Some Single-Valued Neutrosophic Power Heronian Aggregation Operators and Their Application to Multiple-Attribute Group Decision-Making

Abstract

The power Heronian aggregation (PHA) operator can use the advantages of power average and the Heronian mean operator, which together could take into account the interrelationship of the aggregated arguments, and therefore alleviate the effects caused by unreasonable data through considering the support degree between input arguments. However, PHA operators cannot be used to process single-valued neutrosophic numbers (SVNNs), which is significant for extending it to SVNNs. We propose some new PHA operators for SVNNs and introduce a novel MAGDM method on the basis of the proposed operators. Firstly, the definition, properties, comparison method, and operational rules of SVNNs are introduced briefly. Then, some PHA operators are proposed, such as the single-valued neutrosophic power Heronian aggregation (SVNPHA) operator, the single-valued neutrosophic weighted power Heronian aggregation (SVNWPHA) operator, single-valued neutrosophic geometric power Heronian aggregation (SVNGPHA) operator, single-valued neutrosophic weighted geometric power Heronian aggregation (SVNWGPHA) operator. Furthermore, we discuss some properties of these new aggregation operators and several special cases. Moreover, the method to solve the MAGDM problems with SVNNs is proposed, based on the SVNWPHA and SVNWGPHA operators. Lastly, we verified the application and effectiveness of the proposed method by using an example for the MAGDM problem.

1. Introduction

In real decision-making, multiple-attribute group decision-making (MAGDM) methods are extensively used to rank alternatives for relevant attributes. The decision information involved in MAGDM problems is often fuzzy and easily expressed by fuzzy information owing to the limitations of human thinking and the complexity of decision-making problems. Accordingly, Zadeh [1] first proposed fuzzy sets (FSs) to handle fuzzy information. Then, Atanassov [2,3] proposed the intuitionistic fuzzy sets (IFSs) based on FS owing to fuzziness. IFSs can embody the degrees of satisfaction and dissatisfaction to express the judgment of alternative. IFS has immediately gained widespread attention and been extensively studied in the field of MAGDM [4,5,6,7,8].

Although the IFS theory has been generalized, it still cannot solve all uncertain problems in real decision-making, such as those that involve indeterminate and inconsistent information. Therefore, to overcome these shortcomings, neutrosophic sets (NSs) was introduced by Smarandache [9], which can substantially express indeterminate or inconsistent information. The NSs is finer and smoother in describing fuzzy information than FS and IFS. However, the application of NSs is very difficult in practice due to the lack of specific description. Therefore, Wang et al. [10] defined single-valued neutrosophic sets (SVNSs), which is an instance of NSs. Meanwhile, SVNSs have attracted increasing attention from researchers, obtained numerous research achievements [11,12,13,14,15] and been extensively applied in many areas. The concept of distance of the two SVNSs and several similarity measures between SVNSs are defined and proposed by Majumdar and Samanta [16]. Ye [15] introduced the correlation coefficients and decision-making method of SVNSs. Şahin and Küçük [17] introduced a method to deal with SVNSs based on the neutrosophic subsethood measure. Based on the use of Hamming distance, Stanujkić [18] proposed a extend WASPAS method with single-valued intuitionistic fuzzy numbers, and application it in website evaluation. Liu, et al. [19] presented a single valued neutrosophic number Decision-making Trial and Evaluation Laboratory Method (SVNN-DEMATEL) model for evaluating and selecting a transport service provider. Pamučar et al. [20] proposed a Linguistic Neutrosophic Numbers Pairwise-CODAS model, which can eliminate subjective qualitative assessments and assumptions by decision makers in complex decision-making conditions. Pamučar et al. [21] presented a new Linguistic Neutrosophic Numbers Weighted Aggregated Sum Product Assessment (LNN WASPAS) model and further developed LNN CODAS (Combinative Distance-Based Assessment), LNN VIKOR (Multi-criteria Optimization and Compromise Solution) and LNN MABAC (Multi-Attributive Border Approximation Area Comparison) models.

Information aggregation generally became an important topic of MAGDM and received increasing attention from researchers [22,23,24]. A few new extended aggregation operators for NS and SVNS have been proposed previously [25,26,27,28]. Ji, et al. [29] proposed several Frank prioritized Bonferroni mean operators for SVNSs. Ren and Sugumaran [30] proposed some prioritized weighted geometric operators for SVNSs. Ye [31] extended several weighted arithmetic average operators for SNSs. Liu and Wang [32] extended a weighted Bonferroni mean operator for SVNSs. Li, et al. [33] extended Heronian mean (HM) operator for SVNSs. Yang and Li [34] extended the PA operator to SVNSs and developed some single-valued neutrosophic power weighted average (SVNPWA) operators. Wei and Zhang [35] proposed some Bonferroni power aggregation operators for SVNSs.

It is obvious that there are different functions for the different aggregation operators. Yager [36] firstly proposed power average (PA) operator, which can alleviate the effects caused by the unreasonable data through considering the support degree between input data. Xu [37] further defined a power geometric (PG) operator. Beliakov [38] first proposed Heronian mean operator, which can capture the interrelation of the input arguments. As an important generalization of Bonferroni mean, Yu [39] proposed the geometric Heronian mean. To consider the advantages of PA and HM operators together, P. Liu proposed some power Heronian aggregation (PHA) operators through combining the PA operator and HM operator to IVIFNs [40] and Linguistic Neutrosophic Sets (LNSs) [41].

The decision-making problems have increased in complexity, and in order to derive the best alternative for MAGDM problems, we need to consider both the influence of some unreasonable attribute values caused by the preference of decision maker, and the interrelationship and interaction among the attributes. Taking the advantages of the HM and PA operator, the PHA operator can achieve the following two functions: the interrelationship of the aggregated arguments could be taken into account, and the influence of unreasonable data could be eliminated by considering the support degree between input arguments. Besides, up to now, there has been no research on how to use PHA operator to aggregate the SVNNs. So, the goal and motivation of this study is to:

(1)

Establish the single-valued neutrosophic PHA (SVNPHA) operator, single-valued neutrosophic geometric PHA (SVNGPHA) operators and the weighted form of these operators (the form of shorthand is SVNWPHA and SVNWGPHA).

(2)

Discuss their properties and analyze special cases.

(3)

Propose a novel MAGDM method based on the SVNWPHA and SVNWGPHA operators for SVNNs.

(4)

Demonstrate the application and effectiveness of the developed methods.

The remainder of this paper is organized as follows. In Section 2, some definitions of SVNSs and some operational rules of SVNNs are introduced. In Section 3, we propose some new PHA operators, such as SVNPHA, SVNWPHA, SVNGPHA and SVNWGPHA operators. In Section 4, a MAGDM method is proposed based on the above operators in SVNNs environment. In Section 5, we demonstrate the application and effectiveness of the proposed method by using an example for the MAGDM problem. Section 6 presents the conclusion.

2. Preliminaries

2.1. The SVNNs

Definition 1.

[42]. Let X be a space of points with a generic element in X denoted by x. SVNS A in X is as follows:

$$A=\left\{x\left(T_A(x),I_A(x),F_A(x)\right)|x\in X\right\}$$

(1)

where $T_A(x)$ is the truth-membership function, $I_A(x)$ is the indeterminacy-membership function, and $F_A(x)$ is the falsity-membership function. For each point $x$ in $X$, we have $T_A(x)$, $I_A(x)$, $F_A(x)\in \left[0,1\right]$, and $0\le T_A(x),I_A(x),F_A(x)\le 3$.

For convenience, we can use $x=\left(T_A,I_A,F_A\right)$ to represent an element in SVNS and call it an SVNN.

2.2. Operational Rules and Properties of SVNNs

Definition 2.

[32]. Let $x_i=\left(T_i,I_i,F_i\right)$ and $x_j=\left(T_j,I_j,F_j\right)$ be any two SVNNs and $\lambda >0$, the operations are defined as follows:

$$x_i\oplus x_j=\left(T_i+T_j-T_i{T}_j,I_i{I}_j,F_i{F}_j\right)$$

(2)

$$x_i\otimes x_j=\left(T_i{T}_j,I_i+I_j-I_i{I}_j,F_i+F_j-F_i{F}_j\right)$$

(3)

$$\lambda x_i=\left(1-{\left(1-T_i\right)}^{\lambda },I_i^{\lambda },F_i^{\lambda }\right)$$

(4)

$$x_i{}^{\lambda }=\left(T_i^{\lambda },1-{\left(1-I_i\right)}^{\lambda }1-{\left(1-F_i\right)}^{\lambda }\right)$$

(5)

Theorem 1.

