A Novel Distance Measure of Bipolar Neutrosophic Sets with an Application in Pattern Classification

Abstract

A bipolar neutrosophic set (BNS) is designed to handle uncertain information by capturing both supportive and opposing aspects of data. In this paper, the pattern classification method is studied based on the proposed Hamming–Chebyshev hybrid distance measure (HCHDM). First, the HCHDM of the bipolar neutrosophic sets is proposed that not only captures discrete differences, but also reflects the maximum dimensional deviation in a more complex environment. Then, the axiomatic definition of the distance measure is proved and some examples are given to show it can better discriminate between the differences of BNSs. Based on the distance measure, an algorithm to solve the pattern classification problem is given. The numerical examples show that the proposed distance measure method is effective in solving pattern classification problems.

1. Introduction

In order to describe the fuzzy information, Zadeh proposed the fuzzy set to handle the uncertainty, which was a great breakthrough in classical mathematics. It has been extensively employed in intelligent systems, automation control, decision making, data analysis, and many other areas [1,2,3]. Inspired by Zadeh’s work, many extensions have been subsequently developed, such as intuitionistic fuzzy set (IFS), interval-valued intuitionistic fuzzy set (IVIFS), hesitant fuzzy set (HFS), and neutrosophic set (NS) [4,5,6,7,8,9,10,11]. In the future, with the develop of artificial intelligence, the extension of fuzzy sets will develop in three main directions: dynamicization (membership degrees that evolve over time), heterogeneous integration (fusion of multi-source data), and intelligent adaptation (automated learning of extended parameters).

On the framework of fuzzy sets, Smarandache developed the neutrosophic set (NS), which addresses the incomplete, indeterminate, and inconsistent information with the simultaneous consideration of truth, falsity, and indeterminacy membership degrees [7,8]. Majumdar developed the similarity metric and entropy of neutrosophic sets and discussed their relationships [12]. Xu introduced a new distance measure for single-valued neutrosophic sets and proposed an improved method with the known parameter of risk attitude based on TODIM and TOPSIS, which contributed to solving venture capital problems [13]. Ye developed the clustering techniques on the basis of distance-derived similarity measures of single-valued neutrosophic sets [14,15]. Huang proposed a novel distance measure of neutrosophic sets and applied it to multi-attribute policy decision-making problems [16]. Liu introduced a novel method to construct entropy for interval-valued neutrosophic sets (IVNSs) that enhanced the capability to model higher-order uncertainty and built a foundation for decision-making problems [17]. To capture the information from favorable and unfavorable viewpoints, Lee put forward the bipolar set, which broadens the membership degrees to a more comprehensive closed interval [−1, 1] [11]. After that, the bipolar fuzzy set is applied to several aspects, such as theoretical innovation, practical application, and related interdisciplinary research [18,19,20]. Deli extended the bipolar set to the bipolar neutrosophic set, which deals with the uncertain modeling by accounting for positive and negative influences across six distinct degrees [21,22]. Ulucay constructed some similarity measures and developed a multi-attribute decision-making (MADM) method to handle bipolar information [23]. Subsequently, Sahin proposed the Jaccard vector similarity metrics of bipolar neutrosophic sets which defined as the size of their intersection divided by the size of their union and applied it to multi-attribute decision-making problems [24]. Mohamed proposed several similarity measures and analyzed their mathematical properties and further developed two multi-attribute decision-making (MADM) methods in the context of a bipolar neutrosophic set and an interval-valued bipolar neutrosophic set [25]. Dey developed the TOPSIS and ELECTRE-I methods to the complex decision-making problems characterized by uncertainty and bipolarity [26]. Irvanizam integrated the DMAS method with bipolar neutrosophic sets and further extended this improved approach to solve group decision-making problems [27]. Mumtaz introduced the bipolar neutrosophic soft sets (BNSSs) and developed a series of algebraic aggregation operations and established a theoretical foundation for handling uncertain information in complex decision-making scenarios [28]. Surapati proposed cross entropy for single-valued and interval-valued bipolar neutrosophic sets and applied them to multi-attribute decision-making problems to effectively deal with the uncertainty [29]. Lazim employed the decision-making trial and evaluation laboratory (DEMATEL) method to assess 12 key criteria influencing sustainable urban transport. These findings provide decision makers with urban transport planning and management [30]. Memet proposed the distance measure of two BNSs with the consideration of positive membership function and negative membership function of the forward and backward differences. The calculation of distance measures relies on the deviation of membership degrees within the set itself, which is insensitive to differences between two sets [31]. A neutrosophic soft set is a tool for data modeling, which has been the main research in practical applications and theoretical mathematical perspectives. The energy of a bipolar neutrosophic soft set has important applications in multi-criteria decision making and sustainable energy selection [32,33]. Fahmi introduced operational laws for Aczél-Alsina aggregation on the framework of generalized bipolar neutrosophic sets (GBNSs) and tailored them for group decision-making scenarios [34]. Roan proposed an H-Max distance measure in bipolar neutrosophic environments that overcomes the limitations of the related measures by including cross-evaluations. But, it involves several parameters that need to be defined manually or determined through calculation and it has an impact on the distance measure [35]. Based on the aforementioned literature review, we have identified the following main issues in the field of distance measures of bipolar neutrosophic sets. The distance measure proposed by Memet et al. relies on calculating deviations in membership degrees within sets, which lacks sufficient sensitivity to differences between sets. Although the H-Max distance measure introduced by Roan et al. overcomes some limitations through cross-evaluation, it requires manually defining or calculating multiple parameters. The parameter settings may affect the objectivity and stability of the measurement results. With the application of a bipolar neutrosophic set, the limitations of the existing distance measure may lead to decision errors. Currently, there are few studies on the distance measure method for the pattern classification problem under the environment of BNS. This study proposes a novel distance measure of bipolar neutrosophic sets (BNSs) that requires no manual setting of multiple parameters and accounts for both intra-set deviations and inter-set differences, aiming to enhance the objectivity and sensitivity of results.

