Soft Computing

, Volume 23, Issue 14, pp 5835–5848 | Cite as

Distance and similarity measures of hesitant fuzzy sets based on Hausdorff metric with applications to multi-criteria decision making and clustering

  • Miin-Shen YangEmail author
  • Zahid Hussain
Methodologies and Application


Distance and similarity measures play a vital role to differentiate between two sets or objects. Different distance and similarity measures had been proposed for hesitant fuzzy sets (HFSs) in the literature, but either they are insufficient or not reflect desirable results. In this paper, the construction of new distance and similarity measures between HFSs based on Hausdorff metric is proposed. We first present a novel and simple method for calculating a distance between HFSs based on Hausdorff metric in a suitable and intuitive way. Two main features of the proposed approach are: (1) not necessary to add a minimum value, a maximum value or any value to the shorter one of hesitant fuzzy elements (HFEs) for extending it to the larger one of HFEs; and (2) no need to arrange HFEs either in ascending or descending order. This is because adding such values and arrangements of elements will not put any impact on final results. We then extend distance to similarity measure between HFSs. Furthermore, we claim some properties and also compare the proposed distance and similarity measures with existing methods. The comparison results demonstrate that the proposed distance and similarity measure are simpler, intuitive and better than most existing methods. Finally, we apply the proposed distance of HFSs to multi-criteria decision making and the similarity measure of HFSs to clustering.


Fuzzy sets Hesitant fuzzy sets Distance Similarity measure Hausdorff metric Multi-criteria decision making Clustering 



The authors would like to thank the anonymous referees for their helpful comments in improving the presentation of this paper.

Compliance with ethical standards

Conflict of interest

Both the authors declare that they have no conflict of interest.


