Distance Measures of (m,a,n)-Fuzzy Neutrosophic Sets and Their Applications in Decision Making

Abstract

A neutrosophic set is an important tool for handling vagueness and impreciseness in real-world problems, and distance measures are also exhibited in neutrosophic set theory. The (m,a,n)-fuzzy neutrosophic set is more flexible and efficient than the existing extensions of neutrosophic sets when discussing the distance between multiple objects. This paper invents ten distance measures for comparing (m,a,n)-fuzzy neutrosophic sets. Moreover, the created distance measures are applied in pattern classifications and multi-criteria decisions. Additionally, numerical examples demonstrate these distance measures in practical and scientific applications involving classifying materials and investment problems. The comparative analysis, along with graphical interpretations, further illustrates the effectiveness and superiority of the proposed measures.

1. Introduction

Distance measures are foundational concepts in mathematics and data analysis, playing a vital role in pattern recognition, image processing, and decision-making models. Symmetry [1,2] plays a significant role in defining and understanding distance measures. In the context of distance measures, symmetry ensures that the calculated distance between two points or objects remains the same, irrespective of their order. This property is essential for maintaining consistency in evaluating similarities or dissimilarities among data elements. Together, symmetry and distance measures contribute to the robustness and reliability of various computational techniques, particularly in fuzzy systems, clustering, and classification tasks. Distance measures serve as quantitative methods for determining how dissimilar or far apart two entities are. Distance metrics are essential for evaluating the level of variation between elements, thereby enabling effective identification of different patterns in real-world applications. Although the terms “difference” and “distance” are often used interchangeably, distance specifically represents a type of difference, typically reflecting the level of deviation or irregularity between two items. On the other hand, difference more broadly captures any form of contrast or divergence between objects based on multiple attributes. In essence, distance measures help assess how separate or distinct two objects are. This study seeks to deepen our grasp of this key concept and its applications, offering insights that bridge disciplines and foster innovations and practical outcomes.

1.1. Literature Review

The notion of FSs was invented by Zadeh [3] as a generalization of classical sets that handles vagueness and ambiguity in information. Many applications of FS theory in the fields of science, engineering, medicine, management, and social science have been considered by different authors. After the invention of FSs, many researchers such as Helgason and Jobe [4], Saha and Bandyopadhyay [5], Colliota and his coworkers [6], Dubois and Prade [7], Rosenfeld [8], Adlassnig [9], Gerla and Volpe [10], Liu [11], and others created various fuzzy DMs and applied them to developed methods and algorithms for decision making and object recognition.

In 1986, Atanassov [12] invented IFSs as an extension of FSs by considering the PMD and the NMD, whose sum belongs to [0, 1]. After the invention of IFSs, several generalizations of IFSs such as PFS [13], q-ROFS [14], FFS [15], and (m,n)-FS [16,17] appeared in the literature. Researchers like Szmidt and Kacprzyk [18], Wang and Xin [19], Lee, Pedrycz, and Sohn [20], Nguyen [21], Zeng, Li, and Yin [22], Hussian and Yang [23], Peng and his coworkers [24], Ejegwa and his coworkers [25], Aydın [26], Kirişci [27], Ashraf and his coworkers [28], Peng and Liu [29], Thakur and his coworkers [30,31], Sivdas and John [32], and others have proposed different distance measures between the above-mentioned classes of FSs and presented applications in the areas of image processing, medical diagnosis, pattern recognition, and decision making.

In 1995, Smarandache [33] established a mathematical notion called neutrosophic sets by considering the PMD, IMD, and NMD, whose sum belongs to [0, 3]. In the recent past, many researchers such as Biswas, Pramanik, and Giri [34], Zulqarnain and his coworkers [35], Ali, Hussain, and Yang [36], Xu, and Zhao [37], and others created various DMs for NSs and applied them in decision making. In the recent past, PyNS [38], FNS [39], and q-RONS [40,41] were invented to depict vagueness and impreciseness in real-world problems. Recently, Theresa, Puzhakkara, and Shiny [42], Bhuvaneshwari, Sweety, and Broumi [43], and Princy, Radha, and Broumi [44] created some distance measures for FNSs and applied them in clustering, crop forming, and detecting patterns of infection-induced fertility.

1.2. Motivation

In existing neutrosophic set extensions such as PyFSs, FSSs, and q-ROFSs, a symmetric approach is generally adopted when assigning powers to the Positive Membership Degree (PMD), Indeterminacy Membership Degree (IMD), and Negative Membership Degree (NMD) for any attribute in the universe of discourse. This uniform treatment restricts the ability to differentiate the relative importance of the PMD, IMD, and NMD, which becomes particularly evident in systems like PyNSs, FNSs, and q-RONSs. For example, in the q-RONS model, the sum of the $q^{\mathrm{th}}$ powers of the PMD and NMD must not exceed 1, while the $q^{\mathrm{th}}$ power of the IMD must remain within the interval [0, 1]. Suppose we have a triplet (0.7, 0.4, 0.9) representing the PMD, IMD, and NMD, respectively. If a decision-maker wants to assign different levels of priority—say, a power of 2 to the PMD, 6 to the IMD, and 10 to the NMD—this level of customization is not possible within the rigid structure of q-RONSs, which enforces the same power value for all three components.

To address this limitation, Thakur and Jafari [45] introduced the (m,a,n)-FNSs framework, which enables distinct powers to be assigned to the PMD, IMD, and NMD, offering greater modeling flexibility in uncertain and imprecise environments. One of the main objectives of this research is to explore this newly proposed class—(m,a,n)-FNSs—to formulate a comprehensive family of distance measures. These measures provide a more expressive representation of the PMD, IMD, and NMD compared to conventional q-RONS-based models. The flexibility of choosing different powers allows a more context-sensitive assessment of input data by assigning varying significance to the PMD, IMD, and NMD—an important requirement in multi-criteria decision-making (MCDM) problems. Unlike other generalizations of IFSs in a neutrosophic framework such as PyNSs (where power = 2), FNSs (power = 3), and q-RONSs (power = q), which enforce fixed exponents, the (m,a,n)-FNS model accommodates variable exponentiation for each component. This power function-based flexibility extends the applicability of neutrosophic systems to a broader range of decision-making scenarios. As a result, (m,a,n)-FNSs offer a more inclusive modeling structure, capturing uncertainty more effectively than traditional q-RONSs. In particular, (m,a,n)-FNSs, which utilize triple universes, prove to be more adaptive and accurate than their single-exponent counterparts like m-RONSs, a-RONSs, and n-RONSs when assessing distances among multiple objects. In summary, (m,a,n)-FNSs represent a generalization that not only encompasses q-RONS-based methods as special cases but also provides enhanced capabilities for addressing complex MCDM problems.

1.3. Objectives and Contributions

The objectives of this research work were as follows:

1.

To create an axiomatic definition of an (m,a,n)-fuzzy neutrosophic distance measure and introduce a family of DMs for (m,a,n)-fuzzy neutrosophic sets which generalizes the existing distance measures in orthopair fuzzy sets in a neutrosophic environment.

2.

To show the validity of the proposed (m,a,n)-fuzzy neutrosophic distance measures by establishing theorems, remarks, and examples.

3.

To design an algorithm to solve pattern classification problems with the aid of (m,a,n)-fuzzy neutrosophic distance measures.

4.

To design an algorithm for solving multi-criteria decision-making (MCDM) problems using the TOPSIS approach within the (m,a,n)-fuzzy neutrosophic environment.

5.

To implement the designed algorithms in real-world problems relating to material classification and investment strategies.

This paper builds upon previous work by developing distance measures tailored for (m,a,n)-FNSs. These new measures allow for differentiated treatment of the PMD, IMD, and NMD through distinct power parameters, thereby enhancing the evaluation of input data with varying significance, an especially valuable feature in multi-criteria decision-making problems. By employing diverse power function scales, the proposed distance measures extend the applicability of (m,a,n)-FNSs to more complex and nuanced decision-making scenarios. The main contributions of this research are summarized as follows:

1.

This study introduces distance measures between two (m,a,n)-fuzzy neutrosophic numbers ((m,a,n)-FNs) based on classical metrics such as the Hamming, Euclidean, and Hausdorff distances. Additionally, hybrid and arctangent-based distance formulations are proposed to enhance the precision and adaptability of the framework.

2.

Utilizing the defined distance measures for (m,a,n)-FNSs, a new algorithm is constructed for solving pattern classification problems, enabling effective discrimination between categories under uncertainty and indeterminacy.

3.

The proposed classification algorithm is applied to a material recognition problem. A detailed sensitivity analysis of its parameters is conducted, along with a comparative evaluation in comparison with existing methods to demonstrate its robustness and effectiveness.

4.

The well-known TOPSIS technique is customized to handle multi-criteria decision-making (MCDM) problems within the (m,a,n)-fuzzy neutrosophic environment. This is achieved by employing the newly defined weighted distance measures, thus enhancing decision support under complex conditions.

5.

A comprehensive numerical example is presented to showcase the implementation of the proposed TOPSIS algorithm. The results are then compared with existing methods based on aggregation operators in q-RONSs and (m,n)-FS frameworks to validate the efficiency and consistency of the proposed approach.

The remainder of this paper is structured as follows: Section 1 provides an overview of the study, including the background, motivation, and primary objectives. Section 2 introduces the essential concepts and preliminary definitions required to understand the framework of (m,a,n)-FNSs. Section 3 presents the newly developed distance measures tailored for (m,a,n)-FNSs. It also discusses their mathematical properties and offers a comparative analysis with existing distance measures within the neutrosophic environment. Section 4 introduces a pattern classification algorithm utilizing the proposed (m,a,n)-FNS-based distance measures. This section also includes its application to a material recognition problem, along with a discussion on parameter sensitivity and a comparative performance evaluation. Section 5 proposes a multi-criteria decision-making method based on the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), adapted to the (m,a,n)-FNS environment. An illustrative example from the domain of investment decision making is used to demonstrate the method’s practical applicability. Section 6 concludes the paper by summarizing the key contributions and insights and offering some final remarks regarding future work.

2. Preliminaries

The present section revisits the existing concepts of orthopair fuzzy sets and their associated distance measures within the framework of the neutrosophic environment. Throughout this paper, $\mathbb{S}$ denotes the universe of discourse, and $\mathbb{N}$ refers to the set of all natural numbers. The abbreviations used in this paper are listed in

Table 1. Abbreviations and their descriptions.

Abbreviation	Description	Abbreviation	Description
DM	Distance Measure	SM	Similarity Measure
PMD	Positive Membership Degree	${\mho }_{\mathcal{G}}\left(s\right)$	PMD of s to $\mathcal{G}$
IMD	Indeterminacy Degree	${\varpi }_{\mathcal{G}}\left(s\right)$	IMD of s to $\mathcal{G}$
NMD	Negative Membership Degree	${\Omega }_{\mathcal{G}}\left(s\right)$	NMD of s to $\mathcal{G}$
FS	Fuzzy Set	IFS	Intuitionistic Fuzzy Set
PyFS	Pythagorean Fuzzy Set	FFS	Fermatean Fuzzy Set
q-ROFS	q-Rung Orthopair Fuzzy Set	(m,n)-FS	(m,n)-Fuzzy Set
NS	Neutrosophic Set	PyNS	Pythagorean Neutrosophic Set
FNS	Fermatean Neutrosophic Set	q-RONS	q-Rung Orthopair Neutrosophic Set
(m,a,n)-FNS	(m,a,n)-Fuzzy Neutrosophic Set	(m,a,n)-FNS $\left(\mathbb{S}\right)$	Family of (m,a,n)-FNSs on $\mathbb{S}$
(m,a,n)-FNDM	(m,a,n)-Fuzzy Neutrosophic Distance Measure	(m,a,n)-FNWDM	Fuzzy Neutrosophic Weighted Distance Measure

.

2.1. Orthopair Fuzzy Structures and Their Extensions in a Neutrosophic Environment

Definition 1.

An object of the form

$$\begin{array}{c}\hfill \mathcal{G}=\{<s,{\mho }_{\mathcal{G}}\left(s\right),{\Omega }_{\mathcal{G}}\left(s\right)>:s\in \mathbb{S}\}\end{array}$$

is called the following:

(a) 

An IFS [12] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}\left(s\right)+{\Omega }_{\mathcal{G}}\left(s\right)\le 1$, $\forall$$s\in \mathbb{S}$.

(b) 

A PyFS [13] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^2\left(s\right)+{\Omega }_{\mathcal{G}}^2\left(s\right)\le 1$, $\forall$$s\in \mathbb{S}$.

(c) 

An FFS [15] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^3\left(s\right)+{\Omega }_{\mathcal{G}}^3\left(s\right)\le 1$, $\forall$$s\in \mathbb{S}$.

