A Parametric Similarity Measure for Spherical Fuzzy Sets and Its Applications in Medical Equipment Selection

Information extraction and analysis from real-life problems are full of vagueness and uncertainties. Several attempts have been introduced to reduce this uncertainty. A famous way to reduce uncertainty was introduced by Zadeh by introducing the concept of the fuzzy set (FS) [1]. This concept is a generalization of the crisp set, describing the belongingness of an object with the help of the membership degree (MD). By generalizing the concept of the fuzzy set, Atanassov attempted to further reduce uncertainty by introducing the concept of intuitionistic fuzzy sets (IFS) [2], in which he described the belongingness of an object by both MD and non-membership degree (NMD). To achieve greater accuracy during information extraction, Atanassov [3] formalized the interval-valued IFS by considering MD and NMD as intervals from [0,1]. IFS has been used by researchers in various fields, such as pattern recognition [4], decision making [5], and medical diagnosis [6]. IFS had a limited range for assigning values to the $\mathrm{MD} \, \varrho(\lambda)$ and NMD $\gamma^{\prime}(\lambda)$ because the sum of the MD and NMD did not necessarily equal 1. The limitations of IFS were expanded by the ideas of the Pythagorean fuzzy set (PYFS) [7] and the q-rung orthopair fuzzy set (qROFS) [8], respectively.

The applications of IFS, PYFS, and qROFS have great potential in practical scenarios due to their ability to reduce vagueness in information extraction. However, in some instances, these tools could not extract information without some loss, as they account for only two degrees for the description of an element. To address this limitation and describe an object's belongingness with three degrees, Cuong [9] introduced the concept of picture fuzzy sets (PFS), which include an additional degree known as the abstinence degree (AD). The PFS has been used by many scholars for example, in the work [10]. But some time the concept of PFS failed when $0 \leq \varphi u \alpha\left(\varrho(\lambda), i(\lambda), \gamma^{\prime}(\lambda)\right) \leq 1$ violated. For example, the values of the MD, AD and NMD are 0.8, 0.2 and 0.4 respectively. In this case the $\varphi u \alpha\left(\varrho(\lambda), i(\lambda), \gamma^{\prime}(\lambda)\right)=1.4\not \leq 1$. The PFS, while useful, had its limitations. To broaden the scope of PFS, Mahmood et al. [11] introduced the concept of Spherical Fuzzy Sets (SFS) and the Total Spherical Fuzzy Sets (TSFS). TSFS, being the latest framework, is designed to extract information with higher accuracy.

MADM is a compelling technique used to identify the best alternative from a set of options. The introduction of fuzzy theory has significantly transformed and enhanced MADM. Numerous scholars have refined the MADM process utilizing various approaches. Khan et al. [12] applied complex SFS to address MADM problems. Senapati et al. [13] adopted interval-valued Intuitionistic IFS for the same purpose. Jana et al. [14] utilized PyFS in their approach to MADM, while Senapati [15] employed PFS for MADM solutions. TSFS were used in work [12] to tackle MADM challenges. Mahmood and Ali [16] resolved MADM issues using complex single-valued neutrosophic sets (CSVNS), and Riaz and Farid [17] applied complex PFS for MADM problems. Khan et al. [18] again used complex SFS in the context of MADM. Riaz et al. [19] addressed MADM with bipolar Fuzzy Sets, while Garg [20] implemented IFS for MADM solutions. Ashraf et al. [21] utilized interval-valued PFS for MADM, and Garg [22] adopted PyFS for this purpose. Lastly, Riaz et al. [23] and Pamučar et al. [24] both used qROFS to solve MADM problems.

SM is the significant tool for evaluating the similarity between two fuzzy values (FVs). Numerous scholars have introduced various SMs, finding interesting applications in medical diagnosis, pattern recognition, and MADM. Boran and Akay [4] and Du and Hu [25] introduced SMs within the framework of IFS and discussed their application in pattern recognition. Donyatalab et al. [26] introduced an SM for qROFS, while Mohd and Abdullah [27] presented SMs for PyFS, discussing their intriguing applications. Wei [28] introduced Cosine Similarity Measures (CSMs) based on the cosine function and contingent similarity measures based on the contingent function for PFS and applied them to MADM. Wei and Geo [29] developed a Dice SM for PFS. Van Dinh et al. [30] introduced some SMs for PFS and applied them to MADM problems. Singh et al. [31] extended SMs by considering the refusal degree of PFS and applied them to clustering problems. Luo and Zhang [32] introduced SMs based on basic operations for PFS. In the work [33], the concept of SM was introduced for Spherical Fuzzy Sets (SFS) with applications to MADM. Zhao et al. [34] developed SMs for the SFS framework and applied them to pattern recognition and MADM. Shishavan et al. [35] and Khan et al. [36] introduced SMs for SFS, applying them to pattern recognition, while Mahmood et al. [37] applied them to medical diagnosis and pattern recognition.

From the SMs discussed above, we can draw some key points. All the SMs for IFS, PyFS, qROFS, and PFS are outdated because these frameworks have limited capacity to extract information from real-life scenarios. Consequently, decision-makers cannot find the best results due to uncertainty and information loss. Therefore, advanced SMs for SFS should be defined to assess the similarity between FVs with less uncertainty.

Some SMs fail to compute in certain scenarios. For instance, some cannot provide decision results due to division by zero problems. Thus, the major contribution of this study is to enhance the identification ability of SMs and overcome the defects of current SMs, necessitating the proposal of new SMs.

This paper is organized as follows: Section 2 discusses some basic concepts. Section 3 reviews existing SMs and discusses their limitations. Section 4 develops a new SM for SFS, which improves upon and generalizes existing SMs for SFS by using parameters. Section 5 presents the application of the proposed SMs to the MADM problem, and Section 6 summarizes the study.

This section presents some basic concepts for understanding the article.

2.1 Definition [2]: On a set $X$ a IFS is of the shape $I=\left\{\left(\lambda,\left(\varphi, \gamma^{\prime}\right)\right): 0 \leq \operatorname{sum}\left(\varphi(\lambda), \gamma^{\prime}(\lambda)\right) \leq 1\right\}$. Further, $r(\lambda)=1-\operatorname{sum}\left(\varphi, \gamma^{\prime}\right)$ represents the hesitancy degree of $\lambda \in X$ and the pair $\left(\varphi, \gamma^{\prime}\right)$ is termed as an intuitionist FV (IFV).

2.2 Definition [7]: On a set $X$ a PyFS is of the shape $I=\left\{\left(\lambda,\left(\varphi, \gamma^{\prime}\right)\right): 0 \leq \operatorname{sum}\left(\varphi^2(\lambda), \gamma^{\prime 2}(\lambda)\right) \leq 1\right\}$. Further, $r(\lambda)=1-\operatorname{sum}\left(\varphi^2(\lambda), \gamma^{\prime 2}(\lambda)\right)$ represents the hesitancy degree of $\lambda \in X$ and the pair $\left(\varphi, \gamma^{\prime}\right)$ is termed as a Pythagorean FV (PyFV).

