Neural Computing and Applications

, Volume 30, Issue 2, pp 671–682 | Cite as

An improved ranking method for comparing trapezoidal intuitionistic fuzzy numbers and its applications to multicriteria decision making

  • V. Lakshmana Gomathi Nayagam
  • S. JeevarajEmail author
  • P. Dhanasekaran
Original Article


Problems with qualitative, quantitative and uncertain information can be modelled better using trapezoidal intutionistic fuzzy numbers (TrIFNs) than fuzzy numbers. Due to the partial ordering of TrIFNs, many ranking methods are available in the literature for comparing fuzzy and intuitionistic fuzzy numbers (Li in Comput Math Appl 60:1557–1570, 2010; Li and Yang in Math Comput Appl 20(1):25–38, 2015; De and Das in 12th international conference on intelligent systems design and applications (ISDA), Kochi, India, 27–29 Nov, IEEE, 2012; Dubey and Mehara in Proceedings of the seventh conference of the European society for fuzzy logic and technology, pp 563–569, 2011; Nehi in Int J Fuzzy Syst 12:80–86, 2010; Fangwei and Xu in Soft Comput, 2016. doi: 10.1007/s00500-015-1932-x; Jun in Expert Syst Appl 38:11730–11734, 2011, Int J Gen Syst 41(7):729–739, 2009, Group Decis Negot 21:519–530, 2012; Nayagam and Geetha in Appl Soft Comput 11(4):3368–3372, 2011; Nayagam et al. in Expert Syst Appl 38(3):1464–1467, 2011, Soft Comput, 2016. doi: 10.1007/s00500-016-2249-0; Sahin in Soft Comput 20(7):2557–2563, 2016; Shu-Ping and Dong in Appl Soft Comput 29:153–168, 2015; Zeng et al. in Sci World J 1–8, 2014; Ye in Expert Syst Appl 36:6899–6902, 2009; Zhang and Yu in Knowl Based Syst 30:115–120, 2012). In order to overcome the shortcomings and limitations of the existing methods, a new method for comparing TrIFNs based on the concept of improved value index and improved ambiguity index is introduced, and the significance of the proposed method is illustrated using numerical examples. Further to show the applicability of the proposed method, a new algorithm for solving multicriteria decision-making problem is established.


Trapezoidal intuitionistic fuzzy number Improved value index Improved ambiguity index Triangular intuitionistic fuzzy number (TIFN) Interval-valued intuitionistic fuzzy number MCDM IFMCDM 



Authors are grateful to the anonymous reviewers, Editor and Associate editor for their valuable comments. The corresponding author thanks the Council of Scientific and Industrial Research (CSIR-HRDG), India for supporting this research under CSIR SRF.