(d) 

A q-ROFS [14] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^q\left(s\right)+{\Omega }_{\mathcal{G}}^q\left(s\right)\le 1$, $\forall$$s\in \mathbb{S}$ and $q\in \mathbb{N}$.

(e) 

An (m,n)-FS [16,17] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^m\left(s\right)+{\Omega }_{\mathcal{G}}^n\left(s\right)\le 1$, $\forall$$s\in \mathbb{S}$ and $m,n\in \mathbb{N}$.

where ${\mho }_{\mathcal{G}}:\mathbb{S}\to [0,1]$ and ${\Omega }_{\mathcal{G}}:\mathbb{S}\to [0, 1]$.

Definition 2

([33]). An NS $\mathcal{G}$ in $\mathbb{S}$ is a structure

$$\begin{array}{c}\hfill \mathcal{G}=\{<s,{\mho }_{\mathcal{G}}\left(s\right),{\varpi }_{\mathcal{G}}\left(s\right),{\Omega }_{\mathcal{G}}\left(s\right)>:s\in \mathbb{S}\}\end{array}$$

where ${\mho }_{\mathcal{G}}:\mathbb{S}\to {]}^{-}0,1^{+}[$, ${\varpi }_{\mathcal{G}}:\mathbb{S}\to {]}^{-}0,1^{+}[$, and ${\Omega }_{\mathcal{G}}:\mathbb{S}\to {]}^{-}0,1^{+}[$ denotes the PMD, IMD, and NMD of $\mathcal{G}$, which satisfies the condition ${}^{-}0\le {\mho }_{\mathcal{G}}\left(s\right)+{\varpi }_{\mathcal{G}}\left(s\right)+{\Omega }_{\mathcal{G}}\left(s\right)\le 3^{+},\forall s\in \mathbb{S}$.

Definition 3.

An object of the form

$$\begin{array}{c}\hfill \mathcal{G}=\{<s,{\mho }_{\mathcal{G}}\left(s\right),{\varpi }_{\mathcal{G}}\left(s\right),{\Omega }_{\mathcal{G}}\left(s\right)>:s\in \mathbb{S}\}\end{array}$$

is called the following:

(a) 

A PyNS [38] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^2\left(s\right)+{\Omega }_{\mathcal{G}}^2\left(s\right)\le 1$ and $0\le {\varpi }_{\mathcal{G}}^2\left(s\right)\le 1$, $\forall$$s\in \mathbb{S}$.

(b) 

An FNS [39] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^3\left(s\right)+{\Omega }_{\mathcal{G}}^3\left(s\right)\le 1$ and $0\le {\varpi }_{\mathcal{G}}^3\left(s\right)\le 1$ , $\forall$$s\in \mathbb{S}$.

(c) 

A q-RONS [40] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^q\left(s\right)+{\Omega }_{\mathcal{G}}^q\left(s\right)\le 1$, $0\le {\varpi }_{\mathcal{G}}^q\left(s\right)\le 1$, $\forall s\in \mathbb{S}$, and $q\in \mathbb{N}$.

(d) 

An ((m,a,n)-FNS [45] in $\mathbb{S}$ if $0\le {\mho }_{\mathcal{G}}^m\left(s\right)+{\Omega }_{\mathcal{G}}^n\left(s\right)\le 1$, $0\le {\varpi }_{\mathcal{G}}^a\left(s\right)\le 1$, $\forall s\in \mathbb{S}$, and $m,a,n\in \mathbb{N}$.

where ${\mho }_{\mathcal{G}}:\mathbb{S}\to [0, 1]$, ${\varpi }_{\mathcal{G}}:\mathbb{S}\to [0, 1]$, and ${\Omega }_{\mathcal{G}}:\mathbb{S}\to [0, 1]$ denote the PMD, IMD, and NMD of $\mathcal{G}$, respectively.

For simplicity, an (m,a,n)-FNS $\mathcal{G}=\{<s,{\mho }_{\mathcal{G}}\left(s\right),{\varpi }_{\mathcal{G}}\left(s\right),{\Omega }_{\mathcal{G}}\left(s\right)>:s\in \mathbb{S}\}$ in $\mathbb{S}$ is referred to by $({\mho }_{\mathcal{G}},{\varpi }_{\mathcal{G}},{\Omega }_{\mathcal{G}})$.

Proposition 1

([45]). If $\mathcal{G}=\{<s,{\mho }_{\mathcal{G}}\left(s\right),{\varpi }_{\mathcal{G}}\left(s\right),{\Omega }_{\mathcal{G}}\left(s\right)>:s\in \mathbb{S}\}$ is an (m,a,n)-FNS over $\mathbb{S}$, then $0\le {\mho }_{\mathcal{G}}^m\left(s\right)+{\varpi }_{\mathcal{G}}^a\left(s\right)+{\Omega }_{\mathcal{G}}^n\left(s\right)\le 2$, $\forall s\in \mathbb{S}$, and $m,a,n\in \mathbb{N}$.

Remark 1

([45]). The notion of a q-RONS (resp. FNS, PyNS) is a special case of an (m,a,n)-FNS for m=a=n=q (resp. m=a=n=3, m=a=n=2).

Definition 4

([45]). Let $\mathcal{G},{\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, the following apply:

(a) 

${\mathcal{G}}_1\subset {\mathcal{G}}_2\iff {\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_2}$, ${\varpi }_{{\mathcal{G}}_1}\ge {\varpi }_{{\mathcal{G}}_2}$, and ${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_2}$.

(b) 

${\mathcal{G}}_1={\mathcal{G}}_2\iff {\mho }_{{\mathcal{G}}_1}={\mho }_{{\mathcal{G}}_2}$, ${\varpi }_{{\mathcal{G}}_1}={\varpi }_{{\mathcal{G}}_2}$, and ${\Omega }_{{\mathcal{G}}_1}={\Omega }_{{\mathcal{G}}_2}$.

(c) 

${\mathcal{G}}^c=({\Omega }_{\mathcal{G}}^{\frac{n}{m}},1-{\varpi }_{\mathcal{G}},{\mho }_{\mathcal{G}}^{\frac{m}{n}})$.

(d) 

${\mathcal{G}}_1\cup {\mathcal{G}}_2=(max\{{\mho }_{{\mathcal{G}}_1},{\mho }_{{\mathcal{G}}_2}\},max\{{\varpi }_{{\mathcal{G}}_1},{\varpi }_{{\mathcal{G}}_2}\},min\{{\Omega }_{{\mathcal{G}}_1},{\Omega }_{{\mathcal{G}}_2}\})$.

(e) 

${\mathcal{G}}_1\cap {\mathcal{G}}_2=(min\{{\mho }_{{\mathcal{G}}_1},{\mho }_{{\mathcal{G}}_2}\},min\{{\varpi }_{{\mathcal{G}}_1},{\varpi }_{{\mathcal{G}}_2}\},max\{{\Omega }_{{\mathcal{G}}_1},{\Omega }_{{\mathcal{G}}_2}\})$.

2.2. Distance Measures for Orthopair Fuzzy Structures in a Neutrosophic Environment

Definition 5

([46]). Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ be a universe of discourse and ${\mathcal{G}}_1,{\mathcal{G}}_2\in PyNS\left(\mathbb{S}\right)$. Assume that $\omega$ = ${({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ is the weighting vector of the elements $s_i$ (i = 1, 2, …, r), satisfying the conditions ${\sum }_{i=1}^r{\omega }_i=1$, ∀${\omega }_i$∈ [0, 1], and i = 1, 2, …, r.

(i) 

The weighted Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d_1({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3}}\sum _1^r{\omega }_i\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^2\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^2\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^2\left(s_i\right)\end{array}\right)$$

(1)

(ii) 

The weighted Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d_2({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{3}\sum _1^r{\omega }_i}\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^2\left(s_i\right){|}^2\\ +|{\varpi }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^2\left(s_i\right){|}^2\\ +|{\Omega }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^2\left(s_i\right){|}^2\end{array}\right)}$$

(2)

(iii) 

The generalized weighted distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d_p({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[p]{{\displaystyle \frac{1}{3}\sum _1^r{\omega }_i}\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^2\left(s_i\right){|}^p\\ +|{\varpi }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^2\left(s_i\right){|}^p\\ +|{\Omega }_{{\mathcal{G}}_1}^2\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^2\left(s_i\right){|}^p\end{array}\right)}$$

(3)

Definition 6

([47]). Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ be a universe of discourse and ${\mathcal{G}}_1,{\mathcal{G}}_2\in FNS\left(\mathbb{S}\right)$. Assume that $\omega$ = ${({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ is the weighting vector of the elements $s_i$ (i = 1, 2, …, r), satisfying the conditions ${\sum }_{i=1}^r{\omega }_i=1$, ∀${\omega }_i$∈ [0, 1], and i = 1, 2, …, r. Then, the following apply:

(i) 

The weighted Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d_H({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3}}\sum _1^r{\omega }_i\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^3\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^3\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^3\left(s_i\right)\end{array}\right)$$

(4)

(ii) 

The weighted Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d_{E_u}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{3}\sum _1^r{\omega }_i}\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^3\left(s_i\right){|}^2\\ +|{\varpi }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^3\left(s_i\right){|}^2\\ +|{\Omega }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^3\left(s_i\right){|}^2\end{array}\right)}$$

(5)

(iii) 

The generalized weighted distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d_{F_n}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{3}\sum _1^r{\omega }_i}\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^3\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^3\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^3\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^3\left(s_i\right){|}^{\lambda }\end{array}\right)}$$

(6)

3. Distance Measures Between (m,a,n)-Fuzzy Neutrosophic Sets

In this section, several novel distance measures for (m,a,n)-FNSs are introduced. Their fundamental properties are discussed, and a comparative analysis is carried out with existing distance measures for PyNSs and FNSs.

3.1. Definitions and Properties

Definition 7.

Let ${\mathcal{G}}_1,{\mathcal{G}}_2,{\mathcal{G}}_3\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, the DM is a function $d:(m,a,n)-FNS\left(\mathbb{S}\right)\times (m,a,n)-FNS\left(\mathbb{S}\right)\to [0,1]$ that satisfies the following conditions:

(D1) 

$0\le d({\mathcal{G}}_1,{\mathcal{G}}_2)\le 1$.

(D2) 

$d({\mathcal{G}}_1,{\mathcal{G}}_2)\iff {\mathcal{G}}_1={\mathcal{G}}_2$.

(D3) 

$d({\mathcal{G}}_1,{\mathcal{G}}_2)=d({\mathcal{G}}_2,{\mathcal{G}}_1)$.

(D4) 

If ${\mathcal{G}}_1⋐{\mathcal{G}}_2⋐{\mathcal{G}}_3$, then $d({\mathcal{G}}_1,{\mathcal{G}}_2)\le d({\mathcal{G}}_1,{\mathcal{G}}_3)$ and $d({\mathcal{G}}_2,{\mathcal{G}}_3)\le d({\mathcal{G}}_1,{\mathcal{G}}_3)$.

Definition 8.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ be a universe of discourse and ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, the following apply:

(i) 

The Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^H({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)\end{array}\right)$$

(7)

(ii) 

The Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d^E({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{3}}{\sum }_1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}$$

(8)

(iii) 

The generalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d^G({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{3}}{\sum }_1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}$$

(9)

Remark 2.

The measures defined in Equations (7), (8), and (9) are not valid DMs because they do not satisfy all the conditions of Definition 7.

Example 1.

Let $\mathbb{S}=\{s_1,s_2,s_3,s_4\}$ and ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be two (2,1,3)-FNSs over $\left(\mathbb{S}\right)$, where

$$\begin{array}{c}\hfill {\mathcal{G}}_1=\left\{\begin{array}{c}< s_1, 0.3, 0.2, 0.8 >, < s_2, 0.0, 0.0, 1.0 >, < s_3, 0.8, 0.1, 0.5 >, < s_4, 0.0.0.0, 1.0 >\end{array}\right\},\\ \hfill {\mathcal{G}}_2=\left\{\begin{array}{c}< s_1, 0.0, 0.0, 1.0 >, < s_2, 0.8, 0.1, 0.4 >, < s_3, 0.0, 0.0, 1.0 >, < s_4, 0.6, 0.3, 0.6 >\end{array}\right\}.\end{array}$$

Then, from the definitions of $d^H,d^E$, and $d^G$, we obtain that

$$\begin{array}{c}\hfill d^H({\mathcal{G}}_1,{\mathcal{G}}_2)=1.7443,d^E({\mathcal{G}}_1,{\mathcal{G}}_2)=1.0522,and{d}^G({\mathcal{G}}_1,{\mathcal{G}}_2)=1.0522(for\lambda =2).\end{array}$$

It is clear that the Hamming distance, Euclidean distance, and generalized distance do not satisfy condition $\left(D_1\right)$ of Definition 7.

Now, we define the (m,a,n)-FNDMs by considering the PMD, IMD, and NMD of the (m,a,n)-FNSs.

