Some Einstein Geometric Aggregation Operators for q-Rung Orthopair Fuzzy Soft Set With Their Application in MCDM

q-rung orthopair fuzzy soft sets (q-ROFSS) is a progressive form for orthopair fuzzy sets. It is also an appropriate extension of intuitionistic fuzzy soft sets (IFSS) and Pythagorean fuzzy soft sets (PFSS). The strict prerequisite gives assessors too much autonomy to precise their opinions about membership and non-membership values. The q-ROFSS has a wide range of real-life presentations. The q-ROFSS capably contracts with unreliable and ambiguous data equated to the prevailing IFSS and PFSS. It is the most powerful method for amplifying fuzzy data in decision-making. The hybrid form of orthopair q-rung fuzzy sets with soft sets has emerged as a helpful framework in fuzzy mathematics and decision-making. The hybrid structure of q-rung orthopair fuzzy sets with soft sets has occurred as an expedient context in fuzzy mathematics and decision-making. The fundamental impartial of this research is to propose Einstein’s operational laws for q-rung orthopair fuzzy soft numbers (q-ROFSNs). The core objective of this research is to develop some geometric aggregation operators (AOs), such as q-rung orthopair fuzzy soft Einstein weighted geometric (q-ROFSEWG), and q-rung orthopair fuzzy soft Einstein ordered weighted geometric (q-ROFSEOWG) operators. We will discuss the idempotency, boundedness, and homogeneity of the proposed AOs. Multi-criteria decision-making (MCDM) is dynamic in dealing with the density of real-world complications. Still, the prevalent MCDM techniques consistently deliver irreconcilable outcomes. Based on the presented AOs, a strong MCDM technique is deliberate to accommodate the flaws of the prevailing MCDM approaches under the q-ROFSS setting. Moreover, an inclusive comparative analysis is executed to endorse the expediency and usefulness of the suggested method with some previously existing techniques. The outcomes gained through comparative studies spectacle that our established approach is more capable than prevailing methodologies.

SECTION I. Introduction

MCDM has deliberated as the most proper scheme for verdict the adequate alternative since all probable choices, subsequent standards, or features. Most rulings are taken when the intentions and confines are generally unspecified or unclear in real-life surroundings. Zadeh presented the notion of the fuzzy set [1] to overcome such vague and indeterminate facts. It is a fundamental tool to handle the insignificances and hesitations in decision-making. The existing FS cannot deal with the scenarios when the experts consider a membership degree (MD) in intervals during the decision-making procedure. The prevailing fuzzy set cannot deliver information about any alternative’s non-membership degree (NMD). Atanassov [2] overcame the abovementioned limitations and developed the intuitionistic fuzzy set (IFS). Wang and Liu [3] introduced several operations such as Einstein product, Einstein sum, etc., and AOs for IFS. Garg and Kaur [4] protracted the idea of IFS and settled the cubic intuitionistic fuzzy set (CIFS). Still, the prevailing IFS cannot grip the incompatible and ambiguous information because it envisions the linear inequality among the MD and NMD. If the group of professionals elects the MD and NMD so that their sum exceeds 1, such as MD = 0.6 and NMD = 0.7, respectively, then the IFS, as declared formerly, cannot deal with it because ${}\,0.6 + 0.7\geq 1.\,{}$ Yager [5] offered the Pythagorean fuzzy set (PFS) to resolve the inadequacy mentioned above by modifying the elementary state ${\kappa }+{\delta }\leq 1$ to ${\kappa ^{2}}+{\delta ^{2}}{\leq }1$ . He also established the score and accuracy functions to compute the ranking. Rahman et al. [6] constructed a multi-attribute group decision-making (MAGDM) system using their projected Einstein-weighted geometric AOs for PFS.

Zhang and Xu [7] settled some basic operational laws for PFS and protracted the order of preference by similarity to the ideal solution (TOPSIS) to determine MCDM problems. Wei and Lu [8] presented the Pythagorean fuzzy power AOs and reflected their necessary structures. Using their proposed operators, they also established a DM technique to resolve multi-attribute decision-making (MADM). Wang and Li [9] demonstrated the interaction operational laws for Pythagorean fuzzy numbers (PFNs) and settled power Bonferroni mean operators. Zhang [10] projected a unique DM methodology built on similarity measures to determine MCGDM complications under the PFS setting. Yager [11] estimated a generalization of IFS and PFS as q-ROFS, enlarging the modeler’s impartiality to prompt their thoughts about MG and NMG values. For example, if MG = 0.7 and MG = 0.8, then available IFS and PFS fail to cope with this scenario. But, Yager’s q-ROFS capably deal with such situations by modifying the condition $0.7^{3}\,\,+\,\,0.8^{3}\leq 1$ , so employing the q-ROFS to delight such circumstances is proper. Riaz et al. [12], [13], [14] designated AOs such as geometric, hybrid, and Einstein to temper the q-ROFS setting. Sheng [15] presented the Einstein AOs for q-ROFS and developed a MADM technique to resolve decision-making complications. Liu and Wang [16] conferred some new AOs for q-ROFS to contract with DM obstacles. Garg et al. [17] protracted the neutrality AOs for complex q-ROFS and developed the MADM technique to resolve decision-making complications. Liu et al. [18] protracted the complex q-ROFS and introduced some AOs under-considered scenarios to explain the MAGDM obstacles. Ullah et al. [19] presented the competition graph for complex q-ROFS and explored the k-competition, p-competition, neighborhood, and m-step neighborhood graphs for complex q-ROFS. Xu et al. [20] settled the MAGDM technique using the amended AOs for the q-ROFS environment. Farid and Riaz [21] protracted the generalized Einstein interaction AOs for q-ROFS and established an MCDM technique using their developed AOs. Wang et al. [22] demarcated the similarity measures (SM) for q-ROFS employing the cosine function. Ullah et al. [23] introduced the dice SM for T-spherical fuzzy sets and developed the TOPSIS method and entropy measure for T-spherical fuzzy sets. Peng and Liu [24] explored the associations among distance measures and SM of q-ROFS and offered some novel formulations for information measures of q-ROFS. Ali et al. [25] presented the novel AOs for complex T-spherical fuzzy sets and developed a MADM technique based on established AOs.

The above techniques have broad applications, but these theories have some limitations on parametric chemistry due to their ineffectiveness. Molodtsov [26] introduced the soft sets (SS) theory and defined some basic operations with their features to handle misperception and haziness. Maji et al. [27] settled the fuzzy soft set with some desirable properties by including two general concepts, fuzzy set, and SS. Maji et al. [28] protracted the IFSS and some essential operations with their fundamental properties. Arora and Garg [29] presented the AOs for IFSS and planned a DM technique based on their settled operators. Peng et al. [30] anticipated the PFSS by merging two prevailing ideas, PFS and SS. Zulqarnain et al. [31], [32] presented some operational laws for PFSS and prolonged the AOs and interaction AOs for PFSS. They also used their developed operators’ green supplier chain management application to establish the decision-making methodologies. Zulqarnain et al. [33] developed the Einstein AOs for PFSS and employed their established AOs in MAGDM. Zulqarnain et al. [34] prolonged the Einstein-ordered operational laws for PFSS and announced the Einstein-ordered weighted ordered geometric AO for PFSS. They also established a MAGDM technique to solve complex real-life complications. Zulqarnain et al. [35] protracted the Einstein-ordered weighted average aggregation operator for PFSS and offered a decision-making technique using their recognized operator. Zulqarnain et al. [36] settled the TOPSIS technique for PFSS using correlation coefficient (CC) and developed the MADM method to resolve DM obstacles. Hussain et al. [37] planned a syndicated study of SS and q-ROFS entitled q-ROFSS. They developed different average AOs on q-ROFSS and debated their properties. Zulqarnain et al. [38] prolonged the novel MCDM technique by employing the interaction AOs for q-ROFSS. Chinram et al. [39] established the geometric AOs under the q-ROFSS setting and used their protracted AOs in MCDM obstacles. Still, the above techniques are not adequately deal with the uncertain information in some cases.

A. Motivation and Shortcomings of Existing Methods

The q-ROFSS is an amalgam logical configuration of SS, PFSS, and the q-ROFS are dominant scientific tools for allocating anonymous and restricted data. It has been identified that AOs are imperious in decision-making, so collectively assessed facts from unlike causes can be collected in a distinctive valuation. To the unsurpassed of our consideration, Einstein AOs with hybridization with a SS and q-ROFS have no presence in the literature. Still, existing AOs for q-ROFSS cannot expertly deal with uncertain and imprecise information during the decision-making (DM) process. Moreover, the model states that the whole MD (NMD) is self-determining it’s NMD (MD). Hence, agreeing to these replicas, the consequences are not productive, so no proper inclination is indicated for substitutes. So, how to integrate these q-ROFSNs over Einstein operations is a fascinating subject. We will introduce the Einstein AOs for q-ROFSS, such as q-ROFSEWG and q-ROFSEOWG. The developed Einstein geometric AOs are proficient compared to prevailing amalgam organizations of fuzzy sets. The above replicas have inferred that the general MD (NMD) is liberated of its compatible NMD (MD) values. As a result, the consequences of these AOs are inconsistent, and no substitute for alternatives is given. Therefore, incorporating these q-ROFSNs through Einstein AOs is an interesting subject. The methodologies chosen in [39] are inadequate to examine the data with a reflective intelligence for higher notion and correct inferences. For example, $\mathcal {Q=} \left \{{{\mathcal {Q} ^{1}},\,\,{\mathcal {Q} ^{2}}}\right \}{}$ be a set of two experts, and ${\mathsf {e}_{1}},~{\mathsf {e}_{2}}$ denoted the attributes, and $\mathcal {X}$ = $\begin{aligned} {\begin{bmatrix}\left ({0.5,0.6}\right)& \left ({0.3,0.4}\right)\\ \left ({0.0,0.4}\right)& \left ({0.2,0.7}\right)\end{bmatrix}} \end{aligned}$ . Let ${\Omega _{i}}$ = ${\left ({0.4,~0.6}\right)^{T}}$ and ${\gamma _{j}}$ = ${\left ({0.2,~.0.8}\right)^{T}}$ be the weight vectors for specialists and attributes. Then, we found the collected value expending the q-ROFSWG [39] operator is $\left \langle{ 0,~0.9129}\right \rangle$ . This displays that there is no outcome on the mutual consequence ${\mu _{\mathsf {e}}}$ . Because ${\mu _{\mathsf {e}}}$ = ${\mu _{11}}$ = 0.5, ${\mu _{12}}$ = 0.0, ${\mu _{21}}$ = 0.3, and ${\mu _{22}}$ = 0.2, which is unreasoning. A boosted categorization method captivates investigators to crash inexplicable and insufficient facts. Contrary to the exploration effects, q-ROFSS plays a vigorous part in DM by assembling many cradles into a particular value.

B. Contribution

The Einstein geometric AOs are a well-known fascinating conjecture AOs. It has been perceived that the predominant AOs appear apathetic to designing the precise decision over the DM process in some surroundings. To overcome these specific complications, these AOs prerequisite to be modified. So, to stimulate the existing study and boundaries stated above of q-ROFSS, we will state Einstein’s geometric AOs built on weird facts; the essential purposes of the subsequent research are assumed as follows:

1. We determine innovative operational laws constructed on Einstein operations for q-ROFSNs.

2. The q-ROFSS expertly contracts the multifaceted matters, seeing the DM procedure’s attributes. To preserve this improvement in attention, we institute the Einstein geometric AOs for q-ROFSS.

3. The q-ROFSEWG and q-ROFSEOWG operators have been recognized using Einstein operational laws.

4. An innovative MCDM method is built on the projected Einstein geometric AOs to handle DM problems under the q-ROFSS setting.

