Evaluation of Cryptocurrency Markets Based on q-Rung Orthopair Fuzzy Hypersoft Frank Approach

The cryptocurrency, often known as virtual or digital currency, is one of the most significant breakthroughs brought about by digitalization. It has been mentioned recently, an innovative system for stockholders in particular. Cryptocurrencies of all kinds, including Bitcoin, Shib, Dogecoin, and Tether are decentralized. Categorization complicates decision-making (DM) as well as the transfer of uncertainty and digital currency validation. The multi-attribute group decision-making (MAGDM) method is defined in this article using the Frank weighted averaging and Frank weighted geometric aggregation operators (AOs). This research examines the uniqueness of the qRung orthopair fuzzy hypersoft set (qROFHSS) that is responsive to volatilities, vagueness, doubt, and inaccurate data. Additionally, some specific essential qROFHSS operational laws are defined in this study. The framework presented here is the best alternative for understanding electronic money. This research supports the difficulty of situations involving making choices that must take into account a variety of qualities and sub-attributes to select the best choice. We see that Bitcoin has a variety of applications and that cryptocurrencies have an excellent chance to emerge as a prominent benefit lesson in financial choice-making. In this paper, to demonstrate the viability of the suggested methodology we look at a case study on choosing the best currency out of these that are accessible to manage cryptocurrency. Comparative analysis shows that the suggested strategy is stable and good. We conclude that the proposed methodology provides a thorough and systematic approach to analyzing cryptocurrency. Finally, a numerical example is provided to show the method’s efficiency.

SECTION I. Introduction

The idea of cryptocurrencies has been around for a while. Digital currencies utilizing cryptography to verify transactions are referred to as cryptocurrencies. A coin that was created to be a collaborative electronic cash exchange established by Nakamoto [1]. Consequently, Bitcoin, the initial digital currency, was established in 2009. Bitcoins are digital money systems that rely on the technology of blockchain and the technique of cryptography by Natarajan [2]. A difficult activity in the online market is the purchase or sale of a cryptocurrency due to the market’s exponential changes. To address this, we need to analyze the currency exchange using the DM procedure. Making decisions involves selecting the most suitable options from a set of data. A lot of concepts have been provided by various researchers to help people select the best choice. At the start of the era, decisions were made using precise numerical datasets, but this led to inadequate findings that were more useful in actual situations. Various DM models were worked by numerous researchers in several areas of statistics, mathematics, and artificial intelligence. Trading behavior and high volatility, explained by Urquhart [3], are what attract individuals to Bitcoin. However, it was noted that no useful outcomes for predicting volatility could be found in search engines. The reaction sequence between the socio-economic indications in the cryptocurrency budget, yet, was the focus of Garcia et al. [4] research. Additionally, two beneficial feedback cycles were identified using vector auto regression. In 2020, created a model to forecast for the price of bitcoin using Chen logical technique and fuzzy time series of Markov chain. Because this method is devoid of classical assumptions, fuzzy period sequence can simulate a variety of period sequence data patterns.

The traditional idea of a set, Zadeh [5] created the fuzzy set (FS) theory. In their work on FS, Bhattacharya, and Mukherjee [6] created fuzzy relations and fuzzy groupings. Using an FS set, Yager and Filev [7], [8] created AOs fuzzy illustrations, and constructions in 1994. Demirci [9] described fuzzy functions and their basic characteristics. Intuitionist fuzzy sets (IFS) were presented by Atanassov [10] with the restriction that the sum of these two values should not be greater than one. When the sum of the membership value (MV) and non-membership value (NMV) exceeds one, IFS may not have fully satisfied this requirement. (For instance, $0.8 +0.7>1)$ , IFS are invalid. In an extension of IF, Yager [11] presented Pythagorean fuzzy sets (PFS), where the square sum of the MV and NMV is less than or equal to one. The same linear inequalities between MV and NMV are investigated as in the prior study.

However, when the power is increased to $2~0.84+0.721$ is found, indicating that the PFS theory is likewise incorrect. Complex interval-valued PFS (CIVPFS) were employed in green supply chain administration by Ali et al. [12]. As a solution to an issue of DM, Ashraf et al. [13] introduced the interval-valued picture fuzzy Maclaurin symmetric mean operator (IVPFMSMO). The requirements for the membership function and non-membership function are familiar in the situation of the q-rung orthopair fuzzy set (qROFS) [14], [15], [16]. We can somewhat handle the MV and NMV separately, even for very big standards of “q.” As a consequence, the qROFS is better than IFS and PFS in interpreting equivocal data. On the other hand, these theories fail to explain the parameter results of the alternative models. To get over these limitations, Molodtsov [17] introduced the idea of soft set (SS) theory for handling uncertainty in an invariable approach. In direction to solve issues, found several accurate models and put forth the SS theory. Cagman et al. [18] presented fuzzy soft sets (FSS), as well as fuzzy AOs, by utilizing the notions of SS and putting them to use in practical settings. Maji [19], [20] developed a fuzzy soft set (FSS) theory and a neutrosophic soft set (NSS) theory using the SS scheme. The prior work only looked at data gaps brought on by MV and NMV. These presumptions, however, are inadequate to account for the data’s general irregularity and correctness. When features of a group of parameters incorporate extra sub-attributes, previous theories are unable to manage the issue. Smarandache [21] used the SS concept to introduce the hypersoft set (HSS) theme to get over the restriction mentioned above. Saeed et al. [22] then introduced the fundamental concepts of the HSS, including hypersoft subset, complement, and non-set AOs. Numerous researchers have examined various operators and properties that fall under the HSS and its expansions [23], [24], [25], [26], and [27].

