Multi-Criteria Decision-Making Method Based on Simplified Neutrosophic Linguistic Information with Cloud Model

Abstract

This study introduces simplified neutrosophic linguistic numbers (SNLNs) to describe online consumer reviews in an appropriate manner. Considering the defects of studies on SNLNs in handling linguistic information, the cloud model is used to convert linguistic terms in SNLNs to three numerical characteristics. Then, a novel simplified neutrosophic cloud (SNC) concept is presented, and its operations and distance are defined. Next, a series of simplified neutrosophic cloud aggregation operators are investigated, including the simplified neutrosophic clouds Maclaurin symmetric mean (SNCMSM) operator, weighted SNCMSM operator, and generalized weighted SNCMSM operator. Subsequently, a multi-criteria decision-making (MCDM) model is constructed based on the proposed aggregation operators. Finally, a hotel selection problem is presented to verify the effectiveness and validity of our developed approach.

1. Introduction

Nowadays, multi-criteria decision-making (MCDM) problems are attracting more and more attention. Lots of studies suggest that it is difficult to describe decision information completely because the information is usually inconsistent and indeterminate in real-life problems. To address this issue, Smarandache [1] put forward neutrosophic sets (NSs). Now, NSs have been applied to many fields and extended to various forms. Wang et al. [2] presented the concept of single-valued neutrosophic sets (SVNSs) and demonstrated its application, Ye [3] proposed several kinds of projection measures of SVNSs, and Ji et al. [4] proposed Bonferroni mean aggregation operators of SVNSs. Wang et al. [5] used interval numbers to extend SVNSs, and proposed the interval-valued neutrosophic set (IVNS). Ye [6] introduced trapezoidal neutrosophic sets (TrNSs), and proposed a series of trapezoidal neutrosophic aggregation operators. Liang et al. [7] introduced the preference relations into TrNSs. Peng et al. [8] combined the probability distribution with NSs to propose the probability multi-valued neutrosophic sets. Wu et al. [9] further extended this set to probability hesitant interval neutrosophic sets. All of the aforementioned sets are the descriptive tools of quantitative information.

Zhang et al. [10] proposed a method of using NSs to describe online reviews posted by consumers. For example, a consumer evaluates a hotel with the expressions: ‘the location is good’, ‘the service is neither good nor bad’, and ‘the room is in a mess’. Obviously, there is active, neutral, and passive information in this review. According to the NS theory, such review information can be characterized by employing truth, neutrality, and falsity degrees. This information presentation method has been proved to be feasible [11]. However, in practical online reviews, the consumer usually gives a comprehensive evaluation before posting the text reviews. NSs can describe the text reviews, but they cannot represent the comprehensive evaluation. To deal with this issue, many scholars have studied the combination of NSs and linguistic term sets [12,13]. The semantic of linguistic term set provides precedence on a qualitative level, and such precedence is more sensitive for decision-makers than a common ranking due to the expression of absolute benchmarks [14,15,16]. Based on the concepts of NSs and linguistic term sets, Ye [17] proposed interval neutrosophic linguistic sets (INLSs) and interval neutrosophic linguistic numbers (INLNs). Then, many interval neutrosophic linguistic MCDM approaches were developed [18,19]. Subsequently, Tian et al. [20] introduced the concepts of simplified neutrosophic linguistic sets (SNLSs) and simplified neutrosophic linguistic numbers (SNLNs). Wang et al. [21] proposed a series of simplified neutrosophic linguistic Maclaurin symmetric mean aggregation operators and developed a MCDM method. The existed studies on SNLNs simply used the linguistic functions to deal with linguistic variables in SNLNs. This strategy is simple, but it cannot effectively deal with qualitative information because it ignores the randomness of linguistic variables.

The cloud model is originally proposed by Li [22] in the light of probability theory and fuzzy set theory. It characterizes the randomness and fuzziness of a qualitative concept rely on three numerical characters and makes the conversion between qualitative concepts and quantitative values becomes effective. Since the introduction of the cloud model, many scholars have conducted lots of studies and applied it to various fields [23,24,25], such as hotel selection [26], data detection [27], and online recommendation algorithms [28]. Currently, the cloud model is considered as the best way to handle linguistic information and it is used to handle multiple qualitative decision-making problems [29,30,31], such as linguistic intuitionistic problems [32] and Z-numbers problems [33]. Considering the effectiveness of the cloud model in handling qualitative information, we utilize the cloud model to deal with linguistic terms in SNLNs. In this way, we propose a new concept by combining SNLNs and cloud model to solve real-life problems.

The aggregation operator is one of the most important tool of MCDM method [34,35,36,37]. Maclaurin symmetric mean (MSM) operator, defined by Maclaurin [38], possess the prominent advantage of summarizing the interrelations among input variables lying between the maximum value and minimum value. The MSM operator can not only take relationships among criteria into account, but it can also improve the flexibility of aggregation operators in application by adding parameters. Since the MSM operator was proposed, it has been expanded to various fuzzy sets [39,40,41,42,43]. For example, Liu and Zhang [44] proposed many MSM operators to deal with single-valued trapezoidal neutrosophic information, Ju et al. [45] proposed a series of intuitionistic linguistic MSM aggregation operators, and Yu et al. [46] proposed the hesitant fuzzy linguistic weighted MSM operator.

From the above analysis, the motivation of this paper is presented as follows:

• The cloud model is a reliable tool for dealing with linguistic information, and it has been successfully applied to handle multifarious linguistic problems, such as probabilistic linguistic decision-making problems. The existing studies have already proved the effectiveness and feasibility of using the cloud model to process linguistic information. In view of this, this paper introduces the cloud model to process linguistic evaluation information involved in SNLNs.
• As an efficient and applicable aggregation operator, MSM not only takes into account the correlation among criteria, but also adjusts the scope of the operator through the transformation of parameters. Therefore, this paper aims to accommodate the MSM operator to simplified neutrosophic linguistic information environments.

