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International Journal of Fuzzy Systems

, Volume 19, Issue 6, pp 1880–1903 | Cite as

Method of Multiple Attribute Group Decision Making Based on 2-Dimension Interval Type-2 Fuzzy Aggregation Operators with Multi-granularity Linguistic Information

  • Qun Wu
  • Feng Wang
  • Ligang ZhouEmail author
  • Huayou Chen
Article

Abstract

This paper proposes an approach to linguistic multiple attribute group decision-making (MAGDM) problems with 2-dimension multi-granularity linguistic information by adding a subjective imprecise estimation of reliability of the linguistic assessment information for expressing fuzzy information appropriately. To avoid the drawback of traditional transformation functions of multi-granularity linguistic information (MGLI) defined in the basic linguistic term set or 2-tuple linguistic information, in this paper, decision information is fully expressed by using 2-dimension interval type-2 trapezoidal fuzzy number (2DIT2TFN) representation model under multi-granular linguistic contexts. Firstly, the definition, operational laws, and ranking method of 2DIT2TFNs are proposed. Then 2-dimension interval type-2 trapezoidal fuzzy ordered weighted average (2DIT2TFOWA) operator and quasi-2-dimension interval type-2 trapezoidal fuzzy ordered weighted average (quasi-2DIT2TFOWA) operator are put forward. Moreover, we apply new operators to developing approach to MAGDM problem with linguistic term set of different odd cardinalities. Finally, a numerical example related to health management is provided to illustrate the utility and effectiveness of the developed method. We also make comparisons between the methods proposed in this paper and some existing ones to confirm its feasibility and rationality. The main contribution of this paper possesses three points: (1) proposing the 2DIT2TFNs, which can reflect the evaluation on objects more reasonably; (2) developing some new aggregation operators under 2-dimension interval type-2 trapezoidal fuzzy environment; and (3) applying new operators to developing approach to MAGDM problem with MGLI.

Keywords

Multiple attribute group decision making Multi-granularity Aggregation operator Interval type-2 fuzzy sets 2-Dimension interval type-2 trapezoidal fuzzy numbers 

Notes

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions for improving the paper. The work was supported by National Natural Science Foundation of China (Nos. 71301001, 71371011, 71501002, 71272047), Project of Anhui Province for Excellent Young Talents, the Doctoral Scientific Research Foundation of Anhui University, and Anhui provincial philosophy and social science program.

