Skip to main content
SpringerLink
Log in

Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy

  • Foundations
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

Typical hesitant fuzzy sets (THFSs), possessing a finite-set-valued fuzzy membership degrees called typical hesitant fuzzy elements (THFEs), is a special kind of hesitant fuzzy sets. Fuzzy inclusion relationship, as the order structure in fuzzy mathematics, plays an elementary role in the theoretical research and practical applications of fuzzy sets. In this paper, a new partial order for THFEs is defined via the disjunctive semantic meaning of a set, based on which fuzzy inclusion relationship is defined for THFSs. Furthermore, inclusion measures are defined to present the quantitative ranking of every two THFEs and THFSs and different inclusion measures are constructed. The related similarity measure, distance and fuzzy entropy of THFSs are presented and their relationship with inclusion measures are investigated. Finally, an example is given to show that the inclusion measure can be applied effectively in hesitant fuzzy multi-attribute decision making.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    Article  MathSciNet  MATH  Google Scholar 

  • Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31:343–349

    Article  MathSciNet  MATH  Google Scholar 

  • Bedregal B, Reiser R, Bustince H, Lopez-Molina C, Torra V (2014a) Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Inf Sci 255:82–99

  • Bedregal B, Santiago RHN, Bustince H, Paternain D, Reiser R (2014b) Typical hesitant fuzzy negations. Int J Intell Syst 29:525–543

  • Bustince H, Mohedano V, Barrenechea E, Pagola M (2006) Definition and construction of fuzzy DI subsethood measures. Inf Sci 176:3190–3231

    Article  MathSciNet  MATH  Google Scholar 

  • Chen N, Xu ZS, Xia MM (2013) Interval-valued hesitant preference relations and their applications to group decision making. Knowl-Based Syst 37:528–540

    Article  MathSciNet  Google Scholar 

  • Cornelis C, Donck CV, Kerre EE (2003) Sinha–Dougherty approach to the fuzzification of set inclusion revisited. Fuzzy Sets Syst 134:283–295

    Article  MathSciNet  MATH  Google Scholar 

  • Dubois D, Prade H (2012) Gradualness, uncertainty and bipolarity: making sense of fuzzy sets. Fuzzy Sets Syst 192:3–24

    Article  MathSciNet  MATH  Google Scholar 

  • Fan JL, Xie WX, Pei JH (1999) Inclusion measure: new definitions. Fuzzy Sets Syst 106:201–209

    Article  MathSciNet  MATH  Google Scholar 

  • Fan SQ, Zhang WX, Xu W (2006) Fuzzy inference based on fuzzy concept lattice. Fuzzy Sets Syst 157:3177–3187

    Article  MathSciNet  MATH  Google Scholar 

  • Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144

    Article  MathSciNet  MATH  Google Scholar 

  • Farhadinia B (2014a) Distance and similarity measures for higher order hesitant fuzzy sets. Knowl-Based Syst 55:43–48

  • Farhadinia B (2014b) Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int J Intell Syst 29:184–205

  • Grattan-Guiness I (1976) Fuzzy membership mapped onto interval and many-valued quantities. Math Log Q 22:149–160

    Article  MATH  Google Scholar 

  • Grzegorzewski P (2011) On possible and necessary inclusion of intuitionistic fuzzy sets. Inf Sci 181:342–350

    Article  MathSciNet  MATH  Google Scholar 

  • Kitainik LM (1987) Fuzzy inclusions and fuzzy dichotomous decision procedures. In: Kacprzyk J, Orlovski S (eds) Optimization models using fuzzy sets and possibility theory. Reidel, Dordrecht, pp 154–170

    Chapter  Google Scholar 

  • Liu HB, Rodrguez RM (2014) A fuzzy envelope for hesitant fuzzy linguistic term set and its application to multi-criteria decision making. Inf Sci 258:220–238

  • Ma Z, Zhang W, Ma W (1999) Assessment of data redundancy in fuzzy relational databases based on semantic inclusion degree. Inf Process Lett 72:25–29

    Article  MathSciNet  Google Scholar 

  • Qiu GF, Li HZ, Xu LD, Zhang WX (2003) A knowledge processing method for intelligent systems based on inclusion degree. Expert Syst 20:187–195

    Article  Google Scholar 

  • Rodriguez R, Martinez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20:109–119

    Article  Google Scholar 

  • Smets P, Magrez P (1987) Implication in fuzzy logic. Int J Approx Reason 1:327–347

