Cognitive Computation

, Volume 10, Issue 2, pp 242–259 | Cite as

A Novel Picture Fuzzy Linguistic Aggregation Operator and Its Application to Group Decision-making

Article
  • 145 Downloads

Abstract

The intuitionistic fuzzy set (IFS) is an effective tool to express uncertain and incomplete cognitive information with the membership, non-membership, and hesitance degrees. The picture fuzzy set (PFS) is an extension of IFS, and it can contain more information than IFS. Linguistic variables (LVs) can easily describe the complex cognitive information provided by the decision makers. So it is necessary to combine the PFS with the linguistic term and define the picture fuzzy linguistic set (PFLS), then extend Archimedean triangular norms (t-norms) and triangular conorms (s-norms), which are the generalizations of a number of other t-norms and s-norms, such as Einstein, Hamacher, and Frank s-norms and t-norms, to process PFLS. By combining the PFS and the linguistic term, we define the picture fuzzy linguistic set (PFLS) and the operations of the picture fuzzy linguistic numbers (PFLNs). Further, we develop the Archimedean picture fuzzy linguistic weighted arithmetic averaging (A-PFLWAA) operator and discuss a number of properties and special cases of this operator. Besides, we propose an approach to deal with multiple attributes group decision-making (MAGDM) problem on the basis of the developed A-PFLWAA operator. An approach to solve MAGDM problem is proposed on the basis of the A-PFLWAA operator, and the validity and reliability of the proposed method are proven by an illustrative example. The PFLNs are the generalizations of the intuitionistic linguistic numbers (ILNs), and they contain more information than ILNs. In addition, our method cannot only be utilized to solve the problems with ILNs but also be used to deal with the problems with PFLNs, and it is a generalization of a number of the existing methods.

Keywords

Picture fuzzy set Triangular norms and conorms Multiple attribute group decision-making 

Notes

Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.

