Dombi weighted aggregation operators of neutrosophic Z-numbers for multiple attribute decision making in equipment supplier selection

Abstract

Advances in multiple attribute decision making (MADM) require the development of information aggregation operations flexible enough to deal with neutrosophic Z-number (NZN) information. In this situation, new operations of NZNs are needed to aggregate NZNs with different values of operational parameter and to realize the MADM flexibility. Motivated by the Dombi operations, this study proposes the Dombi operations and some Dombi weighted aggregation operators of NZNs to solve a gap of flexible MADM in the setting of NZNs. Thus, the main aims of this article are (i) to propose several Dombi operations of NZNs, (ii) to present the NZN Dombi weighted arithmetic averaging (NZNDWAA) and NZN Dombi weighted geometric averaging (NZNDWGA) operators for aggregating NZN information and their properties, (iii) to establish a MADM approach based on the NZNDWAA and NZNDWGA operators for solving MADM problems under the environment of NZNs, and (iv) to give a MADM example and related comparative analysis on the issue of equipment supplier selection for indicating the applicability and efficiency of the developed MADM approach. However, the proposed MADM approach is more flexible for the selection of decision makers’ preferences and the actual requirements in MADM applications.

Keywords

Neutrosophic Z-number dombi operation neutrosophic Z-number Dombi weighted arithmetic averaging operator neutrosophic Z-number Dombi weighted geometric averaging operator multiple attribute decision-making

1. Introduction

The operations of the Dombi T-norm and T-conorm [1] indicate the advantage of better flexibility due to their changeable operational parameter. In view of the Dombi operators and the Bonferroni mean operators, Liu et al. [2] presented the intuitionistic fuzzy number (IFN) Dombi Bonferroni mean and geometric Bonferroni mean operators and the IFN weighted Dombi Bonferroni mean and Dombi geometric Bonferroni mean operators and their multiple attribute decision-making (MADM) problems. Furthermore, Wu et al. [3] introduced the interval-valued IFN Dombi and weighted Dombi Hamy mean operators and the interval-valued IFN dual and weighted dual Dombi Hamy mean operators for the evaluation of the elderly tourism service quality in the tourism destination. However, these aggregation operators and MADM methods only can deal with MADM problems with IFNs. Then, Chen and Ye [4] put forward single-valued neutrosophic Dombi weighted arithmetic and geometric averaging (SVNDWAA and SVNDWGA) operators and used them for MAMD problems under the single-valued neutrosophic environment.

To make fuzzy information more reliable, Zadeh [5] first introduced the notion of a Z-number, which is characterized by the combination of a fuzzy number with a reliability measure. Its main advantage is that a fuzzy number is related to its reliability measure to enhance the reliability of fuzzy information. Hence, it has been applied to many real-life problems, such as modeling [6], decision making [7], data fusion [8], approximate reasoning [9], optimization [10], environmental evaluation [11], medicine option [12], etc. However, existing (interval-valued) intuitionistic fuzzy sets [13, 14] and neutrosophic sets (containing single-valued and interval-valued or simplified neutrosophic sets) [15, 16, 17, 18] lack their reliability measures in their information expressions. To enhance the reliability measures of the truth, falsity and indeterminacy degrees in the neutrosophic set, Du et al. [19] first proposed the notion of a neutrosophic Z-number (NZN), which is depicted by both a neutrosophic value and a neutrosophic reliability measure (truth, falsity and indeterminacy Z-numbers), and then developed the NZN weighted arithmetic and geometric averaging (NZNWAA and NZNWGA) operators and their MADM method in the setting of NZNs. But the NZNWAA and NZNWGA operators in the existing MADM method [19] lack the operational flexibility, which implies their flaw. However, in the setting of NZNs, there are only NZNWAA and NZNWGA operators in the existing literature. In this case, the existing MADM approach [19] is difficult to reach flexible decision requirements. To realize flexible operations and MADM in the setting of NZNs, we have to develop new aggregation operators and MADM approaches in the environment of NZNs to solve flexible MADM problems with NZN information. Motivated by the operations of the Dombi T-norm and T-conorm, this study needs to propose the Dombi operations and some Dombi weighted aggregation operators of NZNs for solving a gap of flexible MADM in the setting of NZNs. To do so, the main aims of this article are (1) to propose several Dombi operations of NZNs, (2) to present the NZNDWAA and NZNDWGA operators for aggregating NZN information and their properties, (3) to establish a MADM approach based on the NZNDWAA and NZNDWGA operators for solving flexible MADM problems under the environment of NZNs, and (4) to give a MADM example of a selection problem of equipment suppliers and the related comparative analysis for indicating its applicability and efficiency in the setting of NZNs.

The rest of the article is organized by the sections. In Section 2, some notions of NZNs are depicted briefly as some preliminaries of this study. Section 3 presents several new Dombi operations of NZNs. In Section 4, we present the NZNDWAA and NZNDWGA operators and some properties. Section 5 establishes a MADM approach based on the NZNDWAA and NZNDWGA operators and the score function. In Section 6, a MADM example and related comparative analysis on the issue of equipment supplier selection are given to indicate its applicability and efficiency in the NZN setting. Finally, conclusions and future research are summarized in Section 7.

2. Some concepts of NZNs

Based on the truth, falsity and indeterminacy Z-numbers, Du et al. [19] first introduced the notion of a NZN set. A NZN set in a universe set $U$ is denoted as

$\displaystyle Z=\left\{\left\langle u,T(N,M)(u),I(N,M)(u),F(N,M)(u)\right% \rangle|u\in U\right\},$

where the functions $T(N,M)(u)=(T_{N}(u),F_{M}(u))$ , $F(N,M)(u)=(F_{N}(u),F_{M}(u))$ , $I(N,M)(u)=(I_{N}(u),\linebreak I_{M}(u)):U\rightarrow$ [0, 1] ${}^{2}$ are depicted by the ordered pairs of truth, falsity and indeterminacy fuzzy values. In the three functions, the first component $N$ indicates neutrosophic values in $U$ and the second component $M$ indicates neutrosophic reliability measures for $N$ , such that $0\leqslant T_{N}(u)+I_{N}(u)+F_{N}(u)\leqslant 3$ and $0\leqslant T_{M}(u)+I_{M}(u)+F_{M}(u)\leqslant 3$ for $u\in U$ .

For the representative convenience, the NZN $\left\langle u,T(N,M)(u),I(N,M)(u),F(N,M)(u)\right\rangle$ in $Z$ is simply denoted as $z=\left\langle T(N,M),I(N,M),F(N,M)\right\rangle=\left\langle(T_{N},T_{M}),(I_% {N},I_{M}),(F_{N},F_{M})\right\rangle$ .

Regarding two NZNs $z_{1}=\left\langle T_{1}(N,M),I_{1}(N,M),F_{1}(N,M)\right\rangle=\left\langle(% T_{N1},T_{M1}),(I_{N1},I_{M1}),(F_{N1},F_{M1})\right\rangle$ and $z_{2}=\left\langle T_{2}(N,M),I_{2}(N,M),F_{2}(N,M)\right\rangle=\left\langle(% T_{N2},T_{M2}),(I_{N2},I_{M2}),(F_{N2},F_{M2})\right\rangle$ and any real number $\alpha>0$ , Du et al. [19] defined the following operations:

(1)
$z_{1}\supseteq z_{2}\Leftrightarrow T_{N1}\geqslant T_{N2},T_{M1}\geqslant T_{% M2},I_{N1}\leqslant I_{N2},I_{M1}\leqslant I_{M2},F_{N1}\leqslant F_{N2}$ , and $F_{M1}\leqslant F_{M2}$ ;
(2)
$z_{1}=z_{2}\Leftrightarrow z_{1}\supseteq z_{2}$ and $z_{2}\supseteq z_{1}$ ;
(3)
$z_{1}\bigcup z_{2}=\left\langle(T_{N1}\vee T_{N2},T_{M1}\vee T_{M2}),(I_{N1}% \wedge I_{N2},I_{M1}\wedge I_{M2}),(F_{N1}\wedge F_{N2},F_{M1}\wedge F_{M2})\right\rangle$ ;
(4)
$z_{1}\bigcap z_{2}=\left\langle(T_{N1}\wedge T_{N2},T_{M1}\wedge T_{M2}),(I_{N% 1}\vee I_{N2},I_{M1}\vee I_{M2}),(F_{N1}\vee F_{N2},F_{M1}\vee F_{M2})\right\rangle$ ;
(5)
$z_{1}^{c}=\left\langle\left(F_{N1},F_{M1}\right),\left(1-I_{N1},1-I_{M1}\right% ),\left(T_{N1},T_{M1}\right)\right\rangle$ (Complement of $z_{1}$ );
(6)
$z_{1}\oplus z_{2}=\left\langle\left(T_{N1}+T_{N2}-T_{N1}T_{N2},T_{M1}+T_{M2}-T% _{M1}T_{M2}\right),\left(I_{N1}I_{N2},I_{M1}I_{M2}\right),\left(F_{N1}F_{N2},F% _{M1}F_{M2}\right)\right\rangle$ ;
(7)
$z_{1}\otimes z_{2}=\left\langle\begin{array}[]{l}{\left(T_{N1}T_{N2},T_{M1}T_{% M2}\right),\left(I_{N1}+I_{N2}-I_{N1}I_{N2},I_{M1}+I_{M2}-I_{M1}I_{M2}\right),% }\\ {\left(F_{N1}+F_{N2}-F_{N1}F_{N2},F_{M1}+F_{M2}-F_{M1}F_{M2}\right)}\end{array% }\right\rangle$ ;
(8)
$\alpha\cdot z_{1}=\left\langle\left(1-(1-T_{N1})^{\alpha},1-(1-T_{M1})^{\alpha% }\right),\left(I_{N1}^{\alpha},I_{M1}^{\alpha}\right),\left(F_{N1}^{\alpha},F_% {M1}^{\alpha}\right)\right\rangle$ ;
(9)
$z_{1}^{\alpha}=\left\langle\left(T_{N1}^{\alpha},T_{M1}^{\alpha}\right),\left(% 1-(1-I_{N1})^{\alpha},1-(1-I_{M1})^{\alpha}\right),\left(1-(1-F_{N1})^{\alpha}% ,1-(1-F_{M1})^{\alpha}\right)\right\rangle$ .

To rank NZNs $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ ( $j=$ 1, 2), Du et al. [19] defined the score function:

$\displaystyle V(z_{j})=(2+T_{Nj}T_{Mj}-I_{Nj}I_{Mj}-F_{Nj}F_{Mj})/3\text{ for % }V(z_{j})\in[0,1].$ (1)

Thus, the ranking of both is $z_{1}\geqslant z_{2}$ when $V(z_{1})\geqslant V(z_{2})$ .

Suppose $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ ( $j=1,2,\ldots,n$ ) are a group of NZNs with their weights $\alpha_{j}$ for $\alpha_{j}\in$ [0, 1] and $\sum\nolimits_{j=1}^{n}\alpha_{j}=1$ . To aggregate NZNs, Du et al. [19] further proposed the NZNWAA and NZNWGA operators:

$\displaystyle\textit{NZNWAA}(z_{1},z_{2},\cdots,z_{n})=\sum\limits_{j=1}^{n}% \alpha_{j}\cdot z_{j}$ $\displaystyle=\left\langle\left(1-\prod\limits_{j=1}^{n}(1-T_{Nj})^{\alpha_{j}% },1-\prod\limits_{j=1}^{n}(1-T_{Mj})^{\alpha_{j}}\right),\left(\prod\limits_{j% =1}^{n}I_{Nj}^{\alpha_{j}},\prod\limits_{j=1}^{n}I_{Mj}^{\alpha_{j}}\right),% \left(\prod\limits_{j=1}^{n}F_{Nj}^{\alpha_{j}},\prod\limits_{j=1}^{n}F_{Mj}^{% \alpha_{j}}\right)\right\rangle,$ (2) $\displaystyle\textit{NZNWGA}(z_{1},z_{2},\cdots,z_{n})=\prod\limits_{j=1}^{n}z% _{j}^{\alpha_{j}}=\left\langle\begin{array}[]{l}{\left(\prod\limits_{j=1}^{n}T% _{Nj}^{\alpha_{j}},\prod\limits_{j=1}^{n}T_{Mj}^{\alpha_{j}}\right),}\\ {\left(1-\prod\limits_{j=1}^{n}(1-I_{Nj})^{\alpha_{j}},1-\prod\limits_{j=1}^{n% }(1-I_{Mj})^{\alpha_{j}}\right),}\\ {\left(1-\prod\limits_{j=1}^{n}(1-F_{Nj})^{\alpha_{j}},1-\prod\limits_{j=1}^{n% }(1-F_{Mj})^{\alpha_{j}}\right)}\end{array}\right\rangle.$ (3)
3. Dombi weighted aggregation operators of NZNs

Suppose $c$ and $d$ are any two real numbers, then the Dombi T-norm and T-conorm between $c$ and $d$ are defined below [1]:

$\displaystyle D(c,d)=\frac{1}{1+\left\{\left(\frac{1-c}{c}\right)^{q}+\left(% \frac{1-d}{d}\right)^{q}\right\}^{1/q}},$ (4) $\displaystyle D^{c}(c,d)=1-\frac{1}{1+\left\{\left(\frac{c}{1-c}\right)^{q}+% \left(\frac{d}{1-d}\right)^{q}\right\}^{1/q}},$ (5)

where $q\geqslant$ 1 and $c$ , $d\in$ [0, 1].

Regarding Eqs (4) and (5), the Dombi operations of NZNs are defined below.

Definition 1. Set $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ for $j=$ 1, 2 as two NZNs with any real numbers $\alpha>$ 0 and $q\geqslant$ 1. Then, the Dombi T-norm and T-conorm operations of NZNs are defined below:

$\displaystyle(1)z_{1}\oplus z_{2}=\left\langle\begin{array}[]{l}{\left({1}-% \frac{1}{1+\left\{\left(\frac{T_{N1}}{1-T_{N1}}\right)^{q}+\left(\frac{T_{N2}}% {1-T_{N2}}\right)^{q}\right\}^{1/q}},1-\frac{1}{1+\left\{\left(\frac{T_{M1}}{1% -T_{M1}}\right)^{q}+\left(\frac{T_{M2}}{1-T_{M2}}\right)^{q}\right\}^{1/q}}% \right){,}}\\ {\left(\frac{1}{1+\left\{\left(\frac{1-I_{N1}}{I_{N1}}\right)^{q}+\left(\frac{% 1-I_{N2}}{I_{N2}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\left(\frac{1-% I_{M1}}{I_{M1}}\right)^{q}+\left(\frac{1-I_{M2}}{I_{M2}}\right)^{q}\right\}^{1% /q}}\right),}\\ {\left(\frac{1}{1+\left\{\left(\frac{1-F_{N1}}{F_{N1}}\right)^{q}+\left(\frac{% 1-F_{N2}}{F_{N2}}\right)^{q}\right\}^{1/q}},\frac{1}{1+\left\{\left(\frac{1-F_% {M1}}{F_{M1}}\right)^{q}+\left(\frac{1-F_{M2}}{F_{M2}}\right)^{q}\right\}^{1/q% }}\right)}\end{array}\right\rangle;$ $\displaystyle(2)z_{1}\otimes z_{2}=\left\langle\begin{array}[]{l}{\left(\frac{% 1}{1+\left\{\left(\frac{1-T_{N1}}{T_{N1}}\right)^{q}+\left(\frac{1-T_{N2}}{T_{% N2}}\right)^{q}\right\}^{1/q}},\frac{1}{1+\left\{\left(\frac{1-T_{M1}}{T_{M1}}% \right)^{q}+\left(\frac{1-T_{M2}}{T_{M2}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left({1}-\frac{1}{1+\left\{\left(\frac{I_{N1}}{1-I_{N1}}\right)^{q}+\left(% \frac{I_{N2}}{1-I_{N2}}\right)^{q}\right\}^{1/q}},1-\frac{1}{1+\left\{\left(% \frac{I_{M1}}{1-I_{M1}}\right)^{q}+\left(\frac{I_{M2}}{1-I_{M2}}\right)^{q}% \right\}^{1/q}}\right),}\\ {\left({1}-\frac{1}{1+\left\{\left(\frac{F_{N1}}{1-F_{N1}}\right)^{q}+\left(% \frac{F_{N2}}{1-F_{N2}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+\left\{\left(% \frac{F_{M1}}{1-F_{M1}}\right)^{q}+\left(\frac{F_{M2}}{1-F_{M2}}\right)^{q}% \right\}^{1/q}}\right)\,}\end{array}\right\rangle;$ $\displaystyle(3)\alpha\cdot z_{1}=\left\langle\begin{array}[]{l}{\left(1-\frac% {1}{1+\left\{\alpha\left(\frac{T_{N1}}{1-T_{N1}}\right)^{q}\right\}^{1/q}}{,}1% -\frac{1}{1+\left\{\alpha\left(\frac{T_{M1}}{1-T_{M1}}\right)^{q}\right\}^{1/q% }}\right){,}}\\ {\left(\frac{1}{1+\left\{\alpha\left(\frac{1-I_{N1}}{I_{N1}}\right)^{q}\right% \}^{1/q}}{,}\frac{1}{1+\left\{\alpha\left(\frac{1-I_{M1}}{I_{M1}}\right)^{q}% \right\}^{1/q}}\right),}{\left(\frac{1}{1+\left\{\alpha\left(\frac{1-F_{N1}}{F% _{N1}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\alpha\left(\frac{1-F_{M1% }}{F_{M1}}\right)^{q}\right\}^{1/q}}\right)}\end{array}\right\rangle{;}$ $\displaystyle(4)z_{1}{}^{\alpha}=\left\langle\begin{array}[]{l}{\left(\frac{1}% {1+\left\{\alpha\left(\frac{1-T_{N1}}{T_{N1}}\right)^{q}\right\}^{1/q}}{,}% \frac{1}{1+\left\{\alpha\left(\frac{1-T_{M1}}{T_{M1}}\right)^{q}\right\}^{1/q}% }\right),}\\ {\left(1-\frac{1}{1+\left\{\alpha\left(\frac{I_{N1}}{1-I_{N1}}\right)^{q}% \right\}^{1/q}}{,}1-\frac{1}{1+\left\{\alpha\left(\frac{I_{M1}}{1-I_{M1}}% \right)^{q}\right\}^{1/q}}\right),}\\ {\left(1-\frac{1}{1+\left\{\alpha\left(\frac{F_{N1}}{1-F_{N1}}\right)^{q}% \right\}^{1/q}}{,}1-\frac{1}{1+\left\{\alpha\left(\frac{F_{M1}}{1-F_{M1}}% \right)^{q}\right\}^{1/q}}\right)}\end{array}\right\rangle.$

4. Two Dombi weighted aggregation operators of NZNs

This section proposes two Dombi weighted aggregation operators of NZNs according to the Dombi operations of NZNs in Definition 1 and indicates their properties.

Definition 2. Set $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ ( $j=1,2,\linebreak\ldots,n$ ) as a group of NZNs with their corresponding weights $\alpha_{j}$ for $\alpha_{j}\in$ [0, 1] and $\sum\nolimits_{j=1}^{n}\alpha_{j}=1$ . Then, the NZNDWAA and NZNDWGA operators are defined, respectively, as follows:

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2},\ldots,z_{n})=\mathop{\oplus}\limits% _{j=1}^{n}\alpha_{j}\cdot z_{j},$ (6) $\displaystyle\textit{NZNDWGA}(z_{1},z_{2},\ldots,z_{n})=\mathop{\otimes}% \limits_{j=1}^{n}z_{j}^{\alpha_{j}}.$ (7)

Theorem 1. Set $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ ( $j=1,2,\linebreak\ldots,n$ ) as a group of NZNs with their corresponding weights $\alpha_{j}$ for $\alpha_{j}\in$ [0, 1] and $\sum\nolimits_{j=1}^{n}\alpha_{j}=1$ . Then, the aggregated value of the NZNDWAA operator is still a NZN, which is obtained by

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2},\ldots,z_{n})=\mathop{\oplus}\limits% _{j=1}^{n}\alpha_{j}\cdot z_{j}$ $\displaystyle=\left\langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum% \limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^{1% /q}},1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{Mj}}{1-% T_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-I_{Nj}}{% I_{Nj}}\right)^{q}\right\}^{1/q}},\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-F_{Nj}}{% F_{Nj}}\right)^{q}\right\}^{1/q}},\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end{% array}\right\rangle.$ (8)

Thus, Theorem 1 can be verified by the mathematical induction.

Proof. Set $n=$ 2. Based on the Dombi operations of NZNs in Definition 1, we can get the following result:

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2})=\alpha_{1}\cdot z_{1}\oplus\alpha_{% 2}\cdot z_{2}$ $\displaystyle=\left\langle\begin{array}[]{l}{\left({1}-\frac{1}{1+{}\left\{% \alpha_{1}\left(\frac{T_{N1}}{1-T_{N1}}\right)^{q}{+}\alpha_{2}\left(\frac{T_{% N2}}{1-T_{N2}}\right)^{q}{}\right\}^{1/q}}{,1}-\frac{1}{1+\left\{\alpha_{1}% \left(\frac{T_{M1}}{1-T_{M1}}\right)^{q}{+}\alpha_{2}\left(\frac{T_{M2}}{1-T_{% M2}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {{}\left(\frac{1}{1+\left\{\alpha_{1}\left(\frac{1-I_{N1}}{I_{N1}}{}\right)^{q% }{+}\alpha_{2}\left(\frac{1-I_{N2}}{I_{N2}}\right)^{q}\right\}^{1/q}}{,}\frac{% 1}{1+\left\{\alpha_{1}\left(\frac{1-I_{M1}}{I_{M1}}{}\right)^{q}{+}\alpha_{2}% \left(\frac{1-I_{M2}}{I_{M2}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+\left\{\alpha_{1}\left(\frac{1-F_{N1}}{F_{N1}}{}\right)^{q}{% +}\alpha_{2}\left(\frac{1-F_{N2}}{F_{N2}}\right)^{q}\right\}^{1/q}}{,}\frac{1}% {1+\left\{\alpha_{1}\left(\frac{1-F_{M1}}{F_{M1}}{}\right)^{q}{+}\alpha_{2}% \left(\frac{1-F_{M2}}{F_{M2}}\right)^{q}\right\}^{1/q}}\right)}\end{array}\right\rangle$ $\displaystyle\quad=\left\langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{% \sum\limits_{j=1}^{2}\alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right% \}^{1/q}}{,}1-\frac{1}{1+\left\{\sum\limits_{j=1}^{2}\alpha_{j}\left(\frac{T_{% Mj}}{1-T_{Mj}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{2}\alpha_{j}\left(\frac{1-I_{Nj}% }{I_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{2}% \alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{2}\alpha_{j}\left(\frac{1-F_{Nj}% }{F_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{2}% \alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end{% array}\right\rangle.$ (9)

Set $n=k$ . Equation (4) can keep the following equation:

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2},\ldots,z_{k})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{k}\alpha_{j}\cdot z_{j}=\left% \langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{k}% \alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^{1/q}}{,}1-\frac{1}% {1+\left\{\sum\limits_{j=1}^{k}\alpha_{j}\left(\frac{T_{Mj}}{1-T_{Mj}}\right)^% {q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{k}\alpha_{j}\left(\frac{1-I_{Nj}% }{I_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{k}% \alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{k}\alpha_{j}\left(\frac{1-F_{Nj}% }{F_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{k}% \alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end{% array}\right\rangle.$ (10)

Set $n=k+1$ . Based on Eqs (4) and (4), we have the following operation:

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2},\ldots,z_{k},z_{k+1})$ $\displaystyle=\mathop{\oplus}\limits_{j=1}^{k+1}\alpha_{j}\cdot z_{j}d=\left% \langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{k}% \alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^{1/q}}{,}1-\frac{1}% {1+\left\{\sum\limits_{j=1}^{k}\alpha_{j}\left(\frac{T_{Mj}}{1-T_{Mj}}\right)^% {q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{k}\alpha_{j}\left(\frac{1-I_{Nj}}{% I_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{k}% \alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{k}\alpha_{j}\left(\frac{1-F_{Nj}}{% F_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{k}% \alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end{% array}\right\rangle\oplus\alpha_{k+1}\cdot z_{k+1}$ $\displaystyle=\left\langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum% \limits_{j=1}^{k+1}\alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^% {1/q}}{,}1-\frac{1}{1+\left\{\sum\limits_{j=1}^{k+1}\alpha_{j}\left(\frac{T_{% Mj}}{1-T_{Mj}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{k+1}\alpha_{j}\left(\frac{1-I_{Nj}% }{I_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{k+1% }\alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{k+1}\alpha_{j}\left(\frac{1-F_{Nj}% }{F_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{k+1% }\alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end% {array}\right\rangle$

Hence, Eq. (4) exists for all $n$ .

