QNN-MAGDM strategy for E-commerce site selection using quadripartition neutrosophic neutrality aggregative operators

Abstract

In this paper, we define the Quadripartition Neutrosophic Weighted Neutrality Aggregative (QNWNA) operator and Quadripartition Neutrosophic Ordered Weighted Neutrality Aggregative (QNOWNA) operator for solving Multi-Attribute Group Decision Making (MAGDM) problems. The basic properties of both operators are discussed thoroughly. A new MAGDM strategy is developed using these developed operators. A case study of e-commerce site selection in India is discussed to show the applicability of the proposed MADM strategy. Moreover, the performance of the QNWNA and QNOWNA operators are compared with Quadripartition Neutrosophic Weighted Arithmetic Aggregation (QNWAA) operator and Quadripartition Neutrosophic Weighted Geometric Aggregation (QNWGA), Quadripartition Single valued Neutrosophic Dombi Weighted Arithmetic Aggregation (QSVNDWAA) and QSVN Dombi Weighted Geometric Aggregation (QSVNDWAA) operator.

Keywords

MAGDM quadripartition neutrosophic number QNWNA QNOWNA

1. Introduction

Multi-Attribute Group Decision Making (MAGDM) is a popular decision-making strategy. It is a process to identify the most suitable alternative from a set of feasible alternatives under some criteria. The utilization of MAGDM problems [1, 2, 3] is increasing as the intricacies of our daily lives get more complex. It is often difficult for the decision makers to express the rating value as a specific number in various uncertainty situations. To deal with it, Zadeh first defined “Fuzzy Set” (FS) [4] to handle the ambiguities in the information, which utilizes the “Membership Degree” (MD) that reflects uncertainty. The concept of FS was extended to “Intuitionistic Fuzzy Set” (IFS) [5] by Atanassov by adding “Non-Membership Degree” (NMD) along with MD to describe the fuzzy information. The range of IFS information is very narrow with the constraint condition that MDs $(\alpha)$ and NMDs $(\beta)$ should not be greater than one i.e. $\alpha+\beta\leqslant 1$ . Later on, Smarandache defined the “Neutrosophic Set” (NS) [6] which contains “Indeterminacy Membership Degree” (IMD) to handle the decision-making problems more accurately. The NS has three MDs – “Truth Membership Degree (TMD) $\sigma$ , Indeterminacy Membership Degree (IMD) $\lambda$ , and Falsity Membership Degree (FMD) $\tau$ ”, and they hold the condition $0^{-}\leqslant\sigma+\lambda+\tau\leqslant 3^{+}$ . Researchers have intensively investigated neutrosophic theory [7, 8, 9, 10] in dealing with real-life situations containing ambiguity. Even though the hesitation margin of neutrosophic theory is independent of truth or falsity membership, it appears to be more comprehensive than intuitionistic fuzzy sets.

Atanassov et al. [11] recently looked at the relationships between inconsistent IFS, picture fuzzy sets [12], NS [13], and IFS [5]. However, whether the element’s belongingness or non-belongingness causes the indeterminacy associated with a particular element is unclear. This has been noticed by Chatterjee et al. [14] who defined a more generic construction of NS, namely Quadripartitioned Single Valued Neutrosophic Set (QSVNS). The concept of QSVNS is based on Smarandache’s four numerical-valued neutrosophic logic [15] and Belnap’s four-valued logic [16], in which indeterminacy is separated into two independent parts: unknown and contradiction. However, in the context of neutrosophic research, the QSVNS appears to be quite rational. So, researchers are giving more consideration in this theory, and they are using Aggregation Operators(AOs) to solve DMPs. For example, Chatterjee et al. [14] presented the concept of some similarity measures and entropy on QSVNS. Chatterjee et al. [17] developed AOs for Quadripartition Neutrosophic Numbers(QNNs). Mohanasundari and Mohana [18] presented Dombi weighted AOs for MADM. Roy et al. [19] defined similarity measures of Quadripartitioned Single Valued Bipolar Neutrosophic Sets (QSVBNS) and applied them to the MADM problem. Mohanasundari and Mohana [20] presented axiomatic characterizations of Quadripartitioned Single Valued Neutrosophic Rough Sets (QSVNRS).

1.1 Research gap and motivation

From the foregoing studies, it has been seen that researchers have put a lot of efforts into developing different kinds of AOs under SVNSs and QNN environments to handle the uncertainties in the data. No studies have been conducted using a neutrality operator in the QNN environment.

This research gap motivates us to develop a new MAGDM strategy using neutrality operator in the QNN environment. Until now many AOs have been defined in NS [21, 22, 23, 24] and QNNs [17, 18] environments. But the neutral operator is different from all other operators. In neutral operator, we have to define the Probability Sum (PS) function and the proportional distribution rules for each membership function. He et al. [25] defined Neutral Averaging Operators (NAO) using this PS function and the proportional distribution rules of MD and NMD of IFSs and applied it to the MADM problem. Garg [26] defined the NO-based Pythagorean Fuzzy Aggregation Operator (PFAO) and employed them to solve Multiple Attribute Group Decision-Making (MAGDM) processes. Garg and Chen [27] defined the NAO of q-rung orthopair FSs, and applied it for MAGDM. Garg et al. [28] presented NAO for complex q-rung orthopair FSs. Garg [29] presented the novel NAO-based MAGDM method for single-valued neutrosophic numbers. All the above NAOs dealt with two or three membership degrees. They missed the information about CMD and IMD. To get fair treatment towards evaluating these four MDs, we utilize the PS function and the interaction between the MDs to define some new operational laws. For this, we define some new addition and scalar multiplication rules for different QNNs to aggregate different QNNs. Based on these operational laws, some weighted averaging AOs are proposed to explore the process to solve the MADM problems. A literature review for different aggregation operators in neutrosophic and QNN environments is shown in Table 1.

Table 1
Literature review for different aggregation operators in neutrosophic and QNN environments

Operator	Approach	Application
Averaging and geometric AOs	Averaging and geometric AOs under QNNs environment [17]	Air quality evaluation
Harmonic mean AOs	Harmonic mean AOs in neutrosophic number environment [30]	Investment company
Power AOs	Power AOs of simplified neutrosophic sets [31]	Invest problem concerning a financial company
Neutral AOs	NAO under single-valued neutrosophic numbers environment [29]	Identifying suitable company for investment.
Dombi weighted AOs	Dombi weighted aggregation operators of QSVNNs [18]	Investment company
Hamacher AOs	Hamacher aggregation operators in neutrosophic number environment [32]	Air quality evaluation
Bonferroni mean einstein AOs	Bonferroni mean einstein AOsi NS environment [33]	Investment company
Maclaurin symmetric mean operators	Linguistic neutrosophic partitioned maclaurin symmetric mean operators [34]	Logistical park location selection
Heronian AOs	Neutrosophic uncertain linguistic number Heronian mean operators [35]	Investment company

It is found during the study that when decision-makers evaluate the objects equally considering the MDs and NMDs, their aggregated values by addition or multiplication are not equal. As a result, biasness may occur in decision making. By utilizing the probability sum and proportional distribution rules of the membership function, we offer some additional neutrality addition and scalar operations to manage such circumstances and to make a decision more flexible and relevant considering the decision maker’s attitudes.

The main objectives of this paper are as follows:

(1)

To develop the neutral operational laws for QNN based on a process-neutral decision.

(2)

To create a new MADM strategy based on QNWNA and QNOWNA operators in a QNNs environment.

(3)

To present a comparative study with the existing operators in QNNs environment.

(4)

To show the advantages of the developed operators.

The following is the outline of the paper. Section 2 presents some fundamental concepts of QNNs, such as some basic operations on QNNs, weighted arithmetic and geometric aggregation operators, and the entropy measure of QNNs. In Section 3, the neutrality operational law for QNNs is introduced. Then, the QNWNA and QNOWNA operators are developed and some of their characteristics are investigated in Section 4. Section 5 develops an MAGDM strategy using entropy weights and QNWNA and QNOWNA operators. The applicability of the proposed MAGDM strategy is illustrated in Section 6 with the help of a case study of E-commerce site selection in India. In Section 7, we also conduct a comparison with other aggregating operators. Finally, Section 8 concludes the paper and outlines some future research directions.

2. Some preliminaries

Definition 2.1 A quadripartition neutrosophic number [14] is an element of $[0,1]^{4}$ denoted by $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ , where $b_{\varsigma}$ denotes the TMD, $c_{\varsigma}$ denotes the CMD, $d_{\varsigma}$ denotes the IMD, and $e_{\varsigma}$ denotes the FMD. From the definition, $b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma}\in[0,1]$ .

Definition 2.2 Assume that $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and $\gamma=(b_{\gamma},c_{\gamma},d_{\gamma},e_{\gamma})$ be two QNNs. Then the following operations [14] hold:

(1) $\varsigma+\gamma=<b_{\varsigma}+b_{\gamma}-b_{\varsigma}.b_{\gamma},c_{% \varsigma}+c_{\gamma}-c_{\varsigma}.c_{\gamma},d_{\varsigma}.d_{\gamma},e_{% \varsigma}.e_{\gamma}>$ (2) $\varsigma.\gamma=<b_{\varsigma}.b_{\gamma},c_{\varsigma}.c_{\gamma},d_{% \varsigma}+d_{\gamma}-d_{\varsigma}.d_{\gamma},e_{\varsigma}+e_{\gamma}-e_{% \varsigma}.e_{\gamma}>$

Definition 2.3 The score function [14] of a QNN $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ is defined as

$\displaystyle Sc(\varsigma)=(b_{\varsigma}+c_{\varsigma})-(d_{\varsigma}+e_{% \varsigma}),-2\leqslant Sc(\varsigma)\leqslant 2$ (1)

The accuracy functions [14] are defined by

$\displaystyle{Ac_{1}(\varsigma)}=(b_{\varsigma}+c_{\varsigma})+(d_{\varsigma}+% e_{\varsigma}),0\leqslant Ac_{1}(\varsigma)\leqslant 4$ (2) $\displaystyle{Ac_{2}(\varsigma)}=(b_{\varsigma}-c_{\varsigma}),-1\leqslant Ac_% {2}(\varsigma)\leqslant 1$ (3) $\displaystyle{Ac_{3}(\varsigma)}=(d_{\varsigma}-e_{\varsigma}),-1\leqslant Ac_% {3}(\varsigma)\leqslant 1$ (4)

Definition 2.4: Let $\varsigma,\gamma\in\textit{QNN}$ . Then, for any two QNNs $\varsigma$ and $\gamma$ , the idea of comparability [14] in a broader sense is defined as follows:

I. $Sc(\varsigma)>Sc(\gamma)\Rightarrow\varsigma>\gamma$ II. $Sc(\varsigma)<Sc(\gamma)\Rightarrow\varsigma<\gamma$ III. If $Sc(\varsigma)=Sc(\gamma)$ , then

(a) ${Ac_{1}(\varsigma)}<{Ac_{1}(\gamma)}\Rightarrow\varsigma<\gamma$ (b) ${Ac_{1}(\varsigma}={Ac_{1}(\gamma)}$ with ${Ac_{2}(\varsigma)}<{Ac_{2}(\gamma)}\Rightarrow\varsigma<\gamma$ (c) ${Ac_{1}(\varsigma)}={Ac_{1}(\gamma)}$ with ${Ac_{2}(\varsigma)}={Ac_{2}(\gamma)}$ and ${Ac_{3}(\varsigma)}<{Ac_{3}(\gamma)}\Rightarrow\varsigma<\gamma$ (d) ${Ac_{1}(\varsigma)}={Ac_{1}(\gamma)}$ with ${Ac_{2}(\varsigma)}={Ac_{2}(\gamma)}$ and ${Ac_{3}(\varsigma)}={Ac_{3}(\gamma)}\Rightarrow\varsigma=\gamma$ .

