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Soft Computing

, Volume 23, Issue 14, pp 5327–5351 | Cite as

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

  • Li Zhang
  • Jianming ZhanEmail author
  • José Carlos R. Alcantud
Foundations

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.

Keywords

Fuzzy soft \(\beta \)-minimal (maximal) description Fuzzy soft \(\beta \)-neighborhood Fuzzy soft \(\beta \)-covering-based fuzzy rough set Attribute reduction Measure degree Multi-criteria fuzzy group decision making 

Notes

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).

Compliance with ethical standards

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.

References

  1. Alcantud JCR (2002) Revealed indifference and models of choice behavior. J Math Psychol 46:418–430MathSciNetCrossRefzbMATHGoogle Scholar
  2. Alcantud JCR (2016a) A novel algorithm for fuzzy soft set based decision making from multiobserver input parameter data set. Inform Fusion 29:142–148CrossRefGoogle Scholar
  3. Alcantud JCR (2016b) Some formal relationships among soft sets, fuzzy sets and their extensions. Int J Approx Reason 68:45–53MathSciNetCrossRefzbMATHGoogle Scholar
  4. Ali MI (2011) A note on soft sets, rough sets and fuzzy soft sets. Appl Soft Comput 11:3329–3332CrossRefGoogle Scholar
  5. Ali MI, Shabir M, Feng F (2017) Representation of graphs based on neighborhoods and soft sets. Int J Mach Learn Cybern 8:1525–1535CrossRefGoogle Scholar
  6. Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in the rough set theory. Inform Sci 107:149–167MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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
  8. 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 136:159–171CrossRefGoogle Scholar
  9. D’eer L, Cornelis C (2018) A comprehensive study of fuzzy covering-based rough set models: definitions, properties and interrelationships. Fuzzy Sets Syst 336:1–26MathSciNetCrossRefzbMATHGoogle Scholar
  10. D’eer L, Cornelis C, Godo L (2017) Fuzzy neighborhood operators based on fuzzy coverings. Fuzzy Sets Syst 312:17–35MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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 78:62–72MathSciNetCrossRefzbMATHGoogle Scholar
  12. D’eer L, Restrepo M, Cornelis C, Jonatan G (2016b) Neighborhood operators for covering based rough sets. Inform Sci 336:21–44CrossRefzbMATHGoogle Scholar
  13. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefzbMATHGoogle Scholar
  14. Feng F, Li C, Davvaz B, Ali MI (2010) Soft sets combined with fuzzy sets and rough sets: a tentative approach. Soft Comput 14:899–911CrossRefzbMATHGoogle Scholar
  15. Feng F, Liu X, Leoreanu-Fotea V, Jun YB (2011) Soft sets and soft rough sets. Inform Sci 181:1125–1137MathSciNetCrossRefzbMATHGoogle Scholar
  16. Feng T, Zhang SP, Mi JS (2012) The reduction and fusion of fuzzy covering systems based on the evidence theory. Int J Approx Reason 53:87–103MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hao J, Li QG (2011) The relationship between \(L\)-fuzzy rough set and \(L\)-topology. Fuzzy Set Syst 178:74–83MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hu BQ, Wong H (2014) Generalized interval-valued fuzzy variable precision rough sets. Int J Fuzzy Syst 16:554–565MathSciNetGoogle Scholar
  19. Klir G, Yuan B (1995) Fuzzy sets and fuzzy logic, theory and applications. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  20. Li TJ, Ma JM (2007) Fuzzy approximation operators based on coverings. In: Proceedings of joint rough set symposium, pp 55–62Google Scholar
  21. Li TJ, Leung Y, Zhang WX (2008) Generalized fuzzy rough approximation operators based on fuzzy coverings. Int J Approx Reason. 48:836–856MathSciNetCrossRefzbMATHGoogle Scholar
  22. Li Z, Xie N, Wen G (2015) Soft coverings and their parameter reductions. Appl Soft Comput 31:48–60CrossRefGoogle Scholar
  23. Ma L (2006) Two fuzzy covering rough set models and their generalizations over fuzzy lattices. Fuzzy Set Syst 294:59–70MathSciNetGoogle Scholar
  24. Ma L (2012) On some types of neighborhood-related covering rough sets. Int J Approx Reason 53:901–911MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ma ZM, Hu BQ (2013) Topological and lattice structures of \(L\)-fuzzy rough sets determined by lower and upper sets. Inform Sci 218:194–204MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ma X, Liu Q, Zhan J (2017) A survey of decision making methods based on certain hybrid soft set models. Artif Intell Rev 47:507–530CrossRefGoogle Scholar
  27. Maji PK, Roy AR, Biswas R (2002) An application of soft sets in a decision making problem. Comput Math Appl 44:1077–1083MathSciNetCrossRefzbMATHGoogle Scholar
