International Journal of Fuzzy Systems

, Volume 21, Issue 8, pp 2373–2391 | Cite as

Interval-Valued Intuitionistic 2-Tuple Linguistic Bonferroni Mean Operators and Their Applications in Multi-attribute Group Decision Making

  • Kang Du
  • Hongjun YuanEmail author


The main purposes of this paper are to propose some new Bonferroni mean (BM) operators under the interval-valued intuitionistic 2-tuple linguistic environment and apply them to multi-attribute group decision-making (MAGDM) problems. In order to aggregate correlated IVI2TLVs, we propose some new interval-valued intuitionistic 2-tuple linguistic Bonferroni mean operators, including the interval-valued intuitionistic 2-tuple linguistic Bonferroni mean (IVI2TLBM) operator, the interval-valued intuitionistic 2-tuple linguistic weighted Bonferroni mean (IVI2TLWBM) operator, and the combined interval-valued intuitionistic 2-tuple linguistic weighted Bonferroni mean (CIVI2TLWBM) operator. Furthermore, a ranking method is presented to solve MAGDM problems. Finally, a real-life application of the proposed method is given, and the sensitivity analysis and comparative analysis are carried out to verify the effectiveness and feasibility of the method.


Multi-attribute group decision making Interval-valued intuitionistic 2-tuple linguistic variables BM operator IVI2TLBM operator IVI2TLWBM operator CIVI2TLWBM operator 



This work was supported by Youth Projects of National Social Science Foundation of China (NO. 13CTJ006) and National Natural Science Foundation of China (NO. 71803001).