Mehnaz et al. [28] proposed Frank operational laws for T-spherical values using TCN and TCNM depending on the proposed operations. Reference [29] developed qRO fuzzy frank aggregation operators (qROFFAOs) and their application on MADM and AOs depends on the TCN and TCNM. Xia et al. [30] established intuitionistic fuzzy aggregation operators (IFAOs) depending on Archimedean TCN and TCNM and also introduced some operational laws. Silambarasan [31] proposed generalized orthopair fuzzy which depends on the Hamacher TCN and TCNM. Operational laws and their properties, Demorgan laws over complement proposed for qROFS. Reference [32] developed comprehensive geometric AOs that depend on TCN. DM method is established for the multi-criteria decision-making (MCDM) issues. Garg [33] generalized IF geometric interaction operators through Einstein TCN and TCNM on the MCDM. Yang et al. [34] Pythagorean fuzzy (PF) on Benferroni means which depend on the TCN and TCNM. Xing et al. [35] developed AOs for Pythagorean fuzzy numbers (PFN) which are based on the FTCN and FTCNM and also introduced the Choquet Frank averaging and geometric operators on the MADM technique. Khan et al. [36] introduced a theme of some fundamental operations such as qROFHS subset, qROFHS null set, qROFHS absolute set, union, intersection, and complement. We also present “AND” and “OR” operators.

The DM problem modeling detailed consideration of the traits is necessary, and we cannot simply examine or disregard any quality without looking at the other’s significance. It is essential to utilize the principle of the HSS to handle additional attributes. Given that FHSS deals with attributes and sub-attributes uncertainly and that SS only deals with attributes, HSS deals with attributes and sub-attributes. In light of the nature of sub-attributes, this is the reason why we settled on the discipline of HSS. The development of new AOs for a qROFHS environment is the primary objective of our research. We also developed an algorithm to describe situations involving several criteria for decision-making, as well as a mathematical example to demonstrate in what way the recommended method is effective in the qROFHS setting. The process of choosing and evaluating cryptocurrencies is crucial in the digital economy. More studies using MAGDM methodologies in the selection of cryptocurrencies are needed to correctly represent the ambiguity of the cryptocurrency market information in addition to that of the individual making the choice. We propose a set of operational guidelines that are dependent on the outcome method in terms of the qROFHS set. Then, we create two aggregation operators, the qROFHSFWA and qROFHSFWG, using operational ideas. Score and accuracy functions for contrasting the qROFHS set are also offered.

The qROFHSS is a particular case of the IFHSS, PFHSS, and FHSS. It is revealed by these formations. Because our suggested framework, when compared to previous research, provides additional details. It is suggested design can handle ambiguous data in making choices in a relatively straightforward manner. As a result, compared to other fuzzy frameworks, the structure of the qROFHS set is more useful. To address issues of daily living, we use a variety of attribute options in our proposed framework. Compared to the current systems, our proposed structure specifically addressed uncertainties. Section II gathers some essential components that will help in the composition of the article’s remaining section the qROFHS set, the SS, and the HSS. Section III, this section covers the score, accuracy function, qROFHS weighted average, and qROFHS weighted geometric operators for qROFHSVs. We also use created qROFHSVs to describe the fundamental characteristics of qROFHS weighted averaging and qROFHS weighted geometric AOs. In section IV, construct a numerical example. Section V concludes the study.

SECTION II. Preliminaries

In this section, we define fundamental definitions of SS and qROFHSS. dcltmp

Definition 1[17]:

Let °F be a universe set with characteristics set $\aleph$ , and $\Delta \subseteq \aleph$ and a pair $\left ({\beth , \Delta }\right )$ , where $\beth$ is a function that $\beth : \Delta \rightarrow {\beta ^{ ^{\circ }\mathrm {F}}}$ and ${\beta ^{ ^{\circ }\mathrm {F}}}$ denotes the collection of all possible subsets of °F to as a soft set over °F. It is possible to define a pair as $\left ({\beth , \Delta }\right )=\left \{{\left ({{\sigma },\beth \left ({{\sigma }}\right )}\right )|{\sigma } \in \Delta ,\beth \left ({{\sigma }}\right ) \in {\gimel ^{ ^{\circ }\mathrm {F}}}}\right \}{}$ .

Definition 2[21]:

Let °F be a universal set and $~{ }^{\circ }\mathrm {C}$ unique characteristics sets ${c_{1 }},{c_{2 }}, \cdots {c_{n}}$ , whose characteristics values belong to the sets ${\omega _{1 }},{\omega _{2 }}, \cdots {\omega _{n}}$ , as well, with respect to ${\omega _{i}} \cap {\omega _{j}}\mathfrak {=H}$ , $~{\forall },i,j=\left \{{1,2 \cdots m}\right \}{}$ . Over °F a pair $\left ({\beth , \Delta }\right )$ is referred to as a hypersoft, where $\beth$ is a function such that $\beth : \Delta \rightarrow p(u)$ and $\Delta ={\omega _{1 }},{\omega _{2 }}, \cdots {\omega _{n}}$ . The definition of a pair $\left ({\beth , \Delta }\right )=\left \{{\left ({{\varpi }, \Delta \left ({{\varpi }}\right )}\right )|{\varpi } \in \Delta ,\Delta \left ({{\varpi }}\right ) \in p(u)}\right \}{}$ .