2.3 Definition [9]: On a set $X$ a PFS is of the shape $I=\left\{\left(\lambda,\left(\varphi, i, \gamma^{\prime}\right)\right): 0 \leq \operatorname{sum}\left(\varphi(\lambda), i(\lambda), \gamma^{\prime}(\lambda)\right) \leq 1\right\}$. Further, $r(\lambda)=1-\operatorname{sum}\left(\varphi(\lambda), i(\lambda), \gamma^{\prime}(\lambda)\right)$ represents the refusal degree of $\lambda \in X$ and the pair $\left(\varphi, i, \gamma^{\prime}\right)$ is termed as a picture FV (PFV).

2.4 Definition [11]: For any universal set $X$ a SFS is of the form $I=\left\{\left(\lambda,\left(\varphi, i, \gamma^{\prime}\right)\right): \forall \lambda \in X\right\}$. Here $\varphi, i$, and $\gamma^{\prime}$ are mappings form $X \rightarrow[ 0,1]$ denoting $\mathrm{MD}, \mathrm{AD}$, and $\mathrm{ND}$ respectively provided that $0 \leq$ $\operatorname{sum}\left(\varphi^2(\lambda), i^2(\lambda), \gamma^{\prime 2}(\lambda)\right) \leq 1$ and $r(\lambda)=\sqrt{1-\operatorname{sum}\left(\varphi^2(\lambda), i^2(\lambda), \gamma^{\prime 2}(\lambda)\right)}$ is known as the RD of $\lambda$ in $I$. The triplet $\left(\varphi, i, \gamma^{\prime}\right)$ is considered as a spherical FV (SFV).

2.5 Definition [11]: Let $\vartheta=\left\{\left(\lambda, \varphi_{\vartheta}^2(\lambda), i_{\vartheta}^2(\lambda),{\gamma^{\prime}}^2{ }_{\vartheta}(\lambda)\right) \mid \lambda \in X\right\}$ and $\Phi=\left\{\left(\lambda, \varphi_{\Phi}^2(\lambda), i_{\Phi}^2(\lambda), {\gamma^{\prime}}_{\Phi}^2(\lambda)\right) \mid \lambda \in X\right\}$ be any two be SFSs on universe $\lambda$, then

$\vartheta \subseteq \Phi$ If and only if $\varphi_{\vartheta}^2(\lambda) \leq \varphi_{\Phi}^2(\lambda), i_{\vartheta}^2(\lambda) \geq i_{\Phi}^2(\lambda),{\gamma^{\prime}}_{\vartheta}^{2}(\lambda) \geq{\gamma^{\prime}}_{\Phi}^2(\lambda)$. For $\lambda \in X$.

$\vartheta=\Phi$ If and only if $\vartheta \subseteq \Phi$ and $\Phi \subseteq \vartheta$ i.e. $\varphi_{\vartheta}^2(\lambda)=\varphi_{\Phi}^2(\lambda), i_{\vartheta}^2(\lambda)=i_{\Phi}^2(\lambda), \gamma^{\prime 2}{ }_{\vartheta}(\lambda)=\gamma^{\prime 2}{ }_{\Phi}(\lambda)$.

$\vartheta^c=\left\{\left(\lambda, \varphi_{\vartheta}^2(\lambda), i_{\vartheta}^2(\lambda), \gamma_{\vartheta}^{\prime 2}(\lambda)\right) \mid \lambda \in X\right\}$.

2.6 Definition [34]: Let $\vartheta=\left\{\left(\lambda, \varphi_{\vartheta}^2(\lambda), i_{\vartheta}^2(\lambda),{\gamma^{\prime}}_{\vartheta}^2(\lambda)\right) \mid \lambda \in X\right\}$ and $\Phi=\left\{\left(\lambda, \varphi_{\Phi}^2(\lambda), i_{\Phi}^2(\lambda),{\gamma^{\prime}}_{\Phi}^{2}(\lambda)\right) \mid \lambda \in\right.$ $X\}$ be any two be SFSs on universe $X$, then the SM between $\vartheta$ and $\Phi$ is defined as $\varrho(\vartheta, \Phi)$, which satisfies the following axioms:

$\left(\varrho_1\right) 0 \leq \varrho(\vartheta, \Phi) \leq 1$;

$\left(\varrho_2\right) \varrho(\vartheta, \Phi)=1$ Iff $\vartheta=\Phi$;

$\left(\varrho_3\right) \varrho(\vartheta, \Phi)=\varrho(\Phi, \vartheta) ;$

$\left(\varrho_4\right)$ Let $C$ be any SFS such that $\vartheta \subseteq \Phi \subseteq C$, then $\varrho(\vartheta, C) \leq \varrho(\vartheta, \Phi)$ and $\varrho(\vartheta, C) \leq \varrho(\Phi, C)$;

Now, we will review some existing similarity measures for SFSs in the following section.

Let $\vartheta=\left\{\left(\lambda_i, \varphi_{\vartheta}\left(\lambda_i\right), i_{\vartheta}\left(\lambda_i\right), \gamma^{\prime}{ }_{\vartheta}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\}$ and $\Phi=\left\{\left(\lambda_i, \varphi_{\Phi}\left(\lambda_i\right), i_{\Phi}\left(\lambda_i\right), \gamma^{\prime}{ }_{\Phi}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\}$ be any two be SFSs on $X=\left\{\lambda_1, \lambda_2, \ldots, \lambda_2\right\}, \rho_{\vartheta}\left(\lambda_i\right)$ and $\rho_{\Phi}\left(\lambda_i\right)$ be the refusal degrees of element $\lambda_i$ belonging to SFSs A and B respectively, where $\rho_{\vartheta}\left(\lambda_i\right)=1-\varphi_{\vartheta}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right), \gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)$ and $\rho_{\Phi}\left(\lambda_i\right)=1-\varphi_{\Phi}^2\left(\lambda_i\right), i_{\Phi}^2\left(\lambda_i\right), \gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)$. The existing similarity degrees between SFSs $\vartheta$ and $\Phi$ are reviewed as follows:

where, $i=1,2,3 \ldots 2$.

The SMs for SFSs as defined by the work [38] are given as follows:

Ullah et al. [39] introduced the SMs for SFSs based on the cosine function, as provided below.

As a numerical tool for calculating the degree of similarity between objects, SMs have been utilized to solve problems in decision-making, clinical diagnosis, and pattern recognition. Although many SMs for SFSs have been proposed, they can yield unreasonable and counter-intuitive results in practical applications, bringing significant challenges to users. In this section, we comprehensively analyze some existing SMs from an arithmetic perspective, as presented in Table 1 below.