5. DM is a notable feature of medical science as it perceives the concrete criteria of all ingredients. Medical diagnosis is a demanding but noteworthy stage in the creative process.

6. A comparative study of the advanced MCDM process and existing methodologies has been anticipated to measure pragmatism and sovereignty.

This paper’s organization is assumed as follows: the second section of this paper involves some basic notions that support us in developing the structure of the subsequent study. Section 3, some novel operational laws for q-ROFSNs have been designed considering the Einstein operations. Furthermore, the q-ROFSEWG operator is introduced in the same section based on our developed Einstein operational laws with basic properties. Similarly, in section 4, we presented the q-ROFSEOWG operator ad deliberated its essential properties. In section 5, an MCDM approach has been constructed based on the proposed Einstein geometric AOs. A numerical example for medical diagnoses is described in the same section to approve the pragmatism of the presented technique. Furthermore, a brief comparative analysis to endorse the competency of the developed approach is in section 6.

SECTION II. Preliminaries

This section will collect the main descriptions that support us in assembling the consequent article configuration.

A. Definition 1 [26]

Let $\mathfrak {U}$ be a universe of discourse and $\mathcal {E}$ be the set of attributes and $\mathcal {A} \subseteq \mathcal {E}$ . Let $\mathcal {P} (\mathfrak {U})$ be the power set of the universe of discourse $\mathfrak {U}$ . Then a pair is $(\mathcal {F},~\mathcal {A})$ is termed a SS and is expressed through a mapping $\mathcal {F}$ :$\mathcal {A} \rightarrow \mathcal {P} (\mathfrak {U})$ . Mathematically it can be written as $$\begin{equation*}\left ({\mathcal {F,A}}\right)=\left \{{\mathcal {F}\left ({{e}}\right) \mathcal {\in P}\left ({\mathfrak {U}}\right):{e}{\in }\mathcal {E,~F} \left ({{e}}\right)=\,\,{\emptyset }~if~{e}{\not \in }\mathcal {A}}\right \}\end{equation*}$$

B. Definition 2 [5]

Let $\mathfrak {U}$ be a universe of discourse, then the PFS over $\mathfrak {U}$ is specified as $\wp$ = $\left \{{\left ({{\boldsymbol {u}},\,\,{{\mu _{\wp }}\left ({{\boldsymbol {u}}}\right),\,\,{\vartheta }_{\wp }}\left ({{\boldsymbol {u}}}\right)}\right)\vert {\boldsymbol {u}} \in \mathfrak {U}}\right \}{}$ , ${\mu _{\wp }}\left ({{\boldsymbol {u}}}\right)$ and ${\vartheta _{\wp }}\left ({{\boldsymbol {u}}}\right)$ signifies the MD and NMD, which contains the subsequent circumstances $$\begin{align*}&\hspace {-1pc}{{\mu _{\wp }}\left ({{\boldsymbol {u}}}\right),~{\vartheta }_{\wp }}\left ({{\boldsymbol {u}}}\right): \mathfrak {U}\rightarrow \left [{0,~1}\right], {\left ({{\mu _{\wp }}\left ({{\boldsymbol {u}}}\right)}\right)^{2}} \\&+ {\left ({{\vartheta _{\wp }}\left ({{\boldsymbol {u}}}\right)}\right)^{2}}{\leq }1\,{}~\text {for all} ~ {\boldsymbol {u}}\in \mathfrak {U}.\end{align*}$$ where ${\mathfrak {L}_{\wp }} ({\boldsymbol {u}})$ = $\sqrt {1-{\left ({{\mu _{\wp }}\left ({{\boldsymbol {u}}}\right)}\right)^{2}}+{\left ({{\vartheta _{\wp }}\left ({{\boldsymbol {u}}}\right)}\right)^{2}}}$ signifies the degree of hesitancy.

C. Definition 3 [30]

Let $\mathfrak {U}$ a universe of discourse and $\mathcal {E}$ be the set of attributes, then the PFSS over $\mathfrak {U}$ is presented as follows: $$\begin{align*} {\mathcal {F} _{ {\boldsymbol {u}}_{i}}}({e_{j}})=&\left \{{\left ({{ {\boldsymbol {u}}_{i}},~{\mu _{j}}\left ({{ {\boldsymbol {u}}_{i}}}\right),~{\vartheta _{j}}\left ({{ {\boldsymbol {u}}_{i}}}\right)}\right)\vert { {\boldsymbol {u}}_{i}}\in \mathfrak {U}}\right \}{.} \\ \quad \mathcal {F} {:~} \mathcal {E}\rightarrow&P{\mathcal {K} ^{\mathfrak {U}}}\end{align*}$$ $P{\mathcal {K} ^{\mathfrak {U}}}$ symbolizes the assortment of Pythagorean fuzzy subsets of $\mathfrak {U}$ and fulfilled the succeeding circumstance ${\mu ^{2}}+\, {\vartheta ^{2}}{\leq }1.\,{}$

For readers’ suitability, the PFSN can be stated as ${\aleph _{\mathsf {e}_{ij}}}$ = $\left \langle{ {\mu _{ij}},~{\vartheta _{ij}}}\right \rangle$ .

Assume three PFSNs, such as ${\aleph _{\mathsf {e}}}$ = $\left ({{\mu },~{\vartheta }}\right)$ , ${\aleph _{\mathsf {e}_{11}}}$ = $\left ({{\mu _{11}},~{\vartheta _{11}}}\right)$ , and ${\aleph _{\mathsf {e}_{12}}}$ = $\left ({{\mu _{12}},~{\vartheta _{12}}}\right)$ and ${\alpha }>0$ . Then,

1. $$\begin{equation*}~{\aleph _{\mathsf {e}_{11}}}{\oplus }{\aleph _{\mathsf {e}_{12}}}=\left \langle{ \sqrt {{{\mu _{11}}^{2}}+{{\mu _{12}}^{2}}-{{\mu _{11}}^{2}}{{\mu _{12}}^{2}}}, ~{\vartheta _{11}}{\vartheta _{12}}}\right \rangle\end{equation*}$$

2. $$\begin{equation*}~{\aleph _{\mathsf {e}_{11}}}{\otimes }{\aleph _{\mathsf {e}_{12}}}=\left \langle{ {\mu _{11}}{\mu _{12}},~\sqrt {{{\vartheta _{11}}^{2}}+{{\vartheta _{12}}^{2}} -{{\vartheta _{11}}^{2}}{{\vartheta _{12}}^{2}}}}\right \rangle\end{equation*}$$

3. $$\begin{equation*}~{\alpha }{\aleph _{\mathsf {e}}}{\,\,=\,\,}\left \langle{ \sqrt {1-{\left ({1-{\mu ^{2}}}\right)^{\alpha }}},~{\vartheta ^{\alpha }}}\right \rangle {}\end{equation*}$$

4. $$\begin{equation*}~{\aleph _{\mathsf {e}}^{\alpha }}{\,\,=\,\,}\left \langle{ {\mu ^{\alpha }},~\sqrt {1-{\left ({1-{\vartheta ^{2}}}\right)^{\alpha }}}}\right \rangle {}\end{equation*}$$

D. Definition 4 [37]

Let $\mathfrak {U}$ be the universal set and $\mathcal {E}$ be the set of attributes. The q-ROFSS over $\mathfrak {U}$ is defined as follows: $$\begin{align*}\quad {\mathfrak {I}_{\mathsf {e}_{j}}}({ {\boldsymbol {u}}_{i}})=&\left \{{\left ({{ {\boldsymbol {u}}_{i}},~{\mu _{j}^{q}}\left ({{ {\boldsymbol {u}}_{i}}}\right),~{\vartheta _{j}^{q}}\left ({{ {\boldsymbol {u}}_{i}}}\right)}\right)\vert { {\boldsymbol {u}}_{i}}\in \mathfrak {U,~}q{\geq }3}\right \}{.} \\ {}~\mathfrak {I}{:~} \mathcal {F}\rightarrow&{q-ROFS^{\mathfrak {(U)}}}\end{align*}$$ ${q-ROFS^{\mathfrak {(U)}}}$ be a collection of q-ROFSs of $\mathfrak {U}$ and ${\mu _{j}^{q}}\left ({{ {\boldsymbol {u}}_{i}}}\right),\,\,{\vartheta _{j}^{q}}\left ({{ {\boldsymbol {u}}_{i}}}\right)$ denotes the MD and NMD with the subsequent circumstance $0{\leq }\,\,{\left ({{\mu _{j}}\left ({{ {\boldsymbol {u}}_{i}}}\right)}\right)^{q}}+{\left ({{\mu _{j}}\left ({{ {\boldsymbol {u}}_{i}}}\right)}\right)^{q}}{\leq }1$ , $q{\geq }3$ . Basically, ${\mathfrak {I}_{\mathsf {e}_{j}}}({ {\boldsymbol {u}}_{i}})$ = $\left \{{\left ({{ {\boldsymbol {u}}_{i}},\,\,{\mu _{j}^{q}}\left ({{ {\boldsymbol {u}}_{i}}}\right),\,\,{\vartheta _{j}^{q}}\left ({{ {\boldsymbol {u}}_{i}}}\right)}\right)}\right \}{}$ can be transcribed ${\aleph _{\mathsf {e}_{ij}}}$ = $\left ({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ . That is named q-ROFSN. The degree of uncertainty for q-ROFSN can be stated as ${\beth _{\aleph _{\mathsf {e}_{ij}}}} = \sqrt [{q}]{`1-\left ({{\left ({{\mu _{ij}}}\right)^{q}}+{\left ({{\vartheta _{ij}}}\right)^{q}}}\right)}$ .

Suppose ${\aleph _{\mathsf {e}}}$ = $\left ({{\mu },~{\vartheta }}\right)$ , ${\aleph _{\mathsf {e}_{11}}}$ = $\left ({{\mu _{11}},~{\vartheta _{11}}}\right)$ , and ${\aleph _{\mathsf {e}_{12}}}$ = $\left ({{\mu _{12}},~{\vartheta _{12}}}\right)$ be three q-ROFSNs and ${\alpha }>0$ . then some operations for q-ROFSS [37]:

1. $$\begin{align*}~&\hspace {-1pc}{\aleph _{\mathsf {e}_{11}}}{\oplus }{\aleph _{\mathsf {e}_{12}}} \\=&\left \langle{ \sqrt [{q}]{{\left ({{\mu _{11}}}\right)^{q}}+{\left ({{\mu _{12}}}\right)^{q}} -{\left ({{\mu _{11}}}\right)^{q}}{\left ({{\mu _{12}}}\right)^{q}}},~{\vartheta _{11}}{\vartheta _{12}}}\right \rangle {}\end{align*}$$

2. $$\begin{align*}&\hspace {-1pc}{\aleph _{\mathsf {e}_{11}}}{\otimes }{\aleph _{\mathsf {e}_{12}}} \\=&\left \langle{ {\mu _{11}}{\mu _{12}},~\sqrt [{q}]{{\left ({{\vartheta _{11}}}\right)^{q}} +{\left ({{\vartheta _{12}}}\right)^{q}}-{\left ({{\vartheta _{11}}}\right)^{q}}{\left ({{\vartheta _{12}}}\right)^{q}}} }\right \rangle {}\end{align*}$$

3. $$\begin{equation*}~{\alpha }{\aleph _{\mathsf {e}}}{\,\,=\,\,}\left \langle{ \sqrt {1-{\left ({1-{\mu ^{q}}}\right)^{\alpha }}},~{\vartheta ^{\alpha }}}\right \rangle {}\end{equation*}$$