Definition 3[36]:

Let °F be a set with universal and that ${b_{1 }},{b_{2 }}, \cdots {b_{n}}$ be $n$ different characteristics pertaining to °F respectively, whose correspondence characteristics values the sets ${\omega _{1 }},{\omega _{2 }}, \cdots {\omega _{n}}$ such that ${\omega _{i}} \cap {\omega _{j}}={\phi }$ where $i=j$ for each $n>1~and~i,j=\left \{{1,2, \cdots n}\right \}{}$ . A pair $\left ({\beth , \Delta }\right )$ is referred to as qROFHSS, where ${\omega _{1 }},{\omega _{2 }}, \cdots {\omega _{n}}= \Delta =\left \{{{d_{1 }},{d_{2 }}, \cdots {d_{n}}}\right \}{}$ is a collection of sub parameters and $\beth$ is a mapping $\beth : \Delta \rightarrow qROF{S^{ ^{\circ }\mathrm {F}}}$ for qROFS. A pair $\left ({\beth , \Delta }\right )$ can be presented as $\left ({\beth , \Delta }\right )=\left \{{\left ({{\varpi },{\beth _{\Delta }}\left ({{\varpi }}\right )\!:\!{\varpi } \in \Delta ,{\beth _{\Delta }}\left ({{\varpi }}\right ) \in qROF{S^{ ^{\circ }\mathrm {F}}} \in 0,1}\right )}\right \}$ , where $\left ({\beth , \Delta }\right )=\left \{{\left ({y,{\gamma _{\Delta \left ({{\varpi }}\right )}}(y),{\gamma _{\Delta \left ({{\varpi }}\right )}^{\prime }}(y)}\right )|y \in { }^{\circ }\mathrm {F}~and~l{\geq }1}\right \}{}$ , where $qROFHSV$ can be expressed as $\left ({\beth , \Delta }\right )=\left ({{\gamma _{\Delta \left ({{\varpi _{ij}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{ij}}}\right )}^{\prime }}}\right )$ .

Definition 4:

If the following holds we explain $\left ({\beth , \Delta }\right )$ as a qROFHS subset of $\left ({\beth ,{\alpha ^{\prime }}}\right )$ , expressed as $\left ({\beth , \Delta }\right ) \subseteq \left ({\mathrm {Z},{\alpha ^{\prime }}}\right )$ for two qROFHS sets $\left ({\beth , \Delta }\right )$ and $\left ({\mathrm {Z},{\alpha ^{\prime }}}\right )$ through a universe discourse °F.

• $\Delta \subseteq {\alpha ^{\prime }}$

• ${\text {For any}}\varpi \in \Delta ,\beth \left ({\varpi }\right )\subseteq \mathrm {Z}\left ({{a^{\prime }}}\right )$ .

Definition 5[35]:

FTCN and FTCNM are described as two real numbers $\mathcal {H}$ and $\mathfrak {R}$ in the range [0, 1]. $$\begin{align*} {F_{TCN}}\left ({a,b}\right )&=\mathop {\mathrm {log}_{t}}\left ({1+\frac {\left ({{t^{\mathcal {H}}}-1}\right ) ({t^{\mathfrak {R}}}-1)}{t-1}}\right )\\ {F_{TCNM~}}&=1-\mathop {\mathrm {log}_{t}}\left ({1+\frac {\left ({{t^{1 -\mathcal {H}}}-1}\right )\left ({{t^{1 -\mathfrak {R}}}-1}\right )}{t-1}}\right )\end{align*}$$ where $\boldsymbol {}\left ({ \mathcal {H,} \mathfrak {R}}\right ) \in \left [{0,1}\right ]\times [{0,1}]$ and $\mathfrak {~H{\not =}}0$ .

SECTION III. qROFHSF Aggregation Operators Based on qROFHSFWA and qROFHSFWG

This section covers the qROFHSVs score and accuracy function, in addition to the qROFHSF weighted averaging and weighted geometric operators. Additionally, we use constructed qROFHSVs to explore the fundamental characteristics of the qROFHSF weighted averaging and weighted geometric aggregation operators.

Definition 6:

The score function of qROFHSVs is stated as $$\begin{equation*}{\Psi _{\varpi _{ij}}}={\gamma _{\Delta \left ({{\varpi _{ij}}}\right )}^{q}}-{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{ij}}}\right )}^{q}} \tag{1}\end{equation*}$$

Definition 7:

The accuracy function of qROFHVs is stated as $$\begin{equation*}{\alpha }\left ({{\Psi _{\varpi _{ij}}}}\right )={\gamma _{\Delta \left ({{\varpi _{ij}}}\right )}^{q}}+{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{ij}}}\right )}^{q}} \tag{2}\end{equation*}$$ The following laws are divided into categories for qROFHSVs comparison:

1. $P\left ({{\Psi _{\varpi _{ij}}}}\right )>\left ({{\Psi _{\varpi _{ij}}^{\prime }}}\right );{\text {then},}{\Psi _{\varpi _{ij}}}>{\Psi _{\varpi _{ij}}^{\prime }}$

2. $P\left ({{\Psi _{\varpi _{ij}}}}\right )=\left ({{\Psi _{\varpi _{ij}}^{\prime }}}\right );$ then

