A Novel Clark Distance-Based Decision-Making Algorithm on Intuitionistic Fuzzy Sets

Abstract

Fuzzy sets possess remarkable abilities in expressing and handling information uncertainty, which has resulted in their widespread application in various fields. Nevertheless, distance measurement between IFSs for quantitating their differences and levels of differentiation has remained an open problem that deserves attention. Despite the development of various metrics, they either lack intuitive insight or do not satisfy the axioms of distance measurement, leading to counterintuitive results. To address these issues, this paper proposed a distance measurement method based on Clark divergence, which satisfies the distance measurement axioms and exhibits nonlinearity. Numerical examples demonstrate that our method effectively distinguishes different indicators, yielding more reasonable results. Moreover, when comparing relative differences of the results, our method demonstrated superior adaptability to complex environmental decision-making, providing decision-makers with more accurate and confidential judgments. In our numerical and pattern classification application tests, we achieve an accuracy of 98%, a 40% increase in computing time efficiency and a relative diversity improvement of 35%. The pattern classification algorithm designed in this paper will offer a promising solution to inference problems.

1. Introduction

Uncertainty is widespread in real life [1,2,3,4,5,6,7,8,9]. Based on fuzzy mathematics, the fuzzy set theory describes objects that are uncertain, fuzzy or incomplete [1]. In traditional mathematics, an element belongs to a set, whereas in fuzzy set theory, an element can belong to more than one set, and its degree of membership is expressed as a real number between 0 and 1. The concept of fuzzy sets is intuitive and flexible, well-suited for addressing uncertain and vague issues in the real world, such as those encountered in engineering, artificial intelligence, and control systems [10,11,12]. The essence of fuzzy set theory is captured by the concept of membership functions, which perceive the relationship between elements and sets as a gradual process instead of a distinct boundary. This concept allows us to capture the degree of membership between elements and sets, thereby characterizing the level of ambiguity and uncertainty. A fuzzy set is an effective way of dealing with uncertainty problems, but one should define membership functions clearly; otherwise, it may not apply to fuzzy concepts with hard-to-define membership. The intuitive fuzzy set, an extension of fuzzy set theory, expresses fuzzy concepts intuitively without the need for explicit membership functions [13,14]. Briefly, intuitionistic fuzzy sets are fuzzy sets that stem from personal subjective judgments and intuition. The definition of intuitionistic fuzzy sets parallels that of fuzzy sets, indicating the membership degree of elements within a set, yet it does not necessitate explicit membership functions. The primary benefits of intuitionistic fuzzy sets are flexibility and adaptability, allowing them to be expanded and adjusted in line with subjective experience and judgment. Building on this concept, a variety of extensions have been developed, including intuitionistic fuzzy numbers, interval intuitionistic fuzzy sets, and interval intuitionistic fuzzy numbers [15,16,17].

Intuitionistic fuzzy set distances find diverse applications. In image recognition, they are utilized to gauge the similarity between images by converting the intuitionistic fuzzy sets of the two images into feature vectors and assessing their similarity via distance computations. In data mining [18], they are employed to compare the similarities between datasets. Similarly, the intuitionistic fuzzy sets of two datasets can be represented as feature vectors, and their similarity can be evaluated through distance calculation. Pattern recognition [19,20,21,22] also benefits from this technique by comparing the similarities of various patterns. Two patterns’ intuitionistic fuzzy sets can be represented as feature vectors, and their similarity is determined by calculating the distance. In natural language processing [23], it facilitates the comparison of similarities among different texts. The intuitionistic fuzzy sets of two texts can be represented as feature vectors, and their similarity can be assessed by calculating the distance.

Xiao [24] proposed an IFS distance measurement method with nonlinear features based on the Johnson–Shannon divergence. Patel introduced three new distance measurement methods based on Csiszarf-Divergence. Gohain [25] proposed a method for determining the distance between IFS information. Mahanta [26] defined a distance function and verified the axiom definition. It is applicable in pattern recognition and medical diagnosis. Hao [27] defined a measure that uses a nonlinear distance function and additional parameters to control its recognition ability. Liu [15] redefined the distance between IFSs. Ashraf [28] proposed a generalized difference sequence spatial distance measurement method that utilizes summable intuitionistic fuzzy bounded variation (IFBV). Ren [29] defined metric information matrices (MIMs) for IFSs using the metric matrix theory. Saqlain [30] developed distance and similarity measures for intuitionistic fuzzy hypersoft sets (IFHSSs) using aggregate operators. Wan [31] proposed a large-scale group decision-making task based on time series and multicriteria, and involving intuitionistic fuzzy information. Mardani [32] proposed a method named the Composite Proportional Evaluation (COPRAS) based on intuitive fuzzy sets (IFSs). Zadeh [33] proposed a fuzzy theory based on information granularity. Kumar R [34] proposed a method for analyzing the sustainable biomass crop selection (SBCS) problem in intuitionistic fuzzy environments. Wang T [35] established a dual relationship between IFDisM and IFSimM based on fuzzy negation. Khan M S [36] introduced a new method for generating the intuitionistic fuzzy Vietoris–Rips complex (IFVRC) of IFSs using intuitionistic fuzzy distance measures (IFDMs). Khan Z [37] developed various distance measures in complex image blur environments. Finally, we devised an algorithm that utilizes complex image blurring environments and applies them to practical applications in decision-making, medical diagnosis, and pattern recognition problems. Alreshidi N A [38] introduced the similarity and dissimilarity measures of C-IFS with proven basic axioms and revealed features. The axiomatic definition of C-IFS entropy measure is provided. Mishra A R [39] proposes a new distance measure between image fuzzy sets (PFSs) to overcome existing shortcomings. Additionally, a weight determination method is suggested for determining attribute weights using the proposed distance measure between PFS. Patel A [40] introduced a new intuitionistic fuzzy similarity measure (IFSM) that considers the global maximum and minimum differences between members and non-members, as well as their individual differences. Ashraf S [41] proposed the innovative concept of complex intuitionistic hesitant fuzzy sets (CIHFSs), which combines intuitionistic hesitant fuzzy sets with complex fuzzy sets. Huang W [42] studied and utilized the exploratory sum of two most-fuzzy sets, defining a similarity measure using weighted average distances between IFSs. Kumar R [34] proposed a novel ensemble function based on the double normalized multiple-aggregation method to mitigate the aggregation bias of the original CoCoSo method.

Within the existing literature, we observe specific undefined points within the proposed distance formula. These points, particularly those situated on the IFS boundary, carry practical significance. They represent membership, non-membership, or hesitation, respectively. However, the proposed distance formula cannot address these issues. Although the proposed distance formula can yield accurate results in certain complex scenarios, distinguishing the correct outcomes becomes challenging and unique due to numerous slight differences in distance values during the calculation process. The greater the distance value variations calculated by the formula, the stronger its uniqueness, suggesting that it is better suited for making sound decisions in complex situations.

This paper introduces a novel distance measurement method for intuitionistic fuzzy sets (IFSs) utilizing Clark divergence. The method takes into account the three-dimensional representation of IFSs, encompassing their membership, non-membership, and hesitation functions. The findings confirm that the new distance measurement satisfies the essential properties of distance measurement. Furthermore, numerical examples are offered to illustrate the superiority of the proposed method in distinguishing IFSs over existing approaches. The proposed distance measurement algorithm is applicable to medical diagnostic issues.

The structure of this article is as follows: Section 1 provides a brief overview of prior research on this topic. Section 2 reviews some basic concepts of the Clark divergence metric and the existing distance and similarity metrics in the IFS. Section 3 defines and analyzes the Clark distance measurement method of IFSs. Section 4 illustrates the superiority of the proposed method through numerical comparisons and applies the algorithm to pattern classification problems, demonstrating its practicality. Finally, Section 5 summarizes the contributions and findings of this article.

