A novel L q ROPF-Entropy-WASPAS group model based on Frank aggregation operators and improved score function in linguistic q -rung orthopair fuzzy framework

Abstract

Decision analysis plays a crucial role in our everyday actions. Efficient decision-making models rely heavily on accurately representing human cognitive knowledge. The linguistic $q$ -rung orthopair fuzzy sets (L^qROPFSs) offer a versatile means of representing qualitative cognitive information by adapting the parameter $q$ to different scenarios. This study presents a novel scoring function to rank linguistic $q$ -rung orthopair fuzzy numbers (L^qROPFNs) with greater precision compared to the current score function. Next, we present novel arithmetic/geometric aggregation operators (AOs) that utilize new Frank operational rules to combine a finite collection of L^qROPFNs. The work also examines the several desirable characteristics and special cases of the provided AOs. Furthermore, a novel decision-making model called the L^qROPF-Entropy-WASPAS model has been introduced to address the challenges of multiple attribute group decision-making (MAGDM) problems in a linguistic $q$ -rung orthopair fuzzy environment. The model incorporates proposed AOs and a scoring function. The suggested methodology is exemplified by considering a practical decision to select an online teaching platform. The validity of the results is confirmed through an extensive sensitivity analysis and comparative investigation employing various existing MAGDM approaches within the linguistic $q$ -rung orthopair fuzzy framework. The proposed approach offers enhanced flexibility to decision experts, empowering them to analyze decision outcomes across diverse scenarios. This flexibility is achieved by allowing the manipulation of values associated with various parameters, enabling decision-makers to tailor the analysis according to their specific attitudes and requirements. This adaptability ensures a more advanced and personalized analysis of decision outcomes, accommodating decision experts’ distinct viewpoints and preferences in varying situations.

Keywords

Aggregation operators linguistic variable linguistic q-rung orthopair fuzzy set score function WASPAS

1. Introduction

Decision-making is a critical aspect of human life, and it is something that we do on a daily basis. We are faced with choices and decisions every day, ranging from simple ones like what to have for breakfast to complex ones like deciding on a career path or whether to buy a house. Decision-making involves selecting the best option from various alternatives based on the available information, values, and preferences. The process of decision-making is not always straightforward, and it can be influenced by various factors such as emotions, biases, and cognitive limitations. Therefore, it is essential to understand the different aspects of decision-making to make informed and rational choices that align with our goals and values.

Multiple attribute group decision-making (MAGDM) is a comprehensive process that involves the selection of the best alternative among several options based on the evaluation of multiple attributes. It is a common problem in various fields, such as engineering, finance, management, and environmental studies, where a group of decision-makers must make a collective decision based on multiple attributes [1]. The MAGDM process is designed to facilitate consensus among decision-makers by considering the opinions and preferences of all the stakeholders involved. However, decision-making becomes more challenging when information data involves different types of uncertainty and vagueness. This could be due to various reasons, such as incomplete information, conflicting evidence, or unforeseeable events. In such cases, advanced tools are required to solve decision-making problems under uncertainty and vagueness [2]. In general, decision-makers (DMs) have a relatively restricted cognitive framework and rely on their intellect to convey evaluation information.

