Skip to main content
SpringerLink
Log in

Cross Entropy for Discrete Z-numbers and Its Application in Multi-Criteria Decision-Making

  • Published:
International Journal of Fuzzy Systems Aims and scope Submit manuscript

Abstract

Decisions are based on information, and the reliability of information affects the quality of decision-making. Z-number, produced by Zadeh, considers the fuzzy restriction and the reliability restriction of decision information simultaneously. Many scholars have conducted in-depth research on Z-number, and the concept has great application potential in the field of economic management. However, certain problems with the basic operations of Z-number still exist. Entropy is a measure of information uncertainty, and research on entropy and Z-number continues to be rare. This study initially defines the cross entropy of fuzzy restriction and that of the reliability of Z-numbers. On this basis, a comprehensive weighted cross entropy is constructed, which is used to compare two discrete Z-numbers from the perspective of information entropy. Furthermore, one extended Technique for Order Preference by Similarity to Ideal Solution approach is developed to solve a multi-criteria decision-making problem under discrete Z-context. An example of the ranking of job candidates for human resource management is then presented to illustrate the availability of the proposed method along with the sensitivity and comparative analyses for verifying the validity and applicability of the proposed method.

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.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Liang, R., Wang, J.: A linguistic intuitionistic cloud decision support model with sentiment analysis for product selection in E-commerce. Int. J. Fuzzy Syst. 21(3), 963–977 (2019)

    Article  MathSciNet  Google Scholar 

  2. Zhou, H., Wang, J., Zhang, H.: Stochastic multicriteria decision-making approach based on SMAA-ELECTRE with extended gray numbers. Int. Trans. Oper. Res. 26(5), 2032–2052 (2019)

    Article  MathSciNet  Google Scholar 

  3. Peng, H-g, Shen, K-w, He, S-s, Zhang, H-y, Wang, J-q: Investment risk evaluation for new energy resources: an integrated decision support model based on regret theory and ELECTRE III. Energy Convers. Manag. 183, 332–348 (2019)

    Article  Google Scholar 

  4. Ji, P., Zhang, H., Wang, J.: A fuzzy decision support model with sentiment analysis for items comparison in e-commerce: the case study of pconline.com. IEEE Trans. Syst. Man Cybern. Syst. (2018). https://doi.org/10.1109/TSMC.2018.2875163

    Article  Google Scholar 

  5. Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  MATH  Google Scholar 

  6. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Smarandache, F.: Neutrosophic logic—generalization of the intuitionistic fuzzy logic. In: Extractive Metallurgy of Nickel Cobalt & Platinum Group Metals, vol. 269, no. 51, pp. 49–53 (2010)

  8. Tian, Z-p, Wang, J., Wang, J-q, Zhang, H-y: Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development. Group Decis. Negot. 26(3), 597–627 (2017)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  10. Cuong, B.C., Kreinovich, V.: Picture fuzzy sets—a new concept for computational intelligence problems. In: 2013 Third World Congress on Information and Communication Technologies (WICT 2013), pp. 1–6 (2013)

  11. Aliev, R.A., Alizadeh, A.V., Huseynov, O.H.: The arithmetic of discrete Z-numbers. Inf. Sci. 290(C), 134–155 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Qiao, D., Shen, K., Wang, J., Wang, T.: Multi-criteria PROMETHEE method based on possibility degree with Z-numbers under uncertain linguistic environment. J. Ambient Intell. Humaniz. Comput. (2019). https://doi.org/10.1007/s12652-019-01251-z

    Article  Google Scholar 

  13. Shen, K-w, Wang, J-q: Z-VIKOR method based on a new weighted comprehensive distance measure of Z-number and its application. IEEE Trans. Fuzzy Syst. 26(6), 3232–3245 (2018)

    Article  Google Scholar 

  14. Aliev, R.A., Huseynov, O.H., Zeinalova, L.M.: The arithmetic of continuous Z-numbers. Inf. Sci. 373(C), 441–460 (2016)