  1. 1.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)zbMATHGoogle Scholar
  2. 2.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)CrossRefGoogle Scholar
  3. 3.
    Chen, S.M., Tan, J.M.: Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 67(2), 163–172 (1994)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hong, D.H., Choi, C.H.: Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst. 114(1), 103–113 (2000)zbMATHGoogle Scholar
  5. 5.
    Gupta, P., Mehlawat, M.K., Grover, N.: Intuitionistic fuzzy multi-attribute group decision-making with an application to plant location selection based on a new extended VIKOR method. Inf. Sci. 370, 184–203 (2016)Google Scholar
  6. 6.
    Wan, S., Wang, F., Dong, J.: A group decision making method with interval valued fuzzy preference relations based on the geometric consistency. Inf. Fusion 40, 87–100 (2018)Google Scholar
  7. 7.
    Hao, Z., Xu, Z., Zhao, H., Zhang, R.: Novel intuitionistic fuzzy decision making models in the framework of decision field theory. Inf. Fusion 33, 57–70 (2017)Google Scholar
  8. 8.
    Xing, Z., Xiong, W., Liu, H.: A Euclidean approach for ranking intuitionistic fuzzy values. IEEE Trans. Fuzzy Syst. 26(1), 353–365 (2018)Google Scholar
  9. 9.
    Ma, R.F., Liu, S.S., Xu, Z.S., Lei, Q.: The basis and coordinates in intuitionistic fuzzy environment. Int. J. Fuzzy Syst. 20(5), 1483–1494 (2018)MathSciNetGoogle Scholar
  10. 10.
    Zhou, W., Xu, Z.S.: Score-hesitation trade-off and portfolio selection under intuitionistic fuzzy environment. Int. J. Intell. Syst. 34(2), 325–341 (2019)Google Scholar
  11. 11.
    Zadeh, L.A.: The concept of a linguistic variable and its application approximate reasoning I. Inf. Sci. 8, 199–249 (1975)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Jiang, Y.P., Fan, Z.P., Ma, J.: A method for group decision making with multi-granularity linguistic assessment information. Inf. Sci. 178, 1098–1109 (2008)zbMATHGoogle Scholar
  13. 13.
    Herrera, F., Herrera-Viedma, E., Martinez, L.: A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst. 114, 43–58 (2000)zbMATHGoogle Scholar
  14. 14.
    Herrera, F., Martinez, L.: A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 8(6), 746–752 (2000)Google Scholar
  15. 15.
    Ju, Y.B., Liu, X.Y., Wang, A.H.: Some new Shapley 2-tuple linguistic Choquet aggregation operators and their applications to multiple attribute group decision making. Soft Comput. 20(10), 4037–4053 (2016)zbMATHGoogle Scholar
  16. 16.
    Wang, L.D., Wang, Y.J., Pedrycz, W.: Hesitant 2-tuple linguistic Bonferroni operators and their utilization in group decision making. Appl. Soft Comput. 77, 653–664 (2019)Google Scholar
  17. 17.
    Chen, Z.S., Chin, K.S., Tsui, K.L.: Constructing the geometric Bonferroni mean from the generalized Bonferroni mean with several extensions to linguistic 2-tuples for decision-making. Appl. Soft Comput. 78, 595–613 (2019)Google Scholar
  18. 18.
    Li, Y.B., Zhang, J.P.: TOPSIS method for hybrid multiple attribute decision making with 2-tuple linguistic information and its application to computer network security evaluation. J. Intell. Fuzzy Syst. 26, 1563–1569 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cid-López, A., Hornos, M.J., Carrasco, R.A., Herrera-Viedma, E., et al.: Linguistic multi-criteria decision-making model with output variable expressive richness. Expert Syst. Appl. 83, 350–362 (2017)Google Scholar
  20. 20.
    Sohaib, O., Naderpour, M., Hussain, W., Martinez, L.: Cloud computing model selection for e-commerce enterprises using a new 2-tuple fuzzy linguistic decision-making method. Comput. Ind. Eng. 132, 47–58 (2019)Google Scholar
  21. 21.
    Lin, J., Lan, J.B., Lin, Y.H.: A method of multi-attribute group decision-making based on the aggregation operators for interval two-tuple linguistic information. Jilin Norm. Univ. J. 1, 5–9 (2009)Google Scholar
  22. 22.
    Chen, H., Meng, F.Y., Chen, K.: Several generalized interval-valued 2-tuple linguistic weighted distance measures and their application. Int. J. Fuzzy Syst. 19(4), 967–981 (2017)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Wan, S.P., Xu, G.L., Dong, J.Y.: Supplier selection using ANP and ELECTRE II in interval 2-tuple linguistic environment. Inf. Sci. 385, 19–38 (2017)Google Scholar
  24. 24.
    Liu, B.S., Fu, M.Q., Zhang, S.B., Xue, B., et al.: An interval-valued 2-tuple linguistic group decision-making model based on the Choquet integral operator. Int. J. Syst. Sci. 49(2), 407–424 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Qi, K.X., Wang, Q.S., Duan, Q.L., Gong, L., et al.: A multi criteria comprehensive evaluation approach for emergency response capacity with interval 2-tuple linguistic information. Appl. Soft Comput. 72, 419–441 (2018)Google Scholar
  26. 26.
    Liang, Y.Y., Liu, J., Qin, J.D., Tu, Y.: An improved multi-granularity interval 2-tuple TODIM approach and its application to green supplier selection. Int. J. Fuzzy Syst. 21(1), 129–144 (2019)MathSciNetGoogle Scholar
  27. 27.
    Wang, J.Q., Li, H.B.: Multi-criteria decision-making method based on aggregation operators for intuitionistic linguistic fuzzy numbers. Control Decis. 25(10), 1571–1574 (2010)MathSciNetGoogle Scholar
  28. 28.
    Liu, P.D., Wang, Y.M.: Multiple attribute group decision making methods based on intuitionistic linguistic power generalized aggregation operators. Appl. Soft Comput. 17, 90–104 (2014)Google Scholar
