Three-Way Decisions with Intuitionistic Uncertain Linguistic Decision-Theoretic Rough Sets Based on Generalized Maclaurin Symmetric Mean Operators

  • Peide LiuEmail author
  • Hongyu Yang


As a typical model of three-way decisions (3WDs), decision-theoretic rough set (DTRS) has received extensive attention from researchers in the decision-making fields. Intuitionistic uncertain linguistic variables (IULVs) combine the advantages of intuitionistic fuzzy sets (IFSs) and uncertain linguistic variables (ULVs), IULV is more flexible in dealing with uncertain information in decision-making process, and provides a novel means for obtaining loss function (LF) of DTRSs. To get more comprehensive results, a new 3WD model is proposed to solve the multi-attribute group decision-making (MAGDM) problem. First, we gave the LF of DTRSs with IULVs, combined the IULVs and the generalized Maclaurin symmetric mean (GMSM), and proposed the IULGMSM and WIULGMSM operators to aggregate decision information; further, we proposed an intuitionistic uncertain linguistic DTRS model. Then, a method for deducing a new DTRS model is constructed, which can give the corresponding semantic interpretation of the decision results of each alternative. Finally, an example is applied to elaborate the proposed method in detail, and the effects of different conditional probabilities on decision results are discussed.


IULVs GMSM operator 3WDs DTRS 



This paper is supported by the National Natural Science Foundation of China (Nos. 71771140 and 71471172), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), and Shandong Provincial Social Science Planning Project (Nos. 17BGLJ04, 16CGLJ31 and 16CKJJ27).

Compliance with Ethical Standards

Conflicts of interest

We declare that we do not have any commercial or associative interests that represent conflicts of interest in connection with this manuscript. There are no professional or other personal interests that can inappropriately influence our submitted work.

Research Involving Human Participants and/or Animals

This paper does not include any researches with human participants or animals performed by any of the authors.