Definition 9.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ and ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, the following apply:

(i) 

The normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$\begin{array}{c}\hfill d^{NH}({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)\end{array}$$

(10)

(ii) 

The normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d^{NE}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}$$

(11)

(iii) 

The generalized normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}$$

(12)

(iv) 

The normalized Hamming–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{NHH}({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)$$

(13)

(v) 

The normalized Euclidean–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{NEH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}$$

(14)

(vi) 

The generalized normalized Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}$$

(15)

(vii) 

The hybrid normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{HNH}({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)}{2}}\end{array}\right)$$

(16)

(viii) 

The hybrid normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{HNE}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}{2}}\end{array}\right)}$$

(17)

(ix) 

The generalized hybrid normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}$$

(18)

(x) 

The tangent inverse distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)$$

(19)

Remark 3.

For any two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ over a universe of discourse $\mathbb{S}=\{s_1,s_2,...s_r.\}$, the following apply:

(i) 

The generalized normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is reduced to the normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =1$ and the normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =2$.

(ii) 

The generalized normalized Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is reduced to the normalized Hamming–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =1$ and the normalized Euclidean–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =2$.

(iii) 

The generalized hybrid normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is reduced to the hybrid normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =1$ and the hybrid normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =2$.

Example 2.

Let $\mathbb{S}=\{s_1,s_2,s_3,s_4\}$ and ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be two (2,3,4)-FNSs over $\mathbb{S}$, where

$$\begin{array}{c}\hfill {\mathcal{G}}_1=\left\{\begin{array}{c}<s_1,0.8,0.7,0.6>,<s_2,0.6,0.4,0.8>,<s_3,0.7,0.8,0.7>,<s_4,0.9.0.5,0,6>\end{array}\right\},\\ \hfill {\mathcal{G}}_2=\left\{\begin{array}{c}<s_1,0.7,0.5,0.8>,<s_2,0.4,0.6,0.4>,<s_3,0.5,0.4,0.6>,<s_4,0.7,0.4,0.3>\end{array}\right\}.\end{array}$$

Then, from Definition 9, we obtain that

$$\begin{array}{cc}\hfill & d^{NH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2237,d^{NE}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2589,\hfill \\ \hfill & d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2728(\lambda =3),d^{NHH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3580,\hfill \\ \hfill & d^{NEH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3636,d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3691(\lambda =3),\hfill \\ \hfill & d^{HNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2908,d^{HNE}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3120,\hfill \\ \hfill & d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3280(\lambda =3),d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2009.\hfill \end{array}$$

Theorem 1.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ and ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)$ is a DM.

Proof. 

We prove that $d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)$ satisfies the axioms (D1)–(D4).

(D1)

Since ${\mho }_{{\mathcal{G}}_1}\ge 0,{\varpi }_{{\mathcal{G}}_1}\ge 0,{\Omega }_{{\mathcal{G}}_1}\ge 0$, and ${\mho }_{{\mathcal{G}}_2}\ge 0,{\varpi }_{{\mathcal{G}}_2}\ge 0,{\Omega }_{{\mathcal{G}}_2}\ge 0$, we have $|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\ge 0$,  $|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\ge 0$, and $|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\ge 0$. It follows that

$$\begin{array}{c}\hfill d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\ge 0.\end{array}$$

Again, since ${\mho }_{{\mathcal{G}}_1}\le 1,{\varpi }_{{\mathcal{G}}_1}\le 1,{\Omega }_{{\mathcal{G}}_1}\le 1$, and ${\mho }_{{\mathcal{G}}_2}\le 1,{\varpi }_{{\mathcal{G}}_2}\le 1,{\Omega }_{{\mathcal{G}}_2}\le 1$, we have $|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\le 1,|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\le 1,and{|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|}^{\lambda }\le 1$.

It follows that

$$\begin{array}{c}\hfill d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\le 1.\end{array}$$

Consequently, $0\le d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)\le 1$.

(D2)

It follows that

$$\begin{array}{cc}\hfill d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)& =\sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\hfill \\ \hfill & =\sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\hfill \\ \hfill & =d^{GN}({\mathcal{G}}_2,{\mathcal{G}}_1).\hfill \end{array}$$

(D3)

For any two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, we have

$$\begin{array}{cc}\hfill {\mathcal{G}}_1={\mathcal{G}}_2& \iff {\mho }_{{\mathcal{G}}_1}\left(s_i\right)={\mho }_{{\mathcal{G}}_2}\left(s_i\right),{\varpi }_{{\mathcal{G}}_1}\left(s_i\right)={\varpi }_{{\mathcal{G}}_2}\left(s_i\right),\mathrm{and}{\Omega }_{{\mathcal{G}}_1}\left(s_i\right)={\Omega }_{{\mathcal{G}}_2}\left(s_i\right).\hfill \\ \hfill & \iff \sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}=0\hfill \\ \hfill & \iff d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.\hfill \end{array}$$

(D4)

Let ${\mathcal{G}}_3=({\mho }_{{\mathcal{G}}_3},{\varpi }_{{\mathcal{G}}_3},{\Omega }_{{\mathcal{G}}_3})$ be an (m,a,n)-FNS such that ${\mathcal{G}}_1⋐{\mathcal{G}}_2⋐{\mathcal{G}}_3$. Then, we have ${\mathcal{G}}_1⋐{\mathcal{G}}_2$, and ${\mathcal{G}}_1⋐{\mathcal{G}}_3$. It follows that ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_2}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_2}$, ${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_2}$, ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_3}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_3}$, and ${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_3}$. Therefore, for $\forall s_i\in \mathbb{S}(i=1\dots r.)$, we have

$$\begin{array}{cc}\hfill & |{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\le {|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right)|}^{\lambda },\hfill \\ & |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\le {|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right)|}^{\lambda },\hfill \\ & |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\le {|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right)|}^{\lambda }.\hfill \end{array}$$

It implies that

$$\begin{array}{c}\hfill \sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\le \sqrt[\lambda ]{{\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}.\end{array}$$

Consequently, $d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)\le d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_3)$.

Similarly, $d^{GN}({\mathcal{G}}_2,{\mathcal{G}}_3)\le d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_3)$.

   □

Theorem 2.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ and ${\mathcal{G}}_1,({\mho }_{{\mathcal{G}}_2}\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{NH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ and $d^{NE}({\mathcal{G}}_1,{\mathcal{G}}_2)$ are DMs.

Proof. 

Follows from Theorem 1 and Remark 3(i).    □

Theorem 3.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ and ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ is a DM.

Proof. 

We prove that $d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ satisfies axioms (D1)–(D4).

(D1)

Since ${\mho }_{{\mathcal{G}}_1}\ge 0,{\varpi }_{{\mathcal{G}}_1}\ge 0,{\Omega }_{{\mathcal{G}}_1}\ge 0$, and $({\mho }_{{\mathcal{G}}_2}\ge 0,{\varpi }_{{\mathcal{G}}_2}\ge 0,{\Omega }_{{\mathcal{G}}_2}\ge 0$, we have $|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\ge 0,|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\ge 0,\mathrm{and}{|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|}^{\lambda }\ge 0$.

It follows that

$$\begin{array}{c}\hfill d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\ge 0.\end{array}$$

Again, since ${\mho }_{{\mathcal{G}}_1}\le 1,{\varpi }_{{\mathcal{G}}_1}\le 1,{\Omega }_{{\mathcal{G}}_1}\le 1$, and $({\mho }_{{\mathcal{G}}_2}\le 1,{\varpi }_{{\mathcal{G}}_2}\le 1,{\Omega }_{{\mathcal{G}}_2}\le 1$, we have $|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\le 1,|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\le 1,and{|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|}^{\lambda }\le 1$.

It follows that

$$\begin{array}{cc}\hfill d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)& =\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\le 1.\hfill \end{array}$$

Consequently, $0\le d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)\le 1$.

(D2)

It follows that

$$\begin{array}{cc}\hfill d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)& =\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\hfill \\ \hfill & =\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\hfill \\ \hfill & =d^{GNH}({\mathcal{G}}_2,{\mathcal{G}}_1).\hfill \end{array}$$

(D3)

For any two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, we have

$$\begin{array}{cc}\hfill {\mathcal{G}}_1={\mathcal{G}}_2& \iff {\mho }_{{\mathcal{G}}_1}\left(s_i\right)={\mho }_{{\mathcal{G}}_2}\left(s_i\right),{\varpi }_{{\mathcal{G}}_1}\left(s_i\right)={\varpi }_{{\mathcal{G}}_2}\left(s_i\right),\mathrm{and}{\Omega }_{{\mathcal{G}}_1}\left(s_i\right)={\Omega }_{{\mathcal{G}}_2}\left(s_i\right)\hfill \\ \hfill & \iff \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}=0\hfill \\ \hfill & \iff d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.\hfill \end{array}$$

(D4)

Let ${\mathcal{G}}_3=({\mho }_{{\mathcal{G}}_3},{\varpi }_{{\mathcal{G}}_3},{\Omega }_{{\mathcal{G}}_3})$ be an (m,a,n)-FNS such that ${\mathcal{G}}_1⋐{\mathcal{G}}_2⋐{\mathcal{G}}_3$. Then, we have ${\mathcal{G}}_1⋐{\mathcal{G}}_2$, and ${\mathcal{G}}_1⋐{\mathcal{G}}_3$. It follows that ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_2}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_2}$,${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_2}$, ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_3}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_3}$, and ${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_3}$. Therefore, $\forall s_i\in \mathbb{S}(i=1\dots r.)$, and we have

$$\begin{array}{cc}\hfill & |{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\le {|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right)|}^{\lambda },\hfill \\ \hfill & |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\le {|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right)|}^{\lambda },\mathrm{and}\hfill \\ & |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\le {|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right)|}^{\lambda },\hfill \end{array}$$

It implies that

$$\begin{array}{c}\hfill \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\le \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}\end{array}$$

Hence, $d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)\le d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_3)$. Similarly, $d^{GNH}({\mathcal{G}}_2,{\mathcal{G}}_3)\le d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_3)$.

   □

Theorem 4.

Let ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{NHH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ and $d^{NEH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ are valid DMs.

Proof. 

Follows from Theorem 3 and Remark 3(ii).    □

Theorem 5.

Let ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)$ is a valid distance measure.

Proof. 

We prove that $d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)$ satisfies the axioms (D1)–(D4).

(D1)

Since $0\le {\mho }_{{\mathcal{G}}_1},{\mho }_{{\mathcal{G}}_2}\le 1$, we have $0\le |{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\le 1$, $\forall ,s_i\in \mathbb{S}$, and i=1…r. Similarly, $0\le |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\le 1$ and $|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\ge 0$. It follows that

$$\begin{array}{c}\hfill 0\le {\displaystyle \frac{|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }+|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }+{|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|}^{\lambda }}{6}}\le {\displaystyle \frac{1}{2}},\end{array}$$

and

$$\begin{array}{c}\hfill 0\le {\displaystyle \frac{|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\bigvee {|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|}^{\lambda }}{2}}\le {\displaystyle \frac{1}{2}}.\end{array}$$

Therefore,

$$\begin{array}{cc}\hfill 0& \le \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}\hfill \\ \hfill & \le {\displaystyle \frac{1}{r}}({\displaystyle \frac{r}{2}}+{\displaystyle \frac{r}{2}})\hfill \\ \hfill & =1.\hfill \end{array}$$

Hence, $0\le d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)\le 1$.

(D2)

It follows that

$$\begin{array}{cc}\hfill d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)& =\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}\hfill \\ \hfill & =\sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}\hfill \\ \hfill & =d^{GHN}({\mathcal{G}}_2,{\mathcal{G}}_1).\hfill \end{array}$$

(D3)

For any two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, we have

$$\begin{array}{cc}\hfill {\mathcal{G}}_1={\mathcal{G}}_2& \iff {\mho }_{{\mathcal{G}}_1}\left(s_i\right)={\mho }_{{\mathcal{G}}_2}\left(s_i\right),{\varpi }_{{\mathcal{G}}_1}\left(s_i\right)={\varpi }_{{\mathcal{G}}_2}\left(s_i\right),\mathrm{and}{\Omega }_{{\mathcal{G}}_1}\left(s_i\right)={\Omega }_{{\mathcal{G}}_2}\left(s_i\right).\hfill \\ \hfill & \iff \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}=0.\hfill \\ \hfill & \iff d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.\hfill \end{array}$$

(D4)

Let ${\mathcal{G}}_3=({\mho }_{{\mathcal{G}}_3},{\varpi }_{{\mathcal{G}}_3},{\Omega }_{{\mathcal{G}}_3})$ be an (m,a,n)-FNS such that ${\mathcal{G}}_1⋐{\mathcal{G}}_2⋐{\mathcal{G}}_3$. Then, we have ${\mathcal{G}}_1⋐{\mathcal{G}}_2$ and ${\mathcal{G}}_1⋐{\mathcal{G}}_3$. It follows that ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_2}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_2}$,${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_2}$ and ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_3}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_3}$,${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_3}$. Therefore, $\forall s_i\in \mathbb{S}(i=1\dots r.)$, and we have

$$\begin{array}{cc}\hfill & |{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\le {|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right)|}^{\lambda },\hfill \\ & |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\le {|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right)|}^{\lambda },\mathrm{and},\hfill \\ & |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\le {|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right)|}^{\lambda }.\hfill \end{array}$$

It implies that

$$\begin{array}{c}\hfill \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}\\ \hfill \le \sqrt[\lambda ]{{\displaystyle \frac{1}{r}}\sum _1^r\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}.\end{array}$$

Consequently, $d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)\le d^{GhN}({\mathcal{G}}_1,{\mathcal{G}}_3)$. Similarly, $d^{GHN}({\mathcal{G}}_2,{\mathcal{G}}_3)\le d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_3)$.