The axiom $\varrho_2$ is one of the most basic axioms of spherical SMs. By analyzing Table 1, we can easily find that the similarity measures $\varrho_5, \varrho_7, \varrho_8, \varrho_{12}$, and $\varrho_{14}$ do not satisfy this axiom. The detailed discussion is as follows:

(1) Let $\boldsymbol{\vartheta}=\left\{\left(\lambda_i, \varphi_{\vartheta}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right),{\gamma^{\prime}}_{\vartheta}^{2}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\}$ and $\Phi=\left\{\left(\lambda_i, \varphi_{\Phi}^2\left(\lambda_i\right), i_{\Phi}^2\left(\lambda_i\right),{\gamma^{\prime}}_{\Phi}^{2}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\}$ be any two be SFSs on $X=\left\{\lambda_1, \lambda_2, \ldots, \lambda_2\right\}$. For the similarity measure $\varrho_{12}$, there are two cases in which $\varrho_{12}$ does not satisfy the axiom $\left(\varrho_2\right) \varrho(\boldsymbol{\vartheta}, \Phi)=1$ implies $\boldsymbol{\vartheta}=\Phi$ as shown below:

If $\varphi_{\vartheta}^2\left(\lambda_i\right)=i_{\vartheta}^2\left(\lambda_i\right)={\gamma^{\prime}}_{\vartheta}^{ 2}\left(\lambda_i\right) \neq \varphi_{\Phi}^2\left(\lambda_i\right)=i_{\Phi}^2\left(\lambda_i\right)={\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)$

i.e. $\boldsymbol{\vartheta} \neq \boldsymbol{\Phi}$ based on Eq. (12), we have

$\varrho_{12}(\boldsymbol{\vartheta}, \Phi)=\frac{1}{2} \sum_{i=1}^2 \omega_i \frac{\varphi_{\vartheta}^2\left(\lambda_i\right) \varphi_{\Phi}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right) i_{\Phi}^2\left(\lambda_i\right)+\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right) \gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)}{\sqrt{\left(\varphi_{\vartheta}^2\left(\lambda_i\right)\right)^2+\left(i_{\vartheta}^2\left(\lambda_i\right)\right)^2+\left(\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right)^2} \sqrt{\left(\varphi_{\Phi}^2\left(\lambda_i\right)\right)^2+\left(i_{\Phi}^2\left(\lambda_i\right)\right)^2+\left(\gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)\right)^2}}$

$=\omega_i \frac{3 \varphi_{\vartheta}^2\left(\lambda_i\right) \varphi_{\Phi}^2\left(\lambda_i\right)}{\sqrt{3 \varphi_{\vartheta}^2\left(\lambda_i\right)} \sqrt{3 \varphi_{\Phi}^2\left(\lambda_i\right)}}=1$

If $\varphi_{\vartheta}^2\left(\lambda_i\right)=2 \varphi_{\Phi}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right)=2 i_{\Phi}^2\left(\lambda_i\right) a 2 \gamma \gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)=2 \gamma^{\prime }{ }_\phi^2\left(\lambda_i\right)$

i.e., $\boldsymbol{\vartheta} \neq \Phi$ based on Eq. (12), we have

$$\varrho_{12}(\boldsymbol{\vartheta}, \Phi)=\frac{1}{2} \sum_{i=1}^2 \omega_i \frac{\varphi_{\vartheta}^2\left(\lambda_i\right) \varphi_{\Phi}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right) i_{\Phi}^2\left(\lambda_i\right)+\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right) \gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)}{\sqrt{\left(\varphi_{\vartheta}^2\left(\lambda_i\right)\right)^2+\left(i_{\vartheta}^2\left(\lambda_i\right)\right)^2+\left({\gamma^{\prime}}^{2}{ }_{\vartheta}\left(\lambda_i\right)\right)^2} \sqrt{\left(\varphi_{\Phi}^2\left(\lambda_i\right)\right)^2+\left(i_{\Phi}^2\left(\lambda_i\right)\right)^2+\left(\gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)\right)^2}}$$

$$=\omega_i \frac{2\left(\varphi_{\vartheta}^2\left(\lambda_i\right)\right)^2+2\left(i_{\Phi}^2\left(\lambda_i\right)\right)^2+2\left(\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)^2}{\sqrt{4\left(\varphi_{\vartheta}^2\left(\lambda_i\right)\right)^2+4\left(i_{\vartheta}^2\left(\lambda_i\right)\right)^2+4\left(\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)\right)^2} \sqrt{\left(\varphi_{\Phi}^2\left(\lambda_i\right)\right)^2+\left(i_{\Phi}^2\left(\lambda_i\right)\right)^2+\left(\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)^2}}=1$$

Obviously, in the above cases, the $\mathrm{SM} \varrho_{12}$ is invalid.

(2) The similar SMs $\varrho_5, \varrho_7, \varrho_8, \varrho_{12}$, and $\varrho_{14}$ do not satisfy the the axiom $\varrho(\boldsymbol{\vartheta}, \Phi)=1$ implies $\boldsymbol{\vartheta}=\Phi$ and these SMs provide a counter-intuitive result for practical users in this case.

(3) For the SM $\varrho_3, \varrho_7, \varrho_8, \varrho_{11}, \varrho_{12}, \varrho_{13}$ and $\varrho_{14}$, when SFSs $A=\Phi=(\lambda, 0.0,0.0,0.0)$ defined on $X=\{\lambda\}$, we have $\varrho_3(\boldsymbol{\vartheta}, \Phi)=\varrho_7(\boldsymbol{\vartheta}, \Phi)=\varrho_8(\boldsymbol{\vartheta}, \Phi)=\varrho_{11}(\boldsymbol{\vartheta}, \Phi)=\varrho_{12}(\boldsymbol{\vartheta}, \Phi)=\varrho_{13}(\boldsymbol{\vartheta}, \Phi)=\varrho_{14}(\boldsymbol{\vartheta}, \Phi)=\frac{0}{0}$. In this case, these SM are invalid, they do not satisfy the axiom.

(4) The capacity of the SM to recognize the nearness of fuzzy not entirely settled by the articulation structure and the data contained in the articulation. The more data the SM focuses on the more grounded the identification ability. By analyzing Table 1 we find that the SM $\varrho_1$ only considers the difference of positive degree or neutral degree or negative degree or refusal degree between SFSs which brings a big amount of information losing. For example, let $\boldsymbol{\vartheta}=(0.1,0.2,0.1), \Phi=(0.6,0.2,0.1)$ be two SFSs. Since $|0.1-0.1|<|0.2-0.2|<\mid 1-0.1-0.2-$ $0.1|<| 1-0.6-0.2-0.1|<| 0.1-0.6 \mid$, hence, the SM among A and B just considers the distinction of the positive degree between $\mathrm{A}$ and $\mathrm{B}$ by utilizing the $\mathrm{SM} \varrho_1$. For this situation, the $\mathrm{SM}$ will cause a great of data loss in viable application, so that it cannot provide more accurate results for practical users. In addition, in this situation, we also find that the $\operatorname{SMs} \varrho_2, \varrho_4, \varrho_5, \varrho_6, \varrho_{p_3}$ have the same drawback.

