Covering based multigranulation fuzzy rough sets and corresponding applications

Abstract

By combining covering based rough sets, fuzzy rough sets, and multigranulation rough sets, we introduce covering based multigranulation fuzzy rough set models by means of fuzzy \(\beta \)-neighborhoods. We investigate axiomatic characterizations of covering based optimistic, pessimistic and variable precision multigranulation fuzzy rough set models. We propose coverings based \(\alpha \)-optimistic (pessimistic) multigranulation fuzzy rough sets and D-optimistic (pessimistic) multigranulation fuzzy rough sets from fuzzy measures. We examine the relationships among these kinds of coverings based fuzzy rough sets. Finally, we apply the proposed models to solve problems for multi-criteria group decision-making.

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

Fig. 1
Fig. 2

References

  1. Abu-Donia GM (2012) Multi knowledge based rough approximations and applications. Knowl Based Syst 26(1):20–29

    Google Scholar 

  2. Alcantud JCR, de A Calle R (2017) The problem of collective identity in a fuzzy environment. Fuzzy Sets Syst 315:57–75

    MathSciNet  MATH  Google Scholar 

  3. Alcantud JCR, Díaz S (2017) Rational fuzzy and sequential fuzzy choice. Fuzzy Sets Syst 315:76–98

    MathSciNet  MATH  Google Scholar 

  4. Bargiela A, Pedrycz W (2005) Granular mappings. IEEE Trans Syst Man Cybern, Part A 35(2):292–297

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Cabrerizo FJ, Herrera-Viedma E, Pedrycz W (2013) A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts. Eur J Oper Res 230:624–633

    MathSciNet  MATH  Google Scholar 

  7. Chen Y, Kilgour D, Hipel K (2012) A decision rule aggregation approach to multiple criteria-multiple participant sorting. Group Decis Negot 21:727–745

    Google Scholar 

  8. Couso I, Dubois D (2011) Rough sets, coverings and incomplete information. Fund Inform 108:223–247

    MathSciNet  MATH  Google Scholar 

  9. D’eer L, Restrepro M, Cornelis C, Gomez J (2016) Neighborhood operators for coverings based rough sets. Inform Sci 336:21–44

    MATH  Google Scholar 

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

    MathSciNet  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. Inform Sci 177:2308–2326

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  13. Greco S, Matrazzo B, Slowinski R (2001) Rough set theory for multicritera decision analysis. Eur J Oper Res 129:11–47

    Google Scholar 

  14. Hong D, Choi C (2000) Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114:103–113

    MATH  Google Scholar 

  15. Huang B, Guo C, Zhang Y, Li H, Zhou X (2014) Intuitionistic fuzzy multigranulation rough sets. Inform Sci 277:299–320

    MathSciNet  MATH  Google Scholar 

  16. Hwang C, Lin M (1987) Group decision making under multiple criteria, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin

    Google Scholar 

  17. Jensen R, Shen Q (2007) Fuzzy-rough sets assisted attribute selection. IEEE Trans Fuzzy Syst 15(1):73–89

    Google Scholar 

  18. Khan MA, Banerjee M (2008) Formal reasoning with rough sets in multiple-source approximation systems. Int J Approx Reason 49:466–477

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  20. Liang JY, Qian YH (2006) Axiomatic approach of knowledge granulation in information systems. In: LNAI vol 4304, pp 1074–1078

  21. Liang JY, Wang F, Dang CY, Qian YH (2012) An efficient rough feature selsction algorithm with a multi-granulation view. Int J Approx Reason 53(7):1080–1093

    Google Scholar 

  22. Lin GP, Qian YH, Li TJ (2012) NMGS: neighborhood-based multigranulation rough sets. Int J Approx Reason 53(7):1080–1093

    MATH  Google Scholar 

  23. Lin GP, Liang JY, Qian YH (2013) Multigranulation rough sets: from partition to covering. Inform Sci 241:101–118

    MathSciNet  MATH  Google Scholar 

  24. Liu CH, Pedrycz W (2016) Covering-based multi-granulation fuzzy rough sets. J Intell Fuzzy Syst 30:303–318

    MATH  Google Scholar 

  25. Liu GL, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50:521–528

    MathSciNet  MATH  Google Scholar 

  26. Liu CH, Miao DQ, Qian J (2014) On multi-granulation covering rough sets. Int J Approx Reason 55(6):1404–1418

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Ma L (2016) Two fuzzy coverings rough set models and their generalizations over fuzzy lattices. Fuzzy Sets Syst 294:1–17

    MathSciNet  MATH  Google Scholar 

  30. Mardani A, Jusoh A, Zavadskas EK (2015) Fuzzy multiple criteria decision-making techniques and applications—two decades review from 1994 to 2014. Expert Syst Appl 42(8):4126–4148

    Google Scholar 

  31. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11(5):341–356

    MATH  Google Scholar 

  32. Pedrycz W (2002) Relational and directional aspects in the construction of information granules. IEEE Tran Syst Man Cybern, Part A 32(5):605–614

    Google Scholar 

  33. Pedrycz W (2013) Granular computing analysis and design of intelligent systems. CRC Press, Boca Raton

    Google Scholar 

  34. Pedrycz W, Skowron A, Kreinovich V (eds) (2008) Handbook of granular computing. Wiley, New York

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. Qian YH, Liang J, Dang C (2010) Incomplete multigranulation rough sets. IEEE Trans Syst Man Cybern 20:420–431