4. $$\begin{equation*}~{\aleph ^{\alpha }}{\,\,=\,\,}\left \langle{ {\mu ^{\alpha }}~,~\sqrt {1-{\left ({1-{\vartheta ^{q}}}\right)^{\alpha }}}}\right \rangle {}\end{equation*}$$

For q-ROFSNs collection ${\aleph _{\mathsf {e}_{ij}}}=\left ({{\mu _{ij}},\,\,{\vartheta _{ij}}}\right)$ . Let $\Omega$ = ${\left ({{\Omega _{1}},~{\Omega _{1}},\ldots,~{\Omega _{n}}}\right)^{T}}$ be a weight vector for experts such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1 and ${\gamma }$ = ${\left ({{\gamma _{1}},~{\gamma _{2}},~{\gamma _{3}},\ldots,~{\gamma _{m}}}\right)^{T}}$ be a weight vector for parameters such as ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1. Hussain et al. [37] presented average AOs, and Chinram et al. [39] settled the geometric AOs for q-ROFSS. $$\begin{align*}&\hspace {-1pc}q-ROFSWA~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&\left \langle{ \sqrt [{q}]{1-\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({1-{{\mu _{ij}}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}},}\right. \\&\left.{\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({{\vartheta _{ij}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}\right \rangle {} \\&\hspace {-1pc}q-ROFSWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&\left \langle{ \mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({{\mu _{ij}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}},}\right. \\&\left.{\sqrt [{q}]{1-\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{{\vartheta _{ij}}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}\right \rangle {.}\end{align*}$$

E. Definition 5 [37]

Let ${\aleph _{\mathsf {e}_{ij}}}$ = $\left ({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ be a q-ROFSN. Then the score function can be demarcated as follows: $$\begin{align*}~S({\aleph _{\mathsf {e}_{ij}}})=&{\mu _{ij}^{q}}-{\vartheta _{ij}^{q}} +\left ({\frac {e^{{\mu _{ij}^{q}}-{\vartheta _{ij}^{q}}}}{{e^{{\mu _{ij}^{q}} -{\vartheta _{ij}^{q}}}}+1}-\frac {1}{2}}\right){\beth _{\aleph _{\mathsf {e}_{ij}}}^{q}}, \\&\quad {~\text {for}~}q{\geq }3{~\text {and}~}S({\aleph _{\mathsf {e}_{ij}}}){\in }\left [{-1,~1}\right]{. }\tag{1}\end{align*}$$ Let ${\aleph _{\mathsf {e}_{11}}}$ = $\left ({{\mu _{11}},~{\vartheta _{11}}}\right)$ and ${\aleph _{\mathsf {e}_{12}}}$ = $\left ({{\mu _{12}},~{\vartheta _{12}}}\right)$ be two q-ROFSNs. Then

If $S({\aleph _{\mathsf {e}_{11}}})>S({\aleph _{\mathsf {e}_{12}}})$ , then ${\aleph _{\mathsf {e}_{11}}}{\succcurlyeq }{\aleph _{\mathsf {e}_{12}}}$ .

If $S({\aleph _{\mathsf {e}_{11}}}) < S({\aleph _{\mathsf {e}_{12}}})$ , then ${\aleph _{\mathsf {e}_{11}}}{succeq}{\aleph _{\mathsf {e}_{12}}}$ .

If $S\left ({{\aleph _{\mathsf {e}_{11}}}}\right)=S({\aleph _{\mathsf {e}_{12}}})$ , then

If ${\beth _{\aleph _{\mathsf {e}_{11}}}}>{\beth _{\aleph _{12}}}$ , then ${\aleph _{\mathsf {e}_{11}}} < {\aleph _{\mathsf {e}_{12}}}$

If ${\beth _{\aleph _{\mathsf {e}_{11}}}^{q}} = {\beth _{\aleph _{\mathsf {e}_{11}}}^{q}}$ , then ${\aleph _{\mathsf {e}_{11}}}={\aleph _{\mathsf {e}_{12}}}$ .

SECTION III. Einstein Geometric Aggregation Operator for Q- Rung Orthopair Fuzzy Soft Numbers

This section introduces Einstein’s operations under q-ROFSN. Moreover, using the q-ROFSEWG operator with the desired properties is endorsed.

A. Einstein Operational Laws for Q-ROFSNs

1) Definition

Let ${\aleph _{\mathsf {e}}}$ = $\left ({{\mu },~{\vartheta }}\right)$ , ${\aleph _{\mathsf {e}_{11}}}$ = $\left ({{\mu _{11}},~{\vartheta _{11}}}\right)$ , and ${\aleph _{\mathsf {e}_{12}}}$ = $\left ({{\mu _{12}},~{\vartheta _{12}}}\right)$ denote the q-ROFSNs and ${\alpha }>0$ . Then the operational laws ca q-ROFSS considering the Einstein operations are defined as follows:

1. $$\begin{align*}&\hspace {-1pc}{\aleph _{\mathsf {e}_{11}}}{\oplus }{\aleph _{\mathsf {e}_{12}}} \\=&\left \langle{ \frac {\sqrt [{q}]{\left ({1+{\mu _{11}^{q}}}\right)-\,\,\left ({1-{\mu _{12}^{q}}}\right)}} {\sqrt [{q}]{\left ({1+{\mu _{11}^{q}}}\right)+\left ({1-{\mu _{12}^{q}}}\right)}}, \left.{\frac {\sqrt [{q}]{2{\vartheta }_{11}^{q}}}{\sqrt [{q}]{\left ({2-{\vartheta _{11}^{q}}}\right) +{\vartheta _{12}^{q}}}}}\right \rangle }\right.\end{align*}$$

2. $$\begin{align*}&\hspace {-1pc}{\aleph _{\mathsf {e}_{11}}}{\otimes }{\aleph _{\mathsf {e}_{12}}} \\=&\left \langle{ \frac {\sqrt [{q}]{2{\mu }_{11}^{q}}}{\sqrt [{q}]{\left ({2-{\mu _{11}^{q}}}\right) +{\mu _{12}^{q}}}},\left.{\frac {\sqrt [{q}]{\left ({1+{\vartheta _{11}^{q}}}\right) -\left ({1-{\vartheta _{12}^{q}}}\right)}}{\sqrt [{q}]{\left ({1+{\vartheta _{11}^{q}}}\right) +\left ({1-{\vartheta _{12}^{q}}}\right)}}}\right \rangle }\right.\end{align*}$$

3. $$\begin{align*}&\hspace {-1pc}{\alpha }{\aleph _{\mathsf {e}}} \\=&\left \langle{ \! \frac {\sqrt [{q}]{{\left ({1+{\mu ^{q}}}\right)^{\partial }} -{\left ({1-{\mu ^{q}}}\right)^{\partial }}}}{\sqrt [{q}]{{\left ({1+{\mu ^{q}}}\right)^{\partial }} +{\left ({1-{\mu ^{q}}}\right)^{\partial }}}},\!\left.{\frac {\sqrt [{q}]{{2({\vartheta }^{q}})^{\partial }}} {\sqrt [{q}]{{\left ({2-{\vartheta ^{q}}}\right)^{\partial }}+{{({\vartheta }^{q}})^{\partial }}}}\!}\right \rangle }\right.\end{align*}$$

4. $$\begin{align*}&\hspace {-1pc}{\aleph _{\mathsf {e}}^{\alpha }} \\=&\left \langle{ \!\!\frac {\sqrt [{q}]{2\left ({{\mu ^{q}}}\right)^{\partial }}}{\sqrt [{q}]{{\left ({2-{\mu ^{q}}}\right)^{\partial }} +{\left ({{\mu ^{q}}}\right)^{\partial }}}},\!\frac {\sqrt [{q}]{{\left ({1+{\vartheta ^{q}}}\right)^{\partial }} -{\left ({1-{\vartheta ^{q}}}\right)^{\partial }}}}{\sqrt [{q}]{{\left ({1+{\vartheta ^{q}}}\right)^{\partial }} +{\left ({1-{\vartheta ^{q}}}\right)^{\partial }}}}\!}\right \rangle.\end{align*}$$

2) Definition

Let ${\aleph _{\mathsf {e}_{ij}}}$ = $\left ({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ be a collection of q-ROFSNs. If q-ROFSEWG: ${\Delta ^{n}}\rightarrow {\Delta }$ , and $$\begin{align*}&\hspace {-2pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{m}}{\left ({{\otimes _{i=1}^{n}}{\left ({{\aleph _{\mathsf {e}_{ij}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}{.~}\tag{2}\end{align*}$$ where ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1 and ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1 denotes the weights of experts and attributes.

3) Theorem

Let ${\aleph _{\mathsf {e}_{ij}}}$ = $\left ({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ be a collection of q-ROFSNs. Then, the gained aggregated values are also a q-ROFSN and $$\begin{align*}&\hspace {-1pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{m}}{\left ({{\otimes _{i=1}^{n}}{\left ({{\aleph _{\mathsf {e}_{ij}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle \\\tag{3}\end{align*}$$ ${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weight vectors of experts and parameters, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1.

Proof:

Using mathematical induction:

For $n$ = 1, we get ${\Omega _{i}}=1$ $$\begin{align*}&\hspace {-1pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{m}}{\left ({{\otimes _{i=1}^{n}}{\left ({{\aleph _{\mathsf {e}_{ij}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}{~} \\&\hspace {-1pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{m}}{\left ({{\aleph _{\mathsf {e}_{1 j}}}}\right)^{\gamma _{j}}} \\ {}=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{m}{\left ({{\mu _{1 j}^{q}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left ({2-{\mu _{1 j}^{q}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left ({{\mu _{1 j}^{q}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left ({1+{\vartheta _{1 j}^{q}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{m}{\left ({1-{\vartheta _{1 j}^{q}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{m}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\gamma _{j}}}}}}\right \rangle {} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{1} {\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{1}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{1} {\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{1}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{1} {\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{1}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{1} {\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {}\end{align*}$$ For $m$ = 1, we get ${\gamma _{j}}=1$ . $$\begin{align*}&\hspace {-1pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{i=1}^{n}}{\left ({{\aleph _{\mathsf {e}_{i1}}}}\right)^{\Omega _{i}}} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{i=1}^{n}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{i=1}^{n}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}+\mathop {\prod }\nolimits _{i=1}^{n}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}-\,\,\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}+\,\,\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}}}\right \rangle {} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{1}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {}\end{align*}$$ So, equation 3 hold for $n$ = 1 and $m$ = 1.

Assume for $n$ = ${\delta _{2}}$ , $m$ = ${\delta _{1}}+1$ and for $n$ = ${\delta _{2}}+1$ , $m$ = ${\delta _{1}}$ The above equation holds. $$\begin{align*}&\hspace {-2pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{{\delta _{1}}+1}}{\left ({{\otimes _{i=1}^{\delta _{2}}}{\left ({{\aleph _{\mathsf {e}_{ij}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{{\delta _{1}}+1} {\left ({\mathop {\prod }\nolimits _{i=1}^{\delta _{2}}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {} \\&\hspace {-1pc}{\otimes _{j=1}^{\delta _{1}}}{\left ({{\otimes _{i=1}^{{\delta _{2}}+1}}{\left ({{\aleph _{\mathsf {e}_{ij}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{\delta _{1}} {\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{\delta _{1}} {\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{\delta _{1}}{\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{\delta _{1}}{\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{\delta _{1}}{\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{\delta _{1}}{\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{\delta _{1}}{\left ({\mathop {\prod }\nolimits _{i=1}^{{\delta _{2}}+1}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {}\end{align*}$$

Now, for $m$ = ${\delta _{1}}+1$ and $n$ = ${\delta _{2}}+1$ , as shown in the equation at the bottom of the next page.