The Hamming distance measure tends to overlook extreme deviations in main dimensions and the Chebyshev distance tends to ignore cumulative deviations across multiple dimensions. After integration, the anti-interference ability is stronger. Inspired by reference [36], a novel distance metric is proposed that integrates the Hamming distance metric with the Chebyshev distance metric to capture both discrete feature discrepancies and extreme dimensional variations in complex datasets. The hybrid metric is designed to enhance the similarity in scenarios where data exhibit categorical characteristics. It enhanced the robustness of the distance measure, which makes up for the limitations of a single distance metric. In the environment of BNS, the distance measure of BNSs is constructed and the properties are studied. On the basis of the above distance measure, an algorithm is developed to solve the pattern classification problem. The result demonstrates the effectiveness and applicability.

Based on the above analysis, the framework of the current work is arranged in the following way. In Section 2, some preliminaries are given to elicit the concepts and background knowledge needed for this study. In Section 3, the distance measure of IFSs is introduced to prepare for the environment of BNSs. In Section 4, a Hamming–Chebyshev hybrid distance measure (HCHDM) is constructed and some examples are provided to clarify the application of the proposed distance measure. In Section 5, a proposed numerical example for pattern classification is developed in light of the novel distance measure and the comparison with other methods is given. The concluding section synthesizes the findings and contributions.

2. Preliminaries

This section provides the definitions of the basic terms necessary for this work.

Definition 1

([1]). Let $X$ be a collection of points (objects) with its element $\mathrm{x}$. A fuzzy set $A$ over a universe $X$ is denoted by its membership function $\mu \left(\mathrm{x}\right)$.

$$\mu \left(\mathrm{x}\right):X\to \left[\mathrm{0,1}\right]$$

The membership function maps each element to a singleton value in the unit interval [0, 1]. It transcends the binary logic to continuous numerical values within the unit interval [0, 1], which quantifies the strength of an element’s association with a fuzzy set. A fuzzy set $A$ is denoted by the following.

$$A=\left\{⟨\mathrm{x}:\mu \left(\mathrm{x}\right)⟩\left|\mathrm{x}\in \mathrm{X}\right.\right\}$$

People often encounter situations where they are neither affirmative nor negative. In such cases, the membership degree may be insufficient to capture the information of hesitation or uncertainty. The intuitionistic fuzzy set (IFS) is introduced to define non-membership degrees and hesitation degrees.

Definition 2

([4]). Let X be a universe of discourse and $\mathrm{x}$ represent a generic element in $\mathrm{X}$. An intuitionistic fuzzy set $A$ on X is defined by the membership degree $\mu \left(\mathrm{x}\right)$, the non-membership degree $\mathrm{v}\left(\mathrm{x}\right)$, as follows.

$$\mu \left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

$$\mathrm{v}\left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

These satisfy the following constraint.

$$0\le \mu \left(\mathrm{x}\right)+\mathrm{v}\left(\mathrm{x}\right)\le 1$$

For each intuitionistic fuzzy set $A$, $\mathrm{\pi }\left(\mathrm{x}\right)=1-\mu \left(\mathrm{x}\right)-\mathrm{v}\left(\mathrm{x}\right)$ is called the intuitionistic index of $\mathrm{x}$ in $A$, which represents the degree of hesitancy (or uncertainty) of $\mathrm{x}$. An intuitionistic fuzzy set (IFS) $A$ is denoted by the following.

$$A=\left\{⟨\mathrm{x}:\mu \left(\mathrm{x}\right),\mathrm{v}\left(\mathrm{x}\right)⟩\left|\mathrm{x}\in \mathrm{X}\right.\right\}$$

Based on the existed fuzzy set, Lee further extended fuzzy sets to multi-dimensional scenarios. This bidirectional extension not only breaks through the limitations of traditional single-perspective descriptions, but also deepens the ability to characterize the uncertainty.

Definition 3

([22]). Assume $\mathrm{X}$ is a universe of discourse with a generic element $\mathrm{x}$. A bipolar fuzzy set $A$ on $\mathrm{X}$ is characterized by a positive membership degree ${\mu }^{+}\left(\mathrm{x}\right)$ and a negative membership degree ${\mu }^{-}\left(\mathrm{x}\right)$ showing below

$${\mu }^{+}\left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

$${\mu }^{-}\left(\mathrm{x}\right):\mathrm{X}\to \left[-\mathrm{1,0}\right]$$

${\mu }^{+}\left(\mathrm{x}\right)$ represents the satisfaction degree of an element $\mathrm{x}$ to the property corresponding to A, 1 indicates full satisfaction, 0 indicates no satisfaction, and intermediate values represent partial satisfaction. ${\mu }^{-}\left(\mathrm{x}\right)$ represents the dissatisfaction degree of an element $\mathrm{x}$ to some implicit counter-property, −1 indicates full dissatisfaction, 0 indicates no dissatisfaction, and intermediate values represent partial dissatisfaction. A bipolar fuzzy set $A$ is denoted by the following.

$$A=\left\{⟨\mathrm{x}:{\mu }^{+}\left(\mathrm{x}\right),{\mu }^{-}\left(\mathrm{x}\right)⟩\left|\mathrm{x}\in \mathrm{X}\right.\right\}$$

As an extension of the fuzzy set, the neutrosophic set (NS) was proposed by Florentin Smarandache, which was specifically designed to address the limitations of traditional uncertainty in handling incompleteness, indeterminacy, and inconsistency.

Definition 4

([8]). Assume $\mathrm{X}$ is a universe of discourse and $\mathrm{x}$ is a generic element in $\mathrm{X}$. A single-valued neutrosophic set $A$ is characterized by truth membership function ${\mathrm{\mu }}_{\mathrm{A}}\left(\mathrm{x}\right)$, indeterminacy membership function ${\mathrm{\theta }}_{\mathrm{A}}\left(\mathrm{x}\right)$, and falsity membership function ${\mathrm{\vartheta }}_{\mathrm{A}}\left(\mathrm{x}\right)$, and where the following are true.

$${\mathrm{\mu }}_{\mathrm{A}}\left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

$${\mathrm{\theta }}_{\mathrm{A}}\left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

$${\mathrm{\vartheta }}_{\mathrm{A}}\left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

The truth membership degree ($\mu$) represents the degree to which an element belongs to the set, the falsity membership degree ($\vartheta$) indicates the degree to which it does not belong to the set, and the indeterminacy membership degree ($\theta$) quantifies the uncertainty or ambiguity in its membership status. By integrating these three dimensions, neutrosophic sets provide a more comprehensive framework for describing uncertainty, applying for multi-criteria decision-making, pattern recognition, medical diagnosis, and artificial intelligence systems. A single-valued neutrosophic set $A$ is denoted by the following.

$$A=\left\{<\mathrm{x},{\mathrm{\mu }}_{\mathrm{A}}\left(\mathrm{x}\right),\mathrm{\theta }\left(\mathrm{x}\right),{\mathrm{\vartheta }}_{\mathrm{A}}\left(\mathrm{x}\right)>|\mathrm{x}\in \mathrm{X}\right\}$$

The bipolar neutrosophic set (BNS) is proposed by Deli, in which the core innovation lies in the introduction of positive and negative effects and six-dimensional degrees, which enhances the model’s ability to express complex information.

