Soft Computing

, Volume 22, Issue 22, pp 7367–7376 | Cite as

On entropy, similarity measure and cross-entropy of single-valued neutrosophic sets and their application in multi-attribute decision making

  • Haibo Wu
  • Ye Yuan
  • Lijun WeiEmail author
  • Lidan Pei


As a subclass of neutrosophic sets, single-valued neutrosophic sets (SVNS) can be used to represent uncertainty and inconsistent information that exist in real-world situations. Information measures play an important role in SVNS theory, which has received more and more attention in recent years. In this paper, we develop a multi-attribute decision-making (MADM) method based on single-valued neutrosophic information measures. To this end, three axiomatic definitions of information measures are first introduced. These include entropy, similarity measure and cross-entropy. Then, we construct information measure formulas on the basis of the cosine function. The relationship among entropy, similarity measure and cross-entropy as well as their mutual transformations are further discussed. Moreover, an approach to single-valued neutrosophic MADM based on these information measure formulas is presented. Finally, a numerical example of city pollution evaluation is provided. The comparative analysis demonstrates the applicability and effectiveness of the proposed method.


Single-valued neutrosophic sets Entropy Similarity measure Cross-entropy Multi-attribute decision making 



The work was supported by the Natural Science Foundation of Jiangsu Province (BK20170546). The authors are thankful to the anonymous reviewers and the editor for their valuable comments and constructive suggestions that have led to an improved version of this paper.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

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


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hunan Railway Professional Technology CollegeZhuzhouChina
  2. 2.School of Electrical and Information EngineeringJiangsu UniversityZhenjiangChina

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