Definition 2.5: Let $\{\varsigma_{m}\}$ be a set of QNNs where $\varsigma_{m}=(b_{\varsigma_{m}},c_{\varsigma_{m}},d_{\varsigma_{m}},e_{% \varsigma_{m}})(m=1,2,\ldots,s)$ . Then quadripartitioned neutrosophic weighted arithmetic aggregation (QNWAA) operator [14] and quadripartitioned neutrosophic weighted geometric aggregation (QNWGA) operator [14] are defined as

$\displaystyle\textit{QNWAA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})$ (5) $\displaystyle=\left\langle 1-\prod_{m=1}^{s}(1-b_{\varsigma_{m}})^{\nu_{m}},1-% \prod_{m=1}^{s}(1-c_{\varsigma_{m}})^{\nu_{m}},\prod_{m=1}^{s}(d_{\varsigma_{m% }})^{\nu_{m}},\prod_{m=1}^{s}(e_{\varsigma_{m}})^{\nu_{m}}\right\rangle$ $\displaystyle\textit{QNWGA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})$ (6) $\displaystyle=\left\langle\prod_{m=1}^{s}(b_{\varsigma_{m}})^{\nu_{m}},\prod_{% m=1}^{s}(c_{\varsigma_{m}})^{\nu_{m}},1-\prod_{m=1}^{s}(1-d_{\varsigma_{m}})^{% \nu_{m}},1-\prod_{m=1}^{s}(1-e_{\varsigma_{m}})^{\nu_{m}}\right\rangle$

where $\nu_{m}$ is the weight of $\varsigma_{m}$ .

Definition 2.6: The QSVNS $\varsigma_{m}=(b_{\varsigma_{m}},c_{\varsigma_{m}},d_{\varsigma_{m}},e_{% \varsigma_{m}})(m=1,2,\ldots,s)$ has an entropy measure [17] given by

$\displaystyle N_{s}(\varsigma_{m})=1-{\displaystyle\frac{1}{s}}\sum_{m=1}^{s}(% b_{\varsigma_{m}}(x_{m})+e_{\varsigma_{m}}(x_{m})).|c_{\varsigma_{m}}(x_{m})-d% _{\varsigma_{m}}(x_{m})|$ (7)

for all $x_{m}\in X$ .

3. Neutrality operational law

Definition 3.1 Let $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and $\gamma=(b_{\gamma},c_{\gamma},d_{\gamma},e_{\gamma})$ be two QNNs. The neutrality operation of $\varsigma$ and $\gamma$ is defined as

$\displaystyle\varsigma+\gamma=\Bigg{(}\sqrt{{\displaystyle\frac{\textit{TCS}^{% 2}(\varsigma,\gamma)}{\textit{TCS}^{2}(\varsigma,\gamma)+\textit{CCS}^{2}(% \varsigma,\gamma)+\textit{ICS}^{2}(\varsigma,\gamma)+\textit{FCS}^{2}(% \varsigma,\gamma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\gamma}^{2}+c_{\gamma}^{2}+d_{% \gamma}^{2}+e_{\gamma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\gamma)}{% \textit{TCS}^{2}(\varsigma,\gamma)+\textit{CCS}^{2}(\varsigma,\gamma)+\textit{% ICS}^{2}(\varsigma,\gamma)+\textit{FCS}^{2}(\varsigma,\gamma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\gamma}^{2}+c_{\gamma}^{2}+d_{% \gamma}^{2}+e_{\gamma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\gamma)}{% \textit{TCS}^{2}(\varsigma,\gamma)+\textit{CCS}^{2}(\varsigma,\gamma)+\textit{% ICS}^{2}(\varsigma,\gamma)+\textit{FCS}^{2}(\varsigma,\gamma)}}}.$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\gamma}^{2}+c_{\gamma}^{2}+d_{% \gamma}^{2}+e_{\gamma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\gamma)}{% \textit{TCS}^{2}(\varsigma,\gamma)+\textit{CCS}^{2}(\varsigma,\gamma)+\textit{% ICS}^{2}(\varsigma,\gamma)+\textit{FCS}^{2}(\varsigma,\gamma)}}}.$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\gamma}^{2}+c_{\gamma}^{2}+d_{% \gamma}^{2}+e_{\gamma}^{2}}\Big{)}}\Bigg{)}$ (8)

where TCS, CCS, ICS and FCS present truth membership, contradiction membership, ignorance membership, and falsity membership coefficient sum of $\varsigma$ and $\gamma$ , respectively, and $\textit{TCS}(\varsigma,\gamma)=\sqrt{b_{\varsigma}^{2}+b_{\gamma}^{2}}$ , $\textit{CCS}(\varsigma,\gamma)=\sqrt{c_{\varsigma}^{2}+c_{\gamma}^{2}}$ , $\textit{ICS}(\varsigma,\gamma)=\sqrt{d_{\varsigma}^{2}+d_{\gamma}^{2}}$ , $\textit{FCS}(\varsigma,\gamma)=\sqrt{e_{\varsigma}^{2}+e_{\gamma}^{2}}$ and $PS(x,y)=\sqrt{1-(1-x^{2})(1-y^{2})}$ .

Proposition 1 For QNN $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and $\lambda>0$ , we have

$\displaystyle\textit{TCS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{TCS}(% \varsigma),\textit{CCS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{CCS}(% \varsigma),$ $\displaystyle\textit{ICS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{ICS}(% \varsigma),\textit{FCS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{FCS}(\varsigma)$

Proof: If we take $\varsigma=\gamma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ . $\textit{TCS}(\varsigma,\varsigma)=\sqrt{b_{\varsigma}^{2}+b_{\varsigma}^{2}}=% \sqrt{2}\textit{TCS}(\varsigma,\varsigma)$ . By the principle of mathematical induction, we can easily write

$\displaystyle\textit{TCS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{CS(}% \varsigma),\textit{CCS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{CCS}(% \varsigma),$ $\displaystyle\textit{ICS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{ICS}(% \varsigma),\textit{FCS}(\lambda.\varsigma)=\sqrt{\lambda}\textit{FCS}(\varsigma)$

For QNN $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and $\lambda$ , we have

$\displaystyle PS\left(\lambda\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}}\right)$ (9) $\displaystyle=PS\left(\lambda\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}},PS((\lambda-1)\sqrt{b_{\varsigma}^{2}+c_{% \varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}})\right)$

Then we have the following propositions:

Proposition 2 For QNN $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and $\lambda>0$ , we have

$\displaystyle PS\left(\lambda\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}}\right)=\sqrt{1-(\pi_{\varsigma}^{2})^{% \lambda}}$ (10)

where

$\displaystyle\pi^{2}=1-(b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+% e_{\varsigma}^{2})/4$

Definition 3.1 For QNN $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and a real number $\lambda>0$ , scalar neutrality operator is defined by

$\displaystyle\lambda.\varsigma=\Bigg{(}\sqrt{{\displaystyle\frac{\textit{TCS}^% {2}(\lambda.\varsigma)}{\textit{TCS}^{2}(\lambda.\varsigma)+\textit{CCS}^{2}(% \lambda.\varsigma)+\textit{ICS}^{2}(\lambda.\varsigma)+\textit{FCS}^{2}(% \lambda.\varsigma)}}}\times{PS\Big{(}\lambda\sqrt{b_{\varsigma}^{2}+c_{% \varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\lambda.\varsigma)}{% \textit{TCS}^{2}(\lambda.\varsigma)+\textit{CCS}^{2}(\lambda.\varsigma)+% \textit{ICS}^{2}(\lambda.\varsigma)+\textit{FCS}^{2}(\lambda.\varsigma)}}}% \times{PS\Big{(}\lambda\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}% ^{2}+e_{\varsigma}^{2}}\Big{)}},$ (11) $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\lambda.\varsigma)}{% \textit{TCS}^{2}(\lambda.\varsigma)+\textit{CCS}^{2}(\lambda.\varsigma)+% \textit{ICS}^{2}(\lambda.\varsigma)+\textit{FCS}^{2}(\lambda.\varsigma)}}}% \times{PS\Big{(}\lambda\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}% ^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\lambda.\varsigma)}{% \textit{TCS}^{2}(\lambda.\varsigma)+\textit{CCS}^{2}(\lambda.\varsigma)+% \textit{ICS}^{2}(\lambda.\varsigma)+\textit{FCS}^{2}(\lambda.\varsigma)}}}% \times{PS\Big{(}\lambda\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}% ^{2}+e_{\varsigma}^{2}}\Big{)}\Bigg{)}}$

Proposition 3 For QNNs $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ and $\gamma=(b_{\gamma},c_{\gamma},d_{\gamma},e_{\gamma})$ if $b_{\varsigma}=e_{\varsigma}$ and $b_{\gamma}=e_{\gamma}$ then $b_{\varsigma+\gamma}=e_{\varsigma+\gamma}$ . Also, if $b_{\varsigma}=d_{\varsigma},b_{\gamma}=d_{\gamma}$ then $b_{\varsigma+\gamma}=d_{\varsigma+\gamma}$ and if $b_{\varsigma}=c_{\varsigma},b_{\gamma}=c_{\gamma}$ then $b_{\varsigma+\gamma}=c_{\varsigma+\gamma}$ .

Proof: If $b_{\varsigma}=e_{\varsigma}$ and $b_{\gamma}=e_{\gamma}$ then by Eq. (3), we obtain

$\displaystyle{\displaystyle\frac{b_{\varsigma+\gamma}}{e_{\varsigma+\gamma}}}=% \sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\gamma)}{\textit{FCS}^{2}% (\varsigma,\gamma)}}}=\sqrt{{\displaystyle\frac{b_{\varsigma}^{2}+b_{\gamma}^{% 2}}{e_{\varsigma}^{2}+e_{\gamma}^{2}}}}=1$

Thus, $b_{\varsigma+\gamma}=e_{\varsigma+\gamma}$ , which completes the proof.

Similarly, $b_{\varsigma}=d_{\varsigma},b_{\gamma}=d_{\gamma}$ then by Eq. (3), we obtain

$\displaystyle{\displaystyle\frac{b_{\varsigma+\gamma}}{d_{\varsigma+\gamma}}}=% \sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\gamma)}{\textit{FCS}^{2}% (\varsigma,\gamma)}}}=\sqrt{{\displaystyle\frac{b_{\varsigma}^{2}+b_{\gamma}^{% 2}}{d_{\varsigma}^{2}+d_{\gamma}^{2}}}}=1$

$b_{\varsigma+\gamma}=d_{\varsigma+\gamma}$ .

Also, for $b_{\varsigma}=c_{\varsigma},b_{\gamma}=c_{\gamma}$ we have $b_{\varsigma+\gamma}=c_{\varsigma+\gamma}$ .

Remark 1 If $b_{\varsigma}=c_{\varsigma}=d_{\varsigma}=e_{\varsigma}$ and $b_{\gamma}=c_{\gamma}=d_{\gamma}=e_{\gamma}$ , then from Eq. (3), we get $b_{\varsigma+\gamma}\neq c_{\varsigma+\gamma}\neq d_{\varsigma+\gamma}\neq e_{% \varsigma+\gamma}$ . Therefore, from the above proposition, we can say that the proposed operation $\varsigma+\gamma$ presents the neutral nature to the decision maker when the TCS, CCS, ICS and FCS are equal. For this reason, we call Eqs (3) and (3) as neutrality operation and scalar neutrality operation, respectively.

