Quasirung orthopair fuzzy linguistic sets and their application to multi criteria decision making

Abstract

Linguistic term fuzzy sets provide an intuitive way to express preferences, enhancing understanding and communication among decision-makers. In this article, we introduce the novel concept of p,q-quasirung orthopair fuzzy linguistic sets (p,q-QOFLSs), which merge the principles of p,q-quasirung orthopair fuzzy sets (p,q-QOFSs) with linguistic fuzzy sets. This new framework offers a more robust approach to handle uncertain and imprecise information in decision-making processes, characterized by linguistic membership and non-membership degrees. We establish several fundamental operational laws, alongside score and accuracy functions, to facilitate the comparison of p,q-quasirung orthopair fuzzy linguistic numbers. Leveraging these operational laws, we propose a series of weighted averaging and geometric operators under p,q-QOFLSs. Furthermore, we formulate a multi-attribute decision-making methodology using these operators. The significance of the proposed method lies in its ability to model complex decision-making scenarios with enhanced precision. A numerical example validates the practicality and adaptability of the methodology, supported by sensitivity analyses and comparative evaluations, highlighting the innovation and efficiency of the approach.

Introduction

Multi-Criteria decision-making (MCDM) involves selecting the most optimal choice from a limited set of alternatives, typically comprising multiple decision criteria and decision makers (DMs). The theory of MCDM has been widely applied across various fields, including economics¹, management², and engineering and Manufacturing^3,4, among others^5,6. Effectively and reasonably representing the evaluation values of decision makers (DMs) in real-world decision-making scenarios is a critical challenge. Due to the uncertainty, complexity, and variability inherent in decision-making environments and practical problems, attribute values, particularly qualitative ones, often cannot be represented accurately using crisp numbers. To address this issue, Zadeh⁷ introduced fuzzy set theory. Fuzzy sets consist solely of membership degrees ($\vartheta$) that belong to the unit interval $\left[ {0,1} \right]$. Numerous scholars have extensively explored the concept of fuzzy sets and proposed a range of extensions⁸, aggregation operators⁹ and similarity measures^10,11 aimed at enhancing their flexibility, applicability, and accuracy in addressing complex real-world problems. Atanassov¹² introduced intuitionistic fuzzy set (IFS) to assess alternatives in MCDM, considering both membership and non-membership degree ($\zeta$) simultaneously. A notable feature of IFS is that the sum of membership degree and non-membership degree does not exceed 1 ($\vartheta + \zeta { \preccurlyeq }1$). As an extension of IFS, Yager¹³ introduced Pythagorean fuzzy set (PFS), offering an effective approach to address uncertainty and fuzziness by considering both the membership degree and non-membership degree simultaneously. PFS satisfies the constrained condition where $\vartheta^{2} + \zeta^{2} { \preccurlyeq }1$. While both PFS and IFS are effective in describing the uncertainty and fuzziness of MAGDM problems, PFS is notably more effective and accurate in handling these problems compared to IFS. PFS can address certain decision-making challenges that IFS cannot, rendering it more versatile in application. With the rise of the big data era, practical problems and real-life scenarios have become increasingly complex. In certain instances, both PFS and IFS may struggle to effectively manage specific types of data. To address this challenge, Yager¹⁴ relaxed the stringent conditions on membership and non-membership degrees inherent in PFS and introduced the q-rung orthopair fuzzy set (q-ROFS). q-ROFS relaxes the restrictions IFS ($\vartheta + \zeta { \preccurlyeq }1$) and PFS $\vartheta^{2} + \zeta^{2} { \preccurlyeq }1$ to $\vartheta^{q} + \zeta^{q} { \preccurlyeq }1$ ($q{ \succcurlyeq }1$). Subsequently, numerous extensions of the q-ROFS have been developed. For example, Farid and Riaz¹⁵ introduced q-rung orthopair fuzzy aggregation operators utilizing Aczel-Alsina operations. Yang et al.¹⁶ studied interval-valued q-rung orthopair fuzzy soft sets, introducing interval-valued q-rung orthopair fuzzy soft numbers, and investigated corresponding weighted averaging and geometric operators. Joshi and Gegov¹⁷ introduced aggregation operators that integrate the familiarity degree of experts with the evaluated objects within the q-rung orthopair fuzzy environment. Seikh and Mandal¹⁸ introduced q-Rung orthopair fuzzy Archimedean AOs and explored their application in selecting suitable sites for software operating units. Özlü¹⁹ introduced q-rung orthopair fuzzy Aczel–Alsina weighted geometric operators for MCDM problems. Özlü²⁰ also developed q-rung orthopair probabilistic hesitant fuzzy hybrid operators for application in MCDM problems. Researchers have utilized these extensions in various practical applications, as outlined in references^{21,22,23,24,25}.

From the preceding discussion, it is evident that in the $q-$ROF situation, decision-makers are bounded to utilizing identical values of $q$ for both membership and non-membership aspects during the DM process, potentially impacting the entire decision-making procedure. To alleviate these constraints, Seikh and Mandal²⁶ presented the concept of $p,q$-QOFSs, representing a generalization of $q$-ROFSs. Within the realm of $p,q$-quasirung orthopair fuzzy ($p,q$-QOF) environments, decision-makers gain the flexibility to assign different values to the parameters $p$ and $q$, offering nuanced control over the influence of MD and NMD based on the prevailing circumstances, subject to the condition that $\vartheta^{p} + \zeta^{q} { \preccurlyeq }1$ ($p,q{ \succcurlyeq }1$). Several aggregation operators and MCDM techniques have been formulated and are frequently applied across diverse fields, leveraging the principles of $p,q$-QOFSs. Rahim et al.²⁷ introduced a set of confidence level-based aggregation operators within the $p,q$-QOF environment. Expanding on this work, Ali and Naeem²⁸ put forth Aczel–Alsina AOs specifically tailored for $p,q$-QOF numbers ($p,q$-QOFNs). Rahim et al.²⁹ presented a series of AOs by integrating Dombi t-norm, t-conorm and $p,q$-QOF numbers to address MCDM problems in $p,q$-QOF environment. Chu et al.³⁰ presented the notion of cubic $p,q$-QOF sets, further enriching the landscape of FS approaches. Moreover, Rahim et al.³¹ contributed to this line of research by proposing some AOs based on sine trigonometric functions, providing an enhanced toolkit for decision-makers working within the $p,q$-QOF framework. Seikh and Mandal³² proposed 3,4-quasirung fuzzy sets as a generalization of PFSs and q-ROFSs.

The relationship between different extension of fuzzy sets is presented in Fig. 1.

Fig. 1

The interrelation among IFS, PFS, q-ROFS and $p,q$-quasirung orthopair fuzzy sets.

From this illustration, it becomes evident that when both $p$ and $q$ equal $1$ ($\vartheta^{1} + \zeta^{1} { \preccurlyeq }1$), $p,q$-QOFSs converge to an IFS environment; similarly, when $p$ and $q$ are both $2$ ($\vartheta^{2} + \zeta^{2} { \preccurlyeq }1$), they simplify to a PFS environment. Furthermore, when $p$ equals $q$ ($\vartheta^{q} + \zeta^{q} { \preccurlyeq }1$), $p,q$-QOFSs transform into a q-rung orthopair fuzzy set environment. Consequently, $p,q$-QOFSs stand out as the most comprehensive version, accommodating decision makers with the flexibility to employ diverse term levels for membership grades. From this discussion, it’s apparent that the studies are predominantly quantitative in nature.

Linguistic sets: an overview

In real-life scenarios, many decision-making problems involve qualitative aspects that represent uncertainty and imprecise information. For instance, when assessing a person’s "intelligence," decision-makers often prefer using linguistic labels like "very low," "low," "medium," "high," "very high," or "perfect." In such cases, decision-makers can utilize linguistic variables to express their opinions regarding the objects. Zadeh^33,34 responded to this linguistic complexity by introducing fuzzy linguistic methods. As practical problems become increasingly complex, and linguistic evaluation information becomes potentially vague and imprecise, decision-makers may lack comprehensive knowledge and experience in assessing the preference degrees of DM alternatives. Wang and Li³⁵ proposed IF linguistic (IFL) sets (IFLSs), while Liu et al.³⁶ designed a set of aggregation operators for IFLSs to deal with MCDM problems. Xian et al.³⁷ extended the idea of IFLSs and presented interval-valued IFLSs (IVIFLSs). They also, introduced TOPSIS method for IVIFLSs to deal with uncertainties in DM process. Zhang³⁸ proposed a set of AOs by incorporating t-norm and t-conorm to handle IFL information. Pang and Yang³⁹ proposed PF linguistic (PFL) sets and their AOs to solve MCSDM problems. Teng et al.⁴⁰ introduced Maclaurin symmetric AOs based on PFL sets to aggregation in the context of PFL numbers. Du et al.⁴¹ presented the idea of interval-valued PFLSs and their AOs for MCDM problems. Liu et al.⁴² integrated uncertain linguistic variables and PFSs to proposed PF uncertain linguistic sets. Also, they proposed prioritized averaging and Maclaurin symmetric operators to handle PFL information. Liu et al.⁴³ developed partitioned Bonferroni mean operators in PFL environment. Xian et al.⁴⁴ proposed a series of AOs based on entropic and trapezoidal PFL variables. Jana et al.⁴⁵ presented a linguistic q-rung orthopair fuzzy Choquet integral approach for assessing sustainable strategies in urban parcel delivery. Xu⁴⁶ introduced the concept of continuous linguistic term sets to describe it. Zhang³⁸ combined linguistic approach and IFS concepts to define the linguistic IFS (LIFS), allowing the expression of membership and non-membership degrees through linguistic terms. Xu⁴⁷ also introduced linguistic hybrid arithmetic averaging aggregation operators. Garg and Kumar⁴⁸ proposed geometric aggregation operators for LIFS using set pair analysis theory.

Based on the preceding discussion, it’s evident that the studies mentioned rely on IFLS, PFLS, and $q$-ROFLS. However, there are instances where decision-makers may require varying term levels for membership grades. To address this challenge, we extend the concept of q-ROFLS and introduce $p,q$-QOFLS.

$p,q$-QOFLSs were selected as the research focus because they offer a more refined and versatile approach to managing uncertainty and imprecision, particularly in decision-making scenarios involving linguistic terms. Traditional fuzzy sets often fall short when it comes to handling complex or vague information. By combining the strengths of $p,q$-quasirung orthopair fuzzy sets with linguistic fuzzy sets, $p,q$-QOFLSs provide a more effective framework for modeling real-world problems where precise numerical data may not be readily available. The practical relevance of $p,q$-QOFLSs is evident in their ability to enhance decision-making processes in situations where experts’ express opinions using subjective linguistic terms like "high," "moderate," or "low." This is valuable in domains such as risk assessment, resource management, and strategic planning, where decisions are based on incomplete or imprecise information. The proposed approach not only improves decision accuracy but also offers a more intuitive and flexible tool for decision-makers, making it well-suited for a wide range of practical applications.

Motivations

Recently, Seikh and Mandal²⁶ proposed $p,q$-QOFSs which is the extension of IFS, PFS, and q-rung orthopair fuzzy sets. While researchers have extensively utilized theories on the $p,q$-QOFS, their analyses have predominantly focused on the quantitative aspect. Nevertheless, in practical contexts, decision-makers might consider employing linguistic variables, represented as $\mathcalligra{s} = \left( {\mathcalligra{s}_{a} ,\mathcalligra{s}_{b} } \right)$, where $\mathcalligra{s}_{a}$ and $\mathcalligra{s}_{b}$ denote the linguistic variables for membership and non-membership degrees, respectively. This approach offers a more standardized method for expressing uncertainty in the decision-making process.

Drawing inspiration from IFLS, PFLS and linguistic q-rung orthopair fuzzy set, in this paper, we introduce the concept of $p,q$-quasirung orthopair fuzzy linguistic set ($p,q$-QOFLS). In this framework, the membership degree ($\vartheta$) and non-membership degree ($\zeta$) of an element to a linguistic variable are represented by a $p,q$-quasirung orthopair fuzzy number ($p,q$-QOFN), adhering to the condition $\vartheta^{q} + \zeta^{q} { \preccurlyeq }1$ (where $p,q{ \succcurlyeq }1$). The $p,q$-QOFLS offers solutions to certain problems that IFLS, PFLS and linguistic q-rung orthopair fuzzy set cannot address, thus extending the capabilities of these frameworks. Consequently, IFLS, PFLS and linguistic q-rung orthopair fuzzy set emerge as special cases of $p,q$-QOFLS.

To date, there has been a lack of research on $p,q$-QOFLS, highlighting the importance of focusing more attention on this area. Inspired by this realization, the motivation and objective of this paper are:

• 1. Introduce fundamental concepts of $p,q$-QOFLS and formulate novel operational laws, score and accuracy function, and distance measurement techniques for $p.q$-QOFLS.

• 2. Formulate a series of aggregation operators to aggregate $p,q$-QOFL information Also, look into interesting features, and study particular cases and special traits of these operators.

• 3. Introduce a novel approach based on the presented aggregation operators, elucidate the efficacy and flexibility of the presented method through a practical example.

Paper outline

The article is structured as follows: Section “Preliminaries” provides a review of pertinent basic definitions. Section “p,q-quasirung orthopair fuzzy linguistic sets” introduces the $p,q$-QOF linguistic set, delving into detailed discussions on operation laws and novel comparison methods for $p,q$-ROFLS and their aggregation operators. Section “Proposed MCDM for p,q-QOFL information” propose a new approach based on the proposed aggregation operators. Section “Illustrative example” demonstrating the effectiveness and flexibility through a practical example. Finally, Section “Conclusion” serves as the conclusion of the of the proposed work. The step-by-step pathway of the proposed work is depicted in Fig. 2.

