Advertisement

Covering-based intuitionistic fuzzy rough sets and applications in multi-attribute decision-making

  • Jianming ZhanEmail author
  • Bingzhen Sun
Article
  • 94 Downloads

Abstract

Covering based intuitionistic fuzzy (IF) rough set is a generalization of granular computing and covering based rough sets. By combining covering based rough sets, IF sets and fuzzy rough sets, we introduce three classes of coverings based IF rough set models via IF\(\beta \)-neighborhoods and IF complementary \(\beta \)-neighborhood (IFC\(\beta \)-neighborhood). The corresponding axiomatic systems are investigated, respectively. In particular, the rough and precision degrees of covering based IF rough set models are discussed. The relationships among these types of coverings based IF rough set models and covering based IF rough set models proposed by Huang et al. (Knowl Based Syst 107:155–178, 2016). Based on the theoretical analysis for coverings based IF rough set models, we put forward intuitionistic fuzzy TOPSIS (IF-TOPSIS) methodology to multi-attribute decision-making (MADM) problem with the evaluation of IF information problem. An effective example is to illustrate the proposed methodology. Finally, we deal with MADM problem with the evaluation of fuzzy information based on CFRS models. By comparative analysis, we find that it is more effective to deal with MADM problem with the evaluation of IF information based on CIFRS models than the one with the evaluation of fuzzy information based on CFRS models.

Keywords

Covering based IF rough sets IF\(\beta \)-neighborhood IFC\(\beta \)-neighborhood IF-TOPSIS methodology MADM 

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. The first author was supported by the NNSFC (11461025, 11561023). The second author was supported by the NNSFC (71571090), the National Science Foundation of Shaanxi Province of China (2017JM7022), the Key Strategic Project of Fundamental Research Funds for the Central Universities (JBZ170601).