The remainder of this paper is organized as follows. Some basic definitions are introduced in Section 2. In Section 3, we propose a new concept of SNCs and the corresponding operations and distance. In Section 4, we propose some simplified neutrosophic cloud aggregation operators. In Section 5, we put forward a MCDM approach in line with the proposed operators. Then, in Section 6, we provide a practical example concerning hotel selection to verify the validity of the developed method. In Section 7, a conclusion is presented.

2. Preliminaries

This section briefly reviews some basic concepts, including linguistic term sets, linguistic scale function, NSs, SNSs, and cloud model, which will be employed in the subsequent analyses.

2.1. Linguistic Term Sets and Linguistic Scale Function

Definition 1

([47]).Let$H=\left\{h_{\tau }|\tau =1,2,\cdots ,2t+1,t\in N^{\ast }\right\}$be a finite and totally ordered discrete term set, where$N^{\ast }$is a set of positive integers, and$h_{\tau }$is interpreted as the representation of a linguistic variable. Then, the following properties should be satisfied:

(1) 

The linguistic term set is ordered:$h_{\tau }<h_{\upsilon }$if and only if$\tau <\upsilon$, where$(h_{\tau },h_{\upsilon }\in H)$;

(2) 

If a negation operator exists, then$neg(h_{\tau })=h_{(2t+1-\tau )}(\tau ,\upsilon =1,2,\cdots ,2t+1)$.

Definition 2

([48]).Let$h_{\tau }\in H$be a linguistic term. If${\theta }_{\tau }\in [0,1]$is a numerical value, then the linguistic scale function$f$that conducts the mapping from$h_{\tau }$to${\theta }_{\tau }\left(\tau =1,2,\cdots ,2t+1\right)$can be defined as

$$f:{\mathrm{s}}_{\tau }\to {\theta }_{\tau }\left(\tau =1,2,\cdots ,2t+1\right),$$

(1)

where$0\le {\theta }_1<{\theta }_2<\cdots <{\theta }_{2t+1}\le 1$.

Based on the existed studies, three types of linguistic scale functions are described as

$$f_1\left(h_x\right)={\theta }_x=\frac{\mathrm{x}}{2t},\left(x=1,2,\cdots ,2t+1\right),{\theta }_x\in \left[0,1\right];$$

(2)

$$f_2\left(h_y\right)={\theta }_y=\{\begin{array}{l}\frac{{\alpha }^t-{\alpha }^{t-y}}{2{\alpha }^t-2},\left(y=1,2,\cdots ,t+1\right),\\ \frac{{\alpha }^t+{\alpha }^{y-t}-2}{2{\alpha }^t-2},\left(y=t+2,t+3,\cdots ,2t+1\right);\end{array}$$

(3)

$$f_3\left(h_z\right)={\theta }_z=\{\begin{array}{l}\frac{t^{\beta }-{\left(t-z\right)}^{\beta }}{2t^{\beta }},\left(z=1,\cdots ,t+1\right),\\ \frac{t^{\gamma }+{\left(z-t\right)}^{\gamma }}{2t^{\gamma }},\left(z=t+2,\cdots ,2t+1\right).\end{array}$$

(4)

2.2. SNSs and SNLSs

Definition 3

([1]).Let$X$be a space of points (objects), and x be a generic element in$X$. A NS$A$in$X$is characterized by a truth-membership function$T_A\left(x\right)$, a indeterminacy-membership function$I_A\left(x\right)$, and a falsity-membership function$F_A\left(x\right)$. $T_A\left(x\right)$, $I_A\left(x\right)$, and$F_A\left(x\right)$are real standard or nonstandard subsets$\left]0^{-},1^{+}\right[$. That is,$T_A\left(x\right):x\to \left]0^{-},1^{+}\right[$, $I_A\left(x\right):x\to \left]0^{-},1^{+}\right[$, and$F_A\left(x\right):x\to \left]0^{-},1^{+}\right[$. There is no restriction on the sum of$T_A\left(x\right)$, $I_A\left(x\right)$, and$F_A\left(x\right)$, so$0^{-}\le \sup T_A\left(x\right)+\sup I_A\left(x\right)+\sup F_A\left(x\right)\le 3^{+}$.

In fact, NSs are very difficult for application without specification. Given this, Ye [34] introduced SNSs by reducing the non-standard intervals of NSs into a kind of standard intervals.

Definition 4

([17]).Let$X$be a space of points with a generic element$x$. Then, an SNS$B$in$X$can be defined as$B=\left\{\left(x,T_B(x),I_B(x),F_B(x)\right)|x\in X\right\}$, where$T_B(x):X\to [0,1]$, $I_B(x):X\to [0,1]$, and$F_B(x):X\to [0,1]$. In addition, the sum of$T_B(x)$, $I_B(x)$, and$F_B(x)$satisfies$0\le T_B(x)+I_B(x)+F_B(x)\le 3$. For simplicity,$B$can be denoted as$B=\langle T_B(x),I_B(x),F_B(x)\rangle$, which is a subclass of NSs.

Definition 5

([20]).Let$X$be a space of points with a generic element x, and$H=\left\{h_{\tau }|\tau =1,2,\cdots ,2t+1,t\in N^{\ast }\right\}$be a linguistic term set. Then an SNLS$C$in$X$is defined as$C=\left\{\langle x,h_C(x),(T_C(x),I_C(x),F_C(x))\rangle |x\in X\right\}$, where$h_C(x)\in H,T_C(x)\in \left[0,1\right],I_C(x)\in \left[0,1\right]$, $F_C(x)\in \left[0,1\right]$and$0\le T_C(x)+I_C(x)+F_C(x)\le 3$for any$x\in X$. In addition,$T_C(x)$, $I_C(x)$, and$F_C(x)$represent the degree of truth-membership, indeterminacy-membership, and falsity-membership of the element$x$in$X$to the linguistic term$h_C\left(x\right)$, respectively. For simplicity, a SNLN is expressed as$\langle h_C(x),(T_C(x),I_C(x),F_C(x))\rangle$.

