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Top1. Introduction
In complex engineering, economics, and management, multiple attribute group decision making is a very important research topic (Zheng et al. 2018, Lin et al. 2018, Liu et al. 2019a and 2019b). Although fuzzy sets (Zadeh 1965), intuitionistic fuzzy sets (IFSs) (Atanassov 1986), and interval-valued intuitionistic fuzzy sets (IVIFSs) (Atanassov and Gargov 1989) have been developed in vague, incomplete, and uncertain setting, they cannot describe and deal with indeterminate and inconsistent information in various real problems. In this case, Smarandache (1999) proposed the concept of a neutrosophic set as a generalization of the concepts of the classic set, fuzzy set, IFS and IVIFS. In the neutrosophic set, a truth-membership T(x), an indeterminacy-membership I(x) and a falsity-membership F(x) are represented independently and lie within the real standard or nonstandard unit interval ]−0, 1+[. Then, the indeterminacy presented in the neutrosophic set is independent on the truth and falsity values and can include inconsistent information, while the incorporated uncertainty in the IFS is dependent on the degrees of belongingness and non-belongingness and cannot include inconsistent information. Hence, the neutrosophic set can better express incomplete, indeterminate and inconsistent information. However, the neutrosophic set is difficult to be applied in real-life situations due to the nonstandard unit interval ]−0, 1+[ for the range of the three functions T(x), I(x) and F(x). Thus, the range of the functions T(x), I(x) and F(x) can be restrained to the real standard unit interval [0, 1] to be easily applied in real science and engineering problems. Consequently, a single-valued neutrosophic set (SVNS) (Wang et al. 2010), an interval neutrosophic set (INS) (Wang et al. 2005), and a simplified neutrosophic set (SNS) (Ye 2014a) were introduced by some researchers. Then, SNSs are the subclass of neutrosophic sets (Ye 2014a) and include the concepts of SVNSs and INSs.