References

  1. Alcantud JCR (2002) Revealed indifference and models of choice behavior. J Math Psychol 46:418–430MathSciNetCrossRefGoogle Scholar
  2. Alcantud JCR, Torra V (2018) Decomposition theorems and extension principles for hesitant fuzzy sets. Inf Fusion 41:48–56CrossRefGoogle Scholar
  3. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Set Syst 20:87–96CrossRefGoogle Scholar
  4. Atanassov KT, Pasi G, Yager RR (2015) Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision-making. Int J Syst Sci 36:859–868MathSciNetCrossRefGoogle Scholar
  5. Bonikowski Z, Bryniarski E, Wybraniec-Skardowska U (1998) Extensions and intentions in rough set theory. Inf Sci 107:149–167MathSciNetCrossRefGoogle Scholar
  6. Chen CT (2000) Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Set Syst 114:1–9CrossRefGoogle Scholar
  7. Chen DG, Li W, Zhang X, Kwong S (2014) Evidence-theory-based numerical algorithms of attribute reduction with neighborhood-covering rough sets. Int J Approx Reason 55:908–923MathSciNetCrossRefGoogle Scholar
  8. Chen DG, Zhang X, Li W (2015) On measurements of covering rough sets based on granules and evidence theory. Inf Sci 317:329–348MathSciNetCrossRefGoogle Scholar
  9. Chen TY (2011) A comparative analysis of score functions for multiple criteria decision-making in intuitionistic fuzzy settings. Inf Sci 181:3652–3676CrossRefGoogle Scholar
  10. Cornelis C, Cock MD, Kerre EE (2003) Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge. Expert Syst 20(5):260–270CrossRefGoogle Scholar
  11. Dai JH, Hu H, Wu WZ, Qian YH, Huang DB (2018) Maximal discernibility pairs based approach to attribute reduction in fuzzy rough sets. IEEE Trans Fuzzy Syst 26(4):2174–2187CrossRefGoogle Scholar
  12. Dai JH, Wei BJ, Zhang XH, Zhang QH (2017) Uncertainty measurement for incomplete interval-valued information systems based on \(\alpha \)-weak similarity. Knowl Based Syst 136:159–171CrossRefGoogle Scholar
  13. D’eer L, Cornelis C, Godo L (2017) Fuzzy neighborhood operators based on fuzzy coverings. Fuzzy Sets Syst 312:17–35MathSciNetCrossRefGoogle Scholar
  14. D’eer L, Restrepro M, Cornelis C, Gomez J (2016) Neighborhood operators for coverings based rough sets. Inf Sci 336:21–44CrossRefGoogle Scholar
  15. Deng T, Chen Y, Xu W, Dai Q (2007) A novel approach to fuzzy rough sets based on a fuzzy covering. Inf Sci 177:2308–2326MathSciNetCrossRefGoogle Scholar
  16. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209CrossRefGoogle Scholar
  17. Hu BQ (2015) Generalized interval-valued fuzzy variable precision rough sets determined by fuzzy logical operators. Int J Gen Syst 44:849–875MathSciNetCrossRefGoogle Scholar
  18. Hu J, Pedrycz W, Wang G (2016) A roughness measure of fuzzy sets from the perspective of distance. Int J Gen Syst 45:352–367MathSciNetCrossRefGoogle Scholar
  19. Huang B, Guo C, Zhang Y, Li H, Zhou X (2014) Intuitionistic fuzzy multigranulation rough sets. Inf Sci 277:299–320MathSciNetCrossRefGoogle Scholar
  20. Huang B, Guo C, Zhang Y, Li H, Zhou X (2016) An intuitionistic fuzzy graded covering rough sets. Knowl Based Syst 107:155–178CrossRefGoogle Scholar
  21. Hwang CL, Yoon KS (1981) Multiple attibute decision methods and applications. Springer, BerlinGoogle Scholar
  22. Li LQ, Jin Q, Hu K, Zhao FF (2017) The axiomatic characterizations on L-fuzzy covering-based approximation operators. Int J Gen Syst 46:332–353MathSciNetCrossRefGoogle Scholar
  23. Li TJ, Leung Y, Zhang WX (2008) Generalized fuzzy rough approximation operators based on fuzzy covering. Int J Approx Reason 48:836–856MathSciNetCrossRefGoogle Scholar
  24. Liu GL, Sai Y (2009) A comparison of two types of rough sets induced by coverings. Int J Approx Reason 50:521–528MathSciNetCrossRefGoogle Scholar
  25. Luce RD (1956) Semiorders and a theory of utility discrimination. Econometrica 24:178–191MathSciNetCrossRefGoogle Scholar
  26. Ma L (2012) On some types of neighborhood-related covering rough sets. Int J Approx Reason 53:901–911MathSciNetCrossRefGoogle Scholar
  27. Ma L (2015) Some twin approximation operators on covering approximation spaces. Int J Approx Reason 56:59–70MathSciNetCrossRefGoogle Scholar
  28. Ma L (2016) Two fuzzy coverings rough set models and their generalizations over fuzzy lattices. Fuzzy Sets Syst 294:1–17MathSciNetCrossRefGoogle Scholar
  29. 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
  30. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11(5):341–356CrossRefGoogle Scholar
  31. Pomykala JA (1987) Approximation operations in approximation spaces. Bull Pol Acad Sci Math 35:653–662MathSciNetzbMATHGoogle Scholar
  32. She Y, He X (2014) Rough approximation operators on \(R_0\)-algebras (nilpotent minimum algebras) with an application in formal logic \(L^*\). Inf Sci 277:71–89CrossRefGoogle Scholar
  33. She Y, He X, Shi H, Qian Y (2017) A multiple-valued logic approach for multigranulation rough set model. Int J Approx Reason 82:270–284MathSciNetCrossRefGoogle Scholar
  34. She Y, Li J, Yang H (2015) A local approach to rule induction in multi-scale decision tables. Knowl Based Syst 89:398–410CrossRefGoogle Scholar