Considering the reasons for the unsatisfactory results observed in the above analysis (Table 1), we propose an expanded parametric SM for SFSs in this section to address the limitations of existing SMs.

In this section, we introduce a parametric spherical fuzzy SM by developing a paired function. The analysis in Table 1 indicates that the SMs $\varrho_5, \varrho_7, \varrho_8, \varrho_{12}$, and $\varrho_{14}$ have drawbacks. Consequently, we present the parametric SMs in Definition 8, which follows.

3.1.1 Definition 7: Let $\vartheta=\left\{\left(\lambda_i, \varphi_{\vartheta}\left(\lambda_i\right), i_{\vartheta}\left(\lambda_i\right), \gamma^{\prime}{ }_{\vartheta}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\}$ and $\Phi=\left\{\left(\left(\lambda_i, \varphi_{\Phi}\left(\lambda_i\right), i_{\Phi}\left(\lambda_i\right), \gamma_{\Phi}^{\prime}\left(\lambda_i\right)\right)\right) \mid \lambda_i \in X\right\}$ be any two be SFSs on $X=\left\{\lambda_1, \lambda_2, \ldots, \lambda_2\right\}$ then the function $\varrho_\alpha: \operatorname{SFS}(\lambda) \times \operatorname{SFS}(\lambda) \rightarrow[0,1]$ defined by

$\varrho_\alpha(\boldsymbol{\vartheta}, \Phi)$ is a similarity measure between $\boldsymbol{\vartheta}$ and $\Phi$, and $p=2$.

where,

$\begin{aligned} & \Delta_{1 \vartheta \Phi}\left(\lambda_i\right)=\frac{1}{\alpha_1+1}\left|\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right| \alpha_1 \in[0,+\infty), \\ & \Delta_2{ }_{\vartheta \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_2+1}\left|\alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right| \alpha_2 \in[0,+\infty), \\ & \Delta_3{ }_{\vartheta \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_3+1}\left|\alpha_3\left(\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)\right| \alpha_3 \in[0,+\infty), \\ & \end{aligned}$

$\frac{1}{\alpha_1+1}+\frac{1}{2 \alpha_2+1}+\frac{1}{2 \alpha_2+1} \in(0,1]$, and $p$ is any positive integer.

3.1.2 Theorem Let $\boldsymbol{\vartheta}=\left\{\left(\lambda, \varphi_{\vartheta}^2(\lambda), i_{\vartheta}^2(\lambda),{\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2(\lambda)\right) \mid \lambda \in X\right\}$ and $\Phi=\left\{\left(\lambda, \varphi_{\Phi}^2(\lambda), i_{\Phi}^2(\lambda),{\gamma^{\prime}}_{\Phi}^2(\lambda)\right) \mid \lambda \in X\right\}$ be any two be SFSs on universe $X$, then the SM between $\boldsymbol{\vartheta}$ and $\Phi$ is defined as $\varrho_\alpha(\boldsymbol{\vartheta}, \Phi)$, which satisfies the following axioms:

$\left(\varrho_1\right) 0 \leq \varrho_\alpha(\boldsymbol{\vartheta}, \Phi) \leq 1 ;$

$\left(\varrho_2\right) \varrho_\alpha(\boldsymbol{\vartheta}, \Phi)=1$ Iff $\boldsymbol{\vartheta}=\Phi$;

$\left(\varrho_3\right) \varrho_\alpha(\boldsymbol{\vartheta}, \Phi)=\varrho_\alpha(\Phi, \boldsymbol{\vartheta}) ;$

$\left(\varrho_4\right)$ Let $C$ be any SFS such that $\boldsymbol{\vartheta} \subseteq \Phi \subseteq C$, then $\varrho_\alpha(\boldsymbol{\vartheta}, C) \leq \varrho_\alpha(\boldsymbol{\vartheta}, \Phi)$ and $\varrho_\alpha(\boldsymbol{\vartheta}, C) \leq \varrho_\alpha(\Phi, C)$.

Proof: In order to prove that Eq. (15) is a SM, we only need to prove Eq. (15) satisfies axioms $\left(\varrho_1\right)-\left(\varrho_4\right) \boldsymbol{\vartheta}=$ $\left\{\left(\lambda_i, \varphi_{\vartheta}\left(\lambda_i\right), i_{\vartheta}\left(\lambda_i\right), \gamma^{\prime}{ }_{\vartheta}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\} \quad$ and $\quad \Phi=\left\{\left(\left(\lambda_i, \varphi_{\Phi}\left(\lambda_i\right), i_{\Phi}\left(\lambda_i\right), \gamma_{\Phi}^{\prime}\left(\lambda_i\right)\right)\right) \mid \lambda_i \in X\right\} \quad$ and $\quad C=$ $\left\{\left(\left(\lambda_i, \varphi_C\left(\lambda_i\right), i_C\left(\lambda_i\right), \gamma_C^{\prime}{ }\left(\lambda_i\right)\right)\right) \mid \lambda_i \in X\right\}$ be any three SFSs on $X=\left\{\lambda_1, \lambda_2, \ldots, \lambda_2\right\}$.

$\left(\varrho_1\right)$ We can write the following equations:

$\begin{aligned} & \Delta_1{ }_{\vartheta \Phi}\left(\lambda_i\right)= \frac{1}{\alpha_1+1}\left|\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime }{ }_{\Phi}^2\left(\lambda_i\right)\right)\right| \alpha_1 \in[0,+\infty) \\ &=\frac{1}{\alpha_1+1}\left|\left(\alpha_1 \varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)\right)-\left(\alpha_1 \varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right|\end{aligned}$

$\begin{aligned} \Delta_2{ }_{\vartheta \Phi}\left(\lambda_i\right)= & \frac{1}{2 \alpha_2+1}\left|\alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left({\gamma_\vartheta^{\prime}}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right| \alpha_2 \in[0,+\infty) \\ & =\frac{1}{2 \alpha_2+1}\left|\left(\alpha_2 i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)\right)-\left(\alpha_2 i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right|\end{aligned}$