References

  1. 1.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning—I. Inf. Sci. 8, 199–249 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Parreiras, R.O., Ekel, P.Y., Martini, J.S.C., Palhares, R.M.: A flexible consensus scheme for multicriteria group decision making under linguistic assessments. Inf. Sci. 180, 1075–1089 (2010)CrossRefGoogle Scholar
  3. 3.
    Herrera, F., Herrera-Viedma, E.: Linguistic decision analysis: steps for solving decision problems under linguistic information. Fuzzy Sets Syst. 115, 67–82 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets Syst. 97, 33–48 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Xu, Z.S.: A method based on linguistic aggregation operators for group decision making with linguistic preference relation. Inf. Sci. 166, 19–30 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    Merigó, J.M., Casanovas, M., Palacios-Marqués, D.: Linguistic group decision making with induced aggregation operators and probabilistic information. Appl. Soft Comput. 24, 669–678 (2014)CrossRefGoogle Scholar
  8. 8.
    Xu, Z.S.: An approach based on the uncertain LOWG and induced uncertain LOWG operators to group decision making with uncertain multiplicative linguistic preference relations. Decis. Support Syst. 41, 488–499 (2006)CrossRefGoogle Scholar
  9. 9.
    Herrera, F., Martinez, L.: A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 8, 746–752 (2000)CrossRefGoogle Scholar
  10. 10.
    Herrera, F., Herrera-Viedma, E., Martínez, L.: A fuzzy linguistic methodology to deal with unbalanced linguistic term sets. IEEE Trans. Fuzzy Syst. 16, 354–370 (2008)CrossRefGoogle Scholar
  11. 11.
    Dong, Y.C., Li, C.C., Xu, Y.F., Gu, X.: Consensus-based group decision making under multi-granular unbalanced 2-tuple linguistic preference relations. Group Decis. Negot. 24, 217–242 (2015)CrossRefGoogle Scholar
  12. 12.
    Dong, Y.C., Xu, Y.F., Yu, S.: Computing the numerical scale of the linguistic term set for the 2-tuple fuzzy linguistic representation model. IEEE Trans. Fuzzy Syst. 17, 1366–1378 (2009)CrossRefGoogle Scholar
  13. 13.
    Dong, Y.C., Zhang, G.Q., Hong, W.C., Yu, S.: Linguistic computational model based on 2-tuples and intervals. IEEE Trans. Fuzzy Syst. 21, 1006–1018 (2013)CrossRefGoogle Scholar
  14. 14.
    Herrera, F., Herrera-Viedma, E., Martínez, L.: A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets Syst. 114, 43–58 (2000)CrossRefzbMATHGoogle Scholar
  15. 15.
    Herrera, F., Martinez, L.: A model based on linguistic 2-tuples for dealing with multi-granularity hierarchical linguistic contexts in multi-expert decision-making. IEEE Trans. Syst. Man Cybern. 31, 227–234 (2001)CrossRefGoogle Scholar
  16. 16.
    Dong, Y.C., Wu, Y.Z., Zhang, H.J., Zhang, G.Q.: Multi-granular unbalanced linguistic distribution assessments with interval symbolic proportions. Knowl.-Based Syst. 82, 139–151 (2015)CrossRefGoogle Scholar
  17. 17.
    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)CrossRefzbMATHGoogle Scholar
  18. 18.
    Morente-Molinera, J.A., Pérez, I.J., Ureña, M.R., Herrera-Viedma, E.: On multi-granular fuzzy linguistic modeling in group decision making problems: a systematic review and future trends. Knowl.-Based Syst. 74, 49–60 (2015)CrossRefzbMATHGoogle Scholar
  19. 19.
    Liu, S., Chan, F.T.S., Ran, W.X.: Multi-attribute group decision-making with multi-granularity linguistic assessment information: an improved approach based on deviation and TOPSIS. Appl. Math. Model. 37, 10129–10140 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fan, Z.P., Liu, Y.: A method for group decision-making based on multi-granularity uncertain linguistic information. Expert Syst. Appl. 37, 4000–4008 (2010)CrossRefGoogle Scholar
  21. 21.
    Wu, D., Mendel, J.M., Coupland, S.: Enhanced interval approach for encoding words into interval type-2 fuzzy sets and its convergence analysis. IEEE Trans. Fuzzy Syst. 20, 499–513 (2012)CrossRefGoogle Scholar
  22. 22.
    Wu, D., Mendel, J.M.: Enhanced Karnik–Mendel algorithms. IEEE Trans. Fuzzy Syst. 17, 923–934 (2009)CrossRefGoogle Scholar
  23. 23.