2. Related Work

This section reviews some fundamental concepts related to IFSs and the Clark divergence measurement.

2.1. Intuitionistic Fuzzy Sets

In various applications of modern society, information uncertainty is a common problem, and as the amount and complexity of information increase, this uncertainty becomes more prominent. Effective methods must be sought to address the uncertainty issues in information fusion, in order to improve the accuracy and reliability of decision-making. In recent years, various information fusion methods have been proposed and applied, including the negation method, entropy method, belief function method, gray prediction model method, and DEMATEL algorithm, among which fuzzy sets and IFSs are the most commonly used methods. Due to their ability to effectively handle uncertain information and fuzzy problems, these methods are widely adopted in practical application scenarios.

Let X be a non-empty set called the universe of discourse (UOD). x is the X subset, denoted as $x\subseteq X$ and $x_i$ is an element in x, denoted as $x_i\in x$,$x=\left\{x_1,x_2,\dots ,x_n\right\}$. An intuitionistic fuzzy set A on X can be represented as follows:

$$A=\left\{〈x,{\alpha }_A\left(x\right),{\beta }_A\left(x\right)〉\mid x\subseteq X\right\}$$

(1)

It can also be written as

$$A=\left\{〈x_i,{\alpha }_A\left(x_i\right),{\beta }_A\left(x_i\right)〉\mid x_i\in x\right\}$$

(2)

The variables ${\alpha }_A\left(x_i\right)$ and ${\beta }_A\left(x_i\right)$ represent the membership degree and non-membership degree of element X from the UOD A in the intuitionistic fuzzy set x.

$$\begin{array}{cc}\hfill & {\alpha }_A\left(x_i\right):X\to [0,1],x\in X\to {\alpha }_A\left(x_i\right)\in [0,1]\hfill \\ \hfill & {\beta }_A\left(x_i\right):X\to [0,1],x\in X\to {\beta }_A\left(x_i\right)\in [0,1]\hfill \end{array}$$

(3)

Their values satisfy $x\in X,0\le {\alpha }_A\left(x_i\right)+{\beta }_A\left(x_i\right)\le 1,{\gamma }_A\left(x_i\right)=1-{\alpha }_A\left(x_i\right)-{\beta }_A\left(x_i\right)$.

The degree ${\gamma }_A\left(x\right)$ denotes the hesitancy or uncertainty with which X belongs to A, with $0\le {\gamma }_A\left(x\right)\le 1$ for all $x_i$.

Let S and T be two IFSs in X. An intuitionistic fuzzy relation from $\mathrm{S}$ to $\mathrm{T}$, denoted as $IFR(S\to T)$, is a subset of the Cartesian product $S\times T$, and is represented by membership functions ${\alpha }_{IFR}$ and ${\beta }_{IFR}$.

Assume that $\xi =〈x,{\alpha }_{\xi }\left(x_i\right),{\beta }_{\xi }\left(x_i\right)〉$ and $\eta =〈x,{\alpha }_{\eta }\left(x_i\right),{\beta }_{\eta }\left(x_i\right)〉$ are two IFSs in X, then, the following hold:

(1) $\xi \cap \eta =\left\{min\left\{{\alpha }_{\xi }\left(x_i\right),{\alpha }_{\eta }\left(x_i\right)\right\},max\left\{{\beta }_{\xi }\left(x_i\right),{\beta }_{\eta }\left(x_i\right)\right\}\right\}$;

(2) $\xi \cup \eta =\left\{max\left\{{\alpha }_{\xi }\left(x_i\right),{\alpha }_{\eta }\left(x_i\right)\right\},min\left\{{\beta }_{\xi }\left(x_i\right),{\beta }_{\eta }\left(x_i\right)\right\}\right\}$;

(3) $\xi \subseteq \eta \iff \forall x\in X,{\alpha }_{\xi }\left(x_i\right)\le {\alpha }_{\eta }\left(x_i\right),{\beta }_{\xi }\left(x_i\right)\ge {\beta }_{\eta }\left(x_i\right)$;

(4) $\xi =\eta \iff \forall x\in X,{\alpha }_{\xi }\left(x_i\right)={\alpha }_{\eta }\left(x_i\right),{\beta }_{\xi }\left(x_i\right)={\beta }_{\eta }\left(x_i\right)$.

2.2. Clark Distance Algorithm

The Clark distance is widely used in clustering analysis and classification algorithms, particularly for situations with small sample points. It is efficient in identification and segmentation, with a straightforward and rapid calculation method. As a result, it has been applied in many fields [43,44,45].

Let E and F be two probability distributions of the discrete random variable U with $E=\left\{e_1,e_2,\dots ,e_n\right\}$ and $F=\left\{f_1,f_2,\dots ,f_n\right\}$. The Clark distance between E and F is defined as follows:

$$\begin{array}{c}\hfill Clk(E,F)={\left({\displaystyle \frac{|E-F|}{E+F}}\right)}^2\end{array}$$

(4)

where ${\sum }_i{e}_i={\sum }_i{f}_i=1$.

The above question can also be expressed by

$$\begin{array}{c}\hfill Clk(E,F)=\sum _i{\left({\displaystyle \frac{\left|e_i-f_i\right|}{e_i+f_i}}\right)}^2\end{array}$$

(5)

The square root of $Clk$ is calculated by

$$\begin{array}{c}\hfill S{R}_{Clk}=\sqrt{Clk(E,F)}\end{array}$$

(6)

Property 1:

(1) ${\mathrm{SR}}_{\mathrm{Clk}}(E,F)\ge 0$, where ${\mathrm{SR}}_{\mathrm{Clk}}(E,F)=0$ if $E=F$, for $E,F\in U$;

(2) ${\mathrm{SR}}_{\mathrm{Clk}}(E,F)={\mathrm{SR}}_{\mathrm{Clk}}(F,E)$, for $E,F\in U$;

(3) If $E\subseteq F\subseteq \mathrm{K}$, then ${\mathrm{SR}}_{\mathrm{Clk}}(E,F)\le {\mathrm{SR}}_{\mathrm{Clk}}(E,\mathrm{K})$ and ${\mathrm{SR}}_{\mathrm{Clk}}(F,K)\le {\mathrm{SR}}_{\mathrm{Clk}}(F,K)$;

(4) $0\le {\mathrm{SR}}_{\mathrm{Clk}}(E,F)\le 1$, for $E,F\in U$.

3. Clark Distance Algorithm of IFSs

This section is divided into five parts to illustrate the Clark distance algorithm for a comprehensive financing strategy.

3.1. Initialization Procedure

In this section, we propose a new distance metric for IFSs and derive and demonstrate its properties using the Clark distance.

Properties of the Clark distance.

We define $\xi =\left\{x,{\alpha }_{\xi }\left(x\right),{\beta }_{\xi }\left(x\right)\mid x\subseteq X\right\}$ and $\eta =\left\{x,{\alpha }_{\eta }\left(x\right),{\beta }_{\eta }\left(x\right)\mid x\subseteq X\right\}$ as two IFSs. The Clark distance between the IFSs is defined as follows:

$$\begin{array}{c}\hfill Cl{k}_{IFS}=\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x\right)-{\gamma }_{\eta }\left(x\right)\right|}{{\gamma }_{\xi }\left(x\right)+{\gamma }_{\eta }\left(x\right)}}\right)}^2\right]/2\end{array}$$

(7)

Afterward, the intuitionistic fuzzy divergence defines a new IFS distance metric. For two IFSs $\mathrm{P}$ and $\mathrm{Q}$ in the finite set $\mathrm{X}$, the new distance measure between the $\mathrm{IFSs}$ is defined as $d_c(\xi ,\eta )$ and is denoted as

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=\sqrt{Cl{k}_{IFS}(\xi ,\eta )}\end{array}$$

(8)

The properties of $d_c$ are deduced as follows.

