Some q-rung orthopair fuzzy point weighted aggregation operators for multi-attribute decision making

  • Yuping Xing
  • Runtong ZhangEmail author
  • Zhen Zhou
  • Jun Wang
Methodologies and Application


q-Rung orthopair fuzzy sets, originally proposed by Yager, can dynamically adjust the range of indication of decision information by changing a parameter q based on the different hesitation degree, and point operator is a useful aggregation technology that can control the uncertainty of valuating data from some experts and thus get intensive information in the process of decision making. However, the existing point operators are not available for decision-making problems under q-Rung orthopair fuzzy environment. Thus, in this paper, we firstly propose some new point operators to make it conform to q-rung orthopair fuzzy numbers (q-ROFNs). Then, associated with classic arithmetic and geometric operators, we propose a new class of point weighted aggregation operators to aggregate q-rung orthopair fuzzy information. These proposed operators can redistribute the membership and non-membership in q-ROFNs according to different principle. Furthermore, based on these operators, a novel approach to multi-attribute decision making (MADM) in q-rung orthopair fuzzy context is introduced. Finally, we give a practical example to illustrate the applicability of the new approach. The experimental results show that the novel MADM method outperforms the existing MADM methods for dealing with MADM problems.


Multi-attribute decision making q-Rung orthopair fuzzy set q-Rung orthopair fuzzy point operators q-Rung orthopair fuzzy point weighted aggregation operators 



This work was partially supported by a key program of the National Natural Science Foundation of China (No. 71532002) and Fundamental Research Funds for the Central Universities (No. 2017YJS067) and Beijing Logistics Informatics Research Base.