Fig. 2

Layout of the article.

Preliminaries

The following presents a review of some fundamental definitions related to Linguistic Term Sets (LTS), Linguistic Scale Functions (LSF), and $p,q$-QOFS.

Linguistic term set

Consider ${\mathcal{S}} = \left\{ {\mathcalligra{s}_{k} \left| {k = 0,1, \ldots 2\mathcalligra{g} + 1} \right.} \right\}$ be a partial Linguistic Term Set (LTS), where $2\mathcalligra{g} + 1$ represents the cardinality of ${\mathcal{S}}$ and $\mathcalligra{s}_{k}$ symbolizes a potential value of a linguistic term. An example of a partial LTS ${\mathcal{S}}$ with eight cardinalities can be represented as follows: $\begin{gathered} {\mathcal{S}} = \left\{ {\mathcalligra{s}_{0} = {\text{extremely}}\;{\text{low}},\mathcalligra{s}_{1} = {\text{very}}\;{\text{low}},\mathcalligra{s}_{2} = {\text{low}},\mathcalligra{s}_{3} = {\text{neutral}},\mathcalligra{s}_{4} = {\text{moderately}}\;{\text{high}},\mathcalligra{s}_{5} = {\text{high}},\mathcalligra{s}_{6} = {\text{very}}\;{\text{high}},} \right. \hfill \\ \left. {\mathcalligra{s}_{7} = {\text{extremely}}\;{\text{high}}.} \right\} \hfill \\ \end{gathered}$

Typically, a partial LTS adheres to the following situations^49,50:

• 1. Order: $\mathcalligra{s}_{k} \succ \mathcalligra{s}_{l}$ if $k \succ l$,

• 2. Negative operator: ${\text{Negative}}\left( {\mathcalligra{s}_{k} } \right) = \mathcalligra{s}_{l}$, where $l = 2\mathcalligra{g} - k$.

• 3. Minimize operator: $\min \left( {\mathcalligra{s}_{k} ,\mathcalligra{s}_{l} } \right) = \mathcalligra{s}_{k}$, if $\mathcalligra{s}_{k} { \preccurlyeq }\mathcalligra{s}_{l}$,

• 4. Maximize operator: $\max \left( {\mathcalligra{s}_{k} ,\mathcalligra{s}_{l} } \right) = \mathcalligra{s}_{k}$, if $\mathcalligra{s}_{k} { \succcurlyeq }\mathcalligra{s}_{l}$.

In order to mitigate the loss of decision statistics, the discrete LTS underwent generalization to form a successive LTS ${\hat{\mathcal{S}}} = \left\{ {\mathcalligra{s}_{\mathcalligra{x} } \left| {\mathcalligra{s}_{0} { \preccurlyeq }\mathcalligra{s}_{\mathcalligra{x}} { \preccurlyeq }\mathcalligra{s}_{2\mathcalligra{g} } } \right.} \right\}$, where $\mathcalligra{x} \in \left[ {0,2\mathcalligra{g} } \right]$ and $2\mathcalligra{g}$ is a sufficiently large positive integer. For two consecutive linguistic variables (LVs) $\mathcalligra{s}_{a}$, $\mathcalligra{s}_{b} \in {\hat{\mathcal{S}}}$, the operations are stipulated as follows:

• 1. $\mathcalligra{s}_{a} \oplus \mathcalligra{s}_{b} = \mathcalligra{s}_{a + b}$,

• 2. $\mathcalligra{s}_{a} \otimes \mathcalligra{s}_{b} = \mathcalligra{s}_{a \times b}$,

• 3. $\eta \mathcalligra{s}_{a} = \mathcalligra{s}_{\eta a}$,

• 4. $\mathcalligra{s}_{a}^{\eta } = \mathcalligra{s}_{{a^{\eta } }}$.

The distance measure between two consecutive LVs $\mathcalligra{s}_{a}$, $\mathcalligra{s}_{b} \in {\hat{\mathcal{S}}}$ is defined as follows⁵¹:

$$\mathcalligra{d} \left( {\mathcalligra{s}_{a} ,\mathcalligra{s}_{b} } \right) = \frac{{\left| {a - b} \right|}}{2\mathcalligra{g} }$$

(1)

Linguistic scale functions

Linguistic scale functions (LSFs) offer a flexible means of expressing semantics in various situations, making them widely employed in practical decision-making processes.

Definition 1

(Wang et al.⁵²) Consider ${\mathcal{S}} = \left\{ {\mathcalligra{s}_{k} \left| {k = 0,1,2, \ldots ,2\mathcalligra{g} } \right.} \right\}$ be a partial LTS, where $2\mathcalligra{g} + 1$ represents the cardinality of ${\mathcal{S}}$, $\psi_{k} \in \left[ {0,1} \right]$ $\left( {k = 0,1,2, \ldots ,2\mathcalligra{g} } \right)$ is a real number, then LSF is defined as follows:

$${\mathcal{U}}:\mathcalligra{s}_{k} \to \psi_{k} \;\left( {k = 0,1,2, \ldots ,2\mathcalligra{g} } \right)$$

(2)

where $0{ \preccurlyeq }\psi_{0} { \preccurlyeq }\psi_{1} { \preccurlyeq } \ldots { \preccurlyeq }\psi_{2\mathcalligra{g} } { \preccurlyeq }1$, and ${\mathcal{U}}$ represents a strictly monotonically increasing function with respect to the linguistic subscript $k$. The values $\psi_{k}$ (where $k = 0,1,2, \ldots ,2\mathcalligra{g}$) signify the preferences of Decision Makers ($\widetilde{{{\text{DM}}}}$) when applying the linguistic terms $\mathcalligra{s} \in {\mathcal{S}}$. Evidently, the value of the function ${\mathcal{U}}$ captures the semantics of the linguistic terms. Three LSFs can be articulated as follows⁵²:

Linguistic subscript function

$${\mathcal{U}}_{1} \left( {\mathcalligra{s}_{k} } \right) = \psi_{k} = \frac{k}{2\mathcalligra{g} }\;\left( {k = 0,1,2, \ldots ,2\mathcalligra{g} } \right)$$

(3)

In this context, ${\psi }_{k}\in [{0,1}]$ and function $\mathcal{U}$'s value, as mentioned earlier, is computed as an average. While the linguistic subscript function is straightforward and comprehensible, it may fall short of meeting the requirements of progressively intricate practical issues. To tackle intricate DM problems effectively, Wang et al.⁵² introduced a composite function tailored for situations where the absolute for situations where dispersion between adjacent linguistic subscripts progressively increase as the LTS expands from its mid to terminations.

Compound evaluation scale function

$${\mathcal{U}}_{2} \left( {\mathcalligra{s}_{k} } \right) = \left\{ \begin{gathered} \frac{{\mu^{\mathcalligra{g} } - \mu^{\mathcalligra{g} - k} }}{2g - 2},\quad \quad \;\;\left( {k = 0,1,2, \ldots ,\mathcalligra{g} } \right) \hfill \\ \frac{{\mu^{\mathcalligra{g} } + \mu^{k - \mathcalligra{g} } - 2}}{{2\mu^{\mathcalligra{g} } - 2}},\quad \left( {k = \mathcalligra{g} + 1,\mathcalligra{g} + 2, \ldots ,2\mathcalligra{g} } \right) \hfill \\ \end{gathered} \right.$$

(4)

In this context $\psi_{k} \in \left[ {0,1} \right]$. Typically, a $\mu = \sqrt[ \mathcalligra{i} ]{ \mathcalligra{m} }$, where $\mathcalligra{m}$ denotes the ratio signifying that criterion $\mathcalligra{c}_{1}$ is more crucial than criterion $\mathcalligra{c}_{2}$, and $\mathcalligra{i}$ signifies the scale level. The superior boundary value for $\mathcalligra{m}$ is $\mathcalligra{m} = 9$, and existing experimental survey data recommends that a falls within the range $\left[ {1.36,1.4} \right]$. In the case of $\mathcalligra{i} = 7$, $\mu$ is calculated as $\mu = \sqrt[7]{9} = 1.37[7]$${9} = 1.37$. For further details, please refer to literature⁵². Wang et al.⁵² also introduced a reformative evaluation scale function tailored for situations where the absolute dispersion between adjacent linguistic subscripts diminishes as the LTS expands from its middle to endpoints.

Reformative evaluation scale function

$${\mathcal{U}}_{3} \left( {\mathcalligra{s}_{k} } \right) = \psi_{k} = \left\{ \begin{gathered} \frac{{\mathcalligra{g}^{a} - \left( {\mathcalligra{g} - k} \right)^{a} }}{{2\mathcalligra{g}^{a} }}, \quad \left( {k = 0,1,2, \ldots ,\mathcalligra{g} } \right) \hfill \\ \frac{{\mathcalligra{g}^{b} - \left( {k - \mathcalligra{g} } \right)^{b} }}{{2\mathcalligra{g}^{b} }},\quad \;\left( {k = \mathcalligra{g} + 1,\mathcalligra{g} + 2, \ldots ,2\mathcalligra{g} } \right) \hfill \\ \end{gathered} \right.$$

(5)

In this context, $a$ and $b$ belong to the interval $\left[ {0,1} \right]$. If $a = b = 1$, then ${\mathcal{U}}_{3} \left( {\mathcalligra{s}_{k} } \right) = {\mathcal{U}}_{1} \left( {\mathcalligra{s}_{k} } \right)$. To minimize the loss of decision data and facilitate calculation, the aforementioned occupations can be prolonged to ${\mathcal{U}}^{*} :{\hat{\mathcal{S}}} \to {\mathbb{R}}^{ + }$$\left( {{\mathbb{R}}^{ + } = \left\{ {\mathcalligra{r} \left| {\mathcalligra{r} { \succcurlyeq }0, \mathcalligra{r} \in {\mathbb{R}}} \right.} \right\}} \right)$.

Definition 2

The distance measure between two successive LVs, denoted as $\mathcalligra{s}_{b}$, $\mathcalligra{s}_{a} \in {\hat{\mathcal{S}}}$, and belonging is defined based on LSF as follows:

$$\mathcalligra{d} \left( {\mathcalligra{s}_{a} ,\mathcalligra{s}_{b} } \right) = \left| {{\mathcal{U}}^{*} \left( {\mathcalligra{s}_{a} } \right) - {\mathcal{U}}^{*} \left( {\mathcalligra{s}_{b} } \right)} \right|$$

(6)

where ${\mathcal{U}}^{*}$ represents the Linguistic Scale Function.

$\varvec{p},\varvec{q}$-quasirung orthopair fuzzy sets

Definition 3

(Seikh and Mandal²⁶) Let $F$ be any finite set. A $q-$QOF set $R$ for a component $\tau \in F$ in can be expressed as:

$$R = \left\{ {\tau ,\left( {\zeta_{R} \left( \tau \right),\vartheta_{R} \left( \tau \right)} \right)\left| {\tau \in F} \right.} \right\}$$

(7)

where $\zeta_{R} \left( \tau \right) \in \left[ {0,1} \right]$ denotes the MD, and $\vartheta_{R} \left( \tau \right) \in \left[ {0,1} \right]$ denotes the NMD of an element $\tau$ in set $R$, respectively such that $\left( {\zeta_{R} \left( \tau \right)} \right)^{p} + \left( {\vartheta_{R} \left( \tau \right)} \right)^{q} { \preccurlyeq }1$ ($p,q{ \succcurlyeq }1$). A $q$-ROF numbers can be represented as $R = \left( {\zeta_{R} ,\vartheta_{R} } \right)$ such that $\left( {\zeta_{R} } \right)^{p} + \left( {\vartheta_{R} } \right)^{q} { \preccurlyeq }1$ ($p,q{ \succcurlyeq }1$).

Remark 1

The parameters $p$ and $q$ in Definition 5 are positive integers, where $p$ can be less than, equal to, or greater than $q$.

Definition 4

(Seikh and Mandal²⁶) Let $R_{1} = \left( {\zeta_{{R_{1} }} ,\vartheta_{{R_{1} }} } \right)$, $R_{2} = \left( {\zeta_{{R_{2} }} ,\vartheta_{{R_{2} }} } \right)$ and $R = \left( {\zeta_{R} ,\vartheta_{R} } \right)$ be any three $q$-QOFNs, then

• 1. $R_{1} \vee R_{2} = \left( {\max \left( {\zeta_{{R_{1} }} ,\zeta_{{R_{2} }} } \right),\min \left( {\vartheta_{{R_{1} }} ,\vartheta_{{R_{2} }} } \right)} \right)$;

• 2. $R_{1} \wedge R_{2} = \left( {\min \left( {\zeta_{{R_{1} }} ,\zeta_{{R_{2} }} } \right),\max \left( {\vartheta_{{R_{1} }} ,\vartheta_{{R_{2} }} } \right)} \right)$;

• 3. $R_{1} \oplus R_{2} = \left( {\sqrt[p]{{\left( {\zeta_{{R_{1} }} } \right)^{p} + \left( {\zeta_{{R_{2} }} } \right)^{p} - \left( {\zeta_{{R_{1} }} } \right)^{p} \left( {\zeta_{{R_{2} }} } \right)^{p} }},\vartheta_{{R_{1} }} \vartheta_{{R_{2} }} } \right)$;

• 4. $R_{1} \otimes R_{2} = \left( {\zeta_{{R_{1} }} \zeta_{{R_{2} }} ,\sqrt[q]{{\left( {\vartheta_{{R_{1} }} } \right)^{q} + \left( {\vartheta_{{R_{2} }} } \right)^{q} - \left( {\vartheta_{{R_{1} }} } \right)^{q} \left( {\vartheta_{{R_{2} }} } \right)^{q} }}} \right)$;

• 5. $\eta Q_{1} = \left( {\sqrt[p]{{1 - \left( {1 - \left( {\zeta_{{R_{1} }} } \right)^{p} } \right)^{\eta } }},\left( {\vartheta_{{R_{1} }} } \right)^{\eta } } \right)$, $\eta { \succcurlyeq }0$;

• 6. $\left( {Q_{1} } \right)^{\eta } = \left( {\left( {\zeta_{{R_{1} }} } \right)^{\eta } ,\sqrt[q]{{1 - \left( {1 - \left( {\vartheta_{{R_{1} }} } \right)^{q} } \right)^{\eta } }}} \right)$, $\eta { \succcurlyeq }0$.

