International Journal of Fuzzy Systems

, Volume 21, Issue 6, pp 1786–1800 | Cite as

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

  • Dong Qiao
  • Xiao-kang Wang
  • Jian-qiang Wang
  • Ke ChenEmail author


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.


Z-number Cross entropy Maximum entropy method TOPSIS Performance evaluation 



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).


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Copyright information

© Taiwan Fuzzy Systems Association 2019

Authors and Affiliations

  • Dong Qiao
    • 1
  • Xiao-kang Wang
    • 1
  • Jian-qiang Wang
    • 1
  • Ke Chen
    • 1
    Email author
  1. 1.School of BusinessCentral South UniversityChangshaPeople’s Republic of China

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