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.
Similar content being viewed by others
References
Babitha KV, John SJ (2013) Hesitant fuzzy soft sets. J New Results Sci 3:98–107
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
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
Comer S (1991) An algebraic approach to the approximation of information. Fundam Inform 14:492–502
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
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
Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209
Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144
Jena SP, Ghosh SK (2002) Intuitionistic fuzzy rough sets. Notes Intuit Fuzzy Sets 8:1–18
Kortelainen J (1994) On relationship between modified sets, topological space and rough sets. Fuzzy Sets Syst 61:91–95
Li TJ, Zhang WX (2008) Rough fuzzy approximations on two universes of discourse. Inf Sci 178:892–906
Lin TY (1996) A rough logic formalism for fuzzy controllers: a hard and soft computing view. Int J Approx Reason 15:359–414
Liu GL (2013) Using one axiom to characterize rough set and fuzzy rough set approximations. Inf Sci 223:285–296
Mi JS, Zhang WX (2004) An axiomatic characterization of a fuzzy generalized of rough sets. Inf Sci 160:235–249
Morsi NN, Yakout MM (1998) Axiomatics for fuzzy rough sets. Fuzzy Sets Syst 100:327–342
Nanda S, Majumda S (1992) Fuzzy rough sets. Fuzzy Sets Syst 45:157–160
Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:145–172
Pawlak Z (1991) Rough sets—theoretical aspects to reasoning about data. Kluwer, Boston
Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155
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
Samanta SK, Mondal TK (2001) Intuitionistic fuzzy rough sets and rough intuitionistic fuzzy sets. J Fuzzy Math 9:561–582
Skardowska UW (1989) On a generalization of approximation space. Bull Pol Acad Sci Math 37:51–61
Sun BZ, Gong ZT, Chen DG (2008) Fuzzy rough set theory for the interval-valued fuzzy information systems. Inf Sci 178:2794–2815
Thiele H (2000) On axiomatic characterisation of crisp approximation operators. Inf Sci 129:221–226
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
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
Wu WZ, Zhang WX (2002) Neighborhood operator systems and approximations. Inf Sci 144:201–217
Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159:233–254
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
Wu WZ, Mi JS, Zhang WX (2003) Generalized fuzzy rough sets. Inf Sci 151:263–282
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
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
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
Yang XP (2007) Minimization of axiom sets on fuzzy approximation operators. Inf Sci 177:3840–3854
Yang XP, Li TJ (2006) The minimization of axiom sets characterizing generalized approximation operators. Inf Sci 176:887–899
Yang XB, Song XN, Qi YS, Yang JY (2014) Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput 18:1067–1077
Yao YY (1996) Two views of the theory of rough sets on finite universes. Int J Approx Reason 15:291–317
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
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
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
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
Zhang XH, Zhou B, Li P (2012) A general frame for intuitionistic fuzzy rough sets. Inf Sci 216:34–49
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
Zhou L, Wu WZ (2008) On generalized intuitionistic fuzzy approximation operators. Inf Sci 178:2448–2465
Zhou L, Wu WZ (2009) On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf Sci 179:883–898
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
Corresponding author
Additional information
Communicated by F. Marcelloni.
Rights and permissions
About this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-014-1490-7