Skip to main content
SpringerLink
Log in

Novel classes of fuzzy soft \(\beta \)-coverings-based fuzzy rough sets with applications to multi-criteria fuzzy group decision making

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this article, we put forward the concepts of fuzzy soft \(\beta \)-minimal descriptions and fuzzy soft \(\beta \)-maximal descriptions and construct four types of fuzzy soft \(\beta \)-neighborhoods. Secondly, we define five kinds of fuzzy soft \(\beta \)-coverings-based fuzzy rough sets and investigate the relationships among them. Furthermore, we investigate under what conditions two different fuzzy soft \(\beta \)-coverings induce the same lower (upper) approximation operators. Then, we introduce the concepts of intersection and union reducible elements. Finally, we put forward the algorithms with respect to the first and the fifth types of fuzzy soft \(\beta \)-coverings-based fuzzy rough sets, respectively. Through comparison, we find that although these two models are different, the obtained results are the same and the complexity of the algorithm based on the fifth type model is easier than the one based on the first 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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Alcantud JCR (2002) Revealed indifference and models of choice behavior. J Math Psychol 46:418–430

    Article  MathSciNet  MATH  Google Scholar 

  • Alcantud JCR (2016a) A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set. Inform Fusion 29:142–148

    Article  Google Scholar 

  • Alcantud JCR (2016b) Some formal relationships among soft sets, fuzzy sets and their extensions. Int J Approx Reason 68:45–53

    Article  MathSciNet  MATH  Google Scholar 

  • Ali MI (2011) A note on soft sets, rough sets and fuzzy soft sets. Appl Soft Comput 11:3329–3332

    Article  Google Scholar 

  • Ali MI, Shabir M, Feng F (2017) Representation of graphs based on neighborhoods and soft sets. Int J Mach Learn Cybern 8:1525–1535

    Article  Google Scholar 

  • Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inform Sci 107:149–167

    Article  MathSciNet  MATH  Google Scholar 

  • Dai JH, Hu H, Wu WZ, Qian YH, Huang DB (2007a) Maximal discernibility pairs based approach to attribute reduction in fuzzy rough sets. IEEE Tran Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2017.2768044

  • Dai JH, Wei BJ, Zhang XH, Zhang QH (2017b) Uncertainty measurement for incomplete interval-valued information systems based on \(\alpha \)-weak similarity. Knowl Based Syst 136:159–171

    Article  Google Scholar 

  • D’eer L, Cornelis C (2018) A comprehensive study of fuzzy covering-based rough set models: definitions, properties and interrelationships. Fuzzy Sets Syst 336:1–26

    Article  MathSciNet  MATH  Google Scholar 

  • D’eer L, Cornelis C, Godo L (2017) Fuzzy neighborhood operators based on fuzzy coverings. Fuzzy Sets Syst 312:17–35

    Article  MathSciNet  MATH  Google Scholar 

  • D’eer L, Cornelis C, Yao Y (2016a) A semantically sound approach to Pawlak rough sets and covering based rough sets. Int J Approx Reason 78:62–72

    Article  MathSciNet  MATH  Google Scholar 

  • D’eer L, Restrepo M, Cornelis C, Jonatan G (2016b) Neighborhood operators for covering based rough sets. Inform Sci 336:21–44

    Article  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 

  • Feng F, Li C, Davvaz B, Ali MI (2010) Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14:899–911

    Article  MATH  Google Scholar 

  • Feng F, Liu X, Leoreanu-Fotea V, Jun YB (2011) Soft sets and soft rough sets. Inform Sci 181:1125–1137

    Article  MathSciNet  MATH  Google Scholar 

  • Feng T, Zhang SP, Mi JS (2012) The reduction and fusion of fuzzy covering systems based on the evidence theory. Int J Approx Reason 53:87–103

    Article  MathSciNet  MATH  Google Scholar 

  • Hao J, Li QG (2011) The relationship between \(L\)-fuzzy rough set and \(L\)-topology. Fuzzy Set Syst 178:74–83