Definition 5

([21]). Let $\mathrm{X}$ be a universe of discourse and $\mathrm{x}$ represent a generic element. A bipolar neutrosophic set (BNS) $A$ on $\mathrm{X}$ is assigned by the following:

$$\mathrm{A}=\left\{{\mathrm{\mu }}_{\mathrm{A}}^{+}\left(\mathrm{x}\right),{\mathrm{\theta }}_{\mathrm{A}}^{+}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_{\mathrm{A}}^{+}\left(\mathrm{x}\right),{\mathrm{\mu }}_{\mathrm{A}}^{-}\left(\mathrm{x}\right),{\mathrm{\theta }}_{\mathrm{A}}^{-}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_{\mathrm{A}}^{-}\left(\mathrm{x}\right)\right\}$$

where ${\mathrm{\mu }}_{\mathrm{A}}^{+}\left(\mathrm{x}\right),{\mathrm{\theta }}_{\mathrm{A}}^{+}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_{\mathrm{A}}^{+}\left(\mathrm{x}\right)$ are the positive truth membership function, positive indeterminacy membership function, and positive falsity membership function, respectively. ${\mathrm{\mu }}_{\mathrm{A}}^{-}\left(\mathrm{x}\right),{\mathrm{\theta }}_{\mathrm{A}}^{-}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_{\mathrm{A}}^{-}\left(\mathrm{x}\right)$ are negative truth membership function, negative indeterminacy membership function, and negative falsity membership function, respectively. They are defined by the following:

$${\mathrm{U}}_{\mathrm{A}}^{+}\left(\mathrm{x}\right):\mathrm{X}\to \left[\mathrm{0,1}\right]$$

$${\mathrm{U}}_{\mathrm{A}}^{-}\left(\mathrm{x}\right):\mathrm{X}\to \left[-\mathrm{1,0}\right]$$

where $\mathrm{U}=\mathrm{\mu },\mathrm{\theta },\mathrm{\vartheta }$. Specifically, when the set $\mathrm{X}$ contains only a single element, the bipolar neutrosophic set (BNS) simplifies to a bipolar neutrosophic number (BNN).

In the following, the operation laws of the BNSs are given.

Definition 6

([21]). Let ${\mathrm{A}}_1=\left\{{\mathrm{\mu }}_1^{+}\left(\mathrm{x}\right),{\mathrm{\theta }}_1^{+}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_1^{+}\left(\mathrm{x}\right),{\mathrm{\mu }}_1^{-}\left(\mathrm{x}\right),{\mathrm{\theta }}_1^{-}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_1^{-}\left(\mathrm{x}\right)\right\}$, ${\mathrm{A}}_2=\left\{{\mathrm{\mu }}_2^{+}\left(\mathrm{x}\right),{\mathrm{\theta }}_2^{+}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_2^{+}\left(\mathrm{x}\right),{\mathrm{\mu }}_2^{-}\left(\mathrm{x}\right),{\mathrm{\theta }}_2^{-}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_2^{-}\left(\mathrm{x}\right)\right\}$ be two BNNs. The operations are defined as follows:

1. 

${\mathrm{A}}_1\subseteq {\mathrm{A}}_2$ if ${\mathrm{\mu }}_1^{+}\left(\mathrm{x}\right)\le {\mathrm{\mu }}_2^{+}\left(\mathrm{x}\right)$, ${\mathrm{\theta }}_1^{+}\left(\mathrm{x}\right)\le {\mathrm{\theta }}_2^{+}\left(\mathrm{x}\right)$, ${\mathrm{\vartheta }}_1^{+}\left(\mathrm{x}\right)\ge {\mathrm{\vartheta }}_2^{+}\left(\mathrm{x}\right)$ and ${\mathrm{\mu }}_1^{-}\left(\mathrm{x}\right)\ge {\mathrm{\mu }}_2^{-}\left(\mathrm{x}\right)$, ${\mathrm{\theta }}_1^{-}\left(\mathrm{x}\right)\ge {\mathrm{\theta }}_2^{-}\left(\mathrm{x}\right)$, ${\mathrm{\vartheta }}_1^{-}\left(\mathrm{x}\right)\le {\mathrm{\vartheta }}_2^{-}\left(\mathrm{x}\right)$;

2. 

${\mathrm{A}}_1\cup {\mathrm{A}}_2=\left\{\max \left\{{\mathrm{\mu }}_1^{+},{\mathrm{\mu }}_2^{+}\right\},{\displaystyle \frac{{\mathrm{\theta }}_1^{+}+{\mathrm{\theta }}_2^{+}}{2}},\min \left\{{\mathrm{\vartheta }}_1^{+},{\mathrm{\vartheta }}_2^{+}\right\},\min \left\{{\mathrm{\mu }}_1^{-},{\mathrm{\mu }}_2^{-}\right\},{\displaystyle \frac{{\mathrm{\theta }}_1^{-}+{\mathrm{\theta }}_2^{-}}{2}},\max \left\{{\mathrm{\vartheta }}_1^{-},{\mathrm{\vartheta }}_2^{-}\right\}\right\}$;

3. 

