Document Type : Article

**Authors**

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, Shandong, China.

**Abstract**

The Bonferroni mean (BM) has the advantages that it can capture the interrelationship among the input arguments, and the Harmonic mean is a conservative average lying between the max and min operators. The 2-dimension uncertain linguistic variables add a subjective evaluation on the reliability of the evaluation results given by decision makers, so they can better express fuzzy information. In this paper, in order to combine the advantages of them, we first propose the 2-dimensional uncertain linguistic weighted Bonferroni mean (2DULWBM) operator. However, it cannot consider the case when the given arguments are too high or too low. So we further proposed the 2-dimensional uncertain linguistic improved weighted Bonferroni harmonic mean (2DULIWBHM) operator, which combine the 2DULWBM with Harmonic Mean. Furthermore, we study some desirable properties and some special cases of them. Further, we develop a new method to deal with some multi-attribute group decision making (MAGDM) problems under 2-dimension uncertain linguistic environment based on the proposed operators. Finally, an illustrative example is given to testify the validity of the developed method by comparing with the other existing methods.

**Keywords**

- 2-dimension uncertain linguistic
- weighted Bonferroni harmonic mean
- multi-attribute group decision making

**Main Subjects**

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31. Liu, P.D. and Wang, M.H. "An extended VIKOR method for multiple attribute group decision making based on generalized interval-valued trapezoidal fuzzy numbers", Scientific Research and Essays, 6(4), pp. 766-776 (2011).

32. Liu, P.D. and Zhang, X. "Research on the supplier selection of supply chain based on entropy weight and improved ELECTRE-III method", International Journal of Production Research, 49(3), pp. 637-646 (2011).

33. Yu, S.M., Wang, J., and Wang, J.Q. "An extended TODIM approach with intuitionistic linguistic numbers", International Transactions in Operational Research, 25(3), pp. 781-805 (2018).

34. Tian, Z.P., Wang, J., Wang, J.Q., and Zhang, H.Y. "An improved MULTIMOORA approach for multicriteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information", Neural Computing and Applications, 28(sup 1), pp. 585-597 (2017).

35. Xu, Z.S. and Yager, R.R. "Some geometric aggregation operators based on intuitionistic fuzzy sets", International Journal of General Systems, 35(4), pp. 417-433 (2006).

36. Zhang, X., Liu, P.D., and Wang, Y.M. "Multiple attribute group decision making methods based on intuitionistic fuzzy frank power aggregation operators", Journal of Intelligent & Fuzzy Systems, 29(5), pp. 2235-2246 (2015).

37. Wang, F., Zeng, S., and Zhang, C. "A method based on intuitionistic fuzzy dependent aggregation operators for supplier selection", Mathematical Problems in Engineering, 2013, pp. 1-9 (2013).

38. Meng, F., Zhang, Q., and Zhan, J. "The intervalvalued intuitionistic fuzzy geometric choquet aggregation operator based on the generalized banzhaf index and two-additive measure", Technological and Economic Development of Economy, 21(2), pp. 186- 215 (2015).

39. Bonferroni, C. "Sulle medie multiple di potenze", Bolletino Matematica Italiana., 5(3-4), pp. 267-270 (1950).

40. Wei, G.W., Zhao, X.F., and Lin, R. "Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making", Applied Mathematical Modelling, 37(7), pp. 5277-5285 (2013).

41. Liu, P.D., Liu, J.L., and Chen, S.M. "Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making", Journal of the Operational Research Society, DOI: 10.1057/s41274-017-0190-y (2017).

42. Ding, Z.H. and Wu, Y.Y. "An improved intervalvalued hesitant fuzzy multi-criteria group decisionmaking method and applications", Mathematical and Computational Applications, 21(2), pp. 1-12 (2016).

43. Sun, H. and Sun, M. "Generalized Bonferroni harmonic mean operators and their application to multiple attribute decision making", Journal of Computational Information Systems, 8(14), pp. 5717-5724 (2012).

44. Xu, Z.S. "Induced uncertain linguistic OWA operators applied to group decision making", Inf Fusion, 7(2), pp. 231-238 (2006).

45. Xu, Z.S. "Fuzzy harmonic mean operators", International Journal of Intelligent Systems, 24(2), pp. 152- 172 (2009).

46. Liu, P.D. and Chu, Y.C. "Some two-dimensional uncertain linguistic Heronian mean operators and their application in multiple-attribute decision making", Neural Compute & Application, 26(6), pp. 1461-1480 (2015).

2. Chen, T.Y., Chang, C.H., and Lu, J.R. "The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making", Eur J Oper Res, 226(3), pp. 615-625 (2013).