   □

Theorem 6.

Let ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{HNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ and $d^{HNE}({\mathcal{G}}_1,{\mathcal{G}}_2)$ are valid distance measures.

Proof. 

Follows from Theorem 5 and Remark 3(iii).    □

Theorem 7.

Let ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Then, $d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)$ is a valid distance measure.

Proof. 

We prove that $d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)$ satisfies the axioms (D1)–(D4).

(D1)

Since $0\le {\mho }_{{\mathcal{G}}_1}\left(s_i\right),{\mho }_{{\mathcal{G}}_2}\left(s_i\right)\le 1$, for i…r, we have $0\le ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right),ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)\le 1$, and thus $0\le |ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\le 1$. Similarly, we have $0\le |ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\le 1$ and $0\le |ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\le 1$. Therefore, we obtain $0\le d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)\le 1$.

(D2)

It follows that

$$\begin{array}{cc}\hfill d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)& ={\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)|\end{array}\right)\hfill \\ \hfill & =d^{TI}({\mathcal{G}}_2,{\mathcal{G}}_1)\hfill \end{array}$$

(D3)

For any two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, we have

$$\begin{array}{cc}\hfill {\mathcal{G}}_1={\mathcal{G}}_2& \iff {\mho }_{{\mathcal{G}}_1}\left(s_i\right)={\mho }_{{\mathcal{G}}_2}\left(s_i\right),{\varpi }_{{\mathcal{G}}_1}\left(s_i\right)={\varpi }_{{\mathcal{G}}_2}\left(s_i\right),\mathrm{and}{\Omega }_{{\mathcal{G}}_1}\left(s_i\right)={\Omega }_{{\mathcal{G}}_2}\left(s_i\right)\hfill \\ \hfill & \iff {\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)|\end{array}\right)\hfill \\ \hfill & \iff d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.\hfill \end{array}$$

(D4)

Let ${\mathcal{G}}_3=({\mho }_{{\mathcal{G}}_3},{\varpi }_{{\mathcal{G}}_3},{\Omega }_{{\mathcal{G}}_3})$ be an (m,a,n)-FNS such that ${\mathcal{G}}_1⋐{\mathcal{G}}_2⋐{\mathcal{G}}_3$. Then, we have ${\mathcal{G}}_1⋐{\mathcal{G}}_2$ and ${\mathcal{G}}_1⋐{\mathcal{G}}_3$. It follows that ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_2}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_2}$, ${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_2}$, ${\mho }_{{\mathcal{G}}_1}\le {\mho }_{{\mathcal{G}}_3}$, ${\varpi }_{{\mathcal{G}}_1}\le {\varpi }_{{\mathcal{G}}_3}$, and ${\Omega }_{{\mathcal{G}}_1}\ge {\Omega }_{{\mathcal{G}}_3}$. Therefore, $\forall s_i\in \mathbb{S}(i=1\dots r.)$, and we have

$$\begin{array}{cc}\hfill & |ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\le |ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right)|,\hfill \\ \hfill & |ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\le |ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right)|,\mathrm{and}\hfill \\ & |ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\le |ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right)|.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)=& {\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)\hfill \\ \hfill & {\displaystyle \frac{1}{3r}}\sum _1^r\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_3}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_3}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_3}^n\left(s_i\right)|\end{array}\right)\hfill \\ \hfill =d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_3)\end{array}$$

Thus, $d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)\le d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_3)$. Similarly, we have $d^{TI}({\mathcal{G}}_2,{\mathcal{G}}_3)\le d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_3)$.

   □

Now, we define the (m,a,n)-FNWDMs between two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ by considering the weighting vector of the elements in the (m,a,n)-FNSs.

Definition 10.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r.\}$ and ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Assume that $\omega$ = ${({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ is the weighting vector of the elements $s_i$ (i = 1, 2, …, r), satisfying the conditions ${\sum }_{i=1}^r{\omega }_i=1$, ∀${\omega }_i$∈ [0, 1], and i = 1, 2, …, r. Then, the following statements apply:

(i) 

The weighted normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{WNH}({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3}}\sum _1^r{\omega }_i\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)$$

(20)

(ii) 

The weighted normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d^{WNE}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{{\displaystyle \frac{1}{3}}\sum _1^r{\omega }_i\left(\begin{array}{c}{({\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right))}^2\\ +{({\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right))}^2\\ +{({\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right))}^2+\end{array}\right)}$$

(21)

(iii) 

The generalized weighted normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined by the following expression:

$$d^{GWN}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{{\displaystyle \frac{1}{3}}\sum _1^r{\omega }_i\left(\begin{array}{c}{({\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right))}^{\lambda }\\ +{({\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right))}^{\lambda }\\ +{({\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right))}^{\lambda }\end{array}\right)}$$

(22)

(iv) 

The weighted normalized Hamming–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{WNHH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sum _1^r{\omega }_i\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)$$

(23)

(v) 

The weighted normalized Euclidean–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{WNEH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{\sum _1^r{\omega }_i\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2,\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}$$

(24)

(vi) 

The generalized weighted normalized Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{GWNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{\sum _1^r{\omega }_i\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}$$

(25)

(vii) 

The hybrid weighted normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{HWNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sum _1^r{\omega }_i\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)}{2}}\end{array}\right)$$

(26)

(viii) 

The hybrid weighted normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{HWNE}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt{\sum _1^r{\omega }_i\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^2\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^2\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^2\end{array}\right)}{2}}\end{array}\right)}$$

(27)

(ix) 

The generalized hybrid weighted normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{GHWN}({\mathcal{G}}_1,{\mathcal{G}}_2)=\sqrt[\lambda ]{\sum _1^r{\omega }_i\left(\begin{array}{c}{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ +|{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ +|{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{6}}+{\displaystyle \frac{\left(\begin{array}{c}|{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right){|}^{\lambda }\\ \bigvee |{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right){|}^{\lambda }\\ \bigvee |{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right){|}^{\lambda }\end{array}\right)}{2}}\end{array}\right)}$$

(28)

(x) 

The weighted tangent inverse distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is defined as follows:

$$d^{WTI}({\mathcal{G}}_1,{\mathcal{G}}_2)={\displaystyle \frac{1}{3}}\sum _1^r{\omega }_i\left(\begin{array}{c}|ta{n}^{-1}{\mho }_{{\mathcal{G}}_1}^m\left(s_i\right)-ta{n}^{-1}{\mho }_{{\mathcal{G}}_2}^m\left(s_i\right)|\\ +|ta{n}^{-1}{\varpi }_{{\mathcal{G}}_1}^a\left(s_i\right)-ta{n}^{-1}{\varpi }_{{\mathcal{G}}_2}^a\left(s_i\right)|\\ +|ta{n}^{-1}{\Omega }_{{\mathcal{G}}_1}^n\left(s_i\right)-ta{n}^{-1}{\Omega }_{{\mathcal{G}}_2}^n\left(s_i\right)|\end{array}\right)$$

(29)

Remark 4.

For any two (m,a,n)-FNSs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ over a universe of discourse $\mathbb{S}=\{s_1,s_2,\dots s_r\}$, the following apply:

(i) 

The generalized weighted normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is reduced to the weighted normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =1$ and the weighted normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =2$.

(ii) 

The generalized weighted normalized Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is reduced to the weighted normalized Hamming–Hausdorff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =1$ and the weighted normalized Euclidean–Hausdroff distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =2$.

(iii) 

The generalized hybrid weighted normalized distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is reduced to the hybrid weighted normalized Hamming distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =1$ and the hybrid weighted normalized Euclidean distance between ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ for $\lambda =2$.

Remark 5.

When we take the weighting vector ω = ${({\displaystyle \frac{1}{r}},{\displaystyle \frac{1}{r}},\dots ,{\displaystyle \frac{1}{r}})}^T$, then, in the (m,a,n)-FNWDM, the following occur:

(i) 

$d^{WNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{NH}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(ii) 

$d^{WNE}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{NE}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(iii) 

$d^{GWN}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{GN}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(iv) 

$d^{WNHH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{NHH}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(v) 

$d^{WNEH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{NEH}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(vi) 

$d^{GWNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{GNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(vii) 

$d^{HWNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{HNH}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(viii) 

$d^{HWNE}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{HNE}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(ix) 

$d^{GHWN}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{GHN}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

(x) 

$d^{WTI}({\mathcal{G}}_1,{\mathcal{G}}_2)$ reduce to $d^{TI}({\mathcal{G}}_1,{\mathcal{G}}_2)$.

Example 3.

Let ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be two (m,a,n)-FNSs over $\mathbb{S}$, defined in Example 2, and consider the weighting vector ω = ${(0.18,0.25,0.22,0.35)}^T$. Then,

$$\begin{array}{cc}\hfill & d^{WNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2173,d^{WNE}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2530,\hfill \\ \hfill & d^{GWN}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2695(\lambda =3),d^{WNHH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3569,\hfill \\ \hfill & d^{WNEH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3618,d^{GWNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3668(\lambda =3),\hfill \\ \hfill & d^{HWNH}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.2871,d^{HWNE}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3093,\hfill \\ \hfill & d^{GHWN}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.3254(\lambda =3),d^{WTI}({\mathcal{G}}_1,{\mathcal{G}}_2)=0.1937.\hfill \end{array}$$

Theorem 8.

Let $\mathbb{S}=\{s_1,s_2,\dots s_r\}$ and ${\mathcal{G}}_1,{\mathcal{G}}_2\in (m,a,n)-FNS\left(\mathbb{S}\right)$. Assume that $\omega$ = ${({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ is the weighting vector of the elements $s_i$ (i = $1,2,\dots$, r), satisfying the conditions ${\sum }_{i=1}^r{\omega }_i=1$, ∀${\omega }_i\in [0,1]$, and i = $1,2,\dots$, r. Then, the (m,a,n)-FNWDMs proposed in Definition 10 are the valid DMs for the (m,a,n)-FNSs.

Proof. 

This follows by considering a process similar to that used in the corresponding cases of (m,a,n)-FNDMs.    □

3.2. Comparison of the Proposed DMs with the Existing DMs in a Neutrosophic Environment

This section presents a comparative analysis of the proposed (m,a,n)-FNDMs and (m,a,n)-FNWDMs and the existing DMs for PyNSs and FNSs using remarks and numerical examples.

Remark 6.

The weighted DMs $d_1$, $d_2$, and $d_p$ proposed by Rajan and Krishnaswamy [46] for PyNSs are, respectively, the special cases of $d^{WNH}$, $d^{WNE}$, and $d^{GWN}$ for m = a = n = 2.

Remark 7.

The weighted DMs $d_H$, $d_{E_u}$, and $d_{F_n}$ proposed by Saeed and his coworkers [47] for FNSs are, respectively, the special cases of $d^{WNH}$, $d^{WNE}$, and $d^{GWN}$ for m = a = n = 3.

Remark 8.

The following example shows the that the weighted DMs $d_1$, $d_2$, and $d_p$ proposed by Rajan and Krishnaswamy [46] for PyNSs and the weighted DMs $d_H$, $d_{E_u}$, and $d_{F_n}$ proposed by Saeed and his coworkers [47] for FNSs are inconsistent and irrational.

Example 4.

