Soft Computing

, Volume 21, Issue 7, pp 1803–1816 | Cite as

Hesitant fuzzy rough set over two universes and its application in decision making

Methodologies and Application


In this paper, we propose a general decision-making framework based on the HF rough set model over two universes. By a constructive approach, the HF rough set model over two universes is first presented and some properties of this model are further discussed. The union, the intersection and the composition of hesitant fuzzy approximation spaces are proposed, and some properties are also investigated. We then give a new approach of decision making in uncertainty environment using the hesitant fuzzy rough sets over two universes. Finally, two practical applications are provided to illustrate the validity of this approach.


Hesitant fuzzy rough set Two universes Hesitant fuzzy approximation spaces Decision making 


  1. Chakrabarty K, Gedeon T, Koczy L (1998) Intuitionistic fuzzy rough set. In: Proceedings of fourth joint conference on information sciences (JCIS), Durham, pp 211–214Google Scholar
  2. Chen N, Xu ZS, Xia MM (2013) Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl Math Model 37:2197–2211MathSciNetCrossRefMATHGoogle Scholar
  3. Cornelis C, Cock MD, Kerre EE (2003) Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge. Exp. Syst. Appl. 20:260–270CrossRefGoogle Scholar
  4. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefMATHGoogle Scholar
  5. Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144MathSciNetCrossRefMATHGoogle Scholar
  6. Gong ZT, Sun BZ (2008) Probability rough sets model between different universes and its applications. In: International conference on machine learning and cybernetics, pp 561–565Google Scholar
  7. Jena SP, Ghosh SK (2002) Intuitionistic fuzzy rough sets. Notes Intuit Fuzzy Sets 8:1–18MathSciNetMATHGoogle Scholar
  8. Liao HC, Xu ZS (2013) A VIKOR-based method for hesitant fuzzy multi-criteria decision making. Fuzzy Optim Dec Mak 12:373–392Google Scholar
  9. Liu GL (2010) Rough set theory based on two universal sets and its applications. Knowl Based Syst 23(2):110–115CrossRefGoogle Scholar
  10. Li TJ, Zhang WX (2008) Rough fuzzy approximations on two universes of discourse. Inf Sci 178:892–906MathSciNetCrossRefMATHGoogle Scholar
  11. Ma WM, Sun BZ (2012) Probabilistic rough set over two universes and rough entropy. Int J Approx Reas 53:608–619MathSciNetCrossRefMATHGoogle Scholar
  12. Nanda S, Majumda S (1992) Fuzzy rough sets. Fuzzy Sets Syst 45:157–160MathSciNetCrossRefGoogle Scholar
  13. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:145–172CrossRefMATHGoogle Scholar
  14. Pawlak Z (1991) Rough sets-theoretical aspects to reasoning about data. Kluwer Academic Publisher, BostonMATHGoogle Scholar
  15. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155MathSciNetCrossRefMATHGoogle Scholar
  16. Rizvi S, Naqvi HJ, Nadeem D (2002) Rough intuitionistic fuzzy set. In: Proceedings of the sixth joint conference on information sciences (JCIS), Durham, pp 101–104Google Scholar
  17. Rodrguez RM, Martnez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 1:109–119CrossRefGoogle Scholar
  18. Samanta SK, Mondal TK (2001) Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. J Fuzzy Math 9:561–582MathSciNetMATHGoogle Scholar
  19. Shen YH, Wand FX (2011) Variable precision rough set model over two universes and its properties. Soft Comput 15(3):557–567CrossRefMATHGoogle Scholar
  20. Sun BZ, Gong ZT, Chen DG (2008) Fuzzy rough set theory for the interval-valued fuzzy information systems. Inf Sci 178:2794–2815MathSciNetCrossRefMATHGoogle Scholar
  21. Sun BZ, Ma WM, Liu Q (2013) An approach to decision making based on intuitionistic fuzzy rough sets over two universes. J Oper Res Soc 64:1079–1089CrossRefGoogle Scholar
  22. Sun BZ, Ma WM (2011) Fuzzy rough set model on two different universes and its application. Appl Math Model 35:1798–1809MathSciNetCrossRefMATHGoogle Scholar
  23. Thiele H (2001) On aximatic characterisation of fuzzy approximation operators: II. The rough fuzzy set based case. In: Proceeding of the 31st IEEE international symposium on multiple-valued logic, pp 330–335Google Scholar