Meanwhile, given that the degree of membership, non-membership, and hesitation do not exhibit linear correlations with real-world data, and some of these correlations even entail subjectivity, it is imperative to guarantee that the corresponding intuitionistic fuzzy set decision matrix has excellent representativeness prior to the application of this algorithm. Additionally, as there are denominators in the algorithm’s computations, it is advisable to ensure that all three measures are not all zero throughout the calculation process.

Properties: Let $\xi ,\eta ,\vartheta$ be three IFSs in the set X, then, the following hold:

(P1): $d_c(\xi ,\eta )=0$ if $\xi =\eta$, for $\xi ,\eta \in X$;

(P2): $d_c(\xi ,\eta )=d_c(\eta ,\xi )$, for $\xi ,\eta \in X$;

(P3): If $\xi \subseteq \eta \subseteq \vartheta$, then $d_c(\xi ,\eta )\le d_c(\xi ,\vartheta )$ and $d_c(\eta ,\vartheta )\le d_c(\xi ,\vartheta )$;

(P4): $0\le d_c(\xi ,\eta )\le 1$, for $\xi ,\eta \in X$.

3.2. Reflexivity Verification Procedure

The method of proof by contradiction underpins logical reasoning and mathematical proofs. In the process of reasoning and proving, one can often depend on the intrinsic relationships among elements, serving as the basis for numerous proofs.

Providing (P1): Given two IFSs $\mathrm{P}$ and $\mathrm{Q}$ in $\mathrm{X}$, such that $\xi =\eta$, we have

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=0\end{array}$$

(9)

When $d_c(\xi ,\eta )=0$, we have

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=\sqrt{\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x\right)-{\gamma }_{\eta }\left(x\right)\right|}{{\gamma }_{\xi }\left(x\right)+{\gamma }_{\eta }\left(x\right)}}\right)}^2\right]/2}=0\end{array}$$

(10)

Thus, it can be inferred that

$$\begin{array}{c}\hfill {\alpha }_{\xi }\left(x\right)={\alpha }_{\eta }\left(x\right),{\beta }_{\xi }\left(x\right)={\beta }_{\eta }\left(x\right),{\gamma }_{\xi }\left(x\right)={\gamma }_{\eta }\left(x\right)\end{array}$$

(11)

Therefore, the following conclusion can be reached:

$$\begin{array}{c}\hfill \xi =\eta \end{array}$$

(12)

It has been proven that the reflexivity of $d_c$ is satisfied.

Inversion ensures the integrity of a relationship within its domain, representing a fundamental characteristic of the relationship. Through this approach, validation aids in our understanding of various relationship types, thereby revealing and highlighting their similarities and differences.

3.3. Symmetry Verification Procedure

Symmetry verification is one of the important means to test the correctness of mathematical formulas, symmetry can be used to simplify the analysis of complex problems in order to improve the efficiency of problem solving.

Providing (P2): Given $d_c(\xi ,\eta )$, we have

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=\sqrt{\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x\right)-{\gamma }_{\eta }\left(x\right)\right|}{{\gamma }_{\xi }\left(x\right)+{\gamma }_{\eta }\left(x\right)}}\right)}^2\right]/2}\end{array}$$

(13)

With $d_c(\eta ,\xi )$, we have

$$\begin{array}{c}\hfill d_c(\eta ,\xi )=\sqrt{\left[{\left({\displaystyle \frac{\left|{\alpha }_{\eta }\left(x\right)-{\alpha }_{\xi }\left(x\right)\right|}{{\alpha }_{\eta }\left(x\right)+{\alpha }_{\xi }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\eta }\left(x\right)-{\beta }_{\xi }\left(x\right)\right|}{{\beta }_{\eta }\left(x\right)+{\beta }_{\xi }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\gamma }_{\eta }\left(x\right)-{\gamma }_{\xi }\left(x\right)\right|}{{\gamma }_{\eta }\left(x\right)+{\gamma }_{\xi }\left(x\right)}}\right)}^2\right]/2}\end{array}$$

(14)

Thus,

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=d_c(\eta ,\xi )\end{array}$$

(15)

is still valid after variable exchange or parameter transformation.

It has been proven that the symmetry of $d_c$ is satisfied.

The validation of symmetry enhances the accuracy and broad utility of the method in different situations, and also provides an important theoretical basis for further theoretical development and experimental validation.

3.4. Verification Procedure of Triangle Inequality

In vector space or metric space, the measure of distance has to satisfy a triangle inequality. Verifying that the equation satisfies the triangle inequality ensures that the measure of distance is geometrically and mathematically consistent and helps to ensure the accuracy and reliability of the model in describing and predicting real-world problems.

Providing (P3): If $\xi \subseteq \eta \subseteq \vartheta$, then $d_c(\xi ,\eta )\le d_c(\xi ,\vartheta )$ and $d_c(\eta ,\vartheta )\le d_c(\xi ,\vartheta )$. For any $x\in X$, when $\xi \subseteq \eta \subseteq \vartheta$, we then have

$${\alpha }_{\xi }\left(x\right)\le {\alpha }_{\eta }\left(x\right)\le {\alpha }_{\vartheta }\left(x\right),{\beta }_{\vartheta }\left(x\right)\le {\beta }_{\eta }\left(x\right)\le {\beta }_{\xi }\left(x\right)$$

By the universe of discourse (UOD), we can obtain

$$\begin{array}{cc}\hfill & 0\le {\alpha }_{\xi }\left(x\right)\le {\alpha }_{\eta }\left(x\right)\le {\alpha }_{\vartheta }\left(x\right)\le 1\hfill \\ \hfill & 0\le {\beta }_{\vartheta }\left(x\right)\le {\beta }_{\eta }\left(x\right)\le {\beta }_{\xi }\left(x\right)\le 1\hfill \end{array}$$

(16)

For $m,n\in [0,1]$, and $m+n\le 1$, we construct the function $f(x,y)$ as follows:

$$\begin{array}{c}\hfill f(x,y)={\left({\displaystyle \frac{|x-m|}{x+m}}\right)}^2+{\left({\displaystyle \frac{|y-n|}{y+n}}\right)}^2\end{array}$$

(17)

where $x,y\in \{0,1\}$.

Similar to Deng [46], we can determine that $f(x,y)$ is increasing concerning y when $y\ge n$, and decreasing when $y\le n$.

Let $m={\alpha }_{\xi }\left(x\right),n={\beta }_{\xi }\left(x\right)$. When $\xi \subseteq \eta \subseteq \vartheta$, we have $m={\alpha }_{\xi }\left(x\right)\le {\alpha }_{\eta }\left(x\right)\le {\alpha }_{\vartheta }\left(x\right),{\beta }_{\vartheta }\left(x\right)\le {\beta }_{\eta }\left(x\right)\le {\beta }_{\xi }\left(x\right)=n$, and we can thus obtain $f\left({\alpha }_{\eta }\left(x\right),{\beta }_{\eta }\left(x\right)\right)\le f\left({\alpha }_{\vartheta }\left(x\right),{\beta }_{\vartheta }\left(x\right)\right)$. That is,

$$\begin{array}{cc}\hfill f\left({\alpha }_{\eta }\left(x\right),\right.& \left.{\beta }_{\eta }\left(x\right)\right)={\left({\displaystyle \frac{\left|{\alpha }_{\eta }\left(x\right)-{\alpha }_{\xi }\left(x\right)\right|}{{\alpha }_{\eta }\left(x\right)+{\alpha }_{\xi }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\eta }\left(x\right)-{\beta }_{\xi }\left(x\right)\right|}{{\beta }_{\eta }\left(x\right)+{\beta }_{\xi }\left(x\right)}}\right)}^2\hfill \\ \hfill & \le {\left({\displaystyle \frac{\left|{\alpha }_{\vartheta }\left(x\right)-{\alpha }_{\xi }\left(x\right)\right|}{{\alpha }_{\vartheta }\left(x\right)+{\alpha }_{\xi }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\vartheta }\left(x\right)-{\beta }_{\xi }\left(x\right)\right|}{{\beta }_{\vartheta }\left(x\right)+{\beta }_{\xi }\left(x\right)}}\right)}^2=f\left({\alpha }_{\vartheta }\left(x\right),{\beta }_{\vartheta }\left(x\right)\right)\hfill \end{array}$$