Consequently, Theorem 1 is true. The proof is completed.

Then, the NZNDWAA operator implies some properties:

(i) Reducibility: If $\alpha_{j}=1/n(j=1,2,\ldots,n)$ , there exists the following result:

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2},\ldots,z_{n})=\mathop{\oplus}\limits% _{j=1}^{n}\frac{1}{n}z_{j}$ $\displaystyle=\left\langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum% \limits_{j=1}^{n}\frac{1}{n}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^{% 1/q}}{,}1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\frac{1}{n}\left(\frac{T_{Mj}% }{1-T_{Mj}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\frac{1}{n}\left(\frac{1-I_{Nj% }}{I_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \frac{1}{n}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\frac{1}{n}\left(\frac{1-F_{Nj% }}{F_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \frac{1}{n}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end% {array}\right\rangle.$

(ii) Idempotency: Set $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ ( $j=1,2,\ldots,n$ ) as a group of NZNs. If $z_{j}={z}({j}=1,2,\ldots,{n})$ , there exists NZNDWAA $(z_{1},{z}_{2},\ldots,z_{n})=z$ .

(iii) Commutativity: Assume a group of NZNs $(z^{\prime}_{1},z^{\prime}_{2},\ldots,z^{\prime}_{n})$ is any permutation of ( $z_{1},z_{2},\ldots,z_{n}$ ). Then, $\textit{NZNDWAA}(z^{\prime}_{1},z^{\prime}_{2},\ldots,z^{\prime}_{n})=\textit{% NZNDWAA}\left(z_{1},z_{2},\ldots,z_{n}\right)$ exists.

(iv) Boundedness: If the maximum NZN and the minimum NZN are specified, respectively, as follows:

$\displaystyle z_{\max}=\left\langle{\mathop{\max}\limits_{j}}(T_{j}(N,M)),{% \mathop{\min}\limits_{j}}(I_{j}(N,M)),{\mathop{\min}\limits_{j}}(F_{j}(N,M))% \right\rangle=\left\langle\left({\mathop{\max}\limits_{j}}(T_{Nj}),{\mathop{% \max}\limits_{j}}(T_{Mj})\right),\left({\mathop{\min}\limits_{j}}(I_{Nj}),{% \mathop{\min}\limits_{j}}(I_{Mj})\right),\left({\mathop{\min}\limits_{j}}(F_{% Nj}),{\mathop{\min}\limits_{j}}(F_{Mj})\right)\right\rangle,$ $\displaystyle z_{\min}=\left\langle{\mathop{\min}\limits_{j}}(T_{j}(N,M)),{% \mathop{\max}\limits_{j}}(I_{j}(N,M)),{\mathop{\max}\limits_{j}}(F_{j}(N,M))% \right\rangle=\left\langle\left({\mathop{\min}\limits_{j}}(T_{Nj}),{\mathop{% \min}\limits_{j}}(T_{Mj})\right),\left({\mathop{\max}\limits_{j}}(I_{Nj}),{% \mathop{\max}\limits_{j}}(I_{Mj})\right),\left({\mathop{\max}\limits_{j}}(F_{% Nj}),{\mathop{\max}\limits_{j}}(F_{Mj})\right)\right\rangle.$

Then, $z_{\min}{\leqslant}$ NZNDWAA $(z_{1},{z}_{2},\ldots,z_{m}){\leqslant}z_{\max}$ exists.

Proof. (i) For $\alpha_{j}=1/n$ ( $j=1,2,\ldots,n$ ), the property is obtained by Eq. (4).

(ii) If $z_{j}=\left\langle\left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(% F_{Nj},F_{Mj}\right)\right\rangle=\left\langle\left(T_{N},T_{M}\right),\left(I% _{N},I_{M}\right),\left(F_{N},F_{M}\right)\right\rangle=z(j=1,2,\ldots,{n})$ , by Eq. (4) we can yield the result:

$\displaystyle\textit{NZNDWAA}(z_{1},z_{2},\ldots,z_{n}{)}={}\mathop{\oplus}% \limits_{j=1}^{n}\alpha_{j}\cdot z_{j}$ $\displaystyle=\left\langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum% \limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^{1% /q}}{,}1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{Mj}}{% 1-T_{Mj}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-I_{Nj}% }{I_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-F_{Nj}% }{F_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}\end{% array}\right\rangle$ $\displaystyle=\left\langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum% \limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{N}}{1-T_{N}}\right)^{q}\right\}^{1/q% }}{,}1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{M}}{1-T% _{M}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-I_{N}}% {I_{N}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-I_{M}}{I_{M}}\right)^{q}\right\}^{1/q}}\right),}{\left% (\!\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\!\frac{1-F_{N}}{F% _{N}}\!\right)^{q}\right\}^{1/q}}{\!,\!}\frac{1}{1+\left\{\sum\limits_{j=1}^{n% }\alpha_{j}\left(\!\frac{1-F_{M}}{F_{M}}\!\right)^{q}\right\}^{1/q}}\!\right)}% \end{array}\right\rangle$ $\displaystyle=\left\langle\left(1-\frac{1}{1+\frac{T_{N}}{1-T_{N}}}{,}1-\frac{% 1}{1+\frac{T_{M}}{1-T_{M}}}\right){,}\left(\frac{1}{1+{}\frac{1-I_{N}}{I_{N}}}% {,}\frac{1}{1+\frac{1-I_{M}}{I_{M}}}\right),\left(\frac{1}{1+{}\frac{1-F_{N}}{% F_{N}}}{,}\frac{1}{1+\frac{1-F_{M}}{F_{M}}}\right)\right\rangle$ $\displaystyle=\left\langle\left(T_{N},T_{M}\right),\left(I_{N},I_{M}\right),% \left(F_{N},F_{M}\right)\right\rangle=z.$

Hence, NZNDWAA $(z_{1},z_{2},\ldots,z_{n})=z$ exists.

(iii) It is obvious that the property is true.

(iv) Since there exist the inequalities ${\mathop{{\min}}\limits_{j}}(T_{Nj})\leqslant T_{Nj}\leqslant{\mathop{{\max}}% \limits_{j}}(T_{Nj})$ , ${\mathop{{\min}}\limits_{j}}(T_{Mj})\leqslant T_{Mj}\leqslant{\mathop{{\max}}% \limits_{j}}(T_{Mj})$ , ${\mathop{{\min}}\limits_{j}}(I_{Nj})\leqslant I_{Nj}\leqslant{\mathop{{\max}}% \limits_{j}}(I_{Nj})$ , ${\mathop{{\min}}\limits_{j}}(I_{Mj})\leqslant I_{Mj}\leqslant{\mathop{{\max}}% \limits_{j}}(I_{Mj})$ , ${\mathop{{\min}}\limits_{j}}(I_{Nj})\leqslant F_{Nj}\leqslant{\mathop{{\max}}% \limits_{j}}(F_{Nj})$ , and ${\mathop{{\min}}\limits_{j}}(F_{Mj})\leqslant F_{Mj}\leqslant{\mathop{{\max}}% \limits_{j}}(F_{Mj})$ . Then we have the following inequalities:

$\displaystyle 1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{{% \mathop{{\min}}\limits_{j}}(T_{Nj})}{1{-}{\mathop{{\min}}\limits_{j}}(T_{Nj})}% \right)^{q}\right\}^{1/q}}\leqslant 1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{T_{Nj}}{1-T_{Nj}}\right)^{q}\right\}^{1/q}}$ $\displaystyle\leqslant 1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}% \left(\frac{{\mathop{{\max}}\limits_{j}}(T_{Nj})}{1{-}{\mathop{{\max}}\limits_% {j}}(T_{Nj})}\right)^{q}\right\}^{1/q}},1-\frac{1}{1+\left\{\sum\limits_{j=1}^% {n}\alpha_{j}\left(\frac{{\mathop{{\min}}\limits_{j}}(T_{Mj})}{1-{\mathop{{% \min}}\limits_{j}}(T_{Mj})}\right)^{q}\right\}^{1/q}}$ $\displaystyle\leqslant 1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}% \left(\frac{T_{Mj}}{1-T_{Mj}}\right)^{q}\right\}^{1/q}}\leqslant 1-\frac{1}{1+% \left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{{\mathop{{\max}}\limits_{j}}% (T_{Mj})}{1-{\mathop{{\max}}\limits_{j}}(T_{Mj})}\right)^{q}\right\}^{1/q}},$ $\displaystyle\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{% \mathop{{\max}}\limits_{j}}(I_{Nj})}{{\mathop{{\max}}\limits_{j}}(I_{Nj})}% \right)^{q}\right\}^{1/q}}\leqslant\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-I_{Nj}}{I_{Nj}}\right)^{q}\right\}^{1/q}}\leqslant% \frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{\mathop{{\min}% }\limits_{j}}(I_{Nj})}{{\mathop{{\min}}\limits_{j}}(I_{Nj})}\right)^{q}\right% \}^{1/q}},$ $\displaystyle\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{% \mathop{{\max}}\limits_{j}}(I_{Mj})}{{\mathop{{\max}}\limits_{j}}(I_{Mj})}% \right)^{q}\right\}^{1/q}}\leqslant\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-I_{Mj}}{I_{Mj}}\right)^{q}\right\}^{1/q}}\leqslant% \frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{\mathop{{\min}% }\limits_{j}}(I_{Mj})}{{\mathop{{\min}}\limits_{j}}(I_{Mj})}\right)^{q}\right% \}^{1/q}},$ $\displaystyle\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{% \mathop{{\max}}\limits_{j}}(F_{Nj})}{{\mathop{{\max}}\limits_{j}}(F_{Nj})}% \right)^{q}\right\}^{1/q}}\leqslant\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-F_{Nj}}{F_{Nj}}\right)^{q}\right\}^{1/q}}\leqslant% \frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{\mathop{{\min}% }\limits_{j}}(F_{Nj})}{{\mathop{{\min}}\limits_{j}}(F_{Nj})}\right)^{q}\right% \}^{1/q}},$ $\displaystyle\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{% \mathop{{\max}}\limits_{j}}(F_{Mj})}{{\mathop{{\max}}\limits_{j}}(F_{Mj})}% \right)^{q}\right\}^{1/q}}\leqslant\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{1-F_{Mj}}{F_{Mj}}\right)^{q}\right\}^{1/q}}\leqslant% \frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-{\mathop{{\min}% }\limits_{j}}(F_{Mj})}{{\mathop{{\min}}\limits_{j}}(F_{Mj})}\right)^{q}\right% \}^{1/q}}.$

Based on the property (ii) and Eq. (1), we can obtain $V(z_{\min})\leqslant V(z_{j})\leqslant V(z_{\max})$ , then there is $z_{\min}{\leqslant}$ NZNDWAA $(z_{1},z_{2},\ldots,z_{n}){\leqslant}z_{\max}$ .

Theorem 2. Set $z_{j}=\left\langle T_{j}(N,M),I_{j}(N,M),F_{j}(N,M)\right\rangle=\left\langle% \left(T_{Nj},T_{Mj}\right),\left(I_{Nj},I_{Mj}\right),\left(F_{Nj},F_{Mj}% \right)\right\rangle$ ( $j=1,2,\linebreak\ldots,n$ ) as a group of NZNs with their weights $\alpha_{j}$ for $\alpha_{j}\in$ [0, 1] and $\sum\nolimits_{j=1}^{n}\alpha_{j}=1$ . Then, the aggregated value of the NZNDWGA operator is still a NZN, which can be yielded by the following formula:

$\displaystyle\textit{NZNDWGA}{(}z_{1},z_{2},\ldots,z_{n}{)}$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}z_{j}{}^{\alpha_{j}}={}\left% \langle\begin{array}[]{l}{\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_% {j}\left(\frac{1{-}T_{Nj}}{T_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+% \left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1{-}T_{Mj}}{T_{Mj}}\right)^{% q}\right\}^{1/q}}\right){,}}\\ {{}\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{I_{Nj}% }{1{-}I_{Nj}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+\left\{\sum\limits_{j=1% }^{n}\alpha_{j}\left(\frac{I_{Nj}}{1{-}I_{Nj}}\right)^{q}\right\}^{1/q}}\right% ),}\\ {\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{F_{Nj}}{% 1{-}F_{Nj}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+\left\{\sum\limits_{j=1}^% {n}\alpha_{j}\left(\frac{F_{Mj}}{1{-}F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}% \end{array}\right\rangle.$ (11)

Theorem 2 can be verified by a similar proof manner of Theorem 1, which is not repeated.

Similarly, the NZNDWGA operator also implies the following properties:

(i) Reducibility: If $\alpha_{j}=1/n$ ( $j=1,2,\ldots,{n}$ ), there exists the following result:

$\displaystyle\textit{NZNDWGA}{(}z_{1},z_{2},\ldots,z_{n}{)}$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}z_{j}{}^{1/n}={}\left\langle% \begin{array}[]{l}{\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\frac{1}{n}% \left(\frac{1-T_{Nj}}{T_{Nj}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left\{% \sum\limits_{j=1}^{n}\frac{1}{n}\left(\frac{1-T_{Mj}}{T_{Mj}}\right)^{q}\right% \}^{1/q}}\right){,}}\\ {{}\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\frac{1}{n}\left(\frac{I_{Nj% }}{1-I_{Nj}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+\left\{\sum\limits_{j=1}% ^{n}\frac{1}{n}\left(\frac{I_{Nj}}{1-I_{Nj}}\right)^{q}\right\}^{1/q}}\right),% }\\ {\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\frac{1}{n}\left(\frac{F_{Nj}}% {1-F_{Nj}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+\left\{\sum\limits_{j=1}^{% n}\frac{1}{n}\left(\frac{F_{Mj}}{1-F_{Mj}}\right)^{q}\right\}^{1/q}}\right)}% \end{array}\right\rangle.$ (12)

(ii) Idempotency: Set $z_{j}={z}({j}=1,2,\ldots,{n})$ . Then, NZNDWGA $({z}_{1},{z}_{2},\ldots,z_{n})={z}$ exists.

(iii) Commutativity: Assume a group of NZNs $(z^{\prime}_{1},z^{\prime}_{2},\ldots,z^{\prime}_{n})$ is any permutation of ( $z_{1},z_{2},\ldots,z_{n}$ ). Then, there is $\textit{NZNDWGA}{(}z^{\prime}_{1},z^{\prime}_{2},\ldots,z^{\prime}_{n}{)=}% \textit{NZNDWGA}\left(z_{1},z_{2},\ldots,z_{n}\right)$ .

(iv) Boundedness: If the maximum NZN and the minimum NZN are specified, respectively, by

then the inequality $z_{\min}{\leqslant}$ NZNDWGA $(z_{1},{z}_{2},\ldots,{z_{n}}){\leqslant}{z}_{\max}$ exists.

Similarly, these properties can be verified by the same manner as that of Theorem 1, which are omitted here.

5. MADM approach based on the NZNDWAA and NZNDWGA operators

This section establishes a MADM approach based on the proposed NZNDWAA and NZNDWGA operators to carry out MADM problems under the environment of NZNs.

