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Generalized Interval-Valued Fuzzy Rough Set and its Application in Decision Making

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Abstract

This paper presents a general study of generalized interval-valued fuzzy rough sets integrating the rough set theory with the interval-valued fuzzy set theory by constructive and axiomatic approaches. In the constructive approach, by employing an interval-valued fuzzy residual implicator and its dual operator, generalized upper and lower interval-valued fuzzy rough approximation operators with respect to an arbitrary interval-valued fuzzy approximation space are first defined. Then properties of generalized interval-valued fuzzy rough approximation operators are discussed. Furthermore, connections between special types of interval-valued fuzzy relations and properties of generalized interval-valued fuzzy approximation operator are also established. In the axiomatic approach, generalized interval-valued fuzzy rough approximation operators are defined by axioms. We prove that different axiom sets can characterize the essential properties of generalized interval-valued fuzzy rough approximation operators. Also the composition of two approximation spaces is explored. Finally, a practical application is provided to illustrate the efficiency of the generalized interval-valued fuzzy rough set model.

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References

  1. Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ali, M.I., Davvaz, B., Shabir, M.: Some properties of generalized rough sets. Inf. Sci. 224, 170–179 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bustince, H.: Indicator of inclusion grade for interval-valued fuzzy sets, application to approximate reasoning based on interval-valued fuzzy sets. Int. J. Approx. Reason. 23, 137–209 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bustince, H., Burillo, P.: Mathematical analysis of interval-valued fuzzy relations: application to approximate reasoning. Fuzzy Sets Syst. 113, 205–219 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cornelis, C., Deschrijver, G., Kerre, E.E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: constructive, classification, application. Int. J. Approx. Reason. 35, 55–95 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cornelis, C., Cock, M.D., Kerre, E.E.: Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge. Expert Syst. Appl. 20, 260–270 (2003)

    Article  Google Scholar 

  7. Chakrabarty, K., Gedeon, T., Koczy, L.: Intuitionistic fuzzy rough set. In: Proceedings of Fourth Joint Conference on Information Sciences, pp. 211–214, Durham, 1998

  8. Dubois, D., Prade, H.: Rough fuzzy sets and fuzzy rough sets. Int. J. Gen Syst 17, 191–209 (1990)

    Article  MATH  Google Scholar 

  9. Gorzalczany, M.B.: A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 21, 1–17 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gong, Z.T., Sun, B.Z., Chen, D.G.: Rough set theory for the interval-valued fuzzy information systems. Inf. Sci. 178, 1968–1985 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. He, Y.P., Zhang, H.D.: The residual implication of interval-valued fuzzy triangle norm and its properties. Adv. Mater. Res. 282–283, 291–294 (2011)

    Article  Google Scholar 

  12. Liu, G.L.: Using one axiom to characterize rough set and fuzzy rough set approximations. Inf. Sci. 223, 285–296 (2013)

    Article  MATH  Google Scholar 

  13. Jena, S.P., Ghosh, S.K.: Intuitionistic fuzzy rough sets. Notes Intuit. Fuzzy Sets 8, 1–18 (2002)

    MATH  MathSciNet  Google Scholar 

  14. Li, T.J., Zhang, W.X.: Rough fuzzy approximations on two universes of discourse. Inf. Sci. 178, 892–906 (2008)

    Article  MATH  Google Scholar 

  15. Lin, T.Y.: A rough logic formalism for fuzzy controllers: A hard and soft computing view. Int. J. Approx. Reason. 15, 359–414 (1996)

    Google Scholar 

  16. Mi, J.S., Zhang, W.X.: An axiomatic characterization of a fuzzy generalized of rough sets. Inf. Sci. 160, 235–249 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Morsi, N.N., Yakout, M.M.: Axiomatics for fuzzy rough sets. Fuzzy Sets Syst. 100, 327–342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Nanda, S., Majumda, S.: Fuzzy rough sets. Fuzzy Sets Syst. 45, 157–160 (1992)

