## 1 Introduction  
[PERSON] [1; 2] introduced a neutrosophic set (NS) from a philosophical point of view to express indeterminate and inconsistent information. In an NS A, its truth-membership function \(T_{A}(x)\), indeterminacy-membership \(I_{A}(x)\) and falsity-membership function \(F_{A}(x)\) are represented independently, which lie in real standard or nonstandard subsets of \(]^{-}0,1^{+}[\), that is, \(T_{A}(x):X\rightarrow\,]^{-}0,1^{+}[\), \(I_{A}(x):X\rightarrow\,]^{-}0,1^{+}[\) and \(F_{A}(x):X\rightarrow\,]^{-}0,1^{+}[\). The main advantage of NSs is to depict inconsistent and indeterminate information. An NS has more potential power than other fuzzy mathematical modeling tools, such as fuzzy set [3], intuitionistic fuzzy set (IFS) [4] and interval valued neutrosophic fuzzy set (IVIFS) [5]. However, it is not easy to use NSs in solving practical problems. Therefore, [PERSON] [2] and [PERSON] et al. [6; 7] defined the single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS). Hence, SVNSs and INSs can express much more information than fuzzy sets, IFSs and IVIFSs. [PERSON] [8] presented the correlation and correlation coefficient of single-valued neutrosophic sets (SVNSs) based on the extension of the correlation of intuitionistic fuzzy sets and demonstrates that the cosine similarity measure is a special case of the correlation coefficient in SVNS. [PERSON] and [PERSON] [9] investigated the correlation coefficient with interval neutrosophic numbers(INNs). [PERSON] et al. [10] proposed a new approach for multi-attribute group decision-making problems by extending the technique for order preference by similarity to ideal solution to single-valued neutrosophic environment. [PERSON] et al. [11] proposed the generalized neutrosophic number Hamacher weighted averaging (GNNHWA) operator, generalized neutrosophic number Hamacher ordered weighted averaging (GNNHOWA) operator, and generalized neutrosophic number Hamacher hybrid averaging (GNNHHA) operator, and explored some properties of these operators and analyzed some special cases of them. [PERSON] and [PERSON] [12] proposed the maximizing deviation models for solving the multiple attribute decision-makingproblems with the single-valued neutrosophic information or interval neutrosophic information. [PERSON] [13] developed the Hamming and Euclidean distances between interval neutrosophic sets (INSs) and proposed the similarity measures between INSs based on the relationship between similarity measures and distances. [PERSON] et al. [14] developed the interval neutrosophic number weighted averaging (INNWA) operator and interval neutrosophic number weighted geometric (INNWG) operator. [PERSON] [15] proposed a simplified neutrosophic set (SNS) as a more general concept including SVNS and INS. Many researchers have given them attention to SNSs. For example, [PERSON] et al. [16] defined some basic operational laws for SNNs and developed simplified neutrosophic information aggregation operators. Additionally, [PERSON] et al. [17] developed a new outranking approach for multi-criteria decision-making (MCDM) problems in the context of a simplified neutrosophic environment, where the truth-membership degree, indeterminacy-membership degree and falsity-membership degree for each element are singleton subsets in [0, 1], and then [PERSON] et al. [18] gave an extended version of [PERSON]'s approach to interval neutrosophic environment. [PERSON] and [PERSON] [19] developed generalized weighted power operators with SVNNs. [PERSON] and [PERSON] [20] discussed a method to rank SVNNs. [PERSON] et al. [21] introduced the multi-valued neutrosophic sets (MVNSs), which allowed the truth-membership, indeterminacy-membership and falsity-membership degree have a set of crisp values between zero and one, respectively, and then defined the operations of multi-valued neutrosophic numbers (MVNNs) based on Einstein operations, the multi-valued neutrosophic power weighted average (MVNPWA) operator and the multi-valued neutrosophic power weighted geometric (MVNPWG) operator.