Research Involving Human Participants and/or Animals

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

References

  1. 1.
    Alcalde C, Burusco A, Fuentes-González R, Zubia I. The use of linguistic variables and fuzzy propositions in the L-Fuzzy concept theory. Comput Math Appl. 2011;62(8):3111–22.  https://doi.org/10.1016/j.camwa.2011.08.024.CrossRefGoogle Scholar
  2. 2.
    Beliakov G, Bustince H, Goswami DP, Mukherjee UK, Pal NR. On averaging operators for Atanassov’s intuitionistic fuzzy sets. Inf Sci. 2011;181(6):1116–24.  https://doi.org/10.1016/j.ins.2010.11.024.CrossRefGoogle Scholar
  3. 3.
    Beliakov G, Pradera A, Calvo T. Aggregation functions: a guide for practitioners. Studies in fuzziness and Soft Computing. 2007. p. 221–260.Google Scholar
  4. 4.
    Chen J, Wang Z. Multi-attribute decision making method based on intuitionistic uncertain linguistic new aggregation operator, systems engineering. Theory Pract. 2016;36(7):1871–8.Google Scholar
  5. 5.
    Cuong BC. Picture fuzzy sets-first results (part 1). In: Seminar on neuro-fuzzy systems with applications. Preprint 03/2013. Hanoi: Institute of Mathematics; 2013.Google Scholar
  6. 6.
    Cuong BC. Picture fuzzy sets-first results (part 2). In: Seminar on neuro-fuzzy systems with applications. Preprint 04/2013. Hanoi:Institute of Mathematics; 2013.Google Scholar
  7. 7.
    Cuong BC, Hai PV. Some fuzzy logic operators for picture fuzzy sets. 2015 Seventh International Conference on Knowledge and Systems Engineering (KSE), Ho Chi Minh, Vietnam. 2015. p. 132–137.Google Scholar
  8. 8.
    Cuong BC, Kreinovich V. Picture fuzzy sets—a new concept for computational intelligence problems. Proceedings of the Third World Congress on Information and Communication Technologies WICT’2013, Hanoi, Vietnam, December 15-18, 2013; p. 1–6.Google Scholar
  9. 9.
    Czubenko M, Kowalczuk Z, Ordys A. Autonomous driver based on an intelligent system of decision-making. Cogn Comput. 2015;7(5):569–81.  https://doi.org/10.1007/s12559-015-9320-5.CrossRefGoogle Scholar
  10. 10.
    Farhadinia B, Multiple Criteria A. Decision making model with entropy weight in an interval-transformed hesitant fuzzy environment. Cogn Comput. 2017:1–13.Google Scholar
  11. 11.
    Farhadinia B, Xu ZS. Distance and aggregation-based methodologies for hesitant fuzzy decision making. Cogn Comput. 2017;9(8):81–94.  https://doi.org/10.1007/s12559-016-9436-2.CrossRefGoogle Scholar
  12. 12.
    He YD, Chen HY, He Z, Zhou L. Multi-attribute decision making based on neutral averaging operators for intuitionistic fuzzy information. Appl Soft Comput J. 2015;27(C):64–76.  https://doi.org/10.1016/j.asoc.2014.10.039.CrossRefGoogle Scholar
  13. 13.
    He YD, He Z. Extensions of Atanassov’s intuitionistic fuzzy interaction Bonferroni means and their application to multiple attribute decision making. IEEE Trans Fuzzy Syst. 2016;24(3):558–73.  https://doi.org/10.1109/TFUZZ.2015.2460750.CrossRefGoogle Scholar
  14. 14.
    He YD, He Z, Chen HY. Intuitionistic fuzzy interaction Bonferroni means and its application to multiple attribute decision making. IEEE Trans Cybernet. 2015;45(1):116–28.  https://doi.org/10.1109/TCYB.2014.2320910.CrossRefGoogle Scholar
  15. 15.
    Herrera F, Herrera-Viedma E, Verdegay JL. A model of consensus in group decision making under linguistic assessments. Fuzzy Sets Syst. 1996;79(1):73–87.CrossRefGoogle Scholar
  16. 16.
    Hu JH, Li P, Chen XH. An interval neutrosophic projection-based VIKOR method for selecting doctors. Cogn Comput. 2017;  https://doi.org/10.1007/s12559-017-9499-8.
  17. 17.
    Ju Y, Liu X, Ju D. Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making. Soft Comput. 2015;20(11):4521–48.CrossRefGoogle Scholar
  18. 18.
    Le HS. A novel distributed picture fuzzy clustering method on picture fuzzy sets. Expert Syst Appl. 2015;42(1):51–66.CrossRefGoogle Scholar
  19. 19.
    Li J, Wang JQ. Multi-criteria outranking methods with hesitant probabilistic fuzzy sets. Cogn Comput. 2017;9(5):611–25.  https://doi.org/10.1007/s12559-017-9476-2.CrossRefGoogle Scholar
  20. 20.
    Liu PD. Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to group decision making. Appl Math Model. 2014;22(1):83–97.Google Scholar
  21. 21.
    Liu PD. The aggregation operators based on Archimedean t-conorm and t-norm for the single valued neutrosophic numbers and their application to decision making. Int J Fuzzy Syst. 2016;18(5):1–15.CrossRefGoogle Scholar