(18)

From Property (2), we know that $f\left({\alpha }_{\eta }\left(x\right),{\beta }_{\eta }\left(x\right)\right)\ge 0$, and $f\left({\alpha }_{\vartheta }\left(x\right),{\beta }_{\vartheta }\left(x\right)\right)\ge 0$. Accordingly, we can determine that

$$\begin{array}{cc}\hfill & d_c(\xi ,\eta )=\sqrt{{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2\right]}\hfill \\ \hfill & \le \sqrt{{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\theta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\theta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\theta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\theta }\left(x\right)}}\right)}^2\right]}=d_c(\xi ,\vartheta )\hfill \end{array}$$

(19)

Therefore, $d_c(\xi ,\eta )\le d_c(\xi ,\vartheta )$.

In the same way, we also determine that $d_c(\xi ,\vartheta )\le d_c(\eta ,\vartheta )$.

Accordingly, we determine that when $\xi \subseteq \eta \subseteq \vartheta ,d_c(\xi ,\eta )\le d_c(\xi ,\vartheta )$ and $d_c(\xi ,\vartheta )\le d_c(\eta ,\vartheta )$.

It has been proven that the triangle inequality of $d_c$ is satisfied.

The validation of the triangular inequality property of the method not only demonstrates the mathematical and logical rigor of the method, but also ensures the mathematical and applied correctness and practicality of the distance metric. This validation is a strong guarantee of the effectiveness of the method in solving practical problems.

3.5. Non-Negative Verification Procedure

Providing (P4): Given two IFSs $\xi$ and $\eta$ in X, we obtain

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=\sqrt{\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x\right)-{\gamma }_{\eta }\left(x\right)\right|}{{\gamma }_{\xi }\left(x\right)+{\gamma }_{\eta }\left(x\right)}}\right)}^2\right]/2}\end{array}$$

(20)

According to the concept of intuitive fuzzy set and the universe of discourse (UOD), the lower the similarity of $\xi$ and $\eta$, the greater the distance—that is, the membership of $\xi$ and $\eta$—and the greater the difference in the hesitation, the following situations occur:

$$\begin{array}{cc}& \left(1\right){\xi }_1=\left\{〈x_1,1,0,0〉\right\},{\eta }_1=\left\{〈x_2,0,1,0〉\right\},{\eta }_2=\left\{〈x_3,0,0,1〉\right\}\hfill \\ & \left(2\right){\xi }_2=\left\{〈x_1,0,1,0〉\right\},{\eta }_3=\left\{〈x_2,1,0,0〉\right\},{\eta }_4=\left\{〈x_3,0,0,1〉\right\}\hfill \\ & \left(3\right){\xi }_3=\left\{〈x_1,0,0,1〉\right\},{\eta }_5=\left\{〈x_2,1,0,0〉\right\},{\eta }_6=\left\{〈x_3,0,1,0〉\right\}\hfill \end{array}$$

$$\begin{array}{cc}& d_c\left({\xi }_1,{\eta }_1\right)=1,d_c\left({\xi }_1,{\eta }_2\right)=1\hfill \\ & d_c\left({\xi }_2,{\eta }_3\right)=1,d_c\left({\xi }_2,{\eta }_4\right)=1\hfill \\ & d_c\left({\xi }_3,{\eta }_5\right)=1,d_c\left({\xi }_3,{\eta }_6\right)=1\hfill \end{array}$$

Therefore, the maximum new distance is 1.

Because of

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2\ge 0;{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2\ge 0;{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x\right)-{\gamma }_{\eta }\left(x\right)\right|}{{\gamma }_{\xi }\left(x\right)+{\gamma }_{\eta }\left(x\right)}}\right)}^2\ge 0\end{array}$$

(21)

then, it has

$$\begin{array}{c}\hfill d_c(\xi ,\eta )=\sqrt{\left[{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x\right)-{\alpha }_{\eta }\left(x\right)\right|}{{\alpha }_{\xi }\left(x\right)+{\alpha }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x\right)-{\beta }_{\eta }\left(x\right)\right|}{{\beta }_{\xi }\left(x\right)+{\beta }_{\eta }\left(x\right)}}\right)}^2+{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x\right)-{\gamma }_{\eta }\left(x\right)\right|}{{\gamma }_{\xi }\left(x\right)+{\gamma }_{\eta }\left(x\right)}}\right)}^2\right]/2}\ge 0\end{array}$$

(22)

It has been proven that the non-negativity of $d_c$ is satisfied.

Thus, $0\le d_c(\xi ,\eta )\le 1$.

Suppose that $\xi =\left\{x_i,{\alpha }_{\xi }\left(x_i\right),{\beta }_{\xi }\left(x_i\right)|x_i\in x\right\}$ and $\eta =\left\{x_i,{\alpha }_{\eta }\left(x_i\right),{\beta }_{\eta }\left(x_i\right)|x_i\in x\right\}$ are two IFSs in the finite sets $x=\left\{x_1,x_2,\dots ,x_n\right\}$; then, the normalized distance measure between $\mathrm{P}$ and $\mathrm{Q}$ is given by

$$\begin{array}{c}\hfill d_c^{\sim }=\sqrt{\left[\sum _{i=1}^n{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|}{{\alpha }_{\xi }\left(x_i\right)+{\alpha }_{\eta }\left(x_i\right)}}\right)}^2+\sum _{i=1}^n{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|}{{\beta }_{\xi }\left(x_i\right)+{\beta }_{\eta }\left(x_i\right)}}\right)}^2+\sum _{i=1}^n{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|}{{\gamma }_{\xi }\left(x_i\right)+{\gamma }_{\eta }\left(x_i\right)}}\right)}^2\right]/2n}.\end{array}$$

(23)

After that, some numbers are given as examples to check if the new way of measuring distance works correctly.

Example: Adopted from Xiao [24], suppose that in the set X, IFSs A, B, and C $=\left\{x_1,x_2\right\}$

$$\begin{array}{cc}& A=\left\{〈x_1,0.30,0.20〉〈x_2,0.40,0.30〉\right\}\hfill \\ & B=\left\{〈x_1,0.30,0.20〉〈x_2,0.40,0.30〉\right\}\hfill \\ & C=\left\{\left(x_1,0.15,0.25\right.〉〈x_2,0.25,0.35〉\right\}\hfill \end{array}$$

The distance measurements between $\mathrm{A},\mathrm{B}$, and $\mathrm{C}$ are as follows:

$$\begin{array}{cc}{d}_c^{\sim }(\mathrm{A},\mathrm{B})=0.0000,\hfill & d_c^{\sim }(\mathrm{B},\mathrm{A})=0.0000\hfill \\ d_c^{\sim }(\mathrm{A},\mathrm{C})=0.2298,\hfill & d_c^{\sim }(\mathrm{C},\mathrm{A})=0.2298\hfill \\ d_c^{\sim }(\mathrm{C},\mathrm{B})=0.2298,\hfill & d_c^{\sim }(\mathrm{B},\mathrm{C})=0.2298\hfill \end{array}$$

We can thus obtain the following inequality:

$$\begin{array}{c}\hfill d_c^{\sim }(C,A)+d_c^{\sim }(A,B)>d_c^{\sim }(C,B)\end{array}$$

(24)

Based on our findings, we can assert that $d_c^{\sim }$ adheres to three pivotal rules. The first rule states that when comparing two fuzzy sets, if they appear very similar, their distance is 0. The second rule indicates that even when the positions of the fuzzy sets are switched, the distance remains unchanged. The third rule stipulates that when comparing three fuzzy sets, the distance between the first and third sets is always no greater than the combined distance between the first and second sets plus the distance between the second and third sets.