Let $\mathbb{S}=\{s_1,s_2\}$, and (m,a,n)-FNS ${\mathcal{G}}_i$, ${\mathcal{H}}_i$ (i=1..4) be defined as follows:

$$\begin{array}{cc}\hfill {\mathcal{G}}_1=\{(s_1,0.6,0.5,0.4),(s_2,0.4,0.3,0.6)\},& {\mathcal{H}}_1=\{(s_1,0.7,0.6,0.8),(s_2,0.36,0.2,0.8)\}\hfill \\ \hfill {\mathcal{G}}_2=\{(s_1,0.6,0.5,0.4),(s_2,0.4,0.3,0.6)\},& {\mathcal{H}}_2=\{(s_1,0.4,0.35,0.6),(s_2,0.4,0.7,0.8)\}\hfill \\ \hfill {\mathcal{G}}_3=\{(s_1,0.5,0.8,0.6),(s_2,0.7,0.8,0.2)\},& {\mathcal{H}}_3=\{(s_1,0.4,0.3,0.3),(s_2,0.7,0.3,0.2)\}\hfill \\ \hfill {\mathcal{G}}_4=\{(s_1,0.5,0.8,0.6),(s_2,0.7,0.8,0.2)\},& {\mathcal{H}}_4=\{(s_1,0.3,0.4,0.85),(s_2,0.2,0.7,0.3)\}\hfill \end{array}$$

Assume that ${\omega }_1={\omega }_2={\displaystyle \frac{1}{2}}$ is the weight for the elements of $\mathbb{S}$.

Table 2. Comparison of (m,a,n)-FNWDMs.

Weighted DMs	Environment	$({\mathcal{G}}_1,{\mathcal{H}}_1)$	$({\mathcal{G}}_2,{\mathcal{H}}_2)$	$({\mathcal{G}}_3,{\mathcal{H}}_3)$	$({\mathcal{G}}_4,{\mathcal{H}}_4)$
$d^{NH}$	(2,3,4)-FNS	0.1584	0.1664	0.1969	0.2731
$d^{NE}$	(2,3,4)-FNS	0.2051	0.1982	0.2867	0.2051
$d^{GN}$	(2,3,4)-FNS	0.2387	0.2192	0.3375	0.3469
$d^{NHH}$	(2,3,4)-FNS	0.3968	0.3048	0.4850	0.4490
$d^{NEH}$	(2,3,4)-FNS	0.3360	0.2644	0.4850	0.4490
$d^{GNH}$	(2,3,4)-FNS	0.3400	0.2704	0.4850	0.4490
$d^{HNH}$	(2,3,4)-FNS	0.2439	0.2108	0.3410	0.3600
$d^{HNE}$	(2,3,4)-FNS	0.2784	0.2337	0.3984	0.3896
$d^{GHN}$	(2,3,4)-FNS	0.2979	0.2475	0.4241	0.3935
$d^{TI}$	(2,3,4)-FNS	0.1450	0.1558	0.1833	0.2469
$d_H$ [47]	FNS	0.1664	0.1664	0.2033	0.2445
$d_{E_u}$ [47]	FNS	0.2285	0.2002	0.2915	0.2915
$d_{F_n}$ [47]	FNS	0.2704	0.2213	0.3397	0.3199
$d_1$ [46]	PyNS	uncertain	0.2013	0.2433	0.2754
$d_2$ [46]	PyNS	uncertain	0.2362	0.3381	0.2385
$d_p$ [46]	PyNS	uncertain	0.2588	0.3890	0.3499

displays the established (m,a,n)-FNWDMs and the existing weighted DMs for PyNSs and FNSs. In the considered cases, we observe that ${\mathcal{G}}_1={\mathcal{G}}_2$, ${\mathcal{H}}_1\ne {\mathcal{H}}_2$, ${\mathcal{G}}_3={\mathcal{G}}_4$, and ${\mathcal{H}}_1\ne {\mathcal{H}}_2$. Upon examining Table 2, it becomes apparent that the results produced by the Fermatean neutrosophic weighted Hamming distance $d_H$ for the pairs (${\mathcal{G}}_1,4{\mathcal{H}}_1$) and (${\mathcal{G}}_2$${\mathcal{H}}_2$) is identical, which is contradictory and logically inconsistent given the differences in ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$. A similar anomaly is observed in the case of the Fermatean neutrosophic weighted Euclidean distance $d_{E_u}$, where the distance values for the pairs (${\mathcal{G}}_3,{\mathcal{H}}_3$) and (${\mathcal{G}}_4,{\mathcal{H}}_4$) are also the same, despite the dissimilarity between ${\mathcal{H}}_3$ and ${\mathcal{H}}_3$. Furthermore, the distance measures $d_1$, $d_2$, and $d_p$ fail to compute valid results for the pair (${\mathcal{G}}_1$ ,${\mathcal{H}}_1$) because the sum of the squares of the Positive Membership Degree (PMD) and the Negative Membership Degree (NMD) at point $s_1$ of ${\mathcal{H}}_1$ exceeds 1, violating the required constraints. These inconsistencies highlight the limitations of existing distance measures for orthopair fuzzy sets within the neutrosophic environment.

Remark 9.

The (m,a,n)-FNDMs proposed in Definition 9 will be reduced to DMs for q-RONSs if we take m = a = n = q.

Remark 10.

The (m,a,n)-FNWDMs proposed in Definition 10 will be reduced to weighted DMs for q-RONSs if we take m = a = n = q.

Remarks 6–10 and Examples 2–4 reveal that the proposed distance measures for (m,a,n)-FNSs exhibit enhanced sensitivity and discriminatory power, enabling more reliable and meaningful differentiation among data points. Moreover, the developed (m,a,n)-fuzzy neutrosophic-based distance measures (FNDMs) and weighted distance measures (FNWDMs) are versatile enough to be applied in decision-making problems involving PyNSs, FNSs, and q-RONSs. In contrast, the distance measures formulated under these existing frameworks are generally not applicable to the broader structure of (m,a,n)-FNSs. It is worth noting that the effectiveness of the proposed methods may depend on the selection of the parameters m, a, and n, highlighting the need for a comprehensive analysis of their robustness across varying parameter values. Future research may explore the practical implementation of this framework in real-world decision-making scenarios to assess its applicability and effectiveness.

4. (m,a,n)-FNDMs-Based Approach for Pattern Recognition

In pattern recognition, determining the degree of similarity or difference between data objects plays a pivotal role. Classical approaches often struggle to cope with ambiguous or imprecise data. To address this, (m,a,n)-FNDMs have been proposed as flexible alternatives that incorporate PMD, IMD, and NMD information. These measures are particularly advantageous in vague classification scenarios where patterns may not have clearly defined boundaries.

4.1. Problem Description and Solution Procedure

Let $\mathbb{V}=\{v_1,v_2,..,v_p\}$ be a universe of discourse and

$$\begin{array}{c}\hfill \mathbb{P}=\{{\mathcal{P}}_1,{\mathcal{P}}_2,..,{\mathcal{P}}_k\}\end{array}$$

be k patterns represented by (m,a,n)-FNSs denoted as

$$\begin{array}{c}\hfill {\mathcal{P}}_j=\{<v_i,{\mho }_{{\mathcal{P}}_j}\left(v_i\right),{\varpi }_{{\mathcal{P}}_j}\left(v_i\right),{\Omega }_{{\mathcal{P}}_j}\left(v_i\right)>:v_i\in \mathbb{V}\}(j=1,2,..,k).\end{array}$$

Then, form r test samples

$$\begin{array}{c}\hfill \mathbb{S}=\{{\mathcal{S}}_1,{\mathcal{S}}_2,...,{\mathcal{S}}_r\}\end{array}$$

represented by (m,a,n)-FNSs as

$$\begin{array}{c}\hfill {\mathcal{S}}_g=\{<v_i,{\mho }_{{\mathcal{S}}_g}\left(v_i\right),{\varpi }_{{\mathcal{S}}_g}\left(v_i\right),{\Omega }_{{\mathcal{S}}_g}\left(v_i\right)>:v_i\in \mathbb{V}\}(g=1,2,..,r).\end{array}$$

The object is to assign the test sample to the correct category by following the given pattern, ensuring precise classification.

Step 1: Calculate the distance of $S_g$ from each of ${\mathcal{P}}_1,{\mathcal{P}}_2,...,{\mathcal{P}}_k$ via the proposed (m,a,n)-FNDMs.

Step 2: Determine which pattern produces the lowest value in Step 1.

Step 3: Classify the test sample accordingly for each proposed (m,a,n)-FNDM.

Step 4: Calculate the degree of confidence $\left(DoC\right)$ for each (m,a,n)-FNDM using the following formula:

$$DoC=\sum _{j=1,j\ne j_0}^k|\mathbb{D}({\mathcal{P}}_j,{\mathcal{S}}_{\mathcal{g}})-\mathbb{D}({\mathcal{P}}_{j_0},{\mathcal{S}}_{\mathcal{g}})|$$

(30)

where ${\mathcal{P}}_{j_0}$ represents the identified pattern assigned to the test sample ${\mathcal{S}}_{\mathcal{g}}$. A higher value of $DoC$ clearly indicates a greater level of confidence in the prediction made by the corresponding (m,a,n)-FNDM.

Herein, we present Algorithm 1 to classify the material using the proposed (m, a, n)-FNDMs.

Algorithm 1 Pattern Classification

Input: $\mathbb{P}=\{{\mathcal{P}}_1,{\mathcal{P}}_2,\dots ,{\mathcal{P}}_k\}$, $\mathbb{S}=\{{\mathcal{S}}_1,{\mathcal{S}}_2,\dots ,{\mathcal{S}}_r\}\}$ Output: The classification of $S_g$ 1: for $g=1$ to r do 2:     /* Step 1 */ 3:     for $j=1$ to k do 4:         Calculate the distance $\mathbb{D}({\mathcal{P}}_j,{\mathcal{S}}_g)$ using proposed (m,a,n)-FNDMs 5:     end for 6:     /* Step 2 */ 7:     Choose the minimum distance $\mathbb{D}({\mathcal{P}}_{\alpha },S_g)=\underset{1\le j\le k}{min}\mathbb{D}({\mathcal{P}}_j,{\mathcal{S}}_g)$ 8:     /* Step 3 */ 9:     Compute $\alpha =\underset{1\le j\le k}{argmin}\mathbb{D}({\mathcal{P}}_j,{\mathcal{S}}_g)$ ,${\mathcal{P}}_{\alpha }\to {\mathcal{S}}_g$. 10: end for 11: /* Step 4 */ 12: for each (m,a,n)-FNDM do 13:     Compute $DoC={\sum }_{j=1,j\ne j_0}^k|\mathbb{D}({\mathcal{P}}_j,\mathcal{S})-\mathbb{D}({\mathcal{P}}_{j_0},\mathcal{S})|$ 14: end for

Example 5.

A company imported four types of materials, but the label identifying one batch was lost during transit. The goal is to determine which type of material this unlabeled batch belongs to. The four types of materials are represented by (m,a,n)-FNSs (m = 3, a = 4, n = 5), denoted as ${\mathcal{M}}_i$ for $i=1,2,3,4$ in the feature space $V=\{s_1,s_2,s_3,s_4,s_5,s_6,s_7\}$, which are depicted in Table 2. Now, the data for unknown material $\mathcal{M}$ is also listed in

Table 3. (m,a,n)-FN data of materials.

Material	${\mathcal{M}}_1$	${\mathcal{M}}_2$	${\mathcal{M}}_3$	${\mathcal{M}}_4$	$\mathcal{M}$
$s_1$	(0.56,0.47,0.22)	(0.81,0.30,.037)	(0.43,0.43,0.55)	(0.57,0.51,0.39)	(0.34,0.56,0.78)
$s_2$	(0.11,0.11,0.11)	(0.59,0.66,0.66)	(0.91,0.34,0.68)	(0.56,0.76,0.31)	(0.47,0.38,0.84)
$s_3$	(0.35,0.45,0.61)	(0.42,0.56,0.71)	(0.81,0.41,0.35)	(0.27,0.59,0.72)	(0.55,0.44,0.65)
$s_4$	(0.33,0.54,0.31)	(0.59,0.45,0.90)	(0.44,0.55,0.77)	(0.46,0.46,0.45)	(0.76,0.46,0.85)
$s_5$	(0.35,0.20,0.64)	(0.16,0.33,0.42)	(0.55,0.44,0.29)	(0.57,0.66,0.91)	(0.13,0.35,0.57)
$s_6$	(0.47,0.37,0.68)	(0.68,0.46,0.88)	(0.47,0.66,0.75)	(0.41,0.73,0.41)	(0.24,0.54,0.46)
$s_7$	(0.78,0.55,0.03)	(0.49,0.54,0.39)	(0.58,0.34,0.23)	(0.21,0.43,0.13)	(0.82,0.46,0.69)

. The object is to identify which type the unknown material $\mathcal{M}$ belongs to.

Step 1: Established (m,a,n)-FNDMs are utilized to compute the distances between $\mathcal{M}$ and ${\mathcal{M}}_j$. The results are depicted in

Table 4. (m,a,n)-FNDMs for data given in Table 3.