${\mathrm{A}}_1\cap {\mathrm{A}}_2=\left\{\min \left\{{\mathrm{\mu }}_1^{+},{\mathrm{\mu }}_2^{+}\right\},{\displaystyle \frac{{\mathrm{\theta }}_1^{+}+{\mathrm{\theta }}_2^{+}}{2}},\max \left\{{\mathrm{\vartheta }}_1^{+},{\mathrm{\vartheta }}_2^{+}\right\},\max \left\{{\mathrm{\mu }}_1^{-},{\mathrm{\mu }}_2^{-}\right\},{\displaystyle \frac{{\mathrm{\theta }}_1^{-}+{\mathrm{\theta }}_2^{-}}{2}},\min \left\{{\mathrm{\vartheta }}_1^{-},{\mathrm{\vartheta }}_2^{-}\right\}\right\}$;

4. 

The complement of BNS ${\mathrm{A}}_1$is defined as ${\mathrm{A}}_1^{\mathrm{C}}=\left\{{\mathrm{\mu }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{+},{\mathrm{\theta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{+},{\mathrm{\vartheta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{+},{\mathrm{\mu }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{-},{\mathrm{\theta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{-},{\mathrm{\vartheta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{-}\right\}$

where ${\mathrm{\mu }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{+}=1-{\mathrm{\mu }}_1^{+}$, ${\mathrm{\theta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{+}=1-{\mathrm{\theta }}_1^{+}$, ${\mathrm{\vartheta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{+}=1-{\mathrm{\vartheta }}_1^{+}$, ${\mathrm{\mu }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{-}=1-{\mathrm{\mu }}_1^{-}$, ${\mathrm{\theta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{-}=1-{\mathrm{\theta }}_1^{-}$, ${\mathrm{\vartheta }}_{{\mathrm{A}}_1^{\mathrm{C}}}^{-}=1-{\mathrm{\vartheta }}_1^{-}$.

3. Reviewing the Distance Measure of IFSs

This section presents the definition and formula for the distance measure of intuitionistic fuzzy sets as proposed in reference [36]; it paves the way for the groundwork for the subsequent chapter.

Definition 7

([36]). Let ${\mathrm{A}}_1$, ${\mathrm{A}}_2$ be two IFSs and $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)$ be a real valued function. If it satisfies the following conditions:

(D1) 

$0\le d\left(A_1,A_2\right)\le 1$;

(D2) 

$d\left(A_1,A_2\right)=0$ if and only if $A_1=A_2$;

(D3) 

$d\left(A_1,A_2\right)=d\left(A_2,A_1\right)$;

(D4) 

If $A_1\subseteq A_2\subseteq A_3$, $A_1,A_2,A_3$ are three IFSs in X, then $d\left(A_1,A_2\right)\le d\left(A_1,A_3\right)$, $d\left(A_2,A_3\right)\le d\left(A_1,A_3\right)$;

then d is called the distance measure of intuitionistic fuzzy sets.

Theorem 1

([36]). Let ${\mathrm{A}}_1$, ${\mathrm{A}}_2$ be two IFSs in $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_{\mathrm{n}}\right\}$, where ${\mathrm{A}}_1=\left\{⟨{\mathrm{x}}_{\mathrm{i}}:{\mathrm{\mu }}_1\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\nu }}_1\left({\mathrm{x}}_{\mathrm{i}}\right)⟩\left|{\mathrm{x}}_{\mathrm{i}}\in \mathrm{X}\right.\right\}$, ${\mathrm{A}}_2=\left\{⟨{\mathrm{x}}_{\mathrm{i}}:{\mathrm{\mu }}_2\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\nu }}_2\left({\mathrm{x}}_{\mathrm{i}}\right)⟩\left|{\mathrm{x}}_{\mathrm{i}}\in \mathrm{X}\right.\right\}$.

$$d(A_1,A_2)={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left[{\displaystyle \frac{\left|{\mu }_1\left(x_i\right)-{\nu }_1\left(x_i\right)\right|+\left|{\mu }_2\left(x_i\right)-{\nu }_2\left(x_i\right)\right|}{4}}\right]+{\displaystyle \frac{1}{2}}\mathit{max}\{\left|{\mu }_1\left(x_i\right)-{\nu }_1\left(x_i\right)\right|,\left|{\mu }_2\left(x_i\right)-{\nu }_2\left(x_i\right)\right|\}$$

(1)

This is the distance measure of intuitionistic fuzzy sets.

4. The Novel Distance Measures of BNSs

4.1. The Definition of the Distance Measure of BNSs

Definition 8.

Let ${\mathrm{A}}_1$, ${\mathrm{A}}_2$ be two BNSs and $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)$ be a real valued function, which satisfies the following conditions:

(D1) 

$0\le d\left(A_1,A_2\right)\le 1$;

(D2) 

$d\left(A_1,A_2\right)=0$ if and only if $A_1=A_2$;

(D3) 

$d\left(A_1,A_2\right)=d\left(A_2,A_1\right)$;

(D4) 

If $A_1\subseteq A_2\subseteq A_3$, $A_1,A_2,A_3$ are bipolar neutrosophic sets in $\mathrm{X}$, then $d\left(A_1,A_2\right)\le d\left(A_1,A_3\right)$, $d\left(A_2,A_3\right)\le d\left(A_1,A_3\right)$.

then, d is called the distance measure of bipolar neutrosophic sets.

In the following, an introduction to traditional distance metrics is given, which serve as fundamental tools for measuring similarity or dissimilarity between objects.

4.2. Traditional Distance Measures and an Example

In this section, we will elaborate on the mathematical expressions of several typical traditional distance measures, such as Hamming, Euclidean, and Chebyshev distance measures.

Let ${\mathrm{A}}_1=\left\{{<\mathrm{\mu }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\theta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\vartheta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\mu }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\theta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\vartheta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)>|{\mathrm{x}}_{\mathrm{i}}\in \mathrm{X}\right\}$, ${\mathrm{A}}_2=\left\{{<\mathrm{\mu }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\theta }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\vartheta }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\mu }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\theta }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\vartheta }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)>|{\mathrm{x}}_{\mathrm{i}}\in \mathrm{X}\right\}$ be two BNSs in $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_{\mathrm{n}}\right\}$

The Hamming distance measure between ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ is as follows.

$$\mathrm{d}({\mathrm{A}}_1,{\mathrm{A}}_2)={\displaystyle \frac{1}{6\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left[{\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right]$$

(2)

The Hamming distance measure is used to measure the number of differing characters at corresponding positions in two equal-length strings. It can measure the degree of difference for fixed-length binary or character sequences. For example, it is widely used in cryptography and information retrieval.