For any two SVNNs$x_i=\left(T_i,I_i,F_i\right)$and$x_j=\left(T_j,I_j,F_j\right)$, and$\eta ,{\eta }_1,{\eta }_2$ > 0, their operational rules have the following properties:

$$x_i\otimes x_j=x_j\otimes x_i$$

(6)

$$x_i\oplus x_j=x_j\oplus x_i$$

(7)

$$\eta (x_i\oplus x_j)=\eta x_j\oplus \eta x_i$$

(8)

$${\eta }_1{x}_i\oplus {\eta }_2{x}_i=\left({\eta }_1\oplus {\eta }_2\right)x_i$$

(9)

$$x_i{}^{\eta }\otimes x_j{}^{\eta }={(x_i\otimes x_j)}^{\eta }$$

(10)

$$x_i{}^{{\eta }_1}\otimes x_i{}^{{\eta }_2}=x_i{}^{\left({\eta }_1\oplus {\eta }_2\right)}$$

(11)

2.3. Comparison of SVNNs

Definition 3.

[43]. Let $x=\left(T_i,I_i,F_i\right)$ be an SVNN. The score $s(x)$, accuracy $a\left(x\right)$, and certainty $c\left(x\right)$ functions of $x$ can be defined as follows:

$$s(x)=\left(T_i+2-I_i-F_i\right)/3$$

(12)

$$a(x)=T_i-F_i$$

(13)

$$c\left(x\right)=T_i$$

(14)

Definition 4.

[43]. Suppose $x_i=\left(T_i,I_i,F_i\right)$ and $x_j=\left(T_j,I_j,F_j\right)$ be two SVNNs. The comparison method between $x_i$ and $x_j$ can be defined as follows:

(1) If$s(x_i)>s(x_j)$, then$x_i\succ x_j$;

(2) If $s(x_i)=s(x_j)$ and$a\left(x_i\right)>a\left(x_j\right)$, then$x_i\succ x_j$;

(3) If$s(x_i)=s(x_j)$,$a\left(x_i\right)=a\left(x_j\right)$, and$c\left(x_i\right)>c\left(x_j\right)$, then$x_i\succ x_j$;

(4) If $s(x_i)=s(x_j)$, $a\left(x_i\right)=a\left(x_j\right)$ and$c\left(x_i\right)=c\left(x_j\right)$, then$x_i\sim x_j$.

Definition 5.

[44,45]. Let $x_i=\left(T_i,I_i,F_i\right)$ and $x_j=\left(T_j,I_j,F_j\right)$ be any two SVNNs, the normalized Euclidian distance between $x_i$ and $x_j$ are defined as follows:

$$d(x_i,x_j)=\sqrt{\left[{(T_i-T_j)}^2+{(I_i-I_j)}^2+{(F_i-F_j)}^2\right]/3}$$

(15)

3. Some Power Heronian Aggregation Operators with SVNNs

3.1. Single Valued Neutrosophic Power Heronian Aggregation Operators

As an important aggregation operator, the PA operator is first proposed byYager [36], which can overcome the influence of unreasonable arguments by considering the support degree between input arguments. The traditional PA operator is defined as below.

Definition 6.

[36]. Let $p,q\ge 0$, and ${\varphi }_i\left(i=1,2,\cdots ,n\right)$ be a group of nonnegative numbers. If

$$PA({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n)=\frac{{\displaystyle \sum _{i=1}^n((1+T({\varphi }_i)){\varphi }_i}}{{\displaystyle \sum _{i=1}^n((1+T({\varphi }_i))}}$$

(16)

where $T({\varphi }_i)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup({\varphi }_i,{\varphi }_j)}$. We denote $Sup({\varphi }_i,{\varphi }_j)$ as the support degree for ${\varphi }_i$ from ${\varphi }_j$. $Sup({\varphi }_i,{\varphi }_j)$ satisfies the following axioms:

(1) $Sup({\varphi }_i,{\varphi }_j)=Sup({\varphi }_j,{\varphi }_i)$;

(2) $Sup({\varphi }_i,{\varphi }_j)\in \left[0,1\right]$;

(3) $Sup({\varphi }_i,{\varphi }_j)>Sup({\varphi }_l,{\varphi }_k)$, if $\left|{\varphi }_i-{\varphi }_j\right|<\left|{\varphi }_l-{\varphi }_k\right|$.

Then PA is called the power average (PA) operator.

For example, suppose ${\varphi }_1=0.6$, ${\varphi }_2=0.7$, ${\varphi }_3=0.8$ are nonnegative numbers, the $PA({\varphi }_1,{\varphi }_2,{\varphi }_3)$ are calculated as follows:

Step 1. Calculate the $Sup(x_i,x_j)\left(i,j=1,2,3\right)$ Thereafter, we have

$Sup({\varphi }_1,{\varphi }_2)=0.1$, $Sup({\varphi }_1,{\varphi }_3)=0.2$, $Sup({\varphi }_2,{\varphi }_3)=0.1$.

Step 2. Calculate the power weighting vector through Expression (19). Thereafter, we have

$T({\varphi }_1)=Sup({\varphi }_1,{\varphi }_2)+Sup({\varphi }_1,{\varphi }_3)=0.3$,

$T({\varphi }_2)=Sup({\varphi }_2,{\varphi }_1)+Sup({\varphi }_2,{\varphi }_3)=0.2$,

$T({\varphi }_3)=Sup({\varphi }_3,{\varphi }_1)+Sup({\varphi }_3,{\varphi }_2)=Sup({\varphi }_1,{\varphi }_3)+Sup({\varphi }_2,{\varphi }_3)=0.3$.

Step 3. Calculate the $PA({\varphi }_1,{\varphi }_2,{\varphi }_3)$ using the equation 15 (suppose p = q = 1). Thereafter, we can have

$$\begin{array}{l}PA({\varphi }_1,{\varphi }_2,{\varphi }_3)=\frac{{\displaystyle \sum _{i=1}^3((1+T({\varphi }_i)){\varphi }_i}}{{\displaystyle \sum _{i=1}^3((1+T({\varphi }_i))}}\\ =\frac{((1+T({\varphi }_1)){\varphi }_1+((1+T({\varphi }_2)){\varphi }_2+((1+T({\varphi }_3)){\varphi }_3}{(1+T({\varphi }_1))+(1+T({\varphi }_2))+(1+T({\varphi }_3))}\\ =\frac{(1+0.3)0.6+(1+0.2)0.7+(1+0.3)0.8}{(1+0.3)(1+0.2)(1+0.3)}=0.7\end{array}$$

Beliakov [38] first proposed the Heronian mean operator, which can determine the interrelation of the input arguments [46,47,48]. HM is defined as follows.

Definition 7.

[48]. Let $p,q\ge 0$, and ${\varphi }_i\left(i=1,2,\cdots ,n\right)$ be a group of nonnegative numbers. If

$$H{M}^{p,q}({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n)={\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=i}^n{\varphi }_i^p{\varphi }_j^q}}\right)}^{\frac{1}{p+q}}$$

(17)

Then $H{M}^{p,q}$ is called the Heronian mean (HM) operator.

For example, suppose ${\varphi }_1=0.6$, ${\varphi }_2=0.7$, ${\varphi }_3=0.8$ are nonnegative numbers, the $H{M}^{p,q}({\varphi }_1,{\varphi }_2,{\varphi }_3)$ are calculated as follow (suppose p = q = 1):

$$\begin{array}{l}H{M}^{1,1}({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n)={\left(\frac{2}{3(3+1)}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=i}^n{\varphi }_i^1{\varphi }_j^1}}\right)}^{\frac{1}{2}}\\ ={\left(\frac{2}{3(3+1)}{\varphi }_1{\varphi }_1+{\varphi }_1{\varphi }_2+{\varphi }_1{\varphi }_3+{\varphi }_2{\varphi }_2+{\varphi }_2{\varphi }_3+{\varphi }_3{\varphi }_3\right)}^{\frac{1}{2}}\\ ={\left(\frac{1}{6}(0.6\ast 0.6+0.6\ast 0.7+0.6\ast 0.8+0.7\ast 0.7+0.7\ast 0.8+0.8\ast 0.8)\right)}^{\frac{1}{2}}\\ ={\left(\frac{1}{6}\times 2.95\right)}^{\frac{1}{2}}\approx 0.7012\end{array}$$

Next, we shall develop the SVNPHA and SVNWPHA operators based on the operation laws of SVNNs.

Definition 8.

Let $p,q\ge 0$, and $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs. If

$$\begin{array}{l}SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\\ ={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}{x}_i\right)}^p\otimes {\left(\frac{n(1+T(x_j))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}{x}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\end{array}$$

(18)

where $T(x_i)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup(x_i,x_j)}$. We shall denote $Sup(x_i,x_j)$ as the support degree for $x_i$ from $x_j$. $Sup(x_i,x_j)$ satisfies the following three properties:

(1) $Sup(x_i,x_j)\in \left[0,1\right]$;

(2) $Sup(x_i,x_j)=Sup(x_j,x_i)$;

(3) $Sup(x_i,x_j)>Sup(x_l,x_k)$, if $d(x_i,x_j)<d(x_l,x_k)$ in which $d(x_i,x_j)$ is the distance between SVNNs $x_i$ and $x_j$.

Then, $SVNGPH{A}^{p,q}$ is called the single-valued neutrosophic power Heronian aggregation (SVNPHA) operator.