  22. 22.
    Liu PD, Chu Y, Chen Y. Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int J Fuzzy Syst. 2014;16(2):242–55.Google Scholar
  23. 23.
    Liu PD, Li HG. Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making. Cogn Comput. 2017;9(4):494–512.  https://doi.org/10.1007/s12559-017-9453-9.CrossRefGoogle Scholar
  24. 24.
    Liu PD, Tang GL. Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and Choquet integral. Cogn Comput. 2016;8(6):1036–56.  https://doi.org/10.1007/s12559-016-9428-2.CrossRefGoogle Scholar
  25. 25.
    Meng F, Chen X. Correlation coefficients of hesitant fuzzy sets and their application based on fuzzy measures. Cogn Comput. 7(4):445–63.Google Scholar
  26. 26.
    Meng F, Wang C, Chen X. Linguistic interval hesitant fuzzy sets and their application in decision making. Cogn Comput. 2016;8(1):52–68.  https://doi.org/10.1007/s12559-015-9340-1.CrossRefGoogle Scholar
  27. 27.
    Peter EP, Mesiar R, Pap E. On the relationship of associative compensatory operators to triangular norms and conorms. Int J Uncertainty Fuzziness. 1996;04(2):129–44.CrossRefGoogle Scholar
  28. 28.
    Schweizer B, Sklar A. Probabilistic metric spaces. Probab Theory Relat Fields. 1973;26(3):235–9.Google Scholar
  29. 29.
    Thanh ND, Ali M, Le HS, Novel Clustering A. Algorithm in a neutrosophic recommender system for medical diagnosis. Cogn Comput. 2017;9(4):526–44.  https://doi.org/10.1007/s12559-017-9462-8.CrossRefGoogle Scholar
  30. 30.
    Tian ZP, Wang J, Wang JQ, Zhang HY. A likelihood-based qualitative flexible approach with hesitant fuzzy linguistic information. Cogn Comput. 2016;8(4):670–83.  https://doi.org/10.1007/s12559-016-9400-1.CrossRefGoogle Scholar
  31. 31.
    Wang JQ, Kuang JJ, Wang J, Zhang HY. An extended outranking approach to rough stochastic multi-criteria decision-making problems. Cogn Comput. 2016;8(6):1144–60.  https://doi.org/10.1007/s12559-016-9417-5.CrossRefGoogle Scholar
  32. 32.
    Wang X, Wang J, Deng S. Some geometric operators for aggregating intuitionistic linguistic information. Int J Fuzzy Syst. 2015;17(2):268–78.  https://doi.org/10.1007/s40815-015-0007-6.CrossRefGoogle Scholar
  33. 33.
    Wei G. Picture fuzzy cross-entropy for multiple attribute decision making problems. J Bus Econ Manag. 2016;17(4):491–502.  https://doi.org/10.3846/16111699.2016.1197147.CrossRefGoogle Scholar
  34. 34.
    Wei G, Alsaadi FE, Hayat T, Alsaedi A. Picture 2-tuple linguistic aggregation operators in multiple attribute decision making. Soft Comput. 2016;  https://doi.org/10.1007/s00500-016-2403-8.
  35. 35.
    Wei G, Alsaadi FE, Hayat T, Alsaedi A. Projection models for multiple attribute decision making with picture fuzzy information. Int J Mach Learn Cybern. 2016;  https://doi.org/10.1007/s13042-016-0604-1.
  36. 36.
    Xia M, Xu Z, Zhu B. Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm. Knowl-Based Syst. 2012;31(7):78–88.  https://doi.org/10.1016/j.knosys.2012.02.004.CrossRefGoogle Scholar
  37. 37.
    Yang J, Gong L, Tang Y, Yan J, He H, Zhang L, et al. An improved SVM-based cognitive diagnosis algorithm for operation states of distribution grid. Cogn Comput. 2015;7(5):582–93.  https://doi.org/10.1007/s12559-015-9323-2.CrossRefGoogle Scholar
  38. 38.
    Yang Y, Liang C, Ji S, Liu T. Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making. J Intell Fuzzy Syst. 2015;29(4):1711–22.  https://doi.org/10.3233/IFS-151648.CrossRefGoogle Scholar
  39. 39.
    Yu Q, Hou F, Zhai Y, Du Y. Some hesitant fuzzy einstein aggregation operators and their application to multiple attribute group decision making. Int J Intell Syst. 2015;71(3):825–39.Google Scholar
  40. 40.
    Yu SM, Wang J, Wang JQ. An extended TODIM approach with intuitionistic linguistic numbers. Int Trans Oper Res. 2016;  https://doi.org/10.1111/itor.12363.
  41. 41.
    Zadeh LA. Fuzzy sets versus probability. Proc IEEE. 1980;68(3):421–1.  https://doi.org/10.1109/PROC.1980.11659.
  42. 42.
    Zhang HY, Ji P, Wang JQ, Chen XH, Neutrosophic Normal A. Cloud and its application in decision-making. Cogn Comput. 2016;8(4):649–69.  https://doi.org/10.1007/s12559-016-9394-8.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Management Science and EngineeringShandong University of Finance and EconomicsJinanChina

Personalised recommendations