  1. 1.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)CrossRefzbMATHGoogle Scholar
  2. 2.
    Atanassov, K.T.: More on intuitionistic fuzzy sets. Fuzzy Sets Syst. 33, 37–46 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Atanassov, K.T.: Operators over interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 64, 159–174 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Atanassov, K.T., Gargov, G.: Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 3, 343–349 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Duda, R.O., Hart, P.E.: Pattern classification and scene analysis. Wiley, New York (1973)zbMATHGoogle Scholar
  6. 6.
    Garg, H.: Novel intuitionistic fuzzy decision making method based on an improved operation laws and its application. Eng. Appl. Artif. Intell. 60, 164–174 (2017)CrossRefGoogle Scholar
  7. 7.
    Jafarian, E., Razmi, J., Baki, M.F.: A flexible programming approach based on intuitionistic fuzzy optimization and geometric programming for solving multi-objective nonlinear programming problems. Expert Syst. Appl. 93, 245–256 (2018)CrossRefGoogle Scholar
  8. 8.
    Kahraman, C., Onar, S.C., Cebi, S., Oztaysi, B.: Extension of information axiom from ordinary to intuitionistic fuzzy sets: an application to search algorithm selection. Comput. Ind. Eng. 105, 348–361 (2017)CrossRefGoogle Scholar
  9. 9.
    Li, M.Z., Wang, G.Y.: Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts. Knowl.-Based Syst. 91, 165–178 (2016)CrossRefGoogle Scholar
  10. 10.
    Liang, D.C., Liu, D.: A novel risk decision making based on decision-theoretic rough sets under hesitant fuzzy information. IEEE Trans. Fuzzy Syst. 23(2), 237–247 (2015)CrossRefGoogle Scholar
  11. 11.
    Liang, D.C., Liu, D.: Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets. Inf. Sci. 300, 28–48 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liang, D.C., Xu, Z.S., Liu, D.: Three-way decisions based on decision-theoretic rough sets with dual hesitant fuzzy information. Inf. Sci. 396, 127–143 (2017)CrossRefGoogle Scholar
  13. 13.
    Liang, D.C., Xu, Z.S., Liu, D.: Three-way decisions with intuitionistic fuzzy decision-theoretic rough sets based on point operators. Inf. Sci. 375, 183–201 (2017)CrossRefGoogle Scholar
  14. 14.
    Liu, Y., Bi, J.W., Fan, Z.P.: Ranking products through online reviews: a method based on sentiment analysis technique and intuitionistic fuzzy set theory. Inform. Fusion 36, 149–161 (2017)CrossRefGoogle Scholar
  15. 15.
    Liu, P.D., Chen, Y.B., Chu, Y.Y.: Intuitionistic uncertain linguistic weighted Bonferroni owa operator and its application to multiple attribute decision making. Cybern. Syst. 5, 418–438 (2014)CrossRefzbMATHGoogle Scholar
  16. 16.
    Liu, P.D., Liu, Z.M., Zhang, X.: Some Intuitionistic uncertain linguistic Heronian mean operators and their application to group decision making. Appl. Math. Comput. 230, 570–586 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Liu, P.D., Jin, F.: Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making. Inf. Sci. 205, 58–71 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu, J.B., Zhou, X.Z., Huang, B., Li, H.X.: A three-way decision model based on intuitionistic fuzzy decision systems, pp. 249–263. Springer, New York (2017)Google Scholar
  19. 19.
    Maclaurin, C.: A second letter to Martin Folkes, Esq; concerning the roots of equations, with demonstration of other rules of algebra. Philos. Trans. R. Soc. London Ser. A 36(1729), 59–96 (2000)Google Scholar
  20. 20.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inform. Sci. 11(5), 341–356 (1982)CrossRefzbMATHGoogle Scholar
  21. 21.
    Singh, V., Yadav, S.P.: Modeling and optimization of multi-objective programming problems in intuitionistic fuzzy environment: optimistic, pessimistic and mixed approaches. Expert Syst. Appl. 102, 143–157 (2018)CrossRefGoogle Scholar
  22. 22.
    Sun, B.Z., Ma, W.M., Xiao, X.: Three-way group decision making based on multigranulation fuzzy decision-theoretic rough set over two universes. Int. J. Approximate Reasoning 81, 87–102 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tang, Y., Wen, L.L., Wei, G.W.: Approaches to multiple attribute group decision making based on the generalized Dice similarity measures with intuitionistic fuzzy information. Int. J. Knowl. Based Intell. Eng. Syst. 21(2), 85–95 (2017)CrossRefGoogle Scholar
  24. 24.
    Tian, X.L., Xu, Z.S., Gu, J.: How to select a promising enterprise for venture capitalists with prospect theory under intuitionistic fuzzy circumstance? Appl. Soft Comput. 67, 756–763 (2017)CrossRefGoogle Scholar
  25. 25.
    Wang, J.Q., Li, J.J.: The multi-criteria group decision making method based on multi-granularity intuitionistic two semantics. Sci. Technol. Inform. 33, 8–9 (2009)Google Scholar
  26. 26.
    Wang, J.Q., Yang, Y., Li, L.: Multi-criteria decision-making method based on single-valued neutrosophic linguistic Maclaurin symmetric mean operators. Neural Comput. Appl. 30, 1–19 (2016)Google Scholar
  27. 27.
    Xu, Z.S.: Induced uncertain linguistic OWA operators applied to group decision making. Inform. Fusion 7(2), 231–238 (2006)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Xu, Z.S.: Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Inf. Sci. 168, 171–184 (2004)CrossRefzbMATHGoogle Scholar
  29. 29.
    Yao, Y.Y.: The superiority of three-way decisions in probabilistic rough set models. Inf. Sci. 180, 1080–1096 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yao, Y. Y., Wong, S. K. M., Lingras, P.: A decision-theoretic rough set model, Proceeding of ISMIS, pp. 17–25 (1990)Google Scholar
  31. 31.
    Yao, Y. Y., Zhou, B.: Naive Bayesian rough sets, International conference on rough sets and knowledge technology. pp. 719–726 (2010)Google Scholar

Copyright information

© Taiwan Fuzzy Systems Association 2019

Authors and Affiliations

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

Personalised recommendations