   • $~{\text {If}~}{\alpha }\left ({{\Psi _{\varpi _{ij}}}}\right )>{\alpha }\left ({{\Psi _{\varpi _{ij}}^{\prime }}}\right ){,~\text {then},}{\Psi _{\varpi _{ij}}}>{\Psi _{\varpi _{ij}}^{\prime }}$

   • $~{\text {If}~}{\alpha }\left ({{\Psi _{\varpi _{ij}}}}\right )={\alpha }\left ({{\Psi _{\varpi _{ij}}^{\prime }}}\right ),~{\text {then}~}{\Psi _{\varpi _{ij}}}={\Psi _{\varpi _{ij}}^{\prime }}$

Definition 8:

Let ${\Psi _{\varpi _{ij}}}=\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )$ be two qROFHSVs and ${\mu }=\left ({{\mu _{1 }},{\mu _{2 }}, \cdots {\mu _{m}}}\right )$ and ${\pi }=\left ({{\pi _{1 }},{\pi _{2 }}, \cdots {\pi _{n}}}\right )$ be the specialist’s weight and chosen sub-characteristics with the constraints that ${\mu _{i}}>0,{\sum }_{i=1 }^{n}{\mu _{i}}=1,{\pi _{i}}>0,{\sum }_{i=1 }^{m}{\pi _{i}}=1$ . The mapping for the $qROFHSFWA:{\rho ^{n}}\rightarrow {\rho }$ , where ${\rho }$ is the collection of all $qROFHSVs$ , provided as $$\begin{align*}qROFHSFWA&=\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right ) \\ &=\mathop {\mathop {\oplus }\limits _{j=1 }}\limits ^{m}{\pi _{i}}\left ({{\mathop {\mathop {\oplus }\limits _{i=1 }}\limits ^{n}{\mu }_{i}}{\Psi _{\varpi _{ij}}}}\right )\end{align*}$$

Theorem 1:

Let ${\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right ){}$ qROFHSVs. Then the accumulated result for the qROFHSFWA operator is specified (3), as shown at the bottom of the next page. $$\begin{align*}qROFHSFWA&=\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )=\mathop {\mathop {\oplus }\limits _{j=1 }}\limits ^{m}{\pi _{j}}\left ({{\mathop {\mathop {\oplus }\limits _{i=1 }}\limits ^{n}{\mu }_{i}}{\Psi _{\varpi _{ij}}}}\right ) \\ &\begin{matrix}=\left ({\begin{matrix}\displaystyle \sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+\mathop {\prod }\limits _{i=1 }^{n}{\left ({{t^{\left ({1 -{\left ({\sqrt [{q}]{1 -\mathop {\mathrm {log}_{t}}\left ({\begin{matrix}\displaystyle 1 + \prod _{j=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}\right )}}-1 }\right )^{\mu _{j}}}\end{matrix}}\right )}}\right )^{q}}}\right )}}-1}\right )^{\pi _{i}}}}\right )},\\ \displaystyle \mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{i=1 }^{n}{\left ({{t^{\left ({\mathop {\mathrm {log}_{t}}\left ({1 +\left ({\prod _{j=1 }^{m}{\left ({{t^{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}-1 }\right )^{\mu _{j}}}}\right )}\right )}\right )}}-1}\right )^{\pi _{i}}}}\right )}\right )\end{matrix}}\right )\end{matrix} \tag{3}\end{align*}$$

Proof:

We prove this theorem by the formula as follows. First, we have $$\begin{align*} &\mathop {\mathop {\oplus }\limits _{r=1 }}\limits ^{m}{\mu _{j}}\left ({{\Psi _{\varpi _{ij}}}}\right ) \\ &=\left ({\begin{matrix}\displaystyle \sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+\mathop {\prod }\limits _{j=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}\right )}}-1}\right )^{F{\mu _{j}}}}}\right )},\\ \displaystyle \left ({\mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{j=1 }^{m}{\left ({{t^{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}-1}\right )^{\mu _{j}}}}\right )}\right )}\right )\end{matrix}}\right )\\ &=qROFHSFWA\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\end{align*}$$ Hence, $qROFHSFWA\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )$ gives a qROFHSFWA.

Where ${\pi _{j}}=\left \{{1,2, \cdots n}\right \}{}$ and ${\mu _{i}}=\left \{{1,2, \cdots m}\right \}{}$ are the specialist’s weights and sub-attributes of the chosen parameters by the constraints that, ${\pi _{i}}>0,{\sum }_{i=1 }^{n}{\pi _{i}}=1,{\mu _{j}}>0,{\sum }_{j=1 }^{m}{\mu _{j}}=1$ . The mapping for the qROFHSFWA operator is expressed as $qROFHSWA:{\rho ^{n}}\rightarrow {\rho }$ is the set of all qROFHSVs.

Theorem 2:

Let ${\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right ),~i,j=1,2, \cdots n,m$ be a collection of qROFHSV then $$\begin{equation*}{\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )={\Psi _{\varpi }}\end{equation*}$$ Which is called the idempotency of the qROFHSFWA.