    Article  Google Scholar 

  15. Castillo, O., Melin, P.: A review on the design and optimization of interval type-2 fuzzy controllers. Appl. Soft Comput. 12(4), 1267–1278 (2012)

    Article  Google Scholar 

  16. Zadeh, L.A.: A note on Z-numbers. Inf. Sci. 181(14), 2923–2932 (2011)

    Article  MATH  Google Scholar 

  17. Yager, R.R.: On Z-valuations using Zadeh’s Z-numbers. Int. J. Intell. Syst. 27(3), 259–278 (2012)

    Article  Google Scholar 

  18. Aliev, R.A., Pedrycz, W., Huseynov, O.H., Eyupoglu, S.Z.: Approximate reasoning on a basis of Z-number valued If-Then rules. IEEE Trans. Fuzzy Syst. 25(6), 1589–1600 (2017)

    Article  Google Scholar 

  19. Shen, K-w, Wang, X-k, Wang, J-q: Multi-criteria decision-making method based on Smallest Enclosing Circle in incompletely reliable information environment. Comput. Ind. Eng. 130, 1–13 (2019)

    Article  Google Scholar 

  20. Aliev, R.R., Huseynov, O.H., Aliyeva, K.R.: Z-valued T-norm and T-conorm operators-based aggregation of partially reliable Information. Procedia Comput. Sci. 102, 12–17 (2016)

    Article  Google Scholar 

  21. Peng, H-g, Wang, X-k, Wang, T-l, Wang, J-q: Multi-criteria game model based on the pairwise comparisons of strategies with Z-numbers. Appl. Soft Comput. 74, 451–465 (2019)

    Article  Google Scholar 

  22. Peng, H-g, Wang, J-q: A multicriteria group decision-making method based on the normal cloud model With Zadeh’s Z-numbers. IEEE Trans Fuzzy Syst. 26(6), 3246–3260 (2018)

    Article  Google Scholar 

  23. Shen, K-w, Wang, J-q, Wang, T-l: The arithmetic of multidimensional Z-number. J. Intell. Fuzzy Syst. 36(2), 1647–1661 (2019)

    Article  Google Scholar 

  24. Kang, B., Deng, Y., Hewage, K., Sadiq, R.: Generating Z-number based on OWA weights using maximum entropy. Int. J. Intell. Syst. 33(8), 1745–1755 (2018)

    Article  Google Scholar 

  25. Kang, B., Wei, D., Li, Y., Deng, Y.: Decision making using Z-numbers under uncertain environment. J. Comput. Inf. Syst. 8(7), 2807–2814 (2012)

    Google Scholar 

  26. Jiang, W., Xie, C., Luo, Y., Tang, Y.: Ranking Z-numbers with an improved ranking method for generalized fuzzy numbers. J. Intell. Fuzzy Syst. 32(3), 1931–1943 (2017)

    Article  MATH  Google Scholar 

  27. Kang, B., Deng, Y., Sadiq, R.: Total utility of Z-number. Appl. Intell. 48(3), 703–729 (2018)

    Article  Google Scholar 

  28. Kang, B., Wei, D., Li, Y., Deng, Y.: A method of converting Z-number to classical fuzzy number. J. Inf. Comput. Sci. 9(3), 703–709 (2012)

    Google Scholar 

  29. Yaakob, A.M., Gegov, A.: Interactive TOPSIS based group decision making methodology using Z-numbers. Int. J. Comput. Intell. Syst. 9(2), 311–324 (2016)

    Article  Google Scholar 

  30. Aboutorab, H., Saberi, M., Asadabadi, M.R., Hussain, O., Chang, E.: ZBWM: the Z-number extension of best worst method and its application for supplier development. Expert Syst. Appl. 107, 115–125 (2018)

    Article  Google Scholar 

  31. Kang, B., Chhipi-Shrestha, G., Deng, Y., Hewage, K., Sadiq, R.: Stable strategies analysis based on the utility of Z-number in the evolutionary games. Appl. Math. Comput. 324, 202–217 (2018)