The Euclidean distance measure between ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ is as follows.

$$\mathrm{d}({\mathrm{A}}_1,{\mathrm{A}}_2)=\sqrt{{\displaystyle \frac{1}{6\mathrm{n}}}{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left[{{\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2\right]}$$

(3)

The Euclidean distance measure describes the distance between two points in multidimensional space, which corresponds the most to our daily life. It can reflect the absolute differences between data points in algorithms such as clustering and classification.

The Chebyshev distance measure between ${\mathrm{A}}_1$and ${\mathrm{A}}_2$ is as follows:

$$\mathrm{d}({\mathrm{A}}_1,{\mathrm{A}}_2)=\max \{{\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right),{\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\}$$

(4)

where ${\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\mu }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\mu }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|$, ${\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\theta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\theta }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|$, ${\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\vartheta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\vartheta }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|$, ${\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\mu }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\mu }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|$, ${\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\theta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\theta }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|$, ${\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\vartheta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\vartheta }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|$.

The Chebyshev distance measure emphasizes the maximum of element-wise differences, which directly capture the most stringent constraints. It has been applied in path planning and industrial quality control.

Traditional distance measures have their own limitations in different scenarios, which may lead to the misjudgment or underestimation of data discrepancies in complex data scenarios.

Example 1.

Let ${\mathrm{A}}_1, {\mathrm{A}}_2, {\mathrm{A}}_3$ be three bipolar neutrosophic sets on $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2\right\}$, where

$$\begin{gathered} {\mathrm{A}}_1=\left\{⟨0.3,0.4,0.2,-0.6,-0.3,-0.7⟩,⟨0.8,0.6,0.3,-0.2,-0.6,-0.5⟩\right\}, \\ {\mathrm{A}}_2=\left\{⟨0.3,0.6,0.9,-0.5,-0.3,-0.5⟩,⟨0.3,0.3,0.1,-0.5,-0.1,-0.1⟩\right\}, \\ {\mathrm{A}}_3=\left\{⟨0.2,0.5,0.1,-0.5,-0.6,-0.7⟩,⟨0.4,0.7,0.4,-0.3,-0.6,-0.5⟩\right\} \end{gathered}$$

Using Formula (2), we obtain $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)=\mathrm{d}\left({\mathrm{A}}_2,{\mathrm{A}}_3\right)=0.28333$.

From the above example, we found that the distance measure between these two sets is equal. This equivalence indicates that the current distance measure is insufficiently sensitive to capture the disparities between these sets. To overcome this impasse, it is imperative to investigate more sophisticated techniques or alternative measures that can elucidate their nuanced differences.

4.3. A Hybrid Distance Measure and Examples

On the framework of intuitionistic fuzzy sets, Wang proposed a new distance measure in Formula (1) by integrating the respective advantages of the Hamming distance measure and the Chebyshev distance measure [36]. Inspired by this idea, the hybrid distance measure is extended to the environment of BNSs to more precisely characterize the subtle deviation between sets. So, a new distance measure (5) is proposed and is named the Hamming–Chebyshev hybrid distance measure (HCHD) as follows.

$$\begin{gathered} \mathrm{d}({\mathrm{A}}_1,{\mathrm{A}}_2)={\displaystyle \frac{1}{6\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left[{\mathrm{\mu }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}})+{\mathrm{\theta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}})+{\mathrm{\vartheta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}})+{\mathrm{\mu }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}})+{\mathrm{\theta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}})+{\mathrm{\vartheta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}})\right] \\ +{\displaystyle \frac{1}{6}}\max \{{\mathrm{\mu }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\theta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\vartheta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\mu }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\theta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\vartheta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}})\} \end{gathered}$$

(5)

then, we have the following theorem.

Theorem 2.

Formula (5) is the distance measure of bipolar neutrosophic sets ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$.

Proof. 

Obviously, $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)$ satisfies the condition (D1)–(D3) of Section 4.1; we just need to prove (D4).

For any bipolar neutrosophic sets of ${\mathrm{A}}_1$, ${\mathrm{A}}_2$ and ${\mathrm{A}}_3$, suppose ${\mathrm{A}}_1\subseteq {\mathrm{A}}_2\subseteq {\mathrm{A}}_3$; then, we need to prove $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)\le \mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_3\right)$, $\mathrm{d}\left({\mathrm{A}}_2,{\mathrm{A}}_3\right)\le \mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_3\right)$.

If ${\mathrm{A}}_1\subseteq {\mathrm{A}}_2\subseteq {\mathrm{A}}_3$, then

$${\mathrm{\mu }}_1^{+}\left(\mathrm{x}\right)\le {\mathrm{\mu }}_2^{+}\left(\mathrm{x}\right)\le {\mathrm{\mu }}_3^{+}\left(\mathrm{x}\right), {\mathrm{\theta }}_1^{+}\left(\mathrm{x}\right)\le {\mathrm{\theta }}_2^{+}\left(\mathrm{x}\right)\le {\mathrm{\theta }}_3^{+}\left(\mathrm{x}\right), {\mathrm{\vartheta }}_1^{+}\left(\mathrm{x}\right)\ge {\mathrm{\vartheta }}_2^{+}\left(\mathrm{x}\right)\ge {\mathrm{\vartheta }}_3^{+}\left(\mathrm{x}\right),$$

and ${\mathrm{\mu }}_1^{-}\left(\mathrm{x}\right)\ge {\mathrm{\mu }}_2^{-}\left(\mathrm{x}\right)\ge {\mathrm{\mu }}_3^{-}\left(\mathrm{x}\right)$, ${\mathrm{\theta }}_1^{-}\left(\mathrm{x}\right)\ge {\mathrm{\theta }}_2^{-}\left(\mathrm{x}\right)\ge {\mathrm{\theta }}_3^{-}\left(\mathrm{x}\right)$, ${\mathrm{\vartheta }}_1^{-}\left(\mathrm{x}\right)\le {\mathrm{\vartheta }}_2^{-}\left(\mathrm{x}\right)\le {\mathrm{\vartheta }}_3^{-}\left(\mathrm{x}\right)$.

