International Journal of Fuzzy Systems

, Volume 18, Issue 6, pp 1104–1116 | Cite as

Cross-Entropy and Prioritized Aggregation Operator with Simplified Neutrosophic Sets and Their Application in Multi-Criteria Decision-Making Problems

  • Xiao-hui Wu
  • Jian-qiang Wang
  • Juan-juan Peng
  • Xiao-hong Chen


Simplified neutrosophic sets (SNSs) can effectively solve the uncertainty problems, especially those involving the indeterminate and inconsistent information. Considering the advantages of SNSs, a new approach for multi-criteria decision-making (MCDM) problems is developed under the simplified neutrosophic environment. First, the prioritized weighted average operator and prioritized weighted geometric operator for simplified neutrosophic numbers (SNNs) are defined, and the related theorems are also proved. Then two novel effective cross-entropy measures for SNSs are proposed, and their properties are proved as well. Furthermore, based on the proposed prioritized aggregation operators and cross-entropy measures, the ranking methods for SNSs are established in order to solve MCDM problems. Finally, a practical MCDM example for coping with supplier selection of an automotive company is used to demonstrate the effectiveness of the developed methods. Moreover, the same example-based comparison analysis of between the proposed methods and other existing methods is carried out.


Simplified neutrosophic sets Prioritized aggregation operator Cross-entropy Multi-criteria decision-making 



The author would like to thank the editors and the anonymous referees for their valuable and constructive comments and suggestions that greatly help the improvement of this paper. This work is supported by the National Natural Science Foundation of China (Nos 71571193, 71271218, and 71431006).


