Skip to main content
SpringerLink
Log in

Distance measures on intuitionistic hesitant fuzzy set and its application in decision-making

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

Intuitionistic hesitant fuzzy set (IHFS) provides a valid mean for dealing with the uncertainty of complex problems. Information measures on IHFS can measure the uncertain information, so that we shall introduce a method to construct a class of distance measures for IHFS in this paper. For constructing more objective distance measures to reflect the actual situation, we consider the information content and information clarity of IHFS simultaneously and utilize different functions to adjust their contribution. Its superiority is evidenced by an example of pattern recognition that the proposed distance measure improves numerically results obtained with existing distance measures. In particular, we investigate the connection between distance measure, similarity measure, and entropy measure of IHFS, and prove that they can be constructed mutually under this axiomatic framework. On this basis, we apply the proposed entropy measure to determine criteria weights in multi-criteria decision-making problems and design an extended intuitionistic hesitant fuzzy technique for order preference by similarity to an ideal solution (TOPSIS) method, which effectiveness is presented by a practical application.

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 KT (1986) Intuitionistic fuzzy sets. Fuzzy Set Syst 20(1):87–96

    Article  Google Scholar 

  • Bedregal B, Reiser R, Bustince H et al (2014) Aggregation functions for typical hesitant fuzzy elements and the action of automorphisms. Inf Sci 255:82–99

    Article  MathSciNet  Google Scholar 

  • Boran FE, Akay D (2014) A biparametric similarity measure on intuitionistic fuzzy sets with applications to pattern recognition. Inf Sci 255:45–57

    Article  MathSciNet  Google Scholar 

  • Boran FE, Genc S, Kurt M (2009) A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Expert Syst Appl 36(8):11363–11368

    Article  Google Scholar 

  • Bustince H, Burillo P, Soria F (2003) Automorphisms, negations and implication operators. Fuzzy Sets Syst 134(2):209–229

    Article  MathSciNet  Google Scholar 

  • Chen SM (1995) Measures of similarity between vague sets. Fuzzy Sets Syst 74(2):217–223

    Article  MathSciNet  Google Scholar 

  • Chen SM (1997) Similarity measures between vague sets and between elements. IEEE Trans Syst Man Cybern 27(1):153–158

    Article  Google Scholar 

  • Chen SM, Chang CH (2015) A novel similarity measure between Atanassov’s intuitionistic fuzzy sets based on transformation techniques with applications to pattern recognition. Inf Sci 291:96–114

    Article  Google Scholar 

  • Chen TY, Li CH (2010) Determining objective weights with intuitionistic fuzzy entropy measures: a comparative analysis. Inf Sci 180(21):4207–4222

    Article  Google Scholar 

  • Chen MF, Tzeng GH (2004) Combining grey relation and TOPSIS concepts for selecting an expatriate host country. Math Comput Model 40(13):1473–1490

    Article  Google Scholar 

  • Chen JJ, Huang XJ, Tang J (2018) Distance measures for higher order dual hesitant fuzzy sets. Comput Appl Math 37(2):1784–1806

    Article  MathSciNet  Google Scholar 

  • Costa V, Bedregal B (2020) On typical hesitant fuzzy automata. Soft Comput 24:8725–8736

    Article  Google Scholar 

  • Costaa V, Farias A, Bedregal B et al (2018) Combining multiple algorithms in classifier ensembles using generalized mixture functions. Neurocomputing 313:402–414

    Article  Google Scholar 

  • Couso I, Garrido L, Sanchez L (2013) Similarity and dissimilarity measures between fuzzy sets: a formal relational study. Inf Sci 6(229):122–141

    Article  MathSciNet  Google Scholar 

  • De Luca A, Termini S (1972) A definition of a non-probabilistic entropy in the setting of fuzzy sets theory. Inf Control 20:301–312

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Farhadinia B (2013b) A theoretical development on the entropy of interval-valued fuzzy sets based on the intuitionistic distance and its relationship with similarity measure. Knowl Based Syst 39(2):79–84

    Article  MathSciNet  Google Scholar 

  • Farias A, Costa V, Santiago R, Bedregal B (2016) The image reduction process based on generalized mixture functions. In: Annual conference of the North American Fuzzy Information Processing Society (NAFIPS), 2016

  • Grabisch M, Marichal JL, Mesiar R et al (2009) Aggregation functions. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Hatami-Marbini A, Tavana M (2011) An extension of the Electre I method for group decision-making under a fuzzy environment. Omega 39(4):373–386

    Article  Google Scholar 

  • He XX, Li YF, Qin K, Meng D (2020) Distance measures on intuitionistic fuzzy sets based on intuitionistic fuzzy dissimilarity functions. Soft Comput 24:523–541

    Article  Google Scholar 

  • Hong DH, Kim C (1999) A note on similarity measures between vague sets and between elements. Inf Sci 115:83–96

    Article  MathSciNet  Google Scholar 

  • Hussain Z, Yang MS (2018) Entropy for hesitant fuzzy sets based on Hausdorff metric with construction of hesitant fuzzy TOPSIS. Int J Fuzzy Syst 20(8):2517–2533