It is easy to obtain the following.

$${\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\mu }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\mu }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\le \left|{\mathrm{\mu }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\mu }}_3^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|={\mathrm{\mu }}_{13}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

$${\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\theta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\theta }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\le \left|{\mathrm{\theta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\theta }}_3^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|={\mathrm{\theta }}_{13}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

$${\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\vartheta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\vartheta }}_2^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\le \left|{\mathrm{\vartheta }}_1^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\vartheta }}_3^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|={\mathrm{\vartheta }}_{13}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

$${\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\mu }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\mu }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\le \left|{\mathrm{\mu }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\mu }}_3^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|={\mathrm{\mu }}_{13}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

$${\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\theta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\theta }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\le \left|{\mathrm{\theta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\theta }}_3^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|={\mathrm{\theta }}_{13}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

$${\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)=\left|{\mathrm{\vartheta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\vartheta }}_2^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|\le \left|{\mathrm{\vartheta }}_1^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)-{\mathrm{\vartheta }}_3^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right|={\mathrm{\vartheta }}_{13}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)$$

By substituting these inequalities into the Formula (5), we derive the conclusion. Then, we complete the theorem. □

The proposed Hamming–Chebyshev hybrid distance measure (HCHD) strictly satisfies the non-negativity, symmetry, triangle inequality, and mathematical properties. Next, an example is given to show its application.

Example 2.

Let $A_1$ and $A_2$ be two bipolar neutrosophic sets on $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2\right\}$, where

$$A_1=\left\{⟨0.5,0.7,0.2,-0.7,-0.3,-0.6⟩,⟨0.7,0.6,0.3,-0.8,-0.5,-0.6⟩\right\}$$

$$A_2=\left\{⟨0.9,0.8,0.3,-0.8,-0.6,-0.3⟩,⟨0.8,0.7,0.5,-0.4,-0.2,-0.4⟩\right\}$$

Using Formula (5), we obtain $d\left(A_1,A_2\right)=0.14167$.

Example 3.

Next, the proposed distance measure (5) is applied to Example 1. Using Formula (2), $d\left(A_1,A_2\right)=d\left(A_2,A_3\right)=0.28333$ and using Formula (5), $d\left(A_1,A_2\right)=0.4,d\left(A_2,A_3\right)=0.4167$. From the above result, we see that the difference between $A_1$ and $A_2$ is slightly smaller; we conclude that they are closer.

The Hamming–Chebyshev hybrid distance measure we have defined not only considers the differences between elements of two sets, but also takes into account the maximum value of these element differences. In some cases, this approach is more stable and better aligned with the needs of practical problems. In many instances of practical modeling, weight is given to quantify the differences in the importance of different variables. For example, when we evaluate a student’s academic performance, exam scores are considered more important than attendance rates, so higher weights are assigned to exam scores.

To account for the importance of elements, the following weighted distance measure of bipolar neutrosophic numbers is proposed. The weight assigned to an element ${\mathrm{x}}_{\mathrm{i}}\in \mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_{\mathrm{n}}\right\}$ is denoted by ${\mathrm{w}}_{\mathrm{i}}$, $0\le {\mathrm{w}}_{\mathrm{i}}\le 1$, ${\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}=1$, $\left(\mathrm{i}=1,2,\cdots ,\mathrm{n}\right)$.

$$\begin{gathered} {\mathrm{d}}_{\mathrm{w}}({\mathrm{A}}_1,{\mathrm{A}}_2)={\displaystyle \frac{1}{6}}\sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{w}}_{\mathrm{i}}\left[{\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)+{\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)\right] \\ +{\displaystyle \frac{1}{6}}\max \{{\mathrm{\mu }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\theta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\vartheta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\mu }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\theta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\vartheta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}})\} \end{gathered}$$

(6)

Based on the aforementioned Definition 8, the following theorem can be formulated.

Theorem 3.

Formula (6) is the distance measure of bipolar neutrosophic sets ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$.

The proof of this theorem is similar to Theorem 1. Specifically, when ${\mathrm{w}}_{\mathrm{i}}={\displaystyle \frac{1}{\mathrm{n}}}$, ${\mathrm{d}}_{\mathrm{w}}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)$ becomes $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)$.

Example 4.

Consider two bipolar neutrosophic sets ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ on $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2\right\}$ defined in Example 1; the weight of ${\mathrm{x}}_1,{\mathrm{x}}_2$ is ${\mathrm{w}}_1=0.4$, ${\mathrm{w}}_2=0.6$.

Using Formula (6), we obtain $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)=0.14$.

Corollary 1.

Consider two bipolar neutrosophic sets ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ in $\mathrm{X}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_{\mathrm{n}}\right\}$, ${\mathrm{A}}_1^{\mathrm{C}}and{\mathrm{A}}_2^{\mathrm{C}}$ are the complement of ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$; they are denoted as follows:

$$\begin{gathered} {\mathrm{A}}_1=\left\{{\mathrm{\mu }}_1^{+}\left(\mathrm{x}\right),{\mathrm{\theta }}_1^{+}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_1^{+}\left(\mathrm{x}\right),{\mathrm{\mu }}_1^{-}\left(\mathrm{x}\right),{\mathrm{\theta }}_1^{-}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_1^{-}\left(\mathrm{x}\right)\right\} \\ {\mathrm{A}}_2=\left\{{\mathrm{\mu }}_2^{+}\left(\mathrm{x}\right),{\mathrm{\theta }}_2^{+}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_2^{+}\left(\mathrm{x}\right),{\mathrm{\mu }}_2^{-}\left(\mathrm{x}\right),{\mathrm{\theta }}_2^{-}\left(\mathrm{x}\right),{\mathrm{\vartheta }}_2^{-}\left(\mathrm{x}\right)\right\} \\ {\mathrm{A}}_1^{\mathrm{C}}=\left\{{1-\mathrm{\mu }}_1^{+}\left(\mathrm{x}\right),{1-\mathrm{\theta }}_1^{+}\left(\mathrm{x}\right),1-{\mathrm{\vartheta }}_1^{+}\left(\mathrm{x}\right),{1-\mathrm{\mu }}_1^{-}\left(\mathrm{x}\right),1-{\mathrm{\theta }}_1^{-}\left(\mathrm{x}\right),1-{\mathrm{\vartheta }}_1^{-}\left(\mathrm{x}\right)\right\} \\ {\mathrm{A}}_2^{\mathrm{C}}=\left\{{1-\mathrm{\mu }}_2^{+}\left(\mathrm{x}\right),{1-\mathrm{\theta }}_2^{+}\left(\mathrm{x}\right),1-{\mathrm{\vartheta }}_2^{+}\left(\mathrm{x}\right),1-{\mathrm{\mu }}_2^{-}\left(\mathrm{x}\right),{1-\mathrm{\theta }}_2^{-}\left(\mathrm{x}\right),1-{\mathrm{\vartheta }}_2^{-}\left(\mathrm{x}\right)\right\} \end{gathered}$$