2.3. The Cloud Model

Definition 6

([22]).Let$U$be a universe of discourse and$T$be a qualitative concept in$U$. $x\in U$is a random instantiation of the concept$T$, and$x$satisfies$x~N\left(Ex,{\left(E{n}^{\ast }\right)}^2\right)$, where$E{n}^{\ast }~N\left(En,H{e}^2\right)$, and the degree of certainty that$x$belongs to the concept$T$is defined as

$$\mu =e^{-\frac{{\left(x-Ex\right)}^2}{2{\left(E{n}^{\ast }\right)}^2}},$$

then the distribution of$x$in the universe$U$is called a normal cloud, and the cloud$C$is presented as$C=\left(Ex,En,He\right)$.

Definition 7

([33]).Let$M\left(E{x}_1,E{n}_1,H{e}_1\right)$and$N\left(E{x}_2,E{n}_2,H{e}_2\right)$be two clouds, then the operations between them are defined as

(1) 

$M+N=\left(E{x}_1+E{x}_2,\sqrt{E{n_1}^2+E{n_2}^2},\sqrt{H{e_1}^2+H{e_2}^2}\right)$;

(2) 

$M-N=\left(E{x}_1-E{x}_2,\sqrt{E{n_1}^2+E{n_2}^2},\sqrt{H{e_1}^2+H{e_2}^2}\right)$;

(3) 

$M\times N=\left(E{x}_{1}E{x}_2,\sqrt{{\left(E{n}_{1}E{x}_2\right)}^2+{\left(E{n}_{2}E{x}_1\right)}^2},\sqrt{{\left(H{e}_{1}E{x}_2\right)}^2+{\left(H{e}_{2}E{x}_1\right)}^2}\right)$;

(4) 

$\lambda M=\left(\lambda E{x}_1,\sqrt{\lambda }E{n}_1,\sqrt{\lambda }H{e}_1\right)$; and

(5) 

$M^{\lambda }=\left(E{x_1}^{\lambda },\sqrt{\lambda }E{x_1}^{\lambda -1}E{n}_1,\sqrt{\lambda }E{x_1}^{\lambda -1}H{e}_1\right)$.

2.4. Transformation Approach of Clouds

Definition 8

([33]).Let$H_i$be a linguistic term in$H=\left\{H_i|i=1,2,\dots ,2t+1\right\}$, and$f$be a linguistic scale function. Then, the procedures for converting linguistic variables to clouds are presented below.

(1) 

Calculate${\theta }_i$: Map$H_i$to${\theta }_i$employing Equation (2) or (3) or (4).

(2) 

Calculate$E{x}_i$: $E{x}_i=X_{\min }+{\theta }_i\left(X_{\max }-X_{\min }\right)$.

(3) 

Calculate$E{n}_i$: Let$\left(x,y\right)$be a cloud droplet. Since$x~N\left(E{x}_i,E{n}_i^{\prime 2}\right)$, we have$3E{n}_i^{\prime }=\max \left\{X_{\max }-E{x}_i,E{x}_i-X_{\min }\right\}$in the light of$3\sigma$principle of the normal distribution curve. Then,$E{n}_i^{\prime }=\{\begin{array}{l}\frac{\left(1-{\theta }_i\right)\left(X_{\max }-X_{\min }\right)}{3}1\le i\le t+1\\ \frac{{\theta }_i\left(X_{\max }-X_{\min }\right)}{3}t+2\le i\le 2t+1\end{array}$. Thus$E{n}_i=\frac{E{n}_{i-1}^{\prime }+E{n}_i^{\prime }+E{n}_{i+1}^{\prime }}{3},\left(1<i<2t+1\right)$, $E{n}_i=\frac{E{n}_i^{\prime }+E{n}_{i+1}^{\prime }}{2},\left(i=1\right)$and$E{n}_i=\frac{E{n}_{i-1}^{\prime }+E{n}_i^{\prime }}{2},\left(i=2t+1\right)$can be obtained.

(4) 

Calculate$H{e}_i$: $H{e}_i=\frac{\left(E{n}^{\prime +}-E{n}_i\right)}{3}$, where$E{n}^{\prime +}=\max \left\{E{n}_i^{\prime }\right\}$.

3. Simplified Neutrosophic Clouds and the Related Concepts

Based on SNLNs and the cloud transformation method, a novel concept of SNCs is proposed. Motivated by the existing studies, we provide the operations and comparison method for SNCs and investigate the distance measurement of SNCs.

3.1. SNCs and Their Operational Rules

Definition 9.

Let$X$be a space of points with a generic element$x$, $H=\left\{h_{\tau }|\tau =1,2,\cdots ,2t+1,t\in N^{\ast }\right\}$be a linguistic term set, and$\langle h_C(x),(T_C(x),I_C(x),F_C(x))\rangle$be a SNLN. In accordance with the cloud conversion method described in Section 2.4, the linguistic term$h_C(x)\in H$can be converted into the cloud$\langle Ex,En,He\rangle$. Then, a simplified neutrosophic cloud (SNC) is defined as

$$Y=\left(\langle Ex,En,He\rangle ,\langle T,I,F\rangle \right)$$

Definition 10.

Let$a=\langle \left(E{x}_1,E{n}_1,H{e}_1\right),\left(T_1,I_1,F_1\right)\rangle$and$b=\langle \left(E{x}_2,E{n}_2,H{e}_2\right),\left(T_2,I_2,F_2\right)\rangle$be two SNCs, then the operations of SNC are defined as

(1) 

$\begin{array}{l}a\oplus b=(\langle E{x}_1+E{x}_2,\sqrt{E{n_1}^2+E{n_2}^2},\sqrt{H{e_1}^2+H{e_2}^2}\rangle ,\langle \frac{T_1\left(E{x}_1+E{n_1}^2+H{e_1}^2\right)+T_2\left(E{x}_2+E{n_2}^2+H{e_2}^2\right)}{E{x}_1+E{x}_2+E{n_1}^2+H{e_1}^2+E{n_2}^2+H{e_2}^2},\\ \frac{I_1\left(E{x}_1+E{n_1}^2+H{e_1}^2\right)+I_2\left(E{x}_2+E{n_2}^2+H{e_2}^2\right)}{E{x}_1+E{x}_2+E{n_1}^2+H{e_1}^2+E{n_2}^2+H{e_2}^2},\frac{F_1\left(E{x}_1+E{n_1}^2+H{e_1}^2\right)+F_2\left(E{x}_2+E{n_2}^2+H{e_2}^2\right)}{E{x}_1+E{x}_2+E{n_1}^2+H{e_1}^2+E{n_2}^2+H{e_2}^2}\rangle );\end{array}$