$\begin{aligned} \Delta_3{ }_{\vartheta \Phi}\left(\lambda_i\right)= & \frac{1}{2 \alpha_3+1}\left|\alpha_3\left(\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)\right| \alpha_3 \in[0,+\infty) \\ & =\frac{1}{2 \alpha_3+1}\left|\left(\alpha_3{\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\left(\alpha_3 \gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)\right|\end{aligned}$

By $\varphi_{\vartheta}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right), \gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right), \varphi_{\Phi}^2\left(\lambda_i\right), i_{\Phi}^2\left(\lambda_i\right), \gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right) \in[0,1] \quad$ and $\quad \varphi_{\vartheta}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right)+\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right) \leq 1 \varphi_{\Phi}^2\left(\lambda_i\right)+$ $i_{\Phi}^2\left(\lambda_i\right)+\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right) \leq 1$

We have

$\begin{gathered}-1 \leq \alpha_1 \varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right) \leq \alpha_1 \\ -\alpha_1 \leq-\left(\alpha_1 \varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right) \leq 1\end{gathered}$

$0 \leq\left|\left(\alpha_1 \varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)-\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)\right)-\left(\alpha_1 \varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right| \leq \alpha_1+1$

i.e, $0 \leq \Delta_{1 \vartheta \Phi}\left(\lambda_i\right) \leq 1$

Then

$\begin{gathered}-1 \leq \alpha_2 i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)+\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right) \leq 1 \vee \alpha_2 \\ -\left(1 \vee \alpha_2\right) \leq-\left(\alpha_2 i_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)+\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right) \leq 1\end{gathered}$

Similarly, we get the following inequalities:

$\begin{gathered}-1 \leq \alpha_3 \gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right) \leq 1 \vee \alpha_3 \\ -\left(1 \vee \alpha_3\right) \leq-\left(\alpha_3 \gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)+i_{\Phi}^2\left(\lambda_i\right)\right) \leq 1\end{gathered}$

Then we obtain:

$\begin{aligned} & 0 \leq\left|\left(\alpha_2 i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)+\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right)-\left(\alpha_2 i_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)+\gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)\right)\right| \leq 2 \vee \alpha_2 \\ & 0 \leq\left|\left(\alpha_3 \gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right)\right)-\left(\alpha_3 \gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)+i_{\Phi}^2\left(\lambda_i\right)\right)\right| \leq 2 \vee \alpha_3\end{aligned}$

It means that:

$ 0 \leq \Delta_2{ }_{\vartheta \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_2+1}\left|\left(\alpha_2 i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)+\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right)-\left(\alpha_2 i_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)+{\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)\right)\right| \leq \frac{1}{\alpha_2+1} \vee \frac{1}{2} \leq 1$

$0 \leq \Delta_3{ }_{\vartheta \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_3+1}\left|\left(\alpha_3{\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right)\right)-\left(\alpha_3{\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)+i_{\Phi}^2\left(\lambda_i\right)\right)\right| \leq \frac{1}{\alpha_3+1} \vee \frac{1}{2} \leq 1$

Finally, we have:

$0 \leq 1-\left[\frac{1}{32} \sum_{i=1}^2 \Delta_{1 \vartheta \Phi}^p\left(\lambda_i\right)+\Delta_2^p{ }_{\vartheta \Phi}\left(\lambda_i\right)+\Delta_{3 \vartheta \Phi}^p\left(\lambda_i\right)\right]^{\frac{1}{p}} \leq 1$

Therefore,

$\left(\varrho_1\right) 0 \leq \varrho_\alpha(\boldsymbol{\vartheta}, \Phi) \leq 1$

$\left(\varrho_2\right)$ If $\boldsymbol{\vartheta}=\Phi$ then $\varphi_{\vartheta}^2\left(\lambda_i\right)=\varphi_{\Phi}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right)=i_{\Phi}^2\left(\lambda_i\right)$ and ${\gamma^{\prime}}_{\vartheta}^{ 2}\left(\lambda_i\right)={\gamma^{\prime}}_{\Phi}^{ 2}\left(\lambda_i\right)$. Therefore, $\Delta_{1 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right)=0, \Delta_{2 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right)=0 a 2 \gamma, \Delta_{3 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right)=0$ i.e., $\varrho_\alpha(\boldsymbol{\vartheta}, \Phi)=1$

If $\varrho_\alpha(\boldsymbol{\vartheta}, \Phi)=1$ then

$\Delta_{1 \vartheta \Phi}\left(\lambda_i\right)=\frac{1}{\alpha_1+1}\left|\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right|=0$

$\Delta_{2 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_2+1}\left|\alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left(\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)\right|=0$

$\Delta_{3 \vartheta \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_3+1}\left|\alpha_3\left({\gamma^{\prime}}_{\vartheta}^{\ 2}\left(\lambda_i\right)-{\gamma^{\prime}}_{\Phi}^{ 2}\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)\right|=0$

By the definition of absolute value, we have:

$\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)-{\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)\right)=0$

$\alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left({\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)-{\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)\right)=0$

$\alpha_3\left(\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)=0$

i.e.,

$\left(\begin{array}{ccc}\alpha_1 & -1 & -1 \\ -1 & \alpha_2 & 1 \\ -1 & 1 & \alpha_3\end{array}\right) \quad\left(\begin{array}{c}\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right) \\ i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right) \\ \gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\end{array}\right)=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$

Since $\frac{1}{\alpha_1+1}+\frac{1}{2 \alpha_2+1}+\frac{1}{2 \alpha_3+1} \in(0,1]$ then $2 \leq \alpha_1 \alpha_2 \alpha_3-\left(\alpha_1+\alpha_3+\alpha_3\right)$.

By the definition of matrix determinant, we can get:

$\left|\begin{array}{ccc}\alpha_1 & -1 & -1 \\ -1 & \alpha_2 & 1 \\ -1 & 1 & \alpha_3\end{array}\right|=\alpha_1 \alpha_2 \alpha_3+2-\left(\alpha_1+\alpha_3+\alpha_3\right) \geq 4$

Therefore, we have

$\left(\begin{array}{c}\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right) \\ i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right) \\ \gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\end{array}\right)=\left(\begin{array}{ccc}\alpha_1 & -1 & -1 \\ -1 & \alpha_2 & 1 \\ -1 & 1 & \alpha_3\end{array}\right)^{-1}\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)=\left(\begin{array}{l}0 \\ 0 \\ 0\end{array}\right)$