then, $\mathrm{d}\left({\mathrm{A}}_1,{\mathrm{A}}_2\right)=\mathrm{d}\left({\mathrm{A}}_1^{\mathrm{C}},{\mathrm{A}}_2^{\mathrm{C}}\right)$.

The proof of this theorem is obvious.

Analogous to the Hamming–Chebyshev hybrid distance measure mentioned, the Euclidean–Chebyshev hybrid distance measure (ECHD) is proposed. The proposed distance measure in this paper integrates the strengths of two classical distance metrics. While retaining geometric intuitiveness and mathematical rigor, it enhances the ability to characterize cumulative differences and extreme deviations in multi-dimensional data. The practical application of this distance measure is evident in its relevance to real-world problems in sensor data interpretation, fault detection, bio-informatics, complex data, and other engineering challenges.

$$\begin{gathered} \mathrm{d}({\mathrm{A}}_1,{\mathrm{A}}_2)=\sqrt{{\displaystyle \frac{1}{6\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left[{{\mathrm{\mu }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\theta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\vartheta }}_{12}^{+}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\mu }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\theta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2+{{\mathrm{\vartheta }}_{12}^{-}\left({\mathrm{x}}_{\mathrm{i}}\right)}^2\right]} \\ +{\displaystyle \frac{1}{6}}\max \{{\mathrm{\mu }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\theta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\vartheta }}_{12}^{+}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\mu }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\theta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}}),{\mathrm{\vartheta }}_{12}^{-}({\mathrm{x}}_{\mathrm{i}})\} \end{gathered}$$

(7)

There are many similarities between Formula (7) and Formula (5). They will not be elaborated on here.

5. An Application of the HCDM to Pattern Classification

5.1. The HCDM Method

• A pattern classification is stated as follows.

Assuming there are k known categories ${\mathrm{A}}_1,{\mathrm{A}}_2,\dots ,{\mathrm{A}}_{\mathrm{k}}$ in the feature space $C_1,C_2,$ $\dots ,C_{\mathrm{l}}$, and another unknown object B, it is necessary to determine to which of these k schemes the unknown object belongs. The inputs are the symptoms of categories that are expressed by BNSs.

• The HCDM method:

Step 1. Build the pattern classification matrix in the bipolar neutrosophic environment.

Step 2. Calculate the distance measures between the i-th category $A_i$ and the object B.

Step 3. Rank the distance values in ascending order.

Step 4. Based on the ranking result, a conclusion is drawn. A smaller distance measure indicates a higher similarity between the unknown object and the i-th category, and the unknown object can be assigned to the i-th category.

5.2. Numerical Examples for Pattern Classification

To illustrate a practical application, the pattern classification of building materials within a feature space framework. The scenario consists of three material classes, each represented by a bipolar neutrosophic set (BNS) in the feature space. In this section, the data in the numerical example in [31] are used. We assume that there are three types of building materials $A_1$, $A_2$, $A_3$ and the feature space is $C_1$, $C_2$, $C_3$, $C_4$. The weight vector of $C_1$, $C_2$, $C_3$, $C_4$ is ${\mathrm{w}}_1=0.25$, ${\mathrm{w}}_2=0.25$, ${\mathrm{w}}_3=0.25$, ${\mathrm{w}}_4=0.25$. Next, an unknown building material $B$ is given; our aim is to assign the unidentified pattern $B$ to the existing clusters $A_1$, $A_2$, $A_3$.

The new algorithm is listed as follows:

Step 1. The three known classes with respect to the above four criteria are evaluated by the following bipolar neutrosophic decision matrix $\mathrm{D}$:

$$\mathrm{D}=\left[\begin{array}{c}{A}_1\\ A_2\\ A_3\end{array}\right]=\left[\begin{array}{cccc}{\mathrm{\gamma }}_{11}& {\mathrm{\gamma }}_{12}& {\mathrm{\gamma }}_{13}& {\mathrm{\gamma }}_{14}\\ {\mathrm{\gamma }}_{21}& {\mathrm{\gamma }}_{22}& {\mathrm{\gamma }}_{23}& {\mathrm{\gamma }}_{24}\\ {\mathrm{\gamma }}_{31}& {\mathrm{\gamma }}_{32}& {\mathrm{\gamma }}_{33}& {\mathrm{\gamma }}_{34}\end{array}\right]$$

where

$$\begin{gathered} {\mathrm{\gamma }}_{11}=⟨0.5,0.7,0.2,-0.7,-0.3,-0.6⟩,{\mathrm{\gamma }}_{12}=⟨0.6,0.4,0.5,-0.7,-0.8,-0.4⟩ \\ {\mathrm{\gamma }}_{13}=⟨0.7,0.7,0.5,-0.8,-0.7,-0.6⟩,{\mathrm{\gamma }}_{14}=⟨0.1,0.5,0.7,-0.5,-0.2,-0.8⟩ \\ {\mathrm{\gamma }}_{21}=⟨0.8,0.7,0.5,-0.7,-0.7,-0.1⟩,{\mathrm{\gamma }}_{22}=⟨0.7,0.6,0.8,-0.7,-0.5,-0.1⟩ \\ {\mathrm{\gamma }}_{23}=⟨0.9,0.4,0.6,-0.1,-0.7,-0.5⟩,{\mathrm{\gamma }}_{24}=⟨0.5,0.2,0.7,-0.5,-0.1,-0.9⟩ \\ {\mathrm{\gamma }}_{31}=⟨0.3,0.4,0.2,-0.6,-0.3,-0.7⟩,{\mathrm{\gamma }}_{32}=⟨0.2,0.2,0.2,-0.4,-0.7,-0.4⟩ \\ {\mathrm{\gamma }}_{33}=⟨0.9,0.5,0.5,-0.6,-0.5,-0.2⟩,{\mathrm{\gamma }}_{24}=⟨0.7,0.5,0.3,-0.4,-0.2,-0.2⟩ \end{gathered}$$