(2) 

$a\otimes b=\left(\langle E{x}_{1}E{x}_2,E{n}_{1}E{n}_2,H{e}_{1}H{e}_2\rangle ,\langle \langle T_1{T}_2,I_1+I_2-I_1{I}_2,F_1+F_2-F_1{F}_2\rangle \right);$

(3) 

$\lambda a=\left(\langle \lambda E{x}_1,\sqrt{\lambda }E{n}_1,\sqrt{\lambda }H{e}_1\rangle ,\langle T_1,I_1,F_1\rangle \right);$and

(4) 

$a^{\lambda }=\left(\langle E{x_1}^{\lambda },E{n_1}^{\lambda },H{e_1}^{\lambda }\rangle ,\langle {T_1}^{\lambda },1-{\left(1-I_1\right)}^{\lambda },1-{\left(1-F_1\right)}^{\lambda }\rangle \right).$

Theorem 1.

Let$a=\langle \left(E{x}_1,E{n}_1,H{e}_1\right),\left(T_1,I_1,F_1\right)\rangle$, $b=\langle \left(E{x}_2,E{n}_2,H{e}_2\right),\left(T_2,I_2,F_2\right)\rangle$and$c=\langle \left(E{x}_3,E{n}_3,H{e}_3\right),\left(T_3,I_3,F_3\right)\rangle$be three SNCs. Then, the following properties should be satisfied

(1) 

$a+b=b+a$;

(2) 

$\left(a+b\right)+c=a+\left(b+c\right)$;

(3) 

$\lambda a+\lambda b=\lambda \left(a+b\right)$;

(4) 

${\lambda }_{1}a+{\lambda }_{2}a=\left({\lambda }_1+{\lambda }_2\right)a$;

(5) 

$a\times b=b\times a$;

(6) 

$\left(a\times b\right)\times c=a\times \left(b\times c\right)$;

(7) 

$a^{{\lambda }_1}\times a^{{\lambda }_2}=a^{{\lambda }_1+{\lambda }_2}$;

(8) 

${\left(a\times b\right)}^{\lambda }=a^{\lambda }\times b^{\lambda }$.

3.2. Distance for SNCs

Definition 11.

Let$a=\langle \left(E{x}_1,E{n}_1,H{e}_1\right),\left(T_1,I_1,F_1\right)\rangle$and$b=\langle \left(E{x}_2,E{n}_2,H{e}_2\right),\left(T_2,I_2,F_2\right)\rangle$be two SNCs, then the generalized distance between a and b is defined as

$$\begin{array}{l}d\left(a,b\right)=\left|\left(1-{\beta }_1\right)E{x}_1-\left(1-{\beta }_2\right)E{x}_2\right|+(\frac{1}{3}({\left|\left(1-{\beta }_1\right)E{x}_1{T}_1-\left(1-{\beta }_2\right)E{x}_2{T}_2\right|}^{\lambda }+\\ {\left|\left(1-{\beta }_1\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_2\right)E{x}_2\left(1-I_2\right)\right|}^{\lambda }+{{\left|\left(1-{\beta }_1\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_2\right)E{x}_2\left(1-F_2\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }},\end{array}$$

(5)

where${\beta }_1=\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}$and${\beta }_2=\frac{\sqrt{E{n_2}^2+H{e_2}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}$. When λ = 1 and 2, the generalized distance above becomes the Hamming distance and the Euclidean distance, respectively.

Theorem 2.

Let$a=\langle \left(E{x}_1,E{n}_1,H{e}_1\right),\left(T_1,I_1,F_1\right)\rangle$, $b=\langle \left(E{x}_2,E{n}_2,H{e}_2\right),\left(T_2,I_2,F_2\right)\rangle$, and$c=\langle \left(E{x}_3,E{n}_3,H{e}_3\right),\left(T_3,I_3,F_3\right)\rangle$be three SNCs. Then, the distance given in Definition 11 satisfies the following properties:

(1) 

$d\left(a,b\right)\ge 0$;

(2) 

$d\left(a,b\right)=d\left(b,a\right)$; and

(3) 

If$E{x}_1\le E{x}_2\le E{x}_3,E{n}_1\ge E{n}_2\ge E{n}_3,H{e}_1\ge H{e}_2\ge H{e}_3,T_1\le T_2\le T_3,I_1\ge I_2\ge I_3,$and$F_1\ge F_2\ge F_3$, then$d\left(a,b\right)\le d\left(a,c\right)$, and$d\left(b,c\right)\le d\left(a,c\right)$.

Proof. 

It is easy to prove that (1) and (2) in Theorem 2 are true. The proof of (3) in Theorem 2 is depicted in the following.

Let ${\beta }_{(a,b)1}=\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}$, ${\beta }_{(a,b)2}=\frac{\sqrt{E{n_2}^2+H{e_2}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}$, ${\beta }_{(a,c)1}=\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}$, and ${\beta }_{(a,c)2}=\frac{\sqrt{E{n_3}^2+H{e_3}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}$, then there are

$$\begin{array}{l}d\left(a,c\right)=\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\right|\\ +(\frac{1}{3}({\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3{T}_3\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-I_3\right)\right|}^{\lambda }\\ {+{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-F_3\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }},\end{array}$$

$$\begin{array}{l}d\left(a,b\right)=\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\right|\\ +(\frac{1}{3}({\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2{T}_2\right|}^{\lambda }+|\left(1-{\beta }_{(a,b)1}\right)\\ +{E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-I_2\right)|}^{\lambda }\\ {+{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-F_2\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }}.\end{array}$$