Proof:

As we know that $$\begin{equation*}{\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )={\Psi _{\varpi }}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )\end{equation*}$$ Then we have, as shown in the equation at the bottom of the next page. $$\begin{align*} &\hspace {-3pc}qROFHSFWA\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\\ &=\left ({\begin{matrix}\displaystyle \sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+\mathop {\prod }\limits _{j=1 }^{n}{\left ({{t^{\left ({1 -{\left ({\sqrt [{q}]{1 -\mathop {\mathrm {log}_{t}}\left ({1 +\prod _{i=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}\right )}}-1 }\right )^{\mu _{i}}}}\right )}}\right )^{q}}}\right )}}-1}\right )^{\pi _{j}}}}\right )},\\ \displaystyle \mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{j=1 }^{n}{\left ({{t^{\left ({\mathop {\mathrm {log}_{t}}\left ({1 +\left ({\prod _{i=1 }^{m}{\left ({{t^{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}-1 }\right )^{\mu _{i}}}}\right )}\right )}\right )}}-1}\right )^{\pi _{j}}}}\right )}\right )\end{matrix}}\right )\\ &=\left ({\begin{matrix}\displaystyle \sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+\mathop {\prod }\limits _{i=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}\right )}}-1}\right )^{\mu _{i}}}}\right )},\\ \displaystyle \left ({\mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{i=1 }^{m}{\left ({{t^{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}-1}\right )^{\mu _{i}}}}\right )}\right )}\right )\end{matrix}}\right )\\ &=\left ({\sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}\right )}}-1}\right )^{\mu _{i}}}}\right )},\left ({\mathop {\mathrm {log}_{t}}\left ({1+\left ({{\left ({{t^{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}-1}\right )^{\mu _{i}}}}\right )}\right )}\right )}\right )\\ &=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )\end{align*}$$

Theorem 3:

Let ${{\chi _{e} }_{ij}}=\left ({{{\omega _{e} }_{ij}}}\right ),i,j=1,2, \cdots n,m$ be a qROFHSVs. If that ${\chi ^{-}}=\left ({min{{\omega _{e} }_{ij}},max{{\phi _{e} }_{ij}}}\right )$ and ${\chi ^{+}}=\left ({max{{\omega _{e} }_{ij}},min{{\phi _{e} }_{ij}}}\right )$ $$\begin{equation*}{\chi ^{-}}{\leq }qROFHSFWA\left ({\mathfrak {I}{\mathscr {o}_{11 }}\mathfrak {,~I}{\mathscr {o}_{12 }}\mathfrak {, \cdots I}{\mathscr {o}_{nm}}}\right ){\leq }{\chi ^{+}}\end{equation*}$$

Proof:

Proof is skipped.

Theorem 4:

Let ${~{\Psi }_{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )$ and ${\Psi _{\varpi _{ij}}^{\mathrm {a}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{a}},{{\gamma ^{a}}_{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right ){}$ be two collections of qROFHSVs such that ${\Psi _{\varpi _{ij}}}{\leq }{\Psi _{\varpi _{ij}}^{\mathrm {a}}}$ . Then $$\begin{align*}&\hspace {-2pc}qROFHSFWA\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\\ &\qquad \leq qROFHSFWA\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\end{align*}$$

Definition 9:

To define a weighted geometric qROFHSFWG operator is $$\begin{align*}&qROFHSFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right ) \\ &\qquad \qquad \qquad \qquad =\mathop {\mathop {\otimes }\limits _{i=1 }}\limits ^{m}{\left ({\mathop {\mathop {\otimes }\limits _{j=1 }}\limits ^{n}{\left ({{\Psi _{\varpi _{ij}}}}\right )^{\mu _{j}}}}\right )^{\pi _{i}}}\end{align*}$$ where ${\pi _{j}}=\left \{{1,2, \cdots n}\right \},{\mu _{i}}=\left \{{1,2, \cdots m}\right \}{}$ are specialists’ weights and sub-attributes of the chosen parameters by the constraints that, ${\pi _{j}}>0,\sum _{j=1 }^{n}{\pi _{j}}=1$ and ${\mu _{i}}>0,\sum _{i=1 }^{m}{\mu _{i}}=1$ . The mapping of the qROFHSFWA operator is presented as $qROFHSFWA:{\rho ^{n}}\rightarrow {\rho }$ is the set of all qROFHSVs.

Theorem 5:

A weighted average qROFHSFWA is expressed (4), as shown at the bottom of the next page $$\begin{align*}qROFSHFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )&=\mathop {\mathop {\otimes }\limits _{i=1 }}\limits ^{m}{\left ({\mathop {\mathop {\otimes }\limits _{j=1 }}\limits ^{n}{\left ({{\Psi _{\varpi _{ij}}}}\right )^{\mu _{i}}}}\right )^{\pi _{j}}} \\ &\quad \times \left ({\begin{matrix}\displaystyle \mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{j=1 }^{n}{\left ({{t^{\left ({\mathop {\mathrm {log}_{t}}\left ({1 +\left ({\prod _{i=1 }^{m}{\left ({{t^{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}-1 }\right )^{\mu _{i}}}}\right )}\right )}\right )}}-1}\right )^{\pi _{j}}}}\right )}\right ),\\ \displaystyle \sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({\begin{matrix}\displaystyle 1+ \displaystyle \mathop {\prod }\limits _{j=1 }^{n}{\left ({{t^{\left ({1 -{\left ({\sqrt [{q}]{1 -\mathop {\mathrm {log}_{t}}\left ({1 +\prod _{i=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta }}\left ({{\varpi _{k}}}\right )}\right )}}-1 }\right )^{u_{i}}}}\right )}}\right )^{q}}}\right )}}-1}\right )^{\pi _{j}}}\end{matrix}}\right )}\end{matrix}}\right ) \tag{4}\end{align*}$$

Proof:

We prove this theorem by the formula as follows. First, we have $$\begin{align*}&\mathop {\mathop {\otimes }\limits _{j=1 }}\limits ^{n}{\mu _{i}}\left ({{\Psi _{\varpi _{ij}}}}\right ) \\ &=\left ({\begin{matrix}\displaystyle \left ({\mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{i=1 }^{m}{\left ({{t^{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}-1}\right )^{\mu _{i}}}}\right )}\right )}\right ),\\ \displaystyle {\left ({\sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+\mathop {\prod }\limits _{i=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right )}}-1}\right )^{\mu _{i}}}}\right )}}\right )^{q}}\end{matrix}}\right )\\ &=qROFHSFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\end{align*}$$ Hence, $qROFHSFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )$ gives a qROFHSFWG.

