Diminishing Choquet Hesitant 2-Tuple Linguistic Aggregation Operator for Multiple Attributes Group Decision Making

Main Article Content

Ismat Beg
Raja Noshad Jamil
Tabasam Rashid


In this article, we develop a diminishing hesitant 2-tuple averaging operator (DH2TA) for hesitant 2-tuple linguistic arguments. DH2TA work in the way that it aggregate all hesitant 2-tuple linguistic elements and during the aggregation process it also controls the hesitation in translation of the resultant aggregated linguistic term. We develop a scalar product for hesitant 2-tuple linguistic elements and based on the scalar product a weighted diminishing hesitant 2-tuple averaging operator (DWH2TA) is introduced. Moreover, combining Choquet integral with hesitant 2-tuple linguistic information, the diminishing Chouqet hesitant 2-tuple average operator (DCH2TA) is defined. The proposed operators higher reflect the correlations among the elements. After investigating the properties of these operators, a multiple attribute decision making method based on DCH2TA operator is proposed. Finally, an example is given to illustrate the significance and usefulness of proposed method.

Article Details


  1. R. D. Andr ´es, M. Espinilla and L. Mart ´inez, An extended hierarchical linguistic model for managing integral evaluation, Int. J. Comput. Intell. Syst. 3 (4) (2010), 486-500.
  2. R. D. Andr ´es, J.L. Garc ´ia-Lapresta and L. Mart ´inez, A multi-granular linguistic model for management decision-making in performance appraisal, Soft Computing, 14 (1) (2010), 21-34.
  3. C. Araz and I. Ozkarahan, Supplier evaluation and management system for strategic sourcing based on a new multicriteria sorting procedure, Int. J. Production Economics, 106 (2) (2007), 585-606.
  4. I. Beg and T. Rashid, Multi-criteria of bike purchasing using fuzzy choquet integral, J. Fuzzy Math. 22 (3)(2014), 677-694.
  5. I. Beg and T. Rashid, Hesitant 2-tuple linguistic information in multiple attributes group decision making, J. Intell. Fuzzy Syst. 30 (2016), 109-116.
  6. I. Beg and T. Rashid, Modelling uncertainties in multi-criteria decision making using distance measure and TOPSIS for hesitant fuzzy sets, J. Artif. Intell. Soft Comput. Res. 7(2)(2017), 103-109.
  7. C. Bonferroni, Sulle medie multiple di potenze, Bolletino Matematica Italiana, 5 (1950), 267-270.
  8. C. T. Chen, C. T. Lin, and S. F. Huang, A fuzzy approach for supplier evaluation and selection in supply chain management, Int. J. Production Economics, 102 (2) (2006), 289-301.
  9. C.T. Chen, P. F. Pai and W. Z. Hung, An integrated methodology using linguistic promethee and maximum deviation method for third-party logistics supplier selection, Int. J. Comput. Intell. Syst. 3 (4) (2010), 438-451.
  10. Y. Chen, X. Zeng M. Happiette, P. Bruniaux, R. Ng and W. Yu, Optimisation of garment design using fuzzy logic and sensory evaluation techniques, Eng. Appl. Artif. Intell. 22 (2) (2009), 272-282.
  11. G. Choquet, Theory of capacities. Ann. Inst. Fourier, 5 (1953), 131-295.
  12. S. Y. Chou and Y. H. Chang, A decision support system for supplier selection based on a strategy-aligned fuzzy SMART approach, Expert Syst. Appl. 34 (4) (2008), 2241-2253.
  13. R. Degani and G. Bortolan, The problem of linguistic approximation in clinical decision making, Int. J. Approx. Reason. 2 (1988), 143-162.
  14. M. Delgado, J.L. Verdegay and M.A. Vila, On aggregation operations of linguistic labels, Int. J. Intell. Syst. 8 (3) (1993), 351-370.
  15. Y.C. Dong, C.C. Li and F. Herrera, An optimization-based approach to adjusting the unbalanced linguistic preference relations to obtain a required consistency level, Inf. Sci. 292 (2015), 27-38.
  16. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Kluwer Academic, New York, 1980.
  17. C. X. Feng, J. Wang, and J. S. Wang, An optimization model for concurrent selection of tolerances and suppliers, Comput. Ind. Eng. 40 (1-2) (2001), 15-33.
  18. N. Fenton and W. Wang, Risk and confidence analysis for fuzzy multicriteria decision making, Knowl.based Syst. 19 (6) (2006), 430-437.