    Article  MATH  Google Scholar 

  19. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 145–172 (1982)

    Article  MathSciNet  Google Scholar 

  20. Pawlak, Z.: Rough Sets-Theoretical Aspects to Reasoning About Data. Kluwer Academic Publisher, Boston (1991)

    Google Scholar 

  21. Radzikowska, A.M., Kerre, E.E.: A comparative study of fuzzy rough sets. Fuzzy Sets Syst. 126, 137–155 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rizvi, S., Naqvi, H.J., Nadeem, D.: Rough intuitionistic fuzzy set. In: Proceedings of the Sixth Joint Conference on Information Sciences, pp. 101–104, Durham, 2002.

  23. Sun, B.Z., Gong, Z.T., Chen, D.G.: Fuzzy rough set theory for the interval-valued fuzzy information systems. Inf. Sci. 178, 2794–2815 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sun, B.Z., Ma, W.M., Liu, Q.: An approach to decision making based on intuitionistic fuzzy rough sets over two universes. J. Oper. Res. Soc. 64, 1079–1089 (2013)

    Article  Google Scholar 

  25. Sun, B.Z., Ma, W.M.: Soft fuzzy rough sets and its application in decision making. Artif. Intell. Rev. 41, 67–80 (2014)

    Article  Google Scholar 

  26. Sambuc, R.: Functions \(\phi\)-Flous Application a l’aide au Diagnostic en Pathologie Thyroidienne, These de Doctorat en Merseille, 1975

  27. Samanta, S.K., Mondal, T.K.: Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. J. Fuzzy Math. 9, 561–582 (2001)

    MATH  MathSciNet  Google Scholar 

  28. Thiele, H.: On axiomatic characterization of crisp approximation operators. Inf. Sci. 129, 221–226 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Thiele, H.: On axiomatic characterization of fuzzy approximation operators: I. The fuzzy rough set based case. In: RSCTC 2000, Lecture Notes in Computer Science, vol. 205, pp. 239–247. Springer, Berlin (2001).

  30. Thiele, H.: On axiomatic characterization of fuzzy approximation operators: II. The rough fuzzy set based case. In: Proceeding of the 31st IEEE International Symposium on Multiple-Valued Logic, pp. 330–335, 2001

  31. Thiele, H.: On axiomatic characterization of fuzzy approximation operators: III. The fuzzy diamond and fuzzy box case. In: Proceeding of the 10st IEEE International Conference on Fuzzy Systems, vol. 2, pp. 1148–1151, 2001

  32. Turksen, L.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 80, 191–210 (1986)

    Article  MathSciNet  Google Scholar 

  33. Turksen, L.B., Zhong, Z.: An approximate analogical reasoning schema based on similarity measures and interval-valued fuzzy sets. Fuzzy Sets Syst. 34, 323–346 (1990)

    Article  Google Scholar 

  34. Tiwari, S.P., Srivastava, A.K.: Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst. 210, 63–68 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wu, W.Z., Leung, Y., Zhang, W.X.: On generalized rough fuzzy approximation operators. Transactions on Rough sets V, Lecture Notes in Computer Science, vol. 4100, pp. 263–284, 2006

  36. Wu, W.Z., Leung, Y., Mi, J.S.: On characterizations of (\((\mathcal{I},\mathcal{T})-\)) fuzzy rough approximation operators. Fuzzy Sets Syst. 154, 76–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wu, W.Z., Leung, Y., Zhang, W.X.: Connections between rough-set theory and Dempster-Shafer theory of evidence. Int. J. Gen. Syst. 31, 405–430 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wu, W.Z., Mi, J.S., Zhang, W.X.: Generalized fuzzy rough sets. Inf. Sci. 151, 263–282 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wu, W.Z., Zhang, W.X.: Constructive and axiomatic approaches of fuzzy approximation operators. Inf. Sci. 159, 233–254 (2004)