    Article  MathSciNet  Google Scholar 

  • Hwang CL, Yoon K (1981) Multiple attributes decision making methods and applications. Springer, Berlin, pp 58–191. https://doi.org/10.1007/978-3-642-48318-9

  • Lalotra Sumita, Singh Surender (2020) Knowledge measure of hesitant fuzzy set and its application in multi-attribute decision-making. Comput Appl Math 39(2):39–86

    Article  MathSciNet  Google Scholar 

  • Li DF, Cheng CT (2002) New similarity measure of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognit Lett 23(1–3):221–225

    MATH  Google Scholar 

  • Liu XC (1992) Entropy, distance measure and similarity measure of fuzzy sets and their relations. Fuzzy Set Syst 52(3):305–318

    Article  MathSciNet  Google Scholar 

  • Mitchell HB (2003) On the Dengfeng–Chuntian similarity measure and its application to pattern recognition. Pattern Recognit Lett 24(16):3101–3104

    Article  Google Scholar 

  • Onat NC, Gumus S, Kucukvar M (2015) Application of the TOPSIS and intuitionistic fuzzy set approaches for ranking the life cycle sustainability performance of alternative vehicle technologies. Sustain Prod Consum 6:12–25

    Article  Google Scholar 

  • Peng JJ, Wang JQ, Wang J (2015) The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and their application in multi-criteria decision-making. Int J Syst Sci 46(13):2335–2350

    Article  MathSciNet  Google Scholar 

  • Shen F, Ma XS, Li ZY, Xu ZS, Cai DL (2018) An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation. Inf Sci 428:105–119

    Article  MathSciNet  Google Scholar 

  • Singh P (2017) Distance and similarity measures for multiple-attribute decision making with dual hesitant fuzzy sets. Comput Appl Math 36(1):111–126

    Article  MathSciNet  Google Scholar 

  • Suo CF, Li YM, Li ZH (2020a) An (R, S)-norm information measure for hesitant fuzzy sets and its application in decision making. Comput Appl Math. https://doi.org/10.1007/s40314-020-01339-9

  • Suo CF, Li YM, Li ZH (2020b) A series of information measures of hesitant fuzzy soft sets and their application in decision making. Soft Comput. https://doi.org/10.1007/s00500-020-05485-4

  • Suo CF, Li YM, Li ZH (2020c) On \(n\)-polygonal interval-value fuzzy sets and numbers. Fuzzy Set Syst. https://doi.org/10.1016/j.fss.2020.10.014

  • Tan CQ, Zhi S (2018) TOPSIS method based on intuitionistic hesitation fuzzy set. Oper Res Manag Sci 27(3):66–73. https://doi.org/10.12005/orms.2018.0061

    Article  Google Scholar 

  • Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539

    MATH  Google Scholar 

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

    Article  Google Scholar 

  • Xia MM, Xu ZS (2011) Hesitant fuzzy information aggregation in decision making. Int J Appox Reason 52(3):395–407

    Article  MathSciNet  Google Scholar 

  • Xu ZS, Zhang XL (2013) Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information. Knowl Based Syst 52:53–64

    Article  Google Scholar 

  • Yager RR (1979) On the measure of fuzziness and negation part I: membership in the unit interval. Int J Gen Syst 5:221–229

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  Google Scholar 

  • Zeng W, Guo P (2008) Normalized distance, similarity measure, inclusion measure and entropy of interval-valued fuzzy sets and their relationship. Inf Sci 178(5):1334–1342

    Article  MathSciNet  Google Scholar 

  • Zhang Q, Xing H, Liu F, Ye J, Tang P (2014) Some new entropy measures for interval-valued intuitionistic fuzzy sets based on distances and their relationships with similarity and inclusion measures. Inf Sci 283:55–69

    Article  MathSciNet  Google Scholar 

  • Zhu B, Xu Z, Xia M (2012a) Hesitant fuzzy geometric Bonferroni means. Inf Sci 205:72–85

    Article  MathSciNet  Google Scholar 

  • Zhu B, Xu Z, Xia M (2012b) Dual hesitant fuzzy sets. J Appl Math 1–13

Download references

Acknowledgements

The authors would like to thank the anonymous referees for helping them refine the ideas presented in this paper and improve the clarity of the presentation. This paper was supported by National Science Foundation of China (Grant nos. 11671244, 12071271).

Author information

Authors and Affiliations

  1. College of Computer Science, Shaanxi Normal University, Xi’an, 710062, Shaanxi, People’s Republic of China

    Xiang Chen & Yongming Li

  2. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, Shaanxi, People’s Republic of China

    Chunfeng Suo

Authors
  1. Xiang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chunfeng Suo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yongming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongming Li.

Ethics declarations

Conflict of interest

All authors agree to contribute without conflict of interest.

Additional information

Communicated by Regivan Hugo Nunes Santiago.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Suo, C. & Li, Y. Distance measures on intuitionistic hesitant fuzzy set and its application in decision-making. Comp. Appl. Math. 40, 84 (2021). https://doi.org/10.1007/s40314-021-01478-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-021-01478-7

Keywords

Mathematics Subject Classification

Search

Navigation