Now, based on the definition of TCS, CCS, ICS, FCS and PS function, Eq. (3) can be presented as

$\displaystyle\varsigma+\gamma=\Bigg{(}\sqrt{{\displaystyle\frac{b_{\varsigma}^% {2}+b_{\gamma}^{2}}{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{% \varsigma}^{2}+b_{\gamma}^{2}+c_{\gamma}^{2}+d_{\gamma}^{2}+e_{\gamma}^{2}}}% \times{\Big{(}1-\pi_{\varsigma}^{2}\pi_{\gamma}^{2}\Big{)}}},$ $\displaystyle\sqrt{{\displaystyle\frac{c_{\varsigma}^{2}+c_{\gamma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}+b_{\gamma% }^{2}+c_{\gamma}^{2}+d_{\gamma}^{2}+e_{\gamma}^{2}}}\times{\Big{(}1-\pi_{% \varsigma}^{2}\pi_{\gamma}^{2}\Big{)}}},$ (12) $\displaystyle\sqrt{{\displaystyle\frac{d_{\varsigma}^{2}+d_{\gamma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}+b_{\gamma% }^{2}+c_{\gamma}^{2}+d_{\gamma}^{2}+e_{\gamma}^{2}}}\times{\Big{(}1-\pi_{% \varsigma}^{2}\pi_{\gamma}^{2}\Big{)}}},$ $\displaystyle\sqrt{{\displaystyle\frac{e_{\varsigma}^{2}+e_{\gamma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}+b_{\gamma% }^{2}+c_{\gamma}^{2}+d_{\gamma}^{2}+e_{\gamma}^{2}}}\times{\Big{(}1-\pi_{% \varsigma}^{2}\pi_{\gamma}^{2}\Big{)}}}\Bigg{)}$

In the following, we explain the origin of the scalar neutrality operation as given in definition 3.1.

Now, according to Eq. (3), we obtain

$\displaystyle 2\varsigma=\varsigma+\varsigma=\Bigg{(}\sqrt{{\displaystyle\frac% {\textit{TCS}^{2}(\varsigma,\varsigma)}{\textit{TCS}^{2}(\varsigma,\varsigma)+% \textit{CCS}^{2}(\varsigma,\varsigma)+\textit{ICS}^{2}(\varsigma,\varsigma)+% \textit{FCS}^{2}(\varsigma,\varsigma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_% {\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\varsigma)}{% \textit{TCS}^{2}(\varsigma,\varsigma)+\textit{CCS}^{2}(\varsigma,\varsigma)+% \textit{ICS}^{2}(\varsigma,\varsigma)+\textit{FCS}^{2}(\varsigma,\varsigma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_% {\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\varsigma)}{% \textit{TCS}^{2}(\varsigma,\varsigma)+\textit{CCS}^{2}(\varsigma,\varsigma)+% \textit{ICS}^{2}(\varsigma,\varsigma)+\textit{FCS}^{2}(\varsigma,\varsigma)}}}.$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_% {\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(\varsigma,\varsigma)}{% \textit{TCS}^{2}(\varsigma,\varsigma)+\textit{CCS}^{2}(\varsigma,\varsigma)+% \textit{ICS}^{2}(\varsigma,\varsigma)+\textit{FCS}^{2}(\varsigma,\varsigma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{% \varsigma}^{2}+e_{\varsigma}^{2}},\sqrt{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_% {\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}}\Bigg{)}$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{b_{\varsigma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\times{% \Big{(}1-(\pi_{\varsigma}^{2})^{2}\Big{)}}},\sqrt{{\displaystyle\frac{c_{% \varsigma}^{2}}{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{% \varsigma}^{2}}}\times{\Big{(}1-(\pi_{\varsigma}^{2})^{2}\Big{)}}},$ $\displaystyle\sqrt{{\displaystyle\frac{d_{\varsigma}^{2}}{b_{\varsigma}^{2}+c_% {\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\times{\Big{(}1-(\pi_{% \varsigma}^{2})^{2}\Big{)}}},\sqrt{{\displaystyle\frac{e_{\varsigma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\times{% \Big{(}1-(\pi_{\varsigma}^{2})^{2}\Big{)}}}\Bigg{)}$

Similarly,

$\displaystyle 3\varsigma=2\varsigma+\varsigma=\Bigg{(}\sqrt{{\displaystyle% \frac{\textit{TCS}^{2}(2\varsigma,\varsigma)}{\textit{TCS}^{2}(2\varsigma,% \varsigma)+\textit{CCS}^{2}(2\varsigma,\varsigma)+\textit{ICS}^{2}(2\varsigma,% \varsigma)+\textit{FCS}^{2}(2\varsigma,\varsigma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{1-(\pi_{\varsigma}^{2})^{2}},\sqrt{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(2\varsigma,\varsigma)}% {\textit{TCS}^{2}(2\varsigma,\varsigma)+\textit{CCS}^{2}(2\varsigma,\varsigma)% +\textit{ICS}^{2}(2\varsigma,\varsigma)+\textit{FCS}^{2}(2\varsigma,\varsigma)% }}}$ $\displaystyle\times{PS\Big{(}\sqrt{1-(\pi_{\varsigma}^{2})^{2}},\sqrt{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{TCS^{2}(2\varsigma,\varsigma)}{\textit{% TCS}^{2}(2\varsigma,\varsigma)+\textit{CCS}^{2}(2\varsigma,\varsigma)+\textit{% ICS}^{2}(2\varsigma,\varsigma)+\textit{FCS}^{2}(2\varsigma,\varsigma)}}}$ $\displaystyle\times{PS\Big{(}\sqrt{1-(\pi_{\varsigma}^{2})^{2}},\sqrt{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\textit{TCS}^{2}(2\varsigma,\varsigma)}% {\textit{TCS}^{2}(2\varsigma,\varsigma)+\textit{CCS}^{2}(2\varsigma,\varsigma)% +\textit{ICS}^{2}(2\varsigma,\varsigma)+\textit{FCS}^{2}(2\varsigma,\varsigma)% }}}.$ $\displaystyle\times{PS\Big{(}\sqrt{1-(\pi_{\varsigma}^{2})^{2}},\sqrt{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}\Big{)}}% \Bigg{)}$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{2b_{\varsigma}^{2}+b_{% \varsigma}^{2}}{3(b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{% \varsigma}^{2})}}\times{\Big{(}1-(\pi_{\varsigma}^{2})^{2}}.\pi_{\varsigma}^{2% }\Big{)}},\sqrt{{\displaystyle\frac{2c_{\varsigma}^{2}+c_{\varsigma}^{2}}{3(b_% {\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2})}}\times% {\Big{(}1-(\pi_{\varsigma}^{2})^{2}.\pi_{\varsigma}^{2}\Big{)}}},$ $\displaystyle\sqrt{{\displaystyle\frac{2d_{\varsigma}^{2}+d_{\varsigma}^{2}}{3% (b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2})}}% \times{\Big{(}1-(\pi_{\varsigma}^{2})^{2}.\pi_{\varsigma}^{2}\Big{)}}},\sqrt{{% \displaystyle\frac{2e_{\varsigma}^{2}+e_{\varsigma}^{2}}{3(b_{\varsigma}^{2}+c% _{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2})}}\times{\Big{(}1-(\pi_{% \varsigma}^{2})^{2}.\pi_{\varsigma}^{2}\Big{)}}}\Bigg{)}$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{b_{\varsigma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\times{% \Big{(}1-(\pi_{\varsigma}^{2})^{3}\Big{)}}},\sqrt{{\displaystyle\frac{c_{% \varsigma}^{2}}{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{% \varsigma}^{2}}}\times{\Big{(}1-(\pi_{\varsigma}^{2})^{3}\Big{)}}},$ $\displaystyle\sqrt{{\displaystyle\frac{d_{\varsigma}^{2}}{b_{\varsigma}^{2}+c_% {\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\times{\Big{(}1-(\pi_{% \varsigma}^{2})^{3}\Big{)}}},\sqrt{{\displaystyle\frac{e_{\varsigma}^{2}}{b_{% \varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\times{% \Big{(}1-(\pi_{\varsigma}^{2})^{3}\Big{)}}}\Bigg{)}$

By analogy, for any real positive $n$ , we have

$\displaystyle n\varsigma=\Bigg{(}\sqrt{{\displaystyle\frac{b_{\varsigma}^{2}}{% b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\Big{% (}1-(\pi_{\varsigma}^{2})^{n}\Big{)}},\sqrt{{\displaystyle\frac{c_{\varsigma}^% {2}}{b_{\varsigma}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}% \Big{(}1-(\pi_{\varsigma}^{2})^{n}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{d_{\varsigma}^{2}}{b_{\varsigma}^{2}+c_% {\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\Big{(}1-(\pi_{\varsigma}% ^{2})^{n}\Big{)}},\sqrt{{\displaystyle\frac{e_{\varsigma}^{2}}{b_{\varsigma}^{% 2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{\varsigma}^{2}}}\Big{(}1-(\pi_{% \varsigma}^{2})^{n}\Big{)}}\Bigg{)}$

4. Neutrality aggregative operator of QNNs

Definition 4.1 Let $(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})$ be a collection of QNNs. The Quadripartition Neutrosophic Weighted Neutrality Aggregative (QNWNA) operator for QNNs is thus defined as follows:

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})=% \sum_{m=1}^{s}(\mu_{m}\varsigma_{m})$ (13)

Here $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{s})^{T}$ is the weight vector of $\{\varsigma_{m}\},(m=1,2,\ldots,s)$ ; $\mu_{m}\in[0,1]$ and $\sum_{m=1}^{s}{\mu_{m}}=1$ .

Theorem 4.1 For a collection of QNNs, ${\varsigma}_{m}=(b_{{\varsigma}_{m}},c_{{\varsigma}_{m}},d_{{\varsigma}_{m}},e% _{{\varsigma}_{m}})(m=1,2,\ldots,s)$ , we have

Here, $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{s})^{T}$ is the weight of $\{\varsigma_{m}\},(m=1,2,\ldots,s);\mu_{m}\in[0,1]$ and $\sum_{m=1}^{s}{\mu_{m}}=1$ .

Proof: For $\mu_{1}\in\mu$ and $\tilde{b}_{1}\in\textit{QNN}$ , we have

$\displaystyle\textit{QNWNA}(\varsigma_{1})=\mu_{1}\varsigma_{1}=\Bigg{(}\sqrt{% {\displaystyle\frac{\mu_{1}b_{\varsigma_{1}}^{2}}{\mu_{1}(b_{{\varsigma}_{1}}^% {2}+c_{{\varsigma}_{1}}^{2}+d_{{\varsigma}_{1}}^{2}+e_{{\varsigma}_{1}}^{2})}}% \Big{(}1-(\pi_{1}^{2})^{\mu_{1}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\mu_{1}c_{\varsigma_{1}}^{2}}{\mu_{1}(b% _{{\varsigma}_{1}}^{2}+c_{{\varsigma}_{1}}^{2}+d_{{\varsigma}_{1}}^{2}+e_{{% \varsigma}_{1}}^{2})}}\Big{(}1-(\pi_{1}^{2})^{\mu_{1}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\mu_{1}d_{\varsigma_{1}}^{2}}{\mu_{1}(b% _{{\varsigma}_{1}}^{2}+c_{{\varsigma}_{1}}^{2}+d_{{\varsigma}_{1}}^{2}+e_{{% \varsigma}_{1}}^{2})}}\Big{(}1-(\pi_{1}^{2})^{\mu_{1}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\mu_{1}e_{\varsigma_{1}}^{2}}{\mu_{1}(b% _{{\varsigma}_{1}}^{2}+c_{{\varsigma}_{1}}^{2}+d_{{\varsigma}_{1}}^{2}+e_{{% \varsigma}_{1}}^{2})}}\Big{(}1-(\pi_{1}^{2})^{\mu_{1}}\Big{)}}\Bigg{)}$

Thus, the result Eq. (4) is trivially true for $s=1$ .

