Skip to main content

Advertisement

Springer Nature Link
Log in

Fuzzy soft \(\beta\)-covering based fuzzy rough sets and corresponding decision-making applications

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

In this article, the concepts of fuzzy soft \(\beta\)-coverings, fuzzy soft \(\beta\)-neighborhoods and fuzzy soft complement \(\beta\)-neighborhoods are firstly proposed and some related properties are studied. Then four distinct types of fuzzy soft \(\beta\)-coverings based fuzzy rough sets are defined. Furthermore, we explore the relationships among the several types of fuzzy soft \(\beta\)-coverings based fuzzy rough sets. In particular, by means of \(\beta\)-level subsets of fuzzy soft \(\beta\)-coverings, some kinds of soft coverings based rough sets are also investigated. Finally, by means of fuzzy soft \(\beta\)-coverings based fuzzy rough sets, we give a detailed description of an algorithmic procedure of decision-making for the new approach.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Alcantud JCR, Santos-Garcia G (2017) A new criterion for soft set based decision making problems under incomplete information. Int J Comput Int Sys 10:394–404

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. De Cock M, Cornelis C, Kerre EE (2004) Fuzzy rough sets: beyond the obvious, In: Proceedings of the 2004 IEEE International Conference on Fuzzy Systems, vol 1, pp 103-108

  8. 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 

  9. D’eer L, Cornelis C, Yao Y (2016) 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 

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

    Article  MATH  Google Scholar 

  11. Deng T, Chen Y, Xu W, Dai Q (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inf Sci 177:2308–2326

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  13. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. 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 

  16. Greco S, Matarazzo B, Slowinski R (2002) Rough sets methodology for sorting problems in presence of multiple attributes and criteria. Eur J Oper Res 138:247–259

    Article  MathSciNet  MATH  Google Scholar 

  17. 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 

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

    MathSciNet  Google Scholar 

  19. Klir G, Yuan B (1995) Fuzzy sets and Fuzzy logic, theory and applications. Prentice Hall, New Jersey

    MATH  Google Scholar 

  20. Kondo M (2006) On the structure of generalized rough sets. Inf Sci 176:589–600

    Article  MathSciNet  MATH  Google Scholar 

  21. Li J, Mei C, Lv Y (2013) Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction. Int J Approx Reason 54(1):149–165

    Article  MathSciNet  MATH  Google Scholar 

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

  23. 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 

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

    Article  Google Scholar 

  25. Luce RD (1956) Semiorders and a theory of utility discrimination. Econometrica 24:178–191

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma L (2015) Some twin approximation operators on covering approximation spaces. Int J Approx Reason 56:59–70

    Article  MathSciNet  MATH  Google Scholar 

  29. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  35. Pawlak Z (2002) Rough sets and intelligent data analysis. Inf Sci 147:1–12

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  38. Sun B, Ma W, Qian Y (2017) Multigranulation fuzzy rough set over two universes and its application to decision making. Knowl-Based Syst 123:61–74

    Article  Google Scholar 

  39. 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 

  40. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Wu W, Zhang W (2002) Neighborhood operator systems and appoximations. Inf Sci 144:201–217

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  48. Yüksel Ş, Ergül ZG, Tozlu N (2014) Soft covering based rough sets and their application. Sci World J, p 9 (Article 970893)

  49. 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 

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

    MathSciNet  MATH  Google Scholar 

  51. Zhan J, Ali MI, Mehmood N (2017) 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 

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

    Article  Google Scholar 

  53. 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 

  54. 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 

  55. Zhang X, Mei C, Chen D, Li J (2013) Multi-confidence rule acquisition oriented attribute reduction of covering decision systems via combinatorial optimization. Knowl-Based Syst 50:187–197

    Article  Google Scholar 

  56. Zhang X, Mei C, Chen D, Li J (2016) Feature selection in mixed data: a method using a novel fuzzy rough set-based information entropy. Pattern Recogn 56:1–15

    Article  Google Scholar 

  57. 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 

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

    Article  Google Scholar 

  59. Zhang XH, Park C, Wu S (2018) Soft set theoretical approach to pseudo-BCI algebras. J Intell Fuzzy Syst 34(1):559–568

    Article  Google Scholar 

  60. Zhao Z (2016) On some types of covering rough sets from topological points of view. Int J Approx Reason 68:1–14

    Article  MathSciNet  MATH  Google Scholar 

  61. Zhu W (2009) Relationships among basic concepts in covering-based rough sets. Inf Sci 179:2478–2486

    Article  MathSciNet  MATH  Google Scholar 

  62. Zhu W (2007) Generalized rough sets based on relations. Inf Sci 177:4997–5011

    Article  MathSciNet  MATH  Google Scholar 

  63. Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  65. 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 three anonymous referees 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).

Author information

Authors and Affiliations

  1. Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, People’s Republic of China

    Li Zhang & Jianming Zhan

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

Corresponding author

Correspondence to Jianming Zhan.

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. Fuzzy soft \(\beta\)-covering based fuzzy rough sets and corresponding decision-making applications. Int. J. Mach. Learn. & Cyber. 10, 1487–1502 (2019). https://doi.org/10.1007/s13042-018-0828-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-018-0828-3

Keywords

Search

Navigation