  1. 1.
    Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2016) Multiobjective evolutionary design of granular rule-based classifiers. Granul Comput 1(1):37–58CrossRefGoogle Scholar
  2. 2.
    Apolloni B, Bassis S, Rota J, Galliani GL, Gioia M, Ferrari L (2016) A neurofuzzy algorithm for learning from complex granules. Granul Comput 1(4):1–22CrossRefGoogle Scholar
  3. 3.
    Ciucci D (2016) Orthopairs and granular computing. Granul Comput 1(3):159–170CrossRefGoogle Scholar
  4. 4.
    De PK, Das D (2012) Ranking of trapezoidal intuitionistic fuzzy numbers. In: 12th international conference on intelligent systems design and applications (ISDA), Kochi, India, 27–29 Nov, IEEEGoogle Scholar
  5. 5.
    Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkzbMATHGoogle Scholar
  6. 6.
    Dubois D, Prade H (2016) Bridging gaps between several forms of granular computing. Granul Comput 1(2):115–126CrossRefGoogle Scholar
  7. 7.
    Dubey D, Mehara A (2011) Linear programming with triangular intuitionistic fuzzy number. In: Proceedings of the seventh conference of the European society for fuzzy logic and technology, pp 563–569Google Scholar
  8. 8.
    Fangwei Z, Xu S (2016) Remarks to “Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets”. Soft Comput. doi: 10.1007/s00500-015-1932-x
  9. 9.
    Geetha S, Nayagam VLG, Ponalagusamy R (2014) A complete ranking of incomplete interval information. Expert Syst Appl 41(4):1947–1954CrossRefGoogle Scholar
  10. 10.
    Jun Y (2009) Multicriteria group decision-making method using the distances-based similarity measures between intuitionistic trapezoidal fuzzy numbers. Int J Gen Syst 41(7):729–739Google Scholar
  11. 11.
    Jun Y (2011) Expected value method for intuitionistic trapezoidal fuzzy multicriteria decision-making problems. Expert Syst Appl 38:11730–11734Google Scholar
  12. 12.
    Jun Y (2012) Multicriteria group decision-making method using vector similarity measures for trapezoidal intuitionistic fuzzy numbers. Group Decis Negot 21:519–530CrossRefGoogle Scholar
  13. 13.
    Kreinovich V (2016) Solving equations (and systems of equations) under uncertainty: how different practical problems lead to different mathematical and computational formulations. Granul Comput 1(3):171–179CrossRefGoogle Scholar
  14. 14.
    Li DF (2008) A note on using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly. Microelectron Reliab 48:17–41CrossRefGoogle Scholar
  15. 15.
    Li DF (2010) A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems. Comput Math Appl 60:1557–1570MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Li DF, Yang J (2015) A difference-index based ranking method of trapezoidal intuitionistic fuzzy number and its application to multiattribute decision making. Math Comput Appl 20(1):25–38MathSciNetGoogle Scholar
  17. 17.
    Lingras P, Haider F, Triff M (2016) Granular meta-clustering based on hierarchical, network, and temporal connections. Granul Comput 1(1):71–92CrossRefGoogle Scholar
  18. 18.
    Liu H, Gegov A, Cocea M (2016) Rulebased systems: a granular computing perspective. Granul Comput 1(4):1–16CrossRefGoogle Scholar
  19. 19.
    Livi L, Sadeghian A (2016) Granular computing, computational intelligence, and the analysis of non-geometric input spaces. Granul Comput 1(1):13–20CrossRefGoogle Scholar
  20. 20.
    Loia V, D’Aniello G, Gaeta A, Orciuoli F (2016) Enforcing situation awareness with granular computing: a systematic overview and new perspectives. Granul Comput 1(2):127–143CrossRefGoogle Scholar
  21. 21.
    Maciel L, Ballini R, Gomide F (2016) Evolving granular analytics for interval time series forecasting. Granul Comput 1(4):1–12CrossRefGoogle Scholar
  22. 22.
    Mendel JM (2016) A comparison of three approaches for estimating (synthesizing) an interval type 2 fuzzy set model of a linguistic term for computing with words. Granul Comput 1(1):59–69CrossRefGoogle Scholar
  23. 23.
    Min F, Xu J (2016) Semi-greedy heuristics for feature selection with test cost constraints. Granul Comput 1(3):199–211CrossRefGoogle Scholar
  24. 24.
    Nayagam VLG, Geetha S (2011) Ranking of interval-valued intuitionistic fuzzy sets. Appl Soft Comput 11(4):3368–3372CrossRefGoogle Scholar
  25. 25.
    Nayagam VLG, Muralikrishnan S, Geetha S (2011) Multi criteria decision making method based on interval valued intuitionistic fuzzy sets. Expert Syst Appl 38(3):1464–1467CrossRefGoogle Scholar
  26. 26.
    Nayagam VLG, Jeevaraj S, Dhanasekaran P (2016) An intuitionistic fuzzy multi-criteria decision-making method based on non-hesitance score for interval-valued intuitionistic fuzzy sets. Soft Comput. doi: 10.1007/s00500-016-2249-0 zbMATHGoogle Scholar
  27. 27.
    Nehi HM (2010) A new ranking method for intuitionistic fuzzy numbers. Int J Fuzzy Syst 12:80–86MathSciNetGoogle Scholar
  28. 28.
    Pedrycz W, Chen SM (2015) Granular computing and decision-making: interactive and iterative approaches. Springer, HeidelbergCrossRefGoogle Scholar
  29. 29.
    Peters G, Weber R (2016) DCC: a framework for dynamic granular clustering. Granul Comput 1(1):1–11CrossRefGoogle Scholar
  30. 30.
    Sahin R (2016) Fuzzy multicriteria decision making method based on the improved accuracy function for interval-valued intuitionistic fuzzy sets. Soft Comput 20(7):2557–2563CrossRefzbMATHGoogle Scholar
  31. 31.
    Shu-Ping W, Dong J-Y (2015) Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making. Appl Soft Comput 29:153–168CrossRefGoogle Scholar
  32. 32.
    Skowron A, Jankowski A, Dutta S (2016) Interactive granular computing. Granul Comput 1(2):95–113MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Song M, Wang Y (2016) A study of granular computing in the agenda of growth of artificial neural networks. Granul Comput 1(4):1–11CrossRefGoogle Scholar
  34. 34.
    Wilke G, Portmann E (2016) Granular computing as a basis of human–data interaction: a cognitive cities use case. Granul Comput 1(3):181–197CrossRefGoogle Scholar
  35. 35.
    Xu Z, Wang H (2016) Managing multi-granularity linguistic information in qualitative group decision making: an overview. Granul Comput 1(1):21–35CrossRefGoogle Scholar
  36. 36.
    Yao Y (2016) A triarchic theory of granular computing. Granul Comput 1(2):145–157CrossRefGoogle Scholar
  37. 37.
    Ye J (2009) Multicriteria fuzzy decision-making method based on a novel accuracy function under interval-valued intuitionistic fuzzy environment. Expert Syst Appl 36:6899–6902CrossRefGoogle Scholar
  38. 38.
    Zeng X-T, Li D-F, Yu G-F (2014) A value and ambiguity-based ranking method of trapezoidal intuitionistic fuzzy numbers and application to decision making. Sci World J 2014:1–8CrossRefGoogle Scholar
  39. 39.
    Zhang H, Yu L (2012) MADM method based on cross-entropy and extended TOPSIS with interval-valued intuitionistic fuzzy sets. Knowl Based Syst 30:115–120CrossRefGoogle Scholar

Copyright information

© The Natural Computing Applications Forum 2016

Authors and Affiliations

  • V. Lakshmana Gomathi Nayagam
    • 1
  • S. Jeevaraj
    • 1
    Email author
  • P. Dhanasekaran
    • 1
  1. 1.Department of MathematicsNational Institute of Technology TiruchirappalliTiruchirappalliIndia

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