So, it is valid for $m$ = ${\delta _{1}}+1$ and $n$ = ${\delta _{2}}+1$ .

4) Example

Let $\mathcal {U}$ = {${\mathfrak {u}_{1}}$ , ${\mathfrak {u}_{2}}$ , ${\mathfrak {u}_{3}}$ , ${\mathfrak {u}_{4}}$ , ${\mathfrak {u}_{5}}$ } be a group of professionals with the given weights ${\Omega _{i}}$ = ${\left ({0.24,~0.14,~0.111,~0.401,~0.108}\right)^{T}}$ and ${\mathfrak {L}^{\prime }}=\left \{{{e_{1}}=air\,\,conditioner,\,\,{e_{2}}=airbag,{e_{3}}=\,\,\,\,price,}\right. \left.{{e_{4}}=design}\right \}{}$ be a set of attributes with weights ${\gamma _{j}}$ = ${\left ({0.110,~0.234,~0.489,0.167}\right)^{T}}$ . alternatives ranking in the form of q-ROFSNs $\left ({\aleph,~{\mathfrak {L}^{\prime }}}\right)$ = ${\left \langle{ {\mu _{ij}},\,\,{\vartheta _{ij}}}\right \rangle _{5 \times 4}}$ given as: $$\begin{align*}&\hspace {-1pc}\left ({\aleph,~{\mathfrak {L}^{\prime }}}\right) \\=&\left \lceil{ \!\! {\begin{array}{cccc}(0.27,~0.72)& (0.5,~0.2)& \left ({0.76,~0.44}\right)& \left ({0.9,~0.3}\right)\\ \left ({0.53,~0.94}\right)& \left ({0.6,~0.34}\right)& \left ({0.3,~0.95}\right)& \left ({0.6,~0.54}\right)\\ \left ({0.36,~0.6}\right)& \left ({0.8,~0.4}\right)& \left ({0.44,~0.5}\right)& \left ({0.56,~0.71}\right)\\ \left ({0.80,~0.37}\right)& \left ({0.1,~0.7}\right)& \left ({0.27,~0.75}\right)& \left ({0.59,~0.8}\right)\\ (0.91,~0.2)& (0.29,~0.67)& (0.31,~0.67)& (0.6,~0.4)\end{array}}\!\!}\right \rceil\end{align*}$$ For $q=3$ using Equation 3, $$\begin{align*}&\hspace {-1pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n} {\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{m}{\left ({\mathop {\prod }\nolimits _{i=1}^{n}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {} \\&\hspace {-1pc}q-ROFSEWG~\left ({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{54}}}}\right) \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{4} {\left ({\mathop {\prod }\nolimits _{i=1}^{5}{\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{4} {\left ({\mathop {\prod }\nolimits _{i=1}^{5}{\left ({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{4}{\left ({\mathop {\prod }\nolimits _{i=1}^{5} {\left ({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&\left.{\frac {\sqrt [{q}] {\mathop {\prod }\nolimits _{j=1}^{4}{\left ({\mathop {\prod }\nolimits _{i=1}^{5} {\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{4}{\left ({\mathop {\prod }\nolimits _{i=1}^{5} {\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{4}{\left ({\mathop {\prod }\nolimits _{i=1}^{5} {\left ({1+{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{4}{\left ({\mathop {\prod }\nolimits _{i=1}^{5}{\left ({1-{\vartheta _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {} \\{}\end{align*}$$ As shown in the equation at the bottom of page 8.

5) Theorem

Let ${\aleph _{\mathsf {e}_{ij}}}$ = $\left({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ be a collection of q-ROFSNs, then $$\begin{align*}&\hspace {-2pc}q-ROFSWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\{\leq }&q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)\end{align*}$$ ${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weight vectors for specialists and attributes, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1.

Proof:

As we know that $$\begin{align*}&\hspace {-1pc}\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n} {\left({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n} {\left({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}} \\{\leq }&\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\gamma _{j}}\left({\mathop {\prod }\nolimits _{i=1}^{n}{\Omega _{i}}\left({2-{\mu _{ij}^{q}}}\right)}\right)+\mathop {\prod }\nolimits _{j=1}^{m}{\gamma _{j}}\left({\mathop {\prod }\nolimits _{i=1}^{n}{\Omega _{i}}\left({{\mu _{ij}^{q}}}\right)}\right)} \\&\hspace {-1pc}\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\gamma _{j}}\left({\mathop {\prod }\nolimits _{i=1}^{n}{\Omega _{i}}\left({2-{\mu _{ij}^{q}}}\right)}\right)+\mathop {\prod }\nolimits _{j=1}^{m}{\gamma _{j}}\left({\mathop {\prod }\nolimits _{i=1}^{n}{\Omega _{i}}\left({{\mu _{ij}^{q}}}\right)}\right)}=\sqrt [{q}]{2} \\\Rightarrow&\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n} {\left({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}{\leq }~\sqrt [{q}]{2} \\\Rightarrow&\frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{m} {\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m} {\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({2-{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n} {\left({{\mu _{ij}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}} \\{\geq }&\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n} {\left({{\mu _{ij}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}{~}\tag{4}\end{align*}$$

Again (5), as shown at the bottom of page 9.

Let $$\begin{equation*} q-ROFSWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) = \aleph =\left({{\mu _{\aleph }},{\vartheta _{\aleph }}}\right)\end{equation*}$$ and $$\begin{equation*}q-ROFSEWG \left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)={\aleph ^{\varepsilon }}=\left({{\mu _{\aleph ^{\varepsilon }}},{\vartheta _{\aleph ^{\varepsilon }}}}\right){.}\end{equation*}$$ Then, (4) and (5) can be transformed into the consequent forms ${\mu _{\aleph }} {\leq } {\mu _{\aleph ^{\varepsilon }}}$ and ${\vartheta _{\aleph }}{\geq }{\vartheta _{\aleph ^{\varepsilon }}}$ Compatibly.

So, $S\left({\aleph }\right)$ = ${{\mu _{\aleph }}^{q}}-{{\vartheta _{\aleph }}^{q}} {\leq } {{\mu _{\aleph ^{\varepsilon }}}^{q}}-{{\vartheta _{\aleph ^{\varepsilon }}}^{q}}$ = $S ({\aleph ^{\varepsilon }})$ . Hence, $S\left({\aleph }\right) {\leq } S ({\aleph ^{\varepsilon }})$

If $S\left({\aleph }\right) < S ({\aleph ^{\varepsilon }})$ , then $$\begin{align*}&\hspace {-1pc}q-ROFSWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\ < &q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right){~}\tag{6}\end{align*}$$ If $S\left({\aleph }\right)=\,\,S ({\aleph ^{\varepsilon }})$ , then ${{\mu _{\aleph }}^{q}}-{{\vartheta _{\aleph }}^{q}}$ = ${{\mu _{\aleph ^{\varepsilon }}}^{q}}-{{\vartheta _{\aleph ^{\varepsilon }}}^{q}}$ , so ${{\mu _{\aleph }}^{q}} = {{\mu _{\aleph ^{\varepsilon }}}^{q}}$ and ${{\vartheta _{\aleph }}^{q}}={{\vartheta _{\aleph ^{\varepsilon }}}^{q}}$ .

Then, $A\left({\aleph }\right)$ = ${{\mu _{\aleph }}^{q}}+{{\vartheta _{\aleph }}^{q}} = {{\mu _{\aleph ^{\varepsilon }}}^{q}}+{{\vartheta _{\aleph ^{\varepsilon }}}^{q}}$ = $A ({\aleph ^{\varepsilon }})$ . Thus, $$\begin{align*}&\hspace {-2pc}q-ROFSWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right){~}\tag{7}\end{align*}$$

From (6) and (7), we get $$\begin{align*}&\hspace {-2pc}q-ROFSWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\{\leq }&q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right){.}\end{align*}$$

6) Example

Let $\mathcal {U}$ = ${\mathfrak {u}_{1}}$ , ${\mathfrak {u}_{2}}$ , ${\mathfrak {u}_{3}}$ , ${\mathfrak {u}_{4}}$ , ${\mathfrak {u}_{5}}$ be a set of experts with the given weights ${\Omega _{i}}$ = ${\left({0.24,~0.14,~0.111,~0.401,~0.108}\right)^{T}}$ and ${\mathfrak {L}^{\prime }}=\left \{{{\mathsf {e}_{1}}=air\,\,conditioner,\,\,{\mathsf {e}_{2}}=airbag,\,\,{\mathsf {e}_{3}}= \,\,\,\,price,}\right. \left.{{\mathsf {e}_{4}}=design}\right \}{}$ with weights ${\gamma _{j}}$ = $\left({0.110,~0.234,~0.489,}\right. \left.{0.167}\right)^{T}$ . The assumed rating standards for each feature in the form of q-ROFSNs $\left({\aleph,~{\mathfrak {L}^{\prime }}}\right)$ = ${\left \langle{ {\mu _{ij}},\,\,{\vartheta _{ij}}}\right \rangle _{5 \times 4}}$ given as: $$\begin{align*}&\hspace {-1pc}\left({\aleph,~{\mathfrak {L}^{\prime }}}\right) \\=&\left \lceil{ \!\! {\begin{array}{cccc}(0.27,~0.72)& (0.5,~0.2)& \left({0.76,~0.44}\right)& \left({0.9,~0.3}\right)\\ \left({0.53,~0.94}\right)& \left({0.6,~0.34}\right)& \left({0.3,~0.95}\right)& \left({0.6,~0.54}\right)\\ \left({0.36,~0.6}\right)& \left({0.8,~0.4}\right)& \left({0.44,~0.5}\right)& \left({0.56,~0.71}\right)\\ \left({0.80,~0.37}\right)& \left({0.1,~0.7}\right)& \left({0.27,~0.75}\right)& \left({0.59,~0.8}\right)\\ (0.91,~0.2)& (0.29,~0.67)& (0.31,~0.67)& (0.6,~0.4)\end{array}}\!\!}\right \rceil\end{align*}$$ For $q=3$ using Equation q-ROFSWG operator,

So, Examples 4 and 6 show that $$\begin{align*}&\hspace {-2pc}q-ROFSWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right) \\{\leq }&q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right){.}\end{align*}$$

7) Remark

1. If ${{\mu _{ij}}^{q}}+\, {{\vartheta _{ij}}^{q}}{\leq }1\,{}$ and $q=2$ . Then, the q-ROFSEWG operator is reduced to the PFSEWG operator [33].

2. If ${{\mu _{ij}}^{q}}+\, {{\vartheta _{ij}}^{q}}{\leq }1\,{}$ and $q=1$ . Then, the q-ROFSEWG operator is reduced to the IFSEWG operator [3].

B. Properties of Q-ROFSEWG Operator

1) Idempotency

If ${\aleph _{ij}}=\aleph =\,\,\left({{\mu _{ij}},{\vartheta _{ij}}}\right){\forall }i,\,\,j,\quad$ then $q-ROFSEWG\,\,\left({{\aleph _{\mathsf {e}_{11}}},\,\,{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)=\aleph$

Proof:

As we know that, as shown in the equation at the bottom of page 10.

2) Boundedness

If ${\aleph _{ij}}=\left({{\mu _{ij}},{\vartheta _{ij}}}\right)$ be a collection of q-ROFSNs and ${\aleph _{min}}$ = $min\left({{\aleph _{ij}}}\right)$ , ${\aleph _{max}}$ = $max\left({{\aleph _{ij}}}\right)$ . Then, ${\aleph _{min}} {\leq } q-ROFSEWG\,\,\left({{\aleph _{\mathsf {e}_{11}}},\,\,{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) {\leq } {\aleph _{max}}$ .

