Models for MADM with Single-Valued Neutrosophic 2-Tuple Linguistic Muirhead Mean Operators

1 School of Business, Sichuan Normal University, Chengdu 610101, China; JW970326@163.com (J.W.); lujp2002@163.com (J.L.); weicun1990@163.com (C.W.) 2 School of Economics and Management, Chongqing University of Arts and Sciences, Chongqing 402160, China 3 School of Statistics, Southwestern University of Finance and Economics, Chengdu 611130, China * Correspondence: weiguiwu1973@sicnu.edu.cn (G.W.); linrui@cqwu.edu.cn (R.L.)


Introduction
In order to effectively depict the fuzziness and uncertainty information in real multiple attribute decision making (MADM) problems, Smarandache [1,2] proposed the use of neutrosophic sets (NSs), which have attracted the attention of many scholars. The main advantage of NSs is their capacity to denote inconsistent and indeterminate information. An NS has more potential power than any other fuzzy mathematical tool, such as the fuzzy set [3], the intuitionistic fuzzy set (IFS) [4], and the interval-valued neutrosophic fuzzy set (IVIFS) [5]. However, it is hard to use NSs to solve practical MADM problems. Therefore, Wang et al. [6,7] proposed the use of a single-valued neutrosophic set (SVNS) and an interval neutrosophic set (INS), which can include much more information than fuzzy sets, IFSs, and IVIFSs. Ye [8] proposed the use of MADM with the correlation coefficients of SVNSs. Broumi and Smarandache [9] investigated the correlation coefficients of interval neutrosophic numbers (INNs). Biswas et al. [10] proposed the use of single-valued neutrosophic number TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) models. Liu et al. [11] developed the generalized Hamacher operations for SVNSs. Sahin and Liu [12] presented the maximizing deviation method using neutrosophic settings. Ye [13] defined some similarity measures of INSs. Zhang et al. [14] defined some interval neutrosophic information aggregating operators. Ye [15] proposed the use of a simplified neutrosophic set (SNS), which included SVNSs and INSs. Many researchers have given their attention to SNSs. For example, Peng et al. [16] presented some basic operational laws of simplified neutrosophic number (SNNs) and proposed the use of simplified neutrosophic aggregation operators. Additionally, Peng et al. [17] studied an outranking method to handle simplified neutrosophic information, and then Zhang et al. [18] presented an extended version of Peng's method using an interval neutrosophic environment. Liu and Liu [19] developed a generalized weighted power operator with SVNNs. Deli and Subas [20] discussed a method to rank SVNNs. Peng et al. [21] proposed the use of multi-valued neutrosophic sets and defined some power operators for multiple attribute group decision making (MAGDM). Zhang et al. [22] defined the weighted correlation coefficient for INNs. Chen and Ye [23] proposed the use of Dombi operations with SVNNs. Liu and Wang [24] proposed the use of the SVN normalized weighted Bonferroni mean (WBM). Wu et al. [25] proposed the use of prioritized operator and cross-entropy with SNSs in MADM problems. Li et al. [26] developed some SVNN Heronian mean operators in MADM problems. Xu et al. [27] proposed the use of the TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision Making)method for SVN MADM.
Even though SVNSs have been widely used in some areas, all the existing methods are unsuitable for expressing the truth-membership, indeterminacy-membership, and falsity-membership of an element to a 2-tuple linguistic term set, which can affect a decisionmaker's confidence level when they are making evaluations. In order to overcome this limit, Wu et al. [28] defined the basic concept of single-valued neutrosophic 2-tuple linguistic sets (SVN2TLSs) to cope with this problem on the basis of the SVNSs [6] and 2-tuple linguistic term set [29,30]. Therefore, how to aggregate these single-valued neutrosophic 2-tuple linguistic numbers (SVN2TLNs) is an interesting issue. To solve it, we propose the use of some Muirhead mean (MM) operators with SVN2TLNs. In order to do this, the remainder of this paper is presented as follows: In Section 2, we introduce the concept of SVN2TLSs. In Section 3, we develop some MM operators with SVN2TLNs. In Section 4, we develop some MADM models with SVN2TLNs based on these operators. In Section 5, we present a numerical example to select green suppliers with SVN2TLNs in order to illustrate the method proposed. Section 6 finishes this paper with some concluding remarks.

Preliminaries
Wu et al. [28] proposed the use of the concept of SVN2TLSs based on the SVNSs [6] and 2-tuple linguistic term sets [29,30].

Definition 1 ([28]). A SVN2TLS
A in X is given as follows:  2  1  2  1  2  1  2  1  2  1  2   1  1  1  2  1  2   ,  ,  ,  ,  ,  ;   a a  a  a  a a  a  a  a a  a  a   a a  s  s  T T I I  I I Δ is the function of converting the 2-tuple linguistic variables to the exact numbers and Δ is the function of converting the computing results to the 2-tuple linguistic variables.