As for a MADM problem, a set of several alternatives $Z=\{Z_{1},{Z}_{2},\ldots,Z_{s}\}$ is preliminarily provided, and then is evaluated by a set of multiple attributes $U=\{u_{1},{u}_{2},\ldots,{u_{n}}\}$ . Here, the impotence of each attribute $u_{j}(j=1,2,\ldots,{n})$ is specified by its weight $\alpha_{j}$ for $\alpha_{j}\in$ [0, 1] and $\sum\nolimits_{j=1}^{n}\alpha_{j}=1$ . Each alternative $Z_{i}({i}=1,2,\ldots,{s})$ over each attribute $u_{j}({j}=1,2,\ldots,{n})$ is suitably evaluated by the truth, falsity, and indeterminacy Z-numbers ( $T_{\textit{Nij}}$ , $T_{\textit{Mij}}$ ), ( $F_{\textit{Nij}}$ , $F_{\textit{Mij}}$ ), and ( $I_{\textit{Nij}}$ , $I_{\textit{Mij}}$ ), which can compose the NZN $z_{ij}=\left\langle T_{ij}(N,M),I_{ij}(N,M),F_{ij}(N,M)\right\rangle=\left% \langle\left(T_{\textit{Nij}},T_{\textit{Mij}}\right),\left(I_{\textit{Nij}},I% _{\textit{Mij}}\right),\left(F_{\textit{Nij}},F_{\textit{Mij}}\right)\right\rangle$ for $T_{\textit{Nij}}$ , $F_{\textit{Nij}}$ , $I_{\textit{Nij}}\in$ [0, 1] and $T_{\textit{Mij}}$ , $F_{\textit{Mij}}$ , $I_{\textit{Mij}}\in$ [0, 1]. By similar evaluation values, we can establish the decision matrix of NZNs $Z=(z_{ij})_{s\times n}$ . Regarding this MADM problem, the decision steps are addressed below.

Step 1: Using Eqs (4) or (4), the aggregated NZN $z_{i}({i}=1,2,\ldots,{s})$ is yielded by

$\displaystyle z_{i}=\textit{NZNDWAA}{(}z_{i1},z_{i2},\ldots,z_{in}{)}$ $\displaystyle={}\mathop{\oplus}\limits_{j=1}^{n}\alpha_{j}\cdot z_{ij}=\left% \langle\begin{array}[]{l}{\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}% \alpha_{j}\left(\frac{T_{\textit{Nij}}}{1-T_{\textit{Nij}}}\right)^{q}\right\}% ^{1/q}}{,}1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{T_{% \textit{Mij}}}{1-T_{\textit{Mij}}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-I_{% \textit{Nij}}}{I_{\textit{Nij}}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left% \{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-I_{\textit{Mij}}}{I_{\textit{% Mij}}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(\frac{1}{1+{}\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-F_{% \textit{Nij}}}{F_{\textit{Nij}}}\right)^{q}\right\}^{1/q}}{,}\frac{1}{1+\left% \{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-F_{\textit{Mij}}}{F_{\textit{% Mij}}}\right)^{q}\right\}^{1/q}}\right)}\end{array}\right\rangle.$ (13)

$\displaystyle z_{i}=\textit{NZNDWGA}{(}z_{i1},z_{i2},\ldots,z_{in}{)}$ $\displaystyle=\mathop{\otimes}\limits_{j=1}^{n}z_{ij}{}^{\alpha_{j}}={}\left% \langle\begin{array}[]{l}{\left(\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_% {j}\left(\frac{1-T_{\textit{Nij}}}{T_{\textit{Nij}}}\right)^{q}\right\}^{1/q}}% {,}\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{1-T_{\textit{% Mij}}}{T_{\textit{Mij}}}\right)^{q}\right\}^{1/q}}\right){,}}\\ {{}\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{I_{% \textit{Nij}}}{1-I_{\textit{Nij}}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+% \left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{I_{\textit{Mij}}}{1-I_{% \textit{Mij}}}\right)^{q}\right\}^{1/q}}\right),}\\ {\left(1-\frac{1}{1+\left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{F_{% \textit{Nij}}}{1-F_{\textit{Nij}}}\right)^{q}\right\}^{1/q}}{,1}-\frac{1}{1+% \left\{\sum\limits_{j=1}^{n}\alpha_{j}\left(\frac{F_{\textit{Mij}}}{1-F_{% \textit{Mij}}}\right)^{q}\right\}^{1/q}}\right)}\end{array}\right\rangle.$ (14)

Step 2: The score values of $V(z_{i})({i}=1,2,\ldots,{s})$ are calculated by applying Eq. (1).

Step 3: The alternatives are ranked and the best one is selected.

Step 4: End.

6. MADM example and comparison

6.1 MADM example of equipment supplier selection

This subsection provides a MADM example regarding the selection problem of equipment suppliers adopted from [19] for convenient comparison in the environment of NZNs to illustrate the usability and suitability of the established MADM approach.

Assume a manufacturing company wants to select the best supplier among globe equipment suppliers. A group of experts preliminarily selects a set of four equipment suppliers (alternatives) $Z=\{Z_{1},{Z}_{2},{Z}_{3},{Z}_{4}\}$ from globe equipment suppliers. Then, they should satisfy three attributes: the price of equipment ( $u_{1}$ ), the performance and quality of equipment ( $u_{2}$ ), and the service quality of equipment supplier ( $u_{3}$ ). The weights of the three attributes $u_{j}$ for $j=$ 1, 2, 3 are given by $\alpha_{1}=$ 0.33, $\alpha_{2}=$ 0.35, and $\alpha_{3}=$ 0.32 respectively to specify the importance of each attribute. Then, decision makers/experts are requested to evaluate the four alternatives over the three attributes in NZNs $z_{ij}=\left\langle T_{ij}(N,M),I_{ij}(N,M),F_{ij}(N,M)\right\rangle=\left% \langle\left(T_{\textit{Nij}},T_{\textit{Mij}}\right),\left(I_{\textit{Nij}},I% _{\textit{Mij}}\right),\left(F_{\textit{Nij}},F_{\textit{Mij}}\right)\right\rangle$ ( $i=$ 1, 2, 3, 4; $j=$ 1, 2, 3) that consist of their truth, falsity, and indeterminacy Z-numbers (containing neutrosophic values and neutrosophic measure values of reliability). Thus, these NZNs can be established as the decision matrix of NZNs $Z=(z_{ij})_{4\times 3}$ , which is shown in Table 1 [19].

Table 1
The decision matrix of NZNs $Z=(z_{ij})_{4\times 3}$

	$u_{1}$	$u_{2}$	$u_{3}$
$Z_{1}$	$<$ (0.6, 0.8), (0.1, 0.7), (0.2, 0.8) $>$	$<$ (0.7, 0.6), (0.1, 0.8), (0.2, 0.7) $>$	$<$ (0.7, 0.7), (0.2, 0.8), (0.2, 0.9) $>$
${Z}_{2}$	$<$ (0.8, 0.7), (0.1, 0.8), (0.2, 0.6) $>$	$<$ (0.6, 0.7), (0.1, 0.7), (0.2, 0.7) $>$	$<$ (0.8, 0.8), (0.4, 0.7), (0.2, 0.8) $>$
${Z}_{3}$	$<$ (0.6, 0.6), (0.2, 0.6), (0.1, 0.7) $>$	$<$ (0.7, 0.8), (0.2, 0.7), (0.3, 0.8) $>$	$<$ (0.7, 0.7), (0.3, 0.8), (0.6, 0.7) $>$
${Z}_{4}$	$<$ (0.7, 0.8), (0.1, 0.7), (0.1, 0.7) $>$	$<$ (0.6, 0.7), (0.1, 0.7), (0.1, 0.9) $>$	$<$ (0.7, 0.6), (0.2, 0.7), (0.3, 0.8) $>$

