Advertisement

Applied Intelligence

, Volume 46, Issue 4, pp 788–799 | Cite as

Distance measures between type-2 intuitionistic fuzzy sets and their application to multicriteria decision-making process

  • Sukhveer Singh
  • Harish Garg
Article

Abstract

Type-2 fuzzy set (T2FS) is a generalization of the ordinary fuzzy set in which the membership value for each member of the set is itself a fuzzy set. However, it is difficult, in some situations, for the decision-makers to give their preferences towards the object in terms of single or exact number. For handling this, a concept of type-2 intuitionistic fuzzy set (T2IFS) has been proposed and hence under this environment, a family of distance measures based on Hamming, Euclidean and Hausdorff metrics are presented. Some of its desirable properties have also been investigated in details. Finally, based on these measures, a group decision making method has been presented for ranking the alternatives. The proposed measures has been illustrated with a numerical example.

Keywords

Distance measures Type-2 fuzzy set Type-2 intuitionistic fuzzy set Decision-making 

References

  1. 1.
    Attanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96MathSciNetCrossRefGoogle Scholar
  2. 2.
    Burillo P, Bustince H (1996) Entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Fuzzy Sets Syst 78(3):305–316MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen SM, Lee LW (2010a) Fuzzy multiple attributes group decision-making based on the interval type-2 topsis method. Expert Syst Appl 37(4):2790–2798Google Scholar
  4. 4.
    Chen SM, Lee LW (2010b) Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type - 2 fuzzy sets. Expert Syst Appl 37(1):824 – 833Google Scholar
  5. 5.
    Chen SM, Yang MW, Lee LW, Yang SW (2012) Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets. Expert Syst Appl 39(5):5295–5308CrossRefGoogle Scholar
  6. 6.
    Chen TY (2013) A linear assignment method for multiple-criteria decision analysis with interval type-2 fuzzy sets. Appl Soft Comput 13(5):2735–2748CrossRefGoogle Scholar
  7. 7.
    Chen TY, Chang CH, Lu J f R (2013) 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):615–625MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Garg H (2016a) Generalized intuitionistic fuzzy interactive geometric interaction operators using einstein t-norm and t-conorm and their application to decision making. Comput Ind Eng 101:53–69Google Scholar
  9. 9.
    Garg H (2016b) Generalized pythagorean fuzzy geometric aggregation operators using einstein t-norm and t-conorm for multicriteria decision-making process. Int J Intell Syst. doi: 10.1002/int.21860
  10. 10.
    Garg H (2016c) A new generalized improved score function of interval-valued intuitionistic fuzzy sets and applications in expert systems. Appl Soft Comput 38:988–999Google Scholar
  11. 11.
    Garg H (2016d) A new generalized pythagorean fuzzy information aggregation using einstein operations and its application to decision making. Int J Intell Syst 31(9):886–920Google Scholar
  12. 12.
    Garg H (2016e) A novel accuracy function under interval-valued pythagorean fuzzy environment for solving multicriteria decision making problem. J Intell Fuzzy Syst 31(1):529–540Google Scholar
  13. 13.
    Garg H (2016f) A novel correlation coefficients between pythagorean fuzzy sets and its applications to decision-making processes. Int J Intell Syst 31(12):1234–1253Google Scholar
  14. 14.
    Garg H, Agarwal N, Tripathi A (2015) Entropy based multi-criteria decision making method under fuzzy environment and unknown attribute weights. Global Journal of Technology and Optimization 6:13–20Google Scholar
  15. 15.
    Hu J, Zhang Y, Chen X, Liu Y (2013) Multi-criteria decision making method based on possibility degree of interval type-2 fuzzy number. Knowl-Based Syst 43:21–29CrossRefGoogle Scholar
  16. 16.
    Hung WL, Yang MS (2004a) Similarity measures between type-2 fuzzy sets. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12(06):827–841Google Scholar
  17. 17.
    Hung WL, Yang MS (2004b) Similarity measures of intuitionistic fuzzy sets based on hausdorff distance. Pattern Recogn Lett 25:1603–1611Google Scholar
  18. 18.
    Lee LW, Chen SM (2008a) Fuzzy multiple attributes group decision-making based on the extension of topsis method and interval type - 2 fuzzy sets. In: Proceedings of 2008 International Conference on Machine Learning and Cybernetics, vol 1-7. IEEE, pp 3260–3265Google Scholar
  19. 19.
    Lee LW, Chen SM (2008a) A new method for fuzzy multiple attributes group decision-making based on the arithmetic operations of interval type-2 fuzzy sets. In: Proceedings of 2008 International Conference on Machine Learning and Cybernetics, vol 1-7. IEEE, pp 3084–3089Google Scholar
