Skip to main content
SpringerLink
Log in

On interval-valued hesitant fuzzy rough approximation operators

  • Methodologies and Application
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Interval-valued hesitant fuzzy set is a generalization of classical interval-valued fuzzy set by returning a family of the interval-valued membership degrees for each object in the universe. By combining interval-valued hesitant fuzzy set and rough set models, the concept of an interval-valued hesitant fuzzy rough set is explored in this paper. Both constructive and axiomatic approaches are considered for this study. In constructive approach, by employing an interval-valued hesitant fuzzy relation, a pair of lower and upper interval-valued hesitant fuzzy rough approximation operators is first defined. The connections between special interval-valued hesitant fuzzy relations and interval-valued hesitant fuzzy rough approximation operators are further established. In axiomatic approach, an operators-oriented characterization of the interval-valued hesitant fuzzy rough set is presented, that is, interval-valued hesitant fuzzy rough approximation operators are defined by axioms, and then, different axiom sets of lower and upper interval-valued hesitant fuzzy set-theoretic operators guarantee the existence of different types of interval-valued hesitant fuzzy relations producing the same operators. Finally, a practical application is provided to illustrate the validity of the interval-valued hesitant fuzzy rough set model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Babitha KV, John SJ (2013) Hesitant fuzzy soft sets. J New Results Sci 3:98–107

    Google Scholar 

  • Chakrabarty K, Gedeon T, Koczy L (1998) Intuitionistic fuzzy rough set. In: Proceedings of fourth joint conference on information sciences (JCIS), Durham, NC, pp 211–214

  • Chen N, Xu ZS, Xia MM (2013a) Interval-valued hesitant preference relations and their applications to group decision making. Knowl-Based Syst 37:528–540

  • Chen N, Xu ZS, Xia MM (2013b) Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl Math Model 37:2197–2211

  • Chuchro M (1993) A certain conception of rough sets in topological boolean algebras. Bull Sect Logic 22(1):9–12

    MATH  Google Scholar 

  • Chuchro M (1994) On rough sets in topological Boolean algebras. In: Ziarko W (ed) Rough sets, fuzzy sets and knowledge discovery. Springer, Berlin, pp 157–160

    Chapter  Google Scholar 

  • Comer S (1991) An algebraic approach to the approximation of information. Fundam Inform 14:492–502

    MathSciNet  MATH  Google Scholar 

  • Comer S (1993) On connections between information systems, rough sets, and algebraic logic. In: Rauszer C (ed), Algebraic methods in logic and computer science, vol 28. Banach Center Publisher, Polish Academy of Sciences, pp 117–127

  • Cornelis C, Cock MD, Kerre EE (2003) Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge. Expert Syst Appl 20:260–270

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144

    Article  MathSciNet  Google Scholar 

  • Jena SP, Ghosh SK (2002) Intuitionistic fuzzy rough sets. Notes Intuit Fuzzy Sets 8:1–18

    MathSciNet  MATH  Google Scholar 

  • Kortelainen J (1994) On relationship between modified sets, topological space and rough sets. Fuzzy Sets Syst 61:91–95

    Article  MathSciNet  MATH  Google Scholar 

  • Li TJ, Zhang WX (2008) Rough fuzzy approximations on two universes of discourse. Inf Sci 178:892–906

    Article  MATH  Google Scholar 

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

    Google Scholar 

  • Liu GL (2013) Using one axiom to characterize rough set and fuzzy rough set approximations. Inf Sci 223:285–296

    Article  MATH  Google Scholar 

  • Mi JS, Zhang WX (2004) An axiomatic characterization of a fuzzy generalized of rough sets. Inf Sci 160:235–249

    Article  MathSciNet  MATH  Google Scholar 

  • Morsi NN, Yakout MM (1998) Axiomatics for fuzzy rough sets. Fuzzy Sets Syst 100:327–342

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Pawlak Z (1991) Rough sets—theoretical aspects to reasoning about data. Kluwer, Boston

    Google Scholar 

  • Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155

    Article  MathSciNet  MATH  Google Scholar 

  • Rizvi S, Naqvi HJ, Nadeem D (2002) Rough intuitionistic fuzzy set. In: Proceedings of the sixth joint conference on information sciences (JCIS), Durham, NC, pp 101–104

  • Rodrguez RM, Martnez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 1:109–119

    Article  Google Scholar 

  • Samanta SK, Mondal TK (2001) Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. J Fuzzy Math 9:561–582

    MathSciNet  MATH  Google Scholar 

  • Skardowska UW (1989) On a generalization of approximation space. Bull Pol Acad Sci Math 37:51–61