Suppose that, Eq. (4) holds for $s=h$ . Then we get,

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{h})=% \Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}b_{{\varsigma}_{m}}^{2}% }{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d_{{% \varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_{m}% ^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}c_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}},$ (15) $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}d_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}e_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}\Bigg{)}$

Now, for $s=h+1$ where $h\in N$ , we obtain

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{h},% \varsigma_{h+1})=\Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1% -\prod_{m=1}^{h}(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}c_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}d_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h}\mu_{m}e_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}\Bigg{)}$ $\displaystyle+\Bigg{(}\sqrt{{\displaystyle\frac{\mu_{h+1}b_{\varsigma_{h+1}}^{% 2}}{\mu_{h+1}(b_{{\varsigma}_{h+1}}^{2}+c_{{\varsigma}_{h+1}}^{2}+d_{{% \varsigma}_{h+1}}^{2}+e_{{\varsigma}_{h+1}}^{2})}}\Big{(}1-(\pi_{h+1}^{2})^{% \mu_{h+1}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\mu_{h+1}t_{\varsigma_{h+1}}^{2}}{\mu_{% h+1}(t_{\varsigma_{h+1}}^{2}+c_{\varsigma_{h+1}}^{2}+u_{\varsigma_{h+1}}^{2}+f% _{\varsigma_{h+1}}^{2})}}\Big{(}1-(\pi_{h+1}^{2})^{\mu_{h+1}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\mu_{h+1}t_{\varsigma_{h+1}}^{2}}{\mu_{% h+1}(t_{\varsigma_{h+1}}^{2}+c_{\varsigma_{h+1}}^{2}+u_{\varsigma_{h+1}}^{2}+f% _{\varsigma_{h+1}}^{2})}}\Big{(}1-(\pi_{h+1}^{2})^{\mu_{h+1}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\mu_{h+1}t_{\varsigma_{h+1}}^{2}}{\mu_{% h+1}(t_{\varsigma_{h+1}}^{2}+c_{\varsigma_{h+1}}^{2}+u_{\varsigma_{h+1}}^{2}+f% _{\varsigma_{h+1}}^{2})}}\Big{(}1-(\pi_{h+1}^{2})^{\mu_{h+1}}\Big{)}}\Bigg{)}$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{h+1}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{h+1}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1% -\prod_{m=1}^{h+1}(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h+1}\mu_{m}c_{{\varsigma}_{% m}}^{2}}{\sum_{m=1}^{h+1}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{% 2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h+1% }(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h+1}\mu_{m}d_{{\varsigma}_{% m}}^{2}}{\sum_{m=1}^{h}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}% +d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h+1}(% \pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{h+1}\mu_{m}e_{{\varsigma}_{% m}}^{2}}{\sum_{m=1}^{h+1}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{% 2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{h+1% }(\pi_{m}^{2})^{\mu_{m}}\Big{)}}\Bigg{)}$

Therefore, by the principle of mathematical induction, Eq. (4) holds for all $n$ , which completes the proof of the theorem.

Example 4.1 Let $\varsigma_{1}=(0.6,0.4,0.3,0.3)$ , $\varsigma_{2}=(0.7,0.5,0.4,0.2)$ and $\varsigma_{3}=(0.5,0.7,0.5,0.4)$ be three QNNs, and let $\mu=(0.4,0.5,0.3)$ . Then, according to definition 3.1, we obtain: $\pi^{2}_{1}=0.825$ , $\pi^{2}_{2}=0.765$ , $\pi^{2}_{3}=0.7125$ .

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\varsigma_{3})=\Bigg{(% }\sqrt{{\displaystyle\frac{0.4*0.6^{2}+0.5*0.7^{2}+.3*0.5^{2}}{0.4*0.7+0.5*0.4% 7+0.3*0.345}}*(0.2684)},$ $\displaystyle\sqrt{{\displaystyle\frac{0.4*0.4^{2}+0.5*0.5^{2}+0.3*0.7^{2}}{0.% 4*0.7+0.5*0.47+0.3*0.345}}*(0.2684)},$ $\displaystyle\sqrt{{\displaystyle\frac{0.4*0.3^{2}+0.5*0.4^{2}+0.3*0.5^{2}}{0.% 4*0.7+0.5*0.47+0.3*0.345}}*(0.2684)},$ $\displaystyle\sqrt{{\displaystyle\frac{0.4*0.3^{2}+0.5*.2^{2}+0.3*0.4^{2}}{0.4% *0.7+0.5*0.47+0.3*0.345}}*(0.2684)}\Bigg{)}$ $\displaystyle=(0.4487,0.3818,0.2879,0.2124)$

Theorem 4.2 (Idempotency) If $\varsigma_{m}=\varsigma,\forall m=1,2,\ldots,s$ then $\textit{QNWNA}(\varsigma,\varsigma,\ldots,\varsigma)=\varsigma$ .

Proof: We know that

Since $\alpha_{m}=\varsigma\forall m$ and $\sum_{m=1}^{s}{\mu_{m}}=1$ , therefore, the above expression becomes

$\displaystyle\textit{QNWNA}(\varsigma,\varsigma,\ldots,\varsigma)=\Bigg{(}% \sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\varsigma}}^{2}}{\sum_{m=1}% ^{s}\mu_{m}(b_{{\varsigma}}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}+e_{% \varsigma}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{\varsigma}^{2}}% {\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}% +e_{\varsigma}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{\varsigma}^{2}}% {\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}% +e_{\varsigma}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{\varsigma}^{2}}% {\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}}^{2}+c_{\varsigma}^{2}+d_{\varsigma}^{2}% +e_{\varsigma}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}}% \Bigg{)}$ $\displaystyle=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})={\varsigma}$

Hence the theorem.

Theorem 4.3 (Commutativity). Suppose that $(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})$ be a set of QNNs. Then the neutrality operator for QNNs is as follows: $\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})=\textit{QNWNA% }(\varsigma_{s},\varsigma_{s-1},\ldots,\varsigma_{1})$ .

Proof: We have

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},...,\varsigma_{s})=\nu% _{1}\varsigma_{1}+\nu_{2}\varsigma_{2}+\ldots+\nu_{p}\varsigma_{s}$ $\displaystyle=\nu_{s}\varsigma_{s}+\nu_{s-1}\varsigma_{s-1}+\ldots+\nu_{1}% \varsigma_{1}$ (due to the commutativity of QNNs with respect to addition) $\displaystyle=\textit{QNWNA}(\varsigma_{s},\varsigma_{s-1},\ldots,\varsigma_{1})$

Hence the theorem.

Theorem 4.4 (Monotonicity) If ${\varsigma}_{m}=(b_{{\varsigma}_{m}},c_{{\varsigma}_{m}},d_{{\varsigma}_{m}},e% _{{\varsigma}_{m}})(m=1,2,\ldots,s)$ and $\beta_{m}=(b_{\beta_{m}},c_{{\beta}_{m}},d_{{\beta}_{m}},\linebreak e_{{\beta}% _{m}})(m=1,2,\ldots,s)$ and also, ${\varsigma}_{m}\leqslant{\beta}_{m}$ i.e., $b_{{\varsigma}_{m}}\leqslant b_{{\beta}_{m}},c_{{\varsigma}_{m}}\leqslant c_{{% \beta}_{m}},d_{{\varsigma}_{m}}\leqslant d_{{\beta}_{m}},e_{{\varsigma}_{m}}% \leqslant e_{{\beta}_{m}}$ then $\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})\leqslant% \textit{QNWNA}(\beta_{1},\beta_{2},\ldots,\beta_{s})$ .

Proof: Since ${\varsigma}_{m}\leqslant{{\beta}_{m}}(m=1,2,\ldots,s)$ therefore, we have

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})=% \Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\varsigma}_{m}}^{2}% }{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d_{{% \varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}% ^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}\Bigg{)}$ $\displaystyle\textit{QNWNA}(\beta_{1},\beta_{2},\ldots,\beta_{s})=\Bigg{(}% \sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}{\sum_{m=1}% ^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m}}^{2}+e_{{% \beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{{\beta}_{m}}^{2% }}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m% }}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{{\beta}_{m}}^{2% }}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m% }}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{{\beta}_{m}}^{2% }}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m% }}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}}\Bigg{)}$

Case 1: Assume that $\forall m=1,2,\ldots,s,b_{{\varsigma}_{m}}<b_{{\beta}_{m}},c_{{\varsigma}_{m}}% <c_{{\beta}_{m}},d_{{\varsigma}_{m}}<d_{{\beta}_{m}},e_{{\varsigma}_{m}}<e_{{% \beta}_{m}}$ . Then

$\displaystyle b_{{\varsigma}_{m}}<b_{{\beta}_{m}}\Rightarrow{\sum_{m=1}^{s}\mu% _{m}b_{{\varsigma}_{m}}^{2}}<{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}$ $\displaystyle\Rightarrow{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}}$ $\displaystyle<{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}{% \sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m}}^% {2}+e_{{\beta}_{m}}^{2})}}}$ $\displaystyle\Rightarrow{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1% -\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}}$ (16) $\displaystyle<{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}{% \sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m}}^% {2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{% )}}$ $\displaystyle\Rightarrow\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1% -\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}}$ $\displaystyle<\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}}$

Similarly,

$\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}$ (17) $\displaystyle<\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}{\pi_{m}^{2}}{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}$ (18) $\displaystyle<\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}}$ (19) $\displaystyle<\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}}$

From Eqs (4)–(4) we get,

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})<% \textit{QNWNA}(\beta_{1},\beta_{2},\ldots,\beta_{s})$ (20)

Case 2: Suppose $b_{{\varsigma}_{m}}=b_{{\beta}_{m}},c_{{\varsigma}_{m}}=c_{{\beta}_{m}},d_{{% \varsigma}_{m}}=d_{{\beta}_{m}},e_{{\varsigma}_{m}}=e_{{\beta}_{m}},\forall m=% 1,2,\ldots,s$ . Then

$\displaystyle b_{{\varsigma}_{m}}=b_{{\beta}_{m}}\Rightarrow{\sum_{m=1}^{s}\mu% _{m}b_{{\varsigma}_{m}}^{2}}={\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}$ $\displaystyle\Rightarrow{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}}$ $\displaystyle={{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}{% \sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m}}^% {2}+e_{{\beta}_{m}}^{2})}}}$ $\displaystyle\Rightarrow{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1% -\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}}$ (21) $\displaystyle={{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{2}}{% \sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{m}}^% {2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{% )}}$ $\displaystyle\Rightarrow\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{% \varsigma}_{m}}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{% \varsigma}_{m}}^{2}+d_{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1% -\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}\Big{)}}$ $\displaystyle=\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}}$

Similarly,

$\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}$ (22) $\displaystyle=\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}{\pi_{m}^{2}}{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}$ (23) $\displaystyle=\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{{\varsigma}_{m}% }^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\varsigma}_{m}}^{2}+c_{{\varsigma}_{m}}^{2}+d% _{{\varsigma}_{m}}^{2}+e_{{\varsigma}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_% {m}^{2})^{\mu_{m}}\Big{)}}$ (24) $\displaystyle=\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{{\beta}_{m}}^{% 2}}{\sum_{m=1}^{s}\mu_{m}(b_{{\beta}_{m}}^{2}+c_{{\beta}_{m}}^{2}+d_{{\beta}_{% m}}^{2}+e_{{\beta}_{m}}^{2})}}\Big{(}1-\prod_{m=1}^{s}(\pi_{m}^{2})^{\mu_{m}}% \Big{)}}$

From Eqs (4)–(4) we obtain,

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})=% \textit{QNWNA}(\beta_{1},\beta_{2},\ldots,\beta_{s})$ (25)

Finally, from Eqs (20) and (25) we obtain, $\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})\leqslant% \textit{QNWNA}(\beta_{1},\beta_{2},\ldots,\beta_{s})$ .

Hence the proof.

Theorem 4.5 (Boundedness) Let $\{{\varsigma_{m}}\},(m=1,2,\ldots,s)$ , be a set of QNNs, and let ${\varsigma}^{-}=\langle\min_{m}b_{\varsigma_{m}},\linebreak\min_{m}{c_{% \varsigma_{m}}},\max_{m}{d_{\varsigma_{m}}},\max_{m}{e_{\varsigma_{m}}}\rangle$ and ${\varsigma}^{+}=\langle{\max_{m}{b_{\varsigma_{m}}},\max_{m}{c_{\varsigma_{m}}% },\min_{m}{d_{\varsigma_{m}}},\min_{m}{e_{\varsigma_{m}}}}\rangle$ , then ${\varsigma}^{-}\leqslant\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,% \varsigma_{s})\leqslant{\varsigma}^{+}$ .