  35. Sun BZ, Ma W (2011) Fuzzy rough set model on two different universes and its application. Appl Math Model 35(4):1798–1809MathSciNetCrossRefGoogle Scholar
  36. Sun BZ, Ma W (2014) Soft fuzzy rough sets and its application in decision-making. Artif Intell Rev 41(1):67–80CrossRefGoogle Scholar
  37. Sun BZ, Ma W (2016) An approach to evaluation of emergency plans for unconventional emergency events based on soft fuzzy rough set. Kybernetes 45(3):1–26MathSciNetGoogle Scholar
  38. Sun BZ, Ma W, Liu Q (2013) An approach to decision-making based on intuitionistic fuzzy rough sets over two universes. J Oper Res Soc 64:1079–1089CrossRefGoogle Scholar
  39. Sun BZ, Ma W, Xiao X (2017) Three-way group decision-making based on multigranulation fuzzy decision-theoretic rough set over two universes. Int J Approx Reason 81:87–102MathSciNetCrossRefGoogle Scholar
  40. Sun BZ, Ma W, Zhao H (2013) A fuzzy rough set approach to emergrncy matrial demand prediction over two universe. Appl Math Model 37:7062–7070MathSciNetCrossRefGoogle Scholar
  41. Szmidt E, Kacprzyk J (2000) Distances between intuitionistic fuzzy sets. Fuzzy Sets Syst 114:505–518MathSciNetCrossRefGoogle Scholar
  42. Tsang ECC, Chen D, Yeung DS (2008) Approximations and reducts with covering generalized rough sets. Comput Appl Math 56:279–289MathSciNetCrossRefGoogle Scholar
  43. Wang W, Xin X (2005) Distance between intuitionistic fuzzy sets. Pattern Recognit Lett 26:2063–2069CrossRefGoogle Scholar
  44. Xu WH, Zhang WX (2007) Measuring roughness of generalized rough sets induced a covering. Fuzzy Sets Syst 158:2443–2455MathSciNetCrossRefGoogle Scholar
  45. Xu ZS (2007) Intuitionistic fuzzy aggregation operators. IEEE Trans Fuzzy Syst 15:1179–1187CrossRefGoogle Scholar
  46. Xu ZS (2010) A deviation-based approach to intuitionistic fuzzy multiple attribute group decision-making. Group Decis Negot 19:57–76CrossRefGoogle Scholar
  47. Xu ZS (2012) Intuitionistic fuzzy multiattribute decision-making: an interactive method. IEEE Trans Fuzzy Syst 20:514–525CrossRefGoogle Scholar
  48. Xu ZS, Yager RR (2006) Some geometric aggregation operators based on intuitionistic fuzzy sets. Int J Gen Syst 35:417–433MathSciNetCrossRefGoogle Scholar
  49. Xu ZS, Yager RR (2008) Dynamic intuitionistic fuzzy multi-atribute decision-making. Int J Approx Reason 48:246–252CrossRefGoogle Scholar
  50. Xu ZS, Zhao N (2016) Information fusion for intuitionistic fuzzy decision-making: an overview. Inf Fusion 28:10–23CrossRefGoogle Scholar
  51. Yang B, Hu BQ (2016) A fuzzy covering-based rough set model and its generalization over fuzzy lattice. Inf Sci 367–368:463–486CrossRefGoogle Scholar
  52. Yang B, Hu BQ (2017) On some types of fuzzy covering-based on rough sets. Fuzzy Sets Syst 312:36–65MathSciNetCrossRefGoogle Scholar
  53. Yao YY (1998) Relational interpretations of neighborhood operators and rough set approximation operators. Inf Sci 111:239–259MathSciNetCrossRefGoogle Scholar
  54. Yao YY (2010) Three-way decisions with probabilistic rough sets. Inf Sci 180:341–353MathSciNetCrossRefGoogle Scholar
  55. Yao YY (2016) Three-way decisions and cognitive computing. Congit Comput 8(4):543–554Google Scholar
  56. Yao YY, Yao B (2012) Covering based rough set approximations. Inf Sci 200:91–107MathSciNetCrossRefGoogle Scholar
  57. Yue ZL (2014) TOPSIS-based group decision-making methodology in intuitionistic fuzzy setting. Inf Sci 277:141–153MathSciNetCrossRefGoogle Scholar
  58. Żakowski W (1983) Approximations in the space \((U, \Pi )\). Demonstr Math XVI:761–769zbMATHGoogle Scholar
  59. Zhan J, Alcantud JCR (2018) 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
  60. 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
  61. Zhang XY, Miao DQ (2017) Three-way attribute reducts. Int J Approx Reason 88:401–434MathSciNetCrossRefGoogle Scholar
  62. Zhang ZM (2012) Generalized intuitionistic fuzzy rough sets based on intuitionistic fuzzy coverings. Inf Sci 198:186–206MathSciNetCrossRefGoogle Scholar
  63. Zhang ZM (2013) Generalized Atanassov’s intuitionistic fuzzy power geometric operators and their application to multiple attribute group decision-making. Inf Fusion 14:460–486CrossRefGoogle Scholar
  64. Zhou L, Wu WZ (2008) On generalized intuitionistic fuzzy rough approximation operators. Inf Sci 178:2448–2465MathSciNetzbMATHGoogle Scholar
  65. Zhu P (2011) Covering rough sets based on neighborhoods: an approach without using neighborhoods. Int J Approx Reason 52:461–472MathSciNetCrossRefGoogle Scholar
  66. Zhu W (2007) Topological approaches to covering rough sets. Inf Sci 177:1499–1508MathSciNetCrossRefGoogle Scholar
  67. Zhu W (2009) Relationship between generalized rough sets based on binary relation and covering. Inf Sci 179(3):210–225MathSciNetCrossRefGoogle Scholar
  68. Zhu W, Wang F (2007) On three types of covering rough sets. IEEE Trans Knowl Data Eng 19:1131–1144CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsHubei University for NationalitiesEnshiChina
  2. 2.School of Economics and ManagementXidian UniversityXi’anChina

Personalised recommendations