Soft Computing

, Volume 23, Issue 24, pp 13085–13103 | Cite as

Multi-granulation hesitant fuzzy rough sets and corresponding applications

  • Haidong ZhangEmail author
  • Jianming Zhan
  • Yanping He
Methodologies and Application


This paper develops a single-granulation hesitant fuzzy rough set (SGHFRS) model from the perspective of granular computing. In the multi-granulation framework, we propose two types of multi-granulation rough sets model called the optimistic multi-granulation hesitant fuzzy rough sets (OMGHFRSs) and pessimistic multi-granulation hesitant fuzzy rough sets (PMGHFRSs). In the models, the multi-granulation hesitant fuzzy lower and upper approximations are defined based on multiple hesitant fuzzy tolerance relations. The relationships among the SGHFRSs, OMGHFRSs and PMGHFRSs are also established. In order to further measure the uncertainty of multi-granulation hesitant fuzzy rough sets (MGHFRSs), the concepts of rough measure and rough measure about the parameters \(\alpha \) and \(\beta \) are presented and some of their interesting properties are examined. Finally, we give a decision-making method based on the MGHFRSs, and the validity of this approach is illustrated by two practical applications. Compared with the existing results, we also expound its advantages.


Decision-making method Multi-granulation hesitant fuzzy rough set Rough measure Single-granulation hesitant fuzzy rough set 



The authors would like to thank the anonymous referees for their valuable comments and suggestions. This study was funded by the National Natural Science Foundation of China (Nos. 11461025; 11561023), the Natural Science Foundation of Gansu Province (No. 17JR5RA284), the Research Project Funds for Higher Education Institutions of Gansu Province (No. 2016B-005), the Fundamental Research Funds for the Central Universities of Northwest MinZu University (Nos. 31920170010, 31920180116) and the first-class discipline program of Northwest Minzu University.