In order to simply this expression (18). We can define:

$${\tilde{w}}_i=\frac{(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}.$$

(19)

and call $\left({\tilde{w}}_1,{\tilde{w}}_2,\cdots ,{\tilde{w}}_n\right)$ as the power weighting vector with ${\tilde{w}}_i\ge 0,{\displaystyle {\sum }_{i=1}^n{\tilde{w}}_i}=1$

Thereafter, Expression (18) can be expressed as follows:

$$\begin{array}{l}SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\\ ={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\end{array}$$

(20)

Theorem 2.

Let $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a group of SVNNs and $p,q\ge 0$. Then, the result aggregated from SVNPHA operator is still a SVNN, and even

$$\begin{array}{l}SVNPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)\\ =({\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}},\end{array}$$

(21)

$$\begin{gathered} {\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p{\left(1-I_i^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}}, \\ (1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p{\left(1-F_i^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}}) \end{gathered}$$

Proof. 

To prove Equation (21), we first prove that the following equation is right.

$$\begin{array}{l}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\right)\\ =(1-{\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)},\\ {\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p{\left(1-I_j^{n{\tilde{w}}_j}\right)}^q\right)},{\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p{\left(1-F_j^{n{\tilde{w}}_j}\right)}^q\right)})\end{array}$$

(22)

By the operational rules of SVNNs defined in (2–5), we have

$$\begin{array}{l}{\left(n{\tilde{w}}_i{x}_i\right)}^p=\left({\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p,1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p,1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p\right)\\ {\left(n{\tilde{w}}_j{x}_j\right)}^q=\left({\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q,1-{\left(1-I_j^{n{\tilde{w}}_j}\right)}^q,1-{\left(1-F_j^{n{\tilde{w}}_j}\right)}^q\right)\\ {\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\\ =({\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q,1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p{\left(1-I_j^{n{\tilde{w}}_j}\right)}^q,1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p{\left(1-F_j{}^{n{\tilde{w}}_j}\right)}^q)\end{array}$$

(23)

When n = 2, we have

$$\begin{array}{l}\underset{i=1,j=i}{\overset{2}{\oplus }}\left({\left(2{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(2{\tilde{w}}_j{x}_j\right)}^q\right)\\ =\left({\left(2{\tilde{w}}_1{x}_1\right)}^p\otimes {\left(2{\tilde{w}}_1{x}_1\right)}^q\right)\oplus \left({\left(2{\tilde{w}}_1{x}_1\right)}^p\otimes {\left(2{\tilde{w}}_2{x}_2\right)}^q\right)\oplus \left({\left(2{\tilde{w}}_2{x}_2\right)}^p\otimes {\left(2{\tilde{w}}_2{x}_2\right)}^q\right)\\ =({\left(1-{\left(1-T_1\right)}^{2{\tilde{w}}_1}\right)}^p{\left(1-{\left(1-T_1\right)}^{2{\tilde{w}}_1}\right)}^q,1-{\left(1-I_1^{2{\tilde{w}}_1}\right)}^p{\left(1-I_1^{2{\tilde{w}}_1}\right)}^q,1-{\left(1-F_1^{2{\tilde{w}}_1}\right)}^p{\left(1-F_1{}^{2{\tilde{w}}_1}\right)}^q)\\ \oplus ({\left(1-{\left(1-T_1\right)}^{2{\tilde{w}}_1}\right)}^p{\left(1-{\left(1-T_2\right)}^{2{\tilde{w}}_2}\right)}^q,1-{\left(1-I_1^{2{\tilde{w}}_1}\right)}^p{\left(1-I_2^{2{\tilde{w}}_2}\right)}^q,1-{\left(1-F_1^{2{\tilde{w}}_1}\right)}^p{\left(1-F_2{}^{2{\tilde{w}}_2}\right)}^q)\\ \oplus ({\left(1-{\left(1-T_2\right)}^{2{\tilde{w}}_2}\right)}^p{\left(1-{\left(1-T_2\right)}^{2{\tilde{w}}_2}\right)}^q,1-{\left(1-I_2^{2{\tilde{w}}_2}\right)}^p{\left(1-I_2^{2{\tilde{w}}_2}\right)}^q,1-{\left(1-F_2^{2{\tilde{w}}_2}\right)}^p{\left(1-F_2{}^{2{\tilde{w}}_2}\right)}^q)\end{array}$$

By using Equation (2), we get

$$\begin{array}{l}=(1-{\displaystyle \prod _{i=1,j=i}^2\left(1-{\left(1-{\left(1-T_i\right)}^{2{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{2{\tilde{w}}_j}\right)}^q\right)},\\ {\displaystyle \prod _{i=1,j=i}^2\left(1-{\left(1-I_i^{2{\tilde{w}}_i}\right)}^p{\left(1-I_j^{2{\tilde{w}}_j}\right)}^q\right)},{\displaystyle \prod _{i=1,j=i}^2\left(1-{\left(1-F_i^{2{\tilde{w}}_i}\right)}^p{\left(1-F_j^{2{\tilde{w}}_j}\right)}^q\right)})\end{array}$$

(24)

That is, when n = 2, the Equation (22) is right.

Assume n = m, Equation (22) is right:

$$\begin{array}{l}=(1-{\displaystyle \prod _{i=1,j=i}^k\left(1-{\left(1-{\left(1-T_i\right)}^{k{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{k{\tilde{w}}_j}\right)}^q\right)},\\ {\displaystyle \prod _{i=1,j=i}^k\left(1-{\left(1-I_i^{k{\tilde{w}}_i}\right)}^p{\left(1-I_j^{k{\tilde{w}}_j}\right)}^q\right)},{\displaystyle \prod _{i=1,j=i}^k\left(1-{\left(1-F_i^{k{\tilde{w}}_i}\right)}^p{\left(1-F_j^{k{\tilde{w}}_j}\right)}^q\right)})\end{array}$$

(25)

Furthermore, when n = k + 1, we have

$$\begin{array}{l}\underset{i=1,j=i}{\overset{k+1}{\oplus }}\left({\left(\left(k+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_j{x}_j\right)}^q\right)\\ =\underset{i=1,j=i}{\overset{k}{\oplus }}\left({\left(\left(k+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_j{x}_j\right)}^q\right)\underset{i=1}{\overset{k}{\oplus }}\left({\left(\left(k+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_{k+1}{x}_{k+1}\right)}^q\right)\\ \oplus \left({\left(\left(k+1\right){\tilde{w}}_{k+1}{x}_{k+1}\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_{k+1}{x}_{k+1}\right)}^q\right)\end{array}$$

(26)

Firstly, we prove that

$$\begin{array}{l}\underset{i=1}{\overset{k}{\oplus }}\left({\left(\left(k+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_{k+1}{x}_{k+1}\right)}^q\right)\\ =(1-{\displaystyle \prod _{i=1}^k\left(1-{\left(1-{\left(1-T_i\right)}^{(k+1){\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_{(k+1)}\right)}^{(k+1){\tilde{w}}_{(k+1)}}\right)}^q\right)},\\ {\displaystyle \prod _{i=1}^k\left(1-{\left(1-I_i^{(k+1){\tilde{w}}_i}\right)}^p{\left(1-I_{(k+1)}^{(k+1){\tilde{w}}_{(k+1)}}\right)}^q\right)},{\displaystyle \prod _{i=1}^k\left(1-{\left(1-F_i^{(k+1){\tilde{w}}_i}\right)}^p{\left(1-F_{(k+1)}^{(k+1){\tilde{w}}_{(k+1)}}\right)}^q\right)})\end{array}$$

(27)

We shall prove Equation (27) on mathematical induction on k

For k = 2, we have

$$\begin{array}{l}\underset{i=1}{\overset{2}{\oplus }}\left({\left(\left(k+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_{k+1}{x}_{k+1}\right)}^q\right)\\ =\left({\left(3{\tilde{w}}_1{x}_1\right)}^p\otimes {\left(3{\tilde{w}}_3{x}_3\right)}^q\right)\oplus \left({\left(3{\tilde{w}}_2{x}_2\right)}^p\otimes {\left(3{\tilde{w}}_3{x}_3\right)}^q\right)\\ =({\left(1-{\left(1-T_1\right)}^{3{\tilde{w}}_1}\right)}^p{\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)}^q,1-{\left(1-I_1^{3{\tilde{w}}_1}\right)}^p{\left(1-I_3^{3{\tilde{w}}_3}\right)}^q,1-{\left(1-F_1^{3{\tilde{w}}_1}\right)}^p{\left(1-F_3{}^{3{\tilde{w}}_3}\right)}^q)\\ \oplus ({\left(1-{\left(1-T_2\right)}^{3{\tilde{w}}_2}\right)}^p{\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)}^q,1-{\left(1-I_2^{3{\tilde{w}}_2}\right)}^p{\left(1-I_3^{3{\tilde{w}}_3}\right)}^q,1-{\left(1-F_2^{3{\tilde{w}}_2}\right)}^p{\left(1-F_3{}^{3{\tilde{w}}_3}\right)}^q)\\ =(1-{\displaystyle \prod _{i=1}^2\left(1-{\left(1-{\left(1-T_i\right)}^{3{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)}^q\right)},\\ \begin{array}{l}{\displaystyle \prod _{i=1}^2\left(1-{\left(1-I_i^{3{\tilde{w}}_i}\right)}^p{\left(1-I_3^{3{\tilde{w}}_3}\right)}^q\right)},\\ {\displaystyle \prod _{i=1}^2\left(1-{\left(1-F_i^{3{\tilde{w}}_i}\right)}^p{\left(1-F_3^{3{\tilde{w}}_3}\right)}^q\right)}\end{array})\end{array}$$