Proof:

Let $f(x)=\sqrt [{q}]{\frac {2-{x^{q}}}{x^{q}}}$ , $x{\in }\left]{0,~1}\right]$ , then $$\begin{align*} \frac {d}{dx}\left({f(x)}\right)=&-\frac {\left [{\frac {q}{x}+\frac {q}{x^{1 +q}}(2-{x^{q}})}\right]\left [{\frac {1}{\theta ^{q}}{\left({2-{x^{q}}}\right)^{-1 +\frac {1}{q}}}}\right]}{q}\frac {-2}{x^{3}} \\ < &0.\end{align*}$$ So, $f(x)$ is decreasing function on $\left]{0,~1}\right]$ . Since ${\mu _{min}}{\leq }~{\mu _{ij}}~{\leq }{\mu _{max}}~{\forall }i,j$ . Then, $f\left({{\mu _{max}}}\right){\leq }~f\left({{\mu _{ij}}}\right){\leq }f\left({{\mu _{min}}}\right)$ . So, $$\begin{equation*} \sqrt [{q}]{\frac {2-{\mu _{max}^{q}}}{\mu _{max}^{q}}}~{\leq }\sqrt [{q}]{\frac {2-{\mu _{ij}^{q}}}{\mu _{ij}^{q}}} ~{\leq }\sqrt [{q}]{\frac {2-{\mu _{min}^{q}}}{\mu _{min}^{q}}}.\end{equation*}$$ ${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weight vectors for professionals and parameters, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1.

Again, let $\,\,\,\,g(y)=\,\,\sqrt [{q}]{\frac {1-{y^{q}}}{1+{y^{q}}}}$ , $y{\in }\left]{0,~1}\right]$ .

Then, $$\begin{align*} \frac {d}{dy}\left({g(y)}\right)=&-\frac {\left [{\frac {q}{y}+\frac {q}{y^{1 +q}}(2-{y^{q}})}\right]\left [{\frac {1}{\theta ^{q}}{\left({2-{y^{q}}}\right)^{-1 +\frac {1}{q}}}}\right]}{q}\frac {-2}{y^{3}} \\ < &0.\end{align*}$$ So, $g(y)$ is decreasing function on $\left]{0,~1}\right]$ . Thus, ${\vartheta _{min}}{\leq }~{\vartheta _{ij}}{\leq }~{\vartheta _{max}}~{\forall }~i,j$ . So, $g\left({{\vartheta _{max}}}\right){\leq }g\left({{\vartheta _{ij}}}\right)~{\leq }g\left({{\vartheta _{min}}}\right)~{\forall }~i,j$ . $$\begin{equation*}~\Rightarrow \sqrt [{q}]{\frac {1-{\vartheta _{max}^{q}}}{1+{\vartheta _{max}^{q}}}}{\leq }\sqrt [{q}]{\frac {1-{\vartheta _{ij}^{q}}}{1+{\vartheta _{ij}^{q}}}}~{\leq }\sqrt [{q}]{\frac {1-{\vartheta _{min}^{q}}}{1+{\vartheta _{min}^{q}}}}{,}\end{equation*}$$ ${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weight vectors for professionals and parameters, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}} = 1$ , as shown in the equation at the top of page 12.

Let ${\delta }$ = $q-ROFSEWG\,\,\left({{\aleph _{\mathsf {e}_{11}}},\,\,{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)$ =$\left \langle{ {\mu _{\delta }},~{\vartheta _{\delta }}}\right \rangle$ = ${\aleph _{\delta }}$ , then employing Equation 1. $$\begin{align*}&\hspace {-1pc}S({\delta }) \\=&{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,} \left({\frac {e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}}- {{\vartheta _{\delta }}^{q}}}}+1}-\,\,\frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\{\leq }&{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}\!-\!{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\!\left({\!\frac {e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}}\right)^{q}}\!-\!{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}\!+\!1}\!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}} \\=&S({\aleph _{max}}) \\\Rightarrow&S({\delta }){\leq }S({\aleph _{max}})\end{align*}$$ and $$\begin{align*}&\hspace {-1pc}S({\delta }) \\=&{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,} \left({\frac {e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}} -{{\vartheta _{\delta }}^{q}}}}+1}-\,\,\frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\{\geq }&{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}} -{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\! \left({\!\!\frac {e^{{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}\!}\right)^{q}} -{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}} {{e^{{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}} -{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}}\right)^{q}}}}\!\!+\!\!1} \!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{-}}}^{q}} \\=&S({\aleph _{min}}) \\\Rightarrow&S({\delta }){\geq }S({\aleph _{min}})\end{align*}$$ Considering the above procedure, we have the subsequent circumstances:

If $S({\delta }) < S({\aleph _{min}})$ and $S({\delta })>S({\aleph _{min}})$ . Then, we obtain $$\begin{equation*}~{\aleph _{min}} < q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) < {\aleph _{max}}{.}\end{equation*}$$

If $S({\delta })=S({\aleph _{max}})$ , i.e., $$\begin{align*}&\hspace {-1pc}{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,}\left({\frac {e^{{{\mu _{\delta }}^{q}} -{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}+1}-\,\,\frac {1}{2}}\right) {{\pi _{\delta }}^{q}} \\=&{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\!\left({\!\!\frac {e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}\! }\right)^{q}}-{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \} \!}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}}}}\!+\!1}\!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}}\end{align*}$$ using the above inequities, we get

$\begin{aligned} {\mu _{\delta }}\,\,=\,\,\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}{} \end{aligned}$ , and $\begin{aligned} {\vartheta _{\delta }}=\,\,\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}{} \end{aligned}$ . Hence, ${{\pi _{\delta }}^{q}}={{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}}$ then by comparing two q-ROFSNs, we obtain $$\begin{equation*}~q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)={\aleph _{max}}{.}\end{equation*}$$ If $S({\delta })=S({\aleph _{min}})$ , i.e., $$\begin{align*}&\hspace {-1pc}{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,}\left({\frac {e^{{{\mu _{\delta }}^{q}} -{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}+1}- \frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\=&{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}} -{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\left({\!\!\frac {e^{{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}} }\right \}\!}\right)^{q}}-{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i} \end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}\!+\!1}\!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}}\end{align*}$$ using the above inequities, we get

$\begin{aligned} {\mu _{\delta }}\,\,=\,\,\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}{} \end{aligned}$ , and $\begin{aligned} {\vartheta _{\delta }}=\,\,\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}{} \end{aligned}$ . Hence, ${{\pi _{\delta }}^{q}}={{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{-}}}^{q}}$ Then by comparing two q-ROFSNs, we obtain $$\begin{equation*}~q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)={\aleph _{min}}{.}\end{equation*}$$ So, it proved that $$\begin{equation*}~{\aleph _{min}}{\leq }q-ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right){\leq }{\aleph _{max}}{.}\end{equation*}$$

3) Homogeneity

Prove that $$\begin{align*}&\hspace {-2pc}\mathrm {q-ROFSEWG~}\left({\alpha {\aleph _{\mathsf {e}_{11}}},~{\alpha }{\aleph _{\mathsf {e}_{12}}}\ldots,~{\alpha }{\aleph _{\mathsf {e}_{\mathrm {nm}}}}}\right) \\=&\mathrm {\alpha q-}ROFSEWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,~{\aleph _{\mathsf {e}_{nm}}}}\right)~{\alpha }>0.\end{align*}$$

Proof:

Let ${\aleph _{\mathsf {e}_{ij}}}$ be a q-ROFSN and ${\alpha }>0$ . Then, $$\begin{equation*}{\aleph _{\mathsf {e}_{ij}}}=\left \langle{ \frac {\sqrt [{q}]{2\left({{\mu ^{q}}}\right)^{\partial }}}{\sqrt [{q}]{{\left({2-{\mu ^{q}}}\right)^{\partial }}+{\left({{\mu ^{q}}}\right)^{\partial }}}},~\frac {\sqrt [{q}]{{\left({1+{\vartheta ^{q}}}\right)^{\partial }}-{\left({1-{\vartheta ^{q}}}\right)^{\partial }}}}{\sqrt [{q}]{{\left({1+{\vartheta ^{q}}}\right)^{\partial }}+{\left({1-{\vartheta ^{q}}}\right)^{\partial }}}}}\right \rangle\end{equation*}$$ So, as shown in the equation at the top of page 13.

SECTION IV. Einstein Ordered Geometric Aggregation Operator for Q- Rung ORTHOPAIR Fuzzy Soft Numbers

This section presents q-ROFSEOWG operators based on Einstein’s operational laws for q-ROFSNs.

1) Definition

Let ${\aleph _{\mathsf {e}_{ij}}}$ = $\left({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ be a collection of q-ROFSNs. If q-ROFSEOWG: ${\Delta ^{n}}\rightarrow {\Delta }$ , and $$\begin{align*}&\hspace {-2pc}q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{m}}{\left({{\otimes _{i=1}^{n}}{\left({{\aleph _{\mathsf {e}_{\mathfrak {r}(i)\mathfrak {s}(j)}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}{~}\tag{10}\end{align*}$$ Then q-ROFSEOWG is entitled q-rung orthopair fuzzy soft Einstein ordered weighted geometric operator. Where ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1 and ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1 signifies the weights of specialists and parameters. Also, $\mathfrak {r,s}$ are permutations such that ${\aleph _{\mathfrak {r}\left({i-1 }\right)j}}{\geq }{\aleph _{\mathfrak {r}(i)j}}$ and ${\aleph _{i\mathfrak {s}\left({j-1 }\right)}}{\geq }{\aleph _{i\mathfrak {s}(j)}}\,\,{\forall }i,j$ .

2) Theorem

Let ${\aleph _{\mathsf {e}_{ij}}}$ = $\left({{\mu _{ij}},~{\vartheta _{ij}}}\right)$ be an assortment of q-ROFSNs. Then, the conquered collected values are also a q-ROFSN and

$$\begin{align*}&\hspace {-1pc}q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) \\=&{\otimes _{j=1}^{m}}{\left({{\otimes _{i=1}^{n}}{\left({{\aleph _{\mathsf {e}_{\mathfrak {r}(i)\mathfrak {s}(j)}}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}} \\=&\left \langle{ \frac {\sqrt [{q}]{2\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({{\mu _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({2-{\mu _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({{\mu _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right., \\&~\left.{\frac {\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({1+{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}-\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({1-{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}{\sqrt [{q}]{\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({1+{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}+\,\,\mathop {\prod }\nolimits _{j=1}^{m}{\left({\mathop {\prod }\nolimits _{i=1}^{n}{\left({1-{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}\right)^{\Omega _{i}}}}\right)^{\gamma _{j}}}}}}\right \rangle {}\tag{11}\end{align*}$$

${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weight vectors of the specialists and parameters, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1. Also, $\mathfrak {r,s}$ are permutations such that ${\aleph _{\mathfrak {r}\left({i-1 }\right)j}}{\geq }{\aleph _{\mathfrak {r}(i)j}}$ and ${\aleph _{i\mathfrak {s}\left({j-1 }\right)}}{\geq }{\aleph _{i\mathfrak {s}(j)}}\,\,{\forall }i,j$ .

Proof:

Using mathematical induction

For $n$ = 1, we get ${\Omega _{i}}=1$

For $m$ = 1, we get ${\gamma _{j}}=1$ .

So, equation 11 hold for $n$ = 1 and $m$ = 1.

Assume for $n$ = ${\delta _{2}}$ , $m$ = ${\delta _{1}}+1$ and for $n$ = ${\delta _{2}}+1$ , $m$ = ${\delta _{1}}$ The above equation holds.