Proof: From Theorem 4.4, we have

$\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})% \geqslant\textit{QNWNA }(\varsigma_{1}^{-},\varsigma_{2}^{-},\ldots,\varsigma_% {s}^{-})=\varsigma^{-}$ $\displaystyle\textit{QNWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})% \leqslant\textit{QNWNA }(\varsigma_{1}^{+},\varsigma_{2}^{+},\ldots,\varsigma_% {s}^{+})=\varsigma^{+}$

Therefore, we can write

$\displaystyle{\varsigma}^{-}\leqslant\textit{QNWNA}(\varsigma_{1},\varsigma_{2% },\ldots,\varsigma_{s})\leqslant{\varsigma}^{+}$

Hence the theorem.

Definition 4.2 Let $(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})$ be a collection of QNNs. The Quadripartition Neutrosophic Ordered Weighted Neutrality Aggregative (QNOWNA) operator for QNNs is thus defined as follows:

$\displaystyle\textit{QNOWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})% =\sum_{m=1}^{s}{\mu_{m}}\varsigma_{\kappa(m)}$ (26)

Here $\kappa:\{1,2,\ldots,s\}\rightarrow\{1,2,\ldots,s\}$ is a permutation map such that $\varsigma_{\kappa(m-1)}\geqslant\varsigma_{\kappa(m)}$ for $(m=1,2,\ldots,s)$ that is, $\varsigma_{\kappa(m)}$ is the $m$ th largest value of $\varsigma_{m},(m=1,2,\ldots,s)$ and $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{s})^{T}$ is the weight vector of $\{\varsigma_{m}\},(m=1,2,\ldots,s);\mu_{m}\in[0,1]$ and $\sum_{m=1}^{s}{\mu_{m}}=1$ .

Theorem 4.6 For collection of QNNs ${\varsigma}_{m}=(b_{{\varsigma}_{m}},c_{{\varsigma}_{m}},d_{{\varsigma}_{m}},e% _{{\varsigma}_{m}})(m=1,2,\ldots,s)$ , we have

$\displaystyle\textit{QNOWNA}(\varsigma_{1},\varsigma_{2},\ldots,\varsigma_{s})$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}b_{\kappa% (m)}^{2}}{\sum_{m=1}^{s}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(% m)}^{2}+e_{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{s}({\pi_{\kappa(m)}^{2}})^{% \mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}c_{\kappa(m)}^{2}}% {\sum_{m=1}^{s}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(m)}^{2}+e% _{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{s}({\pi_{\kappa(m)}^{2}})^{\mu_{m}}% \Big{)}},$ (27) $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}d_{\kappa(m)}^{2}}% {\sum_{m=1}^{s}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(m)}^{2}+e% _{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{s}({\pi_{\kappa(m)}^{2}})^{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{s}\mu_{m}e_{\kappa(m)}^{2}}% {\sum_{m=1}^{s}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(m)}^{2}+e% _{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{s}({\pi_{\kappa(m)}^{2}})^{\mu_{m}}% \Big{)}}\Bigg{)}$

where, $\mu=(\mu_{1},\mu_{2},\ldots,\mu_{s})^{T}$ is the weight of $\{\varsigma_{m}\},(m=1,2,\ldots,s);\mu_{m}\in[0,1]$ and $\sum_{m=1}^{s}{\mu_{m}}=1$ .

The proof of the theorem is omitted as it is same as Theorem 4.1.

Example 4.2 Let $\varsigma_{1}=(0.6,0.4,0.3,0.3),\varsigma_{2}=(0.7,0.5,0.4,0.2)$ and $\varsigma_{3}=(0.5,0.7,0.5,0.4)$ be three QNNs, and let $\mu_{1}=0.4,\mu_{2}=0.5,\mu_{3}=0.3$ . Then according to definition 2.2 we obtain: $Sc(\varsigma_{1})=0.4,Sc(\varsigma_{2})=0.6,Sc(\varsigma_{3})=0.3$ .

Therefore, $\varsigma_{2}>\varsigma_{1}>\varsigma_{3}$ according to definition 4.2 Then, $\varsigma_{\kappa(1)}=\varsigma_{2},\varsigma_{\kappa(2)}=\varsigma_{1},% \varsigma_{\kappa(3)}=\varsigma_{3},\pi^{2}_{1}=0.765,\pi^{2}_{2}=0.825,\pi^{2% }_{3}=0.7125$

$\displaystyle\textit{QNOWNA}(\varsigma_{1},\varsigma_{2},\varsigma_{3})=\Bigg{% (}\sqrt{{\displaystyle\frac{0.4*0.7^{2}+0.5*0.6^{2}+0.3*.5^{2}}{0.4*0.7+0.5*0.% 47+0.3*0.345}}*(0.2629)},$ $\displaystyle\sqrt{{\displaystyle\frac{0.4*0.5^{2}+0.5*0.4^{2}+0.3*0.7^{2}}{0.% 4*0.7+0.5*0.47+0.3*0.345}}*(0.2629)},$ $\displaystyle\sqrt{{\displaystyle\frac{0.4*0.4^{2}+0.5*0.3^{2}+0.3*0.5^{2}}{0.% 4*0.7+0.5*0.47+0.3*0.345}}*(0.2629)},$ $\displaystyle\sqrt{{\displaystyle\frac{0.4*0.2^{2}+0.5*0.3^{2}+0.3*0.4^{2}}{0.% 4*0.7+0.5*0.47+0.3*0.345}}*(0.2629)}\Bigg{)}$ $\displaystyle=(0.4378,0.3728,0.2796,0.2152)$

Corollary 4.1 The QNOWNA operator has commutativity, idempotent, boundness and monotonicity properties as satisfied by QNOWNA operator given in Theorems 4.2, 4.3, 4.4 and 4.5. The proofs are omitted here for brevity.

5. MAGDM strategy based on QNWNA and QNOWNA operators in the QNN environment

Suppose that $F=\langle{\hat{F}_{1}},{\hat{F}_{2}},\ldots,{\hat{F}_{\lambda}}\rangle$ and $R=\langle{\hat{R}_{1}},{\hat{R_{2}}},\ldots,{\hat{R_{\eta}}}\rangle$ be two sets of $\lambda$ alternatives and $\eta$ attributes, respectively. Assume that the attribute weights are $\nu_{t}\in[0,1],(t=1,2,\ldots,\eta)$ and $\sum_{t=1}^{\eta}\nu_{t}=1$ . Assume that $N=(N^{1},N^{2},\ldots,N^{n}$ ) be the set of $n$ decision makers and $\gamma=(\gamma_{1},\gamma_{2},\ldots,\gamma_{n})$ be the weights of the decision makers and $\sum_{m=1}^{n}\gamma_{m}=1$ . The MAGDM strategy based on QNWNA and QNOWNA operators is developed through the steps given below:

Step 1: Formulate a decision-making matrices.

Let $A=(a_{st})_{\lambda\times\eta}$ be a decision matrix. The details of the alternative $\hat{F_{\lambda}}$ w.r.t. the attribute $\hat{R_{\eta}}$ are included. Then A is defined as follows:

$\displaystyle A=(a_{st})_{\lambda\times\eta}=\left(\begin{array}[]{cccc}a_{11}% &a_{12}&\ldots&a_{1\eta}\\ a_{21}&a_{22}&\ldots&a_{2\eta}\\ \ldots&\ldots&\ldots&\ldots\\ a_{\lambda 1}&a_{\lambda 2}&\ldots&a_{\lambda\eta}\end{array}\right)$ (28)

Step 2: Standardize the decision-making matrices.

Suppose that, in the neutrosophic decision matrices Eq. (28), $a_{st}=\langle b_{a_{st}},c_{a_{st}},d_{a_{st}},e_{a_{st}}\rangle(s=1,2,% \linebreak\ldots,\lambda)(t=1,2,\ldots,\eta)$ is the rating value of alternative $F_{\lambda}$ with regard to attribute $R_{\eta}$ such that

$\displaystyle 0\leqslant{b_{a_{st}}}\leqslant 1,0\leqslant{c_{a_{st}}}% \leqslant 1,0\leqslant{d_{a_{st}}}\leqslant 1,0\leqslant{e_{a_{st}}}\leqslant 1$

The DM $A=(a_{st})_{\lambda\times\eta}$ is standardized to eliminate the effects of various physical measurements. To obtain the standardized DM $A^{*}=(a_{st}^{*})_{\lambda\times\eta}$ , the entry $a_{st}^{*}$ in the matrix $A^{*}$ is to be considered as given in the following:

(i) For benefit criterion,

$\displaystyle a_{st}=\langle b_{a_{st}},c_{a_{st}},d_{a_{st}},e_{a_{st}}% \rangle(s=1,2,\ldots,\lambda)(t=1,2,\ldots,\eta)$ (29)

(ii) For cost criterion,

$\displaystyle a_{st}=\langle e_{a_{st}},d_{a_{st}},c_{a_{st}},b_{a_{st}}% \rangle(s=1,2,\ldots,\lambda)(t=1,2,\ldots,\eta)$ (30)

The following is the standardised A:

$\displaystyle A^{*}=(a_{st}^{*})_{\lambda\times\eta}=\left(\begin{array}[]{% cccc}a_{11}^{*}&a_{12}^{*}&\ldots&a_{1\eta}^{*}\\ a_{21}^{*}&a_{22}^{*}&\ldots&a_{2\eta}^{*}\\ \ldots&\ldots&\ldots&\ldots\\ a_{\lambda 1}^{*}&a_{\lambda 2}^{*}&\ldots&a_{\lambda\eta}^{*}\end{array}\right)$ (31)

Step 3: To obtain aggregate decision matrices, we use QNWNA operator which is presented below:

$\displaystyle\textit{QNWNA}(a_{1t}^{*},a_{2t}^{*},\ldots,a_{st}^{*})=\sum_{m=1% }^{s}(\mu_{m}a^{*}_{mt})$ (32)

where $\mu_{m}$ is the weight of the decision maker.

Step 4: The entropy measure for QSVN is

$\displaystyle N_{t}=1-{\displaystyle\frac{1}{\lambda}}\sum_{s=1}^{\lambda}(b_{% a_{st}^{*}}+e_{a_{st}^{*}}).|c_{a_{st}^{*}}-d_{a_{st}^{*}}|(t=1,2,\ldots,\eta)$ (33)

for all $a_{st}^{*}$ .

The normalized weights of the attributes are given by

$\displaystyle\nu_{t}={\displaystyle\frac{1-N_{t}}{\sum_{t=1}^{\eta}(1-N_{t})}}% ,(t=1,2,\ldots\eta).$ (34)

Figure 1.

Decision-making structure of the QNN-MAGDM strategy.