Thus, we have

$$\begin{array}{l}d\left(a,c\right)-d(a,b)=\left(1-{\beta }_{(a,b)1}\right)E{x}_1-\left(1-{\beta }_{(a,c)1}\right)E{x}_1\\ +\left(1-{\beta }_{(a,c)2}\right)E{x}_3-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\\ +(\frac{1}{3}({\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3{T}_3\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-I_3\right)\right|}^{\lambda }\\ {+{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-F_3\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }}\\ -(\frac{1}{3}({\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2{T}_2\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-I_2\right)\right|}^{\lambda }\\ {+{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-F_2\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }}.\end{array}$$

Let

$$\begin{array}{l}p=\left(1-{\beta }_{(a,b)1}\right)E{x}_1-\left(1-{\beta }_{(a,c)1}\right)E{x}_1+\left(1-{\beta }_{(a,c)2}\right)E{x}_3-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\\ =\left(1-\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}\right)E{x}_1-\left(1-\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}\right)E{x}_1\\ +\left(1-\frac{\sqrt{E{n_3}^2+H{e_3}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}\right)E{x}_3-\left(1-\frac{\sqrt{E{n_2}^2+H{e_2}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}\right)E{x}_2.\end{array}$$

$$\begin{array}{l}q=(\frac{1}{3}({\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3{T}_3\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-I_3\right)\right|}^{\lambda }\\ {+{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-F_3\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }}\\ -(\frac{1}{3}({\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2{T}_2\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-I_2\right)\right|}^{\lambda }\\ {+{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-F_2\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }},\end{array}$$

then $d\left(a,c\right)-d\left(a,b\right)=p+q$.

Simplifying the above equations, the following results can be obtained.

$$\begin{array}{l}p=\frac{\sqrt{E{n_2}^2+H{e_2}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}E{x}_1-\frac{\sqrt{E{n_3}^2+H{e_3}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}E{x}_1\\ +\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}E{x}_3-\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}E{x}_2.\end{array}$$

Since $E{x}_1\le E{x}_2\le E{x}_3$, $E{n}_1\ge E{n}_2\ge E{n}_3$, and $H{e}_1\ge H{e}_2\ge H{e}_3$, we have

$$\frac{\sqrt{E{n_2}^2+H{e_2}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}E{x}_1-\frac{\sqrt{E{n_3}^2+H{e_3}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}E{x}_1\ge 0,$$

$$\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_3}^2+H{e_3}^2}}E{x}_3-\frac{\sqrt{E{n_1}^2+H{e_1}^2}}{\sqrt{E{n_1}^2+H{e_1}^2}+\sqrt{E{n_2}^2+H{e_2}^2}}E{x}_2\ge 0.$$

Thus, $p\ge 0$ is determined.

According to $p=\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\right|-\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\right|\ge 0$, the following inequalities can be deduced.

$$\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\right|\ge \left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\right|,$$

$${\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\right|}^{\lambda }\ge {\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\right|}^{\lambda }.$$

Since $T_1\le T_2\le T_3$, the following inequality is true.

$${\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3{T}_3\right|}^{\lambda }\ge {\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2{T}_2\right|}^{\lambda }.$$

In a similar manner, we can also obtain

$${\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-I_3\right)\right|}^{\lambda }\ge {\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-I_2\right)\right|}^{\lambda },$$

$${\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-F_3\right)\right|}^{\lambda }\ge {\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-F_2\right)\right|}^{\lambda }.$$

Thus, there is

$$\begin{array}{l}q=(\frac{1}{3}({\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,c)2}\right)E{x}_3{T}_3\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-I_3\right)\right|}^{\lambda }\\ +{{\left|\left(1-{\beta }_{(a,c)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,c)2}\right)E{x}_3\left(1-F_3\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }}\\ -(\frac{1}{3}({\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1{T}_1-\left(1-{\beta }_{(a,b)2}\right)E{x}_2{T}_2\right|}^{\lambda }\\ +{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-I_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-I_2\right)\right|}^{\lambda }\\ +{{\left|\left(1-{\beta }_{(a,b)1}\right)E{x}_1\left(1-F_1\right)-\left(1-{\beta }_{(a,b)2}\right)E{x}_2\left(1-F_2\right)\right|}^{\lambda }))}^{\frac{1}{\lambda }}\\ \ge 0.\end{array}$$

Thus, $d\left(a,c\right)-d(a,b)\ge 0\Rightarrow d\left(a,c\right)\ge d(a,b)$. The inequality $d\left(a,c\right)\ge d(b,c)$ can be proved similarly. Hence, the proof of Theorem 2 is completed. □

Example 1.

Let$a=\langle \left(0.5,0.2,0.1\right),\left(0.7,0.3,0.5\right)\rangle$, and$b=\langle \left(0.6,0.1,0.1\right),\left(0.8,0.2,0.4\right)\rangle$be two SNCs. Then, according to Definition 11, the Hamming distance$d_{Ham\min g}\left(a,b\right)$and Euclidean distance$d_{Euclidean}\left(a,b\right)$are calculated as

$$d_{Ham\min g}\left(a,b\right)=0.4304,and{d}_{Euclidean}\left(a,b\right)=0.3224.$$

4. SNCs Aggregation Operators

Maclaurin [38] introduced the MSM aggregation operator firstly. In this section, the MSM operator is expanded to process SNC information, and the SNCMSM operator and the weighted SNCMSM operator are then proposed.

Definition 12

([38]).Let$x_i\left(i=1,2,\cdots ,n\right)$be the set of nonnegative real numbers. A MSM aggregation operator of dimension n is mapping$MS{M}^{\left(m\right)}:{\left(R^{+}\right)}^n\to R^{+}$, and it can be defined as

$$MS{M}^{\left(m\right)}(x_1,x_2,\cdots ,x_n)={\left(\frac{{\displaystyle \sum _{1\le i_1<\cdot \cdot \cdot <i_m\le n}{\displaystyle \prod _{j=1}^m{x}_{i_j}}}}{C_n^m}\right)}^{\frac{1}{m}},$$

(6)

where$\left(i_1,i_2,\cdots ,i_m\right)$traverses all the m-tuple combination of$\left(i=1,2,\cdots ,n\right)$, $C_n^m=\frac{n!}{m!\left(n-m\right)!}$is the binomial coefficient. In the subsequent analysis, assume that$i_1<i_2<,\dots ,<i_m$. In addition,$x_{i_j}$refers to the$i_j$th element in a particular arrangement.