Where ${\pi _{j}}=\left \{{1,2, \cdots n}\right \}{}$ and ${\mu _{i}}=\left \{{1,2, \cdots m}\right \}{}$ are specialists’ weights and sub-attributes of the chosen parameters by the constraints that, ${\pi _{j}}>0,\sum _{j=1 }^{n}{\pi _{j}}=1,{\mu _{i}}>0,\sum _{i=1 }^{m}{\mu _{i}}\!=\!1$ . The mapping for the qROFHSFWG operator is expressed as $qROFHSFWG:{\aleph ^{n}}\rightarrow \aleph$ is the set of all qROFHSFVs.

Theorem 6:

Let ${\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right ){},~i,j=1,2, \cdots n,m$ be a collection of qROFHSV then $$\begin{equation*}{\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )={\Psi _{\varpi }}\end{equation*}$$ Which is called the idempotency of the qROFHSFWA.

Proof:

As we know that $$\begin{equation*}{\Psi _{\varpi _{ij}}}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )={\Psi _{\varpi }}=\left ({\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )}\right )\end{equation*}$$

Then we have, as shown in the equation at the bottom of the next page. $$\begin{align*} &qROFHSFWA\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\\ &=\left ({\begin{matrix}\displaystyle \mathop {\mathrm {log}_{t}}\left ({\begin{matrix}\displaystyle 1+\\ \displaystyle \left ({\mathop {\prod }\limits _{j=1 }^{n}{\left ({\begin{matrix}\displaystyle {t^{\left ({\mathop {\mathrm {log}_{t}}\left ({\begin{matrix}\displaystyle 1 +\\ \displaystyle \left ({{\prod }_{i=1 }^{m}{\left ({{t^{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}-1 }\right )^{\mu _{i}}}}\right )\end{matrix}}\right )}\right )}}\\ \displaystyle -1\end{matrix}}\right )^{\pi _{j}}}}\right )\end{matrix}}\right ),\\ \displaystyle \sqrt [{q}]{\begin{matrix}\displaystyle 1-\\ \displaystyle \mathop {\mathrm {log}_{t}}\left ({\begin{matrix}\displaystyle 1+\\ \displaystyle \mathop {\prod }\limits _{i=1 }^{n}{\left ({{t^{\left ({\begin{matrix}\displaystyle 1 -\\ \displaystyle {\left ({\sqrt [{q}]{\begin{matrix}\displaystyle 1 -\\ \displaystyle \mathop {\mathrm {log}_{t}}\left ({\begin{matrix}\displaystyle 1 +\\ \displaystyle {\prod }_{j=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right )}}-1 }\right )^{\mu _{j}}}\end{matrix}}\right )\end{matrix}}}\right )^{q}}\end{matrix}}\right )}}-1}\right )^{\pi _{i}}}\end{matrix}}\right )\end{matrix}}\end{matrix}}\right )\\ &=\left ({\begin{matrix}\displaystyle \left ({\mathop {\mathrm {log}_{t}}\left ({1+\left ({\mathop {\prod }\limits _{i=1 }^{m}{\left ({{t^{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}-1}\right )^{\mu _{i}}}}\right )}\right )}\right ),\\ \displaystyle \sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+\mathop {\prod }\limits _{i=1 }^{m}{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right )}}-1}\right )^{\mu _{i}}}}\right )}\end{matrix}}\right )\\ &=\left ({\left ({\mathop {\mathrm {log}_{t}}\left ({1+\left ({{\left ({{t^{{\gamma ^{\prime }}_{\Delta \left ({{\varpi _{k}}}\right )}^{q}}}-1}\right )^{\mu _{i}}}}\right )}\right )}\right ),\sqrt [{q}]{1-\mathop {\mathrm {log}_{t}}\left ({1+{\left ({{t^{\left ({1 -{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right )}}-1}\right )^{\mu _{i}}}}\right )}}\right )\\ &=\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right )\end{align*}$$

Theorem 7:

Let ${\Psi _{\varpi _{ij}}}=\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right ),i,j=1,2, \cdots n,m$ be a qROFHSVs. If that ${\Psi ^{-}}=\left ({min{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},max{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}}\right )$ and ${\Psi ^{+}}=\big (max{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}},min{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }}\big )$ $$\begin{equation*} {\Psi ^{-}}{\leq }qROFHSFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right ){\leq }{\Psi ^{+}}\end{equation*}$$

Proof:

Proof is skipped.