Compliance with ethical standards

Conflict of interest

The author declares that there is no conflict of interests.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96zbMATHCrossRefGoogle Scholar
  2. Bao YL, Yang HL, She YH (2018) Using one axiom to characterize L-fuzzy rough approximation operators based on residuated lattices. Fuzzy Sets Syst 336:87–115MathSciNetzbMATHCrossRefGoogle Scholar
  3. Chen N, Xu ZS, Xia MM (2013) Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl Math Model 37:2197–2211MathSciNetzbMATHCrossRefGoogle Scholar
  4. Cock MD, Cornelis C, Kerre EE (2007) Fuzzy rough sets: the forgotten step. IEEE Trans Fuzzy Syst 15(1):121–130CrossRefGoogle Scholar
  5. Deepak D, John SJ (2014) Hesitant fuzzy rough sets through hesitant fuzzy relations. Ann Fuzzy Math Inform 8(1):33–46MathSciNetzbMATHGoogle Scholar
  6. Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications. Academic Press, New YorkzbMATHGoogle Scholar
  7. Dubois D, Prade H (1990) Rough fuzzy sets and fuzzy rough sets. Int J Gen Syst 17:191–209zbMATHCrossRefGoogle Scholar
  8. Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144MathSciNetzbMATHCrossRefGoogle Scholar
  9. Feng T, Mi JS (2016) Variable precision multigranulation decision-theoretic fuzzy rough sets. Knowl Based Syst 91:93–101CrossRefGoogle Scholar
  10. Hu BQ (2014) Three-way decisions space and three-way decisions. Inf Sci 281:21–52MathSciNetzbMATHCrossRefGoogle Scholar
  11. Hu BQ (2015) Generalized interval-valued fuzzy variable precision rough sets determined by fuzzy logical operators. Int J Gen Syst 44:849–875MathSciNetzbMATHCrossRefGoogle Scholar
  12. Hu BQ (2016) Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets. Knowl Based Syst 91:16–31CrossRefGoogle Scholar
  13. Hu BQ (2017) Hesitant sets and hesitant relations. J Intell Fuzzy Syst 33:3629–3640CrossRefGoogle Scholar
  14. Hu BQ, Wong H (2014) Generalized interval-valued fuzzy variable precision rough sets. Int J Fuzzy Syst 16:554–565MathSciNetGoogle Scholar
  15. Huang B, Guo CX, Zhuang YL, Li HX, Zhou XZ (2014) Intuitionistic fuzzy multigranulation rough sets. Inf Sci 277:299–320MathSciNetzbMATHCrossRefGoogle Scholar
  16. Huang B, Li HX, Fuo GF, Zhuang YL (2017) Inclusion measure-based multi-granulation intuitionistic fuzzy decision-theoretic rough sets and their application to ISSA. Knowl Based Syst 138:220–231CrossRefGoogle Scholar
  17. Huang B, Wu WZ, Yan JJ, Li HX, Zhou XZ (2018) Inclusion measure-based multi-granulation decision-theoretic rough sets in multi-scale intuitionistic fuzzy information tables. Inf Sci. CrossRefGoogle Scholar
  18. Jena SP, Ghosh SK (2002) Intuitionistic fuzzy rough sets. Notes Intuitionistic Fuzzy Sets 8:1–18MathSciNetzbMATHGoogle Scholar
  19. Jiang H, Zhan J, Chen D (2018) Covering based variable precision (I, T)-fuzzy rough sets with applications to multi-attribute decision-making. IEEE Trans Fuzzy Syst. CrossRefGoogle Scholar
  20. Ju HR, Li HX, Yang XB, Zhou XZ, Huang B (2017) Cost-sensitive rough set: a multi-granulation approach. Knowl Based Syst 123:137–153CrossRefGoogle Scholar
  21. Kang Y, Wu SX, Li YW, Liu JH, Chen BH (2018) A variable precision grey-based multi-granulation rough set model and attribute reduction. Knowl Based Syst 148:131–145CrossRefGoogle Scholar
  22. Li WT, Xu WH (2015) Multigranulation decision-theoretic rough set in ordered information system. Fundam Inform 139:67–89MathSciNetzbMATHCrossRefGoogle Scholar
  23. Li WT, Zhang XY, Sun WX (2014) Further study of multigranulation \(T\)-fuzzy rough sets. Sci World J 2014:1–18Google Scholar
  24. Li JH, Ren Y, Mei CL, Qian YH, Yang XB (2016) A comparative study of multigranulation rough sets and concept lattices via rule acquisition. Knowl Based Syst 91:152–164CrossRefGoogle Scholar
  25. Liang DC, Liu D (2015) A novel risk decision-making based on decision-theoretic rough sets under hesitant fuzzy information. IEEE Trans Fuzzy Syst 23(2):237–247CrossRefGoogle Scholar
  26. Liang JY, Wang F, Dang CY, Qian YH (2012) An efficient rough feature selection algorithm with a multi-granulation view. Int J Approx Reason 53:912–926MathSciNetCrossRefGoogle Scholar
  27. Liao HC, Xu ZS (2013) A VIKOR-based method for hesitant fuzzy multi-criteria decision making. Fuzzy Optim Decis Mak 12:373–392MathSciNetzbMATHCrossRefGoogle Scholar
  28. Lin GP, Qian YH, Li JJ (2012) NMGRS: neighborhood-based multigranulation rough sets. Int J Approx Reason 53(7):1080–1093MathSciNetzbMATHCrossRefGoogle Scholar