It is clear that$MS{M}^{\left(m\right)}$has the following properties:

(1) 

Idempotency. If$x\ge 0$and$x_i=x$for all$i$, then$MS{M}^{\left(m\right)}\left(x,x,\dots ,x\right)=x$.

(2) 

Monotonicity. If$x_i\le y_i$, for all$i$, $MS{M}^{\left(m\right)}\left(x_1,x_2,\dots ,x_n\right)\le MS{M}^{\left(m\right)}\left(y_1,y_2,\dots ,y_n\right)$, where x_i and y_i are nonnegative real numbers.

(3) 

Boundedness.$MIN\left\{x_1,x_2,\dots ,x_n\right\}\le MS{M}^{\left(m\right)}\left(x_1,x_2,\dots ,x_n\right)\le MAX\left\{x_1,x_2,\dots ,x_n\right\}.$

4.1. SNCMSM Operator

In this subsection, the traditional $MS{M}^{\left(m\right)}$ operator is extended to accommodate the situations where the input variables are made up of SNCs. Then, the SNCMSM operator is developed.

Definition 13.

Let$a_i=\langle \left(E{x}_i,E{n}_i,H{e}_i\right),\left(T_i,I_i,F_i\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of SNCs. Then, the SNCMSM operator can be defined as

$$SNCMS{M}^{(m)}(a_1,a_2,\cdots ,a_n)={\left(\frac{\underset{1\le i_1<\cdot \cdot \cdot <i_m\le n}{\oplus }\left(\underset{j=1}{\overset{m}{\otimes }}{a}_{i_j}\right)}{C_n^m}\right)}^{\frac{1}{m}},$$

(7)

where$m=1,2,\dots ,n$and$\left(i_1,i_2,\cdots ,i_m\right)$traverses all the m-tuple combination of$\left(i=1,2,\cdots ,n\right)$, $C_n^m=\frac{n!}{m!\left(n-m\right)!}$is the binomial coefficient.

In light of the operations of SNCs depicted in Definition 10, Theorem 3 can be acquired.

Theorem 3.

Let$a_i=\langle \left(E{x}_i,E{n}_i,H{e}_i\right),\left(T_i,I_i,F_i\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of SNCs, the aggregated value acquired by the SNCMSM operator is also a SNC and can be expressed as

$$\begin{array}{l}SNCMS{M}^{(m)}(a_1,a_2,\cdots ,a_n)\\ =(\langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}}}{C_n^m}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}}\rangle ,\\ \langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{T}_{i_j^k}}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}},\\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}},\\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-F_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}\rangle ).\end{array}$$

(8)

Proof. 

$$\begin{gathered} a_{i_j^{(k)}}=\left(\langle E{x}_{i_j^{(k)}},E{n}_{i_j^{(k)}},H{e}_{i_j^{(k)}}\rangle ,\langle T_{i_j^{(k)}},I_{i_j^{(k)}},F_{i_j^{(k)}}\rangle \right),\left(\right(j=1,2,\dots ,m). \\ \begin{array}{l}\Rightarrow \underset{j=1}{\overset{m}{\otimes }}{a}_{i_j^{(k)}}=(\langle {\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}},{\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}},{\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\rangle ,\\ \langle {\displaystyle \prod _{j=1}^m{T}_{i_j^{\left(k\right)}}},1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^{\left(k\right)}}\right),1-{\displaystyle \prod _{j=1}^m\left(1-F_{i_j^{\left(k\right)}}\right)}}\rangle )\end{array} \\ \Rightarrow \underset{1\le t_1<\cdot \cdot \cdot <t_m\le n}{\oplus }\left(\underset{j=1}{\overset{m}{\otimes }}{a}_{i_j}\right)=(\langle {\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}},}}\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2}},\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2}}\rangle , \\ \langle \frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{T}_{i_j^k}}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}, \\ \frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}, \\ \frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-F_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\rangle ) \\ \Rightarrow {\left(\frac{\underset{1\le i_1<\cdot \cdot \cdot <i_m\le n}{\oplus }\left(\underset{j=1}{\overset{m}{\otimes }}\left(a_{i_j}\right)\right)}{C_n^m}\right)}^{\frac{1}{m}} \\ =(\langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}}}{C_n^m}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}}\rangle , \\ \langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{T}_{i_j^k}}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-F_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}\rangle ). \end{gathered}$$

The proof of Theorem 3 is completed. □

Theorem 4.

(Idempotency) If$a_i=a=\left(\langle E{x}_a,E{n}_a,H{e}_a\rangle ,\langle T_a,I_a,F_a\rangle \right)$for all$i=1,2,\dots ,n$, then$SNCMS{M}^{(m)}(a,a,\cdots ,a)=a=\left(\langle E{x}_a,E{n}_a,H{e}_a\rangle ,\langle T_a,I_a,F_a\rangle \right)$.

Proof. 

Since $a_i=a$, there are

$$\begin{gathered} SNCMS{M}^{\left(m\right)}\left(a,a,\cdots ,a\right) \\ =(\langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^{m}E{x}_a}}}{C_n^m}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}}\rangle , \\ \langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{T}_a}\left({\displaystyle \prod _{j=1}^{m}E{x}_a}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_a}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_a}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_a}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}E{x}_a}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}E{x}_a}+{\left({\displaystyle \prod _{j=1}^{m}E{n}_a}\right)}^2+{\left({\displaystyle \prod _{j=1}^{m}H{e}_a}\right)}^2\right)}}\right)}^{\frac{1}{m}}\rangle ) \\ =\left(\langle E{x}_a,E{n}_a,H{e}_a\rangle ,\langle T_a,I_a,F_a\rangle \right)=a. \end{gathered}$$

□

Theorem 5 (Commutativity).