Now, for $m$ = ${\delta _{1}}+1$ and $n$ = ${\delta _{2}}+1$

So, it is true for $m$ = ${\delta _{1}}+1$ and $n$ = ${\delta _{2}}+1$ .

3) Example

Let $\mathcal {U}$ = ${\mathfrak {u}_{1}}$ , ${\mathfrak {u}_{2}}$ , ${\mathfrak {u}_{3}}$ be a set of specialists with weights ${\Omega _{i}}$ = ${\left({0.4,~0.3,0.3}\right)^{T}}$ . The team of specialists is working to designate the desirability of an automobile considering the parameters ${\mathfrak {L}^{\prime }}=\left \{{{\mathsf {e}_{1}}=air\,\,conditioner,\,\,{\mathsf {e}_{2}}=airbag,{\mathsf {e}_{3}}=price,}\right. \left.{{\mathsf {e}_{4}}=design}\right \}{}$ with weights ${\gamma _{j}}$ = ${\left({0.3,~0.5,~0.1,~0.1}\right)^{T}}$ . The supposed assessment values for each parameter in the form of q-ROFSNs $(q=3) \left({\aleph,\,\,{\mathfrak {L}^{\prime }}}\right)$ = ${\left \langle{ {\mu _{ij}},\,\,{\vartheta _{ij}}}\right \rangle _{3 \times 4}}$ given as: $$\begin{align*}~\left({\aleph,~{\mathfrak {L}^{\prime }}}\right)=\left \lceil{ {\begin{array}{cccc}\left({0.5,~0.7}\right)& \left({0.5,~0.2}\right)& \left({0.4,~0.6}\right)& (0.4,~0.1)\\ \left({0.5,~0.4}\right)& \left({0.6,~0.3}\right)& \left({0.5,~0.3}\right)& (0.6,~0.5)\\ \left({0.3,~0.5}\right)& \left({0.2,~0.3}\right)& \left({0.8,~0.3}\right)& (0.7,~0.9)\end{array}}}\right \rceil {}\end{align*}$$ Obtain the ordered matrix via the score function, such as: $$\begin{align*}\left({\aleph,~{\mathfrak {L}^{\prime }}}\right)=\,\,\left \lceil{ {\begin{array}{cccc}\left({0.8,~0.3}\right)& \left({0.6,~0.3}\right)& \left({0.5,~0.2}\right)& (0.5,~0.3)\\ \left({0.6,~0.5}\right)& \left({0.2,~0.3}\right)& \left({0.3,~0.5}\right)& (0.5,~0.7)\\ \left({0.5,~0.4}\right)& \left({0.1,~0.4}\right)& \left({0.4,~0.6}\right)& (0.7,~0.9)\end{array}}}\right \rceil\end{align*}$$ For $q=3$ using Equation 11, as shown in the equation at the bottom of page 16.

4) Remark

1. If ${{\mu _{ij}}^{q}}+\, {{\vartheta _{ij}}^{q}}{\leq }1\,{}$ and $q=2$ . Then, the q-ROFSEOWG operator is condensed to the PFSEOWG operator [34].

2. If ${{\mu _{ij}}^{q}}+\, {{\vartheta _{ij}}^{q}}{\leq }1\,{}$ and $q=1$ . Then, the q-ROFSEOWG operator is condensed to the IFSEOWG operator [3].

Now, we will discuss the desired properties for the q-ROFSEOWG operator such as follows:

A. Properties of Q-ROFSEOWG Operator

1) Idempotency

If ${\aleph _{ij}}=\,\,\left({{\mu _{ij}},{\vartheta _{ij}}}\right)$ be collection of q-ROFSNs. If ${\aleph _{\mathfrak {r}(i)\mathfrak {s}(j)}}={\aleph _{ij}}$ are identical, then $$\begin{equation*} q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)=\aleph\end{equation*}$$

Proof:

As we know that

As ${\aleph _{\mathfrak {r}(i)\mathfrak {s}(j)}}={\aleph _{ij}}$

So, $$\begin{equation*} {=}\left \langle{ {\mu _{ij}}}\right.~,~\left.{{\vartheta _{ij}}}\right \rangle {\,\,=\,\,}\aleph {.}\end{equation*}$$

2) Boundedness

If ${\aleph _{ij}}=\left({{\mu _{ij}},{\vartheta _{ij}}}\right)$ be an assortment of q-ROFSNs and ${\aleph _{min}}$ = $min\left({{\aleph _{\mathfrak {r}(i)\mathfrak {s}(j)}}}\right)$ , ${\aleph _{max}}$ = $max\left({{\aleph _{\mathfrak {r}(i)\mathfrak {s}(j)}}}\right)$ . Then, $$\begin{equation*} {\aleph _{min}} {\leq } q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) {\leq } {\aleph _{max}}.\end{equation*}$$

Proof:

Let $f(x)=\sqrt [{q}]{\frac {2-{x^{q}}}{x^{q}}}$ , $x{\in }\left]{0,~1}\right]$ , then $$\begin{align*} \frac {d}{dx}\left({f(x)}\right)=&-\frac {\left [{\frac {q}{x}+\frac {q}{x^{1 +q}}(2-{x^{q}})}\right]\left [{\frac {1}{\theta ^{q}}{\left({2-{x^{q}}}\right)^{-1 +\frac {1}{q}}}}\right]}{q}\frac {-2}{x^{3}} \\ < &0.\end{align*}$$ So, $f(x)$ is decreasing function on $\left]{0,~1}\right]$ . Since ${\mu _{min}}~{\leq }~{\mu _{\mathfrak {r}(i)\mathfrak {s}(j)}}~{\leq }~{\mu _{max}}~{\forall }i,j$ . Then, $$\begin{equation*} f\left({{\mu _{max}}}\right)~{\leq }~f\left({{\mu _{\mathfrak {r}(i)\mathfrak {s}(j)}}}\right)~{\leq }~f\left({{\mu _{min}}}\right).\end{equation*}$$ So, $$\begin{equation*} \sqrt [{q}]{\frac {2-{\mu _{max}^{q}}}{\mu _{max}^{q}}}~{\leq }\sqrt [{q}]{\frac {2-{\mu _{\mathfrak {r}(i) \mathfrak {s}(j)}^{q}}}{\mu _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}~{\leq }\sqrt [{q}]{\frac {2-{\mu _{min}^{q}}}{\mu _{min}^{q}}}.\end{equation*}$$ ${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weights of specialists and parameters, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1.

Again, let $\,\,\,\,g(y)=\,\,\sqrt {\frac {1-{y^{2}}}{1+{y^{2}}}}$ ,$~y{\in }\left]{0,~1}\right]$ .

Then, $$\begin{align*} \frac {d}{dy}\left({g(y)}\right)=&-\frac {\left [{\frac {q}{y}+\frac {q}{y^{1 +q}}(2-{y^{q}})}\right]\left [{\frac {1}{\theta ^{q}}{\left({2-{y^{q}}}\right)^{-1 +\frac {1}{q}}}}\right]}{q}\frac {-2}{y^{3}} \\ < &0.\end{align*}$$ So, $g(y)$ is decreasing function on $\left]{0,~1}\right]$ . Thus, ${\vartheta _{min}}{\leq }~{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}}~{\leq }~{\vartheta _{max}}~{\forall }~i,j$ . So, $g\left({{\vartheta _{max}}}\right)~{\leq }~g\left({{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}}}\right)~ {\leq }g\left({{\vartheta _{min}}}\right)~{\forall }~i,j$ . $$\begin{equation*}~\Rightarrow \sqrt [{q}]{\frac {1-{\vartheta _{max}^{q}}}{1+{\vartheta _{max}^{q}}}}{\leq }\sqrt [{q}]{\frac {1-{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}{1+{\vartheta _{\mathfrak {r}(i)\mathfrak {s}(j)}^{q}}}}~{\leq }~\sqrt [{q}]{\frac {1-{\vartheta _{min}^{q}}}{1+{\vartheta _{min}^{q}}}}{,}\end{equation*}$$ ${\Omega _{i}}$ and ${\gamma _{j}}$ denotes the weights of specialists and parameters, correspondingly, such as ${\Omega _{i}}>0\,{}$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}$ = 1, ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1.

Let ${\delta }$ = $q-ROFSEOWG\,\,\left({{\aleph _{\mathsf {e}_{11}}},\,\,{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)$ =$\left \langle{ {\mu _{\delta }},~{\vartheta _{\delta }}}\right \rangle$ = ${\aleph _{\delta }}$ , then employing Equation 1. $$\begin{align*}&\hspace {-1pc}S({\delta }) \\=&{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,} \left({\frac {e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}}- {{\vartheta _{\delta }}^{q}}}}+1}-\,\,\frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\{\leq }&{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\left({\!\!\frac {e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}\! }\right)^{q}}\!-\!{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}\!\!+\!\!1}\!-\!\frac {1}{2}\!\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}} \\=&S({\aleph _{max}}) \\\Rightarrow&S({\delta }){\leq }S({\aleph _{max}})\end{align*}$$ and $$\begin{align*}&\hspace {-1pc}S({\delta }) \\=&{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,}\left({\frac {e^{{{\mu _{\delta }}^{q}} -{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}+1} -\,\,\frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\{\geq }&{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\!\left({\!\!\frac {e^{{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}\!-\!{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}+1}\!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{-}}}^{q}} \\=&S({\aleph _{min}}) \\\Rightarrow&S({\delta }){\geq }S({\aleph _{min}})\end{align*}$$ Seeing the above practice, we have the subsequent cases:

If $S({\delta }) < S({\aleph _{min}})$ and $S({\delta })>S({\aleph _{min}})$ then, by equating two q-ROFSNs, we attain $$\begin{equation*}~{\aleph _{min}} < q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right) < {\aleph _{max}}{.}\end{equation*}$$ If $S({\delta })=S({\aleph _{max}})$ , i.e., $$\begin{align*}&\hspace {-1pc}{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}{\,\,+\,\,}\left({\frac {e^{{{\mu _{\delta }}^{q}} -{{\vartheta _{\delta }}^{q}}}}{{e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}+1} -\,\,\frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\=&{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\left({\!\!\frac {e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}-{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}-{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}\!+\!1}\!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}}\end{align*}$$ employing the above inequities, we attain $\begin{aligned} {\mu _{\delta }}\,\,=\,\,\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}{} \end{aligned}$ , and $\begin{aligned} {\vartheta _{\delta }}=\,\,\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}{} \end{aligned}$ . Hence, ${{\pi _{\delta }}^{q}}={{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}}$ . Then by equating two q-ROFSNs, we attain $$\begin{equation*}~q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)={\aleph _{max}}{.}\end{equation*}$$ If $S({\delta })=S({\aleph _{min}})$ , i.e., $$\begin{align*}&\hspace {-1pc}{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}} {\,\,+\,\,}\left({\frac {e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}} {{e^{{{\mu _{\delta }}^{q}}-{{\vartheta _{\delta }}^{q}}}}+1}-\,\,\frac {1}{2}}\right){{\pi _{\delta }}^{q}} \\=&{\left({\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}}\right)^{q}}-{\left({\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}}\right)^{q}} \\&+\!\left({\!\!\frac {e^{{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}-{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}{{e^{{\left({\!\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array} \left \{{{\mu _{ij}}}\right \}\!}\right)^{q}}\!-\!{\left({\!\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array} \left \{{{\vartheta _{ij}}}\right \}\!}\right)^{q}}}}\!+\!1}\!-\!\frac {1}{2}\!\!}\right){{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{+}}}^{q}}\end{align*}$$ by employing the above inequities, we attain $\begin{aligned} {\mu _{\delta }}\,\,=\,\,\begin{array}{cc}{min}&{min}\\ {j}&{i}\end{array}\left \{{{\mu _{ij}}}\right \}{} \end{aligned}$ , and $\begin{aligned} {\vartheta _{\delta }}=\,\,\begin{array}{cc}{max}&{max}\\ {j}&{i}\end{array}\left \{{{\vartheta _{ij}}}\right \}{} \end{aligned}$ . Hence, ${{\pi _{\delta }}^{q}}={{\pi _{{\aleph _{\mathsf {e}_{ij}}}^{-}}}^{q}}$ Then by comparing two q-ROFSNs, we obtain $$\begin{equation*}~q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right)={\aleph _{min}}{.}\end{equation*}$$ So, it proved that $$\begin{equation*}~{\aleph _{min}}{\leq }q-ROFSEOWG~\left({{\aleph _{\mathsf {e}_{11}}},~{\aleph _{\mathsf {e}_{12}}},\ldots,{\aleph _{\mathsf {e}_{nm}}}}\right){\leq }{\aleph _{max}}{.}\end{equation*}$$

3) Homogeneity

Prove that $\mathrm {q-ROFSEOWG\,\,}\left({\alpha {\aleph _{\mathsf {e}_{11}}},\,\,{\alpha }{\aleph _{\mathsf {e}_{12}}}\ldots,\,\,{\alpha }{\aleph _{\mathsf {e}_{\mathrm {nm}}}}}\right)$ = $\mathrm {\alpha q-}ROFSEOWG\,\,\left({{\aleph _{\mathsf {e}_{11}}},\,\,{\aleph _{\mathsf {e}_{12}}},\ldots,\,\,{\aleph _{\mathsf {e}_{nm}}}}\right)$ for >0.