It means that $\varphi_{\vartheta}^2\left(\lambda_i\right)=\varphi_{\Phi}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right)=i_{\Phi}^2\left(\lambda_i\right)$ and $\gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)=\gamma^{\prime }{ }_{\Phi}^2\left(\lambda_i\right)$ then $\boldsymbol{\vartheta}=\Phi$ $\left(\varrho_3\right)$. Based on the definition of absolute value, we can get the following equations:

$$\begin{gathered}\Delta_{1 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right)=\frac{1}{\alpha_1+1}\left|\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)-\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)-{\gamma^{\prime }}_{\Phi}^2\left(\lambda_i\right)\right)\right| \\ =\frac{1}{\alpha_1+1}\left|(-1)\left[\alpha_1\left(\varphi_{\Phi}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)\right)-\left(i_{\Phi}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)-{\gamma^{\prime}}_{\vartheta}^2\left(\lambda_i\right)\right)\right]\right| \\ =\frac{1}{\alpha_1+1}\left|\alpha_1\left(\varphi_{\Phi}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)\right)-\left(i_{\Phi}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\left({\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)-{\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)\right)\right| \\ =\Delta_{1 \Phi \vartheta}\left(\lambda_i\right)\end{gathered}$$

$$\begin{gathered}\Delta_{2 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_2+1}\left|\alpha_2\left(i_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left({\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)-{\gamma^{\prime }}_{\Phi}^2\left(\lambda_i\right)\right)\right| \\ =\frac{1}{2 \alpha_2+1}\left|(-1)\left[\alpha_2\left(i_{\Phi}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\left(\varphi_{\Phi}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)\right)+\left({\gamma^{\prime }}_{\Phi}^2\left(\lambda_i\right)-{\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)\right)\right]\right| \\ =\frac{1}{2 \alpha_2+1}\left|\alpha_2\left(i_{\Phi}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\left(\varphi_{\Phi}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)\right)+\left(\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)-\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)\right)\right| \\ =\Delta_{2 \Phi \vartheta}\left(\lambda_i\right)\end{gathered}$$

$$\begin{gathered}\Delta_{3 \vartheta \Phi}\left(\lambda_i\right)=\frac{1}{2 \alpha_3+1}\left|\alpha_3\left({\gamma^{\prime}}_{\boldsymbol{\vartheta}}^{ 2}\left(\lambda_i\right)-{\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)\right)-\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)\right)+\left(i_{\vartheta}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)\right| \\ =\frac{1}{2 \alpha_3+1}\left|(-1)\left[\mid \alpha_3\left({\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)-{\gamma^{\prime}}_{\boldsymbol{\vartheta}}^2\left(\lambda_i\right)\right)-\left(\varphi_{\Phi}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)\right)+\left(i_{\Phi}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)\right]\right| \\ =\frac{1}{2 \alpha_3+1}\left|\alpha_3\left({\gamma^{\prime}}_{\Phi}^2\left(\lambda_i\right)-\gamma_{\vartheta}^{\prime 2}\left(\lambda_i\right)\right)-\left(\varphi_{\Phi}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)\right)+\left(i_{\Phi}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)\right| \\ =\Delta_{3 \Phi \vartheta}\left(\lambda_i\right)\end{gathered}$$

$\varrho_\alpha(\boldsymbol{\vartheta}, \Phi)=\varrho_\alpha(\Phi, \boldsymbol{\vartheta})$

$\left(\varrho_4\right)$ Therefore $\boldsymbol{\vartheta} \subseteq \Phi \subseteq C$ then $\varphi_{\vartheta}^2\left(\lambda_i\right) \leq \varphi_{\Phi}^2\left(\lambda_i\right) \leq \varphi_C^2\left(\lambda_i\right), i_C^2\left(\lambda_i\right) \leq i_{\Phi}^2\left(\lambda_i\right) \leq i_{\vartheta}^2\left(\lambda_i\right)$, $\gamma_C^{\prime 2}\left(\lambda_i\right) \leq \gamma_{\Phi}^{\prime 2}\left(\lambda_i\right) \leq \gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)$

Therefore, we can have

$\begin{aligned} & \alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right) \leq \alpha_1\left(\varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)-\gamma_{\Phi}^{\prime 2}\left(\lambda_i\right)\right) \leq \alpha_1\left(\varphi_C^2\left(\lambda_i\right)-i_C^2\left(\lambda_i\right)-\gamma_C^{\prime 2}\left(\lambda_i\right)\right) \\ & \alpha_2\left(i_C^2\left(\lambda_i\right)-\varphi_C^2\left(\lambda_i\right)-\gamma_C^{\prime 2}\left(\lambda_i\right)\right) \leq \alpha_2\left(i_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)-\gamma^{\prime }{ }_{\Phi}^2\left(\lambda_i\right)\right) \leq \alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right) \\ & \alpha_3\left({\gamma^{\prime}}_C^2\left(\lambda_i\right)-\varphi_C^2\left(\lambda_i\right)-i_C^2\left(\lambda_i\right)\right) \leq \alpha_3\left(\gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right) \leq \alpha_3\left(\gamma^{\prime 2}{ }_{\boldsymbol{\vartheta}}\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)\end{aligned}$

By the property of inequality, we can obtain:

$\begin{aligned} & \left|\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right)-\alpha_1\left(\varphi_\phi^2\left(\lambda_i\right)-i_\phi^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_\phi\left(\lambda_i\right)\right)\right| \\ \leq & \left|\alpha_1\left(\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right)-\alpha_1\left(\varphi_C^2\left(\lambda_i\right)-i_C^2\left(\lambda_i\right)-\gamma_C^{\prime 2}\left(\lambda_i\right)\right)\right|\end{aligned}$

$\begin{aligned} & \left|\alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime}{ }_{\vartheta}^2\left(\lambda_i\right)\right)-\alpha_2\left(i_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\Phi}\left(\lambda_i\right)\right)\right| \\ & \leq\left|\alpha_2\left(i_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-\gamma^{\prime 2}{ }_{\vartheta}\left(\lambda_i\right)\right)-\alpha_2\left(i_C^2\left(\lambda_i\right)-\varphi_C^2\left(\lambda_i\right)-\gamma_C^{\prime 2}\left(\lambda_i\right)\right)\right|\end{aligned}$

$\begin{aligned} & \left|\alpha_3\left(\gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\alpha_3\left(\gamma^{\prime }{ }_{\Phi}^2\left(\lambda_i\right)-\varphi_{\Phi}^2\left(\lambda_i\right)-i_{\Phi}^2\left(\lambda_i\right)\right)\right| \\ & \leq\left|\alpha_3\left(\gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)-\varphi_{\vartheta}^2\left(\lambda_i\right)-i_{\vartheta}^2\left(\lambda_i\right)\right)-\alpha_3\left(\gamma_C^{\prime 2}\left(\lambda_i\right)-\varphi_C^2\left(\lambda_i\right)-i_C^2\left(\lambda_i\right)\right)\right|\end{aligned}$

$\Delta_{2 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right) \leq \Delta_{1 \boldsymbol{\vartheta} C}\left(\lambda_i\right), \Delta_{2 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right) \leq \Delta_{2 \boldsymbol{\vartheta} C}\left(\lambda_i\right), \Delta_{3 \boldsymbol{\vartheta} \Phi}\left(\lambda_i\right) \leq \Delta_{3 \boldsymbol{\vartheta} C}\left(\lambda_i\right)$