    Liu, F.L., Mendel, J.M.: Encoding words into interval type-2 fuzzy sets using an interval approach. IEEE Trans. Fuzzy Syst. 16, 1503–1521 (2008)CrossRefGoogle Scholar
  24. 24.
    Mendel, J.M., John, R.I., Liu, F.: Interval type-2 fuzzy logic systems made simple. IEEE Trans. Fuzzy Syst. 14, 808–821 (2006)CrossRefGoogle Scholar
  25. 25.
    Mendel, J.M.: Type-2 fuzzy sets and systems: an overview. IEEE Comput. Intell. Mag. 2, 20–29 (2007)Google Scholar
  26. 26.
    Chen, T.Y.: A signed-distance-based approach to importance assessment and multi-criteria group decision analysis based on interval type-2 fuzzy set. Know. Inf. Syst. 35, 193–231 (2013)CrossRefGoogle Scholar
  27. 27.
    Qin, J.D., Liu, X.W., Pedrycz, W.: An extended VIKOR method based on prospect theory for multiple attribute decision making under interval type-2 fuzzy environment. Know.-Based Syst. 86, 116–130 (2015)CrossRefGoogle Scholar
  28. 28.
    Chen, S.M., Lee, L.W.: Fuzzy multiple criteria hierarchical group decision-making based on interval type-2 fuzzy sets. IEEE Trans. Syst. Man Cybern. 40, 1120–1128 (2010)CrossRefGoogle Scholar
  29. 29.
    Abdullah, L., Najib, L.: A new type-2 fuzzy set of linguistic variables for the fuzzy analytic hierarchy process. Expert Syst. Appl. 41, 3297–3305 (2014)CrossRefGoogle Scholar
  30. 30.
    Sari, I.U., Kahraman, C.: Interval type-2 fuzzy capital budgeting. Int. J. Fuzzy Syst. 17, 635–646 (2015)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Chen, S.M., Lee, L.W.: Fuzzy multiple attributes group decision-making based on the interval type-2 TOPSIS method. Expert Syst. Appl. 37, 2790–2798 (2010)CrossRefGoogle Scholar
  32. 32.
    Chen, T.Y.: A PROMETHEE-based outranking method for multiple criteria decision analysis with interval type-2 fuzzy sets. Soft. Comput. 18, 923–940 (2014)CrossRefzbMATHGoogle Scholar
  33. 33.
    Chen, T.Y.: An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets. Inf. Sci. 263, 1–21 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Chen, T.Y.: A linear assignment method for multiple-criteria decision analysis with interval type-2 fuzzy sets. Appl. Soft Comput. 13, 2735–2748 (2013)CrossRefGoogle Scholar
  35. 35.
    Ma, X.Y., Wu, P., Zhou, L.G., Chen, H.Y., Zheng, T., Ge, J.Q.: Approaches based on interval type-2 fuzzy aggregation operators for multiple attribute group decision making. Int. J. Fuzzy Syst. 18, 697–715 (2016)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Gong, Y.B.: Fuzzy multi-attribute group decision making method based on interval type-2 fuzzy sets and applications to global supplier selection. Int. J. Fuzzy Syst. 15, 392–400 (2013)MathSciNetGoogle Scholar
  37. 37.
    Han, Z.Q., Wang, J.Q., Zhang, H.Y., Liu, X.X.: Group multi-criteria decision making method with triangular type-2 fuzzy numbers. Int. J. Fuzzy Syst. 18, 673–684 (2016)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zhu, W.D., Zhou, G.Z., Yang, S.L.: An approach to group decision making based on 2-dimension linguistic assessment information. Syst. Eng. 27, 113–118 (2009)Google Scholar
  39. 39.
    Zhu, H., Zhao, J.B., Xu, Y.: 2-dimension linguistic computational model with 2-tuples for multi-attribute group decision making. Knowl.-Based Syst. 103, 132–142 (2016)CrossRefGoogle Scholar
  40. 40.
    Yu, X.H., Xu, Z.S., Liu, S.S., Chen, Q.: Multi-criteria decision making with 2-dimension linguistic aggregation techniques. Int. J. Intell. Syst. 27, 539–562 (2012)CrossRefGoogle Scholar
  41. 41.
    Liu, P.D., Yu, X.C.: 2-Dimension uncertain linguistic power generalized weighted aggregation operator and its application in multiple attribute group decision making. Knowl.-Based Syst. 57, 69–80 (2014)CrossRefGoogle Scholar
  42. 42.
    Dong, Y.C., Xu, Y.F., Li, H.Y., Feng, B.: The OWA-based consensus operator under linguistic representation models using position indexes. Eur. J. Oper. Res. 203, 455–463 (2010)CrossRefzbMATHGoogle Scholar
  43. 43.
    Dong, Y.C., Li, C.C., Herrera, F.: Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its use to deal with hesitant unbalanced linguistic information. Inf. Sci. 367–368, 259–278 (2016)CrossRefGoogle Scholar
  44. 44.
