Soft Computing

, Volume 22, Issue 9, pp 2809–2824 | Cite as

Hesitant intuitionistic fuzzy entropy/cross-entropy and their applications

Foundations

Abstract

In this paper, we introduce a new concept of hesitant intuitionistic fuzzy set (HIFS), which refines the dual hesitant fuzzy set and could be viewed as a more flexible tool to describe the uncertain information in reality. Since the uncertainty in HIFSs may be divided into three facets: fuzziness, intuitionism and hesitancy, we develop a fresh information-theoretic framework of uncertainty measures. We firstly propose the axiomatic principles of hesitant intuitionistic fuzzy entropy and give some distance-based entropy formulas. Then, a hesitant intuitionistic fuzzy cross-entropy is addressed to measure the discrimination of uncertain information between different HIFSs; the relationships between cross-entropy and entropy for HIFSs are also discussed. Moreover, some parameterized cross-entropy and entropy measures of HIFSs are investigated, and the decomposition formula suggests that hesitant intuitionistic fuzzy entropy may be expressed as the weighted average of fuzzy entropy, intuitionistic entropy and hesitant entropy. Finally, we demonstrate the efficiency of the proposed uncertainty measures for medical diagnosis and decision-making approach.

Keywords

Hesitant intuitionistic fuzzy set Hesitant intuitionistic fuzzy entropy Hesitant intuitionistic fuzzy cross-entropy 