  29. 29.
    Ju, Y.B., Liu, X.Y., Ju, D.W.: Some new intuitionistic linguistic aggregation operators based on Maclaurin symmetric mean and their applications to multiple attribute group decision making. Soft Comput. 20(11), 4521–4548 (2016)zbMATHGoogle Scholar
  30. 30.
    Liu, P.D., Wang, P.: Some improved linguistic intuitionistic fuzzy aggregation operators and their applications to multiple-attribute decision making. Int. J. Inf. Technol Decis. Mak. 16(3), 817–850 (2017)Google Scholar
  31. 31.
    Wang, L.D., Wang, Y.J., Sangaiah, A.K., Liao, B.Q.: Intuitionistic linguistic group decision-making methods based on generalized compensative weighted averaging aggregation operators. Soft Comput. 22, 7605–7617 (2018)zbMATHGoogle Scholar
  32. 32.
    Teng, F., Liu, P.D.: Multiple-attribute group decision-making method based on the linguistic intuitionistic fuzzy density hybrid weighted averaging operator. Int. J. Fuzzy Syst. 21(1), 213–231 (2019)MathSciNetGoogle Scholar
  33. 33.
    Zhang, Y., Ma, H.X., Liu, B.H., Liu, J.: Group decision making with 2-tuple intuitionistic fuzzy linguistic preference relations. Soft Comput. 16, 1439–1446 (2012)zbMATHGoogle Scholar
  34. 34.
    Beg, I., Rashid, T.: An intuitionistic 2-tuple linguistic information model and aggregation operators. Int. J. Intell. Syst. 31(6), 569–592 (2016)Google Scholar
  35. 35.
    Liu, P.D., Chen, S.M.: Multiattribute group decision making based on intuitionistic 2-tuple linguistic information. Inf. Sci. 430, 599–619 (2018)MathSciNetGoogle Scholar
  36. 36.
    Zhang, H., Jiao, Z.M., Li, B.Q.: Interval-valued intuitionistic 2-tuple linguistic aggregation operators and their multi-attribute group decision making method. Comput. Eng. Appl. 52(24), 43–49 (2016)Google Scholar
  37. 37.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans. Syst. Man Cybern. 18(1), 183–190 (1988)zbMATHGoogle Scholar
  38. 38.
    Yager, R.R., Xu, Z.S.: The continuous ordered weighted geometric operator and its application to decision making. Fuzzy Sets Syst. 157(10), 1393–1402 (2006)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Yager, R.R.: The power average operator. IEEE Trans. Syst. Man Cybern. 31(6), 724–731 (2001)Google Scholar
  40. 40.
    Bonferroni, C.: Sulle medie multiple di potenze. Bolletino Matematica Italiana 5(3), 267–270 (1950)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Xu, Z.S., Yager, R.R.: Intuitionistic fuzzy Bonferroni means, IEEE Trans. Syst. Man Cybern. Part B: Cybern. 41(2), 568–578 (2011)Google Scholar
  42. 42.
    Zhang, Z.M.: Geometric Bonferroni means of interval-valued intuitionistic fuzzy numbers and their application to multiple attribute group decision making. Neural Comput. Appl. 29(11), 1139–1154 (2018)Google Scholar
  43. 43.
    Wei, G.W., Zhao, X.F., Lin, R., Wang, H.J.: Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making. Appl. Math. Model. 37(7), 5277–5285 (2013)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Zhu, B., Xu, Z.S.: Hesitant fuzzy Bonferroni means for multi-criteria decision making. J. Oper. Res. Soc. 64(12), 1831–1840 (2013)Google Scholar
  45. 45.
    He, Y.D., He, Z., Shi, L.X., Meng, S.S.: Multiple attribute group decision making based on IVHFPBMs and a new ranking method for interval-valued hesitant fuzzy information. Comput. Ind. Eng. 99, 63–77 (2016)Google Scholar
  46. 46.
    Dutta, B., Guha, D.: Partitioned Bonferroni mean based on linguistic 2-tuple for dealing with multi-attribute group decision making. Appl. Soft Comput. 37, 166–179 (2015)Google Scholar
  47. 47.
    Liu, X., Tao, Z.F., Chen, H.Y., Zhou, L.G.: A new interval-valued 2-tuple linguistic Bonferroni mean operator and its application to multiattribute group decision making. Int. J. Fuzzy Syst. 19(1), 86–108 (2017)MathSciNetGoogle Scholar
  48. 48.
    Atanassov, K.T.: Gargov: Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31, 343–349 (1989)zbMATHGoogle Scholar
  49. 49.
    Herrera, F., Herrera-Viedma, E.: Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst. 115(1), 67–82 (2000)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Liu, P.D.: Some geometric aggregation operators based on interval intuitionistic uncertain linguistic variables and their application to group decision making. Appl. Math. Model. 37(4), 2430–2444 (2013)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Wan, S.P.: Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl. Math. Model. 37(6), 4112–4126 (2013)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Narayanamoorthy, S., Geetha, S., Rakkiyappan, R., Joo, Y.H.: Interval-valued intuitionistic hesitant fuzzy entropy based VIKOR method for industrial robots selection. Expert Syst. Appl. 121, 28–37 (2019)Google Scholar
  53. 53.
    Liu, H.C., Ren, M.L., Wu, J., Lin, Q.L.: An interval 2-tuple linguistic MCDM method for robot evaluation and selection. Int. J. Prod. Res. 52(10), 2867–2880 (2014)Google Scholar
  54. 54.
    Singh, A., Gupta, A., Mehra, A.: Energy planning problems with interval-valued 2-tuple linguistic information. Oper. Res. 17(3), 821–848 (2017)Google Scholar

Copyright information

© Taiwan Fuzzy Systems Association 2019

Authors and Affiliations

  1. 1.School of Statistics and Applied MathematicsAnhui University of Finance and EconomicsBengbuChina

Personalised recommendations