DMs	$({\mathcal{M}}_1,\mathcal{M})$	$({\mathcal{M}}_2,\mathcal{M})$	$({\mathcal{M}}_3,\mathcal{M})$	$({\mathcal{M}}_4,\mathcal{M})$	Alternative	DoC
$d^{NH}$	0.1295	0.1691	0.1691	0.2012	${\mathcal{M}}_1$	0.1507
$d^{NE}$	0.1864	0.2299	0.2312	0.2629	${\mathcal{M}}_1$	0.1649
$d^{GN}$	0.2318	0.2742	0.2835	0.3075	${\mathcal{M}}_1$	0.1699
$d^{NHH}$	0.2284	0.2999	0.3349	0.3674	${\mathcal{M}}_1$	0.3168
$d^{NEH}$	0.2699	0.3461	0.3661	0.3972	${\mathcal{M}}_1$	0.2999
$d^{GNH}$	0.3008	0.3753	0.3978	0.4204	${\mathcal{M}}_1$	0.2911
$d^{HNH}$	0.1790	0.2345	0.2520	0.2843	${\mathcal{M}}_1$	0.2338
$d^{HNE}$	0.2505	0.2938	0.3347	0.3368	${\mathcal{M}}_1$	0.1854
$d^{GHN}$	0.2707	0.3324	0.3500	0.3635	${\mathcal{M}}_1$	0.2339
$d^{TI}$	0.1245	0.1579	0.1558	0.1904	${\mathcal{M}}_1$	0.1305

.

Step 2: The minimum distances between $\mathcal{M}$ and ${\mathcal{M}}_j$ are also depicted in Table 4.

Step 3: Assign the material to the category associated with the lowest value identified in Step 2. For each proposed (m,a,n)-FN DM, material ${\mathbb{M}}_1$ consistently has the lowest distance score across all ten proposed (m,a,n)-FNDMs, which shows that the unknown material ${\mathbb{M}}_1$ is most similar to unknown $\mathbb{M}$.

Step 4: Calculate the value of the $DoC$ for each (m,a,n)-FNDM using Equation (30). It is evident that the highest $d^{NHH}$ value corresponds to the highest confidence in the prediction across all proposed (m,a,n)-FNDMs.

4.2. Parameter Sensitivity Analysis

This subsection analyzes how changes in parameters influence the ranking results, aiming to validate the stability of the proposed (m,a,n)-FNDMs-based classification method. To this end, a sensitivity analysis was performed to evaluate the robustness of the pattern recognition approach across varying parameter configurations. This involved making small adjustments to the parameters m, a, and n, which are considered the core elements of the (m,a,n)-FNDMs used in the method. The analysis explores how these variations affect the ranking outcomes.

Table 5. Ranking and classification results for different values of m, a, and n.

m	a	n	DMs	Ranking	Alternative
			$d^{NH}$	$Uncertain$	Unclassified
			$d^{NE}$	$Uncertain$	Unclassified
			$d^{GN}$	$Uncertain$	UNclassified
			$d^{NHH}$	$Uncertain$	Unclassified
$1\le m\le 3$	$1\le a\le 3$	$1\le n\le 3$	$d^{NEH}$	$Uncertain$	Unclassified
			$d^{GNH}$	$Uncertain$	Unclassified
			$d^{HNH}$	$Uncertain$	Unclassified
			$d^{HNE}$	$Uncertain$	Unclassified
			$d^{GHN}$	$Uncertain$	Unclassified
			$d^{TI}$	$Uncertain$	Unclassified
			$d^{NH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NE}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{GN}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NHH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NEH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
3	4	5	$d^{GNH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{HNH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{HNE}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{GHN}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{TI}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NE}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{GN}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NHH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NEH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
5	5	5	$d^{GNH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{HNH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{HNE}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{GHN}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{TI}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}^1$
			$d^{NH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NE}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{GN}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NHH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NEH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
6	8	10	$d^{GNH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{HNH}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{HNE}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{GHN}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{TI}$	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NH}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NE}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{GN}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NHH}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^4<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{NEH}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
10	50	100	$d^{GNH}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{HNH}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{HNE}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{GHN}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$
			$d^{TI}$	${\mathcal{M}}^1<{\mathcal{M}}^4<{\mathcal{M}}^2<{\mathcal{M}}^3$	${\mathcal{M}}^1$

displays the results, showing how the materials are ranked under each of the (m,a,n)-FNDMs with different values of m, a, and n tested in a systematic manner. It is evident from Table 5 that the proposed (m,a,n)-FNDMs are not suitable for handling the data in this example when the combination of m, a, and n lies between 1 and 3. This is because the sum of the $m^{th}$ power of the PMD and the $n^{th}$ power of the NMD exceeds 1 for some attributes in the considered dataset. Consequently, the distance measures $d_1$, $d_2$, and $d_p$ proposed by Rajan and Krishnaswamy [46] for PyNSs (with m = a = n = 2), as well as the distance measures $d_H$, $d_E$, $d_{Eu}$, and $d_{F_n}$ proposed by Saeed and his coworkers [47] for FNSs (with m = a = n = 3), fail to classify the material accurately. It is evident from the table that the ranking and selection of the optimal material remain consistent across all combinations of m, a, and n, particularly when greater emphasis is placed on the PMD compared to the IMD and NMD. The ranking and optimal results may vary if greater priority is assigned to the NMD compared to the PMD and IMD. Overall, it can be concluded that the proposed method offers greater efficiency and flexibility compared to existing approaches in pattern recognition.

4.3. Comparative Study

This subsection aims to compare the outcomes obtained using the proposed (m,a,n)-FNDMs with those derived from distance measures developed within other orthopair fuzzy and neutrosophic frameworks. As discussed in the previous section, the distance measures $d_1$, $d_2$, and $d_p$ introduced by Rajan and Krishnaswamy [46] for PyNSs (with m = a = n = 2), along with the distance measures $d_H$, $d_E$, $d_{Eu}$, and $d_{F_n}$ proposed by Saeed and his coworkers [47] for FNSs (with m = a = n), fail to accurately classify the material in the considered example. To enable a meaningful comparison, the (m,a,n)-FNSs are reduced to (m,n)-fuzzy sets by omitting the Indeterminacy Membership Degree (IMD) in the data. The resulting dataset is then evaluated using the distance measures proposed by Thakur and his associates [30,31] and Sivdas and John [32] for (m,n)-FSs in pattern recognition applications. These methods are implemented on the same dataset using m = 3 and n = 5. The resulting rankings and classification outcomes are presented in

Table 6. Comparison of current and existing methods.

Method	Environment	Ranking	Alternative
Proposed method based on $d^{NH}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{NE}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{GN}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{NHH}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{NEH}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{GNH}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{HNH}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{HNE}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{GHN}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Proposed method based on $d^{TI}$	(3,4,5)-FNS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Method based on $D^L$ [31]	(3,5)-FS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Method based on $d^{TI}$ [31]	(3,5)FS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Method based on $d^1$ [32]	(3,5)-FS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Method based on $d^2$ [32]	(3,5)-FS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Method based on $d^3$ [32]	(3,5)-FS	${\mathcal{M}}^1<{\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4$	${\mathcal{M}}_1$
Method based on $d^{nH}$ [30]	(3,5)-FS	${\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^1<{\mathcal{M}}^4$	${\mathcal{M}}_2$
Method based on $d^{dE}$ [30]	(3,5)-FS	${\mathcal{M}}^2<{\mathcal{M}}^3<{\mathcal{M}}^4<{\mathcal{M}}^1$	${\mathcal{M}}_2$

, allowing for a direct comparison between the proposed method and existing approaches based on (m,n)-FS distance measures. Table 6 indicates that the classification outcomes achieved with the suggested distance measures closely correspond to the results obtained by most of the distance measures for the (m,n)-FSs discussed in [30,31,32]. This confirms the viability, effectiveness, and reliability of the proposed distance measures in pattern recognition problems.

Figure 1 depicts a comparison of our study with others, and it also explains that our results are suitable and more accurate than those of previous work.

5. Multi-Criteria Decision Making with TOPSIS Approach Under (m,a,n)-FNSs

Multi-criteria decision making (MCDM) is a widely recognized area within the field of decision analysis. It involves making optimal choices by assessing, ranking, or selecting among alternatives when multiple, often conflicting, criteria are involved. In this section, we develop a TOPSIS method for multi-criteria decision making in an (m,a,n)-fuzzy neutrosophic environment and demonstrate how the proposed (m,a,n)-FNWDMs effectively address complex MCDM problems involving uncertainty and ambiguity.

5.1. Problem Description and Solution Procedure

Let $\mathcal{C}=\{{\mathcal{C}}^1,{\mathcal{C}}^2,\dots {\mathcal{C}}^q\}$ denote the set of q evaluation criteria, and let $\mathcal{A}=\{{\mathcal{A}}^1,{\mathcal{A}}^2,\dots {\mathcal{A}}^p\}$ represent the set of p possible alternatives. According to the preferences provided by the decision-maker, each criterion ${\mathcal{C}}^j$ is assigned a corresponding weight ${\omega }_j$, forming the weight vector $\omega =\{{\omega }_1,{\omega }_2,..{\omega }_q\}$, where ${\omega }_j\in [0,1]$ for all j and ${\Sigma }_1^q{\omega }_j=1$. Based on the evaluations provided by decision-makers using the (m,a,n)-FN framework, a decision matrix $\mathcal{D}={\left[d_{ij}\right]}_{p\times q}=\left[\right({\mho }_{ij},{\varpi }_{ij}$ is constructed. In this matrix, ${\mho }_{ij}$ indicates the degree to which the alternative ${\mathcal{A}}^i$ satisfies criterion ${\mathcal{C}}^j$, ${\phi }_{ij}$ denotes the extent of neutral satisfaction of ${\mathcal{A}}^i$ with respect to ${\mathcal{C}}^j$, and ${\Omega }_{ij}$ reflects which alternative ${\mathcal{A}}^i$ does not satisfy criterion ${\mathcal{C}}^j$. These values must satisfy $0\le {\mho }_{ij}^m+{\Omega }_{ij}^n\le 1$ and $0\le {\varpi }_{ij}^a\le 1$ for $m,a,n\in \mathbb{N}$. The objective is to identify the most suitable alternative based on this structured evaluation.

Step 1: Construct the decision matrix $\mathcal{D}={\left[d_{ij}\right]}_{p\times q}={\left[({\mho }_{ij},{\varpi }_{ij},{\Omega }_{ij})\right]}_{p\times q}$, where ${\mho }_{ij}$ encapsulates the assessment of each alternative with respect to each criterion. In this structure, ${\mho }_{ij}$ indicates how strongly the alternative ${\mathcal{A}}^i$ satisfies the criteria ${\mathcal{C}}^j$, ${\varpi }_{ij}$ reflects the neutral satisfaction of the alternative ${\mathcal{A}}^i$ with respect to ${\mathcal{C}}^j$, and ${\Omega }_{ij}$ expresses the extent to which the alternative ${\mathcal{A}}^i$ does not meet criterion ${\mathcal{C}}^j$.

Step 2: Formulate the normalized decision matrix ${\mathcal{D}}^{\prime }={\left[d_{ij}^{\prime }\right]}_{p\times q}={\left[({\mho }_{ij}^{\prime },{\varpi }_{ij}^{\prime },{\Omega }_{ij}^{\prime })\right]}_{p\times q}$, where

$$({\mho }_{ij}^{\prime },{\varpi }_{ij}^{\prime },{\Omega }_{ij}^{\prime })=\left\{\begin{array}{cc}({\Omega }_{ij}^{{\displaystyle \frac{n}{m}}},1-{\varpi }_{ij},{\mho }_{ij}^{{\displaystyle \frac{m}{n}}}),\hfill & \mathrm{if}{\mathcal{C}}^j\mathrm{is}\mathrm{the}\cos \mathrm{t}\mathrm{criteria}\hfill \\ ({\mho }_{ij},{\varpi }_{ij},{\Omega }_{ij}),\hfill & \mathrm{if}{\mathcal{C}}^j\mathrm{is}\mathrm{the}\mathrm{benefit}\mathrm{criteria}\hfill \end{array}\right.$$

(31)

Step 3: Determine the (m,a,n)-FN positive ideal solution ${\mathcal{P}}^{+}$ and the (m,a,n)-FN negative ideal solution ${\mathcal{P}}^{-}$ as follows:

$$\begin{array}{c}\hfill {\mathcal{P}}^{+}=\{({\mho }_1^{+},{\varpi }_1^{+},{\Omega }_1^{+}),({\mho }_2^{+},{\varpi }_2^{+},{\Omega }_2^{+}),\dots ({\mho }_q^{+},{\varpi }_q^{+},{\Omega }_q^{+})\}\end{array}$$

where ${\mho }_j^{+}=\underset{i}{M}ax{\mho }_{ij}^{+}$, ${\varpi }_j^{+}=\underset{i}{M}ax{\varpi }_{ij}^{+}$, and ${\Omega }_j^{+}=\underset{i}{M}in{\Omega }_{ij}^{+}$ for j = 1, 2, …, q.

$$\begin{array}{c}\hfill {\mathcal{P}}^{-}=\{({\mho }_1^{-},{\varpi }_1^{-},{\Omega }_1^{-}),({\mho }_2^{-},{\varpi }_2^{-},{\Omega }_2^{-}),\dots ({\mho }_q^{-},{\varpi }_q^{-},{\Omega }_q^{-})\}\end{array}$$

where ${\mho }_j^{-}=\underset{i}{M}in{\mho }_{ij}^{-}$, ${\varpi }_j^{-}=\underset{i}{M}in{\varpi }_{ij}^{-}$, and ${\Omega }_j^{-}=\underset{i}{M}ax{\Omega }_{ij}^{-}$ for j = 1, 2, …, q.

