International Journal of Fuzzy Systems

, Volume 21, Issue 8, pp 2340–2353 | Cite as

Bidirectional Projection Method for Probabilistic Linguistic Multi-criteria Group Decision-Making Based on Power Average Operator

  • Peide LiuEmail author
  • Ying Li
  • Fei Teng


The probabilistic linguistic terms set (PLTS) consists of several possible linguistic terms and their relative probability, and the Power average (PA) operator takes the interrelationships among the attributes into consideration. At the same time, the bidirectional projection (BP) method can consider the distance and the angle of the evaluated alternatives, moreover, it can also take the bidirectional projection magnitude into account. For the sake of fully taking the advantages of PLTS, PA operator and the BP method, in this article, we combine the PA operator with the BP method and extend it to the environment of probabilistic linguistic information (PLI), meanwhile, based on the calculation process of the weighted averaging operator of distribution assessments, the probabilistic linguistic PA (PLPA) operator and the weighted probabilistic linguistic PA (WPLPA) operator are proposed. Simultaneously, we discuss some properties of these operators. Further, we define the BP measures based on PLTS. Based on the combination of the WPLPA operator and the BP method, we develop the approach which can solve the problems of multiple attribute group decision-making with PLI. Lastly, a numerical instance is given to demonstrate the feasibility and the superiority of the proposed method.


Probabilistic linguistic PA Bidirectional projection MAGDM 



This paper is supported by the National Natural Science Foundation of China (Nos. 71771140 and 71471172), and the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).