Let$\left(a{\prime }_1,a{\prime }_2,\cdots ,a{\prime }_n\right)$be any permutation of$\left(a_1,a_2,\cdots ,a_n\right)$. Then,$SNCMS{M}^{\left(m\right)}\left(a{\prime }_1,a{\prime }_2,\cdots ,a{\prime }_n\right)=SNCMS{M}^{\left(m\right)}\left(a_1,a_2,\cdots ,a_n\right)$.

Theorem 5 can be proved easily in accordance with Definition 13 and Theorem 3.

Three special cases of the SNCMSM operator are discussed below by selecting different values for the parameter m.

(1)

If m = 1, then the SNCMSM operator becomes the simplest arithmetic average aggregation operator as follows:

$$\begin{gathered} SNCMS{M}^{\left(1\right)}\left(a_1,a_2,\cdots ,a_n\right)=\frac{{\oplus }_{i=1}^n{a}_i}{n} \\ =(\langle {\displaystyle \sum _{i=1}^{n}E{x}_i},\sqrt{{\displaystyle \sum _{i=1}^{n}E{n_i}^2}},\sqrt{{\displaystyle \sum _{i=1}^{n}H{e_i}^2}}\rangle ,\langle \frac{{\displaystyle \sum _{i=1}^n{T}_i\left(E{x}_i+E{n_i}^2+H{e_i}^2\right)}}{{\displaystyle \sum _{i=1}^n\left(E{x}_i+E{n_i}^2+H{e_i}^2\right)}}, \\ \frac{{\displaystyle \sum _{i=1}^n{I}_i\left(E{x}_i+E{n_i}^2+H{e_i}^2\right)}}{{\displaystyle \sum _{i=1}^n\left(E{x}_i+E{n_i}^2+H{e_i}^2\right)}},\frac{{\displaystyle \sum _{i=1}^n{F}_i\left(E{x}_i+E{n_i}^2+H{e_i}^2\right)}}{{\displaystyle \sum _{i=1}^n\left(E{x}_i+E{n_i}^2+H{e_i}^2\right)}}\rangle ). \end{gathered}$$

(9)

(2)

If m = 2, then the SNCMSM operator is degenerated to the following form:

$$\begin{gathered} SNCMS{M}^{\left(2\right)}\left(a_1,a_2,\cdots ,a_n\right)={\left(\frac{{\oplus }_{i,j=1,i\ne j}^n{a}_i\otimes a_j}{n(n-1)}\right)}^{\frac{1}{2}} \\ =(\langle {\left(\frac{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^{n}E{x}_{i}E{x}_j}}{n(n-1)}\right)}^{\frac{1}{2}},{\left(\sqrt{\frac{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n{\left(E{n}_{i}E{n}_j\right)}^2}}{n(n-1)}}\right)}^{\frac{1}{2}},{\left(\sqrt{\frac{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n{\left(H{e}_{i}H{e}_j\right)}^2}}{n(n-1)}}\right)}^{\frac{1}{2}}\rangle , \\ {\langle \left(\frac{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n{T}_i{T}_j\left(E{x}_{i}E{x}_j+E{n_i}^{2}E{n_j}^2+H{e_i}^{2}H{e_j}^2\right)}}{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left(E{x}_{i}E{x}_j+E{n_i}^{2}E{n_j}^2+H{e_i}^{2}H{e_j}^2\right)}}\right)}^{\frac{1}{2}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left[1-\left(1-I_i\right)\left(1-I_j\right)\right]\left(E{x}_{i}E{x}_j+E{n_i}^{2}E{n_j}^2+H{e_i}^{2}H{e_j}^2\right)}}{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left(E{x}_{i}E{x}_j+E{n_i}^{2}E{n_j}^2+H{e_i}^{2}H{e_j}^2\right)}}\right)}^{\frac{1}{2}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left[1-\left(1-F_i\right)\left(1-F_j\right)\right]\left(E{x}_{i}E{x}_j+E{n_i}^{2}E{n_j}^2+H{e_i}^{2}H{e_j}^2\right)}}{{\displaystyle \sum _{\begin{array}{l}i,j=1\\ i\ne j\end{array}}^n\left(E{x}_{i}E{x}_j+E{n_i}^{2}E{n_j}^2+H{e_i}^{2}H{e_j}^2\right)}}\right)}^{\frac{1}{2}}\rangle ). \end{gathered}$$

(10)

(3)

If m = n, then the SNCMSM operator becomes the geometric average aggregation operator as follows:

$$\begin{array}{l}SNCMS{M}^{\left(n\right)}\left(a_1,a_2,\cdots ,a_n\right)={\left({\otimes }_{i=1}^n{a}_i\right)}^{\frac{1}{n}}\\ =(\langle {\left({\displaystyle \prod _{i=1}^{n}E{x}_i}\right)}^{\frac{1}{n}},{\left({\displaystyle \prod _{i=1}^{n}E{n}_i}\right)}^{\frac{1}{n}},{\left({\displaystyle \prod _{i=1}^{n}H{e}_i}\right)}^{\frac{1}{n}}\rangle ,\\ \langle {\left({\displaystyle \prod _{i=1}^n{T}_i}\right)}^{\frac{1}{n}},{\left(1-{\displaystyle \prod _{i=1}^n\left(1-I_i\right)}\right)}^{\frac{1}{n}},{\left(1-{\displaystyle \prod _{i=1}^n\left(1-F_i\right)}\right)}^{\frac{1}{n}}\rangle ).\end{array}$$

(11)

4.2. Weighted SNCMSM Operator

In this subsection, a weighted SNCMSM operator is investigated. Moreover, some desirable properties of this operator are analyzed.

Definition 14.