  29. Lin GP, Liang JY, Qian YH (2013) Multigranulation rough sets: from partition to covering. Inf Sci 241:101–118MathSciNetzbMATHCrossRefGoogle Scholar
  30. Lin GP, Liang JY, Qian YH (2015) An information fusion approach by combining multigranulation rough sets and evidence theory. Inf Sci 314:184–199MathSciNetzbMATHCrossRefGoogle Scholar
  31. Liu GL (2013) The relationship among different covering approximations. Inf Sci 250:178–183MathSciNetzbMATHCrossRefGoogle Scholar
  32. Liu CH, Wang MZ (2011) Covering fuzzy rough set based on multi-granulations. In: International conference on uncertainty reasoning and knowledge engineering, Indonesia, pp 146–149Google Scholar
  33. Liu CH, Miao DQ, Qian J (2014) On multi-granulation covering rough sets. Int J Approx Reason 55(6):1404–1418MathSciNetzbMATHCrossRefGoogle Scholar
  34. Mandal P, Ranadive AS (2017) Multi-granulation bipolar-valued fuzzy probabilistic rough sets and their corresponding three-way decisions over two universes. Soft Comput. CrossRefzbMATHGoogle Scholar
  35. Mandal P, Ranadive AS (2018) Multi-granulation interval-valued fuzzy probabilistic rough sets and their corresponding three-way decisions based on interval-valued fuzzy preference relations. Granul Comput. CrossRefzbMATHGoogle Scholar
  36. Miyamoto S (2005) Remarks on basics of fuzzy sets and fuzzy multisets. Fuzzy Sets Syst 156:427–431MathSciNetzbMATHCrossRefGoogle Scholar
  37. Nanda S, Majumda S (1992) Fuzzy rough sets. Fuzzy Sets Syst 45:157–160MathSciNetCrossRefGoogle Scholar
  38. Pawlak Z (1982) Rough sets. Int J Comput Inf Sci 11:145–172zbMATHCrossRefGoogle Scholar
  39. Pawlak Z (1991) Rough sets-theoretical aspects to reasoning about data. Kluwer Academic Publisher, BostonzbMATHGoogle Scholar
  40. Pedrycz W (2013) Granular computing: analysis and design of intelligent systems. CRC Press/Francis Taylor, Boca RatonCrossRefGoogle Scholar
  41. Qian YH, Liang JY, Yao YY, Dang CY (2010) MGRS: a multi-granulation rough set. Inf Sci 180:949–970MathSciNetzbMATHCrossRefGoogle Scholar
  42. Qian YH, Liang JY, Pedrycz W, Dang CY (2011) An efficient accelerator for attribute reduction from incomplete data in rough set framework. Pattern Recognit 44:1658–1670zbMATHCrossRefGoogle Scholar
  43. Qian YH, Zhang H, Sang YL, Liang JY (2014) Multigranulation decision-theoretic rough sets. Int J Approx Reason 55:225–237MathSciNetzbMATHCrossRefGoogle Scholar
  44. Qian YH, Liang XY, Lin GP, Guo Q, Liang JY (2017) Local multigranulation decision-theoretic rough sets. Int J Approx Reason 82:119–137MathSciNetzbMATHCrossRefGoogle Scholar
  45. Radzikowska AM, Kerre EE (2002) A comparative study of fuzzy rough sets. Fuzzy Sets Syst 126:137–155MathSciNetzbMATHCrossRefGoogle Scholar
  46. She YH, He XL (2012) On the structure of the multigranulation rough set model. Knowl Based Syst 36:81–92CrossRefGoogle Scholar
  47. She YH, He XL, Shi HX, Qian YH (2017) A multiple-valued logic approach for multigranulation rough set model. Int J Approx Reason 82:270–284MathSciNetzbMATHCrossRefGoogle Scholar
  48. Sun BZ, Ma WM, Qian YH (2017a) Multigranulation fuzzy rough set over two universes and its application to decision making. Knowl Based Syst 123:61–74CrossRefGoogle Scholar
  49. Sun BZ, Ma WM, Xiao X (2017b) Three-way group decision making based on multigranulation fuzzy decision-theoretic rough set over two universes. Int J Approx Reason 81:87–102MathSciNetzbMATHCrossRefGoogle Scholar
  50. Tiwari SP, Srivastava AK (2013) Fuzzy rough sets, fuzzy preorders and fuzzy topologies. Fuzzy Sets Syst 210:63–68MathSciNetzbMATHCrossRefGoogle Scholar
  51. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539zbMATHGoogle Scholar
  52. Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems, Korea, pp 1378–1382Google Scholar
  53. Wu WZ, Zhang WX (2004) Constructive and axiomatic approaches of fuzzy approximation operators. Inf Sci 159:233–254MathSciNetzbMATHCrossRefGoogle Scholar
  54. Wu WZ, Leung Y, Shao MW (2013) Generalized fuzzy rough approximation operators determined by fuzzy implicators. Int J Approx Reason 54:1388–1409MathSciNetzbMATHCrossRefGoogle Scholar
  55. Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407MathSciNetzbMATHCrossRefGoogle Scholar
  56. Xu ZS, Xia MM (2011a) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138MathSciNetzbMATHCrossRefGoogle Scholar
  57. Xu ZS, Xia MM (2011b) On distance and correlation measures of hesitant fuzzy information. Int J Intell Syst 26:410–425zbMATHCrossRefGoogle Scholar