Step 5: Collect the overall assessment values $a_{\lambda\eta}^{*}$ into a global value $a_{s}^{*}$ by utilizing either QNWNA operator as:

$\displaystyle a_{s}^{*}=\textit{QNWNA}(a_{s1}^{*},a_{s2}^{*},\ldots,a_{st}^{*})$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}b_{a_{sm}% ^{*}}^{2}}{\sum_{m=1}^{t}\mu_{m}(b_{a_{sm}^{*}}^{2}+c_{a_{sm}^{*}}^{2}+d_{a_{% sm}^{*}}^{2}+e_{a_{sm}^{*}}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{m}^{2}})^{\mu% _{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}c_{a_{sm}^{*}}^{2}% }{\sum_{m=1}^{t}\mu_{m}(b_{a_{sm}^{*}}^{2}+c_{a_{sm}^{*}}^{2}+d_{a_{sm}^{*}}^{% 2}+e_{a_{sm}^{*}}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{m}^{2}})^{\mu_{m}}\Big{% )}},$ (35) $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}d_{a_{sm}^{*}}^{2}% }{\sum_{m=1}^{t}\mu_{m}(b_{a_{sm}^{*}}^{2}+c_{a_{sm}^{*}}^{2}+d_{a_{sm}^{*}}^{% 2}+e_{a_{sm}^{*}}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{m}^{2}})^{\mu_{m}}\Big{% )}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}e_{a_{sm}^{*}}^{2}% }{\sum_{m=1}^{t}\mu_{m}(b_{a_{sm}^{*}}^{2}+c_{a_{sm}^{*}}^{2}+d_{a_{sm}^{*}}^{% 2}+e_{a_{sm}^{*}}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{m}^{2}})^{\mu_{m}}\Big{% )}}\Bigg{)}$

or, the QNOWNA operator

$\displaystyle a_{s}^{*}=\textit{QNOWNA}(a_{s1}^{*},a_{s2}^{*},\ldots,a_{st}^{*})$ $\displaystyle=\Bigg{(}\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}b_{\kappa% (m)}^{2}}{\sum_{m=1}^{t}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(% m)}^{2}+e_{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{\kappa(m)}^{2}})^{% \mu_{m}}\Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}c_{\kappa(m)}^{2}}% {\sum_{m=1}^{t}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(m)}^{2}+e% _{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{\kappa(m)}^{2}})^{\mu_{m}}% \Big{)}},$ (36) $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}d_{\kappa(m)}^{2}}% {\sum_{m=1}^{t}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(m)}^{2}+e% _{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{\kappa(m)}^{2}})^{\mu_{m}}% \Big{)}},$ $\displaystyle\sqrt{{\displaystyle\frac{\sum_{m=1}^{t}\mu_{m}e_{\kappa(m)}^{2}}% {\sum_{m=1}^{t}\mu_{m}(b_{\kappa(m)}^{2}+c_{\kappa(m)}^{2}+d_{\kappa(m)}^{2}+e% _{\kappa(m)}^{2})}}\Big{(}1-\prod_{m=1}^{t}({\pi_{\kappa(m)}^{2}})^{\mu_{m}}% \Big{)}}\Bigg{)}$

Step 6: Determine the final decision matrix score and accuracy value:

$\displaystyle Sc(a_{s}^{*})=(b_{a_{s}^{*}}+c_{a_{s}^{*}})-(d_{a_{s}^{*}}+e_{a_% {s}^{*}})$ (37) $\displaystyle Ac_{1}(a_{s}^{*})=(b_{a_{s}^{*}}+c_{a_{s}^{*}})+(d_{a_{s}^{*}}+e% _{a_{s}^{*}})$ (38) $\displaystyle Ac_{2}(a_{s}^{*})=(b_{a_{s}^{*}}-c_{a_{s}^{*}})$ (39) $\displaystyle Ac_{3}(a_{s}^{*})=(d_{a_{s}^{*}}-e_{a_{s}^{*}})$ (40)

Step 7: Use the score and accuracy values of the alternatives to rank the options.

The above steps are summarised in Fig. 1.

6. Application of the proposed strategy: A case study of selection of E-commerce site alternatives in India

The e-commerce sites are gradually becoming popular in India. During COVID-19 pandemic situation all over the country, these sites were used as home shopping sources. Here we take five e-commerce sectors

Table 2
Decision matrices

Alternative		$\hat{R}_{1}$	$\hat{R}_{2}$	$\hat{R}_{3}$	$\hat{R}_{4}$	$\hat{R}_{5}$	$\hat{R}_{6}$	$\hat{R}_{7}$
$N^{1}$	$\hat{F}_{1}$	$\langle 0.75,0.33,0.36,0.25\rangle$	$\langle 0.75,0.65,0.35,0.25\rangle$	$\langle 0.55,0.6,0.35,0.26\rangle$	$\langle 0.65,0.55,0.29,0.33\rangle$	$\langle 0.78,0.68,0.45,0.26\rangle$	$\langle 0.45,0.36,0.65,0.76\rangle$	$\langle 0.33,0.42,0.55,0.69\rangle$
	$\hat{F}_{2}$	$\langle 0.75,0.460.36,0.25\rangle$	$\langle 0.35,0.46,0.55,0.65\rangle$	$\langle 0.45,0.48,0.36,0.28\rangle$	$\langle 0.52,0.43,0.3,0.28\rangle$	$\langle 0.42,0.44,0.38,0.35\rangle$	$\langle 0.8,0.78,0.35,0.26\rangle$	$\langle 0.65,0.45,0.39,0.72\rangle$
	$\hat{F}_{3}$	$\langle 0.65,0.55,0.35,0.35\rangle$	$\langle 0.25,0.35,0.65,0.75\rangle$	$\langle 0.7,0.65,0.55,0.3\rangle$	$\langle 0.55,0.45,0.65,0.72\rangle$	$\langle 0.85,0.82,0.25,0.22\rangle$	$\langle 0.82,0.76,0.26,0.28\rangle$	$\langle 0.75,0.65,0.36,0.25\rangle$
	$\hat{F}_{4}$	$\langle 0.68,0.36,0.41,0.33\rangle$	$\langle 0.85,0.75,0.35,0.25\rangle$	$\langle 0.35,0.44,0.67,0.88\rangle$	$\langle 0.35,0.45,0.66,0.8\rangle$	$\langle 0.38,0.45,0.68,0.79\rangle$	$\langle 0.38,0.42,0.78,0.83\rangle$	$\langle 0.7,0.64,0.38,0.33\rangle$
	$\hat{F}_{5}$	$\langle 0.57,0.33,0.46,0.4\rangle$	$\langle 0.25,0.33,0.62,0.72\rangle$	$\langle 0.85,0.65,0.33,0.28\rangle$	$\langle 0.4,0.39,0.61,0.65\rangle$	$\langle 0.36,0.32,0.62,0.79\rangle$	$\langle 0.85,0.76,0.36,0.32\rangle$	$\langle 0.55,0.45,0.6,0.48\rangle$
$N^{2}$	$\hat{F}_{1}$	$\langle 0.78,0.32,0.4,0.22\rangle$	$\langle 0.76,0.62,0.32,0.28\rangle$	$\langle 0.6,0.48,0.33,0.22\rangle$	$\langle 0.68,0.56,0.32,0.31\rangle$	$\langle 0.8,0.69,0.39,0.21\rangle$	$\langle 0.41,0.33,0.66,0.71\rangle$	$\langle 0.3,0.41,0.52,0.7\rangle$
	$\hat{F}_{2}$	$\langle 0.72,0.44,0.33,0.2\rangle$	$\langle 0.33,0.48,0.54,0.68\rangle$	$\langle 0.5,0.41,0.29,0.35\rangle$	$\langle 0.58,0.47,0.33,0.25\rangle$	$\langle 0.43,0.48,0.31,0.32\rangle$	$\langle 0.82,0.71,0.31,0.22\rangle$	$\langle 0.55,0.42,0.4,0.75\rangle$
	$\hat{F}_{3}$	$\langle 0.62,0.49,0.33,0.29\rangle$	$\langle 0.22,0.38,0.68,0.8\rangle$	$\langle 0.75,0.59,0.48,0.28\rangle$	$\langle 0.6,0.39,0.61,0.7\rangle$	$\langle 0.81,0.68,0.21,0.22\rangle$	$\langle 0.83,0.71,0.22,0.21\rangle$	$\langle 0.78,0.62,0.33,0.21\rangle$
	$\hat{F}_{4}$	$\langle 0.7,0.32,0.45,0.3\rangle$	$\langle 0.83,0.73,0.29,0.19\rangle$	$\langle 0.33,0.48,0.61,0.85\rangle$	$\langle 0.32,0.43,0.68,0.81\rangle$	$\langle 0.39,0.41,0.69,0.8\rangle$	$\langle 0.31,0.43,0.75,0.81\rangle$	$\langle 0.75,0.59,0.36,0.31\rangle$
	$\hat{F}_{5}$	$\langle 0.55,0.3,0.41,0.38\rangle$	$\langle 0.22,0.36,0.59,0.76\rangle$	$\langle 0.88,0.69,0.3,0.29\rangle$	$\langle 0.42,0.32,0.68,0.62\rangle$	$\langle 0.39,0.31,0.66,0.81\rangle$	$\langle 0.89,0.72,0.31,0.28\rangle$	$\langle 0.51,0.38,0.62,0.49\rangle$
$N^{3}$	$\hat{F}_{1}$	$\langle 0.81,0.31,0.32,0.22\rangle$	$\langle 0.78,0.66,0.38,0.22\rangle$	$\langle 0.5,0.62,0.34,0.28\rangle$	$\langle 0.68,0.51,0.3,0.3\rangle$	$\langle 0.79,0.6,0.47,0.29\rangle$	$\langle 0.3,0.31,0.68,0.79\rangle$	$\langle 0.33,0.42,0.65,0.71\rangle$
	$\hat{F}_{2}$	$\langle 0.73,0.46,0.31,0.24\rangle$	$\langle 0.37,0.41,0.76,0.61\rangle$	$\langle 0.47,0.4,0.364,0.3\rangle$	$\langle 0.57,0.47,0.31,0.27\rangle$	$\langle 0.47,0.4,0.3,0.4\rangle$	$\langle 0.8,0.79,0.3,0.24\rangle$	$\langle 0.62,0.45,0.35,0.72\rangle$
	$\hat{F}_{3}$	$\langle 0.61,0.5,0.31,0.38\rangle$	$\langle 0.23,0.37,0.68,0.71\rangle$	$\langle 0.77,0.61,0.51,0.32\rangle$	$\langle 0.51,0.55,0.61,0.6\rangle$	$\langle 0.85,0.85,0.25,0.22\rangle$	$\langle 0.89,0.76,0.25,0.27\rangle$	$\langle 0.72,0.67,0.37,0.2\rangle$
	$\hat{F}_{4}$	$\langle 0.67,0.38,0.42,0.32\rangle$	$\langle 0.87,0.78,0.31,0.2\rangle$	$\langle 0.38,0.41,0.67,0.78\rangle$	$\langle 0.37,0.41,0.61,0.8\rangle$	$\langle 0.38,0.45,0.68,0.77\rangle$	$\langle 0.39,0.41,0.78,0.85\rangle$	$\langle 0.7,0.64,0.38,0.33\rangle$
	$\hat{F}_{5}$	$\langle 0.59,0.32,0.41,0.4\rangle$	$\langle 0.21,0.3,0.61,0.72\rangle$	$\langle 0.81,0.6,0.31,0.21\rangle$	$\langle 0.42,0.32,0.6,0.68\rangle$	$\langle 0.37,0.32,0.61,0.8\rangle$	$\langle 0.8,0.79,31,0.28\rangle$	$\langle 0.65,0.4,0.58,0.28\rangle$