Proof:

Let ${\aleph _{\mathsf {e}_{ij}}}$ be a q-ROFSN and ${\alpha }>0$ . Then $$\begin{equation*} {\aleph _{\mathsf {e}_{ij}}}=\left \langle{ \frac {\sqrt [{q}]{2\left({{\mu ^{q}}}\right)^{\partial }}}{\sqrt [{q}]{{\left({2-{\mu ^{q}}}\right)^{\partial }}+{\left({{\mu ^{q}}}\right)^{\partial }}}},~\frac {\sqrt [{q}]{{\left({1+{\vartheta ^{q}}}\right)^{\partial }}-{\left({1-{\vartheta ^{q}}}\right)^{\partial }}}}{\sqrt [{q}]{{\left({1+{\vartheta ^{q}}}\right)^{\partial }}+{\left({1-{\vartheta ^{q}}}\right)^{\partial }}}}}\right \rangle\end{equation*}$$ So,

SECTION V. Multi-Criteria Decision-Making Approach for Q-ROFSEWG and Q-ROFSEOWG Operators

The subsequent section will build an MCDM method operating our established AOs.

A. Proposed MCDM Approach

Let $\mathcal {X}$ = $\left \{{{\mathcal {X} _{1}},~{\mathcal {X} _{2}},~{\mathcal {X} _{3}},\ldots,~{\mathcal {X} _{s}}}\right \}{}$ be a set of $s$ alternatives $\mathcal {Q}$ = $\left \{{{\mathcal {Q} ^{1}},~{\mathcal {Q} ^{2}},~{\mathcal {Q} ^{3}},\ldots,~{\mathcal {Q} ^{n}}}\right \}{}$ be a set $n$ experts. Assume $\Omega$ = ${\left({{\Omega _{1}},~{\Omega _{1}},\ldots,~{\Omega _{n}}}\right)^{T}}$ and ${\left({{\gamma _{1}},~{\gamma _{2}},~{\gamma _{3}},\ldots,~{\gamma _{m}}}\right)^{T}}$ denotes the weights of the specialists and parameters such as ${\Omega _{i}}>0$ , $\mathop {\sum }\nolimits _{i=1}^{n}{\Omega _{i}}= 1$ and ${\gamma _{j}}>0\,{}$ , $\mathop {\sum }\nolimits _{j=1}^{m}{\gamma _{j}}$ = 1. Specialists ${\mathcal {Q} ^{i}}$ : $i$ = 1, 2,$\ldots, n$ assess the considered alternatives ${\mathcal {X} _{\mathrm {z}}}$ : $z$ = 1, 2,$\ldots, s$ in the form of q-ROFSNs such as ${D^{(z)}}$ = ${\left({{\aleph _{\mathsf {e}_{ij}}}}\right)_{n\times m}} = {\left({{\mu _{ij}},\,\,{\vartheta _{ij}}}\right)_{n\times m}}$ , where ${}\,0\leq {\mu _{ij}},~{\vartheta _{ij}}\leq 1\,{}$ and ${}\,0\leq {\mu _{ij}^{2}}+{\vartheta _{ij}^{2}}\leq 1\, {\forall }\,\,i$ , $j$ assumed in Table 1–​Table 4. To use the assessments of senior experts, the collected q-ROFSNs for substitutes ${\mathcal {X} _{s}}$ specified by engaging the intended Einstein AOs. Lastly, applied the score function to investigate the rank of the substitutes. The step-wise process of the above-presented approach is given as follows:

TABLE 1 Q-ROFS Decision Matrix for ${\mathcal{X} _{1}}$

TABLE 2 Q-ROFS Decision Matrix for ${\mathcal{X} _{2}}$

TABLE 3 Q-ROFS Decision Matrix for ${\mathcal{X} _{3}}$

TABLE 4 Q-ROFS Decision Matrix for ${\mathcal{X} _{4}}$

B. Algorithm for Q-ROFSEWG Operator

Step 1. Accumulate the specialist’s assessment information for each alternative to their compatible attributes and then build a decision matrix such as follows: $$\begin{align*}&\hspace {-2pc}{\left({{\mathcal {X} ^{(z)}} \mathcal {,~E} }\right)_{n\times m}} \\=&{\begin{array}{c}\\ {\mathcal {Q} _{1}}\\ {\mathcal {Q} _{2}}\\ \vdots \\ {\mathcal {Q} _{n}}\end{array}}\left({{\begin{array}{cccc}\\ \left({{\mu _{11}^{(z)}},~{\vartheta _{11}^{(z)}}}\right)& \left({{\mu _{12}^{(z)}},~{\vartheta _{12}^{(z)}}}\right)& \cdots & \left({{\mu _{1 m}^{(z)}},~{\vartheta _{1 m}^{(z)}}}\right)\\ \left({{\mu _{21}^{(z)}},~{\vartheta _{21}^{(z)}}}\right)& \left({{\mu _{22}^{(z)}},~{\vartheta _{22}^{(z)}}}\right)& \cdots & \left({{\mu _{2 m}^{(z)}},~{\vartheta _{2 m}^{(z)}}}\right)\\ \vdots & \vdots & \vdots & \vdots \\ \left({{\mu _{n1}^{(z)}},~{\vartheta _{n1}^{(z)}}}\right)& \left({{\mu _{n2}^{(z)}},~{\vartheta _{n2}^{(z)}}}\right)& \cdots & \left({{\mu _{nm}^{(z)}},~{\vartheta _{nm}^{(z)}}}\right)\end{array}}}\right)\end{align*}$$

Step 2. Convert the cost type parameters to benefit type using the normalization rule. $$\begin{align*}~{ {\boldsymbol {h}}_{ij}^{(z)}}=\begin{cases}{\aleph _{\mathsf {e}_{ij}}^{c}}=\left({{\vartheta _{ij}^{(z)}},~{\mu _{ij}^{(z)}}}\right);&\mathrm {cost~type~parameter}\\ {\aleph _{\mathsf {e}_{ij}}}=\left({{\mu _{ij}^{(z)}},{\vartheta _{ij}^{(z)}}}\right);&\mathrm {benefit~type~parameter} \end{cases}\end{align*}$$

Step 3. Conquered a communal decision matrix ${\mathcal {L} _{k}}$ for alternatives $\mathcal {X}$ = $\left \{{{\mathcal {X} _{1}},~{\mathcal {X} _{2}},~{\mathcal {X} _{3}},\ldots,~{\mathcal {X} _{s}}}\right \}{}$ settled q-ROFSEWG operator.

Step 4. Calculate the ranking of the alternatives via the score function.

Step 5. Indicate the extraordinary alternative with a supreme score value ${\mathcal {L} _{k}}$ .

Step 6. Detect the position of the substitutes.

C. Algorithm for Q-ROFSEOWG Operator

Step 1. Accumulate the specialist’s assessment facts for each alternative to their compatible parameters: $$\begin{align*}&\hspace {-2pc}{\left({{\mathcal {X} ^{(z)}} \mathcal {,~E} }\right)_{n\times m}} \\=&{\begin{array}{c}\\ {\mathcal {Q} _{1}}\\ {\mathcal {Q} _{2}}\\ \vdots \\ {\mathcal {Q} _{n}}\end{array}}\left({{\begin{array}{cccc}\\ \left({{\mu _{11}^{(z)}},~{\vartheta _{11}^{(z)}}}\right)& \left({{\mu _{12}^{(z)}},~{\vartheta _{12}^{(z)}}}\right)& \cdots & \left({{\mu _{1 m}^{(z)}},~{\vartheta _{1 m}^{(z)}}}\right)\\ \left({{\mu _{21}^{(z)}},~{\vartheta _{21}^{(z)}}}\right)& \left({{\mu _{22}^{(z)}},~{\vartheta _{22}^{(z)}}}\right)& \cdots & \left({{\mu _{2 m}^{(z)}},~{\vartheta _{2 m}^{(z)}}}\right)\\ \vdots & \vdots & \vdots & \vdots \\ \left({{\mu _{n1}^{(z)}},~{\vartheta _{n1}^{(z)}}}\right)& \left({{\mu _{n2}^{(z)}},~{\vartheta _{n2}^{(z)}}}\right)& \cdots & \left({{\mu _{nm}^{(z)}},~{\vartheta _{nm}^{(z)}}}\right)\end{array}}}\right)\end{align*}$$

Step 2. Calculate the ordered matrix via the score function.

Step 3. Convert the cost type parameters to benefit type using the normalization rule. $$\begin{align*}~{ {\boldsymbol {h}}_{ij}^{(z)}}{\,\,=\,\,}\begin{cases}{\aleph _{\mathsf {e}_{ij}}^{c}}=\left({{\vartheta _{ij}^{(z)}},~{\mu _{ij}^{(z)}}}\right);&\mathrm {cost~type~parameter}\\ {\aleph _{\mathsf {e}_{ij}}}=\left({{\mu _{ij}^{(z)}},~{\vartheta _{ij}^{(z)}}}\right);&\mathrm {benefit~type~parameter} \end{cases}\end{align*}$$

Step 4. Conquered a communal decision matrix ${\mathcal {L} _{k}}$ for alternatives via settled q-ROFSEOWG operator.

Step 5. Calculate the alternatives score values using the score function.

Step 6. Indicate the most suitable alternative with a supreme score value ${\mathcal {L} _{k}}$ .

Step 7. Detect the rank of the substitutes.

SECTION VI. Application of Proposed MCDM Approach and Comparative Studies

In this section, an inarticulate example to demonstrate the productivity and accuracy of the model established through q-ROFSS facts [37], [39]. Furthermore, we present the superiority and comparative analysis of the following: a prearranged method with prevailing approaches.

