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Covering based multigranulation fuzzy rough sets and corresponding applications

  • Jianming ZhanEmail author
  • Xiaohong Zhang
  • Yiyu Yao
Article
  • 6 Downloads

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.

Keywords

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

Notes

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.

References

  1. Abu-Donia GM (2012) Multi knowledge based rough approximations and applications. Knowl Based Syst 26(1):20–29CrossRefGoogle Scholar
  2. Alcantud JCR, de A Calle R (2017) The problem of collective identity in a fuzzy environment. Fuzzy Sets Syst 315:57–75MathSciNetCrossRefzbMATHGoogle Scholar
  3. Alcantud JCR, Díaz S (2017) Rational fuzzy and sequential fuzzy choice. Fuzzy Sets Syst 315:76–98MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bargiela A, Pedrycz W (2005) Granular mappings. IEEE Trans Syst Man Cybern, Part A 35(2):292–297CrossRefzbMATHGoogle Scholar
  5. Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in rough set theory. Inform Sci 107:149–167MathSciNetCrossRefzbMATHGoogle 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 230:624–633MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chen Y, Kilgour D, Hipel K (2012) A decision rule aggregation approach to multiple criteria-multiple participant sorting. Group Decis Negot 21:727–745CrossRefGoogle Scholar
  8. Couso I, Dubois D (2011) Rough sets, coverings and incomplete information. Fund Inform 108:223–247MathSciNetzbMATHGoogle Scholar
  9. D’eer L, Restrepro M, Cornelis C, Gomez J (2016) Neighborhood operators for coverings based rough sets. Inform Sci 336:21–44CrossRefzbMATHGoogle 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. Deng T, Chen Y, Xu W, Dai Q (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inform Sci 177:2308–2326MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefzbMATHGoogle Scholar
  13. Greco S, Matrazzo B, Slowinski R (2001) Rough set theory for multicritera decision analysis. Eur J Oper Res 129:11–47CrossRefGoogle Scholar
  14. Hong D, Choi C (2000) Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Syst 114:103–113CrossRefzbMATHGoogle Scholar
  15. Huang B, Guo C, Zhang Y, Li H, Zhou X (2014) Intuitionistic fuzzy multigranulation rough sets. Inform Sci 277:299–320MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hwang C, Lin M (1987) Group decision making under multiple criteria, Lecture Notes in Economics and Mathematical Systems, Springer, BerlinGoogle Scholar
  17. Jensen R, Shen Q (2007) Fuzzy-rough sets assisted attribute selection. IEEE Trans Fuzzy Syst 15(1):73–89CrossRefGoogle Scholar
  18. Khan MA, Banerjee M (2008) Formal reasoning with rough sets in multiple-source approximation systems. Int J Approx Reason 49:466–477MathSciNetCrossRefzbMATHGoogle Scholar
  19. Li TJ, Leung Y, Zhang WX (2008) Generlizaed fuzzy rough approximation operators based on fuzzy covering. Int J Approx Reason 48:836–856CrossRefzbMATHGoogle Scholar
  20. Liang JY, Qian YH (2006) Axiomatic approach of knowledge granulation in information systems. In: LNAI vol 4304, pp 1074–1078Google Scholar
  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):1080–1093CrossRefGoogle Scholar
  22. Lin GP, Qian YH, Li TJ (2012) NMGS: neighborhood-based multigranulation rough sets. Int J Approx Reason 53(7):1080–1093CrossRefzbMATHGoogle Scholar
  23. Lin GP, Liang JY, Qian YH (2013) Multigranulation rough sets: from partition to covering. Inform Sci 241:101–118MathSciNetCrossRefzbMATHGoogle Scholar
  24. Liu CH, Pedrycz W (2016) Covering-based multi-granulation fuzzy rough sets. J Intell Fuzzy Syst 30:303–318CrossRefzbMATHGoogle Scholar
  25. Liu GL, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50:521–528MathSciNetCrossRefzbMATHGoogle Scholar