  19. J.L. Garc ´ia-Lapresta, B. Llamazares and M. Mart ´inez-Panero, A social choice analysis of the Borda rule in a general linguistic framework, Int. J. Comput. Intell. Syst. 3 (4) (2010), 501-513.
  20. R. E. Gregory, Source selection: a matrix approach, J. Purchas. Mater. Manag. 22 (2) (1986), 24-29.
  21. S. H. Ha and R. Krishnan, A hybrid approach to supplier selection for the maintenance of a competitive supply chain, Expert Syst. Appl. 34 (2) (2008), 1303-1311.
  22. F. Herrera and L. Mart ´inez, A 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans. Fuzzy Syst. 8 (2000), 746-752.
  23. F. Herrera and L. Mart ´inez, The 2-tuple linguistic computational model advantages of its linguistic description, accuracy and consistency, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 9 (2001), 33-48.
  24. F. Herrera, E. Herrera-Viedma and L. Mart ´inez, A fuzzy linguistic methodology to deal with unbalanced linguistic term sets, IEEE Trans. Fuzzy Syst. 16(2) (2008), 354-370.
  25. E. Herrera-Viedma, A.G. L ´opez-Herrera, M. Luque and C. Porcel, A fuzzy linguistic IRS model based on a 2-tuple fuzzy linguistic approach, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 15 (2) (2007), 225-250.
  26. V.N. Huynh and Y. Nakamori, A satisfactory-oriented approach to multi-expert decision-making under linguistic assessments, IEEE Trans. Syst. Man Cybern. 35 (2) (2005), 184-196.
  27. H. Ishibuchi, T. Nakashima and M. Nii, Classification and Modeling with Linguistic Information Granules: Advanced Approaches to Linguistic Data Mining, Springer, Berlin, 2004.
  28. Y.P. Jiang and Z.P. Fan, Property analysis of the aggregation operators for 2-tuple linguistic information, Control Decision, 18(6) (2003), 754-757.
  29. D.K. Joshi, I. Beg and S. Kumar, Hesitant probabilistic fuzzy linguistic sets with applications in multi-criteria group decision making problems, Mathematics 6(4)(2018), Article ID 47.
  30. A. Khalid and I. Beg, Incomplete hesitant fuzzy preference relations in group decision making, Int. J. Fuzzy Syst. 19(3)(2017), 637-645.
  31. G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall PTR, 1995.
  32. C. Labreuche and M. Grabisch, Generalized Choquet-like aggregation functions for handling bipolar scales, Eur. J. Oper. Res. 172 (3) (2006), 931-955.
  33. J. Lawry, A framework for linguistic modelling, Artif. Intell. 155 (1-2) (2004), 1-39.
  34. C. C. Li, Y. Donga, F. Herrera, E. H. Viedm and L. Mart ´inez, Personalized individual semantics in computing with words for supporting linguistic group decision making. An application on consensus reaching, Inf. Fusion, 33 (2017), 29-40.
  35. J. Lu, G. Zhang and F. Wu, Team situation awareness using web-based fuzzy group decision support systems, Int. J. Comput. Intell. Syst. 1 (1) (2008), 51-60.
  36. J. Lu, Y. Zhu, X. Zeng, L. Koehl, J. Ma and G. Zhang, A linguistic multi-criteria group decision support system for fabric hand evaluation, Fuzzy Optim. Decis. Mak. 8 (4) (2009), 395-413.
  37. O. Martin and G.J. Klir, On the problem of retranslation in computing with perceptions, Int. J. General Syst. 35 (6) (2006), 655-674.
  38. L. Mart ´inez, D. Ruan, F. Herrera, E. Herrera-Viedma and P.P. Wang, Linguistic decision making: tools and applications, Inf. Sci. 179 (14) (2009), 2297-2298.
  39. L. Mart ´inez, J. Liu, D. Ruan and J.B. Yang, Dealing with heterogeneous information in engineering evaluation processes, Inf. Sci. 177 (7) (2007), 1533-1542.
  40. L. Mart ´inez, J. Liu, J. B. Yang and F. Herrera, A multigranular hierarchical linguistic model for design evaluation based on safety and cost analysis, Int. J. Intell. Syst. 20 (12) (2005), 1161-1194.
  41. L. Mart ´inez, R. M. Rodr ´iguez and F. Herrera, 2-tuple Linguistic Model Computing with words in Decision Making, Springer (2015).
  42. J.M. Mendel and D. Wu, Perceptual Computing: Aiding People in Making Subjective Judgments, IEEE-Wiley, (2010).
  43. R. Mesiar and A. Kolesarova, On the fuzzy set theory and aggregation functions: Histor and some recent advances, Iran. J. Fuzzy Syst. in press.
  44. W.R.W. Mohd and L. Abdullah, Aggregation methods in group decision making: A decade survey, Informatica 41 (2017), 71-86.