Compliance with ethical standards

Conflict of interest

The authors declare that there are no conflicts of interest to this work.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Arora R, Garg H (2018) Prioritized averaging/geometric aggregation operators under the intuitionistic fuzzy soft set environment. Sci Iran. Google Scholar
  2. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96CrossRefzbMATHGoogle Scholar
  3. Garg H (2016a) A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Appl Soft Comput 38:988–999CrossRefGoogle Scholar
  4. Garg H (2016b) 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 (2016c) 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 (2016d) 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
  7. Garg H (2016e) A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multi-criteria decision making problem. J Intell Fuzzy Syst 31(1):529–540CrossRefzbMATHGoogle Scholar
  8. Garg H (2017a) Generalized Pythagorean fuzzy geometric aggregation operators using Einstein t-norm and t-conorm for multi-criteria decision making process. Int J Intell Syst 32(6):597–630CrossRefGoogle Scholar
  9. Garg H (2017b) A new improved score function of an interval-valued Pythagorean fuzzy set based TOPSIS method. Int J Uncertain Quantif 7(5):463–474CrossRefGoogle Scholar
  10. Garg H (2018a) Some methods for strategic decision-making problems with immediate probabilities in Pythagorean fuzzy environment. Int J Intell Syst 33(4):687–712CrossRefGoogle Scholar
  11. Garg H (2018b) Generalized Pythagorean fuzzy geometric interactive aggregation operators using Einstein operations and their application to decision making. J Exp Theor Artif Int. Google Scholar
  12. Garg H (2018c) New exponential operational laws and their aggregation operators for interval-valued pythagorean fuzzy multi-criteria decision-making. Int J Intell Syst 33(3):653–683CrossRefGoogle Scholar
  13. Garg H (2018d) Linguistic Pythagorean fuzzy sets and its applications in multi attribute decision making process. Int J Intell Syst 33(6):1234–1263CrossRefGoogle Scholar
  14. Garg H (2018e) Hesitant Pythagorean fuzzy sets and their aggregation operators in multiple-attribute decision-making. Int J Uncertain Quantif 8(3):267–289MathSciNetCrossRefGoogle Scholar
  15. Garg H (2018f) A linear programming method based on an improved score function for interval-valued Pythagorean fuzzy numbers and its application to decision-making. Int J Uncertain Fuzziness Knowl-Based Syst 29(1):67–80MathSciNetCrossRefGoogle Scholar
  16. He YD, Chen HY, He Z, Zhou LG (2015) Multi-attribute decision making based on neutral averaging operators for intuitionistic fuzzy information. Appl Soft Comput 27:64–76CrossRefGoogle Scholar
  17. He YD, He Z, Huang H (2017) Decision making with the generalized intuitionistic fuzzy power interaction averaging operators. Soft Comput 21(5):1129–1144CrossRefzbMATHGoogle Scholar
  18. Ju YB, Liu XY, Ju DW (2016) Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making. Soft Comput 20(11):4521–4548CrossRefzbMATHGoogle Scholar
  19. Khaleie S, Fasanghari M (2012) An intuitionistic fuzzy group decision making method using entropy and association coefficient. Soft Comput 16(7):1197–1211CrossRefGoogle Scholar
  20. Le HS (2016) Generalized picture distance measure and applications to picture fuzzy clustering. Appl Soft Comput 46:284–295CrossRefGoogle Scholar
  21. Liang DC, Xu ZS, Darko AP (2017) Projection model for fusing the information of Pythagorean fuzzy multi-criteria group decision making based on geometric Bonferroni mean. Int J Intell Syst 32(9):966–987CrossRefGoogle Scholar
  22. Liu PD, Li DF (2017) Some Muirhead mean operators for intuitionistic fuzzy numbers and their applications to group decision making. PLoS ONE 12(1):e0168767CrossRefGoogle Scholar
  23. Liu PD, Liu JL (2017) Some q-rung orthopai fuzzy Bonferroni mean operators and their application to multi-attribute group decision making. Int J Intell Syst 33(2):315–347MathSciNetCrossRefGoogle Scholar
  24. Liu HW, Wang GJ (2007) Multi-criteria decision-making methods based on intuitionistic fuzzy sets. Eur J Oper Res 179:220–233CrossRefzbMATHGoogle Scholar
  25. Liu PD, Wang P (2017) Some q-rung orthopair fuzzy aggregation operators and their applications to multi-attribute group decision making. Int J Intell Syst 33(2):259–280CrossRefGoogle Scholar
  26. Liu PD, Chen SM, Liu J (2017) Some intuitionistic fuzzy interaction partitioned Bonferroni mean operators and their application to multi-attribute group decision making. Inf Sci 411:98–121CrossRefGoogle Scholar
  27. Ma ZM, Xu ZS (2016) Symmetric Pythagorean fuzzy weighted geometric/averaging operators and their application in multi-criteria decision making problems. Int J Intell Syst 31(12):1198–1219CrossRefGoogle Scholar
  28. Mohammadi SE, Makui A (2017) Multi-attribute group decision making approach based on interval-valued intuitionistic fuzzy sets and evidential reasoning methodology. Soft Comput 21:5061–5080CrossRefzbMATHGoogle Scholar
  29. Peng XD, Yuan HY (2016) Fundamental properties of Pythagorean fuzzy aggregation operators. Fund Inform 147(4):415–446MathSciNetCrossRefzbMATHGoogle Scholar
  30. Peng XD, Dai JG, Garg H (2018) Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function. Int J Intell Syst 33(11):2255–2282CrossRefGoogle Scholar
  31. Wan SP, Yi ZH (2016) Power average of trapezoidal intuitionistic fuzzy numbers using strict t-norms and t-conorms. IEEE Trans Fuzzy Syst 24(5):1035–1047CrossRefGoogle Scholar
  32. Xia MM (2015) Point operators for intuitionistic multiplicative information. J Intell Fuzzy Syst 28:615–620MathSciNetzbMATHGoogle Scholar
  33. Xia MM, Xu ZS (2010) Generalized point operators for aggregating intuitionistic fuzzy information. Int J Intell Syst 25(11):1061–1080zbMATHGoogle Scholar
  34. Xing YP, Zhang RT, Xia MM, Wang J (2017) Generalized point aggregation operators for dual hesitant fuzzy information. J Intell Fuzzy Syst 33(1):515–527CrossRefzbMATHGoogle Scholar
  35. Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15(6):1179–1187CrossRefGoogle Scholar
  36. Yager RR (2014) Pythagorean membership grades in multi-criteria decision making. IEEE Trans Fuzzy Syst 22(4):958–965CrossRefGoogle Scholar
  37. Yager RR (2018) Generalized orthopair fuzzy sets. IEEE Trans Fuzzy Syst 26(5):1222–1230CrossRefGoogle Scholar
  38. Yager RR, Abbasov AM (2013) Pythagorean membership grades, complex numbers, and decision making. Int J Intell Syst 28(5):436–452CrossRefGoogle Scholar
  39. Yager RR, Alajlan N (2017) Approximate reasoning with generalized orthopair fuzzy sets. Inform Fusion 38:65–73CrossRefGoogle Scholar
  40. Zhang X (2016) Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods. Inf Sci 330:104–124CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yuping Xing
    • 1
  • Runtong Zhang
    • 1
    Email author
  • Zhen Zhou
    • 2
  • Jun Wang
    • 1
  1. 1.School of Economics and ManagementBeijing Jiaotong UniversityBeijingChina
  2. 2.School of ManagementCapital Normal UniversityBeijingChina

Personalised recommendations