In order to address the imprecise nature of cognitive information in decision-making, [3] introduced the concept of intuitionistic fuzzy sets (IFS). This approach incorporates a degree of non-membership (DNM) in addition to the degree of membership (DM) of an element, subject to the linear inequality $DM+\textit{DNM}\leqslant 1$ . During the past decades, several researchers have worked on the theory of IFSs and introduced many valuable results, including operational laws [4, 5], information measures [6, 7, 8], and aggregation operators (AOs) [9, 10]. The IFS theory has been successfully employed to deal with several real-world problems such as decision-making [8, 11], clustering analysis [12, 13], medical diagnosis [14, 15] and pattern recognition [16, 17]. In 2013, [18] proposed the concept of Pythagorean fuzzy sets (PFSs) as a new tool for modeling the higher-level uncertain and vague information. In PFS theory, the membership grades of an element can be selected independently and satisfying the condition $DM^{2}+\textit{DNM}^{2}\leqslant 1$ . It confirms that the PFS is more capable of handling the vagueness and impreciseness in complex real-world situations.Many researchers have initiated efforts in the field of PFS theory to broaden its use across other fields. [19] introduced the concept of averaging and geometric aggregation operators in the context of PFSs. [20] provided definitions for division and subtraction operations on Pythagorean fuzzy numbers (PFNs) and conducted a thorough analysis of their properties. [21] proposed some new Pythagorean fuzzy interaction AOs and used them to solve multiple attribute decision-making (MADM) problems. [22] extended the CODAS (Combinative Distance based ASsesment) method with Pythagorean fuzzy information for supplied selection. [23] designed a model for risk evaluation and prevention in hydropower plants in Pythagorean fuzzy context. [24] studied a Pythagorean fuzzy approach for transportation problems. [25] defined some extended set measures for PFSs. [26] developed a set of comprehensive trigonometric similarity measures using Pythagorean fuzzy information. These measures were then utilized to address MAGDM problems. The notion of spherical distance measure was established by [27] to tackle MCDM situations in the Pythagorean fuzzy information framework. [28] used ordered weighted cosine similarity operators to address MAGDM problems. The study specifically focused on adding probabilistic information within the Pythagorean fuzzy context. [29] studied the applications of Pythagorean fuzzy information in resolving green supplier selection challenges in the foof industry based on the TOPSIS (Technique for Order Performance by Similarity to Ideal Solution) approach. [30] introduced centroid distance measure between PFSs and discussed its applications in solving fuzzy clustering and MCDM problems. For more details on Pythagorean fuzzy research, the readers can refer [31].

Consider a scenario where an expert assigns a value of 0.6 to the DM for an alternative satisfying an attribute and a value of 0.9 to the DNM. Nevertheless, it has been noted that the conditions $DM+\textit{DNM}>1$ and $DM^{2}+\textit{DNM}^{2}>1$ fail to meet the essential criteria of IFS and PFS. To address this situation, [32] introduced a generalized set theory called $q$ -rung orthopair fuzzy sets ( ${{}^{q}}$ ROPFSs). The fundamental characteristic of ${{}^{q}}$ ROPFS lies in the fact that the sum of the $q$ -th power of the DM and the $q$ -th power of the DNM must be equal to or less than 1. This modification guarantees that the resultant sets conform to the necessary requirements. Consequently, numerous studies have investigated the utilization of ${{}^{q}}$ ROPFSs in different decision-making situations. For example, [33] proposed some ${{}^{q}}$ ROPF AOs and demonstrated their effectiveness in handling decision-making situations involving ${{}^{q}}$ ROPFSs. [34] defined the ${{}^{q}}$ ROPF partitioned Maclaurin symmetric mean (MSM) and investigated its application MAGDM. [35, 36] applied Heronian mean and MSM operators in the ${{}^{q}}$ ROPF context to evaluate emerging technology commercialization potential. [37] introduced order- $\alpha$ ${{}^{q}}$ ROPF information measures and studied their application in MAGDM problems. These information measures provided valuable insights for decision-making scenarios involving ${{}^{q}}$ ROPFSs. To address decision-making problems comprehensively, [38] proposed a generalization of the MABAC (Multi Attributive Border Approximation Area Comparison) method under the ${{}^{q}}$ ROPF framework. [39] presented a hybrid decision-making approach that combined ${{}^{q}}$ ROPF information with other techniques to solve MAGDM problems. [40] extended the MARCOS (Measurement of Alternatives and Ranking according to the Compromise Solution) method to solid waste management by incorporating ${{}^{q}}$ ROPFSs. [41] developed an innovative method for evaluating mobile medical applications using ${{}^{q}}$ ROPF information. [42] utilized the MULTIMOORA (Multi-Objective Optimization on the basis of a Ratio Analysis plus the full MULTIplicative form) technique for selecting the optimal solid waste disposal method with ${{}^{q}}$ ROPF information. In addition, [43] developed a three-way group analysis method in the ${{}^{q}}$ ROPF framework. [44] introduced new similarity and entropy measure under ${{}^{q}}$ ROPF environment and investigated their utility in addressing decision-making problems.