  28. Mi JS, Zhang WX (2004) An axiomatic characterization of fuzzy generalization of rough sets. Inform Sci 160:235–249MathSciNetCrossRefzbMATHGoogle Scholar
  29. Molodtsov D (1999) Soft set theory-first results. Comput Math Appl 37:19–31MathSciNetCrossRefzbMATHGoogle Scholar
  30. Pawlak Z (1982) Rough sets. Int J Comput Inform Sci 11:341–356CrossRefzbMATHGoogle Scholar
  31. Pomykala JA (1987) Approximation operations in approximation space. Bull Pol Acad Sci 35:653–662MathSciNetzbMATHGoogle Scholar
  32. Shabir M, Ali MI, Shaheen T (2013) Another approach to soft rough sets. Knowl Based Syst 40:72–80CrossRefGoogle Scholar
  33. Sun B, Ma W, Li X (2017) Linguistic value soft set-based approach to multiple criteria group decision-making. Appl Soft Comput 58:285–296CrossRefGoogle Scholar
  34. Tozlu N, Yüksel Ş, Simsekler TH (2016) A topological approach to soft covering approximation space. Int J Math Trends Tech 29:33–38CrossRefGoogle Scholar
  35. Tsang ECC, Chen D, Yeung D (2008) Approximations and reducts with covering generalized rough sets. Comput Math Appl 56:279–289MathSciNetCrossRefzbMATHGoogle Scholar
  36. Wang C, Chen D, Sun B, Hu Q (2012) Communication between information systems with covering based rough sets. Inform Sci 216:17–33MathSciNetCrossRefzbMATHGoogle Scholar
  37. 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):1–22CrossRefGoogle Scholar
  38. Xu W, Zhang W (2007) Measuring roughness of generalized rough sets induced by a covering. Fuzzy Set Syst 158:2443–2455MathSciNetCrossRefzbMATHGoogle Scholar
  39. Yang B, Hu B (2016) A fuzzy covering-based rough set model and its generalization over fuzzy lattice. Inform Sci 367:463–486CrossRefGoogle Scholar
  40. Yang B, Hu B (2017) On some types of fuzzy covering-based rough sets. Fuzzy Set Syst 312:36–65MathSciNetCrossRefzbMATHGoogle Scholar
  41. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inform Sci 101:239–259MathSciNetCrossRefzbMATHGoogle Scholar
  42. Yao YY, Yao B (2012) Covering based rough sets approximations. Inform Sci 200:91–107MathSciNetCrossRefzbMATHGoogle Scholar
  43. Yüksel Ş, Erglü ZG, Tozlu N(2014) Soft covering based rough sets and their application. Sci World J Article 970893Google Scholar
  44. Yüksel Ş, Tozlu N, Dizman TH (2015) An application of multicriteria group decision making by soft covering based rough sets. Filomat 29:209–219MathSciNetCrossRefzbMATHGoogle Scholar
  45. Zakowski W (1983) Approximations in the \((U,\Pi )\)-space. Demonstr Math 16:761–769MathSciNetzbMATHGoogle Scholar
  46. 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
  47. 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
  48. 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
  49. Zhan J, Zhu K (2015) Reviews on decision making methods based on (fuzzy) soft sets and rough soft sets. J Intell Fuzzy Syst 29:1169–1176MathSciNetCrossRefzbMATHGoogle Scholar
  50. Zhan J, Zhu K (2017) A novel soft rough fuzzy set: Z-soft rough fuzzy ideals of hemirings and corresponding decision making. Soft Comput 21:1923–1936CrossRefzbMATHGoogle Scholar
  51. 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 56:446–457CrossRefGoogle Scholar
  52. Zhan J, Liu Q, Herawan T (2017b) A novel soft rough set: soft rough hemirings and corresponding multicriteria group decision making. Appl Soft Comput 54:393–402CrossRefGoogle Scholar
  53. Zhang XH (2017) Fuzzy anti-grouped filters and fuzzy normal filters in pseudo-BCI algebras. J Intell Fuzzy Syst 33:1767–1774CrossRefzbMATHGoogle Scholar
  54. 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
  55. Zhang XH, Miao D, Liu C, Le M (2016) Constructive methods of rough approximation operators and multigranulation rough sets. Knowl Based Syst 91:114–125CrossRefGoogle Scholar
  56. Zhang XH, Park C, Wu SP (2018) Soft set theoretical approach to pseudo-BCI algebras. J Intell Fuzzy Syst 34:559–568CrossRefGoogle Scholar
  57. Zhu W (2007a) Generalized rough sets based on relations. Inform Sci 177:4997–5011MathSciNetCrossRefzbMATHGoogle Scholar
  58. Zhu W (2007b) Topological approaches to covering rough sets. Inform Sci 177:1499–1508MathSciNetCrossRefzbMATHGoogle Scholar
  59. Zhu W, Wang F (2003) Reduction and axiomization of covering generalized rough sets. Inform Sci 152:217–230MathSciNetCrossRefzbMATHGoogle Scholar
  60. Zhu W, Wang F (2007) On three types of covering-based rough sets. IEEE Trans Knowl Data Eng 19:1131–1144CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Li Zhang
    • 1
  • Jianming Zhan
    • 1
    Email author
  • José Carlos R. Alcantud
    • 2
  1. 1.Department of MathematicsHubei University for NationalitiesEnshiChina
  2. 2.BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME)University of SalamancaSalamancaSpain

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