(28)

Suppose k = a, the Equation (27) is right, that is

$$\begin{array}{l}\underset{i=1}{\overset{a}{\oplus }}\left({\left(\left(a+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(a+1\right){\tilde{w}}_{a+1}{x}_{a+1}\right)}^q\right)\\ =(1-{\displaystyle \prod _{i=1}^a\left(1-{\left(1-{\left(1-T_i\right)}^{(a+1){\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_{(a+1)}\right)}^{(a+1){\tilde{w}}_{(a+1)}}\right)}^q\right)},\\ {\displaystyle \prod _{i=1}^a\left(1-{\left(1-I_i^{(a+1){\tilde{w}}_i}\right)}^p{\left(1-I_{(a+1)}^{(a+1){\tilde{w}}_{(a+1)}}\right)}^q\right)},{\displaystyle \prod _{i=1}^a\left(1-{\left(1-F_i^{(a+1){\tilde{w}}_i}\right)}^p{\left(1-F_{(a+1)}^{(a+1){\tilde{w}}_{(a+1)}}\right)}^q\right)})\end{array}$$

Then, when k = a + 1, we have

$$\begin{array}{l}\underset{i=1}{\overset{a+1}{\oplus }}\left({\left(\left(a+2\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(a+2\right){\tilde{w}}_{a+2}{x}_{a+2}\right)}^q\right)\\ =\underset{i=1}{\overset{a}{\oplus }}\left({\left(\left(a+2\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(a+2\right){\tilde{w}}_{a+2}{x}_{a+2}\right)}^q\right)\oplus \left({\left(\left(a+2\right){\tilde{w}}_{a+1}{x}_{a+1}\right)}^p\otimes {\left(\left(a+2\right){\tilde{w}}_{a+2}{x}_{a+2}\right)}^q\right)\\ =(1-{\displaystyle \prod _{i=1}^a\left(1-{\left(1-{\left(1-T_i\right)}^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_{(a+2)}\right)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right)},\\ {\displaystyle \prod _{i=1}^a\left(1-{\left(1-I_i^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-I_{(a+2)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right)},{\displaystyle \prod _{i=1}^a\left(1-{\left(1-F_i^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-F_{(a+2)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right)})\\ \oplus (1-\left(1-{\left(1-{\left(1-T_{(a+1)}\right)}^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_{(a+2)}\right)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right),\\ \left(1-{\left(1-I_{(a+1)}^{(a+2){\tilde{w}}_{(a+1)}}\right)}^p{\left(1-I_{(a+2)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right),\left(1-{\left(1-F_{(a+1)}^{(a+2){\tilde{w}}_{(a+1)}}\right)}^p{\left(1-F_{(a+2)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right))\\ =(1-{\displaystyle \prod _{i=1}^{a+1}\left(1-{\left(1-{\left(1-T_i\right)}^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_{(a+2)}\right)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right)},\\ {\displaystyle \prod _{i=1}^{a+1}\left(1-{\left(1-I_i^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-I_{(a+2)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right)},{\displaystyle \prod _{i=1}^{a+1}\left(1-{\left(1-F_i^{(a+2){\tilde{w}}_i}\right)}^p{\left(1-F_{(a+2)}^{(a+2){\tilde{w}}_{(a+2)}}\right)}^q\right)})\end{array}$$

Therefore, when k = a + 1, the Equation (27) is true. Hence, Equation (27) is established for any k.

Similarly, we can prove the other parts of Equation (26).

So, Equation (26) becomes

$$\begin{array}{l}\underset{i=1,j=i}{\overset{k+1}{\oplus }}\left({\left(\left(k+1\right){\tilde{w}}_i{x}_i\right)}^p\otimes {\left(\left(k+1\right){\tilde{w}}_j{x}_j\right)}^q\right)\\ =(1-{\displaystyle \prod _{i=1,j=i}^{\left(k+1\right)}\left(1-{\left(1-{\left(1-T_i\right)}^{\left(k+1\right){\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{\left(k+1\right){\tilde{w}}_j}\right)}^q\right)},\\ {\displaystyle \prod _{i=1,j=i}^{\left(k+1\right)}\left(1-{\left(1-I_i^{\left(k+1\right){\tilde{w}}_i}\right)}^p{\left(1-I_j^{\left(k+1\right){\tilde{w}}_j}\right)}^q\right)},{\displaystyle \prod _{i=1,j=i}^{\left(k+1\right)}\left(1-{\left(1-F_i^{\left(k+1\right){\tilde{w}}_i}\right)}^p{\left(1-F_j^{\left(k+1\right){\tilde{w}}_j}\right)}^q\right)})\end{array}$$

Therefore, when n = k + 1, the Equation (22) is true. Hence, Equation (22) is established for any n. □

From Equation (22) and the operational rules of SVNNs defined in (2–5), we have

$$\begin{array}{l}\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\right)\\ ={((1-\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\\ {\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p{\left(1-I_j^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}},{\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p{\left(1-F_j^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}})),\end{array}$$

So

$$\begin{array}{l}{\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\left({(1-\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}},\\ 1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p{\left(1-I_i^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}},\\ 1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p{\left(1-F_i^{n{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}})\end{array}$$

Therefore, Equation (21) is right and we complete the proof of the Theorem 2.

To compute the power weight vector ${\tilde{w}}_i$, the support degree between SVNNs should be calculate firstly. Under normal circumstances, we can use the similarity degree between SVNNs to replace the support degree and that is,

$$Sup(x_i,x_j)=1-d(x_i,x_j)$$

(29)

Example 1.

Suppose three SVNNs exist: $x_1=\left(0.260,0.425,0.315\right)$, $x_2=\left(0.220,0.450,0.330\right)$, and $x_3=\left(0.255,0.500,0.245\right)$. Accordingly, we can use SVNPHA to generate a comprehensive value. In the following, the steps are given.

Step 1. Calculate the $Sup(x_i,x_j)\left(i,j=1,2,3\right)$ by using Expressions (15) and (22). Thereafter, we have

$Sup(x_1,x_2)$ = 0.97142 $Sup(x_1,x_3)$ = 0.94070

$Sup(x_2,x_1)$ = 0.97142 $Sup(x_3,x_1)$ = 0.93958

$Sup(x_3,x_1)$ = 0.94070 $Sup(x_3,x_2)$ = 0.93958

Step 2. Calculate the power weighting vector through Expression (19). Thereafter, we have

$T(x_1)$=$Sup(x_1,x_2)$+$Sup(x_1,x_3)$= 1.91212

$T(x_2)$=$Sup(x_2,x_1)$+$Sup(x_2,x_3)$= 1.91100

$T(x_3)$=$Sup(x_3,x_1)$+$Sup(x_3,x_2)$= 1.88028

${\tilde{w}}_1=\frac{1+T(x_1)}{3+T(x_1)+T(x_2)+T(x_3)}=\frac{1+1.91212}{3+1.91212+1.91100+1.88028}=0.33460$

${\tilde{w}}_2=\frac{1+T(x_2)}{3+T(x_1)+T(x_2)+T(x_3)}=\frac{1+1.91100}{3+1.91212+1.91100+1.88028}=0.33447$

${\tilde{w}}_3=\frac{1+T(x_3)}{3+T(x_1)+T(x_2)+T(x_3)}=\frac{1+1.88028}{3+1.91212+1.91100+1.88028}=0.33093$