    Article  MATH  Google Scholar 

  40. Wu, W.Z., Zhang, W.X.: Neighborhood operator systems and approximations. Inf. Sci. 144, 201–217 (2002)

    Article  MATH  Google Scholar 

  41. Yeung, D.S., Chen, D.G., Tsang, E.C.C., Lee, J.W.T., Wang, X.Z.: On the generalization of fuzzy rough sets. IEEE Trans. Fuzzy Syst. 13, 343–361 (2005)

    Article  Google Scholar 

  42. Yang, X.B., Song, X.N., Qi, Y.S., Yang, J.Y.: Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft. Comput. 18, 1067–1077 (2014)

    Article  Google Scholar 

  43. Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Inf. Sci. 109, 21–47 (1998)

    Article  MATH  Google Scholar 

  44. Yao, Y.Y.: Two views of the theory of rough sets on finite universes. Int. J. Approx. Reason. 15, 291–317 (1996)

    Article  MATH  Google Scholar 

  45. Yao, Y.Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Inf. Sci. 111, 239–259 (1998)

    Article  MATH  Google Scholar 

  46. Xu, Z.S., Da, Q.L.: The uncertain OWA operator. Int. J. Intell. Syst. 17, 569–575 (2002)

    Article  MATH  Google Scholar 

  47. Zhang, H.D., Shu, L.: S.L. Liao.: Intuitionistic fuzzy soft rough set and its application in decision making. Abstr. Appl. Anal. 2014, 13 (2014). (Article ID 287314)

    MATH  MathSciNet  Google Scholar 

  48. Zhang, H.D., Shu, L., Liao, S.L.: On interval-valued hesitant fuzzy rough approximation operators. Soft. Comput. (2014). doi:10.1007/s00500-014-1490-7

    Google Scholar 

  49. Zhang, H.Y., Zhang, W.X., Wu, W.-Z.: On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse. Int. J. Approx. Reason. 51, 56–70 (2009)

    Article  MATH  Google Scholar 

  50. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  51. Zadeh, L.A.: The concepts of linguistic variable and its application to approximate reasoning, part I. Inf. Sci. 8, 199–249 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  52. Zhou, L., Wu, W.Z.: On generalized intuitionistic fuzzy approximation operators. Inf. Sci. 178, 2448–2465 (2008)

    MATH  MathSciNet  Google Scholar 

  53. Zhou, L., Wu, W.Z.: On characterization of intuitonistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf. Sci. 179, 883–898 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  54. Zhu, W., Wang, F.Y.: On three types of covering rough sets. IEEE Trans. Knowl. Data Eng. 19, 1131–1144 (2007)

    Article  Google Scholar 

  55. Zhang, X.H., Zhou, B., Li, P.: A general frame for intuitionistic fuzzy rough sets. Inf. Sci. 216, 34–49 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  56. Zhang, Z.M.: Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings. Inf. Sci. 198, 186–206 (2012)

    Article  MATH  Google Scholar 

  57. Zhang, Z.M.: On characterization of generalized interval type-2 fuzzy rough sets. Inf. Sci. 219, 124–150 (2013)

    Article  MATH  Google Scholar 

  58. Zhang, Z.M.: On interval type-2 rough fuzzy sets. Knowl. Based Syst. 35, 1–13 (2012)

    Article  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, People’s Republic of China

    Haidong Zhang & Lan Shu

  2. School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730030, Gansu, People’s Republic of China

    Haidong Zhang

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  1. Haidong Zhang
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  2. Lan Shu
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Correspondence to Haidong Zhang.

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Zhang, H., Shu, L. Generalized Interval-Valued Fuzzy Rough Set and its Application in Decision Making. Int. J. Fuzzy Syst. 17, 279–291 (2015). https://doi.org/10.1007/s40815-015-0012-9

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  • DOI: https://doi.org/10.1007/s40815-015-0012-9

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