3. Chuu, S.J. "Interactive group decision-making using a fuzzy linguistic approach for evaluating the flexibility in a supply chain", Eur J Oper Res, 213(1), pp. 279- 289 (2011).

4. Liu, F., Wang, W.G., and Zhang, Z.X. "A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making", Eur J Oper Res, 218(3), pp. 747- 754 (2012).

5. Ye, J. "Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment", Eur J Oper Res, 205(1), pp. 202- 204 (2010).

6. Zhang, C., Zhou, G.Z., and Zhu, W.D. "Research on peer review system for the national science foundation based on two-dimensional semantics evidence reasoning", China Soft Sci, 2, pp. 176-182 (2011).

7. Liu, P.D. and Tang, G.L. "Multi-criteria group decision-making based on interval neutrosophic uncertain linguistic variables and Choquet integral", Cognitive Computation, 8(6), pp. 1036-1056 (2016).

8. Xu, Z.S. "Intuitionistic fuzzy aggregation operators", IEEE Transactions on Fuzzy Systems, 15(6), pp. 1179- 1187 (2007).

9. Zadeh, L.A. "The concept of a linguistic variable and its application to approximate reasoning-I", Inf Sci, 8(3), pp. 199-249 (1975).

10. Zadeh, L.A. "The concept of a linguistic variable and its application to approximate reasoning-II", Inf Sci, 8(4), pp. 301-357 (1975).

11. Zadeh, L.A. "The concept of a linguistic variable and its application to approximate reasoning-III", Inf Sci, 9(1), pp. 43-80 (1975).

12. Liu, P.D. and Jin, F. "Methods for aggregating intuitionistic uncertain linguistic variables and their application to group decision making", Inf Sci, 205, pp. 58-71 (2012).

13. Wei, G.W. "Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making", International Journal of Machine Learning and Cybernetics, 7(6), pp. 1093-1114 (2016).

14. Lu, M. and Wei, G.W. "Models for multiple attribute decision making with dual hesitant fuzzy uncertain linguistic information", International Journal of Knowledge-Based and Intelligent Engineering Systems, 20(4), pp. 217-227 (2016).

15. Li, Q.X., Zhao, X.F., and Wei, G.W. "Model for software quality evaluation with hesitant fuzzy uncertain linguistic information", Journal of Intelligent and Fuzzy Systems, 26(6), pp. 2639-2647 (2014).

16. Zhou, L.Y., Lin, R., Zhao, X.F., and Wei, G.W. "Uncertain linguistic prioritized aggregation operators and their application to multiple attribute group decision making", International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 21(4), pp. 603-627 (2013).

17. Yu, X.H., Xu, Z.S., Liu, S.S., and Chen, Q. "Multicriteria decision making with two-dimension linguistic aggregation techniques", Int. J. Intell. Syst, 27(6), pp. 539-562 (2012).

18. Zhu, W.D., Zhou, G.Z., and Yang, S.L. "An approach to group decision making based on two-dimension linguistic assessment information", Syst. Eng, 27(2), pp. 113-118 (2009).

19. Liu, P.D. and Zhang, X. "An approach to group decision making based on two-dimension uncertain linguistic assessment information", Technol. Econ. Develop. Econ, 18(3), pp. 424-437 (2012).

20. Liu, P.D. and Yu, X.C. "Two-dimension uncertain linguistic power generalized weighted aggregation operator and its application for multiple attribute group decision making", Knowledge-Based Systems, 57(1), pp. 69-80 (2014).

21. Liu, P.D., Liu, Z.M., and Zhang, X. "Some intuitionistic uncertain linguistic Heronian mean operators and their application to group decision making", Applied Mathematics and Computation, 230(1), pp. 570-586 (2014).

22. Liu, P.D. and Wang, Y.M. "Multiple attribute decision-making method based on single valued neutrosophic normalized weighted Bonferroni mean", Neural Computing and Applications, 25(7-8), pp. 2001-2010 (2014).

23. Zhang, W.C., Xu, Y.J., and Wang, H.M. "A consensus reaching model for two-tuple linguistic multiple attribute group decision making with incomplete weight information", International Journal of Systems Science, 47(2), pp. 389-405 (2016).

24. Ji, P., Wang, J., and Zhang, H. "Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in selecting third party logistics", Neural Computing and Applications, 30(3), pp. 799-823 (2018).

25. Nie, R.X., Wang, J., and Li, L. "2-tuple linguistic intuitionistic preference relation and its application in sustainable location planning voting system", Journal of Intelligent & Fuzzy Systems, 33(2), pp. 885-899 (2017).