Let$a_i=\langle \left(E{x}_i,E{n}_i,H{e}_i\right),\left(T_i,I_i,F_i\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of SNCs, and$w={\left(w_1,w_2,\dots w_n\right)}^T$be the weight vector, with$w_i\in \left[0,1\right]$and${\displaystyle {\sum }_{i=1}^n{w}_i}=1$. Then, the weighted simplified neutrosophic clouds Maclaurin symmetric mean (WSNCMSM) operator is defined as

$$WSNCMS{M_w}^{(m)}(a_1,a_2,\cdots ,a_n)={\left(\frac{\underset{1\le i_1<\cdot \cdot \cdot <i_m\le n}{\oplus }\left(\underset{j=1}{\overset{m}{\otimes }}\left(n{w}_{i_j}\cdot a_{i_j}\right)\right)}{C_n^m}\right)}^{\frac{1}{m}},$$

(12)

where$m=1,2,\dots ,n$and$\left(i_1,i_2,\cdots ,i_m\right)$traverses all the m-tuple combination of$\left(i=1,2,\cdots ,n\right)$, $C_n^m=\frac{n!}{m!\left(n-m\right)!}$is the binomial coefficient.

The specific expression of the WSNCMSM operator can be obtained in accordance with the operations provided in Definition 10.

Theorem 6.

Let$a_i=\langle \left(E{x}_i,E{n}_i,H{e}_i\right),\left(T_i,I_i,F_i\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of SNCs, and$m=1,2,\dots ,n$. Then, the aggregated value acquired by the WSNCMSM operator can be expressed as

$$\begin{gathered} {{WSNCMSM}_w}^{\left(m\right)}\left(a_1,a_2,\cdots ,a_n\right) \\ =(\langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}}}{C_n^m}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}}\rangle , \\ \langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{T}_{i_j^k}}\left({\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-F_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}\rangle ). \end{gathered}$$

(13)

Theorem 6 can be proved similarly according to the proof procedures of Theorem 3.

Theorem 7.

(Reducibility) Let$w={\left(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n}\right)}^T$, then, WSNCMSM_w^(m)(a₁, a₂, …, a_n) = SNCMSM^(m)(a₁, a₂, …, a_n).

Proof. 

When $w={\left(\frac{1}{n},\frac{1}{n},\dots ,\frac{1}{n}\right)}^T$,

$$\begin{gathered} WSNCMS{M_w}^{\left(m\right)}\left(a_1,a_2,\cdots ,a_n\right) \\ =(\langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}}}{C_n^m}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{m}}\rangle , \\ \langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{T}_{i_j^k}}\left({\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-I_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m\left(1-F_{i_j^k}\right)}\right)\left({\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^{m}n\cdot \frac{1}{n}E{x}_{i_j^{\left(k\right)}}}+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}E{n}_{i_j^{\left(k\right)}}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m\sqrt{n\cdot \frac{1}{n}}H{e}_{i_j^{\left(k\right)}}}\right)}^2\right)}}\right)}^{\frac{1}{m}}\rangle ) \\ =SNCMS{M}^{\left(m\right)}\left(a_1,a_2,\cdots ,a_n\right). \end{gathered}$$

The proof of Theorem 7 is completed. □

Definition 15.

Let$a_i=\langle \left(E{x}_i,E{n}_i,H{e}_i\right),\left(T_i,I_i,F_i\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of SNCs, and$w={(w_1,w_2,\dots ,w_n)}^T$be the weight vector, which satisfies${\displaystyle \sum _{i=1}^n{w}_i}=1$, and$w_i>0(i=1,2,\dots ,n)$. Then the generalized weighted simplified neutrosophic clouds Maclaurin symmetric mean (GWSNCMSM) operator is defined as

$$GWSNCMS{M}^{\left(m,p_1,p_2,\dots ,p_m\right)}(a_1,\dots ,a_n)={\left(\frac{{\oplus }_{1\le i_1<\cdot \cdot \cdot <i_m\le n}\left({\otimes }_{j=1}^m{\left(n{w}_{i_j}\otimes a_{i_j}\right)}^{p_j}\right)}{C_n^m}\right)}^{\frac{1}{p_1+\cdot \cdot \cdot +p_m}},$$

(14)

where$m=1,2,\dots ,n$.

The specific expression of the GWSNCMSM operator can be obtained in accordance with the operations provided in Definition 10.

Theorem 8.

Let$a_i=\langle \left(E{x}_i,E{n}_i,H{e}_i\right),\left(T_i,I_i,F_i\right)\rangle \left(i=1,2,\dots ,n\right)$be a collection of SNCs, and$m=1,2,\dots ,n$. Then, the aggregated value acquired by the GWSNCMSM operator can be expressed as

$$\begin{gathered} GWSNCMS{M}^{\left(m,p_1,p_2,\dots ,p_m\right)}\left(a_1,\dots ,a_n\right)=(\langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}{\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}}}{C_n^m}\right)}^{\frac{1}{p_1+\cdot \cdot \cdot +p_m}}, \\ {\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{p_1+\cdot \cdot \cdot +p_m}},{\left(\frac{\sqrt{{\displaystyle \sum _{k=1}^{C_n^m}{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2}}}{\sqrt{C_n^m}}\right)}^{\frac{1}{p_1+\cdot \cdot \cdot +p_m}}\rangle , \\ \langle {\left(\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{\left(T_{i_j^k}\right)}^{P_j}}\left({\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2\right)}}\right)}^{\frac{1}{p_1+\cdot \cdot \cdot +p_m}}, \\ 1-{\left(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m{\left(1-I_{i_j^k}\right)}^{P_j}}\right)\left({\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}\left({\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2\right)}}\right)}^{\frac{1}{p_1+\cdot \cdot \cdot +p_m}}, \\ 1-{(1-\frac{{\displaystyle \sum _{k=1}^{C_n^m}\left(\left(1-{\displaystyle \prod _{j=1}^m{\left(1-F_{i_j^k}\right)}^{P_j}}\right)\left({\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}H{e}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2\right)\right)}}{{\displaystyle \sum _{k=1}^{C_n^m}({\displaystyle \prod _{j=1}^m{\left(n{w}_{i_j}E{x}_{i_j^{\left(k\right)}}\right)}^{P_j}}+{\left({\displaystyle \prod _{j=1}^m{\left(\sqrt{n{w}_{i_j}}E{n}_{i_j^{\left(k\right)}}\right)}^{P_j}}\right)}^2+{({\displaystyle \prod _{j=1}^m{(\sqrt{n{w}_{}}}^{}}}^{}}}}^{} \end{gathered}$$