    MathSciNet  MATH  Google Scholar 

  32. Aliyev, R.R.: Multi-attribute decision making based on Z-valuation. Procedia Comput. Sci. 102(C), 218–222 (2016)

    Article  Google Scholar 

  33. Yang, Y., Wang, J.: SMAA-based model for decision aiding using regret theory in discrete Z-number context. Appl. Soft Comput. 65, 590–602 (2018)

    Article  Google Scholar 

  34. Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106(4), 620–630 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zadeh, L.A.: Probability measures of Fuzzy events. J. Math. Anal. Appl. 23(2), 421–427 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shang, X-g, Jiang, W-s: A note on fuzzy information measures. Pattern Recognit. Lett. 18(5), 425–432 (1997)

    Article  Google Scholar 

  38. Wei, G-w: Picture fuzzy cross-entropy for multiple attribute decision making problems. J. Bus. Econ. Manag. 17(4), 491–502 (2016)

    Article  MathSciNet  Google Scholar 

  39. 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. 47(15), 3598–3608 (2016)

    Article  MATH  Google Scholar 

  40. Wu, X., Wang, J., Peng, J., Chen, X.: Cross-entropy and prioritized aggregation operator with simplified neutrosophic sets and their application in multi-criteria decision-making problems. Int. J. Fuzzy Syst. 18(6), 1104–1116 (2016)

    Article  Google Scholar 

  41. Kang, B., Deng, Y., Hewage, K., Sadiq, R.: A method of measuring uncertainty for Z-number. IEEE Trans. Fuzzy Syst. (2018). https://doi.org/10.1109/TFUZZ.2018.2868496

    Article  Google Scholar 

  42. Casasnovas, J., Riera, J.V.: On the addition of discrete fuzzy numbers. In: Proceedings of the 5th WSEAS international conference on Telecommunications and informatics, World Scientific and Engineering Academy and Society (WSEAS), Istanbul, pp. 432–437 (2006)

  43. Casasnovas, J., Riera, J.V.: Weighted means of subjective evaluations. In: Seising, R., Sanz González, V. (eds.) Soft Computing in Humanities and Social Sciences, pp. 323–345. Springer, Berlin (2012)

    Chapter  Google Scholar 

  44. Voxman, W.: Canonical representations of discrete fuzzy numbers. Fuzzy Sets Syst. 118(3), 457–466 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, G., Wu, C., Zhao, C.: Representation and operations of discrete fuzzy numbers. Southeast Asian Bull. Math. 29(5), 1003–1010 (2005)

    MathSciNet  MATH  Google Scholar 

  46. Kullback, S.: Information theory and statistics. Wiley, New York (1959)

    MATH  Google Scholar 

  47. Shore, J., Johnson, R.: Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory 26(1), 26–37 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  48. Hwang, C.L., Yoon, K.: Multiple attribute decision making. Springer, Berlin (1981)

    Book  MATH  Google Scholar 

  49. Yoon, K.P., Kim, W.K.: The behavioral TOPSIS. Expert Syst. Appl. 89, 266–272 (2017)

    Article  Google Scholar 

  50. Sun, G., Guan, X., Yi, X., Zhou, Z.: An innovative TOPSIS approach based on hesitant fuzzy correlation coefficient and its applications. Appl. Soft Comput. 68, 249–267 (2018)

    Article  Google Scholar 

  51. Wang, X., Wang, J., Zhang, H.: Distance-based multicriteria group decision-making approach with probabilistic linguistic term sets. Expert Syst. 36, e12352 (2019)

    Article  Google Scholar 

  52. Zhao, S., Du, J.: Thirty-two years of development of human resource management in China: review and prospects. Hum Resour Manag Rev. 22(3), 179–188 (2012)

    Article  MathSciNet  Google Scholar 

  53. Siyambalapitiya, J., Zhang, X., Liu, X.: Green human resource management: a proposed model in the context of Sri Lanka’s tourism industry. J. Clean. Prod. (2018). https://doi.org/10.1016/j.jclepro.2018.07.305

    Article  Google Scholar 

  54. Dong, Y., Liu, Y., Liang, H., Chiclana, F., Herrera-Viedma, E.: Strategic weight manipulation in multiple attribute decision making. Omega 75, 154–164 (2018)