Table 3

Standardize decision matrices

Alternative		$\hat{R}_{1}$	$\hat{R}_{2}$	$\hat{R}_{3}$	$\hat{R}_{4}$	$\hat{R}_{5}$	$\hat{R}_{6}$	$\hat{R}_{7}$
$N^{1}$	$\hat{F}_{1}$	$\langle 0.75,0.33,0.36,0.25\rangle$	$\langle 0.25,0.35,0.65,0.75\rangle$	$\langle 0.55,0.6,0.35,0.26\rangle$	$\langle 0.65,0.55,0.29,0.33\rangle$	$\langle 0.33,0.29,0.55,0.65\rangle$	$\langle 0.45,0.36,0.65,0.76\rangle$	$\langle 0.33,0.42,0.55,0.69\rangle$
	$\hat{F}_{2}$	$\langle 0.75,0.460.36,0.25\rangle$	$\langle 0.65,0.55,0.46,0.35\rangle$	$\langle 0.45,0.48,0.36,0.28\rangle$	$\langle 0.28,0.3,0.43,0.52\rangle$	$\langle 0.42,0.44,0.38,0.35\rangle$	$\langle 0.8,0.78,0.35,0.26\rangle$	$\langle 0.65,0.45,0.39,0.72\rangle$
	$\hat{F}_{3}$	$\langle 0.65,0.55,0.35,0.35\rangle$	$\langle 0.75,0.65,0.35,0.25\rangle$	$\langle 0.7,0.65,0.55,0.3\rangle$	$\langle 0.72,0.65,0.45,0.55\rangle$	$\langle 0.85,0.82,0.25,0.22\rangle$	$\langle 0.82,0.76,0.26,0.28\rangle$	$\langle 0.75,0.65,0.36,0.25\rangle$
	$\hat{F}_{4}$	$\langle 0.68,0.36,0.41,0.33\rangle$	$\langle 0.25,0.35,0.75,0.85\rangle$	$\langle 0.35,0.44,0.67,0.88\rangle$	$\langle 0.8,0.66,0.45,0.35\rangle$	$\langle 0.38,0.45,0.68,0.79\rangle$	$\langle 0.38,0.42,0.78,0.83\rangle$	$\langle 0.7,0.64,0.38,0.33\rangle$
	$\hat{F}_{5}$	$\langle 0.57,0.33,0.46,0.4\rangle$	$\langle 0.72,0.62,0.33,0.25\rangle$	$\langle 0.85,0.65,0.33,0.28\rangle$	$\langle 0.65,0.61,0.39,0.4\rangle$	$\langle 0.36,0.32,0.62,0.79\rangle$	$\langle 0.85,0.76,0.36,0.32\rangle$	$\langle 0.55,0.45,0.6,0.48\rangle$
$N^{2}$	$\hat{F}_{1}$	$\langle 0.78,0.32,0.4,0.22\rangle$	$\langle 0.28,0.32,0.62,0.76\rangle$	$\langle 0.6,0.48,0.33,0.22\rangle$	$\langle 0.31,0.32,0.56,0.68\rangle$	$\langle 0.8,0.69,0.39,0.21\rangle$	$\langle 0.41,0.33,0.66,0.71\rangle$	$\langle 0.3,0.41,0.52,0.7\rangle$
	$\hat{F}_{2}$	$\langle 0.72,0.44,0.33,0.2\rangle$	$\langle 0.68,0.54,0.48,0.33\rangle$	$\langle 0.5,0.41,0.29,0.35\rangle$	$\langle 0.0.25,0.33,0.47,0.58\rangle$	$\langle 0.43,0.48,0.31,0.32\rangle$	$\langle 0.82,0.71,0.31,0.22\rangle$	$\langle 0.55,0.42,0.4,0.75\rangle$
	$\hat{F}_{3}$	$\langle 0.62,0.49,0.33,0.29\rangle$	$\langle 0.8,0.68,0.38,0.22\rangle$	$\langle 0.75,0.59,0.48,0.28\rangle$	$\langle 0.7,0.61,0.39,0.6\rangle$	$\langle 0.81,0.68,0.21,0.22\rangle$	$\langle 0.83,0.71,0.22,0.21\rangle$	$\langle 0.78,0.62,0.33,0.21\rangle$
	$\hat{F}_{4}$	$\langle 0.7,0.32,0.45,0.3\rangle$	$\langle 0.19,0.29,0.73,0.83\rangle$	$\langle 0.33,0.48,0.61,0.85\rangle$	$\langle 0.81,0.68,0.43,0.32\rangle$	$\langle 0.39,0.41,0.69,0.8\rangle$	$\langle 0.31,0.43,0.75,0.81\rangle$	$\langle 0.75,0.59,0.36,0.31\rangle$
	$\hat{F}_{5}$	$\langle 0.55,0.3,0.41,0.38\rangle$	$\langle 0.76,0.59,0.36,0.22\rangle$	$\langle 0.88,0.69,0.3,0.29\rangle$	$\langle 0.62,0.68,0.32,0.42\rangle$	$\langle 0.39,0.31,0.66,0.81\rangle$	$\langle 0.89,0.72,0.31,0.28\rangle$	$\langle 0.51,0.38,0.62,0.49\rangle$
$N^{3}$	$\hat{F}_{1}$	$\langle 0.81,0.31,0.32,0.22\rangle$	$\langle 0.22,0.38,0.66,0.78\rangle$	$\langle 0.5,0.62,0.34,0.28\rangle$	$\langle 0.3,0.3,0.51,0.68\rangle$	$\langle 0.79,0.6,0.47,0.29\rangle$	$\langle 0.3,0.31,0.68,0.79\rangle$	$\langle 0.33,0.42,0.65,0.71\rangle$
	$\hat{F}_{2}$	$\langle 0.73,0.46,0.31,0.24\rangle$	$\langle 0.61,0.76,0.41,0.37\rangle$	$\langle 0.47,0.4,0.364,0.3\rangle$	$\langle 0.27,0.31,0.47,0.57\rangle$	$\langle 0.47,0.4,0.3,0.4\rangle$	$\langle 0.8,0.79,0.3,0.24\rangle$	$\langle 0.62,0.45,0.35,0.72\rangle$
	$\hat{F}_{3}$	$\langle 0.61,0.5,0.31,0.38\rangle$	$\langle 0.71,0.68,0.37,0.23\rangle$	$\langle 0.77,0.61,0.51,0.32\rangle$	$\langle 0.6,0.61,0.55,0.51\rangle$	$\langle 0.85,0.85,0.25,0.22\rangle$	$\langle 0.89,0.76,0.25,0.27\rangle$	$\langle 0.72,0.67,0.37,0.2\rangle$
	$\hat{F}_{4}$	$\langle 0.67,0.38,0.42,0.32\rangle$	$\langle 0.2,0.31,0.78,0.87\rangle$	$\langle 0.38,0.41,0.67,0.78\rangle$	$\langle 0.8,0.61,0.41,0.37\rangle$	$\langle 0.38,0.45,0.68,0.77\rangle$	$\langle 0.39,0.41,0.78,0.85\rangle$	$\langle 0.7,0.64,0.38,0.33\rangle$
	$\hat{F}_{5}$	$\langle 0.59,0.32,0.41,0.4\rangle$	$\langle 0.72,0.61,0.3,0.21\rangle$	$\langle 0.81,0.6,0.31,0.21\rangle$	$\langle 0.0.68,0.6,0.32,0.42\rangle$	$\langle 0.37,0.32,0.61,0.8\rangle$	$\langle 0.8,0.79,31,0.28\rangle$	$\langle 0.65,0.4,0.58,0.28\rangle$

Table 4

Aggregated decision matrices

	$\hat{F}_{1}$	$\hat{F}_{2}$	$\hat{F}_{3}$	$\hat{F}_{4}$	$\hat{F}_{5}$
$\hat{R}_{1}$	$\langle 0.1511,0.0258,0.0337,0.0133\rangle$	$\langle 0.1344,0.0511,0.0282,0.0131\rangle$	$\langle 0.0987,0.0662,0.0276,0.0282\rangle$	$\langle 0.1175,0.0306,0.046,0.0249\rangle$	$\langle 0.0805,0.0249,0.0458,0.0384\rangle$
$\hat{R}_{2}$	$\langle 0.0374,0.0458,0.0844,0.1067\rangle$	$\langle 0.1066,0.0918,0.0521,0.0302\rangle$	$\langle 0.1448,0.1122,0.0337,0.0136\rangle$	$\langle 0.0147,0.0252,0.1407,0.1796\rangle$	$\langle 0.1355,0.0917,0.0281,0.0131\rangle$
$\hat{R}_{3}$	$\langle 0.0781,0.0786,0.0288,0.0152\rangle$	$\langle 0.0565,0.047,0.0293,0.0247\rangle$	$\langle 0.1362,0.0951,0.0658,0.0221\rangle$	$\langle 0.0306,0.0505,0.1046,0.1781\rangle$	$\langle 0.1819,0.1072,0.0246,0.018\rangle$
$\hat{R}_{4}$	$\langle 0.0248,0.0232,0.0741,0.1121\rangle$	$\langle 0.0177,0.0248,0.0521,0.0776\rangle$	$\langle 0.1169,0.0975,0.0518,0.0788\rangle$	$\langle 0.1616,0.0872,0.0671,0.0295\rangle$	$\langle 0.1043,0.1013,0.0299,0.0427\rangle$
$\hat{R}_{5}$	$\langle 0.1562,0.1106,0.0467,0.0156\rangle$	$\langle 0.0478,0.0366,0.0279,0.031\rangle$	$\langle 0.1746,0.1507,0.0138,0.0121\rangle$	$\langle 0.0369,0.0472,0.117,0.1557\rangle$	$\langle 0.0351,0.025,0.1005,0.1603\rangle$
$\hat{R}_{6}$	$\langle 0.0402,0.0282,0.1095,0.14\rangle$	$\langle 0.1633,0.1427,0.026,0.0144\rangle$	$\langle 0.1774,0.1372,0.0147,0.0158\rangle$	$\langle 0.0318,0.0444,0.1476,0.1711\rangle$	$\langle 0.1825,0.1414,0.0267,0.0217\rangle$
$\hat{R}_{7}$	$\langle 0.0254,0.0433,00.08,0.1223\rangle$	$\langle 0.0913,0.048,0.037,0.134\rangle$	$\langle 0.1424,0.1035,0.0308,0.0124\rangle$	$\langle 0.1298,0.0962,0.0346,0.0259\rangle$	$\langle 0.0789,0.0422,0.091,0.0491\rangle$

and put seven attributes on them; then, we find out which site is the best to buy products for our daily needs. Here we only consider the clothing selection of the following five e-commerce site: (a) Myntra $(\hat{F}_{1})$ , (b) Flipkart $(\hat{F}_{2})$ , (c) Amazon India $(\hat{F}_{3})$ and (d) Nykaa $(\hat{F}_{4})$ and (e) Meesho $(\hat{F}_{5})$ . For the selection of the site, we consider seven attributes: (1) Availability of the product $(\hat{R}_{1})$ , (2) Product price $(\hat{R}_{2})$ , (3) Quality of the product $(\hat{K}_{3})$ , (4) Delivery time $(\hat{R}_{4})$ , (5) Return policy $(\hat{R}_{5})$ , (6) Discount $(\hat{R}_{6})$ and (7) Customer service $(\hat{R}_{7})$ .

Availability of the product $(\hat{R}_{1})$ , quality of the product $(\hat{R}_{3})$ , return policy $(\hat{R}_{5})$ , Discount $(\hat{R}_{6})$ and Customer service $(\hat{R}_{7})$ are benefit attributes, and product price $(\hat{R}_{2})$ and delivery time $(\hat{R}_{4})$ are cost attributes.

We consider three experts $N^{1},N^{2}$ , and $N^{3}$ to evaluate the given alternatives under each attribute according to the QNN environment. Also, 0.35, 0.4, and 0.25 are respectively the assigned weights of the three experts. We construct the following decision matrices and proceed to do the calculations according to the steps outlined in Section 5.

Step 1: The constructed DM is shown in Table 3.

Step 2: Using Eqs (29) and (30), the standardize DM is then obtained as shown in Table 3.

Step 3: Aggregate the standardize matrices into a signal DM using Eq. (32), which is presented in Table 4.

Step 4: Here, we evaluate the entropy weights of the attributes.

Using Eqs (33) and (34), we obtain the weights of the attributes shown in Table 5.

Table 5

Values of the entropy and weights

	$\hat{R}_{1}$	$\hat{R}_{2}$	$\hat{R}_{3}$	$\hat{R}_{4}$	$\hat{R}_{5}$	$\hat{R}_{6}$	$\hat{R}_{7}$
$N_{j}$	0.9972	0.9889	0.9923	0.9934	0.9869	0.9793	0.993
$v_{j}$	0.0413	0.1605	0.1118	0.0952	0.1897	0.2998	0.1016

Step 5: Using the QNWNA and QNOWNA operators in Eqs (5) and (5), we aggregate the DM and present in Table 6.