  58. Xu ZS, Zhang XL (2013) Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl Based Syst 52:53–64CrossRefGoogle Scholar
  59. Xu WH, Wang QR, Zhang XT (2011) Multi-granulation fuzzy rough sets in a fuzzy tolerance approximation space. Int J Fuzzy Syst 13(4):246–259MathSciNetGoogle Scholar
  60. Xu WH, Sun WX, Zhang XY, Zhang WX (2012) Multiple granulation rough set approach to ordered information systems. Int J Gen Syst 41(5):475–501MathSciNetzbMATHCrossRefGoogle Scholar
  61. Xu WH, Wang QR, Zhang XT (2013) Multi-granulation rough sets based on tolerance relations. Soft Comput 17:1241–1252zbMATHCrossRefGoogle Scholar
  62. Xu WH, Wang QR, Luo SQ (2014) Multi-granulation fuzzy rough sets. J Intell Fuzzy Syst 26:1323–1340MathSciNetzbMATHGoogle Scholar
  63. Yager RR (1986) On the theory of bags. Int J Gen Syst 13:23–37MathSciNetCrossRefGoogle Scholar
  64. Yang HL, Guo ZL (2015) Multigranulation decision-theoretic rough sets in incomplete information systems. Int J Mach Learn Cybern 6:1005–1018CrossRefGoogle Scholar
  65. Yang XB, Song XN, Dou HL (2011) Multi-granulation rough set: from crisp to fuzzy case. Ann Fuzzy Math Inform 1(1):55–70MathSciNetzbMATHGoogle Scholar
  66. Yang T, Li QG, Zhou BL (2013a) Related family: a new method for attribute reduction of covering information systems. Inf Sci 228:175–191MathSciNetzbMATHCrossRefGoogle Scholar
  67. Yang XB, Qi YS, Song XN, Yang JY (2013b) Test cost sensitive multigranulation rough set: model and minimal cost selection. Inf Sci 250:184–199MathSciNetzbMATHCrossRefGoogle Scholar
  68. Yang XB, Song XN, Qi YS, Yang JY (2014) Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput 18:1067–1077zbMATHCrossRefGoogle Scholar
  69. Yeung DS, Chen DG, Tsang ECC, Lee JWT, Wang XZ (2005) On the generalization of fuzzy rough sets. IEEE Trans Fuzzy Syst 13:343–361CrossRefGoogle Scholar
  70. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–352zbMATHCrossRefGoogle Scholar
  71. Zhan JM, Xu WH (2018) Two types of coverings based multigranulation rough fuzzy sets and applications to decision making. Artif Intell Rev. CrossRefGoogle Scholar
  72. Zhan JM, 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
  73. Zhan JM, 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
  74. Zhan JM, Sun BZ, Alcantud JCR (2019) Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making. Inf Sci 476:290–318MathSciNetCrossRefGoogle Scholar
  75. Zhang HD, Shu L (2015) Generalized interval-valued fuzzy rough set and its application in decision making. Int J Fuzzy Syst 17(2):279–291MathSciNetCrossRefGoogle Scholar
  76. Zhang N, Wei G (2013) Extension of VIKOR method for decision making problem based on hesitant fuzzy set. Appl Math Model 37(7):4938–4947MathSciNetzbMATHCrossRefGoogle Scholar
  77. Zhang L, Zhan J (2018) Novel classes of fuzzy soft \(\beta \)-coverings-based fuzzy rough sets with applications to multi-criteria fuzzy group decision making. Soft Comput. CrossRefGoogle Scholar
  78. Zhang XH, Zhou B, Li P (2012) A general frame for intuitionistic fuzzy rough sets. Inf Sci 216:34–49MathSciNetzbMATHCrossRefGoogle Scholar
  79. Zhang HY, Leung Y, Zhou L (2013) Variable-precision-dominance-based rough set approach to interval-valued information systems. Inf Sci 244:75–91MathSciNetzbMATHCrossRefGoogle Scholar
  80. Zhang HD, Shu L, Liao SL (2014) Intuitionistic fuzzy soft rough set and its application in decision making. Abstr Appl Anal 2014:1–13MathSciNetzbMATHGoogle Scholar
  81. Zhang HD, Shu L, Liao SL (2016) On interval-valued hesitant fuzzy rough approximation operators. Soft Comput 20(1):189–209zbMATHCrossRefGoogle Scholar
  82. Zhang HD, Shu L, Liao SL (2017) Hesitant fuzzy rough set over two universes and its application in decision making. Soft Comput 21(7):1803–1816zbMATHCrossRefGoogle Scholar
  83. Zhang L, Zhan J, Xu ZX (2019) Covering-based generalized IF rough sets with applications to multi-attribute decision-making. Inf Sci 478:275–302MathSciNetCrossRefGoogle Scholar
  84. Zhou L, Wu WZ (2008) On generalized intuitionistic fuzzy approximation operators. Inf Sci 178:2448–2465MathSciNetzbMATHGoogle Scholar
  85. Zhou L, Wu WZ (2009) On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators. Inf Sci 179:883–898MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceNorthwest MinZu UniversityLanZhouChina
  2. 2.Department of MathematicsHubei Minzu UniversityEnshiChina
  3. 3.School of Electrical EngineeringNorthwest MinZu UniversityLanZhouChina

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