A. Application of the Developed MCDM Approach

Suppose $\mathcal {Q}$ = $\left \{{{\mathcal {Q} ^{1}},~{\mathcal {Q} ^{2}},~{\mathcal {Q} ^{3}},~{\mathcal {Q} ^{4}},~{\mathcal {Q} ^{5}}}\right \}{}$ be a group of five specialists with weights ${\left({0.18,0.24,0.21,0.15,0.22}\right)^{T}}$ . The panel of experts is going to designate their valuation for four different medical patients $\mathcal {X=} \left \{{{ \mathcal {X} _{1}},\,\,{\mathcal {X} _{2}},\,\,{\mathcal {X} _{3}},\,\,{\mathcal {X} _{4}}}\right \}{}$ . The attribute for patients assessment is given as follows: ${\mathsf {e}_{1}}=$ chest pain, ${e_{2}}=$ fever, ${e_{3}}=$ cough, ${e_{4}}=$ fatigue, ${e_{5}}=$ vomit with weights ${\left({0.26,0.22,0.1,0.27,0.15}\right)^{T}}$ . To judge the optimal alternative, experts deliver their predilections in q-ROFSNs. Subsections B and C offer the technique to bargain the adequate substitute.

B. Q-ROFSEWG Operator

Step-1: The specialists examine the environs and stretch their predilections in q-ROFSN. The score values for each alternative are given in Table 1–​Table 4.

Step 2. All attributes are of the same type. So, no need to normalize.

Step 3. q-ROFSEWG operator can dignify specialists’ assessment of each alternative as follows:

For $q$ =3 $$\begin{align*}~{\mathcal {L} _{1}}=&\left \langle{ \left.{0.2107,0.7135}\right \rangle }\right.{,~}{\mathcal {L} _{2}}{\,\,=\,\,}\left \langle{ \begin{matrix}\left.{\begin{matrix}0.1693,\end{matrix}0.7283}\right \rangle \end{matrix}}\right.{,~} \\ { \mathcal {L} _{3}}=&\left.{\left \langle{ \begin{matrix}0.1619\end{matrix},0.7314}\right.}\right \rangle {,~\text {and}~}{\mathcal {L} _{4}}{=\,\,}\left.{\left \langle{ 0.1602,~0.7329}\right.}\right \rangle\end{align*}$$

Step 4. Calculate the score values using Equation ${}\,1. \mathcal {S(} {\mathcal {L} _{1}})$ = 0.4158, $\mathcal {S(} {\mathcal {L} _{2}})$ = 0.4307, $\mathcal {S(} { \mathcal {L} _{3}})$ = 0.4942, $\mathcal {S(} {\mathcal {L} _{4}})$ = 0.4862.

Step 5.${\mathcal {X} _{3}}$ has the most improbable score value, so ${\mathcal {X} _{3}}$ is the most affected patient.

Step 6. Substitutes ranking using q-ROFSEWG operator given as follows:

$\mathcal {S} ({\mathcal {L} _{3}})> \mathcal {S} ({\mathcal {L} _{4}})> \mathcal {S} ({ \mathcal {L} _{2}})> \mathcal {S} ({\mathcal {L} _{1}})$ . So, ${\mathcal {X} _{3}}>{\mathcal {X} _{4}}>{\mathcal {X} _{2}}>{\mathcal {X} _{1}}$ . So, ${\mathcal {X} _{3}}$ be a most appropriate alternative.

C. For Q-ROFSEOWG Operator

Step 1. Same as above.

Step 2. Acquire the ordered decision matrices given in the following Table 5–​Table 8.

TABLE 5 Q-ROFS Ordered Matrix for ${\mathcal{X} _{1}}$

TABLE 6 Q-ROFS Ordered Matrix for ${\mathcal{X} _{2}}$

TABLE 7 Q-ROFS Ordered Matrix for ${\mathcal{X} _{3}}$

TABLE 8 Q-ROFS Ordered Matrix for ${\mathcal{X} _{4}}$

Step 3. All parameters are of the same type. So, no need to normalize.

Step 4. q-ROFSEOWG operator can dignify experts’ assessment of each patient as follows:

For $q$ = 3 $$\begin{align*}~{\mathcal {L} _{1}}=&\left \langle{ \left.{0.2476,~0.7366}\right \rangle }\right.{,~}{\mathcal {L} _{2}}{\,\,=\,\,}\left \langle{ \left.{\begin{matrix}0.2109,~0.7397\end{matrix}}\right \rangle }\right.{,~} \\ {\mathcal {L} _{3}}=&\left \langle{ \left.{\begin{matrix}0.2085,~0.7638\end{matrix}}\right \rangle }\right.{,~\text {and}~}{\mathcal {L} _{4}}{=\,\,}\left \langle{ \left.{\begin{matrix}0.1856,~0.7952\end{matrix}}\right \rangle }\right.\end{align*}$$

Step 5. Calculate the score values using Equation ${}\,1. \mathcal {S(} {\mathcal {L} _{1}})$ = 0.4251, $\mathcal {S(} {\mathcal {L} _{2}})$ = 0.4467, $\mathcal {S(} { \mathcal {L} _{3}})$ = 0.5138, $\mathcal {S(} {\mathcal {L} _{4}})$ = 0.4568.

Step 6. ${\mathcal {X} _{3}}$ has the most improbable score value, so ${ \mathcal {X} _{3}}$ is the most affected patient.

Step 7. Alternatives ranking by q-ROFSEOWG operator given as follows: $$\begin{equation*}~\mathcal {S} ({\mathcal {L} _{3}})> \mathcal {S} ({\mathcal {L} _{4}})> \mathcal {S} ({\mathcal {L} _{2}})> \mathcal {S} ({\mathcal {L} _{1}}).\end{equation*}$$ So, $$\begin{equation*}{\mathcal {X} _{3}}>{\mathcal {X} _{4}}>{\mathcal {X} _{2}}>{\mathcal {X} _{1}}{.~}\end{equation*}$$

We will be capable of distinguishing variations in the assessments of the two operators. These distinctions are owing to precise conformation methodologies formed by distinct AOs. However, the best and poorest optimal are the same in both cases. The consequences condense the endorsed operators’ cruelty, ability, efficacy, and stability.

D. Benefits and Superiority of the Proposed Method

The projected method is capable and operable; we present a progressive approach under the q-ROFSS background through q-ROFSEWG or q-ROFSEOWG operators. Our anticipated model is more brilliant than predominant structures and can convey the most subtle values of MCDM obstacles. The integrated model is multipurpose and familiar with regulating emergent variations, involvement, and efficiency. Different replicas have specific classification processes, so there are direct modifications among the anticipated method’s statuses to be feasible based on their meditations. From this systematic investigation and assessment, we now determine that the consequences of the prevailing methodology are impulsively associated with the amalgam structure. Moreover, owing to some privileged environments, several fusion organizations of fuzzy set, IFSS, and PFSS have to convert infrequent for q-ROFSS. It is a modest technique of syndicating imperfect and ambiguous facts in the DM method. Information around the purpose can be prepared more rigorously and cogently articulated. It is an ascetically modest implement to assortment imprecise and anxious information in the DM method. So, our scheduled technique will be more capable, substantial, superior, and better than several diversified configurations of fuzzy sets. Table 10 offers a distinctive investigation of the projected methodology with some general studies.

TABLE 9 Patient’s Score Values Using Developed Operators

TABLE 10 Characteristic Analysis of Different Methods

E. Comparative Analysis

To validate the ascendency and influence of the intentional methodology, a reasonable study has been offered between the settled technique and some prevalent mechanisms, based on some AOs. If we contemplate the MD = 0.8 and NMD= 0.5, then $MD+NMD{\leq }1$ and ${MD^{2}}+{NMD^{2}}{\leq }1$ . Thus, the existing PFWSG [31], PFSIWG [32], PFSEWG [33], and PFSEOWG [34] operators cannot grip the situation when the ${MD^{2}}+{NMD^{2}}{\geq }1$ Correspondingly, q-ROFWG [14] can accommodate the scenario if when the ${MD^{2}}+\,\,{NMD^{2}}{\geq }1$ . But, the existing q-ROFWG operator cannot handle the alternatives’ parametric values. On the other hand, our projected operators expertly deal with the parametric values of the alternatives. In some circumstances, specialists’ preferences cannot disturb the communal consequences of the aggregated values. Such as, if $\mathcal {Q=} \left \{{{\mathcal {Q} ^{1}},\,\,{\mathcal {Q} ^{2}}}\right \}{}$ be a set of two specialists and ${\mathsf {e}_{1}},~{\mathsf {e}_{2}}$ are two parameters where $\mathcal {X}$ = $\begin{aligned} \left [{{\begin{array}{cc}\left({0.5,0.6}\right)& \left({0.3,0.4}\right)\\ \left({0.0,0.4}\right)& \left({0.2,0.7}\right)\end{array}}}\right] \end{aligned}$ . Let ${\Omega _{i}}$ = ${\left({0.4,~0.6}\right)^{T}}$ and ${\gamma _{j}}$ = ${\left({0.2,~.0.8}\right)^{T}}$ exemplifies the weights of specialists and parameters, respectively. Then, using the q-ROFSWG [39], we get $\left \langle{ 0,~0.9129}\right \rangle$ . This shows that there is no outcome on the mutual consequence ${\mu _{\mathsf {e}}}$ . Because ${\mu _{\mathsf {e}}}$ = ${\mu _{11}}$ = 0.5, ${\mu _{12}}$ = 0.0, ${\mu _{21}}$ = 0.3, and ${\mu _{22}}$ = 0.2, which is unreasoning. Similarly, we gained the same outcomes using the q-ROFSOWG [39] operator. Meanwhile, our settled q-ROFSEWG and q-ROFSEOWG operators grip such tentative consequences expertly. However, the main advantage of a considered technique is that it consensuses extra information to advocate the DM concerns. It is also an appreciated tool for determining indefinite and unexplainable data in the DM method. The attained ranking by dissimilar models has been offered in Table 11.

TABLE 11 Comparative Analysis With Existing Operators

It is also suitable for furious undetermined and incorrect data in the DM practice. The benefit of the intended technique and allied processes over existing methodologies is to escape interpretations based on offensive reasons. So, it is an appropriate tool for merging inaccurate and unstipulated specifics in the DM method. The graphical demonstration of comparative studies is prearranged in the succeeding Figure 1.

FIGURE 1.

Comparative analysis.

SECTION VII. Conclusion

Decision-making is a pre-planned process of sorting and selecting logical options from among many substitutes. DM is a complex procedure since it can modify from one extract to another. Therefore, it is crucial to identify the characteristics and limitations of the substitutes. Furthermore, DM is a healthy method that increases the probability of identifying the best appropriate alternate. It is essential to differentiate how considerable precise contextual information decision-makers need. The most effective DM method is to focus entirely on your objectives. Molodtsov combines parametric tools with standard sets to study significant SS samples. SS concept does not receive convolution and is an exceptional scientific tool for parameterizing uncertain relationships. The main purpose of this research is to present some new operating rules for q-ROFSS. Einstein geometric AOs have been planned based on the offered concept: q-ROFSEWG and q-ROFSEOWG operators. Furthermore, some basic features of the proposed operator are also discussed. The proposed model poses a medicinal DM delinquent in the q-ROFSS configuration. Next, we use some prevailing techniques to demonstrate the effectiveness of the settled scheme and identify the impact and advantages of existing methods for general research through characterization analysis. The advantage of default concepts is that they can address the complexity of reality by consuming their parametric properties. Therefore, the proven model can resolve the delinquency of DM but not other prevailing operators in the q-ROFSS setting. Future studies will focus on existing DM tools used by several other operators as part of the q-ROFSS. The advocated notion can be pragmatic to various real-life complications comprising the medical profession, robotics, artificial intelligence, pattern recognition, economics, etc. Also, numerous other structures can be established and projected, such as Einstein AOs, Bonferroni mean AOs and dombi AOs, etc., with their DM techniques.

ACKNOWLEDGMENT

This work was supported by the Deanship of Scientific Research at King Khalid University through Large Groups Project under Grant R.G.P. 2/51/43.