    Article  MathSciNet  MATH  Google Scholar 

  • Sinha D, Dougherty ER (1993) Fuzzication of set inclusion: theory and applications. Fuzzy Sets Syst 55:15–42

    Article  MathSciNet  MATH  Google Scholar 

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539

    MATH  Google Scholar 

  • Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. In: The 18th IEEE international conference on fuzzy systems, Jeju Island, Korea, pp 1378–1382

  • Tak\(\acute{{\rm a}}\breve{{\rm c}}\) Z (2013) Inclusion and inclusion measure for interval-valued fuzzy sets and for continuous type-2 fuzzy sets. Fuzzy Sets Syst 224:106–120

  • Wei GW (2012) Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl-Based Syst 31:176–182

    Article  Google Scholar 

  • Wei GW, Zhao XF, Lin R (2013) Some hesitant interval-valued aggregation operators and their applications to multiple attribute decision making. Knowl-Based Syst 46:43–53

    Article  Google Scholar 

  • Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407

    Article  MathSciNet  MATH  Google Scholar 

  • Xu ZS (2014) Hesitant fuzzy set theory. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Xu ZS, Xia MM (2011) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138

    Article  MathSciNet  MATH  Google Scholar 

  • Xu ZB, Liang JY, Chen DG, Chin K (2002) Inclusion degree: a perspective on measures for rough set data analysis. Inf Sci 141:227–236

    Article  MathSciNet  MATH  Google Scholar 

  • Yang SY, Zhang HY, Yue ZW (2013) Inclusion measure and its use in measuring similarity and distance measure between hesitant fuzzy sets. In: IEEE international conference on granular computing (GrC), pp 386–390

  • Yang XB, Song XN, Qi YS (2014) Constructive and axiomatic approaches to hesitant fuzzy rough set. Soft Comput 18:1067–1077

    Article  MATH  Google Scholar 

  • Yao YY, Deng XF (2014) Quantitative rough sets based on inclusion measures. Inf Sci 267:306–322

    Article  MathSciNet  Google Scholar 

  • Ye J (2014) Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making. Appl Math Model 38:659–666

    Article  MathSciNet  Google Scholar 

  • Young RC (1931) The algebra of many-valued quantities. Math Ann 104:270–290

    Article  MathSciNet  Google Scholar 

  • Young VR (1996) Subsethood. Fuzzy Sets Syst 77:371–384

    Article  MathSciNet  MATH  Google Scholar 

  • Yu DJ (2014) Some hesitant fuzzy information aggregation operators based on Einstein operational laws. Int J Intell Syst 00:1–21

    Google Scholar 

  • Zadeh L (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh L (1975) The concept of a linguistic variable and its application to approximate reasoning-I. Inf Sci 8:199–249

    Article  MathSciNet  MATH  Google Scholar 

  • Zeng WY, Li HX (2006) Relationship between similarity measure and entropy of interval-valued fuzzy sets. Fuzzy Sets Syst 157:1477–1484

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang WX, Leung Y (1996) The uncertainty reasoning principles. Xian Jiaotong University Press, Xian

    Google Scholar 

  • Zhang HY, Zhang WX (2009) Hybrid monotonic Inclusion measure and its use in measuring similarity and distance between fuzzy sets. Fuzzy Sets Syst 160:107–118

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang HD, Shu L, Liao SL (2014) On interval-valued hesitant fuzzy rough approximation operators, Soft Comput. doi:10.1007/s00500-014-1490-7

  • Zhao XF, Lin R, Wei GW (2014) Hesitant triangular fuzzy information aggregation based on Einstein operations and their application to multiple attribute decision making. Expert Syst Appl 41:1086–1094

  • Zhu B, Xu ZS (2014) Consistency measures for hesitant fuzzy linguistic preference relations. IEEE Trans Fuzzy Syst 22(1):35–45

    Article  Google Scholar 

  • Zhu B, Xu ZS, Xia MM (2012) Dual hesitant fuzzy sets. J Appl Math 2012:1–13

Download references

Acknowledgments

This work was supported by grants from the National Natural Science Foundation of China (No. 61005042), the Natural Science Foundation of Shaanxi Province (No. 2014JQ8348) and the Fundamental Research Funds for the Central Universities.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Hongying Zhang & Shuyun Yang

Authors
  1. Hongying Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuyun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongying Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by A. Di Nola.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, H., Yang, S. Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy. Soft Comput 20, 1277–1287 (2016). https://doi.org/10.1007/s00500-015-1851-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-015-1851-x

Keywords

Search

Navigation