MM Operators
Muirhead [33] proposed the use of the MM operator. Tang et al. [34] developed some interval-valued Pythagorean fuzzy Muirhead mean operators. Wang et al. [35] proposed the use of some picture fuzzy Muirhead mean operators in MADM problems. (3) Proof. Based on the exponential operation laws of SVN2TLNs, we can derive □ Therefore, by utilizing the multiplication operation laws of SVN2TLNs, the Therefore, according to the addition operation laws of SVN2TLNs, we can obtain Furthermore, based on the scalar-multiplication operation of SVN2TLNs, we can derive Therefore, the aggregated value by using SVN2TLMM operators can be listed as follows: Therefore, Equation (4) is kept. In Equations (4)-(9), the 1 − Δ is the function of converting the 2-tuple linguistic variables to the exact numbers and Δ is the function of converting the computing results to the 2-tuple linguistic variables.
Then, we need to prove that Equation (4) is a SVN2TLN. We need to prove two conditions as follows: Then, That means 0 1 T ≤ ≤ , so ① is maintained. Similarly, we can find that 0 1 ,0      Then, we can identify some properties of the SVN2TLMM operator.

Property 1. (Idempotency) If
Proof. Let , , , , LMM Therefore, T T ≤ , we can also obtain That is a b T T ≤ . Similarly, we can obtain and and and and and and and , , , = SVN2TLMM SVN2TLMM , , , From Property 1: From Property 2:

The Single-Valued Neutrosophic 2-Tuple Linguistic Weighted Muirhead Mean (SVN2TLWMM) Operator
In actual MADM, it is important to consider attribute weights. This section proposes the use of a SVN2TLWMM operator as follows: Proof. From the exponential operation of the SVN2TLNs, we can derive Therefore, by utilizing the scalar-multiplication operation laws of the SVN2TLNs, the Therefore, according to the multiplication operation of the SVN2TLNs, we can obtain Therefore, by utilizing the addition operation of the SVN2TLNs, we can get Furthermore, based on the scalar-multiplication operation of the SVN2TLNs, we can derive Therefore, the aggregated value by using SVN2TLWMM operators can be listed as follows: Therefore, Equation (23)  Then we need to prove that Equation (23) is a SVN2TLN. Then, That means 0 1 T ≤ ≤ , so ① is maintained.
The proof is similar to SVN2TLMM. It is omitted here.

Property 5. (Boundedness) Let
From Property 4, we get: It is obvious that SVN2TLMM operators lacks the property of idempotency.

The Single-Valued Neutrosophic 2-Tuple Linguistic Dual Muirhead Mean (SVN2TLDMM) Operator
Qin and Liu [36] proposed the use of the dual Muirhead mean (DMM) operator.
Proof. From the multiplication operation laws of SVN2TLNs depicted in Definition 3, we can obtain: Therefore, based on the multiplication operation of SVN2TLNs, we can get: Furthermore, by utilizing the exponential operation of SVN2TLNs we can derive: Therefore, the aggregated results using the SVN2TLDMM operator can be shown as: Therefore, (41) is kept. In Equations (41)-(46), the symbol " 1 − Δ " is the function of converting the 2-tuple linguistic variables to the exact numbers and " Δ " is the function of converting the computing results to the 2-tuple linguistic variables.
We now need to prove that (41) is a SVN2TLN. We need to prove the two conditions: Similarly, we can get 0 1,0

The Single-Valued Neutrosophic 2-Tuple Linguistic Weighted Dual Muirhead Mean (SVN2TLWDMM) Operator
In actual MADM, it is important to consider attribute weights. This section proposes the use of a SVN2TLWDMM operator.
Proof. From the exponential operation laws of SVN2TLNs depicted in Definition 3, we can ascertain that .
Then, based on the scalar-multiplication operation laws of SVN2TLNs, we can derive Therefore, according to the addition operation laws of SVN2TLNs, we can get Therefore, by utilizing the multiplication operation laws of SVN2TLNs, we can derive Furthermore, by using the exponential operation laws of SVN2TLNs, we can get Therefore, the fused results using the SVN2TLWDMM operator can be shown as follows: Therefore, Equation (54) is kept. In Equations (54)-(60), Then, we need to prove that Equation (54) is a SVN2TLN. We need to prove the two conditions: That means 0 1 T ≤ ≤ , so ① is maintained.
It is obvious that the SVN2TLWDMM operator lacks the property of idempotency.

Numerical Example
The green supplier selection is a classic MADM problem [37][38][39]. Therefore, in this section we use a numerical example to select green suppliers in green supply chain management with SVN2TLNs in order to show the proposed method. There are five possible green suppliers ( ) 1, 2,3, 4,5 i A i = to be selected. We selected four attributes to assess these possible green suppliers: G1 is the product quality factor, G2 is the environmental factor, G3 is the delivery factor, and G4 is the price factor. These five possible green suppliers ( )  We can now use the approach developed for selecting green suppliers in green supply chain management.
Step 1. According to SVN2TLNs ( )  Table 2.  Step 2. In accordance with the aggregating results in Table 2, the score values of the green suppliers are shown in Table 3. Step 3. According to the score values listed in Table 3, the order of the green suppliers are listed in Table 4. The best green supplier is A4.

Influence of the Parameter on the Final Result
In order to show the effects on the ranking results by altering the parameters of P in the SVN2TLWMM (SVN2TLWDMM) operators, the results are listed in Tables 5 and 6. In future studies, the application of the proposed aggregating operators of SVN2TLNs need to be studied in many other uncertain and fuzzy environments [40][41][42][43] and extended to other application domains [44,45].
Author Contributions: Jie Wang, Jianping Lu, Guiwu Wei, Rui Lin and Cun Wei conceived and worked together to achieve this work, Jie Wang compiled the computing program by Matlab and analyzed the data, Jie Wang and Guiwu Wei wrote the paper. Finally, all the authors have read and approved the final manuscript.

Conflicts of Interest:
The authors declare no conflicts of interest.