Step 3. Calculate the comprehensive value $x=\left(T,I,F\right)$ using the SVNPHA operator (suppose p = q = 1). Thereafter, we can have

$$\begin{gathered} T={\left(1-{\left({\displaystyle \prod _{i=1,j=i}^3\left(1-\left(1-{\left(1-T_i\right)}^{3{\tilde{w}}_i}\right)\left(1-{\left(1-T_j\right)}^{3{\tilde{w}}_j}\right)\right)}\right)}^{\frac{2}{3(3+1)}}\right)}^{\frac{1}{2}} \\ \begin{array}{l}={\left(1-{\left(\begin{array}{l}\left(1-\left(1-{\left(1-T_1\right)}^{3{\tilde{w}}_1}\right)\left(1-{\left(1-T_1\right)}^{3{\tilde{w}}_1}\right)\right)\ast \left(1-\left(1-{\left(1-T_1\right)}^{3{\tilde{w}}_1}\right)\left(1-{\left(1-T_2\right)}^{3{\tilde{w}}_2}\right)\right)\ast \\ \left(1-\left(1-{\left(1-T_1\right)}^{3{\tilde{w}}_1}\right)\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)\right)\ast \left(1-\left(1-{\left(1-T_2\right)}^{3{\tilde{w}}_2}\right)\left(1-{\left(1-T_2\right)}^{3{\tilde{w}}_2}\right)\right)\ast \\ \left(1-\left(1-{\left(1-T_2\right)}^{3{\tilde{w}}_2}\right)\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)\right)\ast \left(1-\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)\left(1-{\left(1-T_3\right)}^{3{\tilde{w}}_3}\right)\right)\end{array}\right)}^{\frac{2}{3(3+1)}}\right)}^{\frac{1}{2}}\\ =0.24518\end{array} \end{gathered}$$

and

$$\begin{gathered} I=1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^3\left(1-{\left(1-I_i^{3{\tilde{w}}_i}\right)}^p{\left(1-I_i^{3{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{3(3+1)}}\right)}^{\frac{1}{2}} \\ \begin{array}{l}=1-{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-I_1^{3{\tilde{w}}_1}\right)}^p{\left(1-I_1^{3{\tilde{w}}_1}\right)}^q\right)\ast \left(1-{\left(1-I_1^{3{\tilde{w}}_1}\right)}^p{\left(1-I_2^{3{\tilde{w}}_2}\right)}^q\right)\ast \\ \left(1-{\left(1-I_1^{3{\tilde{w}}_1}\right)}^p{\left(1-I_3^{3{\tilde{w}}_3}\right)}^q\right)\ast \left(1-{\left(1-I_2^{3{\tilde{w}}_2}\right)}^p{\left(1-I_2^{3{\tilde{w}}_2}\right)}^q\right)\ast \\ \left(1-{\left(1-I_2^{3{\tilde{w}}_2}\right)}^p{\left(1-I_3^{3{\tilde{w}}_3}\right)}^q\right)\ast \left(1-{\left(1-I_3^{3{\tilde{w}}_3}\right)}^p{\left(1-I_3^{3{\tilde{w}}_3}\right)}^q\right)\end{array}\right)}^{\frac{2}{3(3+1)}}\right)}^{\frac{1}{2}}\\ =0.45754\end{array} \end{gathered}$$

and

$$F=1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^3\left(1-{\left(1-F_i^{3{\tilde{w}}_i}\right)}^p{\left(1-F_i^{3{\tilde{w}}_j}\right)}^q\right)}\right)}^{\frac{2}{3(3+1)}}\right)}^{\frac{1}{2}}$$

$$\begin{array}{l}I=1-{\left(1-{\left(\begin{array}{l}\left(1-{\left(1-F_1^{3{\tilde{w}}_1}\right)}^p{\left(1-F_1^{3{\tilde{w}}_1}\right)}^q\right)\ast \left(1-{\left(1-F_1^{3{\tilde{w}}_1}\right)}^p{\left(1-F_2^{3{\tilde{w}}_2}\right)}^q\right)\ast \\ \left(1-{\left(1-F_1^{3{\tilde{w}}_1}\right)}^p{\left(1-F_3^{3{\tilde{w}}_3}\right)}^q\right)\ast \left(1-{\left(1-F_2^{3{\tilde{w}}_2}\right)}^p{\left(1-F_2^{3{\tilde{w}}_2}\right)}^q\right)\ast \\ \left(1-{\left(1-F_2^{3{\tilde{w}}_2}\right)}^p{\left(1-F_3^{3{\tilde{w}}_3}\right)}^q\right)\ast \left(1-{\left(1-F_3^{3{\tilde{w}}_3}\right)}^p{\left(1-F_3^{3{\tilde{w}}_3}\right)}^q\right)\end{array}\right)}^{\frac{2}{3(3+1)}}\right)}^{\frac{1}{2}}\\ =0.29532\end{array}$$

Thus, we can obtain the comprehensive value $x=\left(0.24519,0.45746,0.29532\right)$.

Theorem 3.

(Idempotency). Let $x_i(i=1,2,\cdots ,n)$ be a collection of SVNNs and $x_1=x_2=\cdots =x_n=x$. Hence,

$$SVNPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=x.$$

Proof. 

$$\begin{array}{l}SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_{i}x\right)}^p\otimes {\left(n{\tilde{w}}_{j}x\right)}^q\right)\right)}^{\frac{1}{p+q}}={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n\frac{1}{n}x\right)}^p\otimes {\left(n\frac{1}{n}x\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left(x^{p+q}\right)\right)}^{\frac{1}{p+q}}=x\end{array}$$

thereby completing the proof of Theorem 3. □

Theorem 4.

(Commutativity). Let $\left({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n\right)$ be any permutation of $\left(x_1,x_2,\cdots ,x_n\right)$, then $SVNPH{A}^{p,q}({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n)=SVNPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)$

Proof. 

Since $\left({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n\right)$ be any permutation of $\left(x_1,x_2,\cdots ,x_n\right)$, then

$$\begin{array}{l}SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{\tilde{w}}_i{x}_i\right)}^p\otimes {\left(n{\tilde{w}}_j{x}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\\ ={\left(\frac{2}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(n{{\tilde{w}}^{\prime }}_i{x^{\prime }}_i\right)}^p\otimes {\left(n{{\tilde{w}}^{\prime }}_j{x^{\prime }}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}=SVNPH{A}^{p,q}\left({x^{\prime }}_1,{x^{\prime }}_2,\cdots ,{x^{\prime }}_n\right)\end{array}$$

thereby completing the proof of Theorem 4.□

Theorem 5.

(Boundedness). Let $x_i=\left(T_i,I_i,F_i\right)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs, $x^{-}=\left({\min }_i\left\{T_i\right\},{\max }_i\left\{I_i\right\},{\max }_i\left\{F_i\right\}\right)$, and $x^{+}=\left({\max }_i\left\{T_i\right\},{\min }_i\left\{I_i\right\},{\min }_i\left\{F_i\right\}\right)$. Hence,

$$x^{-}\le SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le x^{+}$$

Proof. 

By the comparison method in Definition 3, we have $x_i\ge x^{-}$, then based on the theorem 2 and 3, we have

$$SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\ge SVNPH{A}^{p,q}\left(x^{-},x^{-},\cdots ,x^{-}\right)=x^{-}$$

Similarly, we can obtain

$$SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le SVNPH{A}^{p,q}\left(x^{+},x^{+},\cdots ,x^{+}\right)=x^{+},\mathrm{then}$$

$$\begin{gathered} SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le SVNPH{A}^{p,q}\left(x^{+},x^{+},\cdots ,x^{+}\right)=x^{+},\mathrm{then} \\ \left(30\right)x^{-}\le SVNPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le x^{+} \end{gathered}$$

thereby completing the proof of Theorem 5.□

In the follow, we can discuss some special cases about $SVNNPH{A}^{p,q}$ operator.

(1) If $q=0$, then the Expression (21) operator is reduced to the single-valued neutrosophic power generalized linear descending weight operator as follows:

$$\begin{array}{l}SVNPH{A}^{p,0}(x_1,x_2,\cdots ,x_n)\\ =({\left(1-{\left({\displaystyle \prod _{i=1,}^n{\left(1-{\left(1-{\left(1-T_i\right)}^{n{\tilde{w}}_i}\right)}^p\right)}^{n+1-i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p}},\\ {\left(1-{\left({{\displaystyle \prod _{i=1}^n\left(1-{\left(1-I_i^{n{\tilde{w}}_i}\right)}^p\right)}}^{n+1-i}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p}},1-{\left(1-{\left({\displaystyle \prod _{i=1}^n{\left(1-{\left(1-F_i^{n{\tilde{w}}_i}\right)}^p\right)}^{n+1-i}}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p}})\end{array}$$

(2) If $p=0$ then the Expression (21) operator is reduced to the single-valued neutrosophic power generalized linear ascending weight operator as follows:

$$\begin{array}{l}SVNPH{A}^q(x_1,x_2,\cdots ,x_n)\\ =({\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n{\left(1-{\left(1-{\left(1-T_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)}^i}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q}},\\ {\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n{\left(1-{\left(1-I_j^{n{\tilde{w}}_j}\right)}^q\right)}^i}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q}},1-{\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n{\left(1-{\left(1-F_j^{n{\tilde{w}}_j}\right)}^q\right)}^i}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{q}})\end{array}$$

(3) If $p=q=\frac{1}{2}$, then the Expression (21) operator is reduced to the single-valued neutrosophic power basic Heronian operator as follows:

$$\begin{array}{l}SVNPH{A}^{\frac{1}{2},\frac{1}{2}}(x_1,x_2,\cdots ,x_n)\\ =({\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-{\left(1-T_i\right)}^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{\frac{1}{2}}{\left(1-{\left(1-T_j\right)}^{\frac{n(1+T(x_j))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{\frac{1}{2}}\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{},\\ {\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-I_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{\frac{1}{2}}{\left(1-I_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{\frac{1}{2}}\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{},\\ 1-\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-F_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{\frac{1}{2}}{\left(1-F_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{\frac{1}{2}}\right)}\right)}^{\frac{2}{n(n+1)}}\right))\end{array}$$

(4) If $p=q=1$, then the operator of Equation (21) is reduced to the single-valued neutrosophic number power line Heronian operator as follows:

$$\begin{array}{l}SVNPH{A}^{1,1}(x_1,x_2,\cdots ,x_n)\\ =({\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n\left(1-\left(1-{\left(1-T_i\right)}^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)\left(1-{\left(1-T_j\right)}^{\frac{n(1+T(x_j))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2}},\\ 1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-I_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{}\left(1-I_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2}},\\ 1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-F_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{}{\left(1-F_i^{\frac{n(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}}\right)}^{}\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{2}})\end{array}$$

In the SVNPHA operators, we only take into account the power weight vector and interrelationship among SVNNs but not the weight of every SVNN. However, in many realistic decision-making, the weights of attributes are also an important parameter. Thus, we propose the single-valued neutrosophic weight PHA (SVNWPHA) operator as follows.