26. Peng, H., Wang, J., and Chen, P. "A linguistic ntuitionistic multi-criteria decision-making method based on the Frank Heronian mean operator and its application in evaluating coal mine safety", International Journal of Machine Learning and Cybernetics, 9(6), pp. 1053-1068 (2018).

27. Peng, J.J., Wang, J., Wu, X., and Tian, C. "Hesitant intuitionistic fuzzy aggregation operators based on the archimedean t-norms and t-conorms", International Journal of Fuzzy Systems, 19(3), pp. 702-714 (2017).

28. Wang, J.Q., Yang, Y., and Li, L. "Multi-criteria decision-making method based on single valued neutrosophic linguistic Maclaurin symmetric mean operators", Neural Computing and Applications, 30(5), pp. 1529-1547 (2018).

29. Jiang, X.P. and Wei, G.W. "Some Bonferroni mean operators with two-tuple linguistic information and their application to multiple attribute decision making", Journal of Intelligent and Fuzzy Systems, 27, pp. 2153-2162 (2014).

30. Liu, P.D. "Multi-attribute decision-making method research based on interval vague set and TOPSIS method", Technological and Economic Development of Economy, 15(3), pp. 453-463 (2009).

31. Liu, P.D. and Wang, M.H. "An extended VIKOR method for multiple attribute group decision making based on generalized interval-valued trapezoidal fuzzy numbers", Scientific Research and Essays, 6(4), pp. 766-776 (2011).

32. Liu, P.D. and Zhang, X. "Research on the supplier selection of supply chain based on entropy weight and improved ELECTRE-III method", International Journal of Production Research, 49(3), pp. 637-646 (2011).

33. Yu, S.M., Wang, J., and Wang, J.Q. "An extended TODIM approach with intuitionistic linguistic numbers", International Transactions in Operational Research, 25(3), pp. 781-805 (2018).

34. Tian, Z.P., Wang, J., Wang, J.Q., and Zhang, H.Y. "An improved MULTIMOORA approach for multicriteria decision-making based on interdependent inputs of simplified neutrosophic linguistic information", Neural Computing and Applications, 28(sup 1), pp. 585-597 (2017).

35. Xu, Z.S. and Yager, R.R. "Some geometric aggregation operators based on intuitionistic fuzzy sets", International Journal of General Systems, 35(4), pp. 417-433 (2006).

36. Zhang, X., Liu, P.D., and Wang, Y.M. "Multiple attribute group decision making methods based on intuitionistic fuzzy frank power aggregation operators", Journal of Intelligent & Fuzzy Systems, 29(5), pp. 2235-2246 (2015).

37. Wang, F., Zeng, S., and Zhang, C. "A method based on intuitionistic fuzzy dependent aggregation operators for supplier selection", Mathematical Problems in Engineering, 2013, pp. 1-9 (2013).

38. Meng, F., Zhang, Q., and Zhan, J. "The intervalvalued intuitionistic fuzzy geometric choquet aggregation operator based on the generalized banzhaf index and two-additive measure", Technological and Economic Development of Economy, 21(2), pp. 186- 215 (2015).

39. Bonferroni, C. "Sulle medie multiple di potenze", Bolletino Matematica Italiana., 5(3-4), pp. 267-270 (1950).

40. Wei, G.W., Zhao, X.F., and Lin, R. "Uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making", Applied Mathematical Modelling, 37(7), pp. 5277-5285 (2013).

41. Liu, P.D., Liu, J.L., and Chen, S.M. "Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making", Journal of the Operational Research Society, DOI: 10.1057/s41274-017-0190-y (2017).

42. Ding, Z.H. and Wu, Y.Y. "An improved intervalvalued hesitant fuzzy multi-criteria group decisionmaking method and applications", Mathematical and Computational Applications, 21(2), pp. 1-12 (2016).

43. Sun, H. and Sun, M. "Generalized Bonferroni harmonic mean operators and their application to multiple attribute decision making", Journal of Computational Information Systems, 8(14), pp. 5717-5724 (2012).

44. Xu, Z.S. "Induced uncertain linguistic OWA operators applied to group decision making", Inf Fusion, 7(2), pp. 231-238 (2006).

45. Xu, Z.S. "Fuzzy harmonic mean operators", International Journal of Intelligent Systems, 24(2), pp. 152- 172 (2009).

46. Liu, P.D. and Chu, Y.C. "Some two-dimensional uncertain linguistic Heronian mean operators and their application in multiple-attribute decision making", Neural Compute & Application, 26(6), pp. 1461-1480 (2015).

Transactions on Industrial Engineering (E)

March and April 2019Pages 975-995