Definition 5

(Seikh and Mandal²⁶) Consider a $p,q$-QOFN denoted as $R = \left( {\zeta_{R} ,\vartheta_{R} } \right)$, the score functions $SCO\left( R \right)$ a can be represented as:

$$SCO\left( R \right) = \frac{{1 + \left( {\zeta_{R} } \right)^{p} - \left( {\vartheta_{R} } \right)^{q} }}{2}$$

(8)

where $0{ \preccurlyeq }SCO\left( R \right){ \preccurlyeq }1$ ($p,q{ \succcurlyeq }1$). A higher value of $SCO\left( R \right)$ a indicates a greater $p,q$-QOFN $R$. If the score values of different $p,q$-QOFNs are equal, then we need to determine the accuracy value by employing an accuracy function to compare any two $p,q$-QOFNs. For example, $R_{1} = \left( {0.40,0.40} \right)$ and $R_{2} = \left( {0.50,0.50} \right)$, then by Eq. (8) $SCO\left( {R_{1} } \right) = 0.50$ and $SCO\left( {R_{2} } \right) = 0.50$.

Definition 6

(Seikh and Mandal²⁶) Consider a $p,q$-QOFN denoted as $R = \left( {\zeta_{R} ,\vartheta_{R} } \right)$, the accuracy functions $ACC\left( R \right)$ a can be represented as:

$$ACC\left( R \right) = \left( {\zeta_{R} } \right)^{p} + \left( {\vartheta_{R} } \right)^{q}$$

(9)

where $0{ \preccurlyeq }ACC\left( R \right){ \preccurlyeq }1$ ($p,q{ \succcurlyeq }1$). A higher value of $ACC\left( R \right)$ a indicates a greater $p,q$-QOFN $R$.

Definition 7

(Seikh and Mandal²⁶) Assume that $R_{1}$ and $R_{2}$ are any two $p,q$-QOFNs, then

• 1. $R_{1} \prec R_{2}$, if $SCO\left( {R_{1} } \right) \prec SCO\left( {R_{2} } \right)$;

• 2. $R_{1} \succ R_{2}$, if $SCO\left( {R_{1} } \right) \succ SCO\left( {R_{2} } \right)$;

• 3. If $SCO(R_{1} ) = SCO\left( {R_{2} } \right)$, then

  • $R_{1} \prec R_{2}$, if $ACC\left( {R_{1} } \right) \prec ACC\left( {R_{2} } \right)$;

  • $R_{1} \succ R_{2}$, if $ACC\left( {R_{1} } \right) \succ ACC\left( {R_{2} } \right)$;

  • If $ACC\left( {R_{1} } \right) = ACC\left( {R_{2} } \right)$, and $SCO\left( {R_{1} } \right) = SCO\left( {R_{2} } \right)$, then $R_{1} = R_{2}$.

Definition 8

(Seikh and Mandal²⁶) Let $R_{1} = \left( {\zeta_{{R_{1} }} ,\vartheta_{{R_{1} }} } \right)$, $R_{2} = \left( {\zeta_{{R_{2} }} ,\vartheta_{{R_{2} }} } \right)$ be any $p,q$-QOFNs. Then Hamming distance (HM) between these to $p,q$-QOFNs can be calculated as follows:

$$\mathcalligra{d} \left( {R_{1} ,R_{2} } \right) = \frac{1}{2}\left[ {\left| {\left( {\zeta_{{R_{1} }} } \right)^{p} - \left( {\zeta_{{R_{2} }} } \right)^{p} } \right| + \left| {\left( {\vartheta_{{R_{1} }} } \right)^{q} - \left( {\vartheta_{{R_{2} }} } \right)^{q} } \right| + \left| {\left( {{\Pi }_{{R_{1} }} } \right)^{\mathcalligra{z} } - \left( {{\Pi }_{{R_{2} }} } \right)^{\mathcalligra{z} } } \right|} \right]$$

(10)

where parameters $p$ and $q$ are any positive integers and $\mathcalligra{z} = LCM\left( {p,q} \right)$. Also, ${\Pi }_{{R_{1} }}$ and ${\Pi }_{{R_{2} }}$ represents the degree of indeterminacy of $\tau$ in relation to $F$ and can be expressed as follows:

$$\begin{gathered} {{\Pi }}_{{R_{1} }} = \sqrt[\exists ]{{\left( {\zeta _{{R_{1} }} } \right)^{\exists } + \left( {\vartheta _{{R_{1} }} } \right)^{\exists } - \left( {\zeta _{{R_{1} }} } \right)^{\exists } \left( {\vartheta _{{R_{1} }} } \right)^{\exists } }}, \hfill \\ {{\Pi }}_{{R_{2} }} = \sqrt[\exists ]{{\left( {\zeta _{{R_{2} }} } \right)^{\exists } + \left( {\vartheta _{{R_{2} }} } \right)^{\exists } - \left( {\zeta _{{R_{2} }} } \right)^{\exists } \left( {\vartheta _{{R_{2} }} } \right)^{\exists } }}\left( {\exists = LCM\left( {p,q} \right)} \right). \hfill \\ \end{gathered}$$

Remark 1

(Seikh and Mandal²⁶) Consider the scenario where we need to determine the minimum values of $p$ and $q$, where $p,q\ge 1$, for a given orthopair $\left({\zeta }_{R},{\vartheta }_{R}\right)$ such that ${{\zeta }_{R}}^{p}+{{\vartheta }_{R}}^{q}\le 1$. Although this problem does not have a closed-form solution, it is always possible to find a unique solution using iterative computational techniques. We define the minimum values of $p$ and $q$ that satisfy ${{\zeta }_{R}}^{p}+{{\vartheta }_{R}}^{q}\le 1$ as the $p,q$-niche of $\left( {\zeta_{R} ,\vartheta_{R} } \right)$. If $p_{o}$ and $q_{o}$ are the $p,q$-niche of $\left( {\zeta_{R} ,\vartheta_{R} } \right)$, then $\left( {\zeta_{R} ,\vartheta_{R} } \right)$ is a valid $p,q$-quasirung membership grade for all $p \ge p_{o}$ and $q \ge q_{o}$.

Let ${\mathcal{X}} = \left\{ {\mathcalligra{x}_{1} ,\mathcalligra{x}_{2} , \ldots ,\mathcalligra{x}_{n} } \right\}$ represent a set of data, and ${\mathcal{Y}}$ be a fuzzy concept. Suppose an expert expresses their preference for each $\mathcalligra{x}_{i} \in {\mathcal{X}}$ as an orthopair. The challenge is to determine the appropriate values of $p$ and $q$ to accurately represent this information. The process can be outlined as follows:

• 1. For each orthopair $\left( {\zeta_{{R_{i} }} ,\vartheta_{{R_{i} }} } \right)$, calculate its corresponding $p,q$-niche, denoted as $p_{i}$ and $q_{i}$.

• 2. Determine the $p^{ \circ }$ and $q^{ \circ } -$niche, where $p^{ \circ } = \max \left( {p_{i} } \right)$ and $q^{ \circ } = \max \left( {q_{i} } \right)$.

• 3. We can then represent ${\mathcal{Y}}$ as a $p^{ \circ } ,q^{ \circ } -$QOFS.

${\varvec{p}},{\varvec{q}}$-quasirung orthopair fuzzy linguistic sets

The In this section, we present fundamental operational laws for $p,q$-quasirung orthopair fuzzy linguistic sets ($p,q$-QOFLSs) along with their features. Building upon these operations, we introduce a set of AOs, namely $p,q$-quasirung orthopair fuzzy linguistic ($p,q$-QOFL) weighted averaging ($p,q$-QOFLWA), $p,q$-QOFL ordered weighted averaging ($p,q$-QOFLOWA), $p,q$-QOFL weighted geometric ($p,q$-QOFLWG), and $p,q$-QOFL ordered weighted geometric ($p,q$-QOFLOWG) operators. Additionally, we delve into a detailed discussion on various properties associated with the proposed operators.

${\varvec{p}},{\varvec{q}}-$QOF linguistic sets and their operational laws

Definition 9

Consider a nonempty set $F$, with ${\hat{\mathcal{S}}}$ being a successive LTS derived from ${\mathcal{S}} = \left\{ {\mathcalligra{s}_{k} \left| {k = 0,1,2, \ldots ,2\mathcalligra{g} } \right.} \right\}.$ In this context, a $p,q$-QOFL set ${\mathcal{I}}$ on $F$ can be represented as follows:

$${\mathcal{I}} = \left\{ {\left\langle {{\mathcal{S}}_{\psi \left( \tau \right)} ,\left( {\zeta_{{\mathcal{I}}} \left( \tau \right),\vartheta_{{\mathcal{I}}} \left( \tau \right)} \right)} \right\rangle \left| {\tau \in F} \right.} \right\}$$

(11)

Here, ${\mathcal{S}}_{\psi \left( \tau \right)} \in {\hat{\mathcal{S}}}$ represents a linguistic variable, while $\zeta_{{\mathcal{I}}} \left( \tau \right)$, $\vartheta_{{\mathcal{I}}} \left( \tau \right) \in \left[ {0,1} \right]$ denote the MD and NDM of $\tau$ to ${\mathcal{S}}_{\psi \left( \tau \right)}$, respectively such that

• 1. $\left( {\zeta_{{\mathcal{I}}} \left( \tau \right)} \right)^{p} + \left( {\vartheta_{{\mathcal{I}}} \left( \tau \right)} \right)^{q} { \preccurlyeq }1$ ($\tau \in F$),

• 2. $p \succ q$, $p \prec q$ or $p = q$.

To simplify, a $p,q$-QOF linguistic number ($p,q$-QOFLN) is denoted as ${\mathcal{I}} = \left( {{\mathcal{S}}_{\psi \left( \tau \right)} ,\left( {\zeta_{{\mathcal{I}}} \left( \tau \right),\vartheta_{{\mathcal{I}}} \left( \tau \right)} \right)} \right)$, where ${\mathcal{S}}$ represents the linguistic variable, $\psi \left( \tau \right)$ a is the linguistic subscript, $\zeta_{{\mathcal{I}}} \left( \tau \right)$ and $\vartheta_{{\mathcal{I}}} \left( \tau \right)$ are the MD and NMD respectively.

Definition 10

Let $\mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ and $\mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right)$ are two $p,q$-QOFLNs, then the operational laws for these $p,q$-QOFLNs are defined as follows:

• 1. $\mathcalligra{u} \oplus \mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right) + \psi \left( \mathcalligra{v} \right)} ,\left( {\sqrt[p]{{\begin{array}{*{20}c} {\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} + \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} } \\ { - \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} } \\ \end{array} }},\vartheta \left( \mathcalligra{u} \right)\vartheta \left( \mathcalligra{v} \right)} \right)} \right);$

• 2. $\mathcalligra{u} \otimes \mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right) \times \psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right)\zeta \left( \mathcalligra{v} \right),\sqrt[q]{{\begin{array}{*{20}c} {\left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} + \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} } \\ { - \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} } \\ \end{array} }}} \right)} \right);$

• 3. $\eta \mathcalligra{u} = \left( {{\mathcal{S}}_{\eta \psi \left( \mathcalligra{u} \right)} ,\left( {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} } \right)^{\eta } }},\left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{\eta } } \right)} \right), \left( {p{ \succcurlyeq }1,\eta \succ 0} \right);$

• 4. $\left( {{\mathcal{S}}_{{\psi \left( \mathcalligra{u} \right)^{\eta } }} ,\left( {\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{\eta } ,\sqrt[q]{{1 - \left( {1 - \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} } \right)^{\eta } }}} \right)} \right), \left( {q{ \succcurlyeq }1,\eta \succ 0} \right);$

• 5. $\mathcalligra{u}^{C} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\vartheta \left( \mathcalligra{u} \right),\zeta \left( \mathcalligra{u} \right)} \right)} \right).$

Definition 11

Let $\mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ be a $p,q$-QOFLN then the score function $SCO$ of $\mathcalligra{u}$ can be defined as follows:

$$SCO\left( \mathcalligra{u} \right) = \left( {\frac{{1 + \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} - \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} }}{2}} \right) \times {\mathcal{U}}^{*} \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} } \right),\;p,q{ \succcurlyeq }1\;{\text{and}}\;SCO\left( \mathcalligra{u} \right) \in \left[ {0,1} \right]$$

(12)

In this context, the linguistic scale function (LSF) is denoted as ${\mathcal{U}}^{*}$. The accuracy function ($ACC$) for $p,q-$QOFLN can be defined as follows:

$$ACC\left( \mathcalligra{u} \right) = \left( {\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} + \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} } \right) \times {\mathcal{U}}^{*} \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} } \right),\;p,q{ \succcurlyeq }1\;{\text{and}}\;ACC\left( \mathcalligra{u} \right) \in \left[ {0,1} \right]$$

(13)

Definition 12

Let $\mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ and $\mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right)$ are two $p,q$-QOFLNs, then

• 1. $\mathcalligra{u} \succ \mathcalligra{v}$ if $SCO\left( \mathcalligra{u} \right) \succ SCO\left( \mathcalligra{v} \right)$;

• 2. $\mathcalligra{u} \prec \mathcalligra{v}$ if $SCO\left( \mathcalligra{u} \right) \prec SCO\left( \mathcalligra{v} \right)$;

• 3. If $SCO\left( \mathcalligra{u} \right) = SCO\left( \mathcalligra{v} \right)$ then

  • $\mathcalligra{u} \succ \mathcalligra{v}$ if $ACC\left( \mathcalligra{u} \right) \succ ACC\left( \mathcalligra{v} \right)$;

  • $\mathcalligra{u} \prec \mathcalligra{v}$ if $ACC\left( \mathcalligra{u} \right) \prec ACC\left( \mathcalligra{v} \right)$;

  • If $SCO\left( \mathcalligra{u} \right) = SCO\left( \mathcalligra{v} \right)$ and $ACC\left( \mathcalligra{u} \right) = ACC\left( \mathcalligra{v} \right)$ then $\mathcalligra{u} = \mathcalligra{v}$.