This result validates the triangle inequality characteristic of the recently introduced distance measure. It is assumed that the IFSs A and B are elements of the set $X=\left\{X\right\}$.

$$\begin{array}{c}\hfill A=\langle x,\alpha ,\beta \rangle ;B=\langle x,\beta ,\alpha \rangle \end{array}$$

(25)

Here, $\alpha$ and $\beta$ represent the memberships and non-memberships within the range of $[0,1]$. It is important to note that $\alpha +\beta$ should be less than or equal to 1, as illustrated in Figure 1. The distance between IFSs $\mathrm{A}$ and $\mathrm{B}$, denoted by $d_c^{\sim }(A,B)$, is evaluated according to the method shown in Figure 2. The value of $d_c^{\sim }(A,B)$ falls within the range of 0 to 1 as the variables $\mu$ and v vary between 0 and 1. In other words, $d_c^{\sim }(A,B)$ lies between 0 and 1, inclusively. It is worth mentioning that $d_c^{\sim }(A,B)$ equals 0 only when $\alpha =\beta$. Furthermore, when $\alpha =0$ and $\beta =1$ (or $\alpha =1$ and $\beta =0$ ), $d_c^{\sim }(\mathrm{A},\mathrm{B})$ equal to 1. These findings confirm that the newly introduced distance measure is bounded.

Assuming IFSs A and B are in the UOD $X=\left\{X\right\}$, then we obtain the following:

$$\begin{array}{c}\hfill A=\langle X,\alpha ,\beta );B=\langle X,\delta ,1-\delta )\end{array}$$

(26)

The memberships and non-memberships of IFS A are set to $\alpha =0.3,\beta =0.7$, and $\alpha =0.5$ and $\beta =0.5,\alpha =0.7,\beta =$ 0.3, respectively, as shown in Figure 3a–c. Meanwhile, the values of $\delta$ and $1-\delta$ of IFS B vary in the range of [0, 1], as shown in Figure 3.

$d_c^{\sim }(\mathrm{A},\mathrm{B})$ is shown in Figure 3, and it is apparent that $d_c^{\sim }$ exhibits nonlinear characteristics in the variations of the memberships and non-memberships of IFSs $\mathrm{A}$ and $\mathrm{B}$ under different conditions of $\alpha$. The nonlinear nature of $d_c^{\sim }$ is demonstrated.

4. Numerical Performance Analysis

This section briefly overviews existing measures for distance and similarity between iterated function systems (IFSs). Given two finite sets of IFSs $\xi$ and $\eta$, where $x=\left\{x_1,x_2,\dots ,x_n\right\}$, we have $\xi =\left\{〈x_i,{\alpha }_{\xi }\left(x_i\right),{\beta }_{\xi }\left(x_i\right)〉\mid x_i\in x\right\}$ and $\eta =$$\left\{\left.〈x_i,{\alpha }_{\eta }\left(x_i\right),{\beta }_{\eta }\left(x_i\right)\right|x_i\in x\right\}.$

Hamming and Euclidean [37] distance measure can be written as

$$\begin{array}{cc}\hfill & d_{\mathrm{Ham}}^{*}(\xi ,\eta )={\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left(\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|+\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|+\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right)\hfill \\ \hfill & d_{\mathrm{Euc}}^{*}(\xi ,\eta )=\sqrt{{\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left({\left({\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right)}^2+{\left({\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right)}^2+{\left({\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right)}^2\right)}\hfill \end{array}$$

(27)

Yang and Chiclana [47] distance measure can be expressed as

$$\begin{array}{c}\hfill d_{\mathrm{YC}}(\xi ,\eta )={\displaystyle \frac{1}{n}}\sum _{i=1}^{n}max\left(\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|,\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|,\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right)\end{array}$$

(28)

Jiang et al. [48] distance measure can be described as

$$\begin{array}{cc}\hfill d_{\mathrm{JI}}(\xi ,\eta )=& {\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left(\left|{\displaystyle \frac{2\left({\alpha }_{\xi }\left(x_i\right){\gamma }_{\eta }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right){\gamma }_{\xi }\left(x_i\right)\right)-4\left({\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right)\mid }{4-{\gamma }_{\xi }\left(x_i\right){\gamma }_{\eta }\left(x_i\right)}}\right|\right.\hfill \\ \hfill & \left.\left|{\displaystyle \frac{4\left({\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right)+2\left({\alpha }_{\xi }\left(x_i\right){\gamma }_{\eta }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right){\gamma }_{\xi }\left(x_i\right)\right)+2\left({\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right)}{4-{\gamma }_{\xi }\left(x_i\right){\gamma }_{\eta }\left(x_i\right)}}\right|\right).\hfill \end{array}$$

(29)

Song et al. [49] distance measure can be obtained as

$$\begin{array}{cc}\hfill d_{48}(\xi ,\eta )=& 1-{\displaystyle \frac{1}{3n}}\sum _{i=1}^n\left[2\sqrt{{\alpha }_{\xi }\left(x_i\right){\alpha }_{\eta }\left(x_i\right)}+2\sqrt{{\beta }_{\xi }\left(x_i\right){\beta }_{\eta }\left(x_i\right)}+\sqrt{{\gamma }_{\xi }\left(x_i\right){\gamma }_{\eta }\left(x_i\right)}\right.\hfill \\ \hfill & \left.+\sqrt{\left(1-{\alpha }_{\xi }\left(x_i\right)\right)\left(1-{\alpha }_{\eta }\left(x_i\right)\right)}+\sqrt{\left(1-{\beta }_{\xi }\left(x_i\right)\right)\left(1-{\beta }_{\eta }\left(x_i\right)\right)}\right].\hfill \end{array}$$

(30)

Xiao [24] distance measure can be described as

$$\begin{array}{cc}\hfill d_{\mathrm{XI}}(\xi ,\eta )=& {\displaystyle \frac{1}{n}}\sum _{i=1}^n\left[{\displaystyle \frac{1}{2}}\left({\alpha }_{\xi }\left(x_i\right)log{\displaystyle \frac{2{\alpha }_{\xi }\left(x_i\right)}{{\alpha }_{\xi }\left(x_i\right)+{\alpha }_{\eta }\left(x_i\right)}}+{\alpha }_{\eta }\left(x_i\right)log{\displaystyle \frac{2{\alpha }_{\eta }\left(x_i\right)}{{\alpha }_{\xi }\left(x_i\right)+{\alpha }_{\eta }\left(x_i\right)}}\right.\right.\hfill \\ & +{\beta }_{\xi }\left(x_i\right)log{\displaystyle \frac{2{\beta }_{\xi }\left(x_i\right)}{{\beta }_{\xi }\left(x_i\right)+{\beta }_{\eta }\left(x_i\right)}}+{\beta }_{\eta }\left(x_i\right)log{\displaystyle \frac{2{\beta }_{\eta }\left(x_i\right)}{{\beta }_{\xi }\left(x_i\right)+{\beta }_{\eta }\left(x_i\right)}}\hfill \\ & {\left.+{\gamma }_{\xi }\left(x_i\right)log{\displaystyle \frac{2{\gamma }_{\xi }\left(x_i\right)}{{\gamma }_{\xi }\left(x_i\right)+{\gamma }_{\eta }\left(x_i\right)}}+{\gamma }_{\eta }\left(x_i\right)log{\displaystyle \frac{2{\gamma }_{\eta }\left(x_i\right)}{{\gamma }_{\xi }\left(x_i\right)+{\gamma }_{\eta }\left(x_i\right)}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \end{array}$$

(31)