    MATH  Google Scholar 

  • Sun BZ, Gong ZT, Chen DG (2008) Fuzzy rough set theory for the interval-valued fuzzy information systems. Inf Sci 178:2794–2815

    Article  MathSciNet  MATH  Google Scholar 

  • Thiele H (2000) On axiomatic characterisation of crisp approximation operators. Inf Sci 129:221–226

    Article  MathSciNet  MATH  Google Scholar 

  • Thiele H (2001a) On axiomatic characterisation of fuzzy approximation operators: I. The fuzzy rough set based case. In: RSCTC 2000, lecture notes in computer science, vol 205, Springer, Berlin, pp 239–247

  • Thiele H (2001b) On axiomatic characterisation of fuzzy approximation operators: II. The rough fuzzy set based case. In: Proceedings of the 31st IEEE international symposium on multiple-valued logic, pp 330–335

  • Thiele H (2001c) On axiomatic characterisation of fuzzy approximation operators: III. The fuzzy diamond and fuzzy box case. In: Proceedings of the 10th IEEE international conference on fuzzy systems, vol 2, pp 1148–1151

  • Tiwari SP, Srivastava AK (2013) Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst 210:63–68

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539

    MATH  Google Scholar 

  • Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems, Jeju Island, Korea, pp 1378–1382

  • Vakarelov D (1991) A modal logic for similarity relations in Pawlak knowledge representation systems. Fundam Inform 15:61C79.

  • Wei GW, Zhao XF, Lin R (2013) Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making. Knowl-Based Syst 46:43–53

    Article  Google Scholar 

  • Wu WZ, Zhang WX (2002) Neighborhood operator systems and approximations. Inf Sci 144:201–217

    Article  MATH  Google Scholar 

  • Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159:233–254

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Wu WZ, Mi JS, Zhang WX (2003) Generalized fuzzy rough sets. Inf Sci 151:263–282

    Article  MathSciNet  MATH  Google Scholar 

  • Wu WZ, Leung Y, Mi JS (2005) On characterizations of \(({\cal I\mathit{},{\cal T}})\)-fuzzy rough approximation operators. Fuzzy Sets Syst 154:76–102

    Article  MathSciNet  MATH  Google Scholar 

  • Wu WZ, Leung Y, Zhang WX (2006) On generalized rough fuzzy approximation operators. Transactions on Rough sets V, Lecture notes in computer science, vol 4100, pp 263–284

  • Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407

    Article  MathSciNet  MATH  Google Scholar 

  • Xu ZS, Da QL (2002) The uncertain OWA operator. Int J Intell Syst 17:569–575

  • Xu ZS, Xia MM (2011) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138

    Article  MathSciNet  MATH  Google Scholar 

  • Yang XP (2007) Minimization of axiom sets on fuzzy approximation operators. Inf Sci 177:3840–3854

    Article  MATH  Google Scholar 

  • Yang XP, Li TJ (2006) The minimization of axiom sets characterizing generalized approximation operators. Inf Sci 176:887–899

    Article  MATH  Google Scholar 

  • Yang XB, Song XN, Qi YS, Yang JY (2014) Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput 18:1067–1077

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Yao YY (1998a) Constructive and algebraic methods of the theory of rough sets. Inf Sci 109:21–47

  • Yao YY (1998b) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111:239–259

  • Yao YY (1998c) Generalized rough set model. In: Polkowski L, Skowron A (eds) Rough sets in knowledge discovery 1. Methodology and applications. Physica-Verlag, Heidelberg, pp 286–318

  • Yao YY, Lin TY (1996) Generalization of rough sets using modal logic. Intell Autom Soft Comput Int J 2:103–120

    Article  Google Scholar 

  • Yeung DS, Chen DG, Tsang ECC, Lee JWT, Wang XZ (2005) On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst 13:343–361

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang HY, Zhang WX, Wu WZ (2009) On characterization of generalized interval-valued fuzzy rough sets on two universes of discourse. Int J Approx Reason 51:56–70

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang ZM (2012a) Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings. Inf Sci 198:186–206

  • Zhang ZM (2012b) On interval type-2 rough fuzzy sets. Knowl-Based Syst 35:1–13

  • Zhang ZM (2013) On characterization of generalized interval type-2 fuzzy rough sets. Inf Sci 219:124–150

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 71261022) and the Fundamental Research Funds for the Central Universities of Northwest University for Nationalities (No. zyz2012076).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, 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 & Shilong Liao

Authors
  1. Haidong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lan Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shilong Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haidong Zhang.

Additional information

Communicated by F. Marcelloni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Shu, L. & Liao, S. On interval-valued hesitant fuzzy rough approximation operators . Soft Comput 20, 189–209 (2016). https://doi.org/10.1007/s00500-014-1490-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-014-1490-7

Keywords

Search

Navigation