Definition 13

Let $\mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ and $\mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right)$ are two $p,q$-QOFLNs, and $\eta$, $\eta_{1}$ and $\eta_{2}$ then

• 1. $\mathcalligra{u} \oplus \mathcalligra{v} = \mathcalligra{v} \oplus \mathcalligra{u}$;

• 2. $\mathcalligra{u} \otimes \mathcalligra{v} = \mathcalligra{v} \otimes \mathcalligra{u}$;

• 3. $\eta \left( {\mathcalligra{u} \oplus \mathcalligra{v} } \right) = \eta \mathcalligra{v} \oplus \eta \mathcalligra{u}$;

• 4. $\left( {\mathcalligra{u} \otimes \mathcalligra{v} } \right)^{\eta } = \mathcalligra{v}^{\eta } \otimes \mathcalligra{u}^{\eta }$;

• 5. $\eta_{1} \mathcalligra{u} \oplus \eta_{2} \mathcalligra{u} = \left( {\eta_{1} + \eta_{2} } \right)\mathcalligra{u}$;

• 6. $\left( \mathcalligra{u} \right)^{{\eta_{1} + \eta_{2} }} = \mathcalligra{u}^{{\eta_{1} }} \otimes \mathcalligra{u}^{{\eta_{2} }}$.

Proof

Let’s demonstrate the validity of Rule 1.

$$\begin{gathered} \mathcalligra{u} \oplus \mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right) \oplus \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right) \hfill \\ \quad \quad \quad = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right) + \psi \left( \mathcalligra{v} \right)} ,\left( {\sqrt[p]{{\begin{array}{*{20}c} {\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} + \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} - } \\ {\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} } \\ \end{array} }},\vartheta \left( \mathcalligra{u} \right)\vartheta \left( \mathcalligra{v} \right)} \right)} \right) = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right) + \psi \left( \mathcalligra{u} \right)} ,\left( {\sqrt[p]{{\begin{array}{*{20}c} {\left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} + \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} - } \\ {\left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} } \\ \end{array} }},\vartheta \left( \mathcalligra{v} \right)\vartheta \left( \mathcalligra{u} \right)} \right)} \right) \hfill \\ \quad \quad \quad = \mathcalligra{v} \oplus \mathcalligra{u} . \hfill \\ \end{gathered}$$

To demonstrate the validity of Rule 2, we get

$$\begin{gathered} \mathcalligra{u} \otimes \mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right) \otimes \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right) \hfill \\ \quad \quad \quad = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right) \times \psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right)\zeta \left( \mathcalligra{v} \right),\sqrt[q]{{\begin{array}{*{20}c} {\left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} + \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} } \\ { - \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} } \\ \end{array} }}} \right)} \right) = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right) \times \psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right)\zeta \left( \mathcalligra{u} \right),\sqrt[q]{{\begin{array}{*{20}c} {\left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} + \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} } \\ { - \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} } \\ \end{array} }}} \right)} \right) \hfill \\ \quad \quad \quad = \mathcalligra{v} \otimes \mathcalligra{u} . \hfill \\ \end{gathered}$$

The remaining rules can be proven in a similar manner.

Definition 14

Let $\mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ and $\mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right)$ are two $p,q$-QOFLNs, then HD distance between $\mathcalligra{u}$ and $\mathcalligra{v}$ can be expressed as follows:

$$\mathcalligra{d}_{1} \left( {\mathcalligra{u} ,\mathcalligra{v} } \right) = \frac{1}{2}\left| {\begin{array}{*{20}c} {\left( {1 + \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} - \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{q} } \right) \times {\mathcal{U}}^{*} \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} } \right)} \\ { - \left( {1 + \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{p} - \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} } \right) \times {\mathcal{U}}^{*} \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} } \right)} \\ \end{array} } \right|$$

(14)

When both $\zeta \left( \mathcalligra{u} \right) = \zeta \left( \mathcalligra{u} \right) = 1$ and $\zeta \left( \mathcalligra{u} \right) = \zeta \left( \mathcalligra{u} \right) = 0$, then $p,q$-QOF linguistic variables $\mathcalligra{u}$ and $\mathcalligra{v}$ can be expressed as linguistic variables ${\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)}$ and ${\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)}$ respectively. In this case, the HD between $\mathcalligra{u}$ and $\mathcalligra{v}$ reduces to Eq. (6), given by:

$$\mathcalligra{d}_{1} \left( {\mathcalligra{u} ,\mathcalligra{v} } \right) = \left| {{\mathcal{U}}^{*} \left( \mathcalligra{u} \right) - {\mathcal{U}}^{*} \left( \mathcalligra{v} \right)} \right|$$

(15)

The HD between variables $\mathcalligra{u}$ and $\mathcalligra{v}$ can alternatively be articulated as:

Definition 15

Let $\mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ and $\mathcalligra{v} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} ,\left( {\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)} \right)$ are two $p,q$-QOFLNs, then HD distance between $\mathcalligra{u}$ and $\mathcalligra{v}$ can be expressed as follows:

$$\mathcalligra{d}_{2} \left( {\mathcalligra{u} ,\mathcalligra{v} } \right) = \mathcalligra{d} \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} } \right) \times \mathcalligra{d} \left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right),\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right)$$

(16)

where $\mathcalligra{d} \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,{\mathcal{S}}_{\psi \left( \mathcalligra{v} \right)} } \right) = \left| {{\mathcal{U}}^{*} \left( {\mathcalligra{s}_{\psi \left( \mathcalligra{u} \right)} } \right) - {\mathcal{U}}^{*} \left( {\mathcalligra{s}_{\psi \left( \mathcalligra{v} \right)} } \right)} \right|$ and

$$\begin{aligned} \mathcalligra{d} \left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right),\zeta \left( \mathcalligra{v} \right),\vartheta \left( \mathcalligra{v} \right)} \right) = & \frac{1}{2}\left| {\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} - \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{q} } \right| + \left| {\left( {{\Pi }\left( \mathcalligra{u} \right)} \right)^{p} - \left( {{\Pi }\left( \mathcalligra{v} \right)} \right)^{q} } \right| \\ \quad + \left| {\left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{p} - \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{q} } \right|\left( {p,q{ \succcurlyeq }1} \right) \\ \end{aligned}$$

(17)

where ${\Pi }\left( \mathcalligra{u} \right)$ and ${\Pi }\left( \mathcalligra{v} \right)$ denote the indeterminacy degree of $\mathcalligra{u}$ and $\mathcalligra{v}$, respectively, and can be computed as follows:

$${\Pi }\left( \mathcalligra{u} \right) = \sqrt[\mathcalligra{z} ]{{\left( {\zeta \left( \mathcalligra{u} \right)} \right)^{\mathcalligra{z} } + \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{\mathcalligra{z} } - \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{\mathcalligra{z} } \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{\mathcalligra{z} } }},\;{\Pi }\left( \mathcalligra{v} \right) = \sqrt[\mathcalligra{z} ]{{\left( {\zeta \left( \mathcalligra{v} \right)} \right)^{\mathcalligra{z} } + \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{\mathcalligra{z} } - \left( {\zeta \left( \mathcalligra{v} \right)} \right)^{\mathcalligra{z} } \left( {\vartheta \left( \mathcalligra{v} \right)} \right)^{\mathcalligra{z} } }}$$

where $\left( {\mathcalligra{z} = LCM\left( {p,q} \right)} \right)$.

${\varvec{p}},{\varvec{q}}-$QOFLWA operator

Definition 16

For any collection of $p,q$-QOFLNs $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ ($k = 1,2, \ldots ,n$), then $p,q$-QOFLWA operator is a mapping $p,q - QOFLWA:{\mathcal{G}}^{n} \to {\mathcal{G}}$ and can be expressed as follows:

$$p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \oplus_{k = 1}^{n} \wp_{k} \mathcalligra{u}_{k}$$

(18)

In this equation, ${\mathcal{G}}$ represents the set of all $p,q$-QOFLNs, and $\wp_{k}$ is the respective weight assigned to $\mathcalligra{u}_{k}$, with $\wp_{k} { \succcurlyeq }0$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$.

Theorem 1

Let $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ be a set of $p,q$-QOFLNs and $\wp_{k}$ be the corresponding weight vector of $\mathcalligra{u}_{k}$, then the aggregated value obtained by $p,q$-QOFLWA Operator is also a $p,q$-QOFLNs and can be represented as follows:

$$p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }},} \\ {\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right)$$

(19)

where $p,q{ \succcurlyeq }1$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$.

Proof

Based on the operational rules, we obtain the following expressions:

Step 1. For $k=2$, we have

$$\begin{gathered} \wp_{1} \mathcalligra{u}_{1} = \left( {{\mathcal{S}}_{{\wp_{1} \psi \left( {\mathcalligra{u}_{1} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{p} } \right)^{{\wp_{1} }} }}} \\ {,\left( {\vartheta \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} } \\ \end{array} } \right)} \right),\;\wp_{2} \mathcalligra{u}_{2} = \left( {{\mathcal{S}}_{{\wp_{2} \psi \left( {\mathcalligra{u}_{2} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{p} } \right)^{{\wp_{2} }} }}} \\ {,\left( {\vartheta \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} } \\ \end{array} } \right)} \right). \hfill \\ \wp_{1} \mathcalligra{u}_{1} \oplus \wp_{2} \mathcalligra{u}_{2} = \left( {{\mathcal{S}}_{{\wp_{1} \psi \left( {\mathcalligra{u}_{1} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{p} } \right)^{{\wp_{1} }} }}} \\ {,\left( {\vartheta \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} } \\ \end{array} } \right)} \right) \oplus \left( {{\mathcal{S}}_{{\wp_{2} \psi \left( {\mathcalligra{u}_{2} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{p} } \right)^{{\wp_{2} }} }},} \\ {\left( {\vartheta \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} } \\ \end{array} } \right)} \right) \hfill \\ \quad = \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\wp_{1} \psi \left( {\mathcalligra{u}_{1} } \right) + \wp_{1} \psi \left( {\mathcalligra{u}_{1} } \right)}} ,\sqrt[p]{{\begin{array}{*{20}c} {1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{p} } \right)^{{\wp_{1} }} } \\ { + 1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{p} } \right)^{{\wp_{2} }} } \\ { - \left( {1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{p} } \right)^{{\wp_{1} }} } \right)} \\ { \times \left( {1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{p} } \right)^{{\wp_{2} }} } \right)} \\ \end{array} }},} \\ {\left( {\vartheta \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} \left( {\vartheta \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} } \\ \end{array} } \right) = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{2} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{2} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }},} \\ {\mathop \prod \limits_{k = 1}^{2} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right). \hfill \\ \end{gathered}$$

Clearly, Theorem 1 is proven for $k=2$.

Step 2. Assuming that Theorem 1 is established for $k=l$, we obtain

$$\begin{gathered} p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{l} } \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{l} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{l} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{l} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right). \hfill \\ \end{gathered}$$

Step 3. When $k=l+1$, in accordance with Definition 11, we acquire

$$\begin{gathered} \wp_{l + 1} \mathcalligra{u}_{l + 1} = \left( {{\mathcal{S}}_{{\wp_{l + 1} \psi \left( {\mathcalligra{u}_{l + 1} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{p} } \right)^{{\wp_{l + 1} }} }}} \\ {,\left( {\vartheta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} } \\ \end{array} } \right)} \right) \hfill \\ p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{l + 1} } \right) = p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{l} } \right) \oplus \wp_{l + 1} \mathcalligra{u}_{l + 1} \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{l} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{l} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{l} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right) \hfill \\ \quad \oplus \left( {{\mathcal{S}}_{{\wp_{l + 1} \psi \left( {\mathcalligra{u}_{l + 1} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{p} } \right)^{{\wp_{l + 1} }} }},} \\ {\left( {\vartheta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} } \\ \end{array} } \right)} \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{l + 1} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{l + 1} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }},} \\ {\mathop \prod \limits_{k = 1}^{l + 1} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right). \hfill \\ \end{gathered}$$

Hence, when $k=l+1$, Theorem 1 is validated.

Example 1

Assume $\mathcalligra{u}_{1} = \left( {{\mathcal{S}}_{2} ,\left( {0.80,0.70} \right)} \right)$, $\mathcalligra{u}_{1} = \left( {{\mathcal{S}}_{1} ,\left( {0.50,0.40} \right)} \right)$, and $\mathcalligra{u}_{1} = \left( {{\mathcal{S}}_{2} ,\left( {0.20,0.80} \right)} \right)$ are three $p,q$-QOFLNs, with corresponding weights $\wp = \left( {0.30,0.40,0.30} \right)$. Applying the $p.q -$QOFLWA operator with $p = q = 3$, we combine these three $p,q$-QOFLNs to obtain a result that is also a $p,q$-QOFLN.

$$\begin{gathered} {\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} = {\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{3} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} = {\mathcal{S}}_{0.30 \times 2 + 0.40 \times 1 + 0.30 \times 3} = {\mathcal{S}}_{1.9} \hfill \\ \sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }} = \sqrt[3]{{1 - \mathop \prod \limits_{k = 1}^{3} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{3} } \right)^{{\wp_{k} }} }} \hfill \\ = \sqrt[3]{{1 - \left( {1 - 0.80^{3} } \right)^{0.30} \times \left( {1 - 0.50^{3} } \right)^{0.40} \left( {1 - 0.20^{3} } \right)^{0.30} }} = 0.6193, \hfill \\ \mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} = \mathop \prod \limits_{k = 1}^{3} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} = \left( {0.70^{0.30} } \right) \times \left( {0.40^{0.40} } \right) \times \left( {0.80^{0.30} } \right) = 0.5823. \hfill \\ \end{gathered}$$

Therefore,

$$\begin{gathered} p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} ,\mathcalligra{u}_{3} } \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{3} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{3} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }},} \\ {\mathop \prod \limits_{k = 1}^{3} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{1.9} , \left( {0.6193,0.5823} \right)} \right). \hfill \\ \end{gathered}$$

${\varvec{p}},{\varvec{q}}-$QOFLOWA operator

Definition 17

For any collection of $p,q$-QOFLNs $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ ($k = 1,2, \ldots ,n$), then $p,q$-QOFLOWA Operator is a mapping $p,q - QOFLOWA:{\mathcal{G}}^{n} \to {\mathcal{G}}$ and can be expressed as follows:

$$p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \oplus_{k = 1}^{n} \wp_{k} \mathcalligra{u}_{\varphi \left( k \right)}$$

(20)

In this equation, ${\mathcal{G}}$ represents the set of all $p,q$-QOFLNs, and $\wp_{k}$ is the respective weight assigned to $\mathcalligra{u}_{\varphi \left( k \right)}$, with $\wp_{k} { \succcurlyeq }0$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$. The term $\varphi \left( k \right)$ represents the $i$th largest element in the set $\left\{ {\mathcalligra{u}_{k} \left| {k = 1,2, \ldots ,n} \right.} \right\}$.

Theorem 2

Let $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ be a set of $p,q$-QOFLNs and $\wp_{k}$ be the corresponding weight vector of $\mathcalligra{u}_{k}$, then the aggregated value obtained by $p,q$-QOFLOWA Operator is also a $p,q$-QOFLNs and can be represented as follows:

$$\begin{gathered} p,q - QOFLOWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) \hfill \\ \quad = \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)}} ,} \\ {\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \\ \end{array} } \right) \hfill \\ \end{gathered}$$

(21)

where $p,q{ \succcurlyeq }1$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$.

Proof

Easy to prove.

${\varvec{p}},{\varvec{q}}-$QOFLWG operator

Definition 18

For any collection of $p,q$-QOFLNs $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ ($k = 1,2, \ldots ,n$), then $p,q$-QOFLWG Operator is a mapping $p,q - QOFLWG:{\mathcal{G}}^{n} \to {\mathcal{G}}$ and can be expressed as follows:

$$p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \otimes_{k = 1}^{n} \left( {\mathcalligra{u}_{k} } \right)^{{\wp_{k} }}$$

(22)

In this equation, ${\mathcal{G}}$ represents the set of all $p,q$-QOFLNs, and $\wp_{k}$ is the respective weight assigned to $\mathcalligra{u}_{k}$, with $\wp_{k} { \succcurlyeq }0$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$.

Theorem 3

Let $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ be a set of $p,q$-QOFLNs and $\wp_{k}$ be the corresponding weight vector of $\mathcalligra{u}_{k}$, then the aggregated value obtained by $p,q$-QOFLWG Operator is also a $p,q$-QOFLNs and can be represented as follows:

$$\begin{gathered} p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) \hfill \\ \quad = \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{n} \left( {\psi \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} }} ,} \\ {\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ {,\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right) \hfill \\ \end{gathered}$$

(23)

where $p,q{ \succcurlyeq }1$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$.

Proof

Based on the operational rules, we obtain the following expressions:

Step 1. For $k=2$, we have

$$\begin{gathered} \left( {\mathcalligra{u}_{1} } \right)^{{\wp_{1} }} = \left( {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} }} ,\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{q} } \right)^{{\wp_{1} }} }}} \\ \end{array} } \right)} \right),\;\left( {\mathcalligra{u}_{2} } \right)^{{\wp_{2} }} = \left( {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} }} ,\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{q} } \right)^{{\wp_{2} }} }}} \\ \end{array} } \right)} \right). \hfill \\ \left( {\mathcalligra{u}_{1} } \right)^{{\wp_{1} }} \oplus \left( {\mathcalligra{u}_{2} } \right)^{{\wp_{2} }} \hfill \\ = \left( {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} }} ,\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{q} } \right)^{{\wp_{1} }} }}} \\ \end{array} } \right)} \right) \oplus \left( {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} }} ,\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{q} } \right)^{{\wp_{2} }} }}} \\ \end{array} } \right)} \right) \hfill \\ = \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} \times \left( {\psi \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} }} ,} \\ {\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{1} } \right)} \right)^{{\wp_{1} }} \left( {\vartheta \left( {\mathcalligra{u}_{2} } \right)} \right)^{{\wp_{2} }} ,} \\ {\sqrt[q]{{\begin{array}{*{20}c} {1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{q} } \right)^{{\wp_{1} }} } \\ { + 1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{q} } \right)^{{\wp_{2} }} } \\ { - \left( {1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{1} } \right)} \right)^{q} } \right)^{{\wp_{1} }} } \right)} \\ { \times \left( {1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{2} } \right)} \right)^{q} } \right)^{{\wp_{2} }} } \right)} \\ \end{array} }}} \\ \end{array} } \right)} \\ \end{array} } \right) = \left( {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{2} \left( {\psi \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} }} ,\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{2} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} ,} \\ {\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{2} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \right). \hfill \\ \end{gathered}$$

Clearly, Theorem 4 is proven for $k=2$.

Step 2. Assuming that Theorem 4 is established for $k=l$, we obtain

$$\begin{gathered} p,q - QOFLWg\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{l} } \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{l} \left( {\psi \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} }} ,\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{l} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} ,} \\ {\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{l} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \right) \hfill \\ \left( {\mathcalligra{u}_{1} } \right)^{{\wp_{l + 1} }} = \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} }} ,} \\ {\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{q} } \right)^{{\wp_{l + 1} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right). \hfill \\ \end{gathered}$$

Step 3. When $k=l+1$, in accordance with Definition 11, we acquire

$$\begin{gathered} \left( {\mathcalligra{u}_{l + 1} } \right)^{{\wp_{l + 1} }} \hfill \\ \quad = \left( {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} }} ,\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{q} } \right)^{{\wp_{l + 1} }} }}} \\ \end{array} } \right)} \right) \hfill \\ p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{l + 1} } \right) = p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{l} } \right) \otimes \left( {\mathcalligra{u}_{l + 1} } \right)^{{\wp_{l + 1} }} \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{l} \left( {\psi \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} }} ,\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{l} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} ,} \\ {\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{l} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \right) \otimes \left( {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} }} ,\left( {\begin{array}{*{20}c} {\left( {\vartheta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{{\wp_{l + 1} }} ,} \\ {\sqrt[q]{{1 - \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{l + 1} } \right)} \right)^{q} } \right)^{{\wp_{l + 1} }} }}} \\ \end{array} } \right)} \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{l + 1} \left( {\psi \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} }} ,\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{l + 1} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} ,} \\ {\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{l + 1} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \right). \hfill \\ \end{gathered}$$

Hence, when $k = l + 1$, Theorem 4 is validated.

${\varvec{p}},{\varvec{q}}$-QOFLOWG operator

Definition 19

For any collection of $p,q$-QOFLNs $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ ($k = 1,2, \ldots ,n$), then $p,q$-QOFLOWG Operator is a mapping $p,q - QOFLOWG:{\mathcal{G}}^{n} \to {\mathcal{G}}$ and can be expressed as follows:

$$p,q - QOFLOWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \otimes_{k = 1}^{n} \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)^{{\wp_{k} }}$$

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In this equation, ${\mathcal{G}}$ represents the set of all $p,q$-QOFLNs, and $\wp_{k}$ is the respective weight assigned to $\mathcalligra{u}_{\varphi \left( k \right)}$, with $\wp_{k} { \succcurlyeq }0$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$. The term $\varphi \left( k \right)$ represents the $i$th largest element in the set $\left\{ {\mathcalligra{u}_{k} \left| {k = 1,2, \ldots ,n} \right.} \right\}$.

Theorem 4

Let $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( {\mathcalligra{u}_{k} } \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ be a set of $p,q$-QOFLNs and $\wp_{k}$ be the corresponding weight vector of $\mathcalligra{u}_{k}$, then the aggregated value obtained by $p,q$-QOFLOWG Operator is also a $p,q$-QOFLNs and can be represented as follows:

$$\begin{gathered} p,q - QOFLOWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) \hfill \\ \quad = \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\left( {\psi \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)} \right)^{{\wp_{k} }} }} ,} \\ {\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)} \right)^{{\wp_{k} }} } \\ {,\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right) \hfill \\ \end{gathered}$$

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where $p,q{ \succcurlyeq }1$ and $\mathop \sum \limits_{k = 1}^{n} \wp_{k} = 1$.

Proof

Easy to prove.

Example 2

Suppose ${\mathcalligra{u} }_{1}=\left({\mathcal{S}}_{2},(\text{0.40,0.30})\right)$, ${\mathcalligra{u} }_{2}=\left({\mathcal{S}}_{3},(\text{0.70,0.60})\right)$ and ${\mathcalligra{u} }_{3}=\left({\mathcal{S}}_{1},(\text{0.50,0.20})\right)$ be three $p.q-$QOFLNs, and $\wp =(\text{0.25,0.35,0.40})$ is the corresponding weights of ${\mathcalligra{u} }_{k}$ $(k=\text{1,2},3)$. We utilize the $p,q-$QOFLWG operator to aggregate the three $p,q-$ROFLNs and obtain a result, which is also a $p,q-$QOFLN (here $p=q=4$).

$$\begin{gathered} {\mathcal{S}}_{{\mathop \prod \limits_{{k = 1}}^{n} \left( {\psi \left( {\mathcalligra{u} _{k} } \right)} \right)^{{\wp _{k} }} }} = {\mathcal{S}}_{{\mathop \prod \limits_{{k = 1}}^{3} \left( {\psi \left( {\mathcalligra{u} _{k} } \right)} \right)^{{\wp _{k} }} }} \hfill \\ = {\mathcal{S}}_{{\left( {2^{{0.25}} } \right) \times \left( {3^{{0.35}} } \right) \times \left( {1^{{0.40}} } \right)}} = {\mathcal{S}}_{{1.7468}} , \hfill \\ \mathop \prod \limits_{{k = 1}}^{n} \left( {\vartheta \left( {\mathcalligra{u} _{k} } \right)} \right)^{{\wp _{k} }} = \mathop \prod \limits_{{k = 1}}^{3} \left( {\vartheta \left( {\mathcalligra{u} _{k} } \right)} \right)^{{\wp _{k} }} = \left( {0.40^{{0.25}} } \right) \times \left( {0.70^{{0.35}} } \right) \times \left( {0.50^{{0.40}} } \right) = 0.5320, \hfill \\ \sqrt[q]{{1 - \mathop \prod \limits_{{k = 1}}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u} _{k} } \right)} \right)^{q} } \right)^{{\wp _{k} }} }} = \sqrt[q]{{1 - \mathop \prod \limits_{{k = 1}}^{3} \left( {1 - \left( {\zeta \left( {\mathcalligra{u} _{k} } \right)} \right)^{q} } \right)^{{\wp _{k} }} }} \hfill \\ \sqrt[4]{{\begin{array}{*{20}c} {1 - \left( {1 - \left( {0.30} \right)^{4} } \right)^{{0.25}} \times \left( {1 - \left( {0.60} \right)^{4} } \right)^{{0.35}} } \\ { \times \left( {1 - \left( {0.20} \right)^{4} } \right)^{{0.40}} } \\ \end{array} }} = 0.4728 \hfill \\ \end{gathered}$$

Thus, $\left({\mathcal{S}}_{{\prod }_{k=1}^{n}{\left(\psi \left({\mathcalligra{u} }_{k}\right)\right)}^{{\wp }_{k}}},\left(\begin{array}{c}{\prod }_{k=1}^{n}{\left(\vartheta \left({\mathcalligra{u} }_{k}\right)\right)}^{{\wp }_{k}},\\ \sqrt[q]{1-{\prod }_{k=1}^{n}{\left(1-{\left(\zeta \left({\mathcalligra{u} }_{k}\right)\right)}^{q}\right)}^{{\wp }_{k}}}\end{array}\right)\right)=({\mathcal{S}}_{1.7468},(\text{0.5320,0.4728}))$.

Some basic properties of the proposed operators

Theorem 5

If $\mathcalligra{u}_{k} = \mathcalligra{u} = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right)$ for all $k = 1,2, \ldots ,n$, then

• 1. $p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \oplus_{k = 1}^{n} \wp_{k} \mathcalligra{u}_{k} = \mathcalligra{u}$.

• 2. $p,q - QOFLOWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \oplus_{k = 1}^{n} \wp_{k} \mathcalligra{u}_{\varphi \left( k \right)} = \mathcalligra{u}$.

• 3. $p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \otimes_{k = 1}^{n} \left( {\mathcalligra{u}_{k} } \right)^{{\wp_{k} }} = \mathcalligra{u}$.

• 4. $p,q - QOFLOWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \otimes_{k = 1}^{n} \left( {\mathcalligra{u}_{\varphi \left( k \right)} } \right)^{{\wp_{k} }} = \mathcalligra{u}$.

Proof

By Theorem 1, we have

$$p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right) = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u}_{k} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right)$$

Since $\mathcalligra{u}_{k} = \mathcalligra{u}$ for all $k = 1,2, \ldots ,n$ it implies that

$$\begin{gathered} p,q - QOFLWA\left( {\mathcalligra{u} ,\mathcalligra{u} , \ldots ,\mathcalligra{u} } \right) = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( \mathcalligra{u} \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( \mathcalligra{u} \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( \mathcalligra{u} \right)} \right)^{p} } \right)^{1} }}} \\ {,\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( \mathcalligra{u} \right)} \right)^{1} } \\ \end{array} } \right)} \right) = \left( {{\mathcal{S}}_{\psi \left( \mathcalligra{u} \right)} ,\left( {\zeta \left( \mathcalligra{u} \right),\vartheta \left( \mathcalligra{u} \right)} \right)} \right) = \mathcalligra{u} . \hfill \\ \end{gathered}$$

The remaining parts can be proven in a similar manner.

Theorem 6

Let $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} ,\left( {\zeta \left( {\mathcalligra{u}_{k} } \right),\vartheta \left( {\mathcalligra{u}_{k} } \right)} \right)} \right)$ and $\mathcalligra{u}_{k} = \left( {{\mathcal{S}}_{{\psi \left( {\widetilde{{\mathcalligra{u}_{k} }}} \right)}} ,\left( {\zeta \left( {\widetilde{{\mathcalligra{u}_{k} }}} \right),\vartheta \left( {\widetilde{{\mathcalligra{u}_{k} }}} \right)} \right)} \right)$ are any $p,q$-ROFLNs, if ${\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{k} } \right)}} { \preccurlyeq \mathcal{S}}_{{\psi \left( {\widetilde{{\mathcalligra{u}_{k} }}} \right)}}$, $\zeta \left( {\mathcalligra{u}_{k} } \right){ \preccurlyeq }\zeta \left( {\widetilde{{\mathcalligra{u}_{k} }}} \right)$ and $\vartheta \left( {\mathcalligra{u}_{k} } \right){ \succcurlyeq }\vartheta \left( {\widetilde{{\mathcalligra{u}_{k} }}} \right)$ for all $k = 1,2, \ldots ,n$ then

• 1. $p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }p,q - QOFLWA\left( {\widetilde{{\mathcalligra{u}_{1} }},\widetilde{{\mathcalligra{u}_{2} }}, \ldots ,\widetilde{{\mathcalligra{u}_{n} }}} \right)$.

• 2. $p,q - QOFLOWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }p,q - QOFLOWA\left( {\widetilde{{\mathcalligra{u}_{1} }},\widetilde{{\mathcalligra{u}_{2} }}, \ldots ,\widetilde{{\mathcalligra{u}_{n} }}} \right)$.

• 3. $p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }p,q - QOFLWG\left( {\widetilde{{\mathcalligra{u}_{1} }},\widetilde{{\mathcalligra{u}_{2} }}, \ldots ,\widetilde{{\mathcalligra{u}_{n} }}} \right)$.

• 4. $p,q - QOFLOWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }p,q - QOFLOWG\left( {\widetilde{{\mathcalligra{u}_{1} }},\widetilde{{\mathcalligra{u}_{2} }}, \ldots ,\widetilde{{\mathcalligra{u}_{n} }}} \right)$.

Proof

Straightforward.

Theorem 7

Let ${\mathcalligra{u} }_{k}=\left({\mathcal{S}}_{\psi ({\mathcalligra{u} }_{k})},\left(\zeta ({\mathcalligra{u} }_{k}),\vartheta ({\mathcalligra{u} }_{k})\right)\right)$ be a group of $p,q-$ROFLNs, if $\mathcalligra{u}_{\inf \left( k \right)} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{\inf \left( k \right)} } \right)}} ,\left( {\begin{array}{*{20}c} {\zeta \left( {\mathcalligra{u}_{\inf \left( k \right)} } \right),} \\ {\vartheta \left( {\mathcalligra{u}_{\sup \left( k \right)} } \right)} \\ \end{array} } \right)} \right)$ and $\mathcalligra{u}_{\sup \left( k \right)} = \left( {{\mathcal{S}}_{{\psi \left( {\mathcalligra{u}_{\sup \left( k \right)} } \right)}} ,\left( {\begin{array}{*{20}c} {\zeta \left( {\mathcalligra{u}_{\sup \left( k \right)} } \right),} \\ {\vartheta \left( {\mathcalligra{u}_{\inf \left( k \right)} } \right)} \\ \end{array} } \right)} \right)$, then

• 1. $\mathcalligra{u}_{\inf \left( k \right)} { \preccurlyeq }p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }\mathcalligra{u}_{\sup \left( k \right)}$.

• 2. $\mathcalligra{u}_{\inf \left( k \right)} { \preccurlyeq }p,q - QOFLOWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }\mathcalligra{u}_{\sup \left( k \right)}$.

• 3. $\mathcalligra{u}_{\inf \left( k \right)} { \preccurlyeq }p,q - QOFLWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }\mathcalligra{u}_{\sup \left( k \right)}$.

• 4. $\mathcalligra{u}_{\inf \left( k \right)} { \preccurlyeq }p,q - QOFLOWG\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} , \ldots ,\mathcalligra{u}_{n} } \right){ \preccurlyeq }\mathcalligra{u}_{\sup \left( k \right)}$.

Proof

Straightforward.

Some special cases

This section delves into specific cases of the proposed operators to demonstrate their originality.

Case 1. When setting the term levels ${\varvec{p}}$ and ${\varvec{q}}$ to $1$, the proposed operators simplify to operators based on linguistic intuitionistic fuzzy sets³⁸.

Case 2. When the term levels ${\varvec{p}}$ and ${\varvec{q}}$ are both set to $2$, the proposed operators converge to operators based on linguistic Pythagorean fuzzy sets⁵³.

Case 3. When the term levels ${\varvec{p}}$ and ${\varvec{q}}$ are both are equal, the proposed operators converge to operators based on linguistic ${\varvec{q}}-$rung orthopair fuzzy sets⁵⁴.

From the preceding discussion, it is evident that the proposed operators are adept at managing information within the linguistic IFSs, linguistic PFSs, and q-rung orthopair fuzzy linguistic sets frameworks. However, it is not obligatory to process information in the $p,q-$QOFLSs framework using the linguistic IFSs, linguistic PFSs, and $q-$rung orthopair fuzzy linguistic sets frameworks.

Proposed MCDM for ${\varvec{p}},{\varvec{q}}-$QOFL information

MCDM approaches offer a vigorous approach to decision investigation by empowering the simultaneous consideration of multiple criteria. This approach enhances decision comprehensiveness, providing a holistic evaluation of alternatives and accommodating both quantitative and qualitative criteria. The transparency of MCDM ensures clear decision pathways, fostering accountability and stakeholder understanding. Its flexibility and adaptability make it appropriate in various frameworks, allowing for the incorporation of individual decisions, conflict determination, and sensitivity analysis. Moreover, MCDM supports efficient DM by scientifically organizing information and simplifying trade-off analysis. The computational nature of the proposed method enables the analysis of complex data, and its sensitivity to uncertainty allows it to be used in wide range of real applications. Finally, MCDM aids in decision efficacy, accuracy, and the capacity to effectively negotiate complex choice situations.

For the construction proposed MCDM, we assume that ${\mathcal{B}} = \left\{ {{\mathcal{B}}_{1} ,{\mathcal{B}}_{2} , \ldots ,{\mathcal{B}}_{ \mathcalligra{m} } } \right\}$ be the set of $\mathcalligra{m}$ alternatives, ${\complement } = \left\{ {{\complement }_{1} ,{\complement }_{2} , \ldots ,{\complement }_{\mathcalligra{n} } } \right\}$ be the set of criteria and $\aleph = \left( {\aleph_{1} ,\aleph_{2} , \ldots ,\aleph_{\mathcalligra{n} } } \right)$ be the associate weights of the criteria ${\complement }_{j}$ ($j = 1,2, \ldots ,\mathcalligra{n}$) such that $0{ \preccurlyeq }\aleph_{j} { \preccurlyeq }1$ and $\mathop \sum \nolimits_{j = 1}^{\mathcalligra{n} } \aleph_{j} = 1$. Suppose that ${\mathcal{E}} = \left\{ {{\mathcal{E}}_{1} ,{\mathcal{E}}_{2} , \ldots ,{\mathcal{E}}_{k} } \right\}$ be the group of experts with the associated weights vector ${\Im } = \left( {{\Im }_{1} ,{\Im }_{2} , \ldots ,{\Im }_{k} } \right)$ such that $0{ \preccurlyeq \Im }_{t} { \preccurlyeq }1$ and $\mathop \sum \nolimits_{t = 1}^{n} {\Im }_{t} = 1$. The rating values of alternative ${\mathcal{B}}_{i}$ with respect to the criteria ${\complement }_{j}$ by the expert ${\mathcal{E}}_{t}$ are represented by $p,q$-QOFLNs and can be expressed as ${\mathcal{L}}_{ij}^{\left( t \right)} = \left( {{\mathcal{S}}_{{\psi \left( {{\mathcal{L}}_{ij}^{\left( k \right)} } \right)}} ,\left( {\zeta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right),\vartheta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right)} \right)} \right)$ ($t = 1,2, \ldots ,k;i = 1,2, \ldots , \mathcalligra{m} ;j = 1,2, \ldots ,\mathcalligra{n}$) with the following conditions:

• 1. ${\mathcal{S}}_{{\psi \left( {{\mathcal{L}}_{ij}^{\left( k \right)} } \right)}} \in {\hat{\mathcal{S}}}\;\text{and}\;\zeta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right),\left( {\vartheta {\mathcal{L}}_{ij}^{\left( t \right)} } \right) \in \left[ {0,1} \right],$

• 2. $\left( {\zeta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right)} \right)^{p} + \left( {\vartheta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right)} \right)^{q} { \preccurlyeq }1\;\text{for}\;\text{all}\;p,q{ \succcurlyeq }1.$

Step 1. The following combinations of data regarding the choices that the decision makers gave can be made in the decision matrix form:

$${\mathfrak{D}} = \left( {\begin{array}{*{20}c} {{\mathcal{L}}_{11}^{\left( t \right)} } & {\begin{array}{*{20}c} {{\mathcal{L}}_{12}^{\left( t \right)} } & \cdots \\ \end{array} } & {{\mathcal{L}}_{1\mathcalligra{n} }^{\left( t \right)} } \\ {\begin{array}{*{20}c} {{\mathcal{L}}_{21}^{\left( t \right)} } \\ \vdots \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\mathcal{L}}_{22}^{\left( t \right)} } \\ \vdots \\ \end{array} } & {\begin{array}{*{20}c} \cdots \\ \ddots \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {{\mathcal{L}}_{2\mathcalligra{n} }^{\left( t \right)} } \\ \vdots \\ \end{array} } \\ {{\mathcal{L}}_{ \mathcalligra{m} 1}^{\left( t \right)} } & {\begin{array}{*{20}c} {{\mathcal{L}}_{ \mathcalligra{m} 2}^{\left( t \right)} } & \cdots \\ \end{array} } & {{\mathcal{L}}_{ \mathcalligra{m} \mathcalligra{n} }^{\left( t \right)} } \\ \end{array} } \right),$$

(26)

The rows of the decision matrix presented in Eq. (26), correspond to the set of alternatives, or ${\mathcal{B}}_{i}$, while the columns display the criteria ${\complement }_{j}$ that are connected to each alternative. Each call in the decision matrix indicates how well an alternative performed or scored in relation to each certain criterion. This configuration offers a systematic design that facilitates carrying out a comprehensive study of choices based on several criteria. While the columns indicate the numerous factors (criteria) that are being taken to the consideration.

Step 2. The selection of alternatives and formation of optimal outcomes are crucial responsibilities of cost criteria (${\mathfrak{C}}_{j}$) and benefit criteria (${\mathfrak{B}}_{j}$), which significantly influence the evaluation of decision-making. To evaluate the financial impacts and resource use of any solution, costs need to be included. The costs of implementing a particular choice, including initial, continuing, and potential long-term expenses, are assessed by decision-makers. On the other hand, ${\mathfrak{B}}_{j}$ comprises the expected profits, or advantages of each alternative. This includes evaluating the potential gains, losses, and contributions of a alternatives. The determination of the optimal cost–benefit ratio has a significant influence on the overall value and feasibility of decision. These elements need to be a carefully taken into the account in order to accomplish successful decision-making. The following method might be applied to convert ${\mathfrak{C}}_{j}$ into ${\mathfrak{B}}_{j}$:

$${\tilde{\mathcal{L}}}_{ij}^{\left( t \right)} = \left\{ {\begin{array}{*{20}c} {\left( {{\mathcal{S}}_{{\psi \left( {{\mathcal{L}}_{ij}^{\left( k \right)} } \right)}} ,\left( {\zeta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right),\vartheta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right)} \right)} \right)\;{\text{if}}\;{\mathfrak{C}}_{j} } \\ {\left( {{\mathcal{S}}_{{\psi \left( {{\mathcal{L}}_{ij}^{\left( k \right)} } \right)}} ,\left( {\vartheta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right),\zeta \left( {{\mathcal{L}}_{ij}^{\left( t \right)} } \right)} \right)} \right)\;i{\text{f}}\;{\mathfrak{B}}_{j} } \\ \end{array} } \right.$$

(27)

The normalized decision matrix can be expressed as follows:

$${\tilde{\mathfrak{D}}} = \left( {\begin{array}{*{20}c} {{\tilde{\mathcal{L}}}_{11}^{\left( t \right)} } & {\begin{array}{*{20}c} {{\tilde{\mathcal{L}}}_{12}^{\left( t \right)} } & \cdots \\ \end{array} } & {{\tilde{\mathcal{L}}}_{1\mathcalligra{n} }^{\left( t \right)} } \\ {\begin{array}{*{20}c} {{\tilde{\mathcal{L}}}_{21}^{\left( t \right)} } \\ \vdots \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{\tilde{\mathcal{L}}}_{22}^{\left( t \right)} } \\ \vdots \\ \end{array} } & {\begin{array}{*{20}c} \cdots \\ \ddots \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {{\tilde{\mathcal{L}}}_{2\mathcalligra{n} }^{\left( t \right)} } \\ \vdots \\ \end{array} } \\ {{\tilde{\mathcal{L}}}_{ \mathcalligra{m} 1}^{\left( t \right)} } & {\begin{array}{*{20}c} {{\tilde{\mathcal{L}}}_{ \mathcalligra{m} 2}^{\left( t \right)} } & \cdots \\ \end{array} } & {{\tilde{\mathcal{L}}}_{ \mathcalligra{m} \mathcalligra{n} }^{\left( t \right)} } \\ \end{array} } \right),$$

(28)

Step 3. Integrate the rating values of alternatives using the proposed AOs listed below:

• 1.
  $$p,q - QOFLWA\left( {{\tilde{\mathcal{L}}}_{ij}^{\left( 1 \right)} ,{\tilde{\mathcal{L}}}_{ij}^{\left( 2 \right)} , \ldots ,{\tilde{\mathcal{L}}}_{ij}^{\left( \mathcalligra{m} \right)} } \right)$$
  $$= \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( {{\tilde{\mathcal{L}}}_{ij}^{\left( k \right)} } \right)}} ,} \\ {\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {{\tilde{\mathcal{L}}}_{ij}^{\left( k \right)} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {{\tilde{\mathcal{L}}}_{ij}^{\left( k \right)} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \\ \end{array} } \right)$$

  (29)
• 2.
  $$p,q - QOFLOWA\left( {{\tilde{\mathcal{L}}}_{ij}^{\left( 1 \right)} ,{\tilde{\mathcal{L}}}_{ij}^{\left( 2 \right)} , \ldots ,{\tilde{\mathcal{L}}}_{ij}^{\left( \mathcalligra{m} \right)} } \right)$$
  $$= \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\mathop \sum \limits_{k = 1}^{n} \wp_{k} \psi \left( {{\tilde{\mathcal{L}}}_{{\varphi \left( {ij} \right)}}^{\left( k \right)} } \right)}} ,} \\ {\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {{\tilde{\mathcal{L}}}_{{\varphi \left( {ij} \right)}}^{\left( k \right)} } \right)} \right)^{p} } \right)^{{\wp_{k} }} }}} \\ {,\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {{\tilde{\mathcal{L}}}_{{\varphi \left( {ij} \right)}}^{\left( k \right)} } \right)} \right)^{{\wp_{k} }} } \\ \end{array} } \right)} \\ \end{array} } \right)$$

  (30)
• 3.
  $$p,q - QOFLWG\left( {{\tilde{\mathcal{L}}}_{ij}^{\left( 1 \right)} ,{\tilde{\mathcal{L}}}_{ij}^{\left( 2 \right)} , \ldots ,{\tilde{\mathcal{L}}}_{ij}^{\left( \mathcalligra{m} \right)} } \right)$$
  $$= \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{n} \left( {\psi \left( {{\tilde{\mathcal{L}}}_{ij}^{\left( k \right)} } \right)} \right)^{{\wp_{k} }} }} ,} \\ {\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {{\tilde{\mathcal{L}}}_{ij}^{\left( k \right)} } \right)} \right)^{{\wp_{k} }} ,} \\ {\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {{\tilde{\mathcal{L}}}_{ij}^{\left( k \right)} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right)$$

  (31)
• 4.
  $$p,q - QOFLOWG\left( {{\tilde{\mathcal{L}}}_{ij}^{\left( 1 \right)} ,{\tilde{\mathcal{L}}}_{ij}^{\left( 2 \right)} , \ldots ,{\tilde{\mathcal{L}}}_{ij}^{\left( \mathcalligra{m} \right)} } \right)$$
  $$= \left( {\begin{array}{*{20}c} {{\mathcal{S}}_{{\mathop \prod \limits_{k = 1}^{n} \left( {\psi \left( {{\tilde{\mathcal{L}}}_{{\varphi \left( {ij} \right)}}^{\left( k \right)} } \right)} \right)^{{\wp_{k} }} }} ,} \\ {\left( {\begin{array}{*{20}c} {\mathop \prod \limits_{k = 1}^{n} \left( {\vartheta \left( {{\tilde{\mathcal{L}}}_{{\varphi \left( {ij} \right)}}^{\left( k \right)} } \right)} \right)^{{\wp_{k} }} ,} \\ {\sqrt[q]{{1 - \mathop \prod \limits_{k = 1}^{n} \left( {1 - \left( {\zeta \left( {{\tilde{\mathcal{L}}}_{{\varphi \left( {ij} \right)}}^{\left( k \right)} } \right)} \right)^{q} } \right)^{{\wp_{k} }} }}} \\ \end{array} } \right)} \\ \end{array} } \right)$$

  (32)

Step 4. By providing a numerical depiction of an alternative’s performance or attractiveness, the score function helps decision-makers evaluate and contrast options according to predetermined standards. This quantitative score streamlines the DM process by providing a numerical foundation for selecting the most favorable alternatives. The score values of $p,q-$QOFLNs are determined using Eq. (8).

Step 5. Place the alternatives based on score values. The step-by-step pathway is shown in Fig. 3.

Fig. 3

Design of the projected MCDM scheme.

The selection of term levels ${\varvec{p}}$ and ${\varvec{q}}$

In this section, our objective is to determine the minimum values of $p$ and $q$ for which the condition ${\vartheta }^{p}+{\zeta }^{q}\preccurlyeq 1$ ($p,q\succcurlyeq 1$) is fulfilled. For this investigation, we provided the following example.

Example 11

Let ${\mathcal{I}}_{1} = \left( {{\mathcal{S}}_{1} ,\left( {0.80,0.60} \right)} \right)$, ${\mathcal{I}}_{2} = \left( {{\mathcal{S}}_{3} ,\left( {0.70,0.80} \right)} \right)$ and ${\mathcal{I}}_{3} = \left( {{\mathcal{S}}_{2} ,\left( {0.75,0.40} \right)} \right)$ be any three $p,q$-QOFLNs, then the term levels for these $p,q$-QOFLNs can be calculated as:

• 1. For $p = q = 1$,

  • 
    

    ${\mathcal{I}}_{1} = \left( {{\mathcal{S}}_{1} ,\left( {\vartheta_{1} ,\zeta_{1} } \right)} \right) = \left( {{\mathcal{S}}_{1} ,\left( {0.80,0.60} \right)} \right)$ and $0.80^{1} + 0.60^{1} = 1.40$,

  • 
    

    ${\mathcal{I}}_{2} = \left( {{\mathcal{S}}_{3} ,\left( {\vartheta_{2} ,\zeta_{2} } \right)} \right) = \left( {{\mathcal{S}}_{3} ,\left( {0.70,0.80} \right)} \right)$ and $0.70^{1} + 0.80^{1} = 1.50$,

  • 
    

    ${\mathcal{I}}_{3} = \left( {{\mathcal{S}}_{2} ,\left( {\vartheta_{3} ,\zeta_{3} } \right)} \right) = \left( {{\mathcal{S}}_{2} ,\left( {0.75,0.40} \right)} \right)$ and $0.75^{1} + 0.40^{1} = 1.15$.

As the condition $\vartheta^{p} + \zeta^{q} { \preccurlyeq }1$ is not fulfilled when $p = q = 1$, it is necessary to elevate the term levels.

• 2. For $p = q = 2$,

  • 
    

    ${\mathcal{I}}_{1} = \left( {{\mathcal{S}}_{1} ,\left( {\vartheta_{1} ,\zeta_{1} } \right)} \right) = \left( {{\mathcal{S}}_{1} ,\left( {0.80,0.60} \right)} \right)$ and $0.80^{2} + 0.60^{2} = 1.00$,

  • 
    

    ${\mathcal{I}}_{2} = \left( {{\mathcal{S}}_{3} ,\left( {\vartheta_{2} ,\zeta_{2} } \right)} \right) = \left( {{\mathcal{S}}_{3} ,\left( {0.70,0.80} \right)} \right)$ and $0.70^{2} + 0.80^{2} = 1.13$,

  • 
    

    ${\mathcal{I}}_{3} = \left( {{\mathcal{S}}_{2} ,\left( {\vartheta_{3} ,\zeta_{3} } \right)} \right) = \left( {{\mathcal{S}}_{2} ,\left( {0.75,0.40} \right)} \right)$ and $0.75^{2} + 0.40^{3} = 0.7225$.

Once more, the condition $\vartheta^{p} + \zeta^{q} { \preccurlyeq }1$ remains unsatisfied when $p = q = 2$.

• 3. For $p = 3$ and $q = 2$

  • 
    

    ${\mathcal{I}}_{1} = \left( {{\mathcal{S}}_{1} ,\left( {\vartheta_{1} ,\zeta_{1} } \right)} \right) = \left( {{\mathcal{S}}_{1} ,\left( {0.80,0.60} \right)} \right)$ and $0.80^{3} + 0.60^{2} = 0.8720$,

  • 
    

    ${\mathcal{I}}_{2} = \left( {{\mathcal{S}}_{3} ,\left( {\vartheta_{2} ,\zeta_{2} } \right)} \right) = \left( {{\mathcal{S}}_{3} ,\left( {0.70,0.80} \right)} \right)$ and $0.70^{3} + 0.80^{2} = 0.9830$,

  • 
    

    ${\mathcal{I}}_{3} = \left( {{\mathcal{S}}_{2} ,\left( {\vartheta_{3} ,\zeta_{3} } \right)} \right) = \left( {{\mathcal{S}}_{2} ,\left( {0.75,0.40} \right)} \right)$ and $0.75^{3} + 0.40^{2} = 0.5818$.

Hence, the condition $\vartheta^{p} + \zeta^{q} { \preccurlyeq }1$ is met when $p = 3$ and $q = 2$. Consequently, the minimum values of the term levels are $2$ and $3$ for $p,q$-QOFLNs ${\mathcal{I}}_{1}$, ${\mathcal{I}}_{2}$ and ${\mathcal{I}}_{3}$.

Illustrative example

SONAR, an acronym for Sound Navigation and Ranging, is a technology that leverages sound waves for underwater navigation, detection, and communication. Active SONAR systems emit sound signals into the water and analyze the returning echoes to ascertain the location and physical appearance of underwater matters, making them effective for precise distance measurement and target tracking. Conversely, passive SONAR systems listen for sounds generated by external sources without emitting their own signals, providing stealth and the facility to notice the existence of submarines or other underwater matters without disclosing the detecting position of system. SONAR proves versatile, offering real-time information, accurate depth measurement, and the detection of underwater objects, rendering it indispensable for applications such as marine navigation, fisheries, environmental monitoring, and military operations. Despite its advantages, concerns regarding about the environmental impact of SONAR, particularly on marine life, have prompted regulatory measures to mitigate potential harm.

The main features for estimating objects beneath the water surface through SONAR (Sound Navigation and Ranging) include:

• 1. Echo return time (${\complement }_{1}$): SONAR systems employ the analysis of distance to underwater objects by measuring the time it takes for sound waves (pings) to travel to the object and back. The echo return time is utilized to regulate the distance between the SONAR system and the object.

• 2. Echo strength or amplitude (${\complement }_{2}$): The strength or amplitude of the returning echo provides information about the characteristics of the object. A stronger echo may indicate a larger or more reflective object, while a weaker echo may suggest a smaller or less reflective object.

• 3. Frequency of sound waves (${\complement }_{3}$): The frequency of the emitted sound waves affects the resolution and penetration capabilities of the SONAR system. High-frequency sound waves provide better resolution but may be absorbed more quickly in water, limiting their range. Low-frequency waves can penetrate greater depths but offer lower resolution.

• 4. Sonar beam width (${\complement }_{4}$): The width of the SONAR beam determines the coverage area and the system’s ability to discriminate between closely spaced objects. Narrow beams provide better target resolution, while wider beams cover a larger area.

Imagine a naval submarine utilizing SONAR to detect nearby rocks and other objects, including potential enemy submarines. Despite having a sea-map indicating the location of possible obstacles, the dynamic nature of the underwater surface, subject to frequent changes, renders complete reliance on the provided sea-map impractical for submarine navigation. To ensure safety, the distances of four potential objects (${\mathcal{B}}_{1}$, ${\mathcal{B}}_{2}$, ${\mathcal{B}}_{3}$, and ${\mathcal{B}}_{4}$) in the vicinity from the submarine need to be assessed. The available data organizes the data in the form of $p,q-$QOFLNs based on four criteria: ${\complement }_{1}$ for ‘Echo Return Time’, ${\complement }_{2}$ for ‘Echo Strength or Amplitude’, ${\complement }_{3}$ for ‘Frequency of Sound Waves’ and ${\complement }_{4}$ for ‘Sonar Beam Width’. Subsequently, the goal of the issue is to identify the nearby object in a manner that ensures an uninterrupted journey for the ship. To achieve this, the actions outlined in Section “Proposed MCDM for p,q-QOFL information” are implemented as follows:

Step 1. The information of experts about the alternatives are summarized in Table 1 within the context of the $p,q-$QOFLNs environment. The associate weights for criteria are $(\text{0.23,0.20,0.32,0.25})$.

Table 1 Rating values of alternatives with respect to criteria.

Step 2. Given that all criteria fall under the category of benefits, there is no requirement for normalization. Additionally, the nature of the criteria being exclusively beneficial negates the necessity for normalization procedures.

Step 3. In this step we utilized the suggested AOs to aggregate the information in Table 1. The accumulated values of alternatives with the suggested operators are listed in Tables 2, 3, 4 and 5. The aggregated values can be calculated as follows:

$$\begin{gathered} p,q - QOFLWA\left( {\mathcalligra{u} _{1} ,\mathcalligra{u} _{2} ,\mathcalligra{u} _{3} ,\mathcalligra{u} _{4} } \right) \hfill \\ \quad = \left( {{\mathcal{S}}_{{\mathop \sum \limits_{{k = 1}}^{n} \wp _{k} \psi \left( {\mathcalligra{u} _{k} } \right)}} ,\left( {\begin{array}{*{20}c} {\sqrt[p]{{1 - \mathop \prod \limits_{{k = 1}}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u} _{k} } \right)} \right)^{p} } \right)^{{\wp _{k} }} }}} \\ {,\mathop \prod \limits_{{k = 1}}^{n} \left( {\vartheta \left( {\mathcalligra{u} _{k} } \right)} \right)^{{\wp _{k} }} } \\ \end{array} } \right)} \right) \hfill \\ {\mathcal{S}}_{{0.23 \times 2 + 0.20 \times 1 + 0.32 \times 3 + 0.25 \times 1}} = {\mathcal{S}}_{{1.8700}} , \hfill \\ \sqrt[p]{{1 - \mathop \prod \limits_{{k = 1}}^{n} \left( {1 - \left( {\zeta \left( {\mathcalligra{u} _{k} } \right)} \right)^{p} } \right)^{{\wp _{k} }} }} = \sqrt[2]{{\begin{array}{*{20}c} {1 - \left( {1 - 0.40^{2} } \right)^{{0.23}} \times \left( {1 - 0.30^{2} } \right)^{{0.23}} } \\ { \times \left( {1 - 0.20^{2} } \right)^{{0.23}} \times \left( {1 - 0.50^{2} } \right)^{{0.23}} } \\ \end{array} }} = 0.3662, \hfill \\ \mathop \prod \limits_{{k = 1}}^{n} \left( {\vartheta \left( {\mathcalligra{u} _{k} } \right)} \right)^{{\wp _{k} }} = \left( {0.50^{{0.23}} } \right) \times \left( {0.45^{{0.20}} } \right) \times \left( {0.60^{{0.32}} } \right) \times \left( {0.45^{{0.25}} } \right) = 0.5055, \hfill \\ \end{gathered}$$

Table 2 Aggregated values of ${\mathcal{B}}_{i}$ using $p,q-QOFLWA$ operator ($p=q=2$).

Table 3 Aggregated values of ${\mathcal{B}}_{i}$ using $p,q-QOFLOWA$ operator.

Table 4 Aggregated values of ${\mathcal{B}}_{i}$ using $p,q-QOFLWG$ operator.

Table 5 Aggregated values of ${\mathcal{B}}_{i}$ using $p,q-QOFLOWG$ operator.

Therefore, $p,q - QOFLWA\left( {\mathcalligra{u}_{1} ,\mathcalligra{u}_{2} ,\mathcalligra{u}_{3} ,\mathcalligra{u}_{4} } \right) = \left( {{\mathcal{S}}_{1.8700} ,\left( {0.3662,0.5055} \right)} \right).$

The graphical depiction in Fig. 4 provides a comprehensive overview of the score values assigned to each alternative. Notably, it highlights ${\mathcal{B}}_{3}$ as the optimal choice among the alternatives. It’s worth noting that, despite variations in the score values resulting from the application of different aggregation methods, the fundamental ranking order remains unchanged. This consistency in ranking underscores the reliability of the evaluation process. As a practical implication, the flexibility to employ any of the proposed aggregation operators during the aggregation process enhances the adaptability and applicability of the methodology across diverse decision-making scenarios. This flexibility allows decision-makers to tailor the aggregation approach based on specific considerations or preferences, further contributing to the robustness and versatility of the decision support framework.

Fig. 4

The Score values obtained by proposed AOs.

Sensitivity analysis

Considering the parameter $q$ ($q=2$) constant, we investigated the impact of parameter $p$ in this section of the study. The evaluation included the use of $p,q-$QOFLWA operators. Figure 5 shows the findings for various values of the parameter $p$. This study aimed to assess the sensitivity of the results to changes in parameter $p$, providing insights into the dynamic performance of the system under discussion.

Fig. 5

The effect of parameter $p$ over ranking order.

Table 6 reveals a noticeable trend: manipulating the parameter $p$ results in a continuous drop in the score values allocated to alternatives. This suggests that pessimistic decision makers may incline toward greater values of the parameter $p$, resulting in a negative trend in aggregate score values. The observed sensitivity to changes in parameter $p$ emphasizes the importance of decision-makers’ viewpoints and preferences in predicting assessment process results. The data provides valuable insights for decision-makers looking to tailor the evaluation strategy to their unique attitudes and concerns.

Table 6 The influence of parameter $p$ on selection findings (where $q=2$).

A detailed inspection of Eq. (19) reveals the $p,q-$QOFLWA’s intrinsic independence from the parameter $q$. As a result, in order to investigate the unique impact of parameter $q$, we will switch our attention to the $p,q-$QOFLWG operator. Table 7 comprehensively presents the comprehensive results, including score values and the related ranking order of alternatives throughout a range value of parameter $q$. This deliberate shift in emphasis permits a more nuanced investigation of the system’s behavior under different parameter $q$ settings, encouraging a deeper grasp of the DM dynamics inherent in the assessment process.

Table 7 The effect of parameter $q$ over ranking order of alternatives (where $p=2$).

Table 7 provides a clear representation illustrating that as parameter $q$ is increased, the accompanying score values climb as well. The study shows that when decision-makers adopt an optimistic viewpoint, selecting a greater value of $q$ during the aggregation process may coincide with their preferences. The encouraging correlation between parameter $q$ and score values emphasizes the system’s versatility to reflect various degrees of optimism in DM. the resulting data is valuable for decision-makers seeking to customize the aggregation process to align with their optimistic opinions and preferences. Figure 6 illustrates the impact of parameters $q$ on decision outcomes.

Fig. 6

The effect of parameter $q$ over decision outcomes.

Comparison

A complete study was performed inside the PF linguistic sets, Fermatean fuzzy linguistic sets (FFLSs), and $q$-ROFLSs environments to demonstrate the supremacy of our provided operators over current techniques^{53,55,56,57,58}. This investigation specifically considered scenarios where $p = q = 2$, $p = q = 3$, and $p = q$, incorporating a weight vector denoted as $\left( {0.23,0.23,0.32,0.25} \right)$. Table 8 presents the concisely assembled resultant optimal score values and the accompanying ranking order of options.

Table 8 Position order of alternatives by employing some existing approaches.

Table 8 makes it clear that the alternatives’ ranking order generally agrees with the suggested ranking, highlighting the consistency of our method. Additionally, the analysis shows that the suggested method has a unique computational process when compared to other approaches that have been used in various contexts. Despite these variations, the outcomes reported in this work are thought to be more representative of DM reality. This is explained by the method’s focus on upholding a constant degree of priority between argument pairings, which improves the decision-making process’s rationality. Finally, it is confirmed that the suggested operators consider the parameters $p$ and $q$ of the decision makers, providing a wider range of options.

The methods currently listed in the Table 8 can effectively manage information within the limitation that the term levels for both membership and non-membership are the same. However, if a decision maker insists on using different term level values for membership (e.g., 3) and non-membership (e.g., 2), then these methods are unable to accommodate such a scenario. Conversely, our proposed approach can handle this situation, offering greater realism and flexibility in addressing real-world decision-making problems. Therefore, the proposed approach represents a more adaptable and practical solution for addressing the complexities inherent in decision-making processes. The graphical view of score values with different existing is presented in Fig. 7.

Fig. 7

The graphical representation score values of choices obtained by different existing approaches.

Furthermore, the results obtained through the proposed approach were compared with those from several existing methods based on $p,q$-QOFSs across different values of $p$ and $q$. The score values and the ranking order of the alternatives, as calculated by some existing approaches, are summarized in Table 9. From this comparison, it can be observed that the ranking of alternatives remains consistent with the results produced by the proposed approach. This alignment demonstrates that, regardless of variations in the parameters $p$ and $q$, the proposed method yields reliable and accurate rankings, reinforcing its robustness in diverse decision-making environments. The consistency in rankings also highlights the validity of the proposed approach as an effective tool for handling uncertainty and imprecise information in multi-criteria decision-making problems.

Table 9 Comparative analysis.

Advantages

The proposed study presents several key advantages

• 1. Enhanced realism in decision-making: The framework recognizes the dynamic nature of decision-makers’ criteria, allowing for a more authentic representation of decision scenarios. By analyzing various score values linked to different parameters $p$ and $q$, the model effectively captures the complexities inherent in real-world decision-making processes. This adaptability ensures that the decision-making model accurately mirrors the multifaceted aspects of practical situations, enhancing its applicability.

• 2. Innovative computational methods: The introduction of a novel computational approach distinguishes this study from previous methodologies across a range of environmental contexts. This innovation not only highlights the uniqueness of the proposed operators but also encourages the exploration of new computational tools within the field of decision science. As a result, the significance and applicability of these operators are enhanced in diverse decision-making environments.

• 3. Flexibility in preference expression: The proposed framework allows decision-makers to articulate their preferences with greater flexibility by offering a wide variety of parametric options for $p$ and $q$. The presented operators create a robust toolkit for evaluating alternatives based on specific characteristics. This flexibility is particularly valuable in complex decision-making scenarios, where decision-makers need versatile tools to address a variety of options and challenges.

Limitations

Despite these advantages, the study has several limitations that warrant consideration

• 1. Dependency on parameter specification: Although the flexibility afforded by parameters $p$ and $q$ can enhance decision-making confidence, it also poses potential risks. The efficacy of the proposed operators is heavily reliant on the accurate specification of these parameters. In practical applications, determining precise values can be challenging, and uncertainty in their definition may lead to inconsistencies in decision outcomes.

• 2. Computational complexity: The computational techniques introduced in this paper, while innovative, may also lead to complexity. While this uniqueness is an asset, it can create challenges related to computational efficiency and ease of use. Decision-makers and practitioners may face difficulties in applying the proposed operators without accessible, user-friendly computational tools, which could limit the practical applicability of the approach in certain contexts.

The proposed framework significantly enhances decision-making through its realism, innovation, and flexibility. However, it is crucial to address the challenges associated with parameter specification and computational complexity. Future research should focus on developing intuitive tools and guidelines that facilitate the practical implementation of these methodologies across various fields. By addressing these limitations, the framework can be refined further, expanding its applicability and ultimately leading to more effective decision-making in real-world scenarios. Enhancing the discussion of these advantages and limitations not only strengthens the overall analysis but also emphasizes the relevance and potential impact of the research.

Conclusion

This article introduces the concept of $p,q-$QOFLSs as an extension of traditional fuzzy sets, such as PFLS and q-ROFLS. The key feature of $p,q-$QOFLSs is that the sum of the pth power of the membership degree and the qth power of the non-membership degree is constrained to be less than or equal to 1, allowing for more flexible representation of uncertainty. This flexibility enables decision-makers to express their evaluations more precisely, making it easier to handle imprecise and ambiguous information in decision-making processes. In this study, we applied the $p,q-$QOFLS framework to a MCDM problem and developed a set of AOs to integrate $p,q-$QOFL information. A numerical example demonstrated the effectiveness of the proposed approach, showing that it can provide clear and consistent rankings of alternatives. The practical impact of parameters $p$ and $q$ on decision outcomes was analyzed in detail, allowing us to adjust these parameters to tailor the decision-making process to specific scenarios. Furthermore, a comparative analysis was performed, showing that the proposed method produces similar or better results compared to existing approaches. This confirms its practicality and reliability in real-world applications. The results highlight the method’s capacity to enhance decision-making in environments characterized by uncertainty, proving it to be a robust tool for handling complex multi-criteria problems. The discussion also identifies the strengths and potential limitations of the proposed approach, providing a balanced perspective on its practical applicability.

In the future, the proposed approach can be extended to address pressing global issues, such as developing effective strategies for managing electronic waste⁶² and determining the most optimal renewable energy sources^63,64. By leveraging its ability to handle uncertainty and imprecise information, the method can provide more accurate and insightful decision-making frameworks in these areas.