Therefore, we have

$1-\left[\frac{1}{3 \eta} \sum_{i=1}^2 \Delta_{1 \vartheta}^p{ }_{\boldsymbol{C} C}\left(\lambda_i\right)+\Delta_2^p{ }_{\vartheta C}\left(\lambda_i\right)+\Delta_{3 \vartheta C}^p\left(\lambda_i\right)\right]^{\frac{1}{p}}$

$=1-\left[\frac{1}{3 \eta} \sum_{i=1}^2 \Delta_{1 \vartheta \Phi}^p\left(\lambda_i\right)+\Delta_{2 \vartheta \Phi}^p\left(\lambda_i\right)+\Delta_{3 \vartheta \Phi}^p\left(\lambda_i\right)\right]^{\frac{1}{p}}$

It means that $\varrho_\alpha(\boldsymbol{\vartheta}, C) \leq \varrho_\alpha(\boldsymbol{\vartheta}, \Phi)$.

Similarity, we have $\varrho_\alpha(\boldsymbol{\vartheta}, C) \leq \varrho_\alpha(\Phi, C)$.

(1) When $\alpha_1=0, \alpha_2=\alpha_3=+\infty$, Eq. (15) can be written as:

(2) When $\alpha_1=\alpha_2=+\infty \alpha_3=0$, Eq. (15) can be written as:

3.1.3 Theorem For any two SFSs $\boldsymbol{\vartheta}=\left\{\left(\lambda_i, \varphi_{\vartheta}\left(\lambda_i\right), i_{\vartheta}\left(\lambda_i\right), \gamma^{\prime}{ }_{\vartheta}\left(\lambda_i\right)\right) \mid \lambda_i \in X\right\} \quad$ and $\Phi=$ $\left\{\left(\left(\lambda_i, \varphi_{\Phi}\left(\lambda_i\right), i_{\Phi}\left(\lambda_i\right), \gamma_{\Phi}^{\prime}\left(\lambda_i\right)\right)\right) \mid \lambda_i \in X\right\}$ on $X=\left\{\lambda_1, \lambda_2, \ldots, \lambda_2\right\} \quad \omega_i \in[0,1]$ and $\sum_{i=1}^2 \omega_i=1$. The function $\varrho_\omega: S F S(\lambda) \times S F S(\lambda) \rightarrow[0,1]$ defined by

$\varrho_\omega(\boldsymbol{\vartheta}, \Phi)=1-\left[\frac{1}{3 \eta} \sum_{i=1}^2 \omega_i\left(\Delta_{1 \vartheta \Phi}^p\left(\lambda_i\right)+\Delta_{2 \vartheta \Phi}^p\left(\lambda_i\right)+\Delta_{3 \vartheta \Phi}^p\left(\lambda_i\right)\right)\right]^{\frac{1}{p}}$

$\varrho_\omega(\boldsymbol{\vartheta}, \Phi)$ is a weighted SM between $\boldsymbol{\vartheta}$ and $\Phi$.

Proof: The proof is similar to Theorem 1 .

In the following, an example is added to clarify more.

3.1.4 Example Let $\boldsymbol{\vartheta}=(\lambda, 0.2,0.4,0.5), \Phi=(\lambda, 0.4,0.3,0.4)$ and $C=(\lambda, 0.5,0.0,0.0)$ are three different SFVs on $X=\{\lambda\} . \boldsymbol{\vartheta}$ Is more similar to $\Phi$ than the $C$ say $\varrho(\boldsymbol{\vartheta}, \Phi)>\varrho(\boldsymbol{\vartheta}, C)$. To prove the accuracy of this view for our proposed SM $\varrho_\alpha$ and the current ones to be specific $\varrho_1, \varrho_2, \varrho_3, \varrho_4, \varrho_5, \varrho_6 \varrho_7, \varrho_9, \varrho_{10} \varrho_{11} \varrho_{12}, \varrho_{13}, \varrho_{14}, \varrho_\alpha$. We can see the obtained values of the SMs in Table 2.

In this section, we apply the proposed the SMs in MADM problems, which show the expected SM is sensible and in accordance with human cognition.

Let $X=\left\{\lambda_1, \lambda_2, \ldots \lambda_\alpha\right\}$ a set of attributes, the $\eta$ alternatives $\boldsymbol{\vartheta}_i=\left(\boldsymbol{\vartheta}_{i j}\right)=\left\{\left(\lambda_j, \varphi_{\boldsymbol{\vartheta} i}\left(\lambda_j\right), i_{\boldsymbol{\vartheta} i}\left(\lambda_j\right), \gamma_{\boldsymbol{\vartheta i}}^{\prime}\left(\lambda_j\right)\right)\right\} \mid \lambda_j \in X$ Where $\lambda_j, \varphi_{\vartheta}^2\left(\lambda_i\right), i_{\vartheta}^2\left(\lambda_i\right), \gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right), \varphi_{\Phi}^2\left(\lambda_i\right), i_{\Phi}^2\left(\lambda_i\right),{\gamma^{\prime}}_{\Phi}^{\ 2}\left(\lambda_i\right) \in[ 0,1]$ and $\varphi_{\vartheta}^2\left(\lambda_i\right)+i_{\vartheta}^2\left(\lambda_i\right)+{\gamma^{\prime}}^{ 2}{ }_{\vartheta}\left(\lambda_i\right) \leq 1, \varphi_{\vartheta}^2\left(\lambda_i\right)$ is a positive degree which is use to alternative $\boldsymbol{\vartheta}_i$ satisfies the $\lambda_j(i=\{1,2, \ldots 2\}, j=\{1,2, \ldots \alpha\}) . i_{\vartheta}^2\left(\lambda_i\right)$ a neutral degree which is use to alternative $\boldsymbol{\vartheta}_i$ does not satisfies the $\lambda_j \cdot \gamma^{\prime }{ }_{\vartheta}^2\left(\lambda_i\right)$, negative degree which is use to alternative $\boldsymbol{\vartheta}_i$ does not satisfies the $\lambda_j$. The decision making is used to choose best alternative steps are following.

Step 1. Standardize decision alternatives.

In this process multi attribute decision making can be divided in to type’s amount type and interest type. The amount type can be changed into interest type by use the formula of decision-making process.