  45. G.A. Miller, The magical number seven, plus or minus two: some limits on our capacity of processing information, Psychol. Rev. 63 (1956), 81-97.
  46. R. F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proc. Edinburgh Math. Soc. 21(3) (1902), 144-162.
  47. R. Narasimhan, An analytic approach to supplier selection, J. Purchas. Supply Manag. 1 (1983), 27-32.
  48. R. L. Nydick and R. P. Hill, Using the Analytic Hierarchy Process to structure the supplier selection procedure, Int. J. Purchas. Mat. Manag. 28 (2) (1992), 31-36.
  49. L. P ´erez-Dom ´inguez, L. Rodr ´iguez-Pic ´on, A. Alvarado-Iniesta, D.L. Cruz and Z.S. Xu, MOORA under Pythagorean fuzzy set for multiple criteria decision making, Complexity, 2018 (2008), Article ID 2602376, 10 pages
  50. I. Saad, S. Hammadi, M. Benrejeb and P. Borne, Choquet integral for criteria aggregation in the flexible job-shop scheduling problems, Math. Comput. Simul. 76 (2008), 447-462.
  51. H. Shi, H. C. Liu, P. Li, X. and G. Xu, An integrated decision making approach for assessing healthcare waste treatment technologies from a multiple stakeholder, Waste Manag. 59 (2017), 508-517.
  52. W. R. Soukup, Supplier selection strategies, J. Purchas. Mat. Manag. 23 (3) (1987), 7-12.
  53. C. Q. Tan and X. H. Chen, Intuitionistic fuzzy Choquet integral operator for multi-criteira decision making, Expert Syst. Appl. 37 (2010), 149-157.
  54. Thompson, Vendor prole analysis, J. Purchas. Mat. Manag. 26 (1) (1990), 11-18.
  55. V. Torra, Hesitant fuzzy sets, Int. J. Intell. Syst. 25 (2010), 529-539.
  56. R. J. Vokurka, J. Choobineh, and L. Vadi, A prototype expert system for the evaluation and selection of potential suppliers, Int. J. Oper. Prod. Manag. 16 (12) (1996), 106-127.
  57. P. Wakker, Additive representations of preferences, Kluwer Academic Publishers, (1999).
  58. Z. Wang and G. Klir, Fuzzy measure theory, New York: Plenum press, (1992).
  59. J.H. Wang and J.Y. Hao, A new version of 2-tuple fuzzy linguistic representation model for computing with words, IEEE Trans. Fuzzy Syst. 14(3) (2006), 435-445.
  60. G.W. Wei, Method for two-tuple linguistic group decision making based on the ET-WG and ET-OWG operators, Expert Syst. Appl. 37 (2010), 7895-7900.
  61. M.M. Xia, Z.S. Xu and B. Zhu, Geometric Bonferroni means with their application in multi-criteria decision making, Knowl. based Syst. 40 (2013), 88-100.
  62. Z.S. Xu, A method based on linguistic aggregation operators for group decision making with linguistic preference relations, Inf. Sci. 166(1-4) (2004), 19-30.
  63. Z.S. Xu, EOWA and EOWG operators for aggregating linguistic labels based on linguistic preference relations, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 12 (2004), 791-810.
  64. Z. Xu, S. Shang, W. Qian and W. Shu, A method for fuzzy risk analysis based on the new similarity of trapezoidal fuzzy numbers, Expert Syst. Appl. 37 (3) (2010), 1920-1927.
  65. R.R. Yager, A new methodology for ordinal multi objective decisions based on fuzzy sets, Decision Sci. 12 (1981), 589-600.
  66. R.R. Yager, Computing with Words and Information/Intelligent Systems 2: Applications, Chapter Approximate Reasoning as a Basis for Computing with Words, Physica Verlag, (1999), 50-77.
  67. R.R. Yager, Induced aggregation operators, Fuzzy Sets Syst. 137 (2003), 59-69.
  68. R.R. Yager, On the retranslation process in Zadeh's paradigm of computing with words, IEEE Transactions on Systems, Man, and Cybernetics - Part B: Cybernetics, 34 (2) (2004), 1184-1195.
  69. R.R. Yager, OWA aggregation of intuitionistic fuzzy sets, Int. J. General Syst. 38 (6) (2009), 617-641.
  70. W. Yang and Z. Chen, New aggregation operators based on the Choquet integral and 2-tuple linguistic information, Expert Syst. Appl. 39 (2012), 2662-2668
  71. L. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338-353.
  72. L. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part III, Inf. Sci. 9 (1) (1975), 43-80.
  73. Y. Zhang and Z.P. Fan, An approach to linguistic multiple attribute decision-making with linguistic information based on ELOWA operator, Syst. Eng. 24(12) (2006), 324-339.
  74. B. Zhu, Z. S. Xu and M. M. Xia, Hesitant fuzzy geometric Bonferroni means, Inf. Sci. 205 (2012), 72-85.