Definition 9.

Let $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a group of SVNNs and $p,q\ge 0$, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$ is the weight vector of $x_i\left(i=1,2,\cdots ,n\right)$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$. If:

$$\begin{array}{l}SVNWPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\\ ={\left(\frac{1}{n(n+1)}\underset{i=1,j=i}{\overset{n}{\oplus }}\left({\left(\frac{n{\tilde{w}}_i{w}_i}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}{x}_i\right)}^p\otimes {\left(\frac{n{\tilde{w}}_j{w}_j}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}{x}_j\right)}^q\right)\right)}^{\frac{1}{p+q}}\end{array}$$

(30)

where ${\tilde{w}}_i=\frac{(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}$ and ${\displaystyle \sum _i^n{\tilde{w}}_i}=1$. $T(x_i)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup(x_i,x_j)}$ we shall denote $Sup(x_i,x_j)$ as the support degree for $x_i$ from $x_j$. $Sup(x_i,x_j)$ satisfies the following three properties:

(1) $Sup(x_i,x_j)\in \left[0,1\right]$;

(2) $Sup(x_i,x_j)=Sup(x_j,x_i)$;

(3) $Sup(x_i,x_j)>Sup(x_l,x_k)$, if $d(x_i,x_j)<d(x_l,x_k)$ in which $d(x_i,x_j)$ is the distance between SVNNs $x_i$ and $x_j$.

Then, $SVNWPH{A}^{p,q}$ is called the single-valued neutrosophic weight power Heronian aggregation (SVNPHA) operator.

Theorem 6.

Let $p,q\ge 0$ and $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$. Then, the result aggregated from SVNWPHA is still a SVNNs, and even

$$\begin{array}{l}SVNWPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)\\ =({\left(1-{\left({\displaystyle \prod _{i=1,j=i}^n\left(1-{\left(1-{\left(1-T_i\right)}^{{}^{\frac{n{\tilde{w}}_i{w}_i}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}}\right)}^p{\left(1-{\left(1-T_j\right)}^{{}^{\frac{n{\tilde{w}}_j{w}_j}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}},\\ {\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-I_i^{\frac{n{\tilde{w}}_i{w}_i}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}\right)}^p{\left(1-I_i^{\frac{n{\tilde{w}}_j{w}_j}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}},\\ 1-{\left(1-{\left({\displaystyle \prod _{i=1.j=i}^n\left(1-{\left(1-F_i^{\frac{n{\tilde{w}}_i{w}_i}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}\right)}^p{\left(1-F_i^{\frac{n{\tilde{w}}_j{w}_j}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}\right)}^q\right)}\right)}^{\frac{2}{n(n+1)}}\right)}^{\frac{1}{p+q}})\end{array}$$

(31)

As with the proof of Theorem 2, it is omitted from this paper.

Obviously, when $W={\left(1/n,1/n,\cdots ,1/n\right)}^T$, the SVNWPHA operator is reduced to the SVNPHA operator.

Similar to the above SVNPHA operator, the SVNWPHA operator also has the same properties.

Theorem 7.

(Idempotency). Let $x_i(i=1,2,\cdots ,n)$ be a collection of SVNNs, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$, and $x_1=x_2=\cdots =x_n=x$. Hence,

$$SVNWPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=x.$$

Theorem 8.

(Commutativity). Let $\left({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n\right)$ be any permutation of $\left(x_1,x_2,\cdots ,x_n\right)$, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$. Hence,

$$SVNWPH{A}^{p,q}({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n)=SVNWPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)$$

.

Theorem 9.

(Boundedness). Let $x_i=\left(T_i,I_i,F_i\right)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$, $x^{-}=\left({\min }_i\left\{T_i\right\},{\max }_i\left\{I_i\right\},{\max }_i\left\{F_i\right\}\right)$, and $x^{+}=\left({\max }_i\left\{T_i\right\},{\min }_i\left\{I_i\right\},{\min }_i\left\{F_i\right\}\right)$. Hence,

$$x^{-}\le SVNWPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le x^{+}$$

3.2. Single Valued Neutrosophic Geometric Power Heronian Aggregation Operators

Based on the PA operator [36] and geometric mean [49], Xu [37] further defined a power geometric (PG) operator:

Definition 10.

[37]. Let $p,q\ge 0$, and ${\varphi }_i\left(i=1,2,\cdots ,n\right)$ be a collection of nonnegative numbers. if:

$$PG({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n)={\displaystyle \prod _{i=1}^n{\varphi }_i{}^{\frac{1+T({\varphi }_i)}{{\displaystyle \sum _{i=1}^n((1+T({\varphi }_i((}}}}$$

(32)

where $T({\varphi }_i)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup({\varphi }_i,{\varphi }_j)}$. We denote $Sup({\varphi }_i,{\varphi }_j)$ as the support degree for ${\varphi }_i$ from ${\varphi }_j$. $Sup({\varphi }_i,{\varphi }_j)>Sup({\varphi }_l,{\varphi }_k)$ satisfies the following axioms:

(1) $Sup({\varphi }_i,{\varphi }_j)\in \left[0,1\right]$;

(2) $Sup({\varphi }_i,{\varphi }_j)Sup({\varphi }_j,{\varphi }_i)$

(3) $Sup({\varphi }_i,{\varphi }_j)>Sup({\varphi }_l,{\varphi }_k)$, if $\left|{\varphi }_i-{\varphi }_j\right|<\left|{\varphi }_l-{\varphi }_k\right|$

Definition 11.

[39]. Let $p,q\ge 0$, and ${\varphi }_i\left(i=1,2,\cdots ,n\right)$ be a collection of nonnegative numbers. If:

$$GH{M}^{p,q}({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n)=\frac{1}{p+q}\left({\displaystyle \prod _{i=1,j=i}^n{\left(p{\varphi }_i+q{\varphi }_j\right)}^{\frac{2}{n(n+1)}}}\right)$$

(33)

Then $GH{M}^{p,q}$ is called the geometric Heronian mean (GHM) operator.

Next, we shall develop the SVNGPHA and SVNWGPHA operators based on the operation laws of SVNNs.

Definition 12.

Let $p,q\ge 0$, and $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of single-valued neutrosophic numbers. if:

$$SVNGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=\frac{1}{p+q}\underset{\begin{array}{l}i=1,\\ j=i\end{array}}{\overset{n}{\otimes }}{\left(p{x}_i{a}_i{}^{n\frac{1+T(x_i)}{{\displaystyle \sum _{i=1}^n((1+T(x_i((}}}\oplus q{x}_j{}^{n\frac{1+T(x_j)}{{\displaystyle \sum _{i=1}^n((1+T(x_j((}}}\right)}^{\frac{2}{n(n+1)}}$$

(34)

where $T(x_i)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup(x_i,x_j)}$. we shall denote $Sup(x_i,x_j)$ as the support degree for $x_i$ from $x_j$. $Sup(x_i,x_j)$ satisfies the following three properties:

(1) $Sup(x_i,x_j)\in \left[0,1\right]$;

(2) $Sup(x_i,x_j)=Sup(x_j,x_i)$

(3) $Sup(x_i,x_j)>Sup(x_j,x_i)$, if $d(x_i,x_j)<d(x_l,x_k)$ in which $d(x_i,x_j)$ is the distance between SVNNs $x_i$ and $x_j$

Then, $SVNGPH{A}^{p,q}$ is called the single-valued neutrosophic geometric power Heronian aggregation (SVNGPHA) operator.

In order to simply this expression X. We can define

$${\tilde{w}}_i=\frac{(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}$$

(35)

and call $\left({\tilde{w}}_1,{\tilde{w}}_2,\cdots ,{\tilde{w}}_n\right)$ as the power weighting vector with $w_i\ge 0,{\displaystyle {\sum }_{i=1}^n{w}_i}=1$.

Then, Expression (18) can be shown as follows:

$$SVNGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=\frac{1}{p+q}\underset{\begin{array}{l}i=1,\\ j=i\end{array}}{\overset{n}{\otimes }}{\left(p{x}_i{}^{n{\tilde{w}}_i}\oplus q{x}_j{}^{n{\tilde{w}}_j}\right)}^{\frac{2}{n(n+1)}}$$

(36)

Theorem 10.