Notes

Acknowledgements

The authors are highly grateful to any anonymous referee for their careful reading and insightful comments, and the views and opinions expressed are those of the authors. The work is supported by the Talent Introduction Project of Anhui University (No. J01006134) and the Natural Science Key Project of Anhui Sanlian University (No. kjzd 2016001).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefMATHGoogle Scholar
  2. Bedregal B et al (2014) Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Inf Sci 255:82–99MathSciNetCrossRefMATHGoogle Scholar
  3. Chen N, Xu ZS, Xia MM (2013) Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl Math Model 37:2197–2211MathSciNetCrossRefMATHGoogle Scholar
  4. Chen YF et al (2014) Approaches to multiple attribute decision making based on the correlation coefficient with dual hesitant fuzzy information. J Intell Fuzzy Syst 26:2547–2556MathSciNetMATHGoogle Scholar
  5. De Luca A, Terini S (1972) A definition of nonprobabilistic entropy in the setting of fuzzy sets theory. Inf Control 20:301–312MathSciNetCrossRefGoogle Scholar
  6. Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144MathSciNetCrossRefMATHGoogle Scholar
  7. Farhadinia B (2014a) A series of score functions for hesitant fuzzy sets. Inf Sci 277:102–110MathSciNetCrossRefMATHGoogle Scholar
  8. Farhadinia B (2014b) Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets. Int J Intell Syst 29:184–205CrossRefGoogle Scholar
  9. Farhadinia B (2014c) Distance and similarity measures for higher order hesitant fuzzy sets. Knowl Based Syst 55:43–48CrossRefMATHGoogle Scholar
  10. Hung WL, Yang MS (2006) Fuzzy entropy on intuitionistic fuzzy sets. Int J Intell Syst 114:443–451CrossRefMATHGoogle Scholar
  11. Mao JJ, Yao DB, Wang CC (2013) A novel cross-entropy and entropy measures of IFSs and their applications. Knowl Based Syst 48:37–45CrossRefGoogle Scholar
  12. Mu ZM, Zeng SZ, Baleentis T (2015) A novel aggregation principle for hesitant fuzzy elements. Knowl Based Syst 84:134–143CrossRefGoogle Scholar
  13. Pal NR et al (2013) Uncertainties with Atanassovs intuitionistic fuzzy sets: fuzziness and lack of knowledge. Inf Sci 228:61–74MathSciNetCrossRefMATHGoogle Scholar
  14. Peng JJ et al (2014) The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making. Int J Syst Sci. doi: 10.1080/00207721.2014.993744 Google Scholar
  15. Quirs P et al (2015) An entropy measure definition for finite interval-valued hesitant fuzzy sets. Knowl Based Syst 84:121–133CrossRefGoogle Scholar
  16. Shang XG, Jiang WS (1997) A note on fuzzy information measures. Pattern Recognit Lett 18:425–432CrossRefGoogle Scholar
  17. Singh P (2014) A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure. Appl Soft Comput 24:559–571CrossRefGoogle Scholar
  18. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539MATHGoogle Scholar
  19. Tyagi SK (2015) Correlation coefficient of dual hesitant fuzzy sets and its applications. Appl Math Model. doi: 10.1016/j.apm.2015.02.046 MathSciNetGoogle Scholar
  20. Vlachos IK, Sergiadis GD (2007) Intuitionistic fuzzy information–applications to pattern recognition. Pattern Recognit Lett 28:197–206CrossRefGoogle Scholar
  21. Wang C, Li Q, Zhou X (2014a) Multiple attribute decision making based on generalized aggregation operators under dual hesitant fuzzy environment. J Appl Math 2014. doi: 10.1155/2014/254271
  22. Wang HJ, Zhao XF, Wei GW (2014b) Dual hesitant fuzzy aggregation operators in multiple attribute decision making. J Intell Fuzzy Syst 26:2281–2290MathSciNetMATHGoogle Scholar
  23. Wang L et al (2014c) Distance and similarity measures of dual hesitant fuzzy sets with their applications to multiple attribute decision making. 2014 International conference on progress in informatics and computing (PIC). IEEE, pp 88–92Google Scholar
  24. Wei Y, Qiu J, Karimi HR (2017) Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults. IEEE Trans Circuits Syst I Regul Pap 64(1):170–181CrossRefGoogle Scholar
  25. Wei Y, Qiu J, Lam HK et al (2016a) Approaches to T–S fuzzy-affine-model-based reliable output feedback control for nonlinear Itô stochastic systems. IEEE Trans Fuzzy Syst 99:1–14Google Scholar
  26. Wei Y, Qiu J, Lam HK (2016b) A novel approach to reliable output feedback control of fuzzy-affine systems with time-delays and sensor faults. IEEE Trans Fuzzy Syst 11:1–15Google Scholar
  27. Wei Y, Qiu J, Shi P et al (2016c) A new design of H-infinity piecewise filtering for discrete-time nonlinear time-varying delay systems via T–S fuzzy affine models. IEEE Trans Syst Man Cybern Syst 99:1–14CrossRefGoogle Scholar
  28. Wei Y, Qiu J, Shi P et al (2016d) Fixed-order piecewise-affine output feedback controller for fuzzy-affine-model-based nonlinear systems with time-varying delay. IEEE Trans Circuits Syst I Regul Pap 12:945–958Google Scholar
  29. Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407MathSciNetCrossRefMATHGoogle Scholar
  30. Xia MM, Xu ZS (2012) Entropy/cross entropy-based group decision making intuitionistic fuzzy environment. Inf Fus 13:31–47CrossRefGoogle Scholar
  31. Xu ZS, Xia MM (2011a) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138MathSciNetCrossRefMATHGoogle Scholar
  32. Xu ZS, Xia MM (2011b) On distance and correlation measures of hesitant fuzzy information. Int J Intell Syst 26:410–425CrossRefMATHGoogle Scholar
  33. Xu ZS, Xia MM (2012) Hesitant fuzzy entropy and cross-entropy and their use in multiattribute decision-making. Int J Intell Syst 27:799–822CrossRefGoogle Scholar
  34. 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
  35. Ye J (2014) Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making. Appl Math Model 38:659–666MathSciNetCrossRefGoogle Scholar
  36. Yu DJ, Zhang WY, Xu YJ (2013) Group decision making under hesitant fuzzy environment with application to personnel evaluation. Knowl Based Syst 52:1–10CrossRefGoogle Scholar
  37. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefMATHGoogle Scholar
  38. Zadeh LA (1968) Probability measures of fuzzy events. J Math Anal Appl 23:421–427MathSciNetCrossRefMATHGoogle Scholar
  39. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning. Inf Sci 8:199–249MathSciNetCrossRefMATHGoogle Scholar
  40. Zhao N, Xu ZS, Liu FJ (2015) Uncertainty measures for Hesitant fuzzy information. Int J Intell Syst 00:1–19Google Scholar
  41. Zhu B, Xu Z, Xia M (2012) Dual hesitant fuzzy sets. J Appl Math 2012. doi: 10.1155/2012/879629
  42. Zhu B, Xu ZS (2014) Some results for dual hesitant fuzzy sets. J Intell Fuzzy Syst 26:1657–1668MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Economics SchoolAnhui UniversityHefeiPeople’s Republic of China
  2. 2.Foundation DepartmentAnhui Sanlian UniversityHefeiPeople’s Republic of China

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