$\boldsymbol{\vartheta}_{i j}^{\prime}=\left\{\begin{array}{c}\boldsymbol{\vartheta}_{i j} \text { for benefit attribute } \lambda_j \\ \boldsymbol{\vartheta}_{i j}^c \text { for cost attribute } \lambda_j\end{array}\right.$

$\boldsymbol{\vartheta}_{i j}^c=\left(\varphi_{\vartheta i}\left(\lambda_j\right), i_{\vartheta i}\left(\lambda_j\right), \gamma^{\prime}{ }_{\vartheta i}\left(\lambda_j\right)\right), i=\{1,2, \ldots \eta\}, j=\{1,2 \ldots \alpha\}$. The above formula based on the alternative $\boldsymbol{\vartheta}_i=\left\{\boldsymbol{\vartheta}_{i j}^{\prime}\right\}$.

Step 2. The SM $\varrho=\left(\boldsymbol{\vartheta}_i, \boldsymbol{\vartheta}\right)(i=1,2,3, \ldots, \eta)$ is calculated where, $\boldsymbol{\vartheta}$ = (0.2,0.4,0.5),(0.2,0.4,0.5),(0.2,0.4,0.5) is a standard provided by the decision maker in the form of the SFV. We find the similarity values with the help of the proposed SM.

Step 3. The maximum one is chosen in $\varrho=\left(\boldsymbol{\vartheta}_{i 0}, \boldsymbol{\vartheta}\right)$ from $\varrho=\left(\boldsymbol{\vartheta}_i, \boldsymbol{\vartheta}\right) i=(i=1,2,3, \ldots \eta)$ i.e $\varrho=\left(\boldsymbol{\vartheta}_{i 0}, \boldsymbol{\vartheta}\right)=$ $\max _{1 \leq i \leq 2}\left\{\varrho\left(\boldsymbol{\vartheta}_i, \boldsymbol{\vartheta}\right)\right\}$. Then the maximum SMs alternative $\boldsymbol{\vartheta}_{i 0}$ according to the principle of maximsum.

In the following example for the similarity measure $\varrho_\alpha, P=3, \alpha_1=\alpha_2=\alpha_3=3$

3.1.5 Example There are three medical equipment $\boldsymbol{\vartheta}_1, \boldsymbol{\vartheta}_2, \boldsymbol{\vartheta}_3$ with four different attributes $\lambda_1, \lambda_2, \lambda_3$ described the SFSs as shown in Table 3. The weight of $\lambda_j(1 \leq j \leq 3)$ are $(0.5,0.3,0.2)$.

In the following, Table 4 shows the values of the SMs of $\boldsymbol{\vartheta}_1, \boldsymbol{\vartheta}_2$, and $\boldsymbol{\vartheta}_3$ with $\boldsymbol{\vartheta}$.

Table 4 displays the values of the SMs for medical equipment based on their attributes. Now, we will determine the ranking of the alternatives using the values obtained from the SMs. The resulting decision rankings for the medical equipment are presented in Table 5.

It is cleared from Table 5, the alternative $\boldsymbol{\vartheta}_3$ is obtained by using the SMs $\varrho_1, \varrho_2, \varrho_3, \varrho_4, \varrho_5, \varrho_6, \varrho_7, \varrho_9, \varrho_{10}, \varrho_{11}, \varrho_{12}$, $\varrho_{13}, \varrho_{14}$ and $\varrho_\alpha$. However, the alternative $\boldsymbol{\vartheta}_3$ is obtained the $\varrho_8$. The ranking of the medical equipment is geometrically represented by Figure 1 as follows.

It is cleared from Figure 1, the alternative $\boldsymbol{\vartheta}_3$ is obtained by using the SMs $\varrho_1, \varrho_2, \varrho_3, \varrho_4, \varrho_5, \varrho_6, \varrho_7, \varrho_9, \varrho_{10}, \varrho_{11}, \varrho_{12}$, $\varrho_{13}, \varrho_{14}$ and $\varrho_\alpha$. However, the alternative $\boldsymbol{\vartheta}_3$ is obtained by the $\varrho_m$.

3.1.6 Example In this example, a MADM problem related to selecting medical equipment is solved. When prioritizing clinical needs assessment, factors such as regulatory compliance, interoperability with current systems, user-friendly interfaces, dependable quality from reputable manufacturers, cost-effectiveness, scalability, patient comfort and safety, evidence-based decision-making, and extensive training and support are crucial. By considering these factors, healthcare facilities can make well-informed decisions that not only improve patient care but also ensure operational effectiveness and regulatory compliance. Based on some attribute i.e., Precision $\left(\lambda_1\right)$, Versatility $\left(\lambda_2\right)$ and Ease of maintenance $\left(\lambda_3\right)$ the weight vector is $\omega=(0.2,0.3,0.5)$. Let there are six candidates to be assessed based on these attributes. Consider $\boldsymbol{\vartheta}=$ $\{(0.11,0.21,0.32),(0.11,0.21,0.32),(0.11,0.21,0.32)\}$ the standard is set in the form of the SFV. The alternative which is more similar to $\boldsymbol{\vartheta}$ is considered as the best employee. After an initial assessment, the employees are assigned the SFVs with respect to the attributes provided in Table 6 in the following.

The similarity of each candidate with the standard is evaluated using both existing and proposed SMs. The results are tabulated in Table 7.

Table 7 displays the values of the SMs for the employees compared to the standard, using both the existing and proposed SMs based on their attributes. The ranking of the medical equipment based on these values is provided in Table 8.

It is cleared from Table 8, the candidate $\boldsymbol{\vartheta}_1$ is obtained by using the SMs $\varrho_1, \varrho_2, \varrho_4, \varrho_5, \varrho_6, \varrho_7, \varrho_9, \varrho_{10}$ the candidate $\boldsymbol{\vartheta}_1$ is obtained by using the SMs $\varrho_\alpha$. Some SM not given the answer. The ranking is also geometrically represented in Figure 2.

In this study, new SMs are defined for SFS to evaluate the similarity between two SFVs. The newly defined SM for SFS generalizes the existing SMs by introducing parameters. The mathematical work and subsequent discussion demonstrate the viability and adaptability of the proposed SM. The limitations of the existing SMs for SFS have also been discussed. The following steps are discussed:

The proposed SM satisfies the axiom (S2), which ensures that the proposed SM avoids counterintuitive situations where $\boldsymbol{\vartheta}=\Phi$ implies $\varrho(\boldsymbol{\vartheta}, \Phi)=1$, a condition not met by some existing SMs.

The proposed SMs are based on the parameters $\alpha_1, \alpha_2$ and $\alpha_3$, giving decision-makers the flexibility to choose the values of these parameters independently. In this scenario, decision-makers can select appropriate values for the parameters $\alpha_1, \alpha_2, \alpha_3$ to obtain a sensible SM that aligns with the current leadership style and decision-making environment.

The proposed SM is capable of providing reliable and sensible decision-making results. It not only has a high level of credibility but can also address dynamic problems that current SMs cannot resolve, yielding reasonable decision outcomes. Therefore, the proposed SM is both practical and adaptable.