The use of ${{}^{q}}$ ROPFSs has proven to be an effective approach for quantifying uncertain and vague information in decision-making. However, there are situations where numerical values fail to express the information accurately, and linguistic variables [45, 46] become more suitable, particularly when dealing with phenomena related to human perception. Therefore, the exploration of decision-making approaches based on linguistic computational models has emerged as a significant and fascinating area of research. Numerous studies in the literature have focused on applying linguistic methods in various information environments, aiming to investigate the effectiveness of linguistic models in decision-making processes [47, 48, 49, 50]. By incorporating linguistic variables, decision-makers can make informed choices that account for the inherent uncertainty and vagueness present in the data. Moreover, employing linguistic models facilitates better communication among decision-makers, reducing the likelihood of misinterpretation and miscommunication. In 2018, [51] proposed the linguistic $q$ -rung orthopair fuzzy sets (L^qROPFSs), which integrate the concept of linguistic terms (LTs) with ${{}^{q}}$ ROPFSs. They extended the power Bonferroni mean operators and Hamming distance measure within the L^qROPF framework. Additionally, [52] defined L^qROPF power Muirhead mean operators to address MAGDM problems. [53] generalized the TOPSIS method by incorporating L^qROPF information and applied it to postgraduate entrance qualification assessments. [54] introduced L^qROPF generalized similarity measures to handle problems in MADM. [55] developed group decision-making models based on Einstein AOs utilizing L^qROPF information Furthermore, [56] proposed several L^qROPF-generalized point-weighted AOs to tackle challenges in group decision-making. [57] proposed arithmetic and geometric AOs for linguistic $q$ -rung orthopair fuzzy numbers (L^qROPFNs) based on the Hamacher $t$ -norm. [58] studied prioritized weighted AOs using L^qROPF information, employing Hamacher operational laws. [57] proposed a novel decision-making technique for wind turbine selection within the L^qROPF framework.The concept of incomplete L^qROPF preference relations was investigated by [59] as a means of addressing decision-making challenges. [60] proposed power AOs for L^qROPF information and applied them to address the MAGDM problem using the DEMATEL (Decision Making Trial and Evaluation Laboratory) approach. These studies have contributed significantly to advancing L^qROPFSs by introducing new AOs, methodologies, and applications to solve decision-making problems.

In 1978, [61] introduced the Frank operations, which include Frank product and Frank sum. The Frank operations have two significant features: (i) they have the same advantages as the algebraic, Einstein, and Hamacher operational laws. (ii) These operational laws are not only a straightforward generalization of the existing operational laws but also provide more flexibility during the operational phase by having an additional parameter to control the power to which the argument values are raised. A review of the existing literature of L^qROPF shows that there is no investigation on the AOs based on Frank product and Frank sum under L^qROPF environment for aggregating a collection of L^qROPFNs. Therefore, to bridge this gap, it is worthwhile to introduce and explore the new AOs based on Frank operations with L^qROPF information.

In numerous real-world decision-making processes involving L^qROPF information, the accurate ranking of L^qROPFNs is of paramount importance for determining the optimal solution. However, the current ranking approach for L^qROPFNs, as developed by [62], often proves computationally infeasible in practical scenarios due to the involvement of score and accuracy functions. It is crucial to note that the existing score function lacks the ability to establish a proper ranking order among the considered L^qROPFNs on its own. The presence of this limitation can lead to inaccuracies during the ranking of alternatives, thereby exerting a substantial impact on the decision-making process. Hence, it is imperative to establish a more resilient score function for L^qROPFNs to address the limitations of the current approach and offer decision experts ample information regarding the alternatives, enabling them to make informed and dependable decisions. By introducing an improved score function, decision experts will have a more efficient and effective tool for ranking L^qROPFNs, thereby improving the quality and accuracy of decision outcomes in various practical domains.

