Soft Computing

, Volume 21, Issue 18, pp 5245–5264 | Cite as

Constructive methods for intuitionistic fuzzy implication operators

Foundations

Abstract

In this paper, two novel constructive methods for fuzzy implication operators are proposed. And they can be used to resolve the questions about how to construct intuitionistic fuzzy implication operators to inference sentence “If x is A, then y is B,” where A and B are intuitionistic fuzzy sets over X and Y, respectively. In the first method, cut sets and representation theorem of intuitionistic fuzzy sets are utilized through triple valued fuzzy implications as well as fuzzy relations to obtain seven intuitionistic fuzzy implication operators. In the second method, the Lebesgue measure of the fuzzy implications with value one and zero is used, and one intuitionistic fuzzy implication operator can be obtained. Finally, properties of those operators are discussed.

Keywords

Intuitionistic fuzzy implication operators Lebesgue measure Cut sets Fuzzy relation Representation theorem 

Notes

Acknowledgments

This work was supported by the National Science Foundation of China (No. 61473327).

Compliance with ethical standards

Conflict of interest

Yiying Shi, Xuehai Yuan and Yuchun Zhang declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

References

  1. Alcalde C, Burusco A, González RF (2005) A constructive method for the definition of interval-valued fuzzy implication operators. Fuzzy Sets Syst 153(2):211–227MathSciNetCrossRefMATHGoogle Scholar
  2. Atanassov K, Gargov G (1998) Elements of intuitionistic fuzzy logic. Part I. Fuzzy Sets Syst 95(1):39–52CrossRefMATHGoogle Scholar
  3. Baczyński M, Jayaram B (2008) (S, N)- and R-implications: a state-of-the-art survey. Fuzzy Sets Syst 159(14):1836–1859MathSciNetCrossRefMATHGoogle Scholar
  4. Baczyński M, Jayaram B (2010) QL-implications: some properties and intersections. Fuzzy Sets Syst 161(2):158–188MathSciNetCrossRefMATHGoogle Scholar
  5. Baczyński M (2014) Distributivity of implication operations over t-representable t-norms in interval-valued fuzzy set theory: The case of nilpotent t-norms. Inf Sci 257(1):388–399MathSciNetCrossRefMATHGoogle Scholar
  6. Bedregal BC, Dimuro GP, Santiago RHN (2010) On interval fuzzy S-implications. Inf Sci 180(8):1373–1389MathSciNetCrossRefMATHGoogle Scholar
  7. Bedregal BC (2010) On interval fuzzy negations. Fuzzy Sets Syst 161(17):2290–2313MathSciNetCrossRefMATHGoogle Scholar
  8. Bustince H, Burillo P (1996) Structures on intuitionistic fuzzy relations. Fuzzy Sets Syst 78(3):293–303MathSciNetCrossRefMATHGoogle Scholar
  9. Bustince H (2000) Construction of intuitionistic fuzzy relations with predetermined properties. Fuzzy Sets Syst 109(3):379–403MathSciNetCrossRefMATHGoogle Scholar
  10. Bustince H, Barrenechea E, Mohedano V (2004) Intuitionistic fuzzy implication operators-an expression and main properties. Int J Unc Fuzz Knowl Based Syst 12(3):387–406MathSciNetCrossRefMATHGoogle Scholar
  11. Carbonell M, Torrens J (2010) Continuous R-implications generated from representable aggregation functions. Fuzzy Sets Syst 161(17):2276–2289MathSciNetCrossRefMATHGoogle Scholar
  12. Cintula P, Metcalfe G (2010) Admissible rules in the implication-negation fragment of intuitionistic logic. Ann Pure Appl Logic 162(2):162–171MathSciNetCrossRefMATHGoogle Scholar
  13. Cornelis C, Deschrijver G, Kerre EE (2004) Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int J Approx Reason 35(1):55–95MathSciNetCrossRefMATHGoogle Scholar