Theorem 8:

Let ${~{\Psi }_{\varpi _{ij}}}=\left ({{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}^{\prime }},{\gamma _{\Delta \left ({{\varpi _{k}}}\right )}}}\right )$ and ${\Psi _{\varpi _{ij}}^{\mathrm {a}}}=\left ({\left ({{\gamma _{~\Delta \left ({{\varpi _{k}}}\right )}^{a}},{{\gamma ^{\prime }}_{~ \Delta \left ({{\varpi _{k}}}\right )}^{a}}}\right )}\right )$ be two collections of qROFHSVs such that ${~{\Psi }_{\varpi _{ij}}}{\leq }{\Psi _{\varpi _{ij}}^{\mathrm {a}}}$ . Then $$\begin{align*}&\hspace {-0.5pc}qROFHSFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\\ &\qquad \qquad \qquad \leq qROFHSFWG\left ({{\Psi _{\varpi _{11 }}},{\Psi _{\varpi _{12 }}}, \cdots {\Psi _{\varpi _{nm}}}}\right )\end{align*}$$

SECTION IV. MAGDM Based on qROFHSFWA and qROFHSFWG

MAGDM is a method developed to select substitutes in different situations. Now we will use our selected method under the qROFHSS environment for multi multi-attribute group decision-making process. Let ${\chi }=\left \{{{\chi ^{1 }},{\chi ^{2 }}, \cdots {\chi ^{s}}}\right \}{}$ be a set of characteristics and $\left \{{{\vartheta _{1 }},{\vartheta _{2 }}, \cdots {\vartheta _{m}}}\right \}{}$ be a set of m specialists. The weights of specialists are ${\pi }={\left \{{{\pi _{1 }},{\pi _{2 }}, \cdots {\pi _{j}}}\right \}^{T}}$ . Let ${\varrho }=\left ({{\varrho _{1 }},{\varrho _{2 }}, \cdots {\varrho _{m}}}\right )$ be the set of characteristics with their corresponding sub-attributes as ${\tau }=\left ({{o_{1 b}},{o_{2 b}}, \cdots {o_{mp}}}\right ),{\forall }~b \in \left \{{1,2, \cdots s}\right \}{}$ with the weights ${\mu }=\left ({{\mu _{1 b}},{\mu _{2 b}}, \cdots {\mu _{mp}}}\right )$ such as ${\mu _{b}}>0,\sum _{p=1 }^{s}{b_{b}}=1$ .

1. Create a decision matrix using parameter sub-attributes

2. Create a decision matrix depending on the expert’s evaluation of each replacement in the form of qROFHSVs. $$\begin{align*} \left ({{\chi },{\pi }}\right )&={\left ({{{\Psi _{\varpi }^{\mathrm {q}}}_{ij}}}\right )_{n\times m}}\\ &=\left ({\begin{matrix}\displaystyle {{\Psi _{\varpi }^{\mathrm {q}}}_{11 }},{{\Psi _{\varpi }^{\mathrm {q}}}_{12 }}, \cdots {{\Psi _{\varpi }^{\mathrm {q}}}_{1 m}}\\ \displaystyle {{\Psi _{\varpi }^{\mathrm {q}}}_{21 }},{{\Psi _{\varpi } }_{22 }}, \cdots {{\Psi _{\varpi } }_{2 m}}\\ \displaystyle .~.~\cdots ~.\\ \displaystyle \quad ~~.\qquad \quad ~~.\qquad \qquad ~\cdots \qquad \quad ~~.\quad ~\\ \displaystyle {\Psi _{\varpi n1 }^{\mathrm {q}}},{\Psi _{\mathrm {\varpi n2 }}^{q}}, \cdots \qquad ~~{\Psi _{\varpi nm}^{q}}\\ \displaystyle \end{matrix}}\right )\end{align*}$$

3. Combine all of the substitutes using the developed aggregation operators, such as the Frank weighted geometric operator and Frank weighted average.

4. Calculate the score for each replacement.

5. By ranking the substitutes in descending order of score value, choose the best option.

To establish how each step of the distinctive DM approach works, we present a practical step-wise method that depends on the following scenario. We want to study the price stability of specific cryptocurrencies denoted as ${\tau }_{1}=Bitcoin, \tau _{2}=Shib, \tau _{3}=Degecoin$ , and $\tau _{4}=Tether$ . A committee of decision makers having weight $\pi _{j}=\left ({ 0.4,0.19,0.11,0.3 }\right )$ and $\mu _{i}=\left ({ 0.41,0.25,0.24,0.1 }\right )$ chooses the best cryptocurrency. The attribute set $\varrho _{1}=\{safety,~development\}$ with their corresponding sub-attributes are given as $\{ o_{11}=strong ~level~of ~safety,o_{12} = low ~level ~of ~safety~\}$ , $\varrho _{2}=development$ , $\{ \varrho _{21}= fast~development,~\varrho _{22}=slow~development \}. \varrho =\varrho _{1}{\times \varrho }_{2}$ Is it a set of sub-characteristics that have 2-tuple elements?

As there does not exist any cost type attribute, hence we do not need to normalize the decision matrix. Hence, we start the aggregation with the help of the developed qROFHSFWA/qROFHSFWG operators. In the following, Table 5 represents the aggregated value of the information obtained from the experts in Tables 1–​4 in the form of the qROFHSVs with the help of the qROFHSFWA/qROFHSFWG operators.

TABLE 1 Decision Matrix for $\tau_{1} =$ Bitcoin

TABLE 2 Decision Matrix for $\tau_{2} =$ Shib

TABLE 3 Decision Matrix for $\tau_{3} =$ Dogecoin

TABLE 4 Decision Matrix for $\tau_{4} =$ Tether

TABLE 5 Decision Matrix for $\tau_{i}$

Table 5 represents the aggregated values of the alternatives obtained by using qROFHSFWA and qROFHSFWG operators. Note that the aggregated values obtained are again in the form of the qROFHSVs. Now, the ranking of the green suppliers of MAGDM tools is obtained by evaluating the score values. The score values and the ranking of the substitutes are provided in Table 6 as follows.