  1. 1.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)zbMATHGoogle Scholar
  2. 2.
    Mizumoto, M., Tanaka, K.: Some properties of fuzzy sets of type 2. Inf. Control 31(4), 312–340 (1976)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)zbMATHGoogle Scholar
  4. 4.
    Yager, R.R.: On the theory of bags. Int. J. Gener. Syst. 13(1), 23–37 (1986)MathSciNetGoogle Scholar
  5. 5.
    Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 6(25), 529–539 (2010)zbMATHGoogle Scholar
  6. 6.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning—I. Inf. Sci. 8(3), 199–249 (1975)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning—II. Inf. Sci. 8, 301–353 (1975)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning—III. Inf. Sci. 9, 43–80 (1975)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Rodriguez, R.M., Martinez, L., Herrera, F.: Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst. 20(1), 109–119 (2012)Google Scholar
  10. 10.
    Pang, Q., Xu, Z.S., Wang, H.: Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci. 369, 128–143 (2016)Google Scholar
  11. 11.
    Gou, X.J., Xu, Z.S.: Novel basic operational laws for linguistic terms, hesitant fuzzy linguistic term sets and probabilistic linguistic term sets. Inf. Sci. 372, 407–427 (2016)Google Scholar
  12. 12.
    Bai, C.Z., Zhang, R., Qian, L.X., Wu, Y.N.: Comparisons of probabilistic linguistic term sets for multi-criteria decision making. Knowl.-Based Syst. 119, 284–291 (2017)Google Scholar
  13. 13.
    Lin, M.W., Xu, Z.S.: Probabilistic linguistic distance measures and their applications in multi-criteria group decision making. In: Soft computing applications for group decision-making and consensus modeling 411–440 (2017)Google Scholar
  14. 14.
    Zhang, X.L., Xing, X.M.: Probabilistic linguistic VIKOR method to evaluate green supply chain initiatives. Sustainability. 9(7), 1231 (2017)Google Scholar
  15. 15.
    Gomes, L.F.A.M., Lima, M.M.P.P.: Todim: basic and application to multicriteria ranking of projects with environmental impacts. Found. Comput. Decis. Sci. 16(1), 113–127 (1991)zbMATHGoogle Scholar
  16. 16.
    Greco, S., Kadziński, M., Mousseau, V., Słowiński, R.: ELECTRE: robust ordinal regression for outranking methods. Eur. J. Oper. Res. 214(1), 118–135 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Brans, J.P., Vincke, P.A.: A preference ranking organization method: the PROMETHEE method for MCDM. Manag. Sci. 31(6), 647–656 (1985)zbMATHGoogle Scholar
  18. 18.
    Wu, X.L., Liao, H.C., Xu, Z.S., Hafezalkotob, A., Herrera, F.: Probabilistic linguistic MULTIMOORA: a multi-criteria decision making method based on the probabilistic linguistic expectation function and the improved borda rule. IEEE Trans. Fuzzy Syst. 26(6), 3688–3702 (2018)Google Scholar
  19. 19.
    Xu, Z.S.: On method for uncertain multiple attribute decision making problems with uncertain multiplicative preference information on alternatives. Fuzzy Optim. Decis. Making 4, 131–139 (2005)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Xu, Z.S., Da, Q.L.: Projection method for uncertain multi-attribute decision making with preference information on alternatives. Int. J. Inf. Technol. Decis. Making 3, 429–434 (2004)Google Scholar
  21. 21.
    Yue, Z.L.: Approach to group decision making based on determining the weights of experts by using projection method. Appl. Math. Model. 36, 2900–2910 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ye, J.: Bidirectional projection method for multiple attribute group decision making with neutrosophic numbers. Neural Comput. Appl. 28(5), 1021–10299 (2017)Google Scholar
  23. 23.
    Ye, J.: Projection and bidirectional projection measures of single-valued neutrosophic sets and their decision-making method for mechanical design schemes. J. Exp. Theor. Artif. Intell. 29(4), 731–740 (2017)Google Scholar
  24. 24.
    Liu, P.D., You, X.L.: Bidirectional projection measure of linguistic neutrosophic numbers and their application to multi-criteria group decision making. Comput. Ind. Eng. 128, 447–457 (2019)Google Scholar
  25. 25.
    Liu, X.D., Zhu, J.J., Liu, S.F.: Bidirectional projection method with hesitant fuzzy information. Syst. Eng. Theory Pract. 34(10), 2637–2644 (2014)Google Scholar
  26. 26.
    Liu, P.D.: Some Hamacher aggregation operators based on the interval-valued intuitionistic fuzzy numbers and their application to Group Decision Making. IEEE Trans. Fuzzy Syst. 22(1), 83–97 (2014)Google Scholar
  27. 27.
    Liu, P.D., Chen, S.M., Wang, P.: Multiple-attribute group decision-making based on q-rung orthopair fuzzy power Maclaurin symmetric mean operators. IEEE Trans. Syst. Man Cybern. Syst. 1, 1 (2018). Google Scholar
  28. 28.
    Liu, P.D., Tang, G.L.: some intuitionistic fuzzy prioritized interactive einstein Choquet operators and their application in decision making. IEEE Access 6, 72357–72371 (2019)Google Scholar
  29. 29.
    Liu, P.D., Wang, P.: Multiple-attribute decision making based on Archimedean Bonferroni operators of q-rung orthopair fuzzy numbers. IEEE Trans. Fuzzy Syst. 27(5), 834–848 (2019)Google Scholar
  30. 30.
    Yager, R.R.: The power average operator. Syst. Man Cybern. Part A Syst. Hum. IEEE Trans. 31(6), 724–731 (2001)Google Scholar
  31. 31.
    Xu, Z.S., Yager, R.R.: Power-geometric operators and their use in group decision making. IEEE Trans. Fuzzy Syst. 18(1), 94–105 (2010)Google Scholar
  32. 32.
    Xu, Y.J., Merigó, J.M., Wang, H.M.: Linguistic power aggregation operators and their application to multiple attribute group decision making. Appl. Math. Model. 36(11), 5427–5444 (2012)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Li, Y., Wang, Y., Liu, P.D.: Multiple attribute group decision-making methods based on trapezoidal fuzzy two-dimension linguistic power generalized aggregation operators. Soft Comput. 20(7), 2689–2704 (2016)zbMATHGoogle Scholar
  34. 34.
    Liu, P.D., Qin, X.Y.: Power average operators of linguistic intuitionistic fuzzy numbers and their application to multiple-attribute decision making. J. Intell. Fuzzy Syst. 32(1), 1029–1043 (2016)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Zhu, C., Zhu, L., Zhang, X.: Linguistic hesitant fuzzy power aggregation operators and their applications in multiple attribute decision-making. Inf. Sci. 367–368, 809–826 (2016)Google Scholar
  36. 36.
    Kobina, A., Liang, D., He, X.: Probabilistic linguistic Power aggregation operators for multi-criteria group decision making. Symmetry 9(12), 320 (2017)Google Scholar
  37. 37.
    Gou, X.J., Xu, Z.S., Liao, H.C.: Multiple criteria decision making based on Bonferroni means with hesitant fuzzy linguistic information. Soft Comput. 21(21), 6515–6529 (2016)zbMATHGoogle Scholar
  38. 38.
    Zhang, G., Dong, Y., Xu, Y.: Consistency and consensus measures for linguistic preference relations based on distribution assessments. Inf. Fusion 17(1), 46–55 (2014)Google Scholar
  39. 39.
    Xu, Z.S.: Deviation measures of linguistic preference relations in group decision making. Omega 33(3), 249–254 (2005)Google Scholar
  40. 40.
    Liang, D., Kobina, A., Quan, W.: Grey relational analysis method for probabilistic linguistic multi-criteria group decision-making based on geometric Bonferroni mean. Int. J. Fuzzy Syst. 20(7), 2234–2244 (2018)Google Scholar
  41. 41.
    Liu, P.D., Zhang, X.H.: Approach to multi-attributes decision making with intuitionistic linguistic information based on Dempster–Shafer evidence theory. IEEE Access 6(1), 52969–52981 (2018)MathSciNetGoogle 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