The unknown building material $\mathrm{B}$ is denoted by $\mathrm{B}=\left({\mathrm{B}}_1,{\mathrm{B}}_2,{\mathrm{B}}_3,{\mathrm{B}}_4\right)$,

$$\begin{gathered} {\mathrm{B}}_1=⟨0.9,0.7,0.2,-0.8,-0.6,-0.1⟩,{\mathrm{B}}_2=⟨0.3,0.5,0.3,-0.5,-0.5,-0.2,⟩ \\ {\mathrm{B}}_3=⟨0.5,0.4,0.5,-0.1,-0.7,-0.2⟩,B_4=⟨0.6,0.2,0.8,-0.5,-0.5,-0.6⟩ \end{gathered}$$

Step 2. The HCHD of $\mathrm{B}$ and ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, ${\mathrm{A}}_3$are calculated by Formula (6) as follows: $\mathrm{d}\left({\mathrm{A}}_1,\mathrm{B}\right)=0.2708$, $\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)=0.1707$, $\mathrm{d}\left({\mathrm{A}}_3,\mathrm{B}\right)=0.2708$.

Step 3. Rank the distance measure by the values; we found $\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)=0.1707$ is the smallest. That means the smaller the difference between the object and the given patterns is, the better their similarity is.

Step 4. The unknown building material $\mathrm{B}$ belongs to ${\mathrm{A}}_2$.

In this subsection, the distance values are rounded to four decimal places. Based on the obtained values, we rank the order to draw a conclusion. Specifically, in this example, we conclude that the object B belongs to ${\mathrm{A}}_2$. However, the rankings are somewhat different depending on the algorithm used, as observed below.

5.3. Comparison Analysis and Discussion

In order to verify the feasibility and effectiveness of the proposed distance measure method, a comparison analysis with the considered methods [31,35] is given based on the same illustrative Example in Section 5.2 in Table 1.

5.3.1. Method of [31]

• Step 1. As presented in Reference [31], the unknown building material $\mathrm{B}$ and $\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{s}{\mathrm{A}}_1$, ${\mathrm{A}}_2$, ${\mathrm{A}}_3$ are as follows.

  $$\mathrm{d}\left({\mathrm{A}}_1,\mathrm{B}\right)=0.0696,\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)=0.0343,\mathrm{d}\left({\mathrm{A}}_3,\mathrm{B}\right)=0.0465$$
• Step 2. Rank the distance measure by the values; $\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)=0.0343\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{s}\mathrm{t}.$
• Step 3. The unknown building material $\mathrm{B}$ belongs to ${\mathrm{A}}_2$.

5.3.2. Method of [35]

• Step 1. Use the H-Max distance measure to calculate the unknown building material $\mathrm{B}$ and ${\mathrm{k}\mathrm{n}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{A}}_1$, ${\mathrm{A}}_2$, ${\mathrm{A}}_3$ as follows: $\mathrm{d}\left({\mathrm{A}}_1,\mathrm{B}\right)=0.2563$, $\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)=0.2688$, $\mathrm{d}\left({\mathrm{A}}_3,\mathrm{B}\right)=0.$2813.
• Step 2. Rank the distance measure by the values; $\mathrm{d}\left({\mathrm{A}}_1,\mathrm{B}\right)=0.2563$ is the smallest.
• Step 3. The unknown building material $\mathrm{B}$ belongs to ${\mathrm{A}}_1$.

Table 1. The comparison of all the different methods.

Method	The Smallest Distance Measure	$\mathbf{B}$ Belongs to Which Category
Method of [31]	$\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)$	${\mathrm{A}}_2$
Method of [35]	$\mathrm{d}\left({\mathrm{A}}_1,\mathrm{B}\right)$	${\mathrm{A}}_1$
The proposed method	$\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)$	${\mathrm{A}}_2$

Table 1. The comparison of all the different methods.

Method	The Smallest Distance Measure	$\mathbf{B}$ Belongs to Which Category
Method of [31]	$\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)$	${\mathrm{A}}_2$
Method of [35]	$\mathrm{d}\left({\mathrm{A}}_1,\mathrm{B}\right)$	${\mathrm{A}}_1$
The proposed method	$\mathrm{d}\left({\mathrm{A}}_2,\mathrm{B}\right)$	${\mathrm{A}}_2$

All the distance measure methods are applicable to complex decision-making problems and pattern classification problems. The result of Reference [31] is consistent with the proposed method, which takes into account the cumulative effect of multi-dimensional differences and the extreme deviations of key dimensions. Neither distance measure contains parameters. In contrast, the method in Reference [35] may involve parameters with similarity and entropy during the data processing stage, which leads to differences in mapping relationships, thereby resulting in divergent results.

6. Conclusions

This paper studied a novel distance measure of bipolar neutrosophic sets (BNSs), which is rigorously constructed to quantify the dissimilarity between two BNSs. The proposed distance measure satisfies all the requisite properties of distance measures, which not only accounts for the pointwise differences between corresponding elements of two bipolar neutrosophic sets, but also incorporates the maximum deviation among these differences and leads to a more nuanced and globally reflective assessment of dissimilarity. In some cases, this approach is more stable and better aligned with the needs of practical problems. The Hamming distance measure captures differences in discrete features and the Chebyshev distance focuses on the maximum deviation of extreme values across dimensions. The integration of the Hamming distance measure and the Chebyshev distance measure combines dual-dimensional detection capabilities to make the distance measure more comprehensive and accurate. On the basis of the distance measure defined above, an algorithm is developed to deal with the problem of pattern classification. The experimental results demonstrate the effectiveness and applicability of the method. In many contexts, distance measure and similarity measure are inversely related: a smaller distance measure often implies a greater similarity measure between objects. In the future, we can study the relationship between them and apply the proposed distance measure to many scenarios, such as natural language processing, computer vision, recommendation systems, and so on.