TABLE 6 The Score Values of the qRFSVs for $\tau_{i}$

A. Effects of Parameters

Since the FTN and FTCN play a significant role in the information fusion due to the parameter $q$ . But it may cause a modification in the found results. Hence, we observe the effects of the modification in the results found in Example 1. Note that Example 1 is solved by $q=3,\sigma =2$ . We change the values of $q,\sigma$ and note the changes in the results found as provided. The outcomes in Tables 7 and 8 in the paragraphs that follow.

TABLE 7 The Aggregate Values in the Form of the qROFHSFVs for $\tau_{i}$

TABLE 8 The Aggregate Values in the Form of the qROFSFVs for $\tau_{i}$

FIGURE 2 shows the ranking of the cryptocurrency MAGDM at the various values of $q$ found in the case of both qROFHSFWA operators. We modified the values of $q,\sigma$ from 3 to 8 and found interesting substances. We can perceive that the $\varsigma _{3}$ is found as the most appropriate cryptocurrency MAGDM tool by the qROFHSFWA operator. The sensitivity of the qROFHSFWG operator is geometrically represented by FIGURE 3 in the following.

FIGURE 1.

Represent the flow chart of the proposed model.

FIGURE 2.

Geometrical representation of the effects of the results obtained from the qROFHSFWA operator.

FIGURE 3.

Geometrical representation of the effects of the results obtained from the qROFHSFWG operator.

FIGURE 3 shows the ranking of the cryptocurrency MAGDM tool at the various values of $q$ found in the case of both qROFHSFWG operators. We modified the values of $q$ from 3 to 8 and found the interesting substances. We can detect that the $\varsigma _{4}$ is obtained as the most appropriate cryptocurrency MAGDM tool by the qROFHSFWG operator.

Table 8 shows the ranking of the cryptocurrency MAGDM tool at the various values of $\sigma$ found in the case of both qROFHSFWA and qROFHSFWG operators. We change the values of $\sigma =\left ({ 3,4,5,6,7,8,9,10,27,30,35,38,40,48,}\right . \left .{ 50 }\right )$ and found the interesting objects. We can observe that the $\varsigma _{4}$ is found as the most suitable tool for the qROFSFWA operator while $\varsigma _{2}$ is found as the most suitable cryptocurrency MAGDM tool obtained by the qROFHSFWG operator at the values of $\sigma$ . Geometrical representation of the effects of $\sigma$ in the results obtained by qROFHSFWA and qROFHSFWG operators are shown as follows in Figures 3 and 4 respectively as follows.

FIGURE 4.

Geometrical representation of the effects of the results obtained from the qROFHSFWA operator.

FIGURE 4 shows the ranking of the cryptocurrency MAGDM at the various values of $\sigma$ found in the case of both qROFHSFWA operators. We modified the values of $\sigma =\left ({ 3,4,5,6,7,9,10,27,30,35,38,40,48,50 }\right )$ and found interesting substances. We can perceive that the $\varsigma _{4}$ is found as the most appropriate cryptocurrency MAGDM tool by the qROFHSFWA operator. The sensitivity of the qROFHSFWG operator is geometrically represented by FIGURE 4.

FIGURE 5. Shows the ranking of the cryptocurrency MAGDM tool at the various values of $\sigma$ found in the case of both qROFHSFWG operators. We modified the values of $\sigma$ and found the interesting substances. We can detect that the $\varsigma _{2}$ is obtained as the most appropriate socio-economic MAGDM tool for the qROFHSFWG operators.

FIGURE 5.

Geometrical representation of the effects of the results obtained from the qROFHSFWG operator.

SECTION V. Conclusion

To manage different kinds of digital currencies, avoid damages, and keep exchanges online, a detailed and accurate analysis of the market for digital currencies is necessary. An examination of the cryptocurrency market showed that financial incentives are only slightly different from safety, reorganization, and claim as the most important factors for Bitcoin investment intentions. High levels of safety, regionalized applications, and rising claims in the Bitcoin market round out the list of sub-factors. Many professors and investigators started working on cryptocurrency as a result. Many academics are using FS theory and its hybrid structures to analyze the marketplace for Bitcoin because uncertainty can be found in practically all systems in the real world. This work develops a revolutionary scholarly tool that reveals factual data in a parametric way. The HSS and the qROFS are two examples of qROFHSS provided by the multi-argument functions. The qROFHSS can be aggregated using weighted mean or weighted geometric methods. The suggested operations and definitions are checked for validity and applicability using pertinent examples. Research plays a crucial role in DM processes. The primary goal of the specialists involved in this work is to use a DM method for numerous qualities and sub-attributes to invest in the Bitcoin market. We aim to develop novel methods to analyze the lite coin market utilizing decision-making problems in the future. Upcoming applications of the structure that is suggested include the Complex Proportional Assessment (COPRAS) method, the Analytic Hierarchy Process (AHP) method, VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method, the TOPSIS method, and the Interactive Multi-Ariteria Decision Making (TODIM) method, which is known in Portuguese. Additionally, we would take into account how the psychological aspects of the experts’ decisions would affect the difficulty of choosing eco-friendly providers. The new approach can potentially be used in several other domains, including the choice of funding initiatives.