    Zhou, L.G., Chen, H.Y.: The induced linguistic continuous ordered weighted geometric operator and its application to group decision making. Comput. Ind. Eng. 66, 222–232 (2013)CrossRefGoogle Scholar
  45. 45.
    Meng, D., Pei, Z.: On weighted unbalanced linguistic aggregation operators in group decision making. Inf. Sci. 223, 31–41 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Park, J.H., Park, J.M., Kwun, Y.C.: 2-Tuple linguistic harmonic operators and their applications in group decision making. Knowl.-Based Syst. 44, 10–19 (2013)CrossRefGoogle Scholar
  47. 47.
    Wan, S.P.: 2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making. Knowl.-Based Syst. 45, 31–40 (2013)CrossRefGoogle Scholar
  48. 48.
    Merigó, J.M., Gil-Lafuente, A.M.: Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making. Inf. Sci. 236, 1–16 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Wang, J., Wang, J.Q., Zhang, H.Y., Chen, X.H.: Multi-criteria group decision-making approach based on 2-tuple linguistic aggregation operators with multi-hesitant fuzzy linguistic information. Int. J. Fuzzy Syst. 18, 81–97 (2016)MathSciNetCrossRefGoogle Scholar
  50. 50.
    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
  51. 51.
    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, 2430–2444 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Meng, F.Y., Chen, X.H., Zhang, Q.: Some interval-valued intuitionistic uncertain linguistic Choquet operators and their application to multi-attribute group decision making. Appl. Math. Model. 38, 2543–2557 (2014)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Wang, X.F., Wang, J.Q., Deng, S.Y.: Some geometric operators for aggregating intuitionstic linguistic information. Int. J. Fuzzy Syst. 17, 268–278 (2015)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Wei, G.W.: Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making. Int. J. Mach. Learn. Cybern. (2015). doi: 10.1007/s13042-015-0433-7 Google Scholar
  55. 55.
    Gou, X.J., Xu, Z.S., Liao, H.C.: Multiple criteria decision making based on Bonferroni means with hesitant fuzzy linguistic information. Soft. Comput. (2016). doi: 10.1007/s00500-016-2211-1 Google Scholar
  56. 56.
    Liu, X.Y., Ju, Y.B., Yang, S.H.: Some generalized interval-valued hesitant uncertain linguistic aggregation operators and their applications to multiple attribute group decision making. Soft. Comput. 20, 495–510 (2016)CrossRefzbMATHGoogle Scholar
  57. 57.
    Qi, X.W., Liang, C.Y., Zhang, J.L.: Multiple attribute group decision making based on generalized power aggregation operators under interval-valued dual hesitant fuzzy linguistic environment. Int. J. Mach. Learn. Cybern. (2015). doi: 10.1007/s13042-015-0445-3 Google Scholar
  58. 58.
    Liu, P.D., He, L., Yu, X.C.: Generalized hybrid aggregation operators based on the 2-dimension uncertain linguistic information for multiple attribute group decision making. Group Decis. Negot. 25, 103–126 (2016)CrossRefGoogle Scholar
  59. 59.
    Li, Y., Wang, Y.M., Liu, P.D.: Multiple attribute group decision-making methods based on trapezoidal fuzzy two-dimension linguistic power generalized aggregation operators. Soft. Comput. 20, 2689–2704 (2016)CrossRefzbMATHGoogle Scholar
  60. 60.
    Herrera, F., Herrera-Viedma, E., Verdegay, J.L.: Direct approach processes in group decision making using linguistic OWA operators. Fuzzy Sets Syst. 79, 175–190 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Qin, J.D., Liu, X.W.: Multi-attribute group decision making using combined ranking value under interval type-2 fuzzy environment. Inf. Sci. 297, 293–315 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Wu, D., Mendel, J.M.: Aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Trans. Fuzzy Syst. 15, 1145–1161 (2007)CrossRefGoogle Scholar
  63. 63.
    Havens, T., Keller, J., Popescu, M.: Computing with words with the ontological self-organizing map. IEEE Trans. Fuzzy Syst. 18, 473–485 (2010)CrossRefGoogle Scholar
  64. 64.
    Kacprzyk, J., Zadrozny, S.: Computing with words is an implementable paradigm: fuzzy queries, linguistic data summaries, and natural-language generation. IEEE Trans. Fuzzy Syst. 18, 461–472 (2010)CrossRefGoogle Scholar