    Google Scholar 

  37. Qian YH, Liang J, Yao YY, Dang C (2010) MGRS: a multi-granulation rough set. Inform Sci 180:949–970

    MathSciNet  MATH  Google Scholar 

  38. Qian YH, Li S, Liang J, Shi Z, Wang F (2014a) Pessimistic rough set based decision: a multigranulation fusion strategy. Inform Sci 264:196–210

    MathSciNet  MATH  Google Scholar 

  39. Qian YH, Zhang H, Sang Y, Liang J (2014b) Multigranulation decision-theoretical rough sets. Int J Approx Reason 55:225–237

    MATH  Google Scholar 

  40. She Y, He X (2012) On the structure of the mulitigranulation rough set model. Knowl-Based Syst 36:81–92

    Google Scholar 

  41. Sun BZ, Ma W (2015a) Multigranulation rough set theory over two universes. J Intell Fuzzy Syst 28:1251–1269

    MathSciNet  MATH  Google Scholar 

  42. Sun BZ, Ma W (2015b) An approach to consenses measurement of linguistic preference relations in multi-attribute group decision making and application. Omega 51:83–92

    Google Scholar 

  43. Sun BZ, Ma W (2017) Fuzzzy rough set over multi-universes and its application in decision making. J Intell Fuzzy Syst 32:1719–1734

    MATH  Google Scholar 

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

    Google Scholar 

  45. Sun BZ, Ma W, Xiao X (2017b) Three-way group decision making based on multigranulation fuzzy decision-theoretic rough set over two universes. Int J Approx Reason 81:87–102

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  47. Wu WZ, Zhang WX (2004) Neighborhood operator systems and approximation operators. Inform Sci 159:233–254

    MathSciNet  Google Scholar 

  48. Xu WH, Leung Y (1998) Theory and applications of granular labed partitions in multi-scale decision tables. Inform Sci 112:67–84

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  50. Xu WH, Wang Q, Zhang X (2011) Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int J Fuzzy Syst 13:246–259

    MathSciNet  Google Scholar 

  51. Xu WH, Sun W, Zhang X (2012) Multiple granulation rough set approach to ordered information systems. Int J Gen Syst 41:475–501

    MathSciNet  MATH  Google Scholar 

  52. Xu WH, Wang Q, Zhang X (2013) Multi-granulation rough sets based on tolerance relations. Soft Comput 17:1241–1252

    MATH  Google Scholar 

  53. Xu WH, Wang Q, Luo S (2014) Multi-granulation fuzzy rough sets. J Intell Fuzzy Syst 26(3):1323–1340

    MathSciNet  MATH  Google Scholar 

  54. Yang B, Hu BQ (2016) A fuzzy covering-based rough set model and its generalization over fuzzy lattice. Inform Sci 367–368:463–486

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  56. Yang X, Song X, Chen Z, Yang J (2012) On multigranulation rough sets in incomplete information system. Int J Mach Learn Cybern 3:223–232

    Google Scholar 

  57. Yao YY (2003) On generalizing rough set theory, RSFDGrC 2003, LNCS (LNAI) 2639, pp 44–51

  58. Yao YY (2005) Perspectives of granular computing. In: Proceeding of 2005 IEEE international conference on granular computing, pp 85–90

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

    MathSciNet  MATH  Google Scholar 

  60. Yao YY (2010) Three-way decisions with probabilistic rough sets. Inform Sci 180:341–353

    MathSciNet  Google Scholar 

  61. Yao YY (2016a) Three-way decisions and cognitive computing. Cogn Comput 8(4):543–554

    Google Scholar 

  62. Yao YY (2016b) A triarchic theory of granular computing. Granul Comput 1:145–157

    Google Scholar 

  63. Yao YY, She YH (2016) Rough set models in multigranulation spaces. Inform Sci 327:40–56

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  65. Yeung DS, Chen D, Lee J, Wang X (2015) On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst 13:343–361

    Google Scholar 

  66. Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasining and fuzzy logic. Fuzzy Sets Syst 90:111–127

    MATH  Google Scholar 

  67. Żakowski W (1983) Approximations in the space \((U, \Pi )\). Demonstr Math XVI:761–769

    MATH  Google Scholar 

  68. 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

    Google Scholar 

  69. 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

    Google Scholar 

  70. Zhang B, Dong YY, Xu Y (2014) Multiple attribute consensus rules with minimum adjustments to support consensus reaching. Knowl-Based Syst 67:35–48

    Google Scholar 

  71. 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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  73. Zhu W (2009a) Relationship between generalized rough sets based on binary relation and covering. Inform Sci 179(3):210–225

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  75. Zhu P (2011) Covering rough sets based on neighborhoods: an approach without using neighborhoods. Int J Approx Reason 52:461–472

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  78. Zhu W, Wang F (2012) The fourth types of covering-based rough sets. Inform Sci 201:80–92

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are extremely grateful to the editor and four anonymous referees for their valuable comments and helpful suggestions which helped to improve the presentation of this paper. This research was partially supported by NNSFC (11461025; 11561023) and a Discovery Grant from NSERC Canada.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jianming Zhan.

Additional information

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

Verify currency and authenticity via CrossMark

Cite this article

Zhan, J., Zhang, X. & Yao, Y. Covering based multigranulation fuzzy rough sets and corresponding applications. Artif Intell Rev 53, 1093–1126 (2020). https://doi.org/10.1007/s10462-019-09690-y

Download citation

Keywords

  • Multigranulation rough set
  • Covering based fuzzy rough set
  • Fuzzy \(\beta \)-neighborhood
  • Decision making