Table 2

The decision results using the NZNDWAA operator

	$z_{i}$	Score value of $V(z_{i})$	Ranking	The best one
$q=$ 1	$z_{1}=$ $<$ (0.6730, 0.7216), (0.1190, 0.4635), (0.2000, 0.6598) $>$ , $z_{2}=$ $<$ (0.7576, 0.7414), (0.1316, 0.5136), (0.2000, 0.5291) $>$ , $z_{3}=$ $<$ (0.6730, 0.7254), (0.2239, 0.5291), (0.2000, 0.4515) $>$ , $z_{4}=$ $<$ (0.6712, 0.7235), (0.1190, 0.4391), (0.1271, 0.4742) $>$	0.7662, 0.7961, 0.7598, 0.7910	$Z_{2}>{Z}_{4}>{Z}_{1}>{Z}_{3}$	${Z}_{2}$
$q=$ 2	${z}_{1}=$ $<$ (0.6769, 0.7365), (0.1142, 0.3656), (0.2000, 0.6131) $>$ , ${z}_{2}=$ $<$ (0.7698, 0.7481), (0.1181, 0.4543), (0.2000, 0.4577) $>$ , ${z}_{3}=$ $<$ (0.6769, 0.7398), (0.2196, 0.4577), (0.1571, 0.3641) $>$ , $z_{4}=$ $<$ (0.6753, 0.7376), (0.1142, 0.3625), (0.1171, 0.3662) $>$	0.7781, 0.8103, 0.7810, 0.8046	$Z_{2}>{Z}_{4}>{Z}_{3}>{Z}_{1}$	${Z}_{2}$
$q=$ 3	${z}_{1}=$ $<$ (0.6802, 0.7485), (0.1108, 0.3253), (0.2000, 0.5848) $>$ , ${z}_{2}=$ $<$ (0.7777, 0.7547), (0.1121, 0.4182), (0.2000, 0.4188) $>$ , ${z}_{3}=$ $<$ (0.6802, 0.7513), (0.2163, 0.4188), (0.1378, 0.3252) $>$ , ${z}_{4}=$ $<$ (0.6788, 0.7492), (0.1108, 0.3250), (0.1119, 0.3254) $>$	0.7854, 0.8187, 0.7919, 0.8120	${Z}_{2}>{Z}_{4}>{Z}_{3}>{Z}_{1}$	${Z}_{2}$
${q}=$ 4	${z}_{1}=$ $<$ (0.6830, 0.7578), (0.1086, 0.3055), (0.2000, 0.5667) $>$ , ${z}_{2}=$ $<$ (0.7827, 0.7607), (0.1090, 0.3972), (0.2000, 0.3973) $>$ , ${z}_{3}=$ $<$ (0.6830, 0.7601), (0.2136, 0.3973), (0.1277, 0.3054) $>$ , ${z}_{4}=$ $<$ (0.6817, 0.7581), (0.1086, 0.3054), (0.1090, 0.3055) $>$	0.7903, 0.8242, 0.7984, 0.8168	${Z}_{2}>{Z}_{4}>{Z}_{3}>{Z}_{1}$	$Z_{2}$
${q}=$ 5	${z}_{1}=$ $<$ (0.6852, 0.7647), (0.1070, 0.2938), (0.2000, 0.5545) $>$ , ${z}_{2}=$ $<$ (0.7860, 0.7659), (0.1072, 0.3842), (0.2000, 0.3843) $>$ , ${z}_{3}=$ $<$ (0.6852, 0.7667), (0.2116, 0.3843), (0.1218, 0.2938) $>$ , ${z}_{4}=$ $<$ (0.6841, 0.7649), (0.1070, 0.2938), (0.1071, 0.2938) $>$	0.7939, 0.8280, 0.8027, 0.8201	${Z}_{2}>{Z}_{4}>{Z}_{3}>{Z}_{1}$	${Z}_{2}$

First, the developed MADM approach using the NZNDWAA or NZNDWAA operator can be used for this MADM problem. Thus, all decision results are given by Eqs (5) or (5) and Eq. (1), which are tabulated in Tables 2 and 3.

Table 3

The decision results using the NZNDWGA operator

	$z_{i}$	Score value of $V(z_{i})$	Ranking	The best one
${q}=$ 1	${z}_{1}=$ $<$ (0.6635, 0.6755), (0.1346, 0.7753), (0.2000, 0.8338) $>$ , ${z}_{2}=$ $<$ (0.7164, 0.7387), (0.2241, 0.7425), (0.2000, 0.7216) $>$ , ${z}_{3}=$ $<$ (0.6635, 0.6922), (0.2350, 0.7216), (0.4000, 0.7447) $>$ , ${z}_{4}=$ $<$ (0.6614, 0.7007), (0.1346, 0.7000), (0.1754, 0.8387) $>$	0.7257, 0.7395, 0.6640, 0.7407	${Z}_{4}>{Z}_{2}>{Z}_{1}>{Z}_{3}$	${Z}_{4}$
${q}=$ 2	${z}_{1}=$ $<$ (0.6582, 0.6662), (0.1442, 0.7796), (0.2000, 0.8519) $>$ , ${z}_{2}=$ $<$ (0.6930, 0.7345), (0.2796, 0.7492), (0.2000, 0.7359) $>$ , ${z}_{3}=$ $<$ (0.6582, 0.6771), (0.2414, 0.7359), (0.4703, 0.7514) $>$ , ${z}_{4}=$ $<$ (0.6560, 0.6933), (0.1442, 0.7000), (0.2058, 0.8559) $>$	0.7186, 0.7175, 0.6382, 0.7259	${Z}_{4}>{Z}_{1}>{Z}_{2}>{Z}_{3}$	${Z}_{4}$
${q}=$ 3	${z}_{1}=$ $<$ (0.6528, 0.6576), (0.1533, 0.7831), (0.2000, 0.8644) $>$ , ${z}_{2}=$ $<$ (0.6735, 0.7308), (0.3139, 0.7558), (0.2000, 0.7477) $>$ , ${z}_{3}=$ $<$ (0.6528, 0.6647), (0.2479, 0.7477), (0.5085, 0.7579) $>$ , ${z}_{4}=$ $<$ (0.6506, 0.6867), (0.1533, 0.7000), (0.2288, 0.8674) $>$	0.7121, 0.7018, 0.6210, 0.7137	${Z}_{4}>{Z}_{1}>{Z}_{2}>{Z}_{3}$	${Z}_{4}$
${q}=$ 4	${z}_{1}=$ $<$ (0.6475, 0.6500), (0.1609, 0.7858), (0.2000, 0.8725) $>$ , ${z}_{2}=$ $<$ (0.6590, 0.7278), (0.3341, 0.7617), (0.2000, 0.7569) $>$ , ${z}_{3}=$ $<$ (0.6475, 0.6547), (0.2540, 0.7569), (0.5306, 0.7636) $>$ , ${z}_{4}=$ $<$ (0.6454, 0.6810), (0.1609, 0.7000), (0.2442, 0.8749) $>$	0.7066, 0.6913, 0.6088, 0.7044	${Z}_{1}>{Z}_{4}>{Z}_{2}>{Z}_{3}$	${Z}_{1}$
${q}=$ 5	${z}_{1}=$ $<$ (0.6427, 0.6436), (0.1670, 0.7880), (0.2000, 0.8779) $>$ , ${z}_{2}=$ $<$ (0.6486, 0.7252), (0.3468, 0.7668), (0.2000, 0.7639) $>$ , ${z}_{3}=$ $<$ (0.6427, 0.6469), (0.2595, 0.7639), (0.5444, 0.7685) $>$ , ${z}_{4}=$ $<$ (0.6407, 0.6762), (0.1670, 0.7000), (0.2545, 0.8798) $>$	0.7021, 0.6839, 0.5997, 0.6975	${Z}_{1}>{Z}_{4}>{Z}_{2}>{Z}_{3}$	${Z}_{1}$

Regarding the ranking orders in Tables 2 and 3, using different operators and values of $q$ in this MADM example can affect their ranking orders. Then the ranking orders based on the NZNDWAA operator are not so sensitive to values of $q$ and then the best alternative is always $Z_{2}$ ; while the ranking orders based on the NZNDWGA operator are sensitive to values of $q$ and then the best alternative is $Z_{1}$ or $Z_{4}$ , which indicates the decision flexibility in the MADM process. From the flexible viewpoint, the NZNDWGA operator is superior to the NZNDWAA operator. In actual MADM problems, however, one of the two operators and a value of $q$ are chosen depending on decision makers’ preferences and actual requirements.