  65. 65.
    Lawry, J.: A methodology for computing with words. Int. J. Approx. Reason. 28, 51–89 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Rubin, S.H.: Computing with words. IEEE Trans. Syst. Man Cybern. 29, 518–524 (1999)CrossRefGoogle Scholar
  67. 67.
    Niewiadomski, A.: On finity, countability, cardinalities, and cylindric extensions of type-2 fuzzy sets in linguistic summarization of databases. IEEE Trans. Fuzzy Syst. 18, 532–545 (2010)CrossRefGoogle Scholar
  68. 68.
    Wu, D., Mendel, J.M.: Linguistic summarization using IF–THEN rules and interval type-2 fuzzy sets. IEEE Trans. Fuzzy Syst. 19, 136–151 (2011)CrossRefGoogle Scholar
  69. 69.
    Mendel, J.M., Wu, D.: Perceptual Computing: Aiding People in Making Subjective Judgments. IEEE-Wiley Press, Piscataway (2010)CrossRefGoogle Scholar
  70. 70.
    Merigó, J.M., Casanovas, M.: The uncertain induced quasi-arithmetic OWA operator. Int. J. Intell. Syst. 26, 1–24 (2011)CrossRefzbMATHGoogle Scholar
  71. 71.
    Liu, X.W.: The orness measure for two compound quasi-arithmetic mean aggregation operators. Int. J. Approx. Reason. 51, 305–334 (2010)CrossRefzbMATHGoogle Scholar
  72. 72.
    Tan, C.Q., Jiang, Z.Z., Chen, X.H.: Some issues on quasi-arithmetic intuitionistic fuzzy OWA operators. Appl. Math. Inf. Sci. 7, 955–961 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Xu, Y.J., Da, Q.L.: Trapezoidal fuzzy ordered weighted averaging operator and its application to decision making. J. Southeast Univ. (Nat. Sci. Edit.) 36, 1034–1038 (2006)zbMATHGoogle Scholar
  74. 74.
    Wang, J.H., Hao, J.: A new version of 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst. 14, 435–445 (2006)CrossRefGoogle Scholar
  75. 75.
    Liu, P.D.: An approach to group decision making based on 2-dimension uncertain linguistic information. Technol. Econ. Dev. Econ. 18, 424–437 (2012)CrossRefGoogle Scholar
  76. 76.
    Liu, P.D., Teng, F.: An extended TODIM method for multiple attribute group decision making based on 2-dimension uncertain linguistic variable. Complexity 21, 20–30 (2016)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Kahneman, D., Tversky, A.: Prospect theory: an analysis of decision under risk. Econometrica 47, 263–291 (1979)CrossRefzbMATHGoogle Scholar
  78. 78.
    Gong, Z.W., Zhang, H.H., Forrest, J., Li, L.S., Xu, X.X.: Two consensus models based on the minimum cost and maximum return regarding either all individuals or one individual. Eur. J. Oper. Res. 240, 183–192 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Gong, Z.W., Xu, X.X., Li, L.S., Xu, C.: Consensus modeling with nonlinear utility and cost constraints: a case study. Knowl.-Based Syst. 88, 210–222 (2015)CrossRefGoogle Scholar
  80. 80.
    Gong, Z.W., Xu, X.X., Lu, F.L., Li, L.S., Xu, C.: On consensus models with utility preferences and limited budget. Appl. Soft Comput. 35, 840–849 (2015)CrossRefGoogle Scholar
  81. 81.
    Gong, Z.W., Forrest, J., Yang, Y.J.: The optimal group consensus models for 2-tuple linguistic preference relations. Knowl.-Based Syst. 37, 427–437 (2013)CrossRefGoogle Scholar
  82. 82.
    Gong, Z.W., Forrest, J., Zhao, Y., Yang, Y.J.: The optimal group consensus deviation measure for multiplicative preference relations. Expert Syst. Appl. 39, 11548–11555 (2012)CrossRefGoogle Scholar
  83. 83.
    Dong, Y.C., Zhang, H.J., Herrera-Viedma, E.: Consensus reaching model in the complex and dynamic MAGDM problem. Knowl.-Based Syst. 106, 206–219 (2016)CrossRefGoogle Scholar
  84. 84.
    Cavdur, F., Kose, M.: A fuzzy logic and binary-goal programming-based approach for solving the exam timetabling problem to create a balanced-exam schedule. Int. J. Fuzzy Syst. 18, 119–129 (2016)CrossRefGoogle Scholar
  85. 85.
    Ghaderi, A., Vasegh, N.: Input-output stabilizing controller synthesis for SISO T–S fuzzy systems by applying large gain theorem. Int. J. Fuzzy Syst. 18, 550–556 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.China Institute of Manufacturing DevelopmentNanjing University of Information Science and TechnologyNanjingChina

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