    Article  Google Scholar 

  55. Liu, Y., Dong, Y., Liang, H., Chiclana, F., Herrera-Viedma, E.: Multiple attribute strategic weight manipulation with minimum cost in a group decision making context with interval attribute weights information. IEEE Trans. Syst Man Cybern. Syst. (2018). https://doi.org/10.1109/TSMC.2018.2874942

    Article  Google Scholar 

  56. Dong, Y., Zhao, S., Zhang, H., Chiclana, F., Herrera-Viedma, E.: A self-management mechanism for non-cooperative behaviors in large-scale group consensus reaching processes. IEEE Trans. Fuzzy Syst. 26(6), 3276–3288 (2018)

    Article  Google Scholar 

  57. Zhang, X., Zhang, H., Wang, J.: Discussing incomplete 2-tuple fuzzy linguistic preference relations in multi-granular linguistic MCGDM with unknown weight information. Soft. Comput. 23(6), 2015–2032 (2019)

    Article  MATH  Google Scholar 

  58. Zhang, H., Dong, Y., Herrera-Viedma, E.: Consensus building for the heterogeneous large-scale GDM with the individual concerns and satisfactions. IEEE Trans. Fuzzy Syst. 26(2), 884–898 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the editors and anonymous referees for their valuable and constructive comments and suggestions that immensely facilitated the improvement of this paper. This work was supported by the National Natural Science Foundation of China (No. 71871228).

Author information

Authors and Affiliations

  1. School of Business, Central South University, Changsha, 410083, People’s Republic of China

    Dong Qiao, Xiao-kang Wang, Jian-qiang Wang & Ke Chen

Authors
  1. Dong Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao-kang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian-qiang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ke Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ke Chen.

Appendix: Proof of the properties of Eq. (16)

Appendix: Proof of the properties of Eq. (16)

Proof

  1. (1)

    In accordance with Definitions 8 and 12, \(H^{F} \left( {A_{1} ,A_{2} } \right) \ge 0\) and \(H^{R} \left( {Z_{1} ,Z_{2} } \right) \ge 0\) exist.

Therefore, \(H^{\omega } \left( {Z_{1} ,Z_{2} } \right) = \omega H^{F} \left( {A_{1} ,A_{2} } \right) + \left( {1 - \omega } \right)H^{R} \left( {Z_{1} ,Z_{2} } \right) \ge 0\) is satisfied.

  1. (2)

    On the basis of Definitions 8 and 12, if \(Z_{1} = Z_{2}\), then \(H^{F} \left( {A_{1} ,A_{2} } \right) = 0\) and \(H^{R} \left( {Z_{1} ,Z_{2} } \right) = 0\) exist.

Therefore, \(H^{\omega } \left( {Z_{1} ,Z_{2} } \right) = \omega H^{F} \left( {A_{1} ,A_{2} } \right) + \left( {1 - \omega } \right)H^{R} \left( {Z_{1} ,Z_{2} } \right) = 0\) is satisfied.

  1. (3)

    According to Definitions 4, 5, 8 and 12, if \(Z_{1} = Z_{2}\), then \(H^{F} \left( {A_{1} ,A_{2} } \right) = H^{F} \left( {A_{2} ,A_{1} } \right)\) and \(H^{R} \left( {Z_{1} ,Z_{2} } \right) = H^{R} \left( {Z_{2} ,Z_{1} } \right)\) exist.

Therefore, \(H^{\omega } \left( {Z_{1} ,Z_{2} } \right) = H^{\omega } \left( {Z_{2} ,Z_{1} } \right)\) is satisfied.\(\hfill\square\)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, D., Wang, Xk., Wang, Jq. et al. Cross Entropy for Discrete Z-numbers and Its Application in Multi-Criteria Decision-Making. Int. J. Fuzzy Syst. 21, 1786–1800 (2019). https://doi.org/10.1007/s40815-019-00674-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40815-019-00674-2

Keywords

Search

Navigation