  14. Deschrijver G, Kerre EE (2005) Implicators based on binary aggregation operators in interval-valued fuzzy set theory. Fuzzy Sets Syst 153(2):229–248MathSciNetCrossRefMATHGoogle Scholar
  15. Dubois D, Prade H (1991) Fuzzy sets in approximate reasoning, Part 1: Inference with possibility distributions. Fuzzy Sets Syst 40(1):143–202CrossRefMATHGoogle Scholar
  16. Dubois D, Hüllermeier E, Prade H (2003) On the representation of fuzzy rules in terms of crisp rules. Inf Sci 151(5):301–326MathSciNetCrossRefMATHGoogle Scholar
  17. Fodor JC (1991) On fuzzy implication operators. Fuzzy Sets Syst 42(3):293–300MathSciNetCrossRefMATHGoogle Scholar
  18. Gorzałczany MB (1989) An interval-valued fuzzy inference method—some basic properties. Fuzzy Sets Syst 31(2):243–251MathSciNetCrossRefGoogle Scholar
  19. Kóczy L, Hirota K (1993) Interpolative reasoning with insufficient evidence in sparse fuzzy rule bases. Inf Sci 71(1–2):169–201MathSciNetCrossRefMATHGoogle Scholar
  20. Li DC, Li YM, Xie YJ (2011) Robustness of interval-valued fuzzy inference. Inf Sci 181(20):4754–4764MathSciNetCrossRefMATHGoogle Scholar
  21. Li DC, Li YM (2012) Algebraic structures of interval-valued fuzzy (S, N) (S, N)-implications. Int J Approx Reason 53(6):892–900MathSciNetCrossRefMATHGoogle Scholar
  22. Luo CZ, Wang PZ (1990) Representation of compositional relations in fuzzy reasoning. Fuzzy Sets Syst 36(1):77–81MathSciNetCrossRefMATHGoogle Scholar
  23. Mas M, Monserrat M, Torrens J, Trillas E (2007) A Survey on fuzzy implication functions. IEEE Trans Fuzzy Syst 15(6):1107–1121CrossRefGoogle Scholar
  24. Pei D (2012) Formalization of implication based fuzzy reasoning method. Int J Approx Reason 53(5):837–846MathSciNetCrossRefMATHGoogle Scholar
  25. Shi YY, Yuan XH (2015) A possibility-based method for ranking fuzzy numbers and applications to decision making. J Intell Fuzzy Syst 29:337–349MathSciNetCrossRefMATHGoogle Scholar
  26. Wang G (2000) Triple I method and interval valued fuzzy reasoning. Sci China Ser E 43(3):242–253MathSciNetCrossRefMATHGoogle Scholar
  27. Wang Z, Xu ZS, Liu SS (2014) Direct clustering analysis based on intuitionistic fuzzy implication. Appl Soft Comput 23(5):1–8Google Scholar
  28. Wu WZ, Leung Y, Shao MW (2013) Generalized fuzzy rough approximation operators determined by fuzzy implicators. Int J Approx Reason 54(9):1388–1409MathSciNetCrossRefMATHGoogle Scholar
  29. Xu Y, Kerre EE, Ruan D, Song Z (2001) Fuzzy reasoning based on the extension principle. Int J Intell Syst 16(4):469–495CrossRefMATHGoogle Scholar
  30. Yao O (2012) On fuzzy implications determined by aggregation operators. Inf Sci 193(11):153–162MathSciNetMATHGoogle Scholar
  31. Yu XH, Xu ZS (2013) Prioritized intuitionistic fuzzy aggregation operators. Inf Fusion 14(1):108–116CrossRefGoogle Scholar
  32. Yuan XH, Li HX, Sun KB (2011) The cut sets, decomposition theorems and representation theorems on intuitionistic fuzzy sets and interval valued fuzzy sets. Sci China Inf Sci 54:91–110MathSciNetCrossRefMATHGoogle Scholar
  33. Zadeh LA (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern 3(1):28–44MathSciNetCrossRefMATHGoogle Scholar
  34. Zheng M, Shi Z, Liu Y (2014) Triple I method of approximate reasoning on Atanassov’s intuitionistic fuzzy sets. Int J Approx Reason 55(6):1369–1382MathSciNetCrossRefMATHGoogle Scholar
  35. Zhou L, Wu WZ, Zhang WX (2009) On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf Sci 179(7):883–898MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of ScienceShenyang Ligong UniversityShenyangChina
  2. 2.School of ScienceDalian University of Technology at PanjinPanjinChina

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