  26. Liu CH, Miao DQ, Qian J (2014) On multi-granulation covering rough sets. Int J Approx Reason 55(6):1404–1418MathSciNetCrossRefzbMATHGoogle Scholar
  27. Ma L (2012) On some types of neighborhood-related covering rough sets. Int J Approx Reason 53:901–911MathSciNetCrossRefzbMATHGoogle Scholar
  28. Ma L (2015) Some twin approximation operators on covering approximation spaces. Int J Approx Reason 56:59–70MathSciNetCrossRefzbMATHGoogle Scholar
  29. Ma L (2016) Two fuzzy coverings rough set models and their generalizations over fuzzy lattices. Fuzzy Sets Syst 294:1–17MathSciNetCrossRefzbMATHGoogle 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):4126–4148CrossRefGoogle Scholar
  31. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11(5):341–356CrossRefzbMATHGoogle Scholar
  32. Pedrycz W (2002) Relational and directional aspects in the construction of information granules. IEEE Tran Syst Man Cybern, Part A 32(5):605–614CrossRefGoogle Scholar
  33. Pedrycz W (2013) Granular computing analysis and design of intelligent systems. CRC Press, Boca RatonCrossRefGoogle Scholar
  34. Pedrycz W, Skowron A, Kreinovich V (eds) (2008) Handbook of granular computing. Wiley, New YorkGoogle Scholar
  35. Pomykala JA (1987) Approximation operations in approximation spaces. Bull Pol Acad Sci Math 35:653–662MathSciNetzbMATHGoogle Scholar
  36. Qian YH, Liang J, Dang C (2010) Incomplete multigranulation rough sets. IEEE Trans Syst Man Cybern 20:420–431CrossRefGoogle Scholar
  37. Qian YH, Liang J, Yao YY, Dang C (2010) MGRS: a multi-granulation rough set. Inform Sci 180:949–970MathSciNetCrossRefzbMATHGoogle Scholar
  38. Qian YH, Li S, Liang J, Shi Z, Wang F (2014a) Pessimistic rough set based decision: a multigranulation fusion strategy. Inform Sci 264:196–210MathSciNetCrossRefzbMATHGoogle Scholar
  39. Qian YH, Zhang H, Sang Y, Liang J (2014b) Multigranulation decision-theoretical rough sets. Int J Approx Reason 55:225–237CrossRefzbMATHGoogle Scholar
  40. She Y, He X (2012) On the structure of the mulitigranulation rough set model. Knowl-Based Syst 36:81–92CrossRefGoogle Scholar
  41. Sun BZ, Ma W (2015a) Multigranulation rough set theory over two universes. J Intell Fuzzy Syst 28:1251–1269MathSciNetzbMATHGoogle 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 51:83–92CrossRefGoogle Scholar
  43. Sun BZ, Ma W (2017) Fuzzzy rough set over multi-universes and its application in decision making. J Intell Fuzzy Syst 32:1719–1734CrossRefzbMATHGoogle 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 123:61–74CrossRefGoogle 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 81:87–102MathSciNetCrossRefzbMATHGoogle Scholar
  46. Tsang ECC, Chen D, Yeung DS (2008) Approximations and reducts with covering generalized rough sets. Comput Appl Math 56:279–289MathSciNetCrossRefzbMATHGoogle Scholar
  47. Wu WZ, Zhang WX (2004) Neighborhood operator systems and approximation operators. Inform Sci 159:233–254MathSciNetCrossRefGoogle Scholar
  48. Xu WH, Leung Y (1998) Theory and applications of granular labed partitions in multi-scale decision tables. Inform Sci 112:67–84CrossRefGoogle Scholar
  49. Xu WH, Zhang WX (2007) Measuring roughness of generalized rough sets induced a covering. Fuzzy Sets Syst 158:2443–2455MathSciNetCrossRefzbMATHGoogle Scholar
  50. Xu WH, Wang Q, Zhang X (2011) Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int J Fuzzy Syst 13:246–259MathSciNetGoogle Scholar
  51. Xu WH, Sun W, Zhang X (2012) Multiple granulation rough set approach to ordered information systems. Int J Gen Syst 41:475–501MathSciNetCrossRefzbMATHGoogle Scholar
  52. Xu WH, Wang Q, Zhang X (2013) Multi-granulation rough sets based on tolerance relations. Soft Comput 17:1241–1252CrossRefzbMATHGoogle Scholar
