A new concept of Cosine similarity measures based on dual hesitant fuzzy sets and its possible applications

  • Yutao Zhang
  • Lei Wang
  • Xiaohan Yu
  • Changhua Yao


Most real-world problems are typical multi-criteria decision making problems which including ambiguity and subjectivity. Dual fuzzy sets (DHFS) are new extensions of fuzzy sets, which can cope with areas of vagueness effectively. In this paper, we propose one new similarity measure called cosine measure which has the ability to model various problems for DHFSs and study some formal relations. We provided three numerical examples including a medical diagnosis problem, an energy projects problem and a weapon selection problem to show the behaviour of the proposed cosine similarity measure.


Cosine similarity measure Pattern recognition Multi-criteria decision making Cluster algorithm Medical diagnosis problem 



The work is supported by the National Natural Science Foundation of China (No. 61702543, 71501186), and the 333 high-level talent training project of Jiangsu Province of China (No. BRA 2016542).


  1. 1.
    Zadeh, L.A.: Fuzzy sets. Inf. Contr. 8, 338–353 (1965)CrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, S.M., Jian, W.S.: Fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups, similarity measures and PSO techniques. Inf. Sci. 391–392, 65–79 (2017)CrossRefGoogle Scholar
  3. 3.
    Cheng, S.H., Chen, S.M., Jian, W.S.: Multicriteria decision making based on the TOPSIS method and similarity measures between intuitionistic fuzzy values. Inf. Sci. 367–368(1), 279–295 (2016)CrossRefGoogle Scholar
  4. 4.
    Martinez-Gil, Jorge: CoTO: A novel approach for fuzzy aggregation of semantic similarity measures. Cogn. Syst. Res. 40, 8–17 (2016)CrossRefGoogle Scholar
  5. 5.
    Chutia, R., Gogoi, M.K.: Fuzzy risk analysis in poultry farming based on a novel similarity measure of fuzzy numbers. Appl. Soft Comput. (2018). Google Scholar
  6. 6.
    Liao, H.C., Xu, Z.S.: Approaches to manage hesitant fuzzy linguistic information based on the cosine distance and similarity measures for HFLTSs and their application in qualitative decision making. Expert Syst. Appl. 42(12), 5328–5336 (2015)CrossRefGoogle Scholar
  7. 7.
    Khorshidi, H.A., Nikfalazar, S.: An improved similarity measure for generalized fuzzy numbers and its application to fuzzy risk analysis. Appl. Soft Comput. 52, 478–486 (2017)CrossRefGoogle Scholar
  8. 8.
    Sen, S., Patra, K., Mondal, S.K.: Fuzzy risk analysis in familial breast cancer using a similarity measure of interval-valued fuzzy numbers. Pac. Sci. A 18, 203–221 (2016)Google Scholar
  9. 9.
    Chen, T.Y.: Remoteness index-based pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis. Inf. Fusion. 41, 129–150 (2018)CrossRefGoogle Scholar
  10. 10.
    Li, J.P., Yao, X.Y., Sun, X.L., Wu, D.S.: Determining the fuzzy measures in multiple criteria decision aiding from the tolerance perspective. Eur. J. Oper. Res. 264(2), 428–439 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Milošević, P., Petrović, B., Jeremić, V.: IFS-IBA similarity measure in machine learning algorithms. Expert Syst. Appl. 89, 296–305 (2017)CrossRefGoogle Scholar
  12. 12.
    Lu, C.Q., Wang, S.P., Wang, X.J.: A multi-source information fusion fault diagnosis for aviation hydraulic pump based on the new evidence similarity distance. Aerosp. Sci. Technol. 71, 392–401 (2017)CrossRefGoogle Scholar
  13. 13.
    Su, P., Shen, Q., Chen, T.H., Shang, C.J.: Ordered weighted aggregation of fuzzy similarity relations and its application to detecting water treatment plant malfunction. Eng. Appl. Artif. Intell. 66, 17–29 (2017)CrossRefGoogle Scholar
  14. 14.
    He, X.X., Li, Y.F., Qin, K.Y., Meng, D.: On the TL-transitivity of fuzzy similarity measures. Fuzzy Set. Syst. 322, 54–69 (2017)CrossRefzbMATHGoogle Scholar
  15. 15.
    Zhu, B., Xu, Z.S., Xia, M.M.: Dual hesitant fuzzy sets. J. Appl. Math. 12, 1–12 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Atanassov, K.: Intuitionistic fuzzy sets. Fuzzy Set. Syst. 20, 87–96 (1986)CrossRefGoogle Scholar