Gohain [25] distance measure can be expressed as

$$\begin{array}{cc}\hfill d_{\omega C}^1(\xi ,\eta )=& {\displaystyle \frac{1}{n}}\sum _{i=1}^n\left[{\displaystyle \frac{1}{2}}\left\{{\displaystyle \frac{\left|{\alpha }_{\zeta }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|+\left|{\beta }_3\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|}{\left(1-{\alpha }_i\left(x_i\right)\right)\left(1-{\alpha }_{\eta }\left(x_i\right)\right)+\left(1+{\beta }_{\zeta }\left(x_i\right)\right)\left(1+{\beta }_{\eta }\left(x_i\right)\right)}}\right\}\right.\hfill \\ \hfill & +{\displaystyle \frac{1}{4}}\left\{\left|min\left\{{\alpha }_3\left(x_i\right),{\beta }_{\eta }\left(x_i\right)\right\}-min\left\{{\alpha }_{\eta }\left(x_i\right),{\beta }_2\left(x_i\right)\right\}\right|+\mid max\left\{{\alpha }_5\left(x_i\right),{\beta }_{\eta }\left(x_i\right)\right\}\right.\hfill \\ \hfill & \left.\left.-max\left\{{\alpha }_{\eta }\left(x_i\right),{\beta }_2\left(x_i\right)\right\}\mid \right\}\right]\hfill \end{array}$$

(32)

Garg and Rani [50] distance measure is obtained as

$$\begin{array}{cc}\hfill d_{GK}^N(\xi ,\eta )=& {\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left[\left|{\alpha }_3\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|+\left|{\displaystyle \frac{1}{1-{\gamma }_{\zeta }\left(x_i\right){\gamma }_{\eta }\left(x_i\right)}}\left({\beta }_3\left(x_1\right)\right.\right.\right.\hfill \\ \hfill & \left.\left.-{\beta }_{\eta }\left(x_i\right)+{\gamma }_2\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)+{\gamma }_{\eta }\left(x_1\right){\beta }_3\left(x_i\right)-{\gamma }_{\zeta }\left(x_i\right){\beta }_{\eta }\left(x_i\right)\right)\mid \right]\hfill \end{array}$$

(33)

Chen and Deng [51] distance measures is defined as

$$\begin{array}{c}\hfill \begin{array}{c}{d}_{\mathrm{Hh}}(\xi ,\eta )={\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left(\left(\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|+\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|\right)\times \left(1-{\displaystyle \frac{1}{2}}\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right)\right)\hfill \\ d_{\mathrm{Hc}}(\xi ,\eta )={\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left(\left(\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|+\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|\right)\times cos\left({\displaystyle \frac{\pi }{6}}\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right)\right)\hfill \\ d_{\mathrm{Eh}}(\xi ,\eta )={\left[{\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left({\left({\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right)}^2+{\left({\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right)}^2\right)\times {\left(1-{\displaystyle \frac{1}{2}}\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right)}^2\right]}^{1/2}\hfill \end{array}\end{array}$$

(34)

Park et al. [52] distance measure can be written as

$$\begin{array}{cc}\hfill d_P(\xi ,\eta )=& {\displaystyle \frac{1}{4n}}\sum _{i=1}^n\left(\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|+\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|+\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right.\hfill \\ \hfill & \left.+2max\left\{\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|\times \left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|\times \left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|\right\}\right).\hfill \end{array}$$

(35)

Several numerical examples are used in this section to demonstrate the proposed distance measurement. Additionally, the effectiveness of this measure is influenced by comparative studies. The steps involved in the process are as follows:

First, obtain the IFS representations of the patterns and the sample.

Determining the relevant parameters of the decision matrix is a key step in fuzzy logic systems, including intuitionistic fuzzy sets and fuzzy sets. There are usually two common methods:

1. Expert judgment: First, clarify the definition and criteria of membership degree, then ask domain experts to evaluate the membership degree of each element based on the defined criteria.

2. Fuzzification function: Normalize the actual data, and use the fuzzification function to convert the normalized data into membership degree values. Common fuzzification functions include triangular functions, trapezoidal functions, and Gaussian functions.

Second, calculate the distance of the sample from each pattern using the equation below:

$$\begin{array}{c}\hfill d_c^{\sim }=\sqrt{\left[\sum _{i=1}^n{\left({\displaystyle \frac{\left|{\alpha }_{\xi }\left(x_i\right)-{\alpha }_{\eta }\left(x_i\right)\right|}{{\alpha }_{\xi }\left(x_i\right)+{\alpha }_{\eta }\left(x_i\right)}}\right)}^2+\sum _{i=1}^n{\left({\displaystyle \frac{\left|{\beta }_{\xi }\left(x_i\right)-{\beta }_{\eta }\left(x_i\right)\right|}{{\beta }_{\xi }\left(x_i\right)+{\beta }_{\eta }\left(x_i\right)}}\right)}^2+\sum _{i=1}^n{\left({\displaystyle \frac{\left|{\gamma }_{\xi }\left(x_i\right)-{\gamma }_{\eta }\left(x_i\right)\right|}{{\gamma }_{\xi }\left(x_i\right)+{\gamma }_{\eta }\left(x_i\right)}}\right)}^2\right]/2n}.\end{array}$$

(36)

Third, classify the test sample as belonging to the pattern from the second point that yields the minimum distance. The detailed algorithm implementation steps are shown in Figure 4.

Fourth, calculate the relative difference using the following formula:

$$\begin{array}{c}\hfill \mathrm{RD}={\displaystyle \frac{\left|d_c^{\sim }(\xi ,\eta )-d_c^{\sim }(\xi ,\vartheta )\right|}{\left(d_c^{\sim }(\xi ,\eta )+d_c^{\sim }(\xi ,\vartheta )\right)}}\end{array}$$

(37)

The larger the $\mathrm{RD}$ value, the higher the relative difference between the two distances. It makes it easier to distinguish categories and is more suitable for complex conditions.

Positive and negative comparison calculation: Adopted from Brindaban Gohain [19], this calculation considers a scenario where a company has a job vacancy and needs to select a suitable candidate based on evaluations from an interview. Let us denote the assessments of two candidates as X and Y, represented as IFSs $X=\{0.5,0.3\}$ and $Y=\{0.5,0.2\}$. The standard for this job vacancy, set by the interviewer, is represented by IFS $O=\{0.4,0.2\}$. Extraversion is denoted by a membership degree of 0, whereas introversion is denoted by a non-membership degree. Based on these evaluations, the following issue arises: how can we select candidates for job vacancies?

Distance measurement is crucial for solving such problems, especially when the distance between 0 and the evaluated candidate is small, which indicates that the candidate is more suitable for the job vacancy.

When we compare how outgoing O is compared to X, we can see that it is the same as how outgoing O is compared to Y. So, by just looking at how outgoing someone is, is not enough to choose who is the best fit. But when we look at how shy or introverted O is compared to X, we see that they are slightly different, but O and Y are the same. It means that the difference in shyness between O and X is bigger than the difference between O and Y. This shows us that candidate Y is a better choice for the job. In simple terms, O is closer to Y in terms of shyness than it is to X.

Table 1. Results of distinct measures in positive and negative comparison calculations.