  1. 1.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Yager, R.R.: Multiple objective decision-making using fuzzy sets. Int. J. Man-Mach. Stud. 9(4), 375–382 (1997)MATHCrossRefGoogle Scholar
  3. 3.
    Khatibi, V., Montazer, G.A.: Intuitionistic fuzzy set vs. fuzzy set application in medical pattern recognition. Artif. Intell. Med. 47(1), 43–52 (2009)CrossRefGoogle Scholar
  4. 4.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23(2), 610–614 (1993)MATHCrossRefGoogle Scholar
  6. 6.
    Liu, H.W., Wang, G.J.: Multi-criteria decision-making methods based on intuitionistic fuzzy sets. Eur. J. Oper. Res. 179(1), 200–233 (2007)MATHCrossRefGoogle Scholar
  7. 7.
    Pei, Z., Zheng, L.: A novel approach to multi-attribute decision making based on intuitionistic fuzzy sets. Expert Syst. Appl. 39(3), 2560–2566 (2012)CrossRefGoogle Scholar
  8. 8.
    Yu, D.J.: Multi-criteria decision making based on generalized prioritized aggregation operators under intuitionistic fuzzy environment. Int. J. Fuzzy Syst. 15(1), 47–54 (2013)MathSciNetGoogle Scholar
  9. 9.
    Tan, C.Q., Chen, X.H.: Dynamic similarity measures between intuitionistic fuzzy sets and its application. Int J Fuzzy Syst. 16(4), 511–519 (2014)MathSciNetGoogle Scholar
  10. 10.
    Tao, Z.F., Chen, H.Y., Zhou, L.G., Liu, J.P.: A generalized multiple attributes group decision making approach based on intuitionistic fuzzy sets. Int. J. Fuzzy Syst. 16(2), 184–195 (2014)MathSciNetGoogle Scholar
  11. 11.
    Wang, J.Q., Zhou, P., Li, K.J., Zhang, H.Y., Chen, X.H.: Multi-criteria decision-making method based on normal intuitionistic fuzzy-induced generalized aggregation operator. TOP 22, 1103–1122 (2014)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Puri, J., Yadav, S.P.: Intuitionistic fuzzy data envelopment analysis: an application to the banking sector in India. Expert Syst. Appl. 42(11), 4982–4998 (2015)CrossRefGoogle Scholar
  13. 13.
    De, S.K., Biswas, R., Roy, A.R.: An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets Syst. 117(2), 209–213 (2001)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Shinoj, T.K., Sunil, J.J.: Intuitionistic fuzzy multisets and its application in medical diagnosis. Int. J. Math. Comput. Sci. 6, 34–37 (2012)Google Scholar
  15. 15.
    Vlachos, I.K., Sergiadis, G.D.: Intuitionistic fuzzy information–applications to pattern recognition. Pattern Recognit. Lett. 28(2), 197–206 (2007)CrossRefGoogle Scholar
  16. 16.
    Li, D.F., Cheng, C.T.: New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognit. Lett. 23(1), 221–225 (2002)MATHGoogle Scholar
  17. 17.
    Joshi, B.P., Kumar, S.: Fuzzy time series model based on intuitionistic fuzzy sets for empirical research in stock market. Int. J. Appl. Evol. Comput. 3(4), 71–84 (2012)CrossRefGoogle Scholar
  18. 18.
    Li, L., Yang, J., Wu, W.: Intuitionistic fuzzy hopfield neural network and its stability. Neural Netw. World 21(5), 461–472 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Khatibi, V., Iranmanesh, H., Keramati, A.: A neuro-IFS intelligent system for marketing strategy selection. Innov. Comput. Technol. 241, 61–70 (2011)CrossRefGoogle Scholar
  20. 20.
    Atanassov, K.T., Gargov, G.: Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Yue, Z., Jia, Y.: An application of soft computing technique in group decision making under interval-valued intuitionistic fuzzy environment. Appl. Soft Comput. 13(5), 2490–2503 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yu, D.J., Merigó, J.M., Zhou, L.G.: Interval-valued multiplicative intuitionistic fuzzy preference relations. Int. J. Fuzzy Syst. 15(4), 412–422 (2013)MathSciNetGoogle Scholar
  23. 23.
    Wang, J.Q., Han, Z.Q., Zhang, H.Y.: Multi-criteria group decision-making method based on intuitionistic interval fuzzy information. Group Decis. Negot. 23(4), 715–733 (2014)CrossRefGoogle Scholar
  24. 24.
    Wei, G.W.: Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information. Int. J. Fuzzy Syst. 17(3), 484–489 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    De Miguel, L., Bustince, H., Fernandez, J., Induráin, E., Kolesárová, A., Mesiar, R.: Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making. Inf. Fusion 27, 189–197 (2016)CrossRefGoogle Scholar
  26. 26.
    Torra, V.: Hesitant fuzzy sets. Int. J. Intell. Syst. 25(6), 529–539 (2010)MATHGoogle Scholar
  27. 27.
    Chen, N., Xu, Z.S., Xia, M.M.: Interval-valued hesitant preference relations and their applications to group decision making. Knowl.-Based Syst. 37, 528–540 (2013)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, J.Q., Wu, J.T., Wang, J., Zhang, H.Y., Chen, X.H.: Multi-criteria decision-making methods based on the Hausdorff distance of hesitant fuzzy linguistic numbers. Soft. Comput. 20(4), 1621–1633 (2016)CrossRefGoogle Scholar
  29. 29.
    Wang, J., Wang, J.Q., Zhang, H.Y., Chen, X.H.: Multi-criteria group decision-making approach based on 2-tuple linguistic aggregation operators with multi-hesitant fuzzy linguistic information. Int. J. Fuzzy Syst. 18(1), 81–97 (2016)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhou, H., Wang, J., Zhang, H.Y., Chen, X.H.: Linguistic hesitant fuzzy multi-criteria decision-making method based on evidential reasoning. Int. J. Syst. Sci. 47(2), 314–327 (2016)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Wang, J., Wang, J.Q., Zhang, H.Y., Chen, X.H.: Multi-criteria decision-making based on hesitant fuzzy linguistic term sets: an outranking approach. Knowl.-Based Syst. 86, 224–236 (2015)CrossRefGoogle Scholar
  32. 32.
