Soft Computing

, Volume 22, Issue 4, pp 1237–1245 | Cite as

New distance measures on hesitant fuzzy sets based on the cardinality theory and their application in pattern recognition

  • Fangwei Zhang
  • Shuyan Chen
  • Jianbo Li
  • Weiwei Huang
Methodologies and Application


As a generalization of fuzzy set, hesitant fuzzy set (HFS) permits the membership of an element to a set having a set of possible values. Distance is one of important tools in measuring the relationship between two HFSs. Based on the cardinality theory, some novel distances which take the cardinal numbers of HFSs into account have been introduced using the concept of “multi-sets.” The main advantage of the distance measures is that they can more objectively and universally measure the relationship between HFSs than the existing methods. Finally, the performance of the proposed distance measures is illustrated through two pattern recognition examples in port enterprise management and transportation infrastructure construction.


Hesitant fuzzy sets Hesitant fuzzy elements Cardinality theory Distance measure Weighted average operator Pattern recognition 



The authors are thankful to the editor Dr. Aniello Castiglione and the anonymous reviewers for their insightful and constructive suggestions and helpful comments in improving this paper. The authors are thankful to Dr. Lihua Luo, Dr. Zhijun Gao, and Associate Prof. Jihong Chen, College of Transport and Communications, Shanghai Maritime University. They have provided useful guidance for this paper in its revised process.

Funding The work of the first author is partially supported in part by the National Natural Science Foundation of China (51508319, 51409157), the research program of the National Special Authorized Social Science Fund of China (07@ZH005) and the Nature and Science Fund from Zhejiang Province Ministry of Education (Y201327642). The second author is partially supported by National Natural Science Foundation of China (61374195). The third author is partially supported by the Humanities and Social Fund of Ministry of Education in China (12YJC910004), the National Natural Science Foundation of China (11201190,11571148) and “Qinglan Project” in Jiangsu Province.

Compliance with ethical standards

Conflict of interest

All the four authors declare that they have no conflict of interest.

Ethical approval

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


  1. Atanassov K (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96CrossRefzbMATHGoogle Scholar
  2. Atanassov K, Gargov G (1989) Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst 31(3):343–349MathSciNetCrossRefzbMATHGoogle Scholar
  3. Farhadinia B (2013) Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf Sci 240:129–144MathSciNetCrossRefzbMATHGoogle Scholar
  4. Goldberg D (1991) What every computer scientist should know about floating-point arithmetic. ACM Comput Surv (CSUR). 23(1):5–48CrossRefGoogle Scholar
  5. Li DQ, Zeng WY, Li JH (2015a) 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
  6. Li DQ, Zeng WY, Zhao YB (2015b) Note on distance measure of hesitant fuzzy sets. Inf Sci 321:103–115MathSciNetCrossRefGoogle Scholar
  7. Ministry of Housing and Urban-Rural Development of the People’s Republic of China (2010) Code for Planning of Intersections on Urban Roads. Standards Press of China, BeijingGoogle Scholar
  8. Ministry of Construction of the People’s Republic of China (1995) Code for transport planning on urban road. Standards Press of China, BeijingGoogle Scholar
  9. Ministry of Housing and Urban-Rural Development of the People’s Republic of China (2006) Code for design of urban road engineering. Standards Press of China, BeijingGoogle Scholar
  10. Nguyen H (2015) A novel similarity/dissimilarity measure for intuitionistic fuzzy sets and its application in pattern recognition. Expert Syst Appl 45:97–107CrossRefGoogle Scholar
  11. Peng DH, Gao CY, Gao ZF (2013) Generalized hesitant fuzzy synergetic weighted distance measures and their application to multiple criteria decision-making. Appl Math Model 37(8):5837–5850MathSciNetCrossRefzbMATHGoogle Scholar
  12. Reuben J, Kittur HM, Shoaib M (2014) A novel clock generation algorithm for system-on-chip based on least common multiple. Comput Electr Eng 40(7):2113–2125CrossRefGoogle Scholar
  13. Rodrguez RM, Martnez L, Torra V (2014) Hesitant fuzzy sets: state of the art and future directions. Int J Intell Syst 29(6):495–524CrossRefGoogle Scholar
  14. 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
  15. Torra V (2010) Hesitant fuzzy sets. Int J Intell Syst 25(6):529–539zbMATHGoogle Scholar
  16. Torra V, Narukawa Y (2009) On hesitant fuzzy sets and decision. The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, pp 1378–1382Google Scholar
  17. Tripathy BC, Baruah A (2009) New type of difference sequence spaces of fuzzy real numbers. Math Modell Analysis 14(3):391–397MathSciNetCrossRefzbMATHGoogle Scholar
  18. Van MI, Tauler R (2013) Multiway and multiset methods. Chemometr Intell Lab 129:1–2CrossRefGoogle Scholar
  19. Vanelslander T, Sys C (2014) Port business-market challenges and management actions. University Press Antwerp, AntwerpGoogle Scholar
  20. Xu ZS, Xia MM (2011a) Distance and similarity measures for hesitant fuzzy sets. Inf Sci 181(11):2128–2138MathSciNetCrossRefzbMATHGoogle Scholar
  21. Xu ZS, Xia MM (2011b) On distance and correlation measures of hesitant fuzzy information. Int J Intell Syst 26(5):410–425CrossRefzbMATHGoogle Scholar
  22. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353CrossRefzbMATHGoogle Scholar
  23. Zhang FW (2016) Several kinds of uncertain multi-attribute decision-making methods and their application in transportation management. People’s Communication Press, Beijing, pp 110–115Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Fangwei Zhang
    • 1
  • Shuyan Chen
    • 2
  • Jianbo Li
    • 3
  • Weiwei Huang
    • 4
  1. 1.College of Transport and CommunicationsShanghai Maritime UniversityShanghaiChina
  2. 2.School of TransportationSoutheast UniversityNanjingChina
  3. 3.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina
  4. 4.College of Teacher EducationZhaoqing UniverstyZhaoqingChina

Personalised recommendations