	$(\mathit{S},\mathit{X})$	$(\mathit{S},\mathit{Y})$	Result	Comment
${\mathrm{d}}_{\mathrm{SH}}$ [55]	0.2	0.1	Y	Reasonable
${\mathrm{d}}_{\mathrm{SE}}$ [55]	0.17	0.1	Y	Reasonable
${\mathrm{d}}_{\mathrm{GH}}$ [53]	0.1	0.1	Both X and Y	Unreasonable
${\mathrm{d}}_{\mathrm{WX}}$ [56]	0.1	0.075	Y	Reasonable
${\mathrm{d}}_{\mathrm{PA}}$ [52]	0.2	0.1	Y	Reasonable
${\mathrm{d}}_{\mathrm{YC}}$ [47]	0.2	0.1	Y	Reasonable
${\mathrm{d}}_{\mathrm{LZ}}$ [57]	0.27	0.13	Y	Reasonable
${\mathrm{d}}_{\mathrm{ZU}}$ [54]	0.17	0.18	X	Unreasonable
${\mathrm{d}}_{\mathrm{NG}}^1$ [58]	0.1	0.067	Y	Reasonable
${\mathrm{d}}_{\mathrm{NG}}^2$ [58]	0.1	0.075	Y	Reasonable
${\mathrm{d}}_{\mathrm{XI}}$ [24]	0.104	0.052	Y	Reasonable
${\mathrm{d}}_{\mathrm{JI}}$ [48]	0.04	0.05	X	Unreasonable
${\mathrm{d}}_{\mathrm{MP}}$ [26]	0.14	0.08	Y	Reasonable
${\mathrm{d}}_{\mathrm{CD}}^1$ [51]	0.09	0.05	Y	Reasonable
${\mathrm{d}}_{\mathrm{CD}}^2$ [51]	0.1	0.05	Y	Reasonable
${\mathrm{d}}_{\mathrm{CD}}^3$ [51]	0.09	0.07	Y	Reasonable
${\mathrm{d}}_{\mathrm{GK}}^{\mathrm{N}}$ [50]	0.06	0.1	X	Unreasonable
${\mathrm{d}}_{\mathrm{GC}}$ [25]	0.104	0.054	Y	Reasonable
${\mathrm{d}}_{\mathrm{c}}^{\sim }$	0.2859	0.1280	Y	Reasonable

and Figure 5 show different ways to measure distances and choose candidates. We can see that some distance measures, like ${\mathrm{d}}_{\mathrm{GH}}$ [53], ${\mathrm{d}}_{\mathrm{ZU}}$ [54], ${\mathrm{d}}_{\mathrm{JI}}$ [48], and ${\mathrm{d}}_{\mathrm{GK}}^{\mathrm{N}}$ [50], make an unfair choice for candidate X. However, the new distance measure makes a fair choice for candidate Y.

From Figure 6, the difference for the newly proposed distance formula is 0.38, greater than that of the other given distance formulas. This suggests that the newly proposed Clark distance formula is better at distinguishing different patterns and making accurate judgments.

More prominent differentiation display: Adopted from Brindaban Gohain [25], consider the evaluation sets IFSs ${\mathrm{S}}_1=\{[0.3,0.2],[0.4,0.3]\},{\mathrm{A}}_1=$$\left\{\right[0.25,0.35],[0.35,0.45\left]\right\}$, and $A_2=\{[0.26,0.36],[0.36,0.46]\}$. The objective is to determine which evaluation set $\left(A_1\right.$ or $\left.A_2\right)$ is closer to $S_1$ and then identify which pattern $\left(A_1\right.$ or $\left.A_2\right)$ is similar to $S_1$. Distance measurement can be employed to achieve this goal.

Table 2. Results of distinct measures in a more prominent differentiation display.

	$({\mathit{S}}_1, {\mathit{A}}_1)$	$({\mathit{S}}_1, {\mathit{A}}_2)$	Classification
${\mathrm{d}}_{\mathrm{SH}}$ [55]	0.15	0.16	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{SE}}$ [55]	0.13	0.14	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{GH}}$ [53]	0.15	0.16	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{WX}}$ [56]	0.125	0.13	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{PA}}$ [52]	0.15	0.16	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{YC}}$ [47]	0.15	0.16	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{LZ}}$ [57]	0.2	0.21	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{ZU}}$ [54]	0.23	0.23	Unrecognizable
${\mathrm{d}}_{\mathrm{NG}}^1$ [58]	0.1	0.1	Unrecognizable
${\mathrm{d}}_{\mathrm{NG}}^2$ [58]	0.1	0.1	Unrecognizable
${\mathrm{d}}_{\mathrm{XI}}$ [24]	0.078	0.085	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{JI}}$ [48]	0.1034	0.1032	${\mathrm{A}}_2$
${\mathrm{d}}_{\mathrm{MP}}$ [26]	0.158	0.155	${\mathrm{A}}_2$
${\mathrm{d}}_{\mathrm{CD}}^1$ [51]	0.095	0.094	${\mathrm{A}}_2$
${\mathrm{d}}_{\mathrm{CD}}^2$ [51]	0.0998	0.0998	Unrecognizable
${\mathrm{d}}_{\mathrm{CD}}^3$ [51]	0.106	0.11	${\mathrm{A}}_1$
${\mathrm{d}}_{\mathrm{GK}}^{\mathrm{N}}$ [50]	0.104	0.097	${\mathrm{A}}_2$
${\mathrm{d}}_{\mathrm{GC}}$ [25]	0.0953	0.0952	${\mathrm{A}}_2$
${\mathrm{d}}_{\mathrm{c}}^{\sim }$	0.2118	0.2318	${\mathrm{A}}_1$

and Figure 7 present various distance measures. It is evident that ${\mathrm{d}}_{\mathrm{ZU}}$ [54], ${\mathrm{d}}_{\mathrm{NG}}^1$ [58], ${\mathrm{d}}_{\mathrm{NG}}^2$ [58], and ${\mathrm{d}}_{\mathrm{CD}}^2$ [51] are unable to distinguish between the patterns effectively. However, the proposed distance measure method successfully differentiates these patterns and identifies that $A_1$ is similar to $S_1$. This result aligns with the majority of existing distance measurement outcomes. Thus, the proposed distance measurement method can effectively apply to pattern recognition problems.

From Figure 8, the margin of error for the newly proposed distance formula is 0.05, which is twice that of other distance formulas. Therefore, in general pattern recognition, the newly proposed distance formula offers superior discriminability and guarantees the accuracy of the results provided.

Effectiveness and rationality under multi-angle conditions: Adopted from Brindaban Gohain [26], the goal of this example is to determine which of the IFSs $A,B$, or C is similar to IFS S. The distances between S and $A,B$, and C are measured using different methods, and the one with the minimum distance will be considered similar to S. The given IFSs are

$$\begin{array}{c}S=\left\{\right[0.37,0.31],[0.23,0.44],[0.04,0.1\left]\right\},\\ A=\left\{\right[0.34,0.34],[0.19,0.48],[0.02,0.12\left]\right\},\\ B=\left\{\right[0.35,0.33],[0.20,0.47],[0.0,0.14\left]\right\},\\ C=\left\{\right[0.33,0.35],[0.21,0.46],[0.01,0.13\left]\right\}.\end{array}$$

Table 3. Results of distinct measures in effectiveness and rationality under multi-angle conditions.

