Some distances, similarity and entropy measures for interval-valued neutrosophic sets and their relationship
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ORIGINAL ARTICLE
Some distances, similarity and entropy measures for intervalvalued neutrosophic sets and their relationship Jun Ye1 · Shigui Du2
Received: 16 August 2017 / Accepted: 23 August 2017 © Springer-Verlag GmbH Germany 2017
Abstract This paper proposes some new distance measures between interval-valued neutrosophic sets (IvNSs) and their similarity measures. Then, some entropy measures of IvNS based on the distances are proposed as the extension of the entropy measures of interval-valued intuitionistic fuzzy sets (IvIFSs). Also, we investigate the relationship between the presented entropy measures and the similarity measures for IvNSs. Finally, the comparison of the new entropy measures with existing entropy measures for IvNSs is given by the numerical and decision-making examples to demonstrate that the proposed new entropy measures for IvNSs are effective and reasonable and more intelligible in representing the degree of fuzziness of IvNSs than the existing ones. Keywords Interval-valued neutrosophic set · Distance measure · Similarity measure · Entropy · Decision making
1 Introduction Entropy is an important tool for measuring uncertain information [11, 12]. Then, Zadeh [31] firstly introduced a fuzzy entropy measure. As a generalization of the fuzzy entropy, Bustince and Burrillo [4] presented an intuitionistic fuzzy entropy measure. Also, Szmidt and Kacprzyk [14] extended the axioms of De Luca and Termini’s [5] non-probabilistic * Jun Ye [email protected] 1
Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, Zhejiang, People’s Republic of China
Department of Civil Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, Zhejiang, People’s Republic of China
2
entropy in fuzzy set setting to the entropy of intuitionistic fuzzy information. Furthermore, Vlachos and Sergiadis [18] put forward intuitionistic fuzzy entropy. Ye [23] proposed the cosine and sine entropy measures of intuitionistic fuzzy sets (IFSs). In interval-valued intuitionistic fuzzy setting, Ye [24] presented the entropy measures for interval-valued intuitionistic fuzzy sets (IvIFSs) and applied them to multiple attribute decision making. Further, Zhang et al. [34] proposed an axiomatical definition of interval-valued intuitionistic fuzzy entropy based on the distance measures of IvIFSs, which is in agreement with the axiomatical definition of fuzzy entropy introduced by De Luca and Termini, and then they introduced some entropy measures for IvIFSs and discussed their relationship with similarity measures and inclusion measures for IvIFSs. Smaradache [13] proposed a neutrosophic set by adding an indeterminacy membership on the basis of an IFS and an IvIFS. Neutrosophic set generalizes the concept of the classic set, fuzzy set [32], interval valued fuzzy set (IvFS) [15], IFS [1], IvIFS [2] etc. A neutrosophic set considers the truth-membership, indeterminacy-membership and falsitymembership functions independently, which are in the real
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