    Tian, Z.P., Wang, J., Wang, J.Q., Chen, X.H.: Multi-criteria decision-making approach based on gray linguistic weighted Bonferroni mean operator. Int. Trans. Oper. Res. (2015). doi: 10.1111/itor.12220 Google Scholar
  33. 33.
    Peng, J.J., Wang, J.Q., Wu, X.H., Zhang, H.Y., Chen, X.H.: The fuzzy cross-entropy for intuitionistic hesitant fuzzy sets and its application in multi-criteria decision-making. Int. J. Syst. Sci. 46(13), 2335–2350 (2015)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Wang, H., Smarandache, F., Zhang, Y.Q., Sunderraman, R.: Single valued neutrosophic sets. Multispace Multistruct 4, 410–413 (2010)MATHGoogle Scholar
  35. 35.
    Smarandache, F.: A unifying field in logics: neutrosophy: neutrosophic probability, set and logics. American Research Press, Rehoboth (1999)MATHGoogle Scholar
  36. 36.
    Wang, H., Smarandache, F., Zhang, Y.Q., Sunderraman, R.: Interval neutrosophic sets and logic: theory and applications in computing. Hexis, Phoenix (2005)MATHGoogle Scholar
  37. 37.
    Liu, P.D., Chu, Y.C., Li, Y.W., Chen, Y.B.: Some generalized neutrosophic number Hamacher aggregation operators and their application to group decision making. Int. J. Fuzzy Syst. 16(2), 242–255 (2014)Google Scholar
  38. 38.
    Liu, P.D., Li, H.G.: Multiple attribute decision-making method based on some normal neutrosophic Bonferroni mean operators. Neural Comput. Appl. (2015). doi: 10.1007/s00521-015-2048-z Google Scholar
  39. 39.
    Maji, P.K.: Weighted neutrosophic soft sets approach in a multi-criteria decision making problem. J. New Theory 5, 1–12 (2015)Google Scholar
  40. 40.
    Peng, J.J., Wang, J.Q., Wu, X.H., Wang, J., Chen, X.H.: Multi-valued neutrosophic sets and power aggregation operators with their Applications in multi-criteria group decision-making problems. Int. J. Comput. Intell. Syst. 8(2), 345–363 (2015)CrossRefGoogle Scholar
  41. 41.
    Tian, Z.P., Wang, J., Zhang, H.Y., Chen, X.H., Wang, J.Q.: Simplified neutrosophic linguistic normalized weighted Bonferroni mean operator and its application to multi-criteria decision-making problems. Filomat (2015)Google Scholar
  42. 42.
    Guo, Y.H., Şengür, A., Tian, J.W.: A novel breast ultrasound image segmentation algorithm based on neutrosophic similarity score and level set. Comput. Methods Programs Biomed. 123, 43–53 (2016)CrossRefGoogle Scholar
  43. 43.
    Ye, J.: A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. J. Intell. Fuzzy Syst. 26(5), 2459–2466 (2014)MathSciNetMATHGoogle Scholar
  44. 44.
    Ye, J.: Single valued neutrosophic cross-entropy for multicriteria decision making problems. Appl. Math. Model. 38(3), 1170–1175 (2014)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Ye, J.: Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Int. J. Gen Syst 42(4), 386–394 (2013)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Tian, Z.P., Zhang, H.Y., Wang, J., Wang, J.Q., Chen, X.H.: Multi-criteria decision-making method based on a cross-entropy with interval neutrosophic sets. Int. J. Syst. Sci. (2015). doi: 10.1080/00207721.2015.1102359 MATHGoogle Scholar
  47. 47.
    Zhang, H.Y., Ji, P., Wang, J., Chen, X.H.: Improved weighted correlation coefficient based on integrated weight for interval neutrosophic sets and its application in multi-criteria decision making problems. Int. J. Comput. Intell. Syst. 8(6), 1027–1043 (2015)CrossRefGoogle Scholar
  48. 48.
    Zhang, H.Y., Wang, J., Chen, X.H.: An outranking approach for multi-criteria decision-making problems with interval-valued neutrosophic sets. Neural Comput. Appl. 27(3), 615–627 (2016)CrossRefGoogle Scholar
  49. 49.
    Ye, J.: Vector similarity measures of simplified neutrosophic sets and their application in multicriteria decision making. Intern. J. Fuzzy Syst. 16(2), 2204–2211 (2014)Google Scholar
  50. 50.
    Peng, J.J., Wang, J., Zhang, H.Y., Chen, X.H.: An outranking approach for multi-criteria decision-making problems with simplified neutrosophic sets. Appl. Soft Comput. 25, 336–346 (2014)CrossRefGoogle Scholar
  51. 51.
    Peng, J.J., Wang, J.Q., Wang, J., Zhang, H.Y., Chen, X.H.: Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. Int. J. Syst. Sci. 47(10), 2342–2358 (2016)MATHCrossRefGoogle Scholar
  52. 52.
    Yager, R.R.: Prioritized aggregation operators. Int. J. Approx. Reason. 48, 263–274 (2008)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Kullback, S.: Information theory and statistics. Wiley, New York (1959)MATHGoogle Scholar
  54. 54.
    Shang, X.G., Jiang, W.S.: A note on fuzzy information measures. Pattern Recognit. Lett. 18, 425–432 (1997)CrossRefGoogle Scholar
  55. 55.
    Ho, W., Xu, X., Dey, P.K.: Multi-criteria decision making approaches for supplier evaluation and selection: a literature review. Eur. J. Oper. Res. 202(1), 16–24 (2010)MATHCrossRefGoogle Scholar
  56. 56.
    Boran, F.E.: A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method. Expert Syst. Appl. 36(8), 11363–11368 (2009)CrossRefGoogle Scholar
  57. 57.
    Liu, P.D., Wang, Y.M.: Multiple attribute decision-making method based on single-valued neutrosophic normalized weighted Bonferroni mean. Neural Comput. Appl. 25(7–8), 2001–2010 (2014)CrossRefGoogle Scholar

Copyright information

© Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Xiao-hui Wu
    • 1
    • 2
  • Jian-qiang Wang
    • 1
  • Juan-juan Peng
    • 1
    • 2
  • Xiao-hong Chen
    • 1
  1. 1.School of BusinessCentral South UniversityChangshaPeople’s Republic of China
  2. 2.School of Economics and ManagementHubei University of Automotive TechnologyShiyanPeople’s Republic of China

Personalised recommendations