	$(\mathit{S}, \mathit{A})$	$(\mathit{S}, \mathit{B})$	$(\mathit{S}, \mathit{C})$	Comment
${\mathrm{d}}_{\mathrm{SH}}$ [55]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{SE}}$ [55]	0.031	0.031	0.031	Unrecognizable
${\mathrm{d}}_{\mathrm{GH}}$ [53]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{WX}}$ [56]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{PA}}$ [52]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{YC}}$ [47]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{LZ}}$ [57]	0.041	0.041	0.041	Unrecognizable
${\mathrm{d}}_{\mathrm{ZU}}$ [54]	0.198	0.198	0.198	Unrecognizable
${\mathrm{d}}_{\mathrm{NG}}^1$ [58]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{NG}}^2$ [58]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{XI}}$ [24]	0.024	NaN	0.028	Unrecognizable
${\mathrm{d}}_{\mathrm{JI}}$ [48]	0.0315	0.0334	0.0328	Similar to A
${\mathrm{d}}_{\mathrm{MP}}$ [26]	0.0822	0.12	0.1009	Similar to A
${\mathrm{d}}_{\mathrm{CD}}^1$ [51]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{CD}}^2$ [51]	0.03	0.03	0.03	Unrecognizable
${\mathrm{d}}_{\mathrm{CD}}^3$ [51]	0.031	0.031	0.031	Unrecognizable
${\mathrm{d}}_{\mathrm{GK}}^{\mathrm{N}}$ [50]	0.056	0.075	0.065	Similar to A
${\mathrm{d}}_{\mathrm{GC}}$ [25]	0.0275	0.0278	0.0281	Similar to A
${\mathrm{d}}_{\mathrm{c}}^{\sim }$	0.2118	0.2318	${\mathrm{A}}_1$	Similar to A

and Figure 9 display the distances measured using different methods. The distances measured by ${\mathrm{d}}_{\mathrm{SH}}$ [55], ${\mathrm{d}}_{\mathrm{GH}}$ [53], ${\mathrm{d}}_{\mathrm{WX}}$ [56], ${\mathrm{d}}_{\mathrm{PA}}$ [52], ${\mathrm{d}}_{\mathrm{YC}}$ [47], ${\mathrm{d}}_{\mathrm{LZ}}$ [57], ${\mathrm{d}}_{\mathrm{ZU}}$ [54], ${\mathrm{d}}_{\mathrm{NG}}$ [58], ${\mathrm{d}}_{\mathrm{XI}}$ [24], and ${\mathrm{d}}_{\mathrm{CD}}$ [51] cannot clearly determine the similarity between S and $A,B$, and C.

However, it determines that A is closer to S. Therefore, A is considered similar to S. This result is consistent with the distance measures ${\mathrm{d}}_{\mathrm{JI}}$ [50], ${\mathrm{d}}_{\mathrm{MP}}$ [26], ${\mathrm{d}}_{GK}^{\mathrm{N}}$ [50], and ${\mathrm{d}}_{GC}$ [25]. Hence, the result is deemed adequate for pattern recognition problems.

Based on the numerical and comparative studies presented above, it can be inferred that the method demonstrates effectiveness and reasonability in determining the similarity between IFSs.

From Figure 10, the difference value of the newly proposed distance formula is approximately three times larger than the difference values of other formulas. It indicates that other formulas cannot clearly distinguish the differences between the IFSs.

The foregoing numerical and comparative studies have substantiated the efficacy and rationality of the proposed distance measurement method. This paper concludes that the method can be effectively applied to decision-making and pattern recognition problems. The subsequent example will examine some potential applications of this distance measurement method.

Decision-making Applications In Integrated Complex Problems: Medical diagnostic problems often involve the use of distance metrics under IFSs. Being able to make the right decisions in complex situations is particularly important. Previous research works have already explored this area extensively [59,60,61,62,63,64,65]. Moreover, fuzzy theory has been widely applied to medical diagnostic problems. Let us consider a group of patients, denoted as $\mathrm{P}=\left\{{\mathrm{A}}_1,{\mathrm{A}}_2,{\mathrm{A}}_3\right\}$, with symptoms $\mathrm{S}=$ {temperature, headache, stomachache, cough, chest pain}, as depicted in

Table 4. Symptoms of the patients presented in IFSs in application in comprehensive and complex problems.

Patient	Temperature	Headache	Stomach Pain	Cough	Chest Pain
${\mathrm{A}}_1$	<0.0, 0.8>	<0.4, 0.4>	<0.6, 0.1>	<0.1, 0.7>	<0.1, 0.8>
${\mathrm{A}}_2$	<0.8, 0.1>	<0.8, 0.1>	<0.0, 0.6>	<0.2, 0.7>	<0.0, 0.5>
${\mathrm{A}}_3$	<0.6, 0.1>	<0.5, 0.4>	<0.3, 0.4>	<0.7, 0.2>	<0.3, 0.4>

.

Table 5. The diseases of the patients manifested by IFSs in application in comprehensive and complex problems.

Disease	Temperature	Headache	Stomach Pain	Cough	Chest Pain
Viral fever (V)	<0.4, 0.0>	<0.3, 0.5>	<0.1, 0.7>	<0.4, 0.3>	<0.1, 0.7>
Malaria (M)	<0.7, 0.0>	<0.2, 0.6>	<0.0, 0.9>	<0.7, 0.0>	<0.1, 0.8>
Typhoid (T)	<0.3, 0.3>	<0.6, 0.1>	<0.2, 0.7>	<0.2, 0.6>	<0.1, 0.9>
Stomach problem (S)	<0.1, 0.7>	<0.2, 0.4>	<0.8, 0.0>	<0.2, 0.7>	<0.2, 0.7>
Chest problem (C)	<0.1, 0.8>	<0.0, 0.8>	<0.2, 0.8>	<0.2, 0.8>	<0.8, 0.1>

shows the diagnostic group D = {viral fever (V), malaria (M), typhoid (T), stomach disease (S), chest disease (C)}. It is important to note that medical diagnostic problems are pattern recognition problems, making distance metrics suitable for diagnosing patients.

Table 6. The relative results of the patients compared to the symptoms in application in comprehensive and complex problems.

Patient	V	M	T	S	C	Diagnosis
${\mathrm{A}}_1$	0.6740	0.7870	0.5950	0.5380	0.8329	Stomach Problem
${\mathrm{A}}_2$	0.7234	0.7545	0.6820	0.7708	0.8854	Typhoid
${\mathrm{A}}_3$	0.4930	0.6782	0.5858	0.6233	0.7889	Viral fever

and Figure 11 illustrate the distances between patients, symptoms, and the corresponding diagnosis. The disease of each patient is determined by identifying the symptom with the minimum distance (highlighted in bold). In this case, patient $A_1$ has stomach disease, $A_2$ has typhoid, and $A_3$ has viral fever.

Furthermore,

Table 7. Result of differences in application in comprehensive and complex problems.

	${\mathit{A}}_1$	${\mathit{A}}_2$	${\mathit{A}}_3$
${\mathrm{d}}_{Ham}$ [66]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{Euc}$ [66]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{YC}$ [47]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{WX}$ [56]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{LZ}$ [57]	Stomach problem	Typhoid	Malaria
${\mathrm{d}}_{JI}$ [48]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{SO}$ [49]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{XI}$ [24]	Cannot identify	Cannot identify	Cannot identify
${\mathrm{d}}_{MP}$ [26]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{CD}$ [51]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_{GK}^N$ [50]	Stomach problem	Typhoid	Malaria
${\mathrm{d}}_{GC}$ [25]	Stomach problem	Typhoid	Viral fever
${\mathrm{d}}_c^{\sim }$	Stomach problem	Typhoid	Viral fever

presents the diagnosis of patients based on existing distance measures. Most of the current distance measures successfully diagnose patients ${\mathrm{A}}_1,{\mathrm{A}}_2$, and ${\mathrm{A}}_3$ with stomach problems, typhoid, and viral fever, respectively.

5. Conclusions

This paper has introduced a novel distance metric method for decision problems, focusing on measuring the difference between IFSs using Clark’s divergence. The main contribution of this research lies in the following:

It introduces a novel method that not only adheres to the axiomatic principles of distance measurement but also exhibits nonlinear characteristics, more accurately reflecting the discriminative nature of distance metrics and avoiding counterintuitive outcomes. A practical comparison with existing formulas reveals that Clark’s formula exhibits a greater relative difference, thereby enhancing uniqueness and offering advantages for making sound decisions in complex scenarios. We have developed a novel pattern classification algorithm and realized it through several case studies. The results demonstrate that the new distance metric outperforms existing ones. The proposed method holds extensive application potential in fields like medical diagnosis, pattern classification, and decision-making.

Future work will involve extending the proposed distance metric to interval-based IFSs in more complex environments. This expansion will further enhance the method’s versatility and applicability, allowing for its utilization in more challenging scenarios.