Researchers have developed different decision-making strategies in the literature to address practical decision-making challenges with greater efficiency. The Weighted Aggregates Sum Product Assessment (WASPAS) approach was proposed by [63] in 2012. The methodology is founded on the amalgamation of two widely recognized decision methodologies for MCDM: the weighted sums model (WSM) and the method of weighted products model (WPM). The WASPAS method has been extensively employed in addressing diverse complex problems in MCDM [64, 65]. In recent years, several advanced versions of the WASPAS approach have been created to address decision-making difficulties in various information frameworks [66, 67, 68, 69, 70]. Nevertheless, the current WASPAS models lack the ability to address decision-making problems using L^qROPF information. Hence, it is imperative to create a novel expansion of the WASPAS model within the L^qROPF framework in order to address this deficiency in the existing body of research.

In MAGDM, determining the weight vector for attributes is a crucial step toward arriving at an appropriate solution. Scholars have proposed various methods to estimate attribute weights [71, 72, 73], which can be classified as either subjective or objective. The entropy method [71] has been widely used in the literature to determine objective weights for attributes. However, its capability to estimate attribute weights in the MAGDM problem under the L^qROPF framework has not been explored yet. In this study, we propose an extension of the entropy method in the L^qROPF framework to determine attribute weights in MAGDM problems. We investigate the method’s potential to estimate objective weights for attributes in the presence of the L^qROPF environment.

The notable research contributions of the paper are summarized as follows:

•
The work addresses shortcomings in the existing score function of L^qROPFNs and introduces an improved score function with elegant properties.
•
Four new AOs, namely the L^qROPF-Frank weighted average (L^qROPFFWA) operator, the L^qROPF-Frank ordered weighted average (L^qROPFFOWA) operator, the L^qROPF-Frank weighted geometric (L^qROPFFWG) operator and the L^qROPF-Frank ordered weighted geometric (L^qROPFFOWG) operator are proposed for aggregating L^qROPFNs based on new frank operational laws, and their desirable properties are discussed.
•
A new group decision-making methodology called L^qROPF-Entropy-WASPAS is introduced to address MAGDM problems in the L^qROPF context. This method leverages the improved score function and Frank AOs proposed in this study. By incorporating these advancements, the developed method offers an effective approach for tackling MAGDM challenges in the L^qROPF domain.
•
A new formula is provided to compute the position weights of the DMs in practical situations to account for the influence of different DMs in the group decision-making process. In addition, an exponential entropy-based approach is also formulated to determine attribute weights within the L^qROPF framework.
•
The practical applicability and performance of the developed L^qROPF-Entropy-WASPAS method are demonstrated through a case study on the selection problem of an online teaching platform. Additionally, the superiority of the proposed method is established through the execution of sensitivity investigation and comparative analysis.

The remaining part of the work is outlined as follows: Section 2 presents fundamental concepts about LVs, ^qROPFSs, L^qROPFSs, and Frank operations. Section 3 examines the limitations of the current score function used in L^qROPFNs and presents a new score function that improves the ranking of L^qROPFNs with a higher efficiency. In Section 4, new algebraic operational rules for L^qROPFNs are defined using Frank operations, and their features are thoroughly discussed. Section 5 introduces four novel AOs: the L^qROPFFWA operator, the L^qROPFFOWA operator, the L^qROPFFWG operator, and the L^qROPFFOWG operator. We also provide rigorous proofs for the various properties associated with these operators. Section 6 introduces the L^qROPF-Entropy-WASPAS technique, which utilizes the proposed AOs and scoring function to address MAