Decision Analysis Algorithm Using Frank Aggregation in the SWARA Framework with ^p,qRung Orthopair Fuzzy Information

Abstract

The present study introduces an innovative approach to multi-criteria decision making (MCDM) aimed at handling decision analysis involving ^p,qrung orthopair fuzzy (^p,qROF) data, where the criteria weights are completely unknown. To achieve this objective, we formulate generalized operational rules referred to as Frank operational rules, tailored for ^p,qROF numbers (^p,qROFNs) utilizing the Frank t-norm and t-conorm. With these newly devised operations as a foundation, we create a variety of ^p,qROF aggregation operators (AOs) to effectively aggregate ^p,qROF information. Furthermore, we examine specific instances of these operators and rigorously establish their desirable properties, including idempotency, monotonicity, boundedness, and symmetry. Subsequently, we adapt the SWARA technique to the realm of ^p,qROF fuzzy data and this adapted technique becomes instrumental in determining criteria weights within the proposed MCDM framework centered around proposed AOs. We furnish a descriptive example to exemplify the practicality of the developed approach. Lastly, the effectiveness and soundness of our approach are underscored through both parameter analysis and a comparative evaluation.

1. Introduction

Multi-criteria decision making (MCDM) is a process that involves evaluating options or alternatives based on multiple criteria or attributes. This approach is widely used across various fields, such as business, engineering, green supplier management, construction development, and public policy. Decision experts (DEs) typically use precise numerical values to evaluate alternatives in accepted MCDM contexts. Cagri Tolga and Basar [1] applied theoretical concepts from MCDM to estimate the rates of population growth and food production resource increases under conditions of uncertainty. Tolga et al. [2] employed a more robust and effective evaluation strategy to select optimal medical devices for enhancement in the health sector, considering type-2 fuzzy environments. Additionally, due to limited data, insufficient information, and time constraints, property estimations—especially for subjective attributes—cannot always be represented by exact numbers. In such cases, fuzzy sets (FSs) often provide a more practical means of representation. Zadeh’s concepts [3] have been applied across various fields, including decision making, design recognition, and clinical diagnosis. However, for any fixed element within the universe of discourse, an FS can only use the membership degree (MD) to express certainty, which does not completely represent the fuzzy problem. To address this limitation, Atanassov and Yager advanced the concept of FSs by introducing the intuitionistic FS (IFS) [4], Pythagorean FS (PFS) [5,6], and q-rung orthopair FS (q-ROFS) [7], each incorporating additional components: non-membership degree (N-MD) and hesitancy degree (HD). The q-ROFS is a generalization of these concepts, with the IFS and PFS being special cases when $q=1$ and $q=2$, respectively. In a q-ROFS, a pair $\left(\mathit{ᵷ},ʎ\right)$ satisfies the constraints $0\le \mathit{ᵷ},ʎ\le 1$ and ${\mathit{ᵷ}}^q+{ʎ}^q\le 1$ (where $q\ge 1$), thus, a larger value of q expands the scope of fuzzy information it can represent.

Upon reviewing the preceding discussion, it is evident that certain limitations constrain existing theories. For example, within q-ROF environments, DEs are restricted by the requirement to assign identical values to the parameter q for both MD and N-MD. This constraint can significantly affect the decision-making process. To address these limitations and enhance the framework’s applicability, a novel approach has been proposed. Seikh and Mandal [8] originated the concept of p,q-rung orthopair fuzzy ^p,qROF sets (^p,qROFSs), which extend q-ROFSs. ^p,qROFSs offer a more flexible structure by incorporating two adjustable parameters, p and q, allowing DEs to tailor MD and N-MD according to specific needs. This enhanced flexibility is better suited for managing complex decision-making scenarios, advancing the field by providing a more nuanced approach. Building on this innovative framework, researchers have developed various methodologies. For instance, Rahim et al. [9] explored sine trigonometric operations and their associated aggregation operators (AOs) for ^p,qROF numbers (^p,qROFNs). Ali and Naeem [10] presented Aczel–Alsina AOs within the ^p,qROF framework to address MCDM problems. Chu et al. [11] investigated cubic ^p,qROFSs and their fundamental operational laws. Additionally, Rahim et al. [12] developed confidence-level-based AOs using ^p,qROFSs. These advancements highlight the substantial potential of ^p,qROFSs in enhancing decision-making processes and broadening the theoretical framework.

The AOs designed by Seikh and Mandal [8] are based on certain t-norms and t-conorms, with the product and probabilistic sum [13] being the most commonly used. These pairs are preferred for their computational simplicity. However, their major drawbacks include a lack of flexibility and robustness. To address these issues, some ^p,qROF AOs [10,14,15] have been introduced using alternative t-norms and t-conorms. These new AOs enhance robustness but still fall short in terms of flexibility, limiting their ability to incorporate all opinions and information necessary for modeling real-world MCDM problems. To bridge this gap, Frank [16] generalized the Łukasiewicz and probabilistic t-norms and t-conorms, proposing the Frank t-norm and t-conorm, which form a more flexible family of triangular norms. The inclusion of adjustable parameters in the Frank t-norm and t-conorm enhances their adaptability in information fusion, making them more suitable for modeling complex decision-making problems. Sarkoci [17] conducted a comparative study between Hamacher and Frank t-norms, concluding that they belong to the same family of t-norms.

Following the introduction of Frank t-norms and t-conorms, mathematicians have developed AOs utilizing this pair of norms. Zhang et al. [18] presented Frank AOs for interval-valued intuitionistic FSs (IFSs) and demonstrated their application in MCGDM problems. Qin et al. [19] explored Frank AOs for triangular interval type-2 fuzzy information based on Frank t-norms and t-conorms, applying them to MCGDM problems. In [20], the development and theory of Frank AOs within probabilistic hesitant fuzzy sets are given, while Tang et al. [21] explored their application in the context of dual hesitant fuzzy information. With the advent of ^p,qROFSs, DEs gained the flexibility to express information using MDs and N-MDs with varying power conditions. This flexibility, combined with the significance of Frank t-norms and t-conorms, makes the concept of Frank AOs within the p,q-ROFS framework particularly noteworthy. Despite their desirable properties, there has been a lack of studies examining the effects of varying parameters within Frank AOs on MCDM problems. This gap in research motivated the author to explore the family of Frank AOs and their properties within the ^p,qROF framework, based on Frank t-norms and t-conorms, and subsequently, apply them to MCDM problems.

The following arguments establish the necessity of this study:

(i)

The ^p,qROFS model’s broad applicability enables it to elucidate complex data in a versatile and comprehensive manner. As a construction-based decision-making strategy, the decision-making mechanism within the ^p,qROFS structure warrants greater attention.

(ii)

The Frank operations are highly adaptable due to the Frank parameter ß, allowing for flexibility in various decision-making environments based on the DE’s risk-taking appetite. By blending realistic elements with a degree of optimism, they cater to different consumer needs while avoiding strict adherence to recklessness. However, the literature lacks these operations in the ^p,qROF setting.

(iii)

The preferences of DEs often shift dynamically based on their pessimistic or optimistic outlooks toward the object being evaluated. Therefore, the AOs necessary for MCDM must be sufficiently comprehensive and flexible to encompass all DEs’ preferences while integrating the evaluation values.

(iv)

In real-world scenarios, many decision-making problems involve distinct processes and information that are often ambiguous and uncertain. As a result, decision-making theory must evolve to develop novel MCDM approaches. Existing methodologies for MCDM [10,12,14,15] in the ^p,qROF context have several limitations, especially when managing decision-making issues where the weights of the criteria are entirely unknown.

(v)

Numerous scholars have developed various MCDM approaches in the existing body of research to address MCDM problems. These approaches aim to determine the most appropriate material selection in different fuzzy contexts. However, there is a notable shortage of scientific studies on managing suitable material selection for construction in a ^p,qROF scenario using aggregation-based stepwise weight assessment ratio analysis (SWARA) techniques.

Building upon the aforementioned motivations, this research has resulted in the development of several noteworthy innovations, which are listed in detail below.

(i)

To establish novel Frank-norm-based operational laws and analyze their properties, followed by numerical examples to elucidate the operational procedures of the proposed operations.

(ii)

To formulate the Frank weighted averaging and weighted geometric operators, along with their ordered weighted and hybrid forms, for aggregating p,q-ROF data.

(iii)

To develop the theory behind Frank operators, including their essential characteristics, limiting cases, and interrelationships.

(iv)

To introduce the ^p,qROF SWARA method for determining the weights of criteria; this allows DEs to express their viewpoints more freely due to the absence of a scale. Compared to the best–worst method, SWARA offers simpler computations and is easier to understand as it does not require solving complex linear objective functions.

(v)

To develop an aggregation-based method within the ^p,qROF framework, utilizing the proposed operators and the SWARA approach, for evaluating and identifying sustainable material choices in the construction industry. By incorporating the SWARA technique into this aggregation-based methodology, we enhance its practicality in decision-making processes where the weight vector is initially unknown.

In Section 2, we revisit some fundamental definitions and preliminaries to provide a foundational understanding of the present study. Section 3 introduces new operational laws for ^p,qROFNs based on the Frank t-norm and t-conorm. Building on these operational laws, Section 4 outlines new AOs along with their related results and special cases. Section 5 presents an algorithm for solving decision-making problems using proposed Frank AOs and the SWARA approach. In Section 6, we validate the proposed method with a real-life example and analyze the impact of parameters and criteria weights on the decision-making outcomes. Section 7 offers a comparison between the results of our method and those of other significant models. Finally, Section 8 concludes the paper.

2. Preliminaries

In this section, we provide a succinct summary of the ^p,qROFS structure.

Definition 1 

([8]). Suppose we have a given set X. A ^p,qROFS denoted by $\mathcal{D}$ over X can be characterized as follows:

$$\mathcal{D}=\left\{\left(\mathsf{ą},\mathit{ᵷ}(\mathsf{ą}),ʎ(\mathsf{ą})\right)|\mathsf{ą}\in X\right\},p,q\ge 1,$$

(1)

Here, the terms $\mathit{ᵷ}(\mathsf{ą})$ and $ʎ(\mathsf{ą})$ exist within the range $\left[0.1\right]$ and represent the MD and N-MD, respectively, assigned to the element $\mathsf{ą}\in X$. This assignment is valid when the condition $0\le {\left(\mathit{ᵷ}(\mathsf{ą})\right)}^p+{\left(ʎ(\mathsf{ą})\right)}^q\le 1$ is met. The degree of indeterminacy is defined as ${\left(\pi (\mathsf{ą})\right)}^{\ell }=1-{\left(\mathit{ᵷ}(\mathsf{ą})\right)}^p+{\left(ʎ(\mathsf{ą})\right)}^q,$ where ℓ represents the least common multiple (lcm) of p and q. For the sake of convenience, in their work [8], Seikh and Mandal introduced the notation $\mathsf{ϝ}=\left(\mathit{ᵷ},ʎ\right)$ as a ^p,qROFN.

Remark 1. 

Consider a scenario where we are tasked with determining the smallest values of p and q greater than or equal to 1 for a given orthopair $\left(\mathit{ᵷ},ʎ\right)$ such that the inequality ${\mathit{ᵷ}}^p+{ʎ}^q\le 1$ holds true. While an explicit mathematical solution in a closed form might not be readily available, it is consistently viable to formulate a distinct solution for these challenges through the utilization of iterative computational techniques.

The minimal values of p and q that satisfy the condition ${\mathit{ᵷ}}^p+{ʎ}^q\le 1$ are termed the $p,q$-niche of the orthopair $\left(\mathit{ᵷ},ʎ\right)$. In cases where $p^{\prime }$ and $q^{\prime }$ represent the $p,q$-niche of $\left(\mathit{ᵷ},ʎ\right)$, it can be deduced that $\left(\mathit{ᵷ},ʎ\right)$ remains valid for all values of p greater than or equal to $p^{\prime }$, and for all values of q greater than or equal to $q^{\prime }$.

Consider a set of provided data $X=\left\{{ą}_1,{ą}_2,\dots ,{ą}_{♭}\right\}$, and let $\mathcal{F}$ represent a fuzzy concept. In this context, an expert contributes their preferences for each element $ą\iota \in X$ in the form of an orthopair $\left(\mathit{ᵷ}(ą\iota ),ʎ(ą\iota )\right)$. The task at hand is to determine suitable values for p and q that effectively capture the characteristics of the data. We can now proceed as outlined below:

• Determine the $p,q$-niche, denoted as $p_{\iota }$ and $q_{\iota }$, for each $p,q$-orthopair $\left(\mathit{ᵷ}(ą\iota ),ʎ(ą\iota )\right)$.
• Define the $p^{\ast },q^{\ast }$-niche as follows: $p^{\ast }=\underset{\iota }{max}{p}_{\iota }$ and $q^{\ast }=\underset{\iota }{max}{q}_{\iota }$.
• Consequently, we can represent the fuzzy notion $\mathcal{F}$ as a $p^{\ast },q^{\ast }$-ROFS.

Remark 2. 

• Definition 1 simplifies to an IFS when we assign $p=q=1.$
• Definition 1 simplifies to a PyFS when we assign $p=q=2.$
• Definition 1 simplifies to an FFS when we assign $p=q=3.$
• Definition 1 simplifies to a ^qROFS when we assign $p=q.$
• Definition 1 simplifies to a ^3,4-quasirung FS when $p=3$, $q=4.$

Definition 2 

([8]). Let ϝ, ${\mathsf{ϝ}}_1$, and ${\mathsf{ϝ}}_2$ be any three ^p,qROFNs and $\nmid >0$, then the basic operational laws on them are listed as:

• ${\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2=\left({\left({\mathit{ᵷ}}_1^{p^{\ast }}+{\mathit{ᵷ}}_2^{p^{\ast }}-{\mathit{ᵷ}}_1^{p^{\ast }}{\mathit{ᵷ}}_2^{p^{\ast }}\right)}^{1/p^{\ast }},{ʎ}_1{ʎ}_2\right);$
• ${\mathsf{ϝ}}_1\otimes {\mathsf{ϝ}}_2=\left({\mathit{ᵷ}}_1{\mathit{ᵷ}}_2,{\left({ʎ}_1^{q^{\ast }}+{ʎ}_2^{q^{\ast }}-{ʎ}_1^{q^{\ast }}{ʎ}_2^{q^{\ast }}\right)}^{1/p^{\ast }}\right);$
• ${\mathsf{ϝ}}^{\nmid }=\left({\mathit{ᵷ}}^{\nmid },{\left(1-{\left(1-{ʎ}^q\right)}^{\nmid }\right)}^{1/q}\right);$
• $\nmid \mathsf{ϝ}=\left({\left(1-{\left(1-{\mathit{ᵷ}}^p\right)}^{\nmid }\right)}^{1/p},{ʎ}^{\nmid }\right);$
• ${\mathsf{ϝ}}^c=\left(ʎ,\mathit{ᵷ}\right)$.

Definition 3 

([8]). Let ϝ be a ^p,qROFN, then the score function can be defined as

$$S\left(\mathsf{ϝ}\right)={\displaystyle \frac{1+{\mathit{ᵷ}}^p-{ʎ}^q}{2}},$$

(2)

where $p,q\in [1,\infty )$, $S\left(\mathsf{ϝ}\right)\in \left[0,1\right]$. The greater the value of $S\left(\mathsf{ϝ}\right)$, the greater the ^p,qROFN ϝ.

Definition 4 

([8]). Let ϝ be a ^p,qROFN, then the degree of accuracy is characterized by

$$A\left(\mathsf{ϝ}\right)={\left(\mathit{ᵷ}\right)}^p+{\left(ʎ\right)}^q;A\left(\mathsf{ϝ}\right)\in \left[0.1\right].$$

(3)

When the computed score values are similar, the larger the degree of accuracy $A\left(\mathsf{ϝ}\right)$, the larger the ^p,qROFN.

When the computed score values are the same, we then examine their accuracy. A higher degree of accuracy $A\left(\mathsf{ϝ}\right)$ corresponds to a larger ^p,qROFN.

Triangular norms (t-norms) have been extensively researched since Zadeh pioneered the max and min operations as examples of triangular norms and their corresponding triangular conorms (t-conorms). Several notable pairs of t-norms and t-conorms include the product t-norm and probabilistic sum t-conorm [22], the Einstein t-norm and t-conorm [23], the Łukasiewicz t-norm and t-conorm [13], and the Hamacher t-norm and t-conorm [24].

Frank operations, such as Frank’s product and Frank’s sum, serve as additional examples of t-norms and t-conorms, respectively. The Frank t-norm $T_F$ and Frank t-conorm $S_F$ are formulated as follows:

$$T_F\left({\AE }_1,{\AE }_2\right)={log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{{\AE }_1}-1\right)\left({ß}^{{\AE }_2}-1\right)}{ß-1}}\right)\forall \left({\AE }_1,{\AE }_2\right)\in {\left[0.1\right]}^2,$$

(4)

$$S_F\left({\AE }_1,{\AE }_2\right)=1-{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{1-{\AE }_1}-1\right)\left({ß}^{1-{\AE }_2}-1\right)}{ß-1}}\right)\forall \left({\AE }_1,{\AE }_2\right)\in {\left[0.1\right]}^2.$$

(5)

According to Wang et al. [25], the Frank t-norm and Frank t-conorm exhibit the following features:

$$T_F\left({\AE }_1,{\AE }_2\right)+S_F\left({\AE }_1,{\AE }_2\right)={\AE }_1+{\AE }_2,$$

(6)

$${\displaystyle \frac{\partial T_F\left({\AE }_1,{\AE }_2\right)}{\partial {\AE }_1}}+{\displaystyle \frac{\partial S_F\left({\AE }_1,{\AE }_2\right)}{\partial {\AE }_1}}=1.$$

(7)

The following findings [25] can be easily verified using limit theory.

(1)

If $ß⟶1$, then $T_F\left({\AE }_1,{\AE }_2\right)⟶{\AE }_1+{\AE }_2-{\AE }_1{\AE }_2$ and $S_F\left({\AE }_1,{\AE }_2\right)⟶{\AE }_1{\AE }_2$; the Frank t-norm and Frank t-conorm are reduced to probabilistic product and probabilistic sum.

(2)

If $ß⟶\infty$, then $T_F\left({\AE }_1,{\AE }_2\right)⟶min\left({\AE }_1+{\AE }_2,1\right)$ and $S_F\left({\AE }_1,{\AE }_2\right)⟶max\left(0,{\AE }_1+{\AE }_2-1\right)$; the Frank t-norm and Frank t-conorm are reduced to the Łukasiewicz product and Łukasiewicz sum, respectively.

3. ^p,qROF Frank Operational Rules

This section introduces the Frank operations of ^p,qROFNs and explores their intriguing characteristics.

Definition 5. 

Let ${\mathsf{ϝ}}_1=\left({\mathit{ᵷ}}_1,{ʎ}_1\right)$ and ${\mathsf{ϝ}}_2=\left({\mathit{ᵷ}}_2,{ʎ}_2\right)$ be two ^p,qROFNs and $\nmid >0$, then:

• ${\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2=\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{\left({\mathcal{ß}}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)\left({\mathcal{ß}}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}{\mathcal{ß}-1}}\right)},& \\ \sqrt[q^{\ast }]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{\left({\mathcal{ß}}^{{ʎ}_1^{q^{\ast }}}-1\right)\left({\mathcal{ß}}^{{ʎ}_2^{q^{\ast }}}-1\right)}{\mathcal{ß}-1}}\right)}\end{array}\right)$;
• ${\mathsf{ϝ}}_1\otimes {\mathsf{ϝ}}_2=\left(\begin{array}{cc}\sqrt[p^{\ast }]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{\left({\mathcal{ß}}^{{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)\left({\mathcal{ß}}^{{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}{\mathcal{ß}-1}}\right)},& \\ \sqrt[q^{\ast }]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{\left({\mathcal{ß}}^{1-{ʎ}_1^{q^{\ast }}}-1\right)\left({\mathcal{ß}}^{1-{ʎ}_2^{q^{\ast }}}-1\right)}{\mathcal{ß}-1}}\right)}\end{array}\right)$;
• ${\mathsf{ϝ}}_1^{\nmid }=\left(\sqrt[p]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{{\left({\mathcal{ß}}^{{\mathit{ᵷ}}_1^p}-1\right)}^{\nmid }}{{\left(\mathcal{ß}-1\right)}^{\nmid -1}}}\right)},\sqrt[q]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{{\left({\mathcal{ß}}^{1-{\vartheta }_1^q}-1\right)}^{\nmid }}{{\left(\mathcal{ß}-1\right)}^{\nmid -1}}}\right)}\right)$;
• $\nmid {\mathsf{ϝ}}_1=\left(\sqrt[p]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{{\left({\mathcal{ß}}^{1-{\mathit{ᵷ}}_1^p}-1\right)}^{\nmid }}{{\left(\mathcal{ß}-1\right)}^{\nmid -1}}}\right)},\sqrt[q]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \frac{{\left({\mathcal{ß}}^{{ʎ}_1^q}-1\right)}^{\nmid }}{{\left(\mathcal{ß}-1\right)}^{\nmid -1}}}\right)}\right)$;
• ${\mathsf{ϝ}}_1^c=\left({ʎ}_1,{\mathit{ᵷ}}_1\right)$.

Theorem 1. 

Let ${\mathsf{ϝ}}_{ȶ}=\left({\mathit{ᵷ}}_{ȶ},{ʎ}_{ȶ}\right)\left(ȶ=1,2\right)$ and $\mathsf{ϝ}=\left(\mathit{ᵷ},ʎ\right)$ be three ^p,qROFNs, and $\nmid ,{\nmid }_1,{\nmid }_2>0$, then: 2

1. ${\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2={\mathsf{ϝ}}_2\oplus {\mathsf{ϝ}}_1$;	5. ${\nmid }_1\mathsf{ϝ}\oplus {\nmid }_2\mathsf{ϝ}=\left({\nmid }_1+{\nmid }_2\right)\mathsf{ϝ}$;
2. ${\mathsf{ϝ}}_1\otimes {\mathsf{ϝ}}_2={\mathsf{ϝ}}_2\otimes {\mathsf{ϝ}}_1$;	6. ${\mathsf{ϝ}}^{{\nmid }_1}\otimes {\mathsf{ϝ}}^{{\nmid }_2}={\mathsf{ϝ}}^{{\nmid }_1+{\nmid }_2}$;
3. $\nmid \left({\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2\right)=\nmid {\mathsf{ϝ}}_1\oplus \nmid {\mathsf{ϝ}}_2$;	7. $\left({\nmid }_1{\nmid }_2\right)\mathsf{ϝ}={\nmid }_1\left({\nmid }_2\mathsf{ϝ}\right)$.
4. ${\left({\mathsf{ϝ}}_1\otimes {\mathsf{ϝ}}_2\right)}^{\nmid }={\mathsf{ϝ}}_1^{\nmid }\otimes {\mathsf{ϝ}}_2^{\nmid }$;

Proof. 

We establish the validity of components 1, 2, 3, and 4, and employ a similar approach for the remaining ones.

• It is obvious.
• ${\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2=\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}{ß-1}}\right)},& \\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}{ß-1}}\right)}& \end{array}\right),$
  Utilizing the Frank operational law (4) as defined in Definition 5, it can be deduced that
  $$\begin{array}{c}\nmid \left({\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2\right)=\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{log}_{ß}\left(1+\frac{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}{ß-1}\right)}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}}\right)},& \\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{log}_{ß}\left(1+\frac{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}{ß-1}\right)}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}}\right)}\end{array}\right)\\ =\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{\nmid }{\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{2\nmid -1}}}\right)},& \\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{\nmid }{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{2\nmid -1}}}\right)}\end{array}\right).\end{array}$$

  (8)
  Now,

  $$\begin{array}{c}\nmid {\mathsf{ϝ}}_1\oplus \nmid {\mathsf{ϝ}}_2=\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}}\right)}& \end{array}\right)\\ \oplus \left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}}\right)}& \end{array}\right)\\ =\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}\right)}-1\right)\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}\right)}-1\right)}{ß-1}}\right)},& \\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}\right)}-1\right)\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{\nmid -1}}\right)}-1\right)}{ß-1}}\right)}& \end{array}\right)\\ =\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{\nmid }{\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{2\nmid -1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{\nmid }{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{\nmid }}{{\left(ß-1\right)}^{2\nmid -1}}}\right)}& \end{array}\right).\end{array}$$

  (9)
  From Equations (8) and (9), we obtain $\nmid \left({\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2\right)=\nmid {\mathsf{ϝ}}_1\oplus \nmid {\mathsf{ϝ}}_2$.
• $$\begin{array}{l}{\nmid }_1\mathsf{ϝ}\oplus {\nmid }_2\mathsf{ϝ}=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_1}}{{\left(ß-1\right)}^{{\nmid }_1-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_1}}{{\left(ß-1\right)}^{{\nmid }_1-1}}}\right)}\end{array}\right)\\ \oplus \left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_1}}{{\left(ß-1\right)}^{{\nmid }_1-1}}\right)}-1\right)\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}\right)}-1\right)}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_1}}{{\left(ß-1\right)}^{{\nmid }_1-1}}\right)}-1\right)\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}\right)}-1\right)}{ß-1}}\right)}\end{array}\right)\\ =\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_1+{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_1+{\nmid }_2-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_1+{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_1+{\nmid }_2-1}}}\right)}& \end{array}\right).\end{array}$$

  (10)

  $$\left({\nmid }_1+{\nmid }_2\right)\mathsf{ϝ}=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_1+{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_1+{\nmid }_2-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_1+{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_1+{\nmid }_2-1}}}\right)}\end{array}\right).$$

  (11)
  Consequently, by utilizing Equations (10) and (11), the desired outcome is obtained.
• Since ${\nmid }_2\mathsf{ϝ}=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}}\right)}\end{array}\right).$
  Building upon this, we can additionally formulate
  ${\nmid }_1\left({\nmid }_2\mathsf{ϝ}\right)=\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}\right)}-1\right)}^{{\nmid }_1}}{{\left(ß-1\right)}^{{\nmid }_1-1}}}\right)},& \\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_2-1}}\right)}-1\right)}^{{\nmid }_1}}{{\left(ß-1\right)}^{{\nmid }_1-1}}}\right)}& \end{array}\right)$
  $=\left(\begin{array}{cc}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}^{p^{\ast }}}-1\right)}^{{\nmid }_1{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_1{\nmid }_2-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}^{q^{\ast }}}-1\right)}^{{\nmid }_1{\nmid }_2}}{{\left(ß-1\right)}^{{\nmid }_1{\nmid }_2-1}}}\right)}& \end{array}\right)$
   $\begin{array}{cc}=\left({\nmid }_1{\nmid }_2\right)\mathsf{ϝ}.& \end{array}$

□

Theorem 2. 

Let ${\mathsf{ϝ}}_1=\left({\mathit{ᵷ}}_1,{ʎ}_1\right)$ and ${\mathsf{ϝ}}_2=\left({\mathit{ᵷ}}_2,{ʎ}_2\right)$ be two ^p,qROFNs, then:

• ${\left({\mathsf{ϝ}}_1\oplus {\mathsf{ϝ}}_2\right)}^c={\mathsf{ϝ}}_1^c\otimes {\mathsf{ϝ}}_2^c$;
• ${\left({\mathsf{ϝ}}_1\otimes {\mathsf{ϝ}}_2\right)}^c={\mathsf{ϝ}}_1^c\oplus {\mathsf{ϝ}}_2^c$.

Proof. 

The proof is straightforward, and thus, omitted in this context. □

4. ^p,qROF Frank Aggregation Operators

Utilizing the presented Frank operation rules, we proceed to introduce a set of weighted aggregation operators for ^p,qROFNs.

4.1. ^p,qROF Frank Averaging Aggregation Operators

Definition 6. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs, then the ^p,qROF Frank weighted averaging operator (^p,qROFWA) is

$${}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\oplus }_{g=1}^j\left({⨇}_g{\mathsf{ϝ}}_g\right),$$

(12)

where $⨇={\left({⨇}_1,{⨇}_2,\dots ,{⨇}_j\right)}^T$ represents the weight vector associated with ${\mathsf{ϝ}}_g\left(g=1,2,..,j\right)$, satisfying the conditions ${⨇}_g>0$ and ${\displaystyle \sum _{g=1}^j}{⨇}_g=1$. In particular, when $⨇={\left({\displaystyle \frac{1}{j}},{\displaystyle \frac{1}{j}},\dots ,{\displaystyle \frac{1}{j}}\right)}^T$, the ^p,qROF Frank weighted averaging (^p,qROFFWA) operator simplifies to the ^p,qROF fuzzy Frank averaging (^p,qROFFA) operator of dimension j, given by

$${}^{p,q}ROFFA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\displaystyle \frac{1}{j}}{\oplus }_{g=1}^j\left({\mathsf{ϝ}}_g\right).$$

(13)

Theorem 3. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a collection of ^p,qROFNs, then the resultant output by employing the ^p,qROFFWA operator remains a ^p,qROFN, and

$${}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\sqrt[q^{\ast }]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\right).$$

(14)

Proof. 

We verify this through a mathematical induction process based on j.

For $j=2$, we have

$\begin{array}{c}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2\right)={⨇}_1{\mathsf{ϝ}}_1\oplus {⨇}_2{\mathsf{ϝ}}_2\end{array}$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}\right)\right)}-1\right)\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_2^t}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}\right)\right)}-1\right)}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}\right)\right)}-1\right)\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}\right)\right)}-1\right)}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}-1\right)\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}-1\right)}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left(1+\frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}-1\right)\left(1+\frac{{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}-1\right)}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}-1\right)\left(1+\frac{{\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}-1\right)}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left(1+\frac{{\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}-1\right)\left(1+\frac{{\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}-1\right)}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+\left({\left({ß}^{1-{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}\right)\left({\left({ß}^{1-{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{{⨇}_2}\right)\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+\left({\left({ß}^{{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}\right)\left({\left({ß}^{{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}\right)\right)}\end{array}\right).$

Hence, the outcome is valid when $j=2$.

Assuming that Equation (14) is true for $j=l$, let us examine the case where $j=l+1$, yielding

$\begin{array}{c}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_{l+1}\right){=}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_l\right)\oplus {⨇}_{l+1}{\mathsf{ϝ}}_{l+1}\end{array}$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)$

$\oplus \left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\frac{{\left({ß}^{1-{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\frac{{\left({ß}^{{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{1-{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{\displaystyle \sum _{g=1}^l}{⨇}_{\left(g\right)}-1}\left(ß-1\right){\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\displaystyle \sum _{g=1}^l}{\left(ß-1\right)}^{{⨇}_{\left(g\right)}-1}\left(ß-1\right){\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{1-{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{\displaystyle \sum _{g=1}^{l+1}}{⨇}_{\left(g\right)}-1}}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\displaystyle \sum _{g=1}^{l+1}}{\left(ß-1\right)}^{{⨇}_{\left(g\right)}-1}}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{1-{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}\right)}\end{array}\right)$

Thus, the outcome remains valid for $j=l+1$. As a result, using the approach of mathematical induction, the assertion presented in Equation (14) is applicable to all positive integer values of j. □

Example 1. 

Let ${\mathsf{ϝ}}_1=\left(0.5,0.4\right),$ ${\mathsf{ϝ}}_2=\left(0.4,0.3\right),$ and ${\mathsf{ϝ}}_3=\left(0.9,0.5\right)$ be three ^p,qROFNs, and $⨇={\left(0.4,0.3,0.3\right)}^T$ be the weight vector of ${\mathsf{ϝ}}_g\left(g=1,2,3\right)$. Suppose $\mathcal{ß}=3$, then by Definition 6 and Theorem 3, we can obtain ($p=3,q=2$)

$\begin{array}{c}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,{\mathsf{ϝ}}_3\right)=\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{1-{\mathit{ᵷ}}_g^3}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[2]{{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{{ʎ}_g^2}-1\right)}^{{⨇}_g}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\left({3^{1-0.5}}^3-1\right)}^{0.4}{\left({3^{1-0.4}}^3-1\right)}^{0.3}{\left({3^{1-0.9}}^3-1\right)}^{0.3}\right)},\\ \sqrt[2]{{log}_3\left(1+{\left({3^{0.4}}^2-1\right)}^{0.4}{\left({3^{0.3}}^2-1\right)}^{0.3}{\left({3^{0.5}}^2-1\right)}^{0.3}\right)}\end{array}\right)\\ =\left(0.7023,0.3940\right)\end{array}$

Theorem 4. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs, and let $\mathcal{ß}>1$. As $\mathcal{ß}⟶1$, the ^p,qROFFWA operator tends to the following limit:

$$\underset{\mathcal{ß}⟶1}{lim}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)}^{{⨇}_g}},\sqrt[q^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({ʎ}_g^{q^{\ast }}\right)}^{{⨇}_g}}\right).$$

(15)

Proof. 

As $ß⟶1$, then $\left({\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g},{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)⟶\left(0,0\right)$

Using the logarithmic property and the principle of infinitesimal changes, we obtain

${log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)={\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}⟶{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}$

${log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)={\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}⟶{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}$

Utilizing Taylor’s expansion formula, we obtain

${ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}=1+\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß+{\displaystyle \frac{{\left((1-{\mathit{ᵷ}}_g^{p^{\ast }})lnß\right)}^2}{2!}}+\dots$

${ß}^{{ʎ}_g^{q^{\ast }}}=1+\left({ʎ}_g^{q^{\ast }}\right)lnß+{\displaystyle \frac{{\left(\left({ʎ}_g^{q^{\ast }}\right)lnß\right)}^2}{2!}}+\dots$

Also, since $ß>1$, then $lnß>0,{ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}=1+\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß+O\left(lnß\right),$

${ß}^{{ʎ}_g^{q^{\ast }}}=1+\left({ʎ}_g^{q^{\ast }}\right)lnß+O\left(lnß\right).$

It follows that ${\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}⟶{\left(\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß\right)}^{{⨇}_g}$

${\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}⟶{\displaystyle \prod _{g=1}^j}\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right){\displaystyle \prod _{g=1}^j}{\left(lnß\right)}^{{⨇}_g}$

${\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}⟶{\displaystyle \prod _{g=1}^j}\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)ln{\left(ß\right)}^{{\displaystyle \sum _{g=1}^j}{⨇}_g}$

${\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}⟶{\displaystyle \prod _{g=1}^j}\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right).$

In a similar fashion, we can obtain ${\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}⟶{\displaystyle \prod _{g=1}^j}\left({ʎ}_g^{q^{\ast }}\right)$.

Then, we have

$\begin{array}{c}\underset{ß⟶1}{lim}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\underset{ß⟶1}{lim}\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)\\ =\underset{ß⟶1}{lim}\left(\sqrt[p^{\ast }]{1-{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}},\right.\left.\sqrt[q^{\ast }]{{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}}\right)\\ =\underset{ß⟶1}{lim}\left(\sqrt[p^{\ast }]{1-{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}},\right.\left.\sqrt[q^{\ast }]{{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}}\right)\\ =\left(\sqrt[p^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)}^{{⨇}_g}},\sqrt[q^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({ʎ}_g^{q^{\ast }}\right)}^{{⨇}_g}}\right)\end{array}$

which completes the proof. □

Theorem 5. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a collection of ^p,qROFNs, and let $ß>1$. As $ß⟶\infty$, the ^p,qROFFWA operator tends to the following limit:

$$\underset{ß⟶\infty }{lim}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right)\right)}\right).$$

(16)

Proof. 

In accordance with Theorem 3, we obtain

$\begin{array}{c}\underset{ß⟶\infty }{lim}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\end{array}$

$=\left(\begin{array}{c}\underset{ß⟶\infty }{lim}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \underset{ß⟶\infty }{lim}\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)$

Applying limit rules, the logarithmic transformation, and L’Hôpital’s rule, we can deduce that

$\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}},\\ \sqrt[q^{\ast }]{\underset{ß⟶\infty }{lim}{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{\frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)\frac{{ß}^{-{\mathit{ᵷ}}_g^{p^{\ast }}}}{{ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1}\right)}{\frac{1}{ß}}}},\\ \sqrt[q^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{\frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right)\frac{{ß}^{{ʎ}_g^{q^{\ast }}-1}}{{ß}^{{ʎ}_g^{q^{\ast }}}-1}\right)}{\frac{1}{ß}}}},\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right){\displaystyle \frac{{ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}}{{ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1}}\right)},\\ \sqrt[q^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right){\displaystyle \frac{{ß}^{{ʎ}_g^{q^{\ast }}}}{{ß}^{{ʎ}_g^{q^{\ast }}}-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left(1-{\mathit{ᵷ}}_g^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right)\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right)\right)}\end{array}\right)$

which ends the validation of Theorem 5. □

Theorem 6. 

(Idempotency) Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ be a series of ^p,qROFNs; if ${\mathsf{ϝ}}_g={\mathsf{ϝ}}_0\forall g$, then

$${}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\mathsf{ϝ}}_0.$$

(17)

Proof. 

Since for all g${\mathsf{ϝ}}_g={\mathsf{ϝ}}_0=\left({\mathit{ᵷ}}_0,{ʎ}_0\right)$, and ${\displaystyle \sum _{g=1}^j}{⨇}_g=1$, according to Theorem 3, we obtain

$\begin{array}{c}{}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\end{array}$

$\left(\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_0^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_0^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)$

$=\left(\sqrt[p^{\ast }]{1-{log}_{ß}{ß}^{1-{\mathit{ᵷ}}_0^{p^{\ast }}}},\sqrt[q^{\ast }]{{log}_{ß}{ß}^{{ʎ}_0^{q^{\ast }}}}\right)=\left({\mathit{ᵷ}}_0,{ʎ}_0\right)={\mathsf{ϝ}}_0.$

Hence, the proof is finalized. □

Theorem 7. 

(Monotonicity) Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ and ${\dot{\mathsf{ϝ}}}_g=\left({\dot{\mathit{ᵷ}}}_g,{\dot{ʎ}}_g\right)\left(g=1,2,\dots ,j\right)$ be two families of ^p,qROFNs such that ${\mathit{ᵷ}}_g\ge {\dot{\mathit{ᵷ}}}_g,$ and ${ʎ}_g\le {\dot{ʎ}}_g\forall g$, then

$${}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right){\ge }^{p,q}ROFFWA\left({\dot{\mathsf{ϝ}}}_1,{\dot{\mathsf{ϝ}}}_2,\dots ,{\dot{\mathsf{ϝ}}}_j\right).$$

(18)

Proof. 

As per Definition 3, when and ${ʎ}_g\le {\dot{ʎ}}_g\forall g$, then

$\begin{array}{c}\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}\ge \sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\dot{\mathit{ᵷ}}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}\\ \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\le \sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\dot{ʎ}}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}$

Thus, $S\left({}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\right)\ge S\left({}^{p,q}ROFFWA\left({\dot{\mathsf{ϝ}}}_1,{\dot{\mathsf{ϝ}}}_2,\dots ,{\dot{\mathsf{ϝ}}}_j\right)\right)$

Hence, ${}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right){\ge }^{p,q}ROFFWA\left({\dot{\mathsf{ϝ}}}_1,{\dot{\mathsf{ϝ}}}_2,\dots ,{\dot{\mathsf{ϝ}}}_j\right).$ □

Theorem 8. 

(Boundedness) Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ be a series of ^p,qROFNs, and let ${\mathsf{ϝ}}^{-}=\left(\underset{1\le g\le j}{min}{\mathit{ᵷ}}_g,\underset{1\le g\le j}{max}{ʎ}_g\right)$, ${\mathsf{ϝ}}^{+}=\left(\underset{1\le g\le j}{max}{\mathit{ᵷ}}_g,\underset{1\le g\le j}{min}{ʎ}_g\right)$, then

$${\mathsf{ϝ}}^{-}{\le }^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\le {\mathsf{ϝ}}^{+}.$$

(19)

Proof. 

Since for all $g,$ $\underset{1\le g\le j}{min}{\mathit{ᵷ}}_g\le {\mathit{ᵷ}}_g\le \underset{1\le g\le j}{max}{\mathit{ᵷ}}_g$, and $\underset{1\le g\le j}{min}{ʎ}_g\le {ʎ}_g\le \underset{1\le g\le j}{max}{ʎ}_g$, thus, relying on idempotency and monotonicity, we obtain

${\mathsf{ϝ}}^{-}{\le }^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\le {\mathsf{ϝ}}^{+}$. □

Theorem 9. 

(Shift-invariance) Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs and $\dot{\mathsf{ϝ}}=\left(\dot{\mathit{ᵷ}},\dot{ʎ}\right)$ as another ^p,qROFN. In this context, we have

$${}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1\oplus \dot{\mathsf{ϝ}},{\mathsf{ϝ}}_2\oplus \dot{\mathsf{ϝ}},\dots ,{\mathsf{ϝ}}_j\oplus \dot{\mathsf{ϝ}}\right){=}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\oplus \dot{\mathsf{ϝ}}.$$

(20)

Theorem 10. 

(Homogeneity) Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs, and let $\nmid >0$ be any real number. Then, we have

$${}^{p,q}ROFFWA\left(\nmid {\mathsf{ϝ}}_1,\nmid {\mathsf{ϝ}}_2,\dots ,\nmid {\mathsf{ϝ}}_j\right)={\nmid }^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right).$$

(21)

The derivation of the aforementioned two theorems can be straightforwardly obtained through the presented Frank operational rules of ^p,qROFNs. However, due to space limitations, we omit the proof here.

Using the operational rules outlined in Definition 5, we analyze the subsequent results.

Definition 7. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs. Then, the ^p,qROF fuzzy Frank ordered weighted averaging (^p,qROFFOWA) operator is defined as follows:

$${}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\oplus }_{g=1}^j\left({\propto }_g{\mathsf{ϝ}}_{\delta \left(g\right)}\right),$$

(22)

where $\propto ={\left({\propto }_1,{\propto }_2,\dots ,{\propto }_j\right)}^T$ is the position weights of ${\mathsf{ϝ}}_g\left(g=1,2,..,j\right)$ satisfying ${\propto }_g>0$ and ${\displaystyle \sum _{g=1}^j}{\propto }_g=1$. $\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(j\right)\right)$ is a permutation of $\left(1,2,3,\dots ,j\right)$, so that ${\mathsf{ϝ}}_{\rho (g-1)}\ge {\mathsf{ϝ}}_{\rho \left(g\right)}$ for $g=2,3..,j.$

Theorem 11. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs. Then, the result attained through the ^p,qROFFOWA operator remains a ^p,qROFN, and

$${}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\mathit{ᵷ}}_{\rho \left(g\right)}^{p^{\ast }}}-1\right)}^{{\propto }_g}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{ʎ}_{\rho \left(g\right)}^{q^{\ast }}}-1\right)}^{{\propto }_g}\right)}\right).$$

(23)

Proof. 

The proof of this outcome follows an analogous approach to that of Theorem 3, and is therefore omitted here. □

Example 2. 

Let ${\mathsf{ϝ}}_1=\left(0.5,0.4\right),$ ${\mathsf{ϝ}}_2=\left(0.4,0.3\right),$ and ${\mathsf{ϝ}}_3=\left(0.9,0.5\right)$ represent three ^p,qROFNs. In accordance with Definition 3, when $p=3,q=2$,

$S\left({\mathsf{ϝ}}_1\right)=0.4825.\mathrm{}$, $S\left({\mathsf{ϝ}}_2\right)=0.4870,$$S\left({\mathsf{ϝ}}_3\right)=0.7395$

Since $S\left({\mathsf{ϝ}}_3\right)>S\left({\mathsf{ϝ}}_2\right)>S\left({\mathsf{ϝ}}_1\right)$, we have ${\mathsf{ϝ}}_{\rho \left(1\right)}=\left(0.9,0.5\right)$, ${\mathsf{ϝ}}_{\rho \left(2\right)}=\left(0.4,0.3\right)$, and ${\mathsf{ϝ}}_{\rho \left(3\right)}=\left(0.5,0.4\right)$, and $\propto ={\left(0.3,0.4,0.3\right)}^T$ is the weight vector associated with the ^p,qROFFOWA operator. Assuming $ß=3$, then based on Definition 7 and Theorem 11, we can derive

$\begin{array}{c}{}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_{\rho \left(1\right)},{\mathsf{ϝ}}_{\rho \left(2\right)},{\mathsf{ϝ}}_{\rho \left(3\right)}\right)=\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{1-{\mathit{ᵷ}}_{\rho \left(g\right)}^3}-1\right)}^{{\propto }_g}\right)},\\ \sqrt[2]{{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{{ʎ}_{\rho \left(g\right)}^2}-1\right)}^{{\propto }_g}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\left({3^{1-0.9}}^3-1\right)}^{0.3}{\left({3^{1-0.4}}^3-1\right)}^{0.4}{\left({3^{1-0.5}}^3-1\right)}^{0.3}\right)},\\ \sqrt[2]{{log}_3\left(1+{\left({3^{0.5}}^2-1\right)}^{0.3}{\left({3^{0.3}}^2-1\right)}^{0.4}{\left({3^{0.4}}^2-1\right)}^{0.3}\right)}\end{array}\right)\\ =\left(0.6987,0.3820\right).\end{array}$

Theorem 12. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs, and let $ß>1$. As $ß⟶1$, the ^p,qROFFOWA operator tends towards the following limit:

$$\underset{\mathcal{ß}⟶1}{lim}{}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{\mathit{ᵷ}}_{\rho \left(g\right)}^{p^{\ast }}\right)}^{{\propto }_g}},\sqrt[q^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({ʎ}_{\rho \left(g\right)}^{q^{\ast }}\right)}^{{\propto }_g}}\right).$$

(24)

Theorem 13. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs, and let $\mathcal{ß}>1$. As $\mathcal{ß}⟶\infty$, the ^p,qROFFOWA operator converges to the following limit:

$$\underset{\mathcal{ß}⟶\infty }{lim}{}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({\mathit{ᵷ}}_{\rho \left(g\right)}^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({ʎ}_{\rho \left(g\right)}^{q^{\ast }}\right)\right)}\right).$$

(25)

Just like the ^p,qROFFWA operator, the ^p,qROFFOWA operator exhibits properties of boundedness, idempotency, monotonicity, shift-invariance, and homogeneity. In addition to these properties, the ^p,qROFFOWA operator presents the following desired outcomes.

Theorem 14. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs. Then, the following holds:

(i) 

If $\propto ={\left(1,0,\dots ,0\right)}^T$, then ${}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=max\left\{{\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right\}$.

(ii) 

If $\propto ={\left(0,0,\dots ,1\right)}^T$, then ${}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=min\left\{{\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right\}$.

(iii) 

If ${\propto }_g=1$ and ${⨇}_i=0\left(i\ne g\right)$, then ${}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\mathsf{ϝ}}_{\rho \left(g\right)}$, where ${\mathsf{ϝ}}_{\rho \left(g\right)}$ is the gth largest of ${\mathsf{ϝ}}_g,\left(g=1,2,\dots ,j\right)$.

Considering the definitions of the ^p,qROFFWA and ^p,qROFFOWA operators, it becomes evident that the ^p,qROFFWA operator solely assigns weights to the ^p,qROFNs, while the ^p,qROFFOWA operator exclusively focuses on the ordered positions of ^p,qROFNs. In practical real-world scenarios, both aspects should be simultaneously considered. To overcome this limitation, we introduce a solution by defining a hybrid averaging operator [26] based on the Frank t-norm and t-conorm. This hybrid operator takes into account both the provided ^p,qROFNs and their corresponding ordered positions.

Definition 8. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a set of ^p,qROFNs. In this context, the ^p,qROF fuzzy Frank hybrid averaging (^p,qROFFHA) operator is defined as follows:

$${}^{p,q}ROFFHA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\oplus }_{g=1}^j\left({\propto }_g{\widehat{\mathsf{ϝ}}}_{\rho \left(g\right)}\right),$$

(26)

where $\propto ={\left({\propto }_1,{\propto }_2,\dots ,{\propto }_j\right)}^T$ is the weight vector associated with ^p,qROFFHA such that ${\propto }_g>0$ and ${\displaystyle \sum _{g=1}^j}{\propto }_g=1$, $⨇={\left({⨇}_1,{⨇}_2,\dots ,{⨇}_j\right)}^T$ is the weight vector of ${\mathsf{ϝ}}_g\left(g=1,2,..,j\right)$ such that ${⨇}_g>0$ and ${\displaystyle \sum _{g=1}^j}{⨇}_g=1$. ${\widehat{\mathsf{ϝ}}}_{\rho \left(g\right)}$ is the gth largest of the weighted ^p,qROFNs ${\widehat{\mathsf{ϝ}}}_g\left({\widehat{\mathsf{ϝ}}}_g=\left(j{⨇}_g\right){\mathsf{ϝ}}_g,g=1,2,..,j\right)$ and j is the balancing coefficient.

Theorem 15. 

Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ be a series of ^p,qROFNs, then the result obtained by using the ^p,qROFFHA operator is still a ^p,qROFN, and

$${}^{p,q}ROFFHA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^{p^{\ast }}}-1\right)}^{{\propto }_g}\right)},\sqrt[q^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\widehat{ʎ}}_{\rho \left(g\right)}^{q^{\ast }}}-1\right)}^{{\propto }_g}\right)}\right),$$

(27)

Proof. 

The proof needed for this result resembles that of Theorem 3, thus we skip it here. □

Example 3. 

Let ${\mathsf{ϝ}}_1=\left(0.5,0.4\right),$ ${\mathsf{ϝ}}_2=\left(0.4,0.3\right)$, and ${\mathsf{ϝ}}_3=\left(0.9,0.5\right),$ be three ^p,qROFNs (p = 3, q = 2), and $⨇={\left(0.4,0.4,0.2\right)}^T$ be the weight vector of ${\mathsf{ϝ}}_g\left(g=1,2,3\right)$. Assume $ß=3$, then, based on Definition 5, we can compute the weighted ^p,qROFNs:

$\begin{array}{c}{\widehat{\mathsf{ϝ}}}_1=3\times 0.4\times {\mathsf{ϝ}}_1=\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\displaystyle \frac{{\left(3^{1-{0.5}^4}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)},\\ \sqrt[2]{{log}_3\left(1+{\displaystyle \frac{{\left(3^{{0.4}^4}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)}\end{array}\right)=\left(0.4210,0.1042\right);\\ {\widehat{\mathsf{ϝ}}}_2=3\times 0.4\times {\mathsf{ϝ}}_2=\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\displaystyle \frac{{\left(3^{1-{0.4}^4}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)},\\ \sqrt[2]{{log}_3\left(1+{\displaystyle \frac{{\left(3^{{0.3}^4}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)}\end{array}\right)=\left(0.3124,0.0522\right);\end{array}$

$\begin{array}{c}{\widehat{\mathsf{ϝ}}}_3=3\times 0.2\times {\mathsf{ϝ}}_3=\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\displaystyle \frac{{\left(3^{1-{0.9}^4}-1\right)}^{3\times 0.2}}{{\left(3-1\right)}^{3\times 0.2-1}}}\right)},\\ \sqrt[2]{{log}_3\left(1+{\displaystyle \frac{{\left(3^{{0.5}^4}-1\right)}^{3\times 0.2}}{{\left(3-1\right)}^{3\times 0.2-1}}}\right)}\end{array}\right)=\left(0.7671,0.4665\right).\end{array}$

As per Definition 3, we can acquire the score of ${\widehat{\mathsf{ϝ}}}_g\left(g=1,2,3\right):$

$S\left({\widehat{\mathsf{ϝ}}}_1\right)=0.5319$, $S\left({\widehat{\mathsf{ϝ}}}_2\right)=0.5139,$$S\left(\widehat{{\mathsf{ϝ}}_3}\right)=0.6169.$

Since $S\left({\widehat{\mathsf{ϝ}}}_3\right)>S\left({\widehat{\mathsf{ϝ}}}_2\right)>S\left({\widehat{\mathsf{ϝ}}}_1\right)$, we have ${\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}=\left(0.7671,0.4665\right),$ ${\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}=\left(0.3124,0.0522\right),$ and ${\widehat{\mathsf{ϝ}}}_{\rho \left(3\right)}=\left(0.4210,0.1042\right).$ Suppose $\propto ={\left(0.3,0.4,0.3\right)}^T$ is the weight vector associated with the ^p,qROFFHA operator, we can deduce from Definition 8 and Theorem 15 the following outcome:

$\begin{array}{c}{}^{p,q}ROFFHA\left({\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)},{\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)},{\widehat{\mathsf{ϝ}}}_{\rho \left(3\right)}\right)=\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{1-{\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^3}-1\right)}^{{\propto }_g}\right)},\\ \sqrt[2]{{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{{\widehat{ʎ}}_{\rho \left(g\right)}^2}-1\right)}^{{\propto }_g}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[3]{1-{log}_3\left(1+{\left({3^{1-0.7671}}^3-1\right)}^{0.3}{\left({3^{1-0.3124}}^3-1\right)}^{0.4}{\left({3^{1-0.4210}}^3-1\right)}^{0.3}\right)},\\ \sqrt[2]{{log}_3\left(1+{\left({3^{0.4665}}^2-1\right)}^{0.3}{\left({3^{0.0522}}^2-1\right)}^{0.4}{\left({3^{0.1042}}^2-1\right)}^{0.3}\right)}\end{array}\right)\\ =\left(0.5702,0.1274\right).\end{array}$

Theorem 16. 

Consider a series of ^p,qROFNs denoted as ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$, where $g=1,2,\dots ,j$, and let $ß>1$. As ß approaches 1, the ^p,qROFFHA operator tends towards the following limit:

$$\underset{ß⟶1}{lim}{}^{p,q}ROFFHA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^{p^{\ast }}\right)}^{{\propto }_g}},\sqrt[q^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({\widehat{ʎ}}_{\rho \left(g\right)}^{q^{\ast }}\right)}^{{\propto }_g}}\right).$$

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Theorem 17. 

Consider the set ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$, where $g=1,2,\dots ,j$, constitutes a set of ^p,qROFNs. Additionally, let $ß>1$. When ß tends towards infinity, the ^p,qROFFHA operator converges to the following limit:

$$\underset{ß⟶\infty }{lim}{}^{p,q}ROFFHA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({\widehat{ʎ}}_{\rho \left(g\right)}^{q^{\ast }}\right)\right)}\right).$$

(29)

Just like the properties observed in the ^p,qROFFWA operator, the ^p,qROFFHA operator exhibits characteristics of boundedness, idempotency, monotonicity, shift-invariance, and homogeneity. In addition to these traits, the ^p,qROFFHA operator presents the following special cases:

Corollary 1. 

The ^p,qROFFWA operator is a special case of the ^p,qROFFHA operator.

Proof. 

Let $\propto ={\left({\displaystyle \frac{1}{j}},{\displaystyle \frac{1}{j}},\dots ,{\displaystyle \frac{1}{j}}\right)}^T,$ then

 $\begin{array}{c}{}^{p,q}ROFFHA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\propto }_1{\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}\oplus {\propto }_2{\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}\oplus \dots \oplus {\propto }_j{\widehat{\mathsf{ϝ}}}_{\rho \left(j\right)}\\ ={\displaystyle \frac{1}{j}}\left({\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}\oplus {\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}\oplus \dots \oplus {\widehat{\mathsf{ϝ}}}_{\rho \left(j\right)}\right)={⨇}_1{\mathsf{ϝ}}_1\oplus {⨇}_2{\mathsf{ϝ}}_2\oplus \dots \oplus {⨇}_j{\mathsf{ϝ}}_j\\ {=}^{p,q}ROFFWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right).\end{array}$□

Corollary 2. 

The ^p,qROFFOWA operator is a special case of the ^p,qROFFHA operator.

Proof. 

Let $⨇={\left({\displaystyle \frac{1}{j}},{\displaystyle \frac{1}{j}},\dots ,{\displaystyle \frac{1}{j}}\right)}^T,$ then

  $\begin{array}{c}{}^{p,q}ROFFHA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\propto }_1{\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}\oplus {\propto }_2{\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}\oplus \dots \oplus {\propto }_j{\widehat{\mathsf{ϝ}}}_{\rho \left(j\right)}\\ ={\propto }_1{\mathsf{ϝ}}_{\rho \left(1\right)}\oplus {\propto }_2{\mathsf{ϝ}}_{\rho \left(2\right)}\oplus \dots \oplus {\propto }_j{\mathsf{ϝ}}_{\rho \left(j\right)}{=}^{p,q}ROFFOWA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right).\end{array}$□

4.2. ^p,qROF Frank Geometric Aggregation Operators

The present section is dedicated to presenting a sequence of ^p,qROF fuzzy Frank geometric aggregation operators, which stem from the introduced Frank operations. Within the realm of geometric aggregation operators, we delve into an exploration of the ^p,qROFFWG, ^p,qROFFOWG, and ^p,qROFFHWG operators. Additionally, we provide fundamental definitions, observations, and outcomes, including corollaries, associated with these operators that are grounded in the utilization of the Frank t-norm and t-conorm.

Definition 9. 

Consider the collection ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$, where $g=1,2,\dots ,j$ represents a set of ^p,qROFNs. In this context, the ^p,qROF fuzzy Frank weighted geometric operator, denoted as (^p,qROFFWG), is defined as follows:

$${}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\otimes }_{g=1}^j{\left({\mathsf{ϝ}}_g\right)}^{{⨇}_g},$$

(30)

where $⨇={\left({⨇}_1,{⨇}_2,\dots ,{⨇}_j\right)}^T$ represents the weight vector of ${\mathsf{ϝ}}_g\left(g=1,2,..,j\right)$, wherein ${⨇}_g>0$ and the sum over g from 1 to j yields ${\displaystyle \sum _{g=1}^j}{⨇}_g=1$. Notably, if $⨇={\left({\displaystyle \frac{1}{j}},{\displaystyle \frac{1}{j}},\dots ,{\displaystyle \frac{1}{j}}\right)}^T$, then the ^p,qROFFWG operator simplifies to the ^p,qROF fuzzy Frank geometric (^p,qROFFG) operator of dimension j, and its formulation is as follows:

$${}^{p,q}ROFFA\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\otimes }_{g=1}^j{\left({\mathsf{ϝ}}_g\right)}^{{\displaystyle \frac{1}{j}}}.$$

(31)

Theorem 18. 

Consider the set ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$, where $g=1,2,\dots ,j$ denotes a series of ^p,qROFNs. Then, the result yielded by utilizing the ^p,qROFFWG operator remains a ^p,qROFN, and this outcome can be expressed as

$${}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\sqrt[q^{\ast }]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\right).$$

(32)

Proof. 

We establish its validity through mathematical induction based on the value of j.

When $j=2$, we observe that

$\begin{array}{c}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2\right)={\mathsf{ϝ}}_1^{{⨇}_1}\otimes {\mathsf{ϝ}}_2^{{⨇}_2}\end{array}$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}\right)\right)}-1\right)\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}\right)\right)}-1\right)}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{1-{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}\right)\right)}-1\right)\left({ß}^{1-\left(1-{log}_{ß}\left(1+\frac{{\left({ß}^{1-{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}\right)\right)}-1\right)}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{\left(1+\frac{{\left({ß}^{{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}-1\right)\left(1+\frac{{\left({ß}^{{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}-1\right)}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{\left(1+\frac{{\left({ß}^{1-{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}}{{\left(ß-1\right)}^{{⨇}_1-1}}-1\right)\left(1+\frac{{\left({ß}^{1-{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}}{{\left(ß-1\right)}^{{⨇}_2-1}}-1\right)}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+\left({\left({ß}^{{\mathit{ᵷ}}_1^{p^{\ast }}}-1\right)}^{{⨇}_1}\right)\left({\left({ß}^{{\mathit{ᵷ}}_2^{p^{\ast }}}-1\right)}^{{⨇}_2}\right)\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+\left({\left({ß}^{1-{ʎ}_1^{q^{\ast }}}-1\right)}^{{⨇}_1}\right)\left({\left({ß}^{1-{ʎ}_2^{q^{\ast }}}-1\right)}^{{⨇}_2}\right)\right)}\end{array}\right).$

Hence, the result remains valid for $j=2$.

Assuming that Equation (32) is true for $j=l$, we can demonstrate that for $j=l+1$ the following applies:

$\begin{array}{c}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_{l+1}\right){=}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_l\right)\otimes {\mathsf{ϝ}}_{l+1}^{{⨇}_{l+1}}\end{array}$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)$$\otimes \left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\left({ß}^{1-{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\frac{{\left({ß}^{{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}{ß-1}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\frac{{\left({ß}^{1-{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}{ß-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{\displaystyle \sum _{g=1}^l}{⨇}_{\left(g\right)}-1}\left(ß-1\right){\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{1-{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{\displaystyle \sum _{g=1}^l}{⨇}_{\left(g\right)}-1}\left(ß-1\right){\left(ß-1\right)}^{{⨇}_{(l+1)}-1}}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{\displaystyle \sum _{g=1}^{l+1}}{⨇}_{\left(g\right)}-1}}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \frac{{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{1-{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}}{{\left(ß-1\right)}^{{\displaystyle \sum _{g=1}^{l+1}}{⨇}_{\left(g\right)}-1}}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{{\mathit{ᵷ}}_{(l+1)}^{p^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^l}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}{\left({ß}^{1-{ʎ}_{(l+1)}^{q^{\ast }}}-1\right)}^{{⨇}_{(l+1)}}\right)}\end{array}\right).$

Therefore, the outcome is valid for $j=l+1$. Consequently, by employing the approach of mathematical induction, the result as presented in Equation (32) is established for all positive integers j. □

Example 4. 

(Continued from Example 1) In accordance with Definition 9 and Theorem 18, it can be deduced that

$\begin{array}{c}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,{\mathsf{ϝ}}_3\right)=\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\prod }_{g=1}^3{\left(3^{{\mathit{ᵷ}}_g^3}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\prod }_{g=1}^3{\left(3^{1-{ʎ}_g^2}-1\right)}^{{⨇}_g}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\left({3^{0.5}}^3-1\right)}^{0.4}{\left({3^{0.4}}^3-1\right)}^{0.3}{\left({3^{0.9}}^3-1\right)}^{0.3}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\left({3^{1-0.4}}^2-1\right)}^{0.4}{\left({3^{1-0.3}}^2-1\right)}^{0.3}{\left({3^{1-0.5}}^2-1\right)}^{0.3}\right)}\end{array}\right)=\left(0.5694,0.4095\right).\end{array}$

Theorem 19. 

Consider the set ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$ for $g=1,2,\dots ,j$, which constitutes a series of ^p,qROFNs. Additionally, let $ß>1$. As ß approaches 1, the ^p,qROFFWG operator converges towards the following limit:

$$\underset{ß⟶1}{lim}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)}^{{⨇}_g}},\sqrt[q^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{ʎ}_g^{q^{\ast }}\right)}^{{⨇}_g}}\right).$$

(33)

Proof. 

As ß approaches 1, the expression $\left({\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}^{p^{\ast }}g}-1\right)}^{{⨇}_g},{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}^{q^{\ast }}g}-1\right)}^{{⨇}_g}\right)$ tends to converge to the point $\left(0.0\right)$ due to the logarithmic property and the application of the rule of infinitesimal changes. This can be expressed as

${log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)={\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}⟶{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}$

${log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)={\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}⟶{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}$

Utilizing Taylor’s expansion formula, we obtain

${ß}^{{\vartheta }_g^t}=1+\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß+{\displaystyle \frac{{\left(\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß\right)}^2}{2!}}+\dots$

${ß}^{1-{\mathit{ᵷ}}_g^{p^{\ast }}}=1+\left(1-{ʎ}_g^{q^{\ast }}\right)lnß+{\displaystyle \frac{{\left((1-{ʎ}_g^{q^{\ast }})lnß\right)}^2}{2!}}+\dots$

Also, since $ß>1$, then $lnß>0,{ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}=1+\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß+O\left(lnß\right),$ ${ß}^{1-{ʎ}_g^{q^{\ast }}}=1+\left(1-{ʎ}_g^{q^{\ast }}\right)lnß+O\left(lnß\right).$

It follows that ${\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}⟶{\left(\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)lnß\right)}^{{⨇}_g}$

${\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}⟶{\displaystyle \prod _{g=1}^j}\left({\mathit{ᵷ}}_g^{p^{\ast }}\right){\displaystyle \prod _{g=1}^j}{\left(lnß\right)}^{{⨇}_g}$

${\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}⟶{\displaystyle \prod _{g=1}^j}\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)ln{\left(ß\right)}^{{\displaystyle \sum _{g=1}^j}{⨇}_g}$

${\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}⟶{\displaystyle \prod _{g=1}^j}\left({\mathit{ᵷ}}_g^{p^{\ast }}\right).$

Similarly, we can acquire

${\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}⟶{\displaystyle \prod _{g=1}^j}\left(1-{ʎ}_g^{q^{\ast }}\right).$

Then, we have

$\begin{array}{c}\underset{ß⟶1}{lim}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\underset{ß⟶1}{lim}\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)\\ =\underset{ß⟶1}{lim}\left(\sqrt[p^{\ast }]{{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}},\right.\left.\sqrt[q^{\ast }]{1-{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}}\right)\\ =\underset{ß⟶1}{lim}\left(\sqrt[p^{\ast }]{{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}},\right.\left.\sqrt[q^{\ast }]{1-{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\vartheta }_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{lnß}}}\right)\\ =\left(\sqrt[p^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)}^{{⨇}_g}},\sqrt[q^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{ʎ}_g^{q^{\ast }}\right)}^{{⨇}_g}}\right)\end{array}$

which ends the proof. □

Theorem 20. 

Consider the set ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$ for $g=1,2,\dots ,j$, forming a series of ^p,qROFNs. Additionally, let $\mathcal{ß}>1$. As $\mathcal{ß}⟶\infty$, the ^p,qROFFWG operator approaches the following limit:

$$\underset{\mathcal{ß}⟶\infty }{lim}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right)\right)}\right).$$

(34)

Proof. 

In accordance with Theorem 18, it follows that

$\begin{array}{c}\underset{ß⟶\infty }{lim}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\end{array}$$\left(\begin{array}{c}\underset{ß⟶\infty }{lim}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \underset{ß⟶\infty }{lim}\sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right).$

Utilizing limit principles, logarithmic transform, and L’Hôpital’s rule, it can be deduced that

$\left(\begin{array}{c}\sqrt[p^{\ast }]{\underset{ß⟶\infty }{lim}{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}},\\ \sqrt[q^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{ln\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}{lnß}}}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{\frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\vartheta }_g^t\right)\frac{{ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}-1}}{{ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1}\right)}{\frac{1}{ß}}}},\\ \sqrt[q^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{\frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left(1-{ʎ}_g^{q^{\ast }}\right)\frac{{ß}^{-{ʎ}_g^{q^{\ast }}}}{{ß}^{1-{ʎ}_g^{q^{\ast }}}-1}\right)}{\frac{1}{ß}}}},\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\mathit{ᵷ}}_g^{p^{\ast }}\right){\displaystyle \frac{{ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}}{{ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1}}\right)},\\ \sqrt[q^{\ast }]{1-\underset{ß⟶\infty }{lim}{\displaystyle \frac{{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}{1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}}}\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left(1-{ʎ}_g^{q^{\ast }}\right){\displaystyle \frac{{ß}^{1-{ʎ}_g^{q^{\ast }}}}{{ß}^{1-{ʎ}_g^{q^{\ast }}}-1}}\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left(1-{ʎ}_g^{q^{\ast }}\right)\right)}\end{array}\right)$

$=\left(\begin{array}{c}\sqrt[p^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({\mathit{ᵷ}}_g^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{⨇}_g\left({ʎ}_g^{q^{\ast }}\right)\right)}\end{array}\right)$

which ends the proof of Theorem 18. □

Theorem 21. 

(Idempotency) Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)$, where $g=1,2,\dots ,j$ is a set of ^p,qROFNs. If ${\mathsf{ϝ}}_g={\mathsf{ϝ}}_0$ for all values of g, then

$${}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\mathsf{ϝ}}_0.$$

(35)

Proof. 

Given that ${\mathsf{ϝ}}_g={\mathsf{ϝ}}_0=\left({\mathit{ᵷ}}_0,{ʎ}_0\right)$ holds for all values of g, and ${\displaystyle \sum _{g=1}^j}{⨇}_g=1$, it follows from Theorem 18 that

$\begin{array}{c}{}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\end{array}$$\left(\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_0^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)},\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_0^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}\right)$

           $\begin{array}{cc}& =\left(\sqrt[p^{\ast }]{{log}_{ß}{ß}^{{\mathit{ᵷ}}_0^{p^{\ast }}}},\sqrt[q^{\ast }]{1-{log}_{ß}{ß}^{1-{ʎ}_0^{q^{\ast }}}}\right)=\left({\mathit{ᵷ}}_0,{ʎ}_0\right)={\mathsf{ϝ}}_0.\end{array}$

Hence, the proof is concluded. □

Theorem 22. 

(Monotonicity) Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ and ${\dot{\mathsf{ϝ}}}_g=\left({\dot{\mathit{ᵷ}}}_g,{\dot{ʎ}}_g\right)$ $\left(g=1,2,\dots ,j\right)$, two classes of ^p,qROFNs, so that ${\mathit{ᵷ}}_g\ge {\dot{\mathit{ᵷ}}}_g$ and ${ʎ}_g\le {\dot{ʎ}}_g\forall g$, then

$${}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\ge {}^{p,q}ROFFWG\left({\dot{\mathsf{ϝ}}}_1,{\dot{\mathsf{ϝ}}}_2,\dots ,{\dot{\mathsf{ϝ}}}_j\right).$$

(36)

Proof. 

As per Definition 3, when ${\mathit{ᵷ}}_g\ge {\dot{\mathit{ᵷ}}}_g$ and ${ʎ}_g\le {\dot{ʎ}}_g\forall g$, then

$\begin{array}{c}\sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\mathit{ᵷ}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}\le \sqrt[p^{\ast }]{{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{{\dot{\mathit{ᵷ}}}_g^{p^{\ast }}}-1\right)}^{{⨇}_g}\right)}\\ \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{ʎ}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\ge \sqrt[q^{\ast }]{1-{log}_{ß}\left(1+{\displaystyle \prod _{g=1}^j}{\left({ß}^{1-{\dot{ʎ}}_g^{q^{\ast }}}-1\right)}^{{⨇}_g}\right)}\end{array}$

Thus, $S\left({}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\right)\ge S\left({}^{p,q}ROFFWG\left({\dot{\mathsf{ϝ}}}_1,{\dot{\mathsf{ϝ}}}_2,\dots ,{\dot{\mathsf{ϝ}}}_j\right)\right)$.

Hence, ${}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right){\ge }^{p,q}ROFFWG\left({\dot{\mathsf{ϝ}}}_1,{\dot{\mathsf{ϝ}}}_2,\dots ,{\dot{\mathsf{ϝ}}}_j\right).$ □

Theorem 23. 

(Boundedness) Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ is a class of ^p,qROFNs, and let ${\mathsf{ϝ}}^{-}=\left(\underset{1\le g\le j}{min}{\mathit{ᵷ}}_g,\underset{1\le g\le j}{max}{ʎ}_g\right)$, ${\mathsf{ϝ}}^{+}=\left(\underset{1\le g\le j}{max}{\mathit{ᵷ}}_g,\underset{1\le g\le j}{min}{ʎ}_g\right)$, then

$${\mathsf{ϝ}}^{-}{\le }^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\le {\mathsf{ϝ}}^{+}.$$

(37)

Proof. 

Since for all $g,$ $\underset{1\le g\le j}{min}{\mathit{ᵷ}}_g\le {\mathit{ᵷ}}_g\le \underset{1\le g\le j}{max}{\mathit{ᵷ}}_g$, and $\underset{1\le g\le j}{min}{ʎ}_g\le {ʎ}_g\le \underset{1\le g\le j}{max}{ʎ}_g$, based on the principles of idempotency and monotonicity, we obtain

${\mathsf{ϝ}}^{-}{\le }^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\le {\mathsf{ϝ}}^{+}$. □

Theorem 24. 

(Shift-invariance) Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ be a series of ^p,qROFNs and $\dot{\mathsf{ϝ}}=\left(\dot{\mathit{ᵷ}},\dot{ʎ}\right)$ be any other ^p,qROFNs, then

$${}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1\otimes \dot{\mathsf{ϝ}},{\mathsf{ϝ}}_2\otimes \dot{\mathsf{ϝ}},\dots ,{\mathsf{ϝ}}_j\otimes \dot{\mathsf{ϝ}}\right){=}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)\otimes \dot{\mathsf{ϝ}}.$$

(38)

Theorem 25. 

(Homogeneity) Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ be a series of ^p,qROFNs and $\nmid >0$ be any real number, then

$${}^{p,q}ROFFWG\left(\nmid {\mathsf{ϝ}}_1,\nmid {\mathsf{ϝ}}_2,\dots ,\nmid {\mathsf{ϝ}}_j\right)=\nmid {}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right).$$

(39)

The validation of the aforementioned pair of theorems can be readily deduced from the presented operational rules of ^p,qROFNs. However, due to limitations in space, we abstain from presenting it here.

Definition 10. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ to be a class of ^p,qROFNs. In this context, the ^p,qROF fuzzy Frank ordered weighted geometric (^p,qROFFOWG) operator can be expressed as

$${}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\otimes }_{g=1}^j\left({\mathsf{ϝ}}_{\rho \left(g\right)}^{{\propto }_g}\right),$$

(40)

where $\propto ={\left({\propto }_1,{\propto }_2,\dots ,{\propto }_j\right)}^T$ is the position weights of ${\mathsf{ϝ}}_g\left(g=1,2,..,j\right)$ satisfying ${\propto }_g>0$ and ${\displaystyle \sum _{g=1}^j}{\propto }_g=1$. $\left(\rho \left(1\right),\rho \left(2\right),\dots ,\rho \left(j\right)\right)$ is a permutation of $\left(1,2,3,\dots ,j\right)$ such that ${\mathsf{ϝ}}_{\rho (g-1)}\ge {\mathsf{ϝ}}_{\rho \left(g\right)}$ for $g=2,3..,j.$

Theorem 26. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, then the result yielded by utilizing the ^p,qROFFOWG operator is still a ^p,qROFN, and

$${}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{{\mathit{ᵷ}}_{\rho \left(g\right)}^{p^{\ast }}}-1\right)}^{{\propto }_g}\right)},\sqrt[q^{\ast }]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{1-{ʎ}_{\rho \left(g\right)}^{q^{\ast }}}-1\right)}^{{\propto }_g}\right)}\right).$$

(41)

Proof. 

The demonstration of this result follows an analogous approach to that of Theorem 18, and thus, we exclude it here. □

Example 5. 

(Continued from Example 2) As per Definition 10 and Theorem 18, we can obtain

$\begin{array}{c}{}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_{\rho \left(1\right)},{\mathsf{ϝ}}_{\rho \left(2\right)},{\mathsf{ϝ}}_{\rho \left(3\right)}\right)\\ =\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{{\mathit{ᵷ}}_{\rho \left(g\right)}^3}-1\right)}^{{\propto }_g}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{1-{ʎ}_{\rho \left(g\right)}^2}-1\right)}^{{\propto }_g}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\left({3^{0.7}}^3-1\right)}^{0.3}{\left({3^{0.3}}^3-1\right)}^{0.4}{\left({3^{0.6}}^3-1\right)}^{0.3}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\left({3^{1-0.5}}^2-1\right)}^{0.3}{\left({3^{1-0.5}}^2-1\right)}^{0.4}{\left({3^{1-0.8}}^2-1\right)}^{0.3}\right)}\end{array}\right)\\ =\left(0.5577,0.4005\right).\end{array}$

Theorem 27. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, and $\mathcal{ß}>1$. As $\mathcal{ß}⟶1$, the ^p,qROFFOWG operator tends towards the following limit:

$$\underset{\mathcal{ß}⟶1}{lim}{}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({\mathit{ᵷ}}_{\rho \left(g\right)}^{p^{\ast }}\right)}^{{\propto }_g}},\sqrt[q^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{ʎ}_{\rho \left(g\right)}^{q^{\ast }}\right)}^{{\propto }_g}}\right).$$

(42)

Theorem 28. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, and $\mathcal{ß}>1$. As $\mathcal{ß}⟶\infty$, the ^p,qROFFOWG operator tends towards the following limit:

$$\underset{\mathcal{ß}⟶\infty }{lim}{}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({\mathit{ᵷ}}_{\rho \left(g\right)}^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({ʎ}_{\rho \left(g\right)}^{q^{\ast }}\right)\right)}\right).$$

(43)

Just like the properties exhibited by the ^p,qROFFWG operator, the ^p,qROFFOWG operator displays characteristics such as boundedness, idempotency, monotonicity, shift-invariance, and homogeneity. Beyond these established properties, the ^p,qROFFOWG operator has the following results.

Theorem 29. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, then we have the following:

(i) 

If $\propto ={\left(1,0,\dots ,0\right)}^T$, then ${}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=max\left\{{\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right\}$.

(ii) 

If $\propto ={\left(0,0,\dots ,1\right)}^T$, then ${}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=min\left\{{\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right\}$.

(iii) 

If ${\propto }_g=1$ and ${⨇}_i=0\left(i\ne g\right)$, then ${}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\mathsf{ϝ}}_{\rho \left(g\right)}$, where ${\mathsf{ϝ}}_{\rho \left(g\right)}$ is the gth largest of ${\mathsf{ϝ}}_g,\left(g=1,2,\dots ,j\right)$.

By examining the definitions of the ^p,qROFFWG and ^p,qROFFOWG operators, it becomes evident that the ^p,qROFFWG operator exclusively assigns weights to the ^p,qROFNs, whereas the ^p,qROFFOWG operator solely assigns weights to the sequential placement of ^p,qROFNs. In practical real-world scenarios, it is imperative to consider both aspects simultaneously. To address this limitation, we introduce the hybrid geometric operator [26], which is based on the Frank t-norm and t-conorm. This operator assigns weights to the given ^p,qROFNs and their respective ordered positions.

Definition 11. 

Let ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ be a series of ^p,qROFNs, then the ^p,qROF fuzzy Frank hybrid geometric (^p,qROFFHG) operator is

$${}^{p,q}ROFFHG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\otimes }_{g=1}^j{\left({\widehat{\mathsf{ϝ}}}_{\rho \left(g\right)}\right)}^{{\propto }_g},$$

(44)

where $\propto ={\left({\propto }_1,{\propto }_2,\dots ,{\propto }_j\right)}^T$ is the weight vector associated with ^p,qROFFHG such that ${\propto }_g>0$ and ${\displaystyle \sum _{g=1}^j}{\propto }_g=1$, $⨇={\left({⨇}_1,{⨇}_2,\dots ,{⨇}_j\right)}^T$ is the weight vector of ${\mathsf{ϝ}}_g\left(g=1,2,..,j\right)$ such that ${⨇}_g>0$ and ${\displaystyle \sum _{g=1}^j}{⨇}_g=1$. ${\widehat{\mathsf{ϝ}}}_{\rho \left(g\right)}$ is the gth largest of the weighted ^p,qROFNs ${\widehat{\mathsf{ϝ}}}_g\left({\widehat{\mathsf{ϝ}}}_g={\left({\mathsf{ϝ}}_g\right)}^{j{⨇}_g},g=1,2,..,j\right)$ and j is the balancing coefficient.

Theorem 30. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, then the result derived by utilizing the ^p,qROFFHG operator is still a ^p,qROFN, and

$${}^{p,q}ROFFHG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{{\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^{p^{\ast }}}-1\right)}^{{\propto }_g}\right)},\sqrt[q^{\ast }]{1-{log}_{\mathcal{ß}}\left(1+{\displaystyle \prod _{g=1}^j}{\left({\mathcal{ß}}^{1-{\widehat{ʎ}}_{\rho \left(g\right)}^{q^{\ast }}}-1\right)}^{{\propto }_g}\right)}\right),$$

(45)

Proof. 

The verification of this result follows a pattern akin to the proof presented in Theorem 18, and hence, we exclude it here. □

Example 6. 

(Continued from Example 3) Utilizing Definition 5, we can derive the weighted ^p,qROFNs as follows:

$\begin{array}{c}{\widehat{\mathsf{ϝ}}}_1={\mathsf{ϝ}}_1^{3\times 0.4}=\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\displaystyle \frac{{\left(3^{{0.5}^3}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\displaystyle \frac{{\left(3^{1-{0.4}^2}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)}\end{array}\right)=\left(0.3181,0.1746\right);\end{array}$

$\begin{array}{c}{\widehat{\mathsf{ϝ}}}_2={\mathsf{ϝ}}_2^{3\times 0.4}=\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\displaystyle \frac{{\left(3^{{0.4}^3}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\displaystyle \frac{{\left(3^{1-{0.3}^2}-1\right)}^{3\times 0.4}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)}\end{array}\right)=\left(0.2215,0.0989\right);\end{array}$

$\begin{array}{c}{\widehat{\mathsf{ϝ}}}_3={\mathsf{ϝ}}_3^{3\times 0.2}=\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\displaystyle \frac{{\left(3^{{0.9}^3}-1\right)}^{3\times 0.2}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\displaystyle \frac{{\left(3^{1-{0.5}^2}-1\right)}^{3\times 0.2}}{{\left(3-1\right)}^{3\times 0.4-1}}}\right)}\end{array}\right)=\left(0.9216,0.1944\right).\end{array}$

Using Definition 3, we obtain the score of ${\widehat{\mathsf{ϝ}}}_g\left(g=1,2,3\right):$

$S\left({\widehat{\mathsf{ϝ}}}_1\right)=0.5009$, $S\left({\widehat{\mathsf{ϝ}}}_2\right)=0.5005,$$S\left(\widehat{{\mathsf{ϝ}}_3}\right)=0.8725.$

Since $S\left({\widehat{\mathsf{ϝ}}}_3\right)>S\left({\widehat{\mathsf{ϝ}}}_1\right)>S\left({\widehat{\mathsf{ϝ}}}_2\right)$, we have ${\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}=\left(0.9216,0.1944\right),$ ${\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}=\left(0.3181,0.1746\right),$ and ${\widehat{\mathsf{ϝ}}}_{\rho \left(3\right)}=\left(0.2215,0.0989\right).$ Assume $\propto ={\left(0.3,0.4,0.3\right)}^T$ is the weight vector associated with the ^p,qROFFHG operator. Then, by Definition 11 and Theorem 30, the following can be deduced:

$\begin{array}{l}{}^{p,q}ROFFHG\left({\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)},{\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)},{\widehat{\mathsf{ϝ}}}_{\rho \left(3\right)}\right)=\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{{\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^3}-1\right)}^{{\propto }_g}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\displaystyle \prod _{g=1}^3}{\left(3^{1-{\widehat{ʎ}}_{\rho \left(g\right)}^2}-1\right)}^{{\propto }_g}\right)}\end{array}\right)\\ =\left(\begin{array}{c}\sqrt[3]{{log}_3\left(1+{\left({3^{0.9216}}^3-1\right)}^{0.3}{\left({3^{0.3181}}^3-1\right)}^{0.4}{\left({3^{0.2215}}^3-1\right)}^{0.3}\right)},\\ \sqrt[2]{1-{log}_3\left(1+{\left({3^{1-0.1944}}^2-1\right)}^{0.3}{\left({3^{1-0.1746}}^2-1\right)}^{0.4}{\left({3^{1-0.0989}}^2-1\right)}^{0.3}\right)}\end{array}\right)\\ =\left(0.4072,0.1652\right).\end{array}$

Theorem 31. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, and $\mathcal{ß}>1$. As $\mathcal{ß}⟶1$, the ^p,qROFFHG operator tends towards the following limit:

$$\underset{\mathcal{ß}⟶1}{lim}{}^{p,q}ROFFHG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{{\displaystyle \prod _{g=1}^j}{\left({\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^{p^{\ast }}\right)}^{{\propto }_g}},\sqrt[q^{\ast }]{1-{\displaystyle \prod _{g=1}^j}{\left(1-{\widehat{ʎ}}_{\rho \left(g\right)}^{q^{\ast }}\right)}^{{\propto }_g}}\right).$$

(46)

Theorem 32. 

Consider ${\mathsf{ϝ}}_g=\left({\mathit{ᵷ}}_g,{ʎ}_g\right)\left(g=1,2,\dots ,j\right)$ as a class of ^p,qROFNs, and $\mathcal{ß}>1$. As $\mathcal{ß}⟶\infty$, the ^p,qROFFHG operator converges to the following limit:

$$\underset{\mathcal{ß}⟶\infty }{lim}{}^{p,q}ROFFHG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\left(\sqrt[p^{\ast }]{1-\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({\widehat{\mathit{ᵷ}}}_{\rho \left(g\right)}^{p^{\ast }}\right)\right)},\sqrt[q^{\ast }]{\left({\displaystyle \sum _{g=1}^j}{\propto }_g\left({\widehat{ʎ}}_{\rho \left(g\right)}^{q^{\ast }}\right)\right)}\right).$$

(47)

Similar to the properties exhibited by the ^p,qROFFWG operator, the ^p,qROFFHG operator demonstrates boundedness, idempotency, monotonicity, shift-invariance, and homogeneity. In addition to these properties, the ^p,qROFFHG operator also encompasses the following special cases.

Corollary 3. 

The ^p,qROFFWG operator is a particular case of the ^p,qROFFHG operator.

Proof. 

Let $\propto ={\left({\displaystyle \frac{1}{j}},{\displaystyle \frac{1}{j}},\dots ,{\displaystyle \frac{1}{j}}\right)}^T,$ then

$\begin{array}{c}{}^{p,q}ROFFHG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)=\\ {\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}^{{\propto }_1}\otimes {\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}^{{\propto }_2}\otimes \dots \otimes {\widehat{\mathsf{ϝ}}}_{\rho \left(j\right)}^{{\propto }_j}={\left({\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}\otimes {\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}\otimes \dots \otimes {\widehat{\mathsf{ϝ}}}_{\rho \left(j\right)}\right)}^{{\displaystyle \frac{1}{j}}}\\ ={\mathsf{ϝ}}_1^{{⨇}_1}\otimes {\mathsf{ϝ}}_2^{{⨇}_2}\otimes \dots \otimes {\mathsf{ϝ}}_j^{{⨇}_j}{=}^{p,q}ROFFWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right).\end{array}$□

Corollary 4. 

The ^p,qROFFOWG operator is a particular case of the ^p,qROFFHG operator.

Proof. 

Let $⨇={\left({\displaystyle \frac{1}{j}},{\displaystyle \frac{1}{j}},\dots ,{\displaystyle \frac{1}{j}}\right)}^T,$ then

  $\begin{array}{c}{}^{p,q}ROFFHG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right)={\widehat{\mathsf{ϝ}}}_{\rho \left(1\right)}^{{\propto }_1}\otimes {\widehat{\mathsf{ϝ}}}_{\rho \left(2\right)}^{{\propto }_2}\otimes \dots \otimes {\widehat{\mathsf{ϝ}}}_{\rho \left(j\right)}^{{\propto }_j}\\ ={\mathsf{ϝ}}_{\rho \left(1\right)}^{{\propto }_1}\otimes {\mathsf{ϝ}}_{\rho \left(2\right)}^{{\propto }_2}\otimes \dots \otimes {\mathsf{ϝ}}_{\rho \left(j\right)}^{{\propto }_j}{=}^{p,q}ROFFOWG\left({\mathsf{ϝ}}_1,{\mathsf{ϝ}}_2,\dots ,{\mathsf{ϝ}}_j\right).\end{array}$□

5. Proposed MCDM Method

This section aims to construct a sentiment analysis algorithm by employing the ^p,qROF Frank operators in conjunction with the stepwise methodology of the SWARA method.

Decision-Making Method

In order to address decision-making challenges effectively, both Frank AOs and the SWARA method have demonstrated their practical utility. Consequently, it becomes imperative to devise a novel approach that extends the application of the Frank AOs and the SWARA method to accommodate complex assessment information within the context of ^p,qROFNs. Therefore, this subsection undertakes the development of a sentiment analysis algorithm, leveraging the adaptability of ^p,qROFNs and integrating the Frank AOs and the SWARA method. The specified MCDM algorithm incorporating ^p,qROF information can be described as follows.

Let $\mathcal{M}=\left\{{\mathcal{M}}_1,{\mathcal{M}}_2,\dots ,{\mathcal{M}}_m\right\}$ be a set of alternatives, and $\mathrm{\Xi }=\left\{{\mathrm{\Xi }}_1,{\mathrm{\Xi }}_2,\dots ,{\mathrm{\Xi }}_j\right\}$ be the set of criteria whose weight vector is unknown. The characteristics of each alternative ${\mathcal{M}}_{\iota }\left(\iota =1,2,..,m\right)$ with respect to each criteria are delineated in terms of ^p,qROFNs ${\mathsf{ϝ}}_{\iota g}=\left({\mathit{ᵷ}}_{\iota g},{ʎ}_{\iota g}\right)$; $0\le {\mathit{ᵷ}}_{\iota g}^p+{ʎ}_{\iota g}^q\le 1$. Then, we proceed with the creation of an MCDM algorithm, which primarily encompasses the following steps.

Step 1:

Development of data matrix:

Take the ^p,qROF data from DEs concerning the finite set of alternatives evaluated against the criteria, and organize it into the matrix M represented as $\left[{\mathsf{ϝ}}_{\iota g}\right]$. Additionally, determine the minimum values of p and q required to ensure that all pairs of the provided data conform to the context of ^p,qROFNs.

Step 2:

Normalization:

Convert the data matrix $D=\left[{\mathsf{ϝ}}_{\iota g}\right]$ into its normalized version $\tilde{D}=\left[{\widetilde{\mathsf{ϝ}}}_{\iota g}\right]$ using the rule given in Equation (48).

$${\widetilde{\mathsf{ϝ}}}_{\iota g}=\left\{\begin{array}{cc}{\mathsf{ϝ}}_{\iota g},\hfill & if{\mathrm{\Xi }}_g\mathrm{is}\mathrm{benefit}\mathrm{type}\hfill \\ {\left({\mathsf{ϝ}}_{\iota g}\right)}^c,\hfill & if{\mathrm{\Xi }}_g\mathrm{is}\cos \mathrm{t}\mathrm{type},\hfill \end{array}\right.$$

(48)

where ${\left({\mathsf{ϝ}}_{\iota g}\right)}^c$ denotes the complement of ${\mathsf{ϝ}}_{\iota g}$.

Step 3:

Criteria weight determination:

Step 3.1:

Determine the score value of the DE’s opinions (^p,qROFNs) regarding the considered criteria according to Equation (2).

Step 3.2:

Quantify the priority classification of the criteria in accordance with the DE’s priorities, listing the criteria from greatest to smallest.

Step 3.3:

Evaluate the level of relative importance, starting from the second criterion, as follows: The relative importance of criterion ${\mathrm{\Xi }}_g$ in relation to the previous criterion ${\mathrm{\Xi }}_{g-1}$ is $S\left({\mathrm{\Xi }}_g\right)-S\left({\mathrm{\Xi }}_{g-1}\right)$. This difference is called the comparative significance of the mean value and is labeled ${\perp }_g$.

Step 3.4:

Assess the comparative coefficient in light of Equation (49).

$${\wp }_g=\left\{\begin{array}{cc}1,\hfill & ifg=1\hfill \\ {\perp }_g+1,\hfill & ifg>1.\hfill \end{array}\right.$$

(49)

Step 3.5:

Calculate the weight of the gth criterion following Equation (50).

$${\mathrm{\Omega }}_g=\left\{\begin{array}{cc}1,\hfill & ifg=1\hfill \\ {\displaystyle \frac{{\mathrm{\Omega }}_{g-1}}{{\wp }_g}},\hfill & ifg>1.\hfill \end{array}\right.$$

(50)

Step 3.6:

Determine the normalized weight of the gth criterion on the basis of Equation (51).

$${⨇}_g={\displaystyle \frac{{\mathrm{\Omega }}_g}{{\displaystyle \sum _{g=1}^j}{\mathrm{\Omega }}_g}};g=1,2,\dots ,j.$$

(51)

Step 4:

Aggregation:

Aggregate the ^p,qROFNs ${\mathsf{ϝ}}_{\iota g}\left(g=1,2,\dots ,j\right)$ for each alternative $o_{\iota }\left(\iota =1,2,\dots ,j\right)$ into the overall preference value ${\mathsf{ϝ}}_{\iota }$ by using either the diagnosed ^p,qROFFWA or ^p,qROFFWG operators. Mathematically,

$${\mathsf{ϝ}}_{\iota }{=}^{p,q}ROFFW{A}_{⨇}\left({\mathsf{ϝ}}_{\iota 1},{\mathsf{ϝ}}_{\iota 2},\dots ,{\mathsf{ϝ}}_{\iota m}\right),$$

(52)

$${\mathsf{ϝ}}_{\iota }{=}^{p,q}ROFFW{G}_{⨇}\left({\mathsf{ϝ}}_{\iota 1},{\mathsf{ϝ}}_{\iota 2},\dots ,{\mathsf{ϝ}}_{\iota m}\right),$$

(53)

where $⨇=\left({⨇}_1,{⨇}_2,\dots ,{⨇}_j\right)$ is the weight vector of the criteria calculated in step 3.

Step 5:

Score values:

Find the score values of $S\left({\mathsf{ϝ}}_{\iota }\right)\left(\iota =1,2,\dots ,m\right)$ of the overall values ${\mathsf{ϝ}}_{\iota }\left(\iota =1,2,\dots ,m\right)$.

Step 6:

Ranking:

Sort the alternatives ${\mathcal{M}}_{\iota }\left(\iota =1,2,\dots ,m\right)$ based on the score values $S\left({\mathsf{ϝ}}_{\iota }\right)$ and choose the optimal one.

6. Case Study

This section initially utilizes a case study to show the operation of the proposed MCDM method, followed by a series of experiments to investigate the impact of various parameter values on the aggregation results.

6.1. Problem Description

In the present part, a detailed case study is given to illustrate the practical implementation of the devised technique in real-world scenarios involving numerous criteria and alternatives. This case study provides decision guidance for sustainable choice of materials in the construction industry. The choice of building materials in any construction project is crucial, as it significantly impacts the built environment’s resilience, safety, and sustainability. The following points underscore the importance of material selection:

Safety: The safety of a building is directly influenced by the materials used. Selecting substandard materials can lead to structural failures, endangering occupants. Thus, it is imperative to choose materials that comply with safety norms and have passed rigorous durability and strength tests [27].

Resilience: The resilience of the built environment is also determined by the materials used. Resilience refers to a structure’s ability to withstand natural disasters such as floods, earthquakes, and hurricanes. Using durable materials can prevent degradation and reduce the demand for maintenance and rebuilding [28].

Sustainability: The sustainability of the built environment is affected by the selected materials. Sustainable materials are reusable or recycled, with a low environmental impact. Using eco-friendly materials can reduce the building’s carbon footprint and promote environmental responsibility [29].

Cost: The cost of materials is a crucial consideration in building project planning. Selecting the right materials can reduce operational costs in the long run. For example, using durable materials can minimize the need for repairs and maintenance, leading to financial savings over time.

Regarding the choice of sustainable materials, incorporating DEs’ evaluations into the proposed approach is vital for addressing complex MCDM problems with numerous criteria and alternatives. DEs with diverse backgrounds and viewpoints can provide valuable insights to raise the standard of the outcomes. Therefore, a panel of DEs with various experiences should be assembled to offer data on the most significant assessment criteria, the weight of each criterion, and the performance evaluations of material choices against these criteria.

Initially, a set of criteria was compiled to address sustainability in material choice. After consulting with DEs, ten criteria were selected to cover the three dimensions, incorporating sustainability in material selection decisions. The impact of these criteria was subsequently assessed. The impact illustrates how each criterion affects overall sustainability. A cost criterion decreases the sustainability level as it increases, whereas a benefit criterion increases the sustainability level as it rises. Once the criteria are established, DEs must evaluate how the alternatives perform against each criterion. This evaluation can be conducted through questionnaires, where DEs use ^p,qROFNs to express their performance ratings for each alternative.

To avoid any misunderstandings regarding the criteria, each question should include a brief description explaining what the criterion entails. Additionally, the identified criteria for the sustainable material selection case are detailed in Table 1.

Table 1. The chosen criteria for selecting sustainable materials [30,31,32].

Criteria Group	Criterion	Explanation	Impact
Economic	${\mathrm{\Xi }}_1:$ Initial price	The amount required for the production or	Cost
		procurement of materials.
	${\mathrm{\Xi }}_2:$ Maintenance price	The funds that need to be allocated for maintenance	Cost
		throughout its useful lifespan.
	${\mathrm{\Xi }}_3:$ Price of disposal	The cost required for the disposal of the material at	Cost
		the end of its lifecycle.
Social	${\mathrm{\Xi }}_4:$ Health and safety	The material must be resilient to disruptions and	Benefit
		ensure user safety and well-being throughout its lifespan.
	${\mathrm{\Xi }}_5:$ Durability against decay	The ability to withstand erosion, corrosion,	Benefit
		and similar factors.
	${\mathrm{\Xi }}_6:$ Fireproof	The ability to withstand fire.	Benefit
Environment	${\mathrm{\Xi }}_7:$ Water consumption	Water usage throughout the material’s lifecycle.	Cost
	${\mathrm{\Xi }}_8:$ CO2 emission	Emissions of CO2 over the material’s usage period.	Cost
	${\mathrm{\Xi }}_9:$ Energy saving	Total energy conserved by the substance.	Benefit
	${\mathrm{\Xi }}_{10}:$ Reuse and recyclability	The material’s potential for recycling and reuse.	Benefit

Consider that our objective is to rank six material alternatives, denoted as ${\mathcal{M}}_i$ $(i=1.2,\dots ,6)$, and select the most sustainable material from among them. We consider ten criteria, as detailed in Table 1.

For evaluating the performance of each alternative against the criteria, we employ the ^p,qROFSs methodology. The parameter p is set to 3, and q is set to 2, representing the performance of each alternative in response to the criteria outlined in Table 2. This method allows us to systematically assess and rank the material alternatives based on their sustainability across the specified criteria.

Table 2. DEs’ opinions about the considered criteria.

Criteria	p,qROFNs
${\mathrm{\Xi }}_1$	$\left(0.55,0.30\right)$
${\mathrm{\Xi }}_2$	$\left(0.44,0.25\right)$
${\mathrm{\Xi }}_3$	$\left(0.60,0.55\right)$
${\mathrm{\Xi }}_4$	$\left(0.70,0.48\right)$
${\mathrm{\Xi }}_5$	$\left(0.38,0.20\right)$
${\mathrm{\Xi }}_6$	$\left(0.52,0.40\right)$
${\mathrm{\Xi }}_7$	$\left(0.50,0.50\right)$
${\mathrm{\Xi }}_8$	$\left(0.72,0.42\right)$
${\mathrm{\Xi }}_9$	$\left(0.50,0.30\right)$
${\mathrm{\Xi }}_{10}$	$\left(0.68,0.48\right)$

6.2. Model Implementation

The following section outlines the stepwise process for calculating the optimal material selection.

Step 1: The ^p,qROF data provided by the DEs for the ten materials, evaluated against six criteria, is tabulated in Table 3.

Table 3. ^p,qROF decision matrix provided by DEs.

	${\mathbf{\Xi }}_1$	${\mathbf{\Xi }}_2$	${\mathbf{\Xi }}_3$	${\mathbf{\Xi }}_4$	${\mathbf{\Xi }}_5$
${\mathcal{M}}_1$	$\left(0.30,0.74\right)$	$\left(0.20,0.48\right)$	$\left(0.44,0.68\right)$	$\left(0.70,0.37\right)$	$\left(0.60,0.35\right)$
${\mathcal{M}}_2$	$\left(0.18,0.37\right)$	$\left(0.20,0.48\right)$	$\left(0.18,0.67\right)$	$\left(0.45,0.30\right)$	$\left(0.40,0.40\right)$
${\mathcal{M}}_3$	$\left(0.56,0.34\right)$	$\left(0.26,0.16\right)$	$\left(0.66,0.47\right)$	$\left(0.38,0.36\right)$	$\left(0.28,0.26\right)$
${\mathcal{M}}_4$	$\left(0.65,0.52\right)$	$\left(0.38,0.44\right)$	$\left(0.50,0.40\right)$	$\left(0.45,0.45\right)$	$\left(0.70,0.55\right)$
${\mathcal{M}}_5$	$\left(0.50,0.60\right)$	$\left(0.55,0.38\right)$	$\left(0.67,0.34\right)$	$\left(0.64,0.16\right)$	$\left(0.72,0.36\right)$
${\mathcal{M}}_6$	$\left(0.50,0.50\right)$	$\left(0.54,0.47\right)$	$\left(0.72,0.59\right)$	$\left(0.37,0.25\right)$	$\left(0.64,0.38\right)$
	${\mathrm{\Xi }}_6$	${\mathrm{\Xi }}_7$	${\mathrm{\Xi }}_8$	${\mathrm{\Xi }}_9$	${\mathrm{\Xi }}_{10}$
${\mathcal{M}}_1$	$\left(0.30,0.17\right)$	$\left(0.45,0.35\right)$	$\left(0.34,0.28\right)$	$\left(0.45,0.50\right)$	$\left(0.75,0.68\right)$
${\mathcal{M}}_2$	$\left(0.34,0.24\right)$	$\left(0.67,0.62\right)$	$\left(0.58,0.47\right)$	$\left(0.35,0.40\right)$	$\left(0.55,0.50\right)$
${\mathcal{M}}_3$	$\left(0.60,0.55\right)$	$\left(0.31,0.28\right)$	$\left(0.48,0.40\right)$	$\left(0.25,0.25\right)$	$\left(0.35,0.30\right)$
${\mathcal{M}}_4$	$\left(0.68,0.64\right)$	$\left(0.26,0.18\right)$	$\left(0.42,0.24\right)$	$\left(0.57,0.65\right)$	$\left(0.30,0.32\right)$
${\mathcal{M}}_5$	$\left(0.55,0.40\right)$	$\left(0.72,0.50\right)$	$\left(0.65,0.42\right)$	$\left(0.50,0.58\right)$	$\left(0.64,0.46\right)$
${\mathcal{M}}_6$	$\left(0.48,0.36\right)$	$\left(0.74,0.52\right)$	$\left(0.66,0.48\right)$	$\left(0.62,0.40\right)$	$\left(0.72,0.48\right)$

Step 2: Based on the nature of the criteria, the data shown in Table 3 are normalized and are presented in Table 4.

Table 4. ^p,qROF normalized decision matrix.

	${\mathbf{\Xi }}_1$	${\mathbf{\Xi }}_2$	${\mathbf{\Xi }}_3$	${\mathbf{\Xi }}_4$	${\mathbf{\Xi }}_5$
${\mathcal{M}}_1$	$\left(0.74,0.30\right)$	$\left(0.48,0.20\right)$	$\left(0.68,0.44\right)$	$\left(0.70,0.37\right)$	$\left(0.60,0.35\right)$
${\mathcal{M}}_2$	$\left(0.37,0.18\right)$	$\left(0.48,0.20\right)$	$\left(0.67,0.18\right)$	$\left(0.45,0.30\right)$	$\left(0.40,0.40\right)$
${\mathcal{M}}_3$	$\left(0.34,0.56\right)$	$\left(0.16,0.26\right)$	$\left(0.47,0.66\right)$	$\left(0.38,0.36\right)$	$\left(0.28,0.26\right)$
${\mathcal{M}}_4$	$\left(0.52,0.65\right)$	$\left(0.44,0.38\right)$	$\left(0.40,0.50\right)$	$\left(0.45,0.45\right)$	$\left(0.70,0.55\right)$
${\mathcal{M}}_5$	$\left(0.60,0.50\right)$	$\left(0.38,0.55\right)$	$\left(0.34,0.67\right)$	$\left(0.64,0.16\right)$	$\left(0.72,0.36\right)$
${\mathcal{M}}_6$	$\left(0.50,0.50\right)$	$\left(0.47,0.54\right)$	$\left(0.59,0.72\right)$	$\left(0.37,0.25\right)$	$\left(0.64,0.38\right)$
	${\mathrm{\Xi }}_6$	${\mathrm{\Xi }}_7$	${\mathrm{\Xi }}_8$	${\mathrm{\Xi }}_9$	${\mathrm{\Xi }}_{10}$
${\mathcal{M}}_1$	$\left(0.30,0.17\right)$	$\left(0.35,0.45\right)$	$\left(0.28,0.34\right)$	$\left(0.45,0.50\right)$	$\left(0.75,0.68\right)$
${\mathcal{M}}_2$	$\left(0.34,0.24\right)$	$\left(0.62,0.67\right)$	$\left(0.47,0.58\right)$	$\left(0.35,0.40\right)$	$\left(0.55,0.50\right)$
${\mathcal{M}}_3$	$\left(0.60,0.55\right)$	$\left(0.28,0.31\right)$	$\left(0.40,0.48\right)$	$\left(0.25,0.25\right)$	$\left(0.35,0.30\right)$
${\mathcal{M}}_4$	$\left(0.68,0.64\right)$	$\left(0.18,0.26\right)$	$\left(0.24,0.42\right)$	$\left(0.57,0.65\right)$	$\left(0.30,0.32\right)$
${\mathcal{M}}_5$	$\left(0.55,0.40\right)$	$\left(0.50,0.72\right)$	$\left(0.42,0.65\right)$	$\left(0.50,0.58\right)$	$\left(0.64,0.46\right)$
${\mathcal{M}}_6$	$\left(0.48,0.36\right)$	$\left(0.52,0.74\right)$	$\left(0.48,0.66\right)$	$\left(0.62,0.40\right)$	$\left(0.72,0.48\right)$

Step 3: Following the proposed SWARA approach, the weights of the criteria are determined as follows:

Step 3.1: The score values of the DEs’ opinions on the ten criteria are calculated according to Equation (2), as shown below:

$S\left({\mathrm{\Xi }}_1\right)=0.2564$, $S\left({\mathrm{\Xi }}_2\right)=0.1477$, $S\left({\mathrm{\Xi }}_3\right)=0.5185$, $S\left({\mathrm{\Xi }}_4\right)=0.5734$, $S\left({\mathrm{\Xi }}_5\right)=0.0949$, $S\left({\mathrm{\Xi }}_6\right)=0.3006$, $S\left({\mathrm{\Xi }}_7\right)=0.3750$, $S\left({\mathrm{\Xi }}_8\right)=0.5496$, $S\left({\mathrm{\Xi }}_9\right)=0.2150$, $S\left({\mathrm{\Xi }}_{10}\right)=0.5448$.

Step 3.2: In accordance with the score values of the criteria, they are arranged in descending order as follows:

$S\left({\mathrm{\Xi }}_4\right)>$, $S\left({\mathrm{\Xi }}_8\right)>$, $S\left({\mathrm{\Xi }}_{10}\right)>$, $S\left({\mathrm{\Xi }}_3\right)>$, $S\left({\mathrm{\Xi }}_7\right)>$, $S\left({\mathrm{\Xi }}_6\right)>$, $S\left({\mathrm{\Xi }}_1\right)>$, $S\left({\mathrm{\Xi }}_9\right)>$, $S\left({\mathrm{\Xi }}_2\right)>$, $S\left({\mathrm{\Xi }}_5\right)$.

Step 3.3: The comparative significance of the mean values is quantified and the results are shown below.

${\perp }_1=0.0442$, ${\perp }_2=0.0673$, ${\perp }_3=0.0263$, ${\perp }_4=0.0000$, ${\perp }_5=0.0528$, ${\perp }_6=0.0744$, ${\perp }_7=0.1435$, ${\perp }_8=0.0238$, ${\perp }_9=0.0414$, ${\perp }_{10}=0.0048$,

Step 3.4: With the aid of Equation (49), the comparative coefficient of each criteria is obtained as follows:

${\wp }_1=1.0442$, ${\wp }_2=1.0673$, ${\wp }_3=1.0263$, ${\wp }_4=1.0000$, ${\wp }_5=1.0528$, ${\wp }_6=1.0744$, ${\wp }_7=1.1435$, ${\wp }_8=1.0238$, ${\wp }_9=1.0414$, ${\wp }_{10}=1.0048$.

Step 3.5: Based on Equation (50), the weight vector of criteria is estimated as

$\mathrm{\Omega }=\left(0.0896,0.0806,0.1150,0.1214,0.0766,0.0936,0.1005,0.1186,0.0861,0.1180\right)$.

Step 3.5: Following Equation (51), the normalized weight vector of criteria is estimated as

$⨇=\left(0.0861,0.1150,0.0766,0.1214,0.0806,0.1186,0.1005,0.0936,0.1180,0.0896\right)$.

Step 4: Employing the formulation of the ^p,qROFFWA and ^p,qROFFWG operators, the data of Table 4 is aggregated, and is summarized in Table 5.

Table 5. Aggregated results of alternatives.

	p,qROFFWA	p,qROFFWG
${\mathcal{M}}_1$	$\left(0.5819,0.3496\right)$	$\left(0.4987,0.4055\right)$
${\mathcal{M}}_2$	$\left(0.4886,0.3338\right)$	$\left(0.4537,0.4063\right)$
${\mathcal{M}}_3$	$\left(0.3933,0.3731\right)$	$\left(0.3303,0.4208\right)$
${\mathcal{M}}_4$	$\left(0.5105,0.4692\right)$	$\left(0.4233,0.5086\right)$
${\mathcal{M}}_5$	$\left(0.5545,0.4681\right)$	$\left(0.5193,0.5317\right)$
${\mathcal{M}}_6$	$\left(0.5515,0.4724\right)$	$\left(0.5243,0.5256\right)$

Steps 5–6: Using Equation (2), the score values ${\mathcal{M}}_{\iota }\left(\iota =1,2,\dots ,6\right)$ and the ordering of the alternatives based on these score values are derived in Table 6.

Table 6. Score values and ranking of materials.

Operator	Score Values	Ranking
p,qROFFWA	$S\left({\mathcal{M}}_1\right)=0.5374$, $S\left({\mathcal{M}}_2\right)=0.5026$, $S\left({\mathcal{M}}_3\right)=0.4608$,	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
	$S\left({\mathcal{M}}_4\right)=0.4565$, $S\left({\mathcal{M}}_5\right)=0.4757$, $S\left({\mathcal{M}}_6\right)=0.4723$.
p,qROFFWG	$S\left({\mathcal{M}}_1\right)=0.4798$, $S\left({\mathcal{M}}_2\right)=0.4642$, $S\left({\mathcal{M}}_3\right)=0.4295$,	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
	$S\left({\mathcal{M}}_4\right)=0.4086$, $S\left({\mathcal{M}}_5\right)=0.4287$, $S\left({\mathcal{M}}_6\right)=0.4339$.

6.3. Sensitivity Analysis

In this section, we primarily examine the impact of parameters $ß,$ $p^{\ast }$ and $q^{\ast }$, as well as variations in criteria weights, on the final ranking of alternatives by considering various scenarios.

6.3.1. Sensitivity Regarding Parameters

The three parameters $p^{\ast }$, $q^{\ast }$, and ß associated with the established model certainly affect the final outcomes; therefore, we attempt to examine their effects.

To investigate the influence of parameter p on the ranking of alternative materials, we assign different values to p while the parameters q and ß are kept constant at 2 and 4, respectively. Table 7 presents the resulting rankings for each p value. Notably, for p values between 03 and 09, material ${\mathcal{M}}_1$ emerges as the superior choice, followed by ${\mathcal{M}}_2$. However, this trend reverses when p exceeds 09, with ${\mathcal{M}}_2$ taking precedence. Additionally, ${\mathcal{M}}_4$ exhibits the lowest performance for p values of 03 and 05, whereas ${\mathcal{M}}_6$ consistently ranks last for all other p values. In addition, ${\mathcal{M}}_3$ initially occupies the fifth position for $p=03$ but subsequently secures a consistent third place for all higher p values. These observations underscore the significant impact of parameter p on the material selection process.

Table 7. Results for ${}^{p,q}ROFFWA$ operator for various values of p.

p	$\mathit{S}\left({\mathcal{M}}_1\right)$	$\mathit{S}\left({\mathcal{M}}_2\right)$	$\mathit{S}\left({\mathcal{M}}_3\right)$	$\mathit{S}\left({\mathcal{M}}_4\right)$	$\mathit{S}\left({\mathcal{M}}_5\right)$	$\mathit{S}\left({\mathcal{M}}_6\right)$	Ranking
$p=03$	0.5374	0.5026	0.4608	0.4565	0.4757	0.4723	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$p=05$	0.4819	0.4617	0.4369	0.4140	0.4193	0.4159	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$
$p=07$	0.4590	0.4499	0.4313	0.4002	0.4001	0.3968	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$
$p=09$	0.4483	0.4461	0.4298	0.3949	0.3931	0.3899	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$
$p=11$	0.4430	0.4448	0.4293	0.3926	0.3903	0.3872	${\mathcal{M}}_2$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$
$p=15$	0.4391	0.4442	0.4289	0.3910	0.3888	0.3857	${\mathcal{M}}_2$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$
$p=20$	0.4378	0.4441	0.4288	0.3906	0.3884	0.3854	${\mathcal{M}}_2$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$
$p=25$	0.4376	0.4441	0.4288	0.3905	0.3884	0.3853	${\mathcal{M}}_2$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$

Similarly to parameter p, we now illustrate the impact of parameter q on the ranking of selection materials. Various values for q are considered, such as 2, 3, 5, 7, 9, 11, 15, 20, and 25, while $p=3$ and $ß=4$ are kept constant. As shown in Table 8, the best material remains ${\mathcal{M}}_1$ across all cases. For $q=2$ and $q=3$, the second-best alternative is ${\mathcal{M}}_2$. However, for all other values of q, ${\mathcal{M}}_5$ moves to the second position, previously occupying the third position in the cases where $p=2$ and $p=3$. Additionally, it is observed that ${\mathcal{M}}_4$ is the worst material in the first two cases, but ${\mathcal{M}}_3$ takes this position in all other cases. This observation indicates that the ranking results are sensitive to changes in parameter q, though to a lesser extent than p, as the top alternative, ${\mathcal{M}}_1$, remains consistent across all scenarios.

Table 8. Results for ${}^{p,q}ROFFWA$ operator for various values of q.

q	$\mathit{S}\left({\mathcal{M}}_1\right)$	$\mathit{S}\left({\mathcal{M}}_2\right)$	$\mathit{S}\left({\mathcal{M}}_3\right)$	$\mathit{S}\left({\mathcal{M}}_4\right)$	$\mathit{S}\left({\mathcal{M}}_5\right)$	$\mathit{S}\left({\mathcal{M}}_6\right)$	Ranking
$q=02$	0.5374	0.5026	0.4608	0.4565	0.4757	0.4723	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$q=03$	0.5786	0.5406	0.5046	0.5154	0.5353	0.5323	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$q=05$	0.5987	0.5571	0.5270	0.5559	0.5760	0.5739	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_3$
$q=07$	0.6012	0.5589	0.5302	0.5646	0.5844	0.5828	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$
$q=09$	0.6015	0.5591	0.5306	0.5665	0.5862	0.5847	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$
$q=11$	0.6015	0.5592	0.5307	0.5669	0.5866	0.5851	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$
$q=15$	0.6016	0.5592	0.5307	0.5670	0.5867	0.5852	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$
$q=20$	0.6016	0.5592	0.5307	0.5670	0.5867	0.5852	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$
$q=25$	0.6016	0.5592	0.5307	0.5670	0.5867	0.5852	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$

To assess the sensitivity of our proposed operator to the Frank parameter ß, we revisited the material selection problem and evaluated it across a range of ß values. Table 9 presents the resulting final scores and rankings for each ß value. Notably, the ranking of materials remained consistent (${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$) regardless of the specific ß value employed, except for $ß\to 01$. This observation suggests that the developed methodology exhibits robustness with respect to variations in the Frank parameter ß.

Table 9. Results for ${}^{p,q}ROFFWA$ operator for various values of ß.

ß	$\mathit{S}\left({\mathcal{M}}_1\right)$	$\mathit{S}\left({\mathcal{M}}_2\right)$	$\mathit{S}\left({\mathcal{M}}_3\right)$	$\mathit{S}\left({\mathcal{M}}_4\right)$	$\mathit{S}\left({\mathcal{M}}_5\right)$	$\mathit{S}\left({\mathcal{M}}_6\right)$	Ranking
$ß\to 01$	0.4747	0.4606	0.4270	0.4047	0.4240	0.4288	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_4$
$ß=02$	0.5397	0.5040	0.4619	0.4586	0.4785	0.4746	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=03$	0.5383	0.5032	0.4613	0.4573	0.4768	0.4732	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=04$	0.5374	0.5026	0.4608	0.4565	0.4757	0.4723	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=08$	0.5354	0.5013	0.4598	0.4546	0.4731	0.4702	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=15$	0.5339	0.5002	0.4589	0.4531	0.4710	0.4684	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=20$	0.5332	0.4998	0.4585	0.4524	0.4701	0.4677	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=25$	0.5328	0.4994	0.4583	0.4520	0.4694	0.4671	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=30$	0.5324	0.4991	0.4580	0.4516	0.4688	0.4666	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=40$	0.5318	0.4987	0.4577	0.4510	0.4680	0.4660	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
$ß=50$	0.5314	0.4983	0.4574	0.4506	0.4674	0.4654	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$

6.3.2. Sensitivity Regarding Criteria Weights

In this part, several simulation experiments were conducted to assess the impact of varying the criteria weights with the highest importance (i.e., ${⨇}_4$) on the final outcomes. Building on the study by [33], ten scenarios were derived using Equation (54).

$${\stackrel{⌢}{⨇}}_{other}=\left({\displaystyle \frac{1-\alpha {⨇}_{max}}{1-{⨇}_{max}}}\right)\times {⨇}_{other}$$

(54)

where ${{\displaystyle \stackrel{⌢}{⨇}}}_{other}$ represents the updated values of all criteria except ${⨇}_4$; $\alpha$ is the reduction coefficient, which can be subjectively determined by DEs; ${⨇}_{other}$ indicates the original weight of the criteria excluding ${⨇}_4$; and ${⨇}_{max}$ denotes the original weight of criteria ${⨇}_4$. In this study, the value of $\alpha$ is set at 10 %, 20 %, …, 100%, allowing for the execution of 10 simulation experiments across 10 different scenarios. The variations in the criteria weights ${⨇}_j$ under these different scenarios are illustrated in Table 10. Using these 10 sets of criteria weights, the corresponding outcomes for the alternatives are obtained, as shown in Table 10.

Table 10. Results for ${}^{p,q}ROFFWA$ operator for various weight vectors.

Scenario	Weight Vector	Score Values						Ranking
		$\mathbf{S}\left({\mathcal{M}}_{\mathbf{1}}\right)$	$\mathbf{S}\left({\mathcal{M}}_{\mathbf{2}}\right)$	$\mathbf{S}\left({\mathcal{M}}_{\mathbf{3}}\right)$	$\mathbf{S}\left({\mathcal{M}}_{\mathbf{4}}\right)$	$\mathbf{S}\left({\mathcal{M}}_{\mathbf{5}}\right)$	$\mathbf{S}\left({\mathcal{M}}_{\mathbf{6}}\right)$
(i) $\alpha =10\%$	$⨇=\left(\begin{array}{c}0.0968,0.1294,0.0862,0.01214,0.0906,\\ 0.1333,0.1130,0.1052,0.1327,0.1007\end{array}\right)$	0.5224	0.4978	0.4567	0.4513	0.4346	0.4557	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$
(ii) $\alpha =20\%$	$⨇=\left(\begin{array}{c}0.0956,0.1277,0.0852,0.02428,0.0895,\\ 0.1317,0.1116,0.1039,0.1310,0.0995\end{array}\right)$	0.5234	0.4979	0.4568	0.4513	0.4384	0.4567	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$
(iii) $\alpha =30\%$	$⨇=\left(\begin{array}{c}0.0944,0.1261,0.0840,0.03642,0.0884,\\ 0.1301,0.1102,0.1027,0.1294,0.0983\end{array}\right)$	0.5244	0.4979	0.4568	0.4512	0.4422	0.4580	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$
(iv) $\alpha =40\%$	$⨇=\left(\begin{array}{c}0.0932,0.1245,0.0830,0.04856,0.0873,\\ 0.1284,0.1088,0.1014,0.1278,0.0970\end{array}\right)$	0.5254	0.4980	0.4569	0.4511	0.4459	0.4591	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$
(v) $\alpha =50\%$	$⨇=\left(\begin{array}{c}0.0920,0.1229,0.0819,0.06070,0.0862,\\ 0.1268,0.1074,0.1001,0.1262,0.0958\end{array}\right)$	0.5264	0.4980	0.4570	0.4510	0.4496	0.4602	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$
(vi) $\alpha =60\%$	$⨇=\left(\begin{array}{c}0.0909,0.1214,0.0808,0.07284,0.0851,\\ 0.1252,0.1061,0.0988,0.1245,0.0946\end{array}\right)$	0.5273	0.4980	0.4570	0.4508	0.4531	0.4612	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_4$
(vii) $\alpha =70\%$	$⨇=\left(\begin{array}{c}0.0897,0.1198,0.0798,0.08498,0.0839,\\ 0.1235,0.1047,0.0975,0.1229,0.0933\end{array}\right)$	0.5284	0.4981	0.4571	0.4508	0.4567	0.4623	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_4$
(viii) $\alpha =80\%$	$⨇=\left(\begin{array}{c}0.0885,0.1182,0.0787,0.09712,0.0828,\\ 0.1219,0.1033,0.0962,0.1213,0.0921\end{array}\right)$	0.5294	0.4982	0.4572	0.4507	0.4603	0.4633	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
(ix) $\alpha =90\%$	$⨇=\left(\begin{array}{c}0.0873,0.1166,0.0777,0.1093,0.0817,\\ 0.1202,0.1019,0.0949,0.1196,0.0908\end{array}\right)$	0.5305	0.4983	0.4573	0.4507	0.4639	0.4644	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
(x) $\alpha =100\%$	$⨇=\left(\begin{array}{c}0.0861,0.1150,0.0766,0.1214,0.0806,\\ 0.1186,0.1005,0.0936,0.1180,0.0896\end{array}\right)$	0.5315	0.4983	0.4574	0.4506	0.4674	0.4654	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$

From Table 10, it is evident that variations in the value of the maximum weight criteria, ${⨇}_4$, lead to slight changes in the ranking of certain materials. However, the top-ranked alternative, ${\mathcal{M}}_1$, and the second-best alternative, ${\mathcal{M}}_2$, consistently maintain their positions across all 10 scenarios. In the first five scenarios, the worst alternative is ${\mathcal{M}}_5$. However, in the latter five scenarios, ${\mathcal{M}}_4$ is ranked as the least favorable, with ${\mathcal{M}}_5$ being more preferred in comparison.

This analysis underscores the significant role that criteria weights play in the selection process. The variations in these weights result in some changes in the overall ranking of the materials. The stability of the top alternatives, ${\mathcal{M}}_1$ and ${\mathcal{M}}_2$, across different scenarios suggests a robustness in their selection, whereas the shifting positions of some materials highlight the sensitivity of alternatives to changes in criteria weights. This insight is crucial for DEs to understand the impact of weight adjustments on material rankings, ensuring more informed and reliable decisions.

After implementing the framed approach to the problem, the following managerial implications may be drawn.

The study’s managerial implications provide useful suggestions for enhancing decision making in construction material selection. Managers can make better-informed and precise decisions by using a structured, data-driven strategy that includes MCDM approaches and q-rung orthopair fuzzy sets, thereby reducing biases and enhancing results. This strategy highlights the importance of safety, resilience, and sustainability by ensuring that selected materials fulfill high durability and compliance criteria, lowering the danger of structural failure and legal liability. Incorporating environmental factors like $C{O}_2$ emissions and recyclability can improve a company’s reputation and long-term competitiveness by promoting sustainable practices.

Furthermore, the framework promotes cost efficiency by comparing initial charges against long-term savings in maintenance and disposal, so encouraging more strategic resource allocation. The participation of different DEs ensures a more thorough evaluation process, resulting in decisions that consider multiple stakeholder views. Finally, the approach’s versatility enables managers to modify it to specific project requirements while assuring regulatory compliance and stimulating innovation in sustainable construction, ultimately contributing to future-proofing and alignment with green building trends.

7. Comparative Study

A comparative analysis with established approaches is conducted in this section to benchmark the efficacy and generalizability of the proposed ^p,qROF operators. Prior studies [8,10,12,14,15,34] serve as the basis for comparison. Table 11 presents the ranking outcomes obtained by applying these existing AOs to a previously explored material selection problem. A comprehensive evaluation of its relative strengths can be detailed as follows by juxtaposing these results with those achieved by our ^p,qROF operators.

Table 11. Comparison with existing aggregation operators.

p,qROFWA [8]	0.4747	0.4606	0.4270	0.4047	0.4240	0.4288	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_4$
p,qROFWG [8]	0.5424	0.5055	0.4631	0.4610	0.4815	0.4771	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
CLp,qROFWA [12]	0.4744	0.4706	0.4566	0.4736	0.4834	0.4832	${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$
CLp,qROFWG [12]	0.5392	0.5465	0.5443	0.5297	0.5281	0.5276	${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$
p,qROFHWA [14]	0.1537	0.1529	0.1274	0.0837	0.0887	0.0909	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_4$
p,qROFHWG [14]	0.9610	0.9535	0.9094	0.9160	0.9261	0.9255	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_3$
p,qROFDWA [15]	0.3832	0.4225	0.3542	0.3321	0.3976	0.4094	${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
p,qROFDWG [15]	0.6321	0.5662	0.5032	0.5355	0.5995	0.5496	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_3$
p,qROFAAWA [10]	0.3850	0.3867	0.3690	0.3465	0.3601	0.3538	${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_1$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
p,qROFAAWG [10]	0.6170	0.5596	0.5151	0.5387	0.5698	0.5458	${\mathcal{M}}_1$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_3$
q-ROFWPM [34]	0.6295	0.5738	0.5404	0.5652	0.5731	0.5690	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_4$ > ${\mathcal{M}}_3$
Proposed p,qROFFWA operator	0.5374	0.5026	0.4608	0.4565	0.4757	0.4723	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$
Proposed p,qROFFWG operator	0.4798	0.4642	0.4295	0.4086	0.4287	0.4339	${\mathcal{M}}_1$ > ${\mathcal{M}}_2$ > ${\mathcal{M}}_6$ > ${\mathcal{M}}_5$ > ${\mathcal{M}}_3$ > ${\mathcal{M}}_4$

CL^p,qROFWA: confidence-level-based ^p,qROF weighted averaging; CL^p,qROFWG: confidence-level-based ^p,qROF weighted geometric; ^p,qROFHWA: ^p,qROF Hamacher weighted averaging; ^p,qROFHWG: ^p,qROF Hamacher weighted geometric; ^p,qROFDWA: ^p,qROF Dombi weighted averaging; ^p,qROFDWG: ^p,qROF Dombi weighted geometric; ^p,qROFAAWA: ^p,qROF Aczel–Alsina weighted averaging; ^p,qROFAAWG: ^p,qROF Aczel–Alsina weighted geometric; q-ROFWPM: q-rung orthopair fuzzy weighted power mean.

(i)

The ranking outcomes of the six materials utilizing the ^p,qROFWA operator [8] and the proposed ^p,qROFFWG operator are identical; however, they exhibit slight differences when compared to the results obtained using the proposed ^p,qROFFWA operator. It is also notable that the results derived through the ^p,qROFWG operator [8] are almost identical to those obtained via the devised ^p,qROFFWG operator, with the only discrepancy being the ranking order between ${\mathcal{M}}_5$ and ${\mathcal{M}}_6$. Both the ^p,qROFWA and ^p,qROFWG operators are special cases of the proposed ^p,qROFFWA and ^p,qROFFWG operators, respectively, and can be derived when the Frank parameter $ß\to 1$. Hence, our method demonstrates greater generality. Furthermore, the proposed operators, based on Frank t-norms, exhibit increased robustness and effectively capture the relationships between the arguments. The slight differences in material ranking are acceptable, as the ranking of alternatives depends on the parameter values and can change up to a specific threshold. Additionally, when $ß\to \infty$, the proposed operators reduce to those based on the Łukasiewicz product and Łukasiewicz sum [25]. Therefore, the proposed Frank-based AOs can encompass almost all arithmetic and geometric AOs for ^p,qROFS, depending on the value of parameter ß.

(ii)

To apply the Rahim et al. [12] confidence-level operators to the presented problem, equal confidence levels were assigned to all DEs and criteria due to the unavailability of data on varying confidence levels. The data presented in Table 11 indicate that the CL^p,qROFWA operator suggested by Rahim et al. ranks material ${\mathcal{M}}_5$ as the best choice. However, their CL^p,qROFWG operator ranks ${\mathcal{M}}_5$ as the second-worst alternative. While differing rankings due to the nature of the operators are not uncommon, the substantial difference between being the best and the second-worst choice is notable. Notably, the developed operators offer greater generalization by including a Frank parameter ß. However, existing operators also have the advantage over the deployed ones of incorporating confidence levels, which can be beneficial in real-world scenarios where confidence in DEs may vary.

(iii)

In comparison with the ^p,qROFHWA and ^p,qROFHWG operators of [14], and the ^p,qROFDWA and ^p,qROFDWG operators of [15], it is evident that, similar to the proposed operators, the best alternative identified by these operators, except for the ^p,qROFDWA operator, is also ${\mathcal{M}}_1$. This consistency verifies the validity of the developed framework. Like the developed operators, these existing operators incorporate parameters—specifically Hamacher and Dombi—which adjust the aggregate value based on specific decision-making needs. However, based on the Frank norm, the proposed operators offer a significant advantage. The Frank t-norm and t-conorm provide greater flexibility than the Dombi and Hamacher norms, thus the ^p,qROFFWA and ^p,qROFFWG operators exhibit higher generality and flexibility in data aggregation. Furthermore, these operators can accommodate the relationships between various arguments, enhancing their applicability in diverse decision-making scenarios.

(iv)

From Ref. [10], it can be verified that, similar to the designed Frank operators, the Aczel–Alsina-based operators also possess the advantage of monotonicity. This property allows DEs to select appropriate values according to their risk preferences. As shown in Table 11, the most appropriate material identified by both the ^p,qROFAAWG [10] and q-ROFWPM [34] operators is ${\mathcal{M}}_1$, consistent with the results derived from the formulated operators. However, the main advantage of the developed method over these existing approaches is the incorporation of the proposed SWARA technique for determining criteria weights. The derived weights are then utilized in the decision-making process, enhancing the overall robustness and accuracy of the outcomes.

In light of the discussion, the proposed approach has the following key merits:

(i)

The operators introduced in this study are more general than those found in Ref. [8]. Unlike the p,q-ROFWA and p,q-ROFWG operators, which are based on algebraic t-norms and t-conorms, the operators developed here utilize Frank t-norms and t-conorms. Notably, as $ß\to 1$, these operators simplify to the Seikh and Mandal operators [8].

(ii)

The developed method effectively captures the relationship between the arguments and includes a parameter ß that can be adjusted to tailor the aggregate value according to specific decision-making needs.

(iii)

Unlike the existing methods in [10,12,14,15], the deployed approach is capable of functioning even when criteria weights are unknown. This capability is enabled by combining the SWARA method for determining weights with the proposed AOs for ranking alternatives within the ^p,qROF environment.

Our research offers valuable insights into the theory and practical use of ^p,qROFSs and Frank AOs; however, it is essential to recognize some limitations within the current study.

(i)

The introduced AOs may exhibit context-specific applicability, and their performance might vary across different problem domains. Further experimentation and evaluation across varied scenarios are required to assess the broader utility of these approaches.

(ii)

Although the proposed operations and AOs address certain complexities in decision making, they may sometimes not fully account for all relevant factors or uncertainties. Our models need ongoing refinement and expansion to ensure their applicability to a wider range of real-world challenges.

8. Conclusions

In this paper, we first extended the Frank t-norm and t-conorm to ^p,qROF environments, introduced several new operational laws for ^p,qROFNs, and analyzed their properties and interrelationships. Building on these operational laws, we then developed a range of new AOs, including ^p,qROFFWA, ^p,qROFFOWA, ^p,qROFFHA, ^p,qROFFWG, ^p,qROFFOWG, and ^p,qROFFHA, specifically designed for scenarios where the arguments are ^p,qROFNs. We explored these operators’ desirable properties and special cases in detail and examined their interconnections. Furthermore, utilizing the ^p,qROFFWA and ^p,qROFFWG operators, we proposed an approach to solving ^p,qROF MCDM problems. We incorporated the SWARA method into the MCDM framework to determine the criteria weights. Subsequently, we discussed a real-world MCDM problem concerning the ’sustainable choice of materials in the construction industry’ to demonstrate our framed methodology’s applicability, feasibility, and superiority. The illustrated example shows that our decision-making procedure is more flexible and effective than existing methods due to the inclusion of operational parameters p, q, and ß, which can be adjusted based on the DEs’ preferences.

In the future, it is possible to continue the developed theory for other FSs such as r, s, and t-spherical FS [35], q-rung picture FS [36], circular intuitionistic FS [37], (a,b)-fuzzy soft set [38], and complemental FS [39]. Various other criteria weight determination techniques, such as best–worst, ITARA, OPA, etc., can also be extended to the ^p,qROF context to solve some other real-life decision-making problems.