  1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20:87–96CrossRefzbMATHGoogle Scholar
  2. Cabrerizo FJ, Al-Hmouz R, Morfeq A, Balamash AS, Martínez MA, Herrera-Viedma E (2017) Soft consensus measures in group decision making using unbalanced fuzzy linguistic information. Soft Comput 21(11):3037–3050CrossRefzbMATHGoogle Scholar
  3. Chaudhuri BB, Rosenfeld A (1999) A modified Hausdorff distance between fuzzy sets. Inf Sci 118:59–171MathSciNetzbMATHGoogle Scholar
  4. Del Moral MJ, Chiclana F, Tapia JM, Herrera-Viedma E (2018) A comparative study on consensus measures in group decision making. Int J Intell Syst. Google Scholar
  5. Diamond P, Kloeden P (1994) Metric spaces of fuzzy sets: theory and applications. World Scientific Publishing, SingaporeCrossRefzbMATHGoogle Scholar
  6. Dong Y, Zhao S, Zhang H, Chiclana F, Herrera-Viedma E (2018) A self-management mechanism for non-cooperative behaviors in large-scale group consensus reaching processes. IEEE Trans Fuzzy Syst. Google Scholar
  7. Dügenci M (2016) A new distance measure for interval valued intuitionistic fuzzy sets and its application to group decision making problems with incomplete weights information. Appl Soft Comput 41:120–134CrossRefGoogle Scholar
  8. Farhadinia B (2013a) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144MathSciNetCrossRefzbMATHGoogle Scholar
  9. Farhadinia B (2013b) A novel method of ranking hesitant fuzzy values for multiple attribute decision-making problems. Int J Intell Syst 28:752–767CrossRefGoogle Scholar
  10. Farhadinia B (2014) Distance and similarity measures for higher order hesitant fuzzy sets. Knowl-Based Syst 55:3–48CrossRefGoogle Scholar
  11. Farhadinia B, Herrera-Viedma E (2018) Sorting of decision making methods based on their outcomes using dominance-vector hesitant fuzzy based distance. Soft Comput. Google Scholar
  12. Gitinavard H, Mousavi SM, Vahdani B (2017) Soft computing-based new interval- valued hesitant fuzzy multi-criteria group assessment method with last aggregation to industrial decision problems. Soft Comput 21:3247–3265CrossRefzbMATHGoogle Scholar
  13. Gorzalczany B (1983) Approximate inference with interval valued fuzzy sets-an outline. In: Proceedings of the Polish symposium on interval and fuzzy mathematics, Poznan, pp 89–95Google Scholar
  14. Gou X, Xu Z, Liao H (2017) Multiple criteria decision making based on Bonferroni means with hesitant fuzzy linguistic information. Soft Comput 21:6515–6529CrossRefzbMATHGoogle Scholar
  15. Grzegorzewski P (2004) Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets Syst 148:319–328MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hung WL, Yang MS (2004) Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance. Pattern Recognit Lett 25:1603–1611CrossRefGoogle Scholar
  17. Huttenlocher DP, Klanderman GA, Rucklidge WJ (1993) Comparing images using the Hausdorff distance. IEEE Trans Pattern Anal Mach Intell 15:850–863CrossRefGoogle Scholar
  18. Hwang CM, Yang MS (2014) New similarity measures between generalized trapezoidal fuzzy numbers using the Jaccard index. Int J Uncertain Fuzziness Knowl-Based Syst 22:831–844MathSciNetCrossRefzbMATHGoogle Scholar
  19. Hwang CL, Yoon K (1981) Multiple attribute decision making: methods and applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
  20. Hwang CM, Yang MS, Hung WL (2012) On similarity, inclusion measure and entropy between type-2 fuzzy sets. Int J Uncertain Fuzziness Knowl-Based Syst 20:433–449MathSciNetCrossRefzbMATHGoogle Scholar
  21. Li D, Zeng W, Li J (2015) New distance and similarity measures on hesitant fuzzy sets and their applications in multiple criteria decision making. Eng Appl Artif Intell 40:11–16CrossRefGoogle Scholar
  22. Liao HC, Xu Z, Xia MM (2014) Distance & similarity measures for hesitant fuzzy linguistic term sets and their application in multi-criteria decision making. Inf Sci 271:125–142MathSciNetCrossRefzbMATHGoogle Scholar
  23. Liao H, Xu Z, Herrera-Viedma E, Herrera F (2018) Hesitant fuzzy linguistic term set and its application in decision making: a state-of-the-art survey. Int J Fuzzy Syst. MathSciNetGoogle Scholar
  24. Meng FY, Chen XH, Zhang Q (2014) Multi-attribute decision analysis under a linguistic hesitant fuzzy environment. Inf Sci 267:287–305MathSciNetCrossRefzbMATHGoogle Scholar
  25. Nalder SB Jr (1978) Hyperspaces of sets. Marcel Dekker, New YorkGoogle Scholar
  26. Olson CF (1998) A probabilistic formulation for Hausdorff matching. IEEE Comput Soc Conf Comput Vis Pattern Recognit 1998:150–156Google Scholar
  27. Onar SC, Büyüközkan G, Öztaysi B, Kahraman C (2016) A new hesitant fuzzy QFD approach: an application to computer workstation selection. Appl Soft Comput 46:1–16CrossRefGoogle Scholar
  28. Rodríguez RM, Martínez L, Herrera F (2012) Hesitant fuzzy linguistic term sets for decision making. IEEE Trans Fuzzy Syst 20:109–119CrossRefGoogle Scholar
  29. Rodríguez RM, Martínez L, Herrera F (2013) A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets. Inf Sci 241:28–42MathSciNetCrossRefzbMATHGoogle Scholar
  30. Rodríguez RM, Martínez L, Torra V, Xu ZS, Herrera F (2014) Hesitant fuzzy sets: state of arts and future directions. Int J Intell Syst 29:495–524CrossRefGoogle Scholar
  31. Rote G (1991) Computing the minimum Hausdorff distance between two point sets on a line under translation. Inf Process Lett 38:123–127MathSciNetCrossRefzbMATHGoogle Scholar
  32. Tong X, Yu L (2016) MADM based on distance and correlation coefficient measures with decision- maker preferences under a hesitant fuzzy environment. Soft Comput 20:4449–4461CrossRefGoogle Scholar
  33. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25:529–539zbMATHGoogle Scholar
  34. Turksen IB (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20:191–210MathSciNetCrossRefzbMATHGoogle Scholar
  35. Wu J, Chiclana F, Herrera-Viedma E (2015) Trust based consensus model for social network in an incomplete linguistic information context. Appl Soft Comput 35:827–839CrossRefGoogle Scholar
  36. Xu Z, Xia MM (2011a) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181:2128–2138MathSciNetCrossRefzbMATHGoogle Scholar
  37. Xu Z, Xia MM (2011b) Hesitant fuzzy information aggregation in decision making. Int J Approx Reason 52:395–407MathSciNetCrossRefzbMATHGoogle Scholar
  38. Yager RR (1986) On the theory of bags. Int J Gen Syst 13:23–37MathSciNetCrossRefGoogle Scholar
  39. Yager RR (1988) On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Trans Syst Man Cybern 18:183–190CrossRefzbMATHGoogle Scholar
  40. Ye J (2014) Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making. Appl Math Model 38:659–666MathSciNetCrossRefzbMATHGoogle Scholar
  41. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353CrossRefzbMATHGoogle Scholar
  42. Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning. Inf Sci 8:199–249MathSciNetCrossRefzbMATHGoogle Scholar
  43. Zeng W, Li D, Yin Q (2016) Distance and similarity measures between hesitant fuzzy sets and their applications in pattern recognitions. Pattern Recognit Lett 84:267–271CrossRefGoogle Scholar
  44. Zhang X, Xu Z (2015) Novel distance and similarity measures on hesitant fuzzy sets with application on cluster analysis. J Intell Fuzzy Syst 28:2279–2296Google Scholar
  45. Zhang H, Yang S (2016) Inclusion measure for typical hesitant fuzzy sets, the relative similarity measure and fuzzy entropy. Soft Comput 20:1277–1287CrossRefzbMATHGoogle Scholar
  46. Zhao H, Xu ZS, Wang H, Liu S (2017) Hesitant fuzzy multi-attribute decision-making based on the minimum deviation method. Soft Comput 21:3439–3459CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsChung Yuan Christian UniversityChung-LiTaiwan
  2. 2.Department of MathematicsKarakoram International UniversityGilgit-BaltistanPakistan

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