    Article  MathSciNet  MATH  Google Scholar 

  • Hu BQ, Wong H (2014) Generalized interval-valued fuzzy variable precision rough sets. Int J Fuzzy Syst 16:554–565

    MathSciNet  Google Scholar 

  • Klir G, Yuan B (1995) Fuzzy sets and fuzzy logic, theory and applications. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  • Li TJ, Ma JM (2007) Fuzzy approximation operators based on coverings. In: Proceedings of joint rough set symposium, pp 55–62

  • Li TJ, Leung Y, Zhang WX (2008) Generalized fuzzy rough approximation operators based on fuzzy coverings. Int J Approx Reason. 48:836–856

    Article  MathSciNet  MATH  Google Scholar 

  • Li Z, Xie N, Wen G (2015) Soft coverings and their parameter reductions. Appl Soft Comput 31:48–60

    Article  Google Scholar 

  • Ma L (2006) Two fuzzy covering rough set models and their generalizations over fuzzy lattices. Fuzzy Set Syst 294:59–70

    MathSciNet  Google Scholar 

  • Ma L (2012) On some types of neighborhood-related covering rough sets. Int J Approx Reason 53:901–911

    Article  MathSciNet  MATH  Google Scholar 

  • Ma ZM, Hu BQ (2013) Topological and lattice structures of \(L\)-fuzzy rough sets determined by lower and upper sets. Inform Sci 218:194–204

    Article  MathSciNet  MATH  Google Scholar 

  • Ma X, Liu Q, Zhan J (2017) A survey of decision making methods based on certain hybrid soft set models. Artif Intell Rev 47:507–530

    Article  Google Scholar 

  • Maji PK, Roy AR, Biswas R (2002) An application of soft sets in a decision making problem. Comput Math Appl 44:1077–1083

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37:19–31

    Article  MathSciNet  MATH  Google Scholar 

  • Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11:341–356

    Article  MATH  Google Scholar 

  • Pomykala JA (1987) Approximation operations in approximation space. Bull Pol Acad Sci 35:653–662

    MathSciNet  MATH  Google Scholar 

  • Shabir M, Ali MI, Shaheen T (2013) Another approach to soft rough sets. Knowl Based Syst 40:72–80

    Article  Google Scholar 

  • Sun B, Ma W, Li X (2017) Linguistic value soft set-based approach to multiple criteria group decision-making. Appl Soft Comput 58:285–296

    Article  Google Scholar 

  • Tozlu N, Yüksel Ş, Simsekler TH (2016) A topological approach to soft covering approximation space. Int J Math Trends Tech 29:33–38

    Article  Google Scholar 

  • Tsang ECC, Chen D, Yeung D (2008) Approximations and reducts with covering generalized rough sets. Comput Math Appl 56:279–289

    Article  MathSciNet  MATH  Google Scholar 

  • Wang C, Chen D, Sun B, Hu Q (2012) Communication between information systems with covering based rough sets. Inform Sci 216:17–33

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Q, Zhan J, Ali MI, Mehmood N (2018) A study on Z-soft rough fuzzy semigroups and its decision-makings. Int J Uncertain Quan 8(1):1–22

    Article  Google Scholar 

  • Xu W, Zhang W (2007) Measuring roughness of generalized rough sets induced by a covering. Fuzzy Set Syst 158:2443–2455

    Article  MathSciNet  MATH  Google Scholar 

  • Yang B, Hu B (2016) A fuzzy covering-based rough set model and its generalization over fuzzy lattice. Inform Sci 367:463–486

    Article  Google Scholar 

  • Yang B, Hu B (2017) On some types of fuzzy covering-based rough sets. Fuzzy Set Syst 312:36–65

    Article  MathSciNet  MATH  Google Scholar 

  • Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inform Sci 101:239–259

    Article  MathSciNet  MATH  Google Scholar 

  • Yao YY, Yao B (2012) Covering based rough sets approximations. Inform Sci 200:91–107