6.2 Comparative analysis

To compare the proposed MADM approach with the existing MADM approach [19] in the setting of NZNs, we introduce all the decision results based on the NZNWAA and NZNWGA operators of Eqs (2) and (3) and the score function of Eq. (1) [19], which are shown in Table 4.

Table 4
The decision results corresponding to the NZNWAA and NZNWGA operators of Eqs (2) and (3) and the score function of Eq. (1) [19]

MADM method	$z_{i}$	Score value of $V(z_{i})$	Ranking	The best one
MADM method based on the NZNWAA operator [19]	${z}_{1}=$ $<$ (0.6701, 0.7098), (0.1248, 0.7655), (0.2000, 0.7928) $>$ , ${z}_{2}=$ $<$ (0.7451, 0.7365), (0.1558, 0.7315), (0.2000, 0.6943) $>$ , ${z}_{3}=$ $<$ (0.6701, 0.7138), (0.2277, 0.6943), (0.2606, 0.7335) $>$ , ${z}_{4}=$ $<$ (0.6682, 0.7123), (0.1248, 0.7000), (0.1421, 0.7977) $>$ .	0.7405, 0.7653, 0.7097, 0.7584	${Z}_{2}>{Z}_{4}>{Z}_{1}>{Z}_{3}$	${Z}_{2}$
MADM method based on the NZNWGA operator [19]	${z}_{1}=$ $<$ (0.6653, 0.6931), (0.1333, 0.7714), (0.2000, 0.8154) $>$ , ${z}_{2}=$ $<$ (0.7234, 0.7306), (0.2095, 0.7376), (0.2000, 0.7103) $>$ , ${z}_{3}=$ $<$ (0.665, 0.6971), (0.2335, 0.7103), (0.3642, 0.7397) $>$ , ${z}_{4}=$ $<$ (0.6632, 0.6963), (0.1333, 0.7000), (0.1695, 0.8206) $>$	0.7317, 0.7440, 0.6762, 0.7431	${Z}_{2}>{Z}_{4}>{Z}_{1}>{Z}_{3}$	${Z}_{2}$

In Table 4, the two ranking orders of the four alternatives and the best one are identical, which are the same as those of the proposed MADM approach based on the NZNDWAA operator for $q=$ 1. Then, the existing MADM approach [19] lacks its decision flexibility owing to using the NZNWAA and NZNWAA operators of Eqs (2) and (3) [19], while the proposed MADM approach using the NZNDWAA and NZNDWAA operators can indicate the main advantage of flexible MADM in actual applications since the NZNDWAA and NZNDWAA operators contain flexible aggregation algorithms corresponding to different values of $q$ . Therefore, the proposed MADM approach is more suitable for decision makers’ preference selection and actual requirements in real MADM applications.

7. Conclusion

Based on the Dombi T-norm and T-conorm operations, this article proposed several Dombi operations of NZNs and the NZNDWAA and NZNDWGA operators in the setting of NZNs. Further, a MADM approach was established by utilizing the NZNDWAA or NZNDWGA operator to solve MADM problems under the environment of NZNs, where the algorithms of the NZNDWAA or NZNDWAA operator and the score function were utilized to determinate the ranking order of alternatives and the best one(s) based on the score values regarding the different values of $q$ . Finally, a MADM example on the problem of equipment supplier selection was presented to indicate the applicability and efficiency of the proposed MADM approach under the environment of NZNs. Then, the decision results and comparative analysis indicated the main highlights of the proposed MADM approach: (i) the NZNDWAA and NZNDWGA operators provide new aggregation algorithms of NZNs for MADM problems under the environment of NZNs; (ii) the decision results based on the NZNDWGA and NZNDWAA operators corresponding to different values of $q$ can affect the ranking of alternatives in the decision process; (iii) the proposed MADM method shows better flexibility due to the selection of different $q$ values related to the preferences and/or actual requirements of the decision makers.

Regarding future research, we shall further develop new NZN aggregation operators, such as Hamacher aggregation operators, Bonferroni mean operators, and Dombi Bonferroni mean operators in the setting of NZNs, and utilize them for these areas such as information fusion, group decision making, medical diagnosis, and image processing.

Footnotes

Conflict of interest

The author declares no conflict of interest.

References

Dombi

. A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets and Systems. 1982; 8: 149-163.

Liu

Chen

. Some intuitionistic fuzzy Dombi Bonferonni operators and their application to multi-attribute group decision-making. J Oper Res Soc. 2018; 69(1): 1-24.

Wei

Gao

Wei

. Some interval-valued intuitionistic fuzzy Dombi Hamy mean operators and their application for evaluating the elderly tourism service quality in tourism destination. Mathematics, 2018; 6(12): 294. doi: 10.3390/math6120294.

Chen

. Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision-making. Symmetry. 2017; 9(6): 82. doi: 10.3390/sym9060082.

Zadeh

. A note on z-numbers. Inf. Sci. 2011; 181(14): 2923-2932.

Patel

Khorasani

Rahimi

. Modeling and implementation of z-number. Soft Comput. 2016; 20(4): 1341-1364.

Kang

Deng

Zhou

. A new methodology of multicriteria decision-making in supplier selection based on z-numbers. Math. Probl. Eng. 2016; (2016): ID 8475987. doi: 10.1155/2016/8475987.

Jiang

Xie

Zhuang

Shou

Tang

. Sensor data fusion with z-numbers and its application in fault diagnosis. Sensors. 2016; 16(9): 1-22.

Aliev

Pedrycz

Huseynov

Eyupoglu

. Approximate reasoning on a basis of z-number-valued if-then rules. IEEE Trans. Fuzzy Syst. 2017; 25(6): 1589-1600.

10.

Qiu

Dong

Chen

. On an optimization method based on Z-numbers and the multi-objective evolutionary algorithm. IntellIgent AutomAtIon & Soft ComputIng. 2018; 24(1): 147-150.

11.

Kang

Zhang

Gao

Chhipi-Shrestha

Hewage

Sadiq

. Environmental assessment under uncertainty using Dempster-Shafer theory and Z-numbers. Journal of Ambient Intelligence and Humanized Computing. 2020; 11(5): 2041-2060.

12.

Ren

Liao

Liu

. Generalized Z-numbers with hesitant fuzzy linguistic information and its application to medicine selection for the patients with mild symptoms of the COVID-19. Computers & Industrial Engineering. 2020; 145. Article Number: UNSP 106517. doi: 10.1016/j.cie.2020.106517.

13.

Atanassov

. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20: 87-96.

14.

Atanassov

Gargov

. Interval-valued intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1989; 31: 343-349.

15.

Smarandache

. Neutrosophy: Neutrosophic probability, set, and logic, American Research Press, Rehoboth, USA; 1998.

16.

Wang

Smarandache

Zhang

Sunderraman

. Single valued neutrosophic sets. Multispace Multistructure. 2010; 4: 410-413.

17.

Wang

Smarandache

Zhang

Sunderraman

. Interval neutrosophic sets and logic: Theory and applications in computing, Hexis, Phoenix, AZ; 2005.

18.

. A multicriteria decision-making method using aggregation operators for simplified neutrosophic sets. Journal of Intelligent & Fuzzy Systems. 2014; 26: 2459-2466.

19.

Yong

Zhang

. Some aggregation operators of neutrosophic Z-numbers and their multicriteria decision making method. Complex & Intelligent Systems. 2021; 7(1): 429-438.

Dombi weighted aggregation operators of neutrosophic Z-numbers for multiple attribute decision making in equipment supplier selection

Abstract

Keywords

1. Introduction

2. Some concepts of NZNs

6.1 MADM example of equipment supplier selection

Table 1 The decision matrix of NZNs Z = ( z i ⁢ j ) 4 × 3

Table 4 The decision results corresponding to the NZNWAA and NZNWGA operators of Eqs (2) and (3) and the score function of Eq. (1) [19]

Footnotes

Conflict of interest

References

Table 1
The decision matrix of NZNs $Z=(z_{ij})_{4\times 3}$

Table 4
The decision results corresponding to the NZNWAA and NZNWGA operators of Eqs (2) and (3) and the score function of Eq. (1) [19]