Table 6

Values of QNWNA and QNOWNA operators

Alternatives	QNWNA	QNOWNA
$\hat{F_{1}}$	$\langle 0.019,0.0198,0.0256,0.0327\rangle$	$\langle 0.014,0.0083,0.1409,0.0179\rangle$
$\hat{F_{2}}$	$\langle 0.0264,0.0362,0.0144,0.0212\rangle$	$\langle 0.0167,0.0103,0.1114,0.014\rangle$
$\hat{F_{3}}$	$\langle 0.0353,0.0396,0.0109,0.0092\rangle$	$\langle 0.0256,0.0177,0.0125,0.0059\rangle$
$\hat{F_{4}}$	$\langle 0.0165,0.0178,0.0381,0.0477\rangle$	$\langle 0.0183,0.011,0.0228,0.0517\rangle$
$\hat{F_{5}}$	$\langle 0.0316,0.0342,0.0198,0.0256\rangle$	$\langle 0.021,0.0134,0.012,0.0112\rangle$

Step 6: The score and accuracy values of the aggregative DM of QNWNA and QNOWNA operators are obtained by using Eqs (37)–(40) as shown in Table 7.

Table 7

The score and accuracy values of the alternatives obtained using the QNWNA and QNOWNA operators

Alternative	Sc. Value	$Ac_{1}$	$Ac_{2}$	$Ac_{3}$	Sc. Value	$Ac_{1}$	$Ac_{2}$	$Ac_{3}$
	QNWNA				QNOWNA
$\hat{F}_{1}$	$-$ 0.0195	0.0971	$-$ 0.0008	$-$ 0.007	$-$ 0.1365	0.1812	0.0057	0.123
$\hat{F}_{2}$	0.0271	0.0982	$-$ 0.0098	$-$ 0.0069	$-$ 0.0985	0.1524	0.0064	0.0974
$\hat{F}_{3}$	0.0547	0.095	$-$ 0.0042	0.0017	0.025	0.0617	0.0079	0.0066
$\hat{F}_{4}$	$-$ 0.0514	0.1202	$-$ 0.0013	$-$ 0.0097	$-$ 0.0452	0.3540	0.0072	$-$ 0.0288
$\hat{F}_{5}$	0.0204	0.1112	$-$ 0.0026	$-$ 0.0057	0.0113	0.0577	0.0076	0.0008

Step 7: The ranking order of the alternatives is determined using the QNWNA operator as follows: $\hat{F_{3}}>\hat{F_{2}}>\hat{F_{5}}>\hat{F_{1}}>\hat{F_{4}}$

Also, the rank of the alternatives is generated using the QNOWNA operator as follows: $\hat{F_{3}}>\hat{F_{5}}>\hat{F_{4}}>\hat{F_{2}}>\hat{F_{1}}$

The above shows that $\hat{F}_{3}$ i.e. selecting the Amazon India for shopping cloth is the most suitable alternative.

Figure 2.

Hierarchical structure of the decision-making problem.

Figure 3.

Graphical presentation of QNWNA, QNOWNA, QNWAA, QNWGA, QSVNDWAA and QSVNDWGA operators.

7. Comparative analysis

To verify the consistency of the proposed strategy with the existing approach [17, 18] presented in the literature under the QNN environment, we make an analysis with some of the AOs; namely QN Weighted Arithmetic Aggregation (QNWAA) [17], QN Weighted Geometric Aggregation (QNWGA) [17], QSVN Dombi Weighted Arithmetic Aggregation (QSVNDWAA) [18], QSVN Dombi Weighted Geometric Aggregation (QSVNDWGA) [18]. The aggregative values of QNWNA, QNWGA, QNWAA, QNWGA, QSVNDWAA, and QSVNDWGA operators are shown in Table 8. Based on these aggregative QNNs $\varsigma=(b_{\varsigma},c_{\varsigma},d_{\varsigma},e_{\varsigma})$ value, we evaluate score value of each of the alternative using $Sc(\varsigma)=(b_{\varsigma}+c_{\varsigma})-(d_{\varsigma}+e_{\varsigma})$ and result is provided in Table 9 corresponding each of the alternative. Based on AOs process, the best alternative $\hat{F_{3}}$ remains the best alternative, while the other alternatives have changed their relevance based on the AOs process. Furthermore, the existing AOs and the suggested ones use different computational processes. A graphical representation of the alternative can be visualized in Fig. 3.

Table 8
Aggregated values of QNWNA, QNWGA, QNWAA, QNWGA, QSVNDWAA and QSVNDWGA operators

	QNWNA	QNOWNA	QNWAA
$\hat{F_{1}}$	$\langle 0.019,0.0198,0.0256,0.0327\rangle$	$\langle 0.014,0.0083,0.1409,0.0179\rangle$	$\langle 0.069,0.0538,0.0765,0.0649\rangle$
$\hat{F_{2}}$	$\langle 0.0264,0.0362,0.0144,0.0212\rangle$	$\langle 0.0167,0.0103,0.1114,0.014\rangle$	$\langle 0.0994,0.0802,0.035,0.0298\rangle$
$\hat{F_{3}}$	$\langle 0.0353,0.0396,0.0109,0.0092\rangle$	$\langle 0.0256,0.0177,0.0125,0.0059\rangle$	$\langle 0.1548,0.1212,0.0255,0.02\rangle$
$\hat{F_{4}}$	$\langle 0.0165,0.0178,0.0381,0.0477\rangle$	$\langle 0.0183,0.011,0.0228,0.0517\rangle$	$\langle 0.057,0.0516,0.0948,0.1034\rangle$
$\hat{F_{5}}$	$\langle 0.0316,0.0342,0.0198,0.0256\rangle$	$\langle 0.021,0.0134,0.012,0.0112\rangle$	$\langle 0.1266,0.0899,0.0475,0.0352\rangle$
	QNWGA	QSVNDWAA	QSVNDWGA
$\hat{F_{1}}$	$\langle 0.0533,0.0453,0.0755,0.089\rangle$	$\langle 0.0705,0.0544,0.0611,0.0356\rangle$	$\langle 00.0444,0.0395,0.0759,0.0904\rangle$
$\hat{F_{2}}$	$\langle 0.0812,0.0659,0.0347,0.0401\rangle$	$\langle 0.1009,0.0814,0.032,0.0238\rangle$	$\langle 0.0621,0.055,0.0347,0.0408\rangle$
$\hat{F_{3}}$	$\langle 0.1525,0.1184,0.0292,0.0218\rangle$	$\langle 0.1551,0.1215,0.0212,0.016\rangle$	$\langle 0.1503,0.1158,0.0293,0.022\rangle$
$\hat{F_{4}}$	$\langle 0.0409,0.0473,0.1134,0.1381\rangle$	$\langle 0.0584,0.0518,0.09,0.0752\rangle$	$\langle 0.0324,0.0438,0.1142,0.1401\rangle$
$\hat{F_{5}}$	$\langle 0.1071,0.0734,0.0489,0.0533\rangle$	$\langle 0.1284,0.091,0.0351,0.0257\rangle$	$\langle 0.0865,0.0571,0.0495,0.055\rangle$

Table 9

The QNWNA, QNOWNA, QNWAA, QNWGA, QSVNDWAA and QSVNDWGA operators are compared in terms of score values and ranking of the alternatives

Operator	Sc. values of the alternatives					Ranking
	$\hat{F_{1}}$	$\hat{F_{2}}$	$\hat{F_{3}}$	$\hat{F_{4}}$	$\hat{F_{5}}$
QNWNA	$-$ 0.0195	0.0271	0.0547	$-$ 0.0514	0.0204	$\hat{F_{3}}>\hat{F_{2}}>\hat{F_{5}}>\hat{F_{1}}>\hat{F_{4}}$
QNOWNA	$-$ 0.1365	$-$ 0.0985	0.025	$-$ 0.0452	0.0113	$\hat{F_{3}}>\hat{F_{5}}>\hat{F_{4}}>\hat{F_{2}}>\hat{F_{1}}$
QNWAA	$-$ 0.0185	0.1149	0.2304	$-$ 0.0896	0.1338	$\hat{F_{3}}>\hat{F_{2}}>\hat{F_{5}}>\hat{F_{1}}>\hat{F_{4}}$
QNWGA	$-$ 0.0658	0.0723	0.22	$-$ 0.1633	0.0783	$\hat{F_{3}}>\hat{F_{5}}>\hat{F_{2}}>\hat{F_{1}}>\hat{F_{4}}$
QSVNDWAA	0.0279	0.1265	0.2394	$-$ 0.055	0.1585	$\hat{F_{3}}>\hat{F_{5}}>\hat{F_{2}}>\hat{F_{1}}>\hat{F_{4}}$
QSVNDWGA	$-$ 0.0824	0.0416	0.2147	$-$ 0.1781	0.0391	$\hat{F_{3}}>\hat{F_{2}}>\hat{F_{5}}>\hat{F_{1}}>\hat{F_{4}}$

The superiority as well as the advantages of the suggested AOs over these existing AOs are demonstrated in the following.

(i)

The QNWNA and QNOWNA operators seem to be more generic than the QNWAA and QNWGA operators. The QNWNA and QNOWNA operators have several advantages, the most important of which is that they present a neutral nature to decision-makers, conforming to their preferences for equal membership and non-membership degrees. This functionality, however, does not satisfy QNWAA and QNWGA operators.

(ii)

The approaches presented in [25, 26, 27, 29] under IFSs, PFSs, q-rung orthopair FSs, complex q-rung orthopair FS, and SVNS are bounded by the area of their MD and NMD. The key benefit of adopting QNNs is that they represent a more generalised version of the above-mentioned set. As a result, these sets are unable to explore data such as (0.8, 0.2, 0.3, 0.7), but QNNs can handle such data.

8. Conclusion

By using the decision-makers attitude characteristics in the analysis under the QNN environment, we have suggested several new weighted averaging aggregation operators. The key benefits of the suggested operators are that they present a neutral nature to the decision-makers in line with their preferences for equal degrees of membership and non-membership. This functionality, however, does not satisfy the numerous current operators. To manage this feature, some additional neutral addition and scalar multiplication operational laws have been proposed for the aggregated QNN data. In this article, we aim to assist with more ambiguities in the structure that represents QNN in the MAGDM problem. This paper has introduced two important operators viz. QNWNA and QNOWNA operators are on NAOs. The fundamental features of these operators are thoroughly examined. The proposed operators have several advantages, the most important of which is that they present a neutral nature to decision-makers. The MAGDM strategy is developed using the QNWNA and QNOWNA operators. The e-commerce site selection is made with the proposed strategy, and the ranking of the alternatives is obtained. Also, the obtained results are compared with the results for QNWAA, QNWGA, QSVNDWAA, and QSVNDWGA operators results. This study’s limitations stem from the fact that these operators have complex mathematical structures and are difficult to compute. QNNs theory can be further explored in the future, and various forms of aggregation operators can be studied. One can also expand the work done in this paper by addressing various types of practical problems in other fields such as uncertain theory [36], pattern recognition [37], cluster analysis [38], selection of renewable energy alternatives [39] etc.

Footnotes

Acknowledgments

The authors would like to express their sincere thanks to the anonymous reviewers for providing helpful comments and suggestions on the earlier version of the manuscript, which have helped a lot in improving the quality of the original manuscript.

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QNN-MAGDM strategy for E-commerce site selection using quadripartition neutrosophic neutrality aggregative operators

Abstract

Keywords

1. Introduction

1.1 Research gap and motivation

Table 1 Literature review for different aggregation operators in neutrosophic and QNN environments

Table 2 Decision matrices

Table 8 Aggregated values of QNWNA, QNWGA, QNWAA, QNWGA, QSVNDWAA and QSVNDWGA operators

Footnotes

Acknowledgments

References

Table 1
Literature review for different aggregation operators in neutrosophic and QNN environments

Table 2
Decision matrices

Table 8
Aggregated values of QNWNA, QNWGA, QNWAA, QNWGA, QSVNDWAA and QSVNDWGA operators