  17. 17.
    Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25, 529–539 (2010)zbMATHGoogle Scholar
  18. 18.
    Torra, V., Narukawa, Y.: On hesitant fuzzy sets and decision. In: The 18th IEEE International Conference on Fuzzy Systems, pp. 1378–1382. Jeju Island, Korea (2009)Google Scholar
  19. 19.
    Wang, P.Z.: Fuzzy Sets and its Applications. Shanghai Science and Technology Press, Shanghai (1983)zbMATHGoogle Scholar
  20. 20.
    Kang, Y., Wu, S., Cao, D., Weng, W.: New hesitation-based distance and similarity measures on intuitionistic fuzzy sets and their applications. Int. J. Syst. Sci. 12, 1–17 (2018)MathSciNetGoogle Scholar
  21. 21.
    Wang, L., Ni, M.F., Zhu, L.: Correlation measures of dual hesitant fuzzy sets. J. Appl. Math. 11, 1–12 (2013)Google Scholar
  22. 22.
    Ye, J.: Improved cosine similarity measures of simplified neutrosophic sets for medical diagnoses. Artif. Intell. Med. 63, 171–179 (2015)CrossRefGoogle Scholar
  23. 23.
    Wang, L., Zheng, X., Zhang, L., Yue, Q.: Notes on distance and similarity measures of dual hesitant fuzzy sets. IAENG Int. J. Appl. Math. 46(4), 488–494 (2016)MathSciNetGoogle Scholar
  24. 24.
    Bustince, H., Burillo, P.: Correlation of interval-valued intuitionistic fuzzy sets. Fuzzy Set. Syst. 74, 237–244 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Grzegorzewski, P.: Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Set. Syst. 148, 319–328 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Xu, Z.S., Xia, M.M.: Some similarity measures of intuitionistic fuzzy sets and their application to multiple attribute decision making. Fuzzy Optim. Decis. Mak. 6, 109–121 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Szmidt, E., Kacprzyk, J.: Distances between intuitionistic fuzzy sets. Fuzzy Set. Syst. 114, 505–518 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xu, Z.S., Xia, M.M.: On distance and correlation measures of hesitant fuzzy information. Int. J. Intell. Syst. 26(5), 410–425 (2011)CrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, L., Shen, Q.G., Zhu, L.: Dual hesitant fuzzy power aggregation operators based on Archimedean t-conorm and t-norm and their application to multiple attribute group decision making. Appl. Soft Comput. 38, 23–50 (2016)CrossRefGoogle Scholar
  30. 30.
    Liu, H.W., Wang, G.J.: Multi-criteria decision making methods based on intuitionistic fuzzy sets. Eur. J. Oper. Res. 197, 220–233 (2007)CrossRefzbMATHGoogle Scholar
  31. 31.
    Merigó, J.M., Gil-Lafuente, A.M.: The induced generalized OWA operator. Inf. Sci. 179, 729–741 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Xu, Z.S., Chen, J., Wu, J.J.: Clustering algorithm for intuitionistic fuzzy sets. Inf. Sci. 78, 3775–3790 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhao, H., Xu, Z.S., Wang, Z.: Intuitionistic fuzzy clustering algorithm based on boolean matrix and association measure. Int. J. Inf. Technol. Decis. 12(1), 95–118 (2013)CrossRefGoogle Scholar
  34. 34.
    Chen, N., Xu, Z.S., Xia, M.M.: Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 37, 2197–2211 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kahraman, C., Kaya, I.: A fuzzy multicriteria methodology for selection among energy alternatives. Expert Syst. Appl. 37, 6270–6281 (2010)CrossRefGoogle Scholar
  36. 36.
    Metin, D., Serkan, Y., Nevzat, K.: Weapon selection using the AHP and TOPSIS methods under fuzzy environment. Expert Syst. Appl. 36, 8143–8151 (2009)CrossRefGoogle Scholar
  37. 37.
    Xia, M.M., Xu, Z.S.: Hesitant fuzzy information aggregation in decision making. Int. J. Approx. Reason. 52(3), 395–407 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Economics and ManagementNanjing University of Science and TechnologyNanjingChina
  2. 2.Post-Doctoral Research CenterNanjing General HospitalNanjingChina
  3. 3.College of Communication EngineeringArmy Engineering University of PLANanjingChina
  4. 4.College of Command and Control EngineeringArmy Engineering University of PLANanjingChina

Personalised recommendations