    Article  MathSciNet  MATH  Google Scholar 

  • Yüksel Ş, Erglü ZG, Tozlu N(2014) Soft covering based rough sets and their application. Sci World J Article 970893

  • Yüksel Ş, Tozlu N, Dizman TH (2015) An application of multicriteria group decision making by soft covering based rough sets. Filomat 29:209–219

    Article  MathSciNet  MATH  Google Scholar 

  • Zakowski W (1983) Approximations in the \((U,\Pi )\)-space. Demonstr Math 16:761–769

    MathSciNet  MATH  Google Scholar 

  • Zhan J, Alcantud JCR (2018a) A novel type of soft rough covering and its application to multicriteria group decision making. Artif Intell Rev. https://doi.org/10.1007/s10462-018-9617-3

  • Zhan J, Alcantud JCR (2018b) A survey of parameter reduction of soft sets and corresponding algorithms. Artif Intell Rev. https://doi.org/10.1007/s10462-017-9592-0

  • Zhan J, Wang Q (2018) Certain types of soft coverings based rough sets with applications. Int J Mach Learn Cybern. https://doi.org/10.1007/s13042-018-0785-x

    Google Scholar 

  • Zhan J, Zhu K (2015) Reviews on decision making methods based on (fuzzy) soft sets and rough soft sets. J Intell Fuzzy Syst 29:1169–1176

    Article  MathSciNet  MATH  Google Scholar 

  • Zhan J, Zhu K (2017) A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making. Soft Comput 21:1923–1936

    Article  MATH  Google Scholar 

  • Zhan J, Ali MI, Mehmood N (2017a) On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods. Appl Soft Comput 56:446–457

    Article  Google Scholar 

  • Zhan J, Liu Q, Herawan T (2017b) A novel soft rough set: soft rough hemirings and corresponding multicriteria group decision making. Appl Soft Comput 54:393–402

    Article  Google Scholar 

  • Zhang XH (2017) Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras. J Intell Fuzzy Syst 33:1767–1774

    Article  MATH  Google Scholar 

  • Zhang L, Zhan J (2018) Fuzzy soft \(\beta \)-covering based fuzzy rough sets and corresponding decision-making applications. Int J Mach Learn Cybern. https://doi.org/10.1007/s13042-018-0828-3

    Google Scholar 

  • Zhang XH, Miao D, Liu C, Le M (2016) Constructive methods of rough approximation operators and multigranulation rough sets. Knowl Based Syst 91:114–125

    Article  Google Scholar 

  • Zhang XH, Park C, Wu SP (2018) Soft set theoretical approach to pseudo-BCI algebras. J Intell Fuzzy Syst 34:559–568

    Article  Google Scholar 

  • Zhu W (2007a) Generalized rough sets based on relations. Inform Sci 177:4997–5011

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W (2007b) Topological approaches to covering rough sets. Inform Sci 177:1499–1508

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu W, Wang F (2003) Reduction and axiomization of covering generalized rough sets. Inform Sci 152:217–230

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgements

The authors are extremely grateful to the editor and the anonymous referee for their valuable comments and helpful suggestions which helped to improve the presentation of this paper. This research is partially supported by NNSFC (11461025; 11561023; 61866011).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, China

    Li Zhang & Jianming Zhan

  2. BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, 37007, Salamanca, Spain

    José Carlos R. Alcantud

Authors
  1. Li Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianming Zhan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José Carlos R. Alcantud
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianming Zhan.

Ethics declarations

Conflict of interest

All authors declare that there is no conflict of interest regarding the publication of this manuscript.

Ethical approval

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

Additional information

Communicated by A. Di Nola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, L., Zhan, J. & Alcantud, J.C.R. Novel classes of fuzzy soft \(\beta \)-coverings-based fuzzy rough sets with applications to multi-criteria fuzzy group decision making. Soft Comput 23, 5327–5351 (2019). https://doi.org/10.1007/s00500-018-3470-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-018-3470-9

Keywords

Search

Navigation