Step 4: Compute the separations $D({\mathcal{A}}^i,{\mathcal{P}}^{+})$ and $D({\mathcal{A}}^i,{\mathcal{P}}^{-})$ for each alternative ${\mathcal{A}}^i$ (i = 1, 2, … p) from the (m,a,n)-FN positive ideal solution ${\mathcal{P}}^{+}$ and the (m,a,n)-FN negative ideal solution ${\mathcal{P}}^{-}$ using the suggested (m,a,n)-FNWDMs.

Step 5: Compute the closeness index ${\gamma }_i$ connected to the ${\mathcal{A}}^i$ (i = 1, 2, … p) as follows:

$${\gamma }_i={\displaystyle \frac{D({\mathcal{A}}^i,{\mathcal{P}}^{-})}{D({\mathcal{A}}^i,{\mathcal{P}}^{+})+D({\mathcal{A}}^i,{\mathcal{P}}^{-})}}$$

(32)

Step 6: Rank the alternatives according to their ${\gamma }_i$ values in descending order. The alternative ${\mathcal{A}}^i$ with the highest value of ${\gamma }_i$ is deemed the optimal alternative.

Herein, we present Algorithm 2 for the proposed TOPSIS method under the (m,a,n)-FNSs environment.

Algorithm 2 TOPSIS for MCDM under (m,a,n)-FNSs

Input: A set of alternatives $\mathcal{A}=\{{\mathcal{A}}^1,{\mathcal{A}}^2,\dots ,{\mathcal{A}}^p\}$, A set of criteria $\mathcal{C}=\{{\mathcal{C}}^1,{\mathcal{C}}^2,\dots ,{\mathcal{C}}^q\}$,with Cost and benefit criteria, (m,a,n)-FN decision matrix $\mathcal{D}={\left[d_{ij}\right]}_{p\times q}={\left[({\mho }_{ij},{\varpi }_{ij},{\Omega }_{ij})\right]}_{p\times q}$, Weight vector $\omega =\{{\omega }_1,{\omega }_2,\dots ,{\omega }_q\}$, with ${\omega }_j\in [0,1]$ and ${\Sigma }_1^q{\omega }_j=1$. Output: Ranking of alternatives. 1: /* Step 1: Construct (m,a,n)-FN normalized Decision Matrix ${\mathcal{D}}^{\prime }={\left[d_{ij}^{\prime }\right]}_{p\times q}$*/ 2: for each $i=1$ to p do 3:     for each $j=1$ to q do 4:         if ${\mathcal{C}}^j$ is cost criteria then 5:            Compute $d_{ij}^{\prime }=({\mho }_{ij}^{\prime },{\varpi }_{ij}^{\prime },{\Omega }_{ij}^{\prime })=({\Omega }_{ij}^{\frac{n}{m}},1-{\varpi }_{ij},{\mho }_{ij}^{\frac{m}{n}})$ 6:         else 7:            Compute $d_{ij}^{\prime }=({\mho }_{ij}^{\prime },{\varpi }_{ij}^{\prime },{\Omega }_{ij}^{\prime })=({\mho }_{ij},{\varpi }_{ij},{\Omega }_{ij})$ 8:         end if 9:     end for 10: end for 11: /* Step 2: Determine the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS) */ 12: for each criteria ${\mathcal{C}}^j$ do 13:     Compute ${\mathcal{P}}_j^{+}=({\mho }_j^{+},{\varpi }_j^{+},{\Omega }_j^{+})=(\underset{i}{M}ax{\mho }_{ij}^{+},\underset{i}{M}ax{\varpi }_{ij}^{+},\underset{i}{M}in{\Omega }_{ij}^{+})$ 14:     Compute ${\mathcal{P}}_j^{-}=({\mho }_j^{-},{\varpi }_j^{-},{\Omega }_j^{-})=(\underset{i}{M}in{\mho }_{ij}^{-},\underset{i}{M}in{\varpi }_{ij}^{-},\underset{i}{M}ax{\Omega }_{ij}^{-})$ 15: end for 16: /* Step 3: Calculate Separation Measures */ 17: for each alternative ${\mathcal{A}}^i$ do 18:     Compute $D({\mathcal{A}}^i,{\mathcal{P}}^{+})$ distance from PIS ${\mathcal{P}}^{+}$ using the suggested (m,a,n)-FNWDMs. 19:     Compute $D({\mathcal{A}}^i,{\mathcal{P}}^{-})$ distance from NIS ${\mathcal{P}}^{-}$ using the same (m,a,n)-FNWDMs metric. 20: end for 21: /* Step 4: Calculate Closeness Coefficient */ 22: for each alternative ${\mathcal{A}}_i$ do 23:     ${\gamma }_i=\frac{D({\mathcal{A}}^i,{\mathcal{P}}^{-})}{D({\mathcal{A}}^i,{\mathcal{P}}^{+})+D({\mathcal{A}}^i,{\mathcal{P}}^{-})}$ 24: end for 25: /* Step 5: Rank Alternatives */ 26: Rank the alternatives based on descending order of ${\gamma }_i$

Example 6.

Suppose a multinational corporation is preparing a financial plan for the upcoming year in alignment with its strategic goals. As part of this process, four clearly defined investment options have been identified and are denoted as ${\mathcal{A}}^1$: investment in the “Information Technology sector”; ${\mathcal{A}}^2$: investment in the “Energy sector”; ${\mathcal{A}}^3$: investment in the “Financial sector”; and ${\mathcal{A}}^4$: investment in the “Heathcare sector”. Following an initial assessment, four key criteria have been selected for the evaluation process: ${\mathcal{C}}^1$, representing “growth”, ${\mathcal{C}}^2$, denoting “risk analysis”, ${\mathcal{C}}^3$, referring to “socio-political impact”, and ${\mathcal{C}}^4$, covering “environmental and related factors”. Based on the company’s strategic financial priorities, the corresponding weight vector is defined as $\omega ={(0.2,0.3,0.1,0.4)}^T$. For simplicity in this illustrative example, the parameter values are set as m = 2, a = 3, and n = 4. The following steps detail the computational procedure used to solve this problem using the proposed method.

Step 1: The information for the four investment alternatives, evaluated against the previously defined criteria in the (m,a,n)-FN framework with parameters m = 2, a = 3, and n = 4, are depicted in

Table 7. (m,a,n)-FN decision matrix.

	${\mathcal{C}}^1$	${\mathcal{C}}^2$	${\mathcal{C}}^3$	${\mathcal{C}}^4$
${\mathcal{A}}^1$	(0.5,0.4,0.8)	(0.8,0.3,0.3)	(0.7,0.4,0.8)	(0.7,0.2,0.3)
${\mathcal{A}}^2$	(0.7,0.1,0.4)	(0.5,0.6,0.6)	(0.9,0.3,0.4)	(0.6,0.6,0.8)
${\mathcal{A}}^3$	(0.8,0.5,0.9)	(0.4,0.5,0.7)	(0.8,0.1,0.3)	(0.2,0.5,0.7)
${\mathcal{A}}^4$	(0.7,0.4,0.9)	(0.5,0.4,0.9)	(0.4,0.2,0.7)	(0.6,0.4,0.4)

.

Step 2: Considering that ${\mathcal{C}}^2$ and ${\mathcal{C}}^3$ are cost criteria and ${\mathcal{C}}^1$ and ${\mathcal{C}}^4$ are benefit criteria, the normalized decision matrix is constructed using Equation (31). The resulting values are shown in

Table 8. (m,a,n)-FN normalized decision matrix for m,a,n = 2,3,4.

	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
${\mathcal{A}}^1$	(0.5,0.4,0.8)	(0.09,0.7,0.894)	(0.64,0.6,0.837)	(0.7,0.2,0.3)
${\mathcal{A}}^2$	(0.7,0.1,0.4)	(0.36,0.4,0.707)	(0.16,0.7,0.949)	(0.6,0.6,0.8)
${\mathcal{A}}^3$	(0.8,0.5,0.9)	(0.49,0.5,0.632)	(0.09,0.9,0.894)	(0.2,0.5,0.7)
${\mathcal{A}}^4$	(0.7,0.4,0.9)	(0.81,0.6,0.707)	(0.49,0.8,0.632)	(0.6,0.4,0.4)

.

Step 3: Now, we will find the (m,a,n)-FN positive ideal solution ${\mathcal{P}}^{+}$ and (m,a,n)-FN negative ideal solution ${\mathcal{P}}^{-}$. These solutions are as follows:

$$\begin{array}{c}\hfill {\mathcal{P}}^{+}=\{(0.8,0.5,0.4),(0.81,0.7,0.632),(0.64,0.9,0.632),(0.7,0.6,0.3)\}\\ \hfill {\mathcal{P}}^{-}=\{(0.5,0.1,0.9),(0.09,0.4,0.894),(0.09,0.6,0.949),(0.2,0.2,0.8)\}\end{array}$$

Step 4: We use the proposed (m,a,n)-FNWDMs to compute the separation of each alternative ${\mathcal{A}}^i$ (i = 1, 2 …p) between the (m,a,n)-FN positive ideal solution ${\mathcal{P}}^{+}$ and the (m,a,n)-FN negative ideal solution ${\mathcal{P}}^{-}$. The results are shown in

Table 9. Separation of alternatives and positive ideal solution.

Weighted DMs	$\mathit{D}({\mathcal{A}}^1,{\mathcal{P}}^{+})$	$\mathit{D}({\mathcal{A}}^2,{\mathcal{P}}^{+})$	$\mathit{D}({\mathcal{A}}^3,{\mathcal{P}}^{+})$	$\mathit{D}({\mathcal{A}}^4,{\mathcal{P}}^{+})$
$d^{WNH}$	0.2302	0.2667	0.2454	0.1319
$d^{WNE}$	0.3236	0.3385	0.3190	0.1942
$d^{GWN}$	0.3743	0.3837	0.3606	0.2584
$d^{WNHH}$	0.4179	0.4425	0.4804	0.2462
$d^{WNEH}$	0.4532	0.4724	0.4860	0.3066
$d^{GWNH}$	0.4814	0.4941	0.4922	0.3662
$d^{HWNH}$	0.3240	0.3252	0.3163	0.1750
$d^{HWNE}$	0.3938	0.3856	0.3650	0.2401
$d^{GHWN}$	0.4344	0.4457	0.4363	0.3213
$d^{WTI}$	0.2031	0.2341	0.2226	0.1157

and

Table 10. Separation of alternatives and negative ideal solution.

Weighted DMs	$\mathit{D}({\mathcal{A}}^1,{\mathcal{P}}^{-})$	$\mathit{D}({\mathcal{A}}^2,{\mathcal{P}}^{-})$	$\mathit{D}({\mathcal{A}}^3,{\mathcal{P}}^{-})$	$\mathit{D}({\mathcal{A}}^4,{\mathcal{P}}^{-})$
$d^{WNH}$	0.1941	0.1576	0.1789	0.2924
$d^{WNE}$	0.2686	0.2412	0.2458	0.3525
$d^{GWN}$	0.3039	0.2955	0.2922	0.3964
$d^{WNHH}$	0.3600	0.3266	0.3622	0.4829
$d^{WNEH}$	0.3698	0.3812	0.4860	0.5102
$d^{GWNH}$	0.3786	0.4147	0.4113	0.5327
$d^{HWNH}$	0.2770	0.2003	0.2584	0.3364
$d^{HWNE}$	0.3232	0.2809	0.3222	0.3933
$d^{GHWN}$	0.3453	0.3648	0.3616	0.4743
$d^{WTI}$	0.1748	0.1438	0.1554	0.2622

.

Step 5: Compute the closeness index ${\gamma }_i$ connected to the ${\mathcal{A}}^i$ (i = 1, 2, …p) using Equation (32) and rank the alternatives according to their ${\gamma }_i$ values in descending order. Choose the best alternative ${\mathcal{A}}^i$ with the highest value of ${\gamma }_i$, as depicted in

Table 11. Closeness index connected to the alternatives and their ranking.