Let $p,q\ge 0$, and $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs. Then, the result aggregated from SVNGPHA is still a SVNN, and even

$$\begin{array}{l}SVNNPGH{A}^{p,q}(x_1,x_2,\cdots ,x_n)\\ =(1-{\left(1-{\displaystyle \prod _{i=1,j=i}^n{\left(1-{\left(1-T_i^{n{\tilde{w}}_i}\right)}^p{\left(1-T_j^{n{\tilde{w}}_j}\right)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{\frac{1}{p+q}},\\ {\left(1-{\displaystyle \prod _{i=1.j=i}^n{\left(1-{\left(1-{\left(1-I_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-I_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{\frac{1}{p+q}},\\ {\left(1-{\displaystyle \prod _{i=1.j=i}^n{\left(1-{\left(1-{\left(1-F_i\right)}^{n{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-F_j\right)}^{n{\tilde{w}}_j}\right)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{\frac{1}{p+q}})\end{array}$$

(37)

Similar to the above SVNPHA operator, the SVNGPHA operator also has the same properties.

Theorem 11.

(Idempotency). Let $x_i(i=1,2,\cdots ,n)$ be a collection of SVNNs, and $x_1=x_2=\cdots =x_n=x$, then

$$SVNGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=x.$$

(38)

Theorem 12.

(Commutativity). Let $\left({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n\right)$ be any permutation of $\left(x_1,x_2,\cdots ,x_n\right)$, then

$$SVNGPH{A}^{p,q}({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n)=SVNGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)$$

(39)

Theorem 13.

(Boundedness). Let $x_i=\left(T_i,I_i,F_i\right)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs, and $x^{-}=\left({\min }_i\left\{T_i\right\},{\max }_i\left\{I_i\right\},{\max }_i\left\{F_i\right\}\right)$, $x^{+}=\left({\max }_i\left\{T_i\right\},{\min }_i\left\{I_i\right\},{\min }_i\left\{F_i\right\}\right)$ then

$$x^{-}\le SVNGPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le x^{+}$$

(40)

In the SVNGPHA operators, we only take into account the power weight vector and interrelationship among SVNNs but not the weight of every SVNN. However, in many realistic decision-making, the weights of attributes are also an important parameter. Thus, we propose the single-valued neutrosophic numbers weight geometric power Heronian aggregation (SVNWGPHA) operator as follows.

Definition 13.

Let $p,q\ge 0$, and $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of single-valued neutrosophic numbers, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$ is the weight vector of $x_i\left(i=1,2,\cdots ,n\right)$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$. If:

$$SVNWGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=\frac{1}{p+q}\underset{\begin{array}{l}i=1,\\ j=i\end{array}}{\overset{n}{\otimes }}{\left(p{x}_i{}^{\frac{n{\tilde{w}}_i{w}_i}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}\oplus q{x}_j{}^{\frac{n{\tilde{w}}_j{w}_j}{{\displaystyle {\sum }_{t=1}^n{\tilde{w}}_t{w}_t}}}\right)}^{\frac{2}{n(n+1)}}$$

(41)

where ${\tilde{w}}_i=\frac{(1+T(x_i))}{{\displaystyle {\sum }_{t=1}^n(1+T(x_t))}}$ and ${\displaystyle \sum _i^n{\tilde{w}}_i}=1$. $T(x_i)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup(x_i,x_j)}$. We shall denote $Sup(x_i,x_j)$ as the support degree for $x_i$ from $x_j$. $Sup(x_i,x_j)$ satisfies the following three properties:

(1) $Sup(x_i,x_j)\in \left[0,1\right]$;

(2) $Sup(x_i,x_j)=Sup(x_j,x_i)$;

(3) $Sup(x_i,x_j)>Sup(x_l,x_k)$, if $d(x_i,x_j)<d(x_l,x_k)$ in which $d(x_i,x_j)$ is the distance between SVNNs $x_i$ and $x_j$.

Then, $SVNWGPH{A}^{p,q}$ is called the single-valued neutrosophic weight geometric power Heronian aggregation (SVNWGPHA) operator.

Theorem 14.

Let $p,q\ge 0$, and $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$. Then, the result aggregated from SVNWGPHA is still a SVNNs, and even

$$\begin{array}{l}SVNWGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)\\ =(1-{\left(1-{\displaystyle \prod _{i=1,j=i}^n{\left(1-{\left(1-T_i^{w_i{\tilde{w}}_i}\right)}^p{\left(1-T_j^{w_j{\tilde{w}}_j}\right)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{\frac{1}{p+q}},\\ {\left(1-{\displaystyle \prod _{i=1.j=i}^n{\left(1-{\left(1-{\left(1-I_i\right)}^{w_i{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-I_j\right)}^{w_j{\tilde{w}}_j}\right)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{\frac{1}{p+q}},\\ {\left(1-{\displaystyle \prod _{i=1.j=i}^n{\left(1-{\left(1-{\left(1-F_i\right)}^{w_i{\tilde{w}}_i}\right)}^p{\left(1-{\left(1-F_j\right)}^{w_j{\tilde{w}}_j}\right)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{\frac{1}{p+q}})\end{array}$$

(42)

Similar to the proof of Theorem 2 it is omitted in this study.

Obviously, when $W={\left(1/n,1/n,\cdots ,1/n\right)}^T$, the SVNWGPHA operator is reduced to the SVNGPHA operator.

Similar to the SVNWPHA operator, the SVNWGPHA operator has the same properties.

Theorem 15.

(Idempotency). Let $x_i(i=1,2,\cdots ,n)$ be a collection of SVNNs, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$, and $x_1=x_2=\cdots =x_n=x$, then $SVNWGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)=x$.

Theorem 16.

(Commutativity). Let $\left({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n\right)$ be any permutation of $\left(x_1,x_2,\cdots ,x_n\right)$ $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ and ${\displaystyle \sum _i^n{w}_i}=1$ then

$$SVNWGPH{A}^{p,q}({x^{\prime }}_1,{x^{\prime }}_2,\cdots {x^{\prime }}_n)=SVNWGPH{A}^{p,q}(x_1,x_2,\cdots ,x_n)$$

Theorem 17.

(Boundedness). Let $x_i=(T_i,I_i,F_i)\left(i=1,2,\cdots ,n\right)$ be a collection of SVNNs, $W={\left(w_1,w_2,\cdots ,w_n\right)}^T$, satisfying $w_i\in \left[0,1\right]$ And ${\displaystyle \sum _i^n{w}_i}=1$ and $x^{-}=\left({\min }_i\left\{T_i\right\},{\max }_i\left\{I_i\right\},{\max }_i\left\{F_i\right\}\right)$, $x^{+}=\left({\max }_i\left\{T_i\right\},{\min }_i\left\{I_i\right\},{\min }_i\left\{F_i\right\}\right)$ then

$$x^{-}\le SVNWGPH{A}^{p,q}\left(x_1,x_2,\cdots ,x_n\right)\le x^{+}$$

4. MAGDM Method Based on the SVNWPHA or SVNWGPHA Operator

In this part, we introduce the application of the SVNWPHA or SVNWGPHA operator in MAGDM. Given the MAGDM problems based on SVNNs, let $E=\left\{E_1,E_2,\cdots ,E_m\right\}$ be the set of alternatives and $G=\left\{G_1,G_2,\cdots ,G_n\right\}$ be the set of attributes respectively. The is the weight of the attributes $G_j\left(j=1,2,\cdots ,n\right)$, where $0\le w_j\le 1\left(j=1,2,\cdots ,n\right)$ and ${\displaystyle {\sum }_{k=1}^t{\gamma }_k}=1$. Suppose that the set of decision makers is $D=\left\{D_1,D_2,\cdots ,D_t\right\}$ and ${\gamma }_k\left(k=1,2,\cdots ,t\right)$ represents a weight of decision maker $D_k$ with $0\le {\gamma }_k\le 1\left(k=1,2,\cdots ,t\right)$, ${\displaystyle {\sum }_{k=1}^t{\gamma }_k}=1$. Suppose that $H^k={\left[{\phi }_{ij}^k\right]}_{m\times n}$ is the decision matrix, where ${\phi }_{ij}^k=\left(T_{ij}^k,I_{ij}^k,F_{ij}^k\right)$ takes the form of SVNN, $T_{ij}^k,I_{ij}^k,F_{ij}^k\in \left[0,1\right]$, and $0\le T_{ij}^k+I_{ij}^k+F_{ij}^k\le 3$, which describes the decision-making information of the attributes $G_j$ in terms of the alternative $E_i$ provided by the decision maker $D_k$. Accordingly, the rank of the alternatives based on the decision information given by decision makers could be attained.

The method includes the following steps:

Step 1: Calculate the supports $Sup({\phi }_{ij}^k,{\phi }_{ij}^h)\left(k,h=1,2,3,\cdots ,t\right)$ by

$$Sup({\phi }_{ij}^k,{\phi }_{ij}^h)=1-d({\phi }_{ij}^k,{\phi }_{ij}^h)$$

(43)

where $d({\phi }_{ij}^k,{\phi }_{ij}^h)$ is the normalized Euclidian distance between two SVNNs is ${\phi }_{ij}^k$ and ${\phi }_{ij}^h$, which is given in Definition 5.