  24. Tiwari SP, Arun K (2013) Srivastava, fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst 210:63–68Google Scholar
  25. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539MATHGoogle Scholar
  26. Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems. Jeju Island, Korea, pp 1378–1382Google Scholar
  27. Wu WZ, Leung Y, Mi JS (2005) On characterizations of (I, T)-fuzzy rough approximation operators. Fuzzy Sets Syst 154:76–102Google Scholar
  28. Wu WZ, Mi JS, Zhang WX (2003) Generalized fuzzy rough sets. Inf Sci 151:263–282MathSciNetCrossRefMATHGoogle Scholar
  29. Wu WZ, Leung Y, Zhang WX (2006) On generalized rough fuzzy approximation operators. Trans Rough sets V Lect Notes Comput Sci 4100:263–284MathSciNetCrossRefMATHGoogle Scholar
  30. Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159:233–254MathSciNetCrossRefMATHGoogle Scholar
  31. Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reas 52:395–407MathSciNetCrossRefMATHGoogle Scholar
  32. Xu ZS, Xia MM (2011) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138MathSciNetCrossRefMATHGoogle Scholar
  33. Xu ZS, Xia MM (2011) On distance and correlation measures of hesitant fuzzy information. Int J Intell Syst 26:410–425CrossRefMATHGoogle Scholar
  34. Xu ZS, Zhang X (2013) Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl Based Syst 52:53–64CrossRefGoogle Scholar
  35. Yan RX, Zheng JG, Liu JL, Zhai YM (2010) Research on the model of rough set over dual-universes. Knowl-Based Syst 23(8):817–822CrossRefGoogle Scholar
  36. Yang HL, Li SG, Wang SY, Wang J (2012) Bipolar fuzzy rough set model on two different universes and its application. Knowl-Based Syst 35:94–101CrossRefGoogle Scholar
  37. Yang HL, Li SG, Guo ZL, Ma CH (2012) Transformation of bipolar fuzzy rough set models. Knowl-Based Syst 27:60–68CrossRefGoogle Scholar
  38. Yang XB, Song XN, Qi YS, Yang JY (2014) Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput 18:1067–1077CrossRefMATHGoogle Scholar
  39. Yao YY (1998) Generalized rough set model. In: Polkowski L, Skowron A (eds) Rough sets in knowledge discovery. 1. In: Methodology and applications. Physica-Verlag, Berlin, pp 286-318Google Scholar
  40. Yeung DS, Chen DG, Tsang ECC, Lee JWT, Wang XZ (2005) On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst 13:343–361CrossRefGoogle Scholar
  41. Zadeh LA (1965) Fuzzy sets. Inf Control 8:352–378CrossRefMATHGoogle Scholar
  42. Zhang ZM (2012a) Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings. Inf Sci 198:186–206Google Scholar
  43. Zhang ZM (2012b) On interval type-2 rough fuzzy sets. Knowl Based Syst 35:1–13Google Scholar
  44. Zhang ZM (2013) On characterization of generalized interval type-2 fuzzy rough sets. Inf Sci 219:124–150MathSciNetCrossRefMATHGoogle Scholar
  45. Zhang N, Wei G (2013) Extension of VIKOR method for decision making problem based on hesitant fuzzy set. Appl Math Model 37(7):4938–4947MathSciNetCrossRefGoogle Scholar
  46. Zhang HY, Zhang WX, Wu WZ (2009) On characterization of genernalized interval-valued fuzzy rough sets on two universes of discourse. Int J Approx Reas 51:56–70CrossRefMATHGoogle Scholar
  47. Zhang XH, Zhou B, Li P (2012a) A general frame for intuitionistic fuzzy rough sets. Inf Sci 216:34–49MathSciNetCrossRefMATHGoogle Scholar
  48. Zhang HD, Shu L, Liao SL (2014) On interval-valued hesitant fuzzy rough approximation operators. Soft Comput. doi:10.1007/s00500-014-1490-7
  49. Zhang HD, Shu L, Liao SL (2015) On a novel hesitant fuzzy rough set. Inf Sci (in revision) Google Scholar
  50. Zhou L, Wu WZ (2008) On genernalized intuitionistic fuzzy approximation operators. Inf Sci 178:2448–2465MATHGoogle Scholar
  51. Zhou L, Wu WZ (2009) On characterization of intuitonistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf Sci 179:883–898CrossRefMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China
  2. 2.School of Mathematics and Computer ScienceNorthwest University for NationalitiesLanzhouPeople’s Republic of China

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