Weighted DMs	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	${\mathit{\gamma }}_3$	${\mathit{\gamma }}_4$	Ranking	Alternative
$d^{WNH}$	0.4574	0.3715	0.4216	0.6892	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{WNE}$	0.4536	0.4161	0.4352	0.6448	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{GWN}$	0.4481	0.4350	0.4476	0.6053	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{WNHH}$	0.4628	0.4246	0.4298	0.6624	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{WNEH}$	0.4493	0.4465	0.5000	0.6246	${\mathcal{A}}^4>{\mathcal{A}}^3>{\mathcal{A}}^1>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{GWNH}$	0.4402	0.4563	0.4553	0.5926	${\mathcal{A}}^4>{\mathcal{A}}^2>{\mathcal{A}}^3>{\mathcal{A}}^1$	${\mathcal{A}}^4$
$d^{HWNH}$	0.4609	0.3812	0.4496	0.6578	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{HWNE}$	0.4508	0.4214	0.4689	0.6209	${\mathcal{A}}^4>{\mathcal{A}}^3>{\mathcal{A}}^1>{\mathcal{A}}^2$	${\mathcal{A}}^4$
$d^{GHWN}$	0.4428	0.4501	0.4532	0.5961	${\mathcal{A}}^4>{\mathcal{A}}^3>{\mathcal{A}}^2>{\mathcal{A}}^1$	${\mathcal{A}}^4$
$d^{WTI}$	0.4626	0.3805	0.4111	0.6938	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$

.

As shown in Table 11, although the rankings produced by different (m,a,n)-FNWDMs may differ, the top-ranked alternative is consistently ${\mathcal{A}}^4$. Therefore, it can be concluded that ${\mathcal{A}}^4$ represents the most suitable choice across all proposed (m,a,n)-FNWDM approaches. This consistency highlights the reliability of our proposed methods in delivering stable results when applied to multi-criteria decision-making problems.

5.2. Comparative Study

In this section, we compare the proposed MCDM method with existing decision-making methods within the framework of q-RNSs and (m,n)-FSs to validate its effectiveness. For the comparative analysis, the information and data of alternatives and criteria in numerical Example 6 are considered. This comparison allows for an evaluation of the effectiveness and consistency of the proposed method relative to existing techniques within the reduced neutrosophic and fuzzy environment.

5.2.1. Comparison with q-RSNWAO- and q-RSNWGO-Based Approaches [41]

In this section, we compare the proposed MCDM method with approaches based on the q-RSNWAO and q-RSNWGO aggregation operators under the framework of q-RONSs. To facilitate this comparison, the (m,a,n)-FNSs are transformed into q-RONSs by taking the maximum value among m, a, and n for each component. Al-Quran, Al-Sharqi, and Djaouti [41] proposed the q-RSNWAO- and q-RSNWGO-based methods to address MCDM problems. We implement their approach on the considered dataset for q = 4, and the procedural steps are outlined as follows:

Step 1: The decision matrix of q-RONNs based on alternative and criteria information is depicted in Table 7.

Step 2: Create the normalized decision matrix by taking the complements of cost criteria ${\mathcal{C}}^2$ and ${\mathcal{C}}^2$. The resulting normalized decision matrix is shown in

Table 12. 4-RONS normalized decision matrix.

Alternative	${\mathcal{C}}^1$	${\mathcal{C}}^2$	${\mathcal{C}}^3$	${\mathcal{C}}^4$
${\mathcal{A}}^1$	(0.5,0.4,0.8)	(0.3,0.7,0.8)	(0.8,0.6,0.7)	(0.7,0.2,0.3)
${\mathcal{A}}^2$	(0.7,0.1,0.4)	(0.6,0.4,0.5)	(0.4,0.7,0.9)	(0.6,0.6,0.8)
${\mathcal{A}}^3$	(0.8,0.5,0.9)	(0.7,0.5,0.4)	(0.3,0.9,0.8)	(0.2,0.5,0.7)
${\mathcal{A}}^4$	(0.7,0.4,0.9)	(0.9,0.6,0.5)	(0.7,0.8,0.4)	(0.6,0.4,0.4)

.

Step 3: Compute the aggregated values of each alternative using q-RSNWAO and q-RSNWGO. These values are depicted in

Table 13. Aggregated and score values of alternatives using q-RSNWAO for q = 4.

Alternative	Aggregated Value	Score Value
${\mathcal{A}}^1$	(0.6353,0.3734,0.5332)	0.5696
${\mathcal{A}}^2$	(0.5551,0.2430,0.6120)	0.5706
${\mathcal{A}}^3$	(0.6444,0.5303,0.6307)	0.4946
${\mathcal{A}}^4$	(0.7786,0.4842,0.5030)	0.6064

and

Table 14. Aggregated and score values of alternatives using q-RSNWGO for q = 4.

Alternative	Aggregated Value	Score Value
${\mathcal{A}}^1$	(0.5144, 0.4749, 0.7105)	0.4467
${\mathcal{A}}^2$	(0.3829, 0.3189, 0.7336)	0.4710
${\mathcal{A}}^3$	(0.4002, 0.5743, 0.7514)	0.3775
${\mathcal{A}}^4$	(0.7097, 0.5240, 0.6832)	0.5039

, respectively.

Step 4: Calculate the score value for the aggregated values computed in Step 3 of each alternative, which are also shown in Table 13 and Table 14, respectively.

Step 5: Rank the alternative corresponding to the obtained score values in descending order. The ranking of the alternatives is stated below:

$$\begin{array}{c}\hfill \mathbf{q}-\mathbf{RSNWGO}:{\mathcal{A}}^4>{\mathcal{A}}^2>{\mathcal{A}}^1>{\mathcal{A}}^3.\\ \hfill \mathbf{q}-\mathbf{RSNWGO}:{\mathcal{A}}^4>{\mathcal{A}}^2>{\mathcal{A}}^1>{\mathcal{A}}^3.\end{array}$$

In both methods, the optimum alternative with the highest ranking is ${\mathcal{A}}^4$.

5.2.2. Comparison with (m,n)-FWPA- and (m,n)-FWPG-Operators-Based Approaches [17]

The $(m,a,n)$-FNSs are reduced to $(m,n)$-fuzzy sets by assuming the Indeterminacy Membership Degree (IMD) to be zero. By neglecting the neutral component, the data from Example 6 is simplified to the $(m,n)$-FS framework. Under this setting, a comparative analysis is conducted of the proposed MCDM method and existing approaches based on the $(m,n)$-FWPA and $(m,n)$-FWPG aggregation operators introduced by Al-Shami and Mhemdi [17]. The procedural steps involved in implementing the $(m,n)$-FWPA- and $(m,n)$-FWPG-based algorithms [17] are outlined below:

Step 1: Create the decision matrix of (m,n)-fuzzy data by neglecting the IMD based on alternatives and criteria information. The decision matrix is depicted in

Table 15. (m,n)-fuzzy decision matrix for m = 2, n = 4.

	${\mathcal{C}}^1$	${\mathcal{C}}^2$	${\mathcal{C}}^3$	${\mathcal{C}}^4$
${\mathcal{A}}^1$	(0.5,0.8)	(0.8,0.3)	(0.7,0.8)	(0.7,0.3)
${\mathcal{A}}^2$	(0.7,0.4)	(0.5,0.6)	(0.9,0.4)	(0.6,0.8)
${\mathcal{A}}^3$	(0.8,0.9)	(0.4,0.7)	(0.8,0.3)	(0.2,0.7)
${\mathcal{A}}^4$	(0.7,0.9)	(0.5,0.9)	(0.4,0.7)	(0.6,0.4)

.

Step-2: The normalized decision matrix is constructed by considering the (m,n)-fuzzy complement in cost criteria ${\mathcal{C}}^2$ and ${\mathcal{C}}^3$. The resulting values are shown in

Table 16. (m,n)-fuzzy normalized decision matrix for m = 2, n = 4.

	${\mathcal{C}}_1$	${\mathcal{C}}_2$	${\mathcal{C}}_3$	${\mathcal{C}}_4$
${\mathcal{A}}^1$	(0.5,0.8)	(0.09,0.894)	(0.64,0.837)	(0.7,0.3)
${\mathcal{A}}^2$	(0.7,0.4)	(0.36,0.707)	(0.16,0.949)	(0.6,0.8)
${\mathcal{A}}^3$	(0.8,0.9)	(0.49,0.632)	(0.09,0.894)	(0.2,0.7)
${\mathcal{A}}^4$	(0.7,0.9)	(0.81,0.707)	(0.49,0.632)	(0.6,0.4)

.

Step 3: Compute the aggregated values of each alternative using $(m,n)$-FWPA and $(m,n)$-FWPG operators. These values are depicted in

Table 17. Aggregated and score values of alternatives using (m,n)-FWPA for m = 2, n = 4.

Alternative	Aggregated Value	Score Value
${\mathcal{A}}^1$	(0.5379, 0.7536)	−0.0331
${\mathcal{A}}^2$	(0.3943, 0.7551)	−0.1696
${\mathcal{A}}^3$	(0.4657, 0.7887)	−0.1222
${\mathcal{A}}^4$	(0.6803, 0.6943)	0.2305

and

Table 18. Aggregated and score values of alternatives using (m,n)-FWPG for m = 2, n = 4.

Alternative	Aggregated Value	Score Value
${\mathcal{A}}^1$	(0.5635, 0.7833)	−0.0589
${\mathcal{A}}^2$	(0.4211, 0.7822)	−0.1969
${\mathcal{A}}^3$	(0.5119, 0.7849)	−0.1175
${\mathcal{A}}^4$	(0.6954, 0.7269)	0.2043

, respectively.

Step 4: Calculate the score value for the aggregated values computed in Step 3 of each alternative, which are also shown in Table 17 and Table 18, respectively.

Step 5: Based on the obtained score values, the alternatives are ranked in descending order of their performance. The ranking of the alternatives is stated below:

$$\begin{array}{c}\hfill (\mathbf{m},\mathbf{n})-\mathbf{FWPA}:{\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2.\\ \hfill (\mathbf{m},\mathbf{n})-\mathbf{FWPG}:{\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2.\end{array}$$

In both methods, the optimum alternative with the highest ranking is clearly identified as ${\mathcal{A}}^4$.

The rankings of the alternatives according to the proposed and compared MCDM methods are presented in

Table 19. Comparative analysis.

Method	Environment	Ranking	Alternative
Proposed method based on $d^{WNH}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{WNE}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{GWN}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{WNHH}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{WNEH}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^3>{\mathcal{A}}^1>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{GWNH}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^2>{\mathcal{A}}^3>{\mathcal{A}}^1$	${\mathcal{A}}^4$
Proposed method based on $d^{HWNH}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{HWNE}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^3>{\mathcal{A}}^1>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Proposed method based on $d^{GHWN}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^3>{\mathcal{A}}^2>{\mathcal{A}}^1$	${\mathcal{A}}^4$
Proposed method based on $d^{WTI}$	(2,3,4)-FNS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Method based on $q-RSNWAO$ [41]	4-RNS	${\mathcal{A}}^4>{\mathcal{A}}^2>{\mathcal{A}}^1>{\mathcal{A}}^3$	${\mathcal{A}}^4$
Method based on $q-RSNWGO$ [41]	4-RNS	${\mathcal{A}}^4>{\mathcal{A}}^2>{\mathcal{A}}^1>{\mathcal{A}}^3$	${\mathcal{A}}^4$
Method based on $(m,n)-FWPAO$ [17]	(2,4)-FS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$
Method based on $(m,n)-FWPGO$ [17]	(2,4)-FS	${\mathcal{A}}^4>{\mathcal{A}}^1>{\mathcal{A}}^3>{\mathcal{A}}^2$	${\mathcal{A}}^4$

. Furthermore, the ranking outcomes are depicted graphically in Figure 2 for visual comparison. Considering all the methods under comparison, the optimum alternative is identified as ${\mathcal{A}}^4$. The consistency in rankings across the different methods, along with the agreement on the top-ranked alternative, demonstrates the robustness and reliability of the proposed method.

6. Conclusions

This study explored distance measures for (m,a,n)-FNSs, which generalize multiple forms of neutrosophic set theories. By introducing three distinct functions, the PMD, IMD, and NMD, the proposed framework offers a versatile approach for managing uncertainty and ambiguity in decision-making contexts. The novel DMs developed for(m,a,n)-FNSs serve as effective tools for quantifying differences between sets, thereby improving the precision of decision-making models. Using illustrative examples such as material classification and investment strategy selection, the proposed measures were shown to be consistent and high-performing. The comparative analysis, along with graphical interpretations, further illustrates the effectiveness and superiority of the proposed measures. In summary, this research advances the field of distance measures by presenting new methodologies and insights for addressing complex decision-making challenges. The approaches outlined here have broad applicability and can support more informed and dependable decisions across a range of disciplines. In the future, researchers can study aggregation operators, graph structure, similarity measures, divergence measures, entropy, inclusion measures, and knowledge measures for (m,a,n)-fuzzy neutrosophic sets and employ them in MCDM methodologies like TOPSIS, VIKOR, and SAR. We hope that this paper will be a reference point for the researchers who study decision-making.