  53. Xu WH, Wang Q, Luo S (2014) Multi-granulation fuzzy rough sets. J Intell Fuzzy Syst 26(3):1323–1340MathSciNetzbMATHGoogle Scholar
  54. Yang B, Hu BQ (2016) A fuzzy covering-based rough set model and its generalization over fuzzy lattice. Inform Sci 367–368:463–486CrossRefGoogle Scholar
  55. Yang B, Hu BQ (2017) On some types of fuzzy covering-based on rough sets. Fuzzy Sets Syst 312:36–65MathSciNetCrossRefzbMATHGoogle Scholar
  56. Yang X, Song X, Chen Z, Yang J (2012) On multigranulation rough sets in incomplete information system. Int J Mach Learn Cybern 3:223–232CrossRefGoogle Scholar
  57. Yao YY (2003) On generalizing rough set theory, RSFDGrC 2003, LNCS (LNAI) 2639, pp 44–51Google Scholar
  58. Yao YY (2005) Perspectives of granular computing. In: Proceeding of 2005 IEEE international conference on granular computing, pp 85–90Google Scholar
  59. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inform Sci 111:239–259MathSciNetCrossRefzbMATHGoogle Scholar
  60. Yao YY (2010) Three-way decisions with probabilistic rough sets. Inform Sci 180:341–353MathSciNetCrossRefGoogle Scholar
  61. Yao YY (2016a) Three-way decisions and cognitive computing. Cogn Comput 8(4):543–554CrossRefGoogle Scholar
  62. Yao YY (2016b) A triarchic theory of granular computing. Granul Comput 1:145–157CrossRefGoogle Scholar
  63. Yao YY, She YH (2016) Rough set models in multigranulation spaces. Inform Sci 327:40–56MathSciNetCrossRefzbMATHGoogle Scholar
  64. Yao YY, Yao B (2012) Covering based rough set approximations. Inform Sci 200:91–107MathSciNetCrossRefzbMATHGoogle Scholar
  65. Yeung DS, Chen D, Lee J, Wang X (2015) On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst 13:343–361CrossRefGoogle Scholar
  66. Zadeh LA (1997) Toward a theory of fuzzy information granulation and its centrality in human reasining and fuzzy logic. Fuzzy Sets Syst 90:111–127CrossRefzbMATHGoogle Scholar
  67. Żakowski W (1983) Approximations in the space \((U, \Pi )\). Demonstr Math XVI:761–769zbMATHGoogle 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 56:446–457CrossRefGoogle 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 54:393–402CrossRefGoogle Scholar
  70. Zhang B, Dong YY, Xu Y (2014) Multiple attribute consensus rules with minimum adjustments to support consensus reaching. Knowl-Based Syst 67:35–48CrossRefGoogle Scholar
  71. Zhang XH, Miao D, Liu C, Le M (2016) Constructive methods of rough approximation operators and multigranulation rough sets. Knowl-Based Syst 91:114–125CrossRefGoogle Scholar
  72. Zhu W (2007) Topological approaches to covering rough sets. Inform Sci 177:1499–1508MathSciNetCrossRefzbMATHGoogle Scholar
  73. Zhu W (2009a) Relationship between generalized rough sets based on binary relation and covering. Inform Sci 179(3):210–225MathSciNetCrossRefzbMATHGoogle Scholar
  74. Zhu W (2009b) Relationships among basic concepts in covering-based rough sets. Inform Sci 179:2478–2486MathSciNetCrossRefzbMATHGoogle Scholar
  75. Zhu P (2011) Covering rough sets based on neighborhoods: an approach without using neighborhoods. Int J Approx Reason 52:461–472MathSciNetCrossRefzbMATHGoogle Scholar
  76. Zhu W, Wang F (2003) Reduction and axiomization of covering generalized rough sets. Inform Sci 152:217–230MathSciNetCrossRefzbMATHGoogle Scholar
  77. Zhu W, Wang F (2007) On three types of covering rough sets. IEEE Trans Knowl Data Eng 19:1131–1144CrossRefGoogle Scholar
  78. Zhu W, Wang F (2012) The fourth types of covering-based rough sets. Inform Sci 201:80–92MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsHubei Minzu UniversityEnshiChina
  2. 2.School of Arts and SciencesShaanxi University of Science and TechnologyXi’anChina
  3. 3.Department of Computer ScienceUniversity of ReginaReginaCanada

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