrossRefGoogle Scholar
  21. Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433CrossRefGoogle Scholar
  22. Yager RR (1988) On ordered weighted avergaing aggregation operators in multi-criteria decision making. IEEE Trans Syst Man Cybern 18(1):183–190CrossRefGoogle Scholar
  23. Yager RR (2013) Pythagorean fuzzy subsets. In Procedding Joint IFSA World Congress and NAFIPS Annual Meeting. Edmonton, Canada, pp 57–61Google Scholar
  24. Yager RR (2014) Pythagorean membership grades in multicriteria decision making. IEEE Trans Fuzzy Syst 22:958–965CrossRefGoogle Scholar
  25. Yager RR, Abbasov AM (2013) Pythagorean membeship grades, complex numbers and decision making. Int J Intell Syst 28:436–452CrossRefGoogle Scholar
  26. Yager RR, Kacprzyk J (1997) The ordered weighted averaging operators: theory and applications. Kluwer Acadenmic Publisher, BostonCrossRefGoogle Scholar
  27. Ye J (2007) Improved method of multi criteria fuzzy decision making based on vague sets. Comput Aided Des 39(2):164–169CrossRefGoogle Scholar
  28. Ye J (2009) Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Expert Syst Appl 36:6809–6902CrossRefGoogle Scholar
  29. Yu D (2014) Intuitionistic fuzzy information aggregation under confidence levels. Appl Soft Comput 19:147–160CrossRefGoogle Scholar
  30. Yu D (2015) A scientometrics review on aggregation operator research. Scientometrics 105(1):115–133CrossRefGoogle Scholar
  31. Yu D, Shi S (2015) Researching the development of atanassov intuitionistic fuzzy set: using a citation network analysis. Appl Soft Comput 32:189–198CrossRefGoogle Scholar
  32. Zhang XL, Xu ZS (2014) Extension of TOPSIS to multi-criteria decision making with pythagorean fuzzy sets. Int J Intell Syst 29:1061–1078CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of MathematicsThapar UniversityPatialaIndia

Personalised recommendations