  20. 20.
    Lin CW, Hong TP (2013) A survey of fuzzy web mining. Data Min Knowl Disc 3(3):190–199CrossRefGoogle Scholar
  21. 21.
    Lin JCW, Li T, Fournier-Viger P, Hong TP, Wu JMT, Zhan J (2016a) Efficient mining of multiple fuzzy frequent itemsets. Intern J Fuzzy Syst:1–9. doi: 10.1007/s40815-016-0246-1
  22. 22.
    Lin JCW, Lv X, Fournier-Viger P, Wu TY, Hong TP (2016b) Efficient Mining of Fuzzy Frequent Itemsets with Type-2 Membership Functions. Springer, Berlin, pp 191–200Google Scholar
  23. 23.
    Mendel JM (2001) Uncertain rule-based fuzzy logic system: introduction and new directionsGoogle Scholar
  24. 24.
    Mendel JM, Wu H (2006) Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 1, forward problems. IEEE Trans Fuzzy Syst 14(6):781–792CrossRefGoogle Scholar
  25. 25.
    Mendel JM, Wu H (2007) Type-2 fuzzistics for symmetric interval type-2 fuzzy sets: Part 2, inverse problems. IEEE Trans Fuzzy Syst 15(2):301–308CrossRefGoogle Scholar
  26. 26.
    Mendel JM, John RI, LIu F (2006) Interval type-2 fuzzy logic systems made simple. IEEE Trans Fuzzy Syst 14(6):808–821CrossRefGoogle Scholar
  27. 27.
    Qin J, Liu X (2014) Frank aggregation operators for triangular interval type-2 fuzzy set and its application in multiple attribute group decision making. J Appl Math 2014:Article ID 923,213 24 pagesGoogle Scholar
  28. 28.
    Singh P (2014) Some new distance measures for type-2 fuzzy sets and distance measure based ranking for group decision making problems. Frontiers of Comput Sci 8(5):741–752MathSciNetCrossRefGoogle Scholar
  29. 29.
    Szmidt E, Kacprzyk J (2001) Entropy for intuitionistic fuzzy sets. Fuzzy Sets Syst 118(3):467–477MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wang W, Liu X, Qin Y (2012) Multi-attribute group decision making models under interval type-2 fuzzy environment. Knowl-Based Syst 30:121–128CrossRefGoogle Scholar
  31. 31.
    Wei CP, Wang P, Zhang YZ (2011) Entropy, similarity measure of interval-valued intuitionistic fuzzy sets and their applications. Inf Sci 181:4273–4286MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wu D, Mendel JM (2007) Aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Trans Fuzzy Syst 15(6):1145–1161CrossRefGoogle Scholar
  33. 33.
    Wu D, Mendel JM (2008) A vector similarity measure for linguistic approximation: Interval type-2 and type-1 fuzzy sets. Inf Sci 178(2):381–402MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Wu D, Mendel JM (2009) A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets. Inf Sci 179(8):1169–1192MathSciNetCrossRefGoogle Scholar
  35. 35.
    Xu Z, Chen J (2007) Approach to group decision making based on interval valued intuitionistic judgment matrices. Systems Engineering - Theory and Practice 27(4):126–133CrossRefGoogle Scholar
  36. 36.
    Xu ZS (2007a) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187Google Scholar
  37. 37.
    Xu ZS (2007b) Methods for aggregating interval-valued intuitionistic fuzzy information and their application to decision making. Control and Decision 22(2):215–219Google Scholar
  38. 38.
    Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Yang MS, Lin DC (2009) On similarity and inclusion measures between type-2 fuzzy sets with an application to clustering. Computers & Mathematics with Applications 57(6):896–907MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle Scholar
  41. 41.
    Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning: Part-1. Inf Sci 8:199–251CrossRefMATHGoogle Scholar
  42. 42.
    Zeng W, Li H (2006) Relationship between similarity measure and entropy of interval-valued fuzzy sets. Fuzzy Sets Syst 157(11):1477–1484MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zeng WY, Guo P (2008) NorMalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship. Inf Sci 178:1334– 1342MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Zhang Z, Zhang S (2013) A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets. Appl Math Model 37(7):4948–4971MathSciNetCrossRefGoogle Scholar
  45. 45.
    Zhou SM, Chiclana F, John RI, Garibaldi JM (2008) Type-2 owa operators: aggregating type-2 fuzzy sets in soft decison making. In: Procedding of the IEEE International Conference on Fuzzy Systems (FUZZ’08), vol 1-5, pp 625–630Google Scholar
  46. 46.
    Zhou SM, John RI, Chiclana F, Garibaldi JM (2010) On aggregating uncertain information by type-2 owa operators for soft decision making. Int J Intell Syst 25(6):540–558MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of MathematicsThapar UniversityPatialaIndia

Personalised recommendations