Step 2: Calculate

$$T({\phi }_{ij}^k)={\displaystyle \sum _{j=1,j\ne i}^{n}Sup({\phi }_{ij}^k,{\phi }_{ij}^h)}\left(k,h=1,2,3,\cdots ,t\right)$$

(44)

Step 3: Calculate the weight vector ${\tilde{w}}_i^k$ of power operator associated with the SVNNs ${\phi }_{ij}^k$

$${\tilde{w}}_{ij}^k=\frac{(1+T({\phi }_{ij}^k))}{{\displaystyle {\sum }_{k=1}^t(1+T({\phi }_{ij}^k))}}\left(k=1,2,\cdots ,t\right)$$

(45)

Step 4: Aggregate and fuse the decision information given by each decision maker $D_k\left(k=1,2,\cdots ,t\right)$ by

$${\overline{\phi }}_{ij}=\left(T_{ij},I_{ij},F_{ij}\right)=SVNWPH{A}^{p,q}\left({\phi }_{ij}^1,{\phi }_{ij}^2,\cdots ,{\phi }_{ij}^t\right)$$

(46)

$$\mathrm{Or}{\overline{\phi }}_{ij}=\left(T_{ij},I_{ij},F_{ij}\right)=SVNWGPH{A}^{p,q}\left({\phi }_{ij}^1,{\phi }_{ij}^2,\cdots ,{\phi }_{ij}^t\right)$$

(47)

in order to get the collective decision matrix $H={\left[{\overline{\phi }}_{ij}^{}\right]}_{m\times n}\left(i=1,2,\cdots ,m;j=1,2,\cdots ,n\right)$.

Step 5: Calculate the supports $Sup({\phi }_{ik},{\phi }_{ij})\left(i=1,2,\cdots ,m;h,j=1,2,\cdots ,n\right)$ by

$$Sup({\phi }_{ik},{\phi }_{ij})=1-d({\phi }_{ik},{\phi }_{ij})$$

(48)

where $d({\phi }_{ik},{\phi }_{ij})$ is the normalized Euclidian distance between two SVNNs is ${\phi }_{ik}$ and ${\phi }_{ij}$, which is given in Definition 5.

Step 6: Calculate

$$T({\tilde{\phi }}_{ij})={\displaystyle \sum _{k=1,k\ne j}^{n}Sup({\phi }_{ij},{\phi }_{ik})}\left(k,j=1,2,3,\cdots ,n\right).$$

(49)

Step 7: Calculate the weight vector ${\tilde{w}}_i^{}$ of power operator associated with the SVNNs ${\phi }_{ij}^k$

$${\tilde{w}}_{ij}^{}=\frac{(1+T({\phi }_{ij}^{}))}{{\displaystyle {\sum }_{j=1}^n(1+T({\phi }_{ij}^{}))}}\left(j=1,2,\cdots ,n\right)$$

(50)

Step 8: Compute the comprehensive value of each alternative by

$${\phi }_i=SVNWPH{A}^{p,q}({\overline{\phi }}_{i1},{\overline{\phi }}_{i2},\cdots ,{\overline{\phi }}_{in})$$

(51)

$$\mathrm{or}{\phi }_i=SVNWGPH{A}^{p,q}({\overline{\phi }}_{i1},{\overline{\phi }}_{i2},\cdots ,{\overline{\phi }}_{in})$$

(52)

where $\left(i=1,2,3,\cdots ,m\right)$

Step 9: Obtain the score $S({\phi }_i)$, accuracy $H\left({\phi }_i\right)$, and certainty $c\left({\phi }_i\right)$ functions based on Definition 4.

Step 10: Ranking ${\phi }_i\left(i=1,2,\cdots ,n\right)$ by the comparison method in Definition 5.

Step 11: End.

5. Illustrative Example

This section verified the application and effectiveness of the proposed method by using an example for the MAGDM problem.

Example 2.

Assuming that the air quality in Guangzhou needs to be assessed, and the air quality data in Guangzhou for 2006–2009 was collected as a series of alternatives that is $\left\{E_1,E_2,E_3,E_4\right\}=\left\{2006,2007,2008,2009\right\}$ [32]. Accordingly, three attributes are considered: (1) $S{O}_2(G_1)$ (2) $N{O}_2(G_2)$ (3) $P{H}_{10}(G_3)$. Importance degree of the measured attributes is $W=(0.40,0.20,0.40{)}^T$. Three air quality monitoring stations assessed as experts are expressed by $\left\{D_1,D_2,D_3\right\}$ and the importance of these experts is $\gamma =(0.314,0.355,0.331{)}^T$, respectively. Suppose that $H^k={\left[{\phi }_{ij}^k\right]}_{m\times n}$ is the decision matrix, where ${\phi }_{ij}^k=\left(T_{ij}^k,I_{ij}^k,F_{ij}^k\right)$ takes the form of SVNN, $T_{ij}^k,I_{ij}^k,F_{ij}^k\in \left[0,1\right]$, and $0\le T_{ij}^k+I_{ij}^k+F_{ij}^k\le 3$, which describes the decision-making information of the attributes $G_j$ in terms of the alternative $E_i$ provided by the decision maker $D_k$. Table 1, Table 2, and Table 3 show the three normalized standardized decision matrices from three experts by the SVNNs, respectively.

5.1. Decision-Making Steps

Use SVNWPHA operator to solve the problem involves the following steps:

Step 1: Calculate the supports $Sup({\phi }_{ij}^k,{\phi }_{ij}^h)\left(k,h=1,2,3,\cdots ,t\right)$ by formulas (40) (for simplicity, we denote $Sup({\phi }_{ij}^k,{\phi }_{ij}^h)$ with $S_{ij}^{kh}$), and we get

$S_{11}^{12}=S_{11}^{21}=0.89292$ $S_{11}^{13}=S_{11}^{31}=0.94070$ $S_{11}^{23}=S_{11}^{32}=0.90279$

$S_{12}^{12}=S_{12}^{21}=0.91958$ $S_{12}^{13}=S_{12}^{31}=0.92211$ $S_{12}^{23}=S_{12}^{32}=0.97551$

$S_{13}^{12}=S_{13}^{21}=0.80315$ $S_{13}^{13}=S_{13}^{31}=0.93462$ $S_{13}^{23}=S_{13}^{32}=0.75818$

$S_{21}^{12}=S_{21}^{21}=0.94021$ $S_{21}^{13}=S_{21}^{31}=0.91102$ $S_{21}^{23}=S_{21}^{32}=0.94607$

$S_{22}^{12}=S_{22}^{21}=0.90726$ $S_{22}^{13}=S_{22}^{31}=0.86841$ $S_{22}^{23}=S_{22}^{32}=0.88077$

$S_{23}^{12}=S_{23}^{21}=0.80605$ $S_{23}^{13}=S_{23}^{31}=0.85834$ $S_{23}^{23}=S_{23}^{32}=0.91140$

$S_{31}^{12}=S_{31}^{21}=0.94285$ $S_{31}^{13}=S_{31}^{31}=0.87938$ $S_{31}^{23}=S_{31}^{32}=0.93584$

$S_{32}^{12}=S_{32}^{21}=0.82205$ $S_{32}^{13}=S_{32}^{31}=0.80121$ $S_{32}^{23}=S_{32}^{32}=0.93662$

$S_{33}^{12}=S_{33}^{21}=0.89323$ $S_{33}^{13}=S_{33}^{31}=0.92897$ $S_{33}^{23}=S_{33}^{32}=0.89696$

$S_{41}^{12}=S_{41}^{21}=0.94693$ $S_{41}^{13}=S_{41}^{31}=0.96734$ $S_{41}^{23}=S_{41}^{32}=0.97959$

$S_{42}^{12}=S_{42}^{21}=0.92351$ $S_{42}^{13}=S_{42}^{31}=0.93946$ $S_{42}^{23}=S_{42}^{32}=0.88812$

$S_{43}^{12}=S_{43}^{21}=0.91663$ $S_{43}^{13}=S_{43}^{31}=0.88504$ $S_{43}^{23}=S_{43}^{32}=0.96734$

Step 2: Calculate the $T({\phi }_{ij}^k)$ by formulas (41), and we get

$T({\phi }_{11}^1)=1.83362$ $T({\phi }_{11}^2)=1.79571$ $T({\phi }_{11}^3)=1.84349$

$T({\phi }_{12}^1)=1.84170$ $T({\phi }_{12}^2)=1.89509$ $T({\phi }_{12}^3)=1.89762$

$T({\phi }_{13}^1)=1.73777$ $T({\phi }_{13}^2)=1.55367$ $T({\phi }_{13}^3)=1.68543$

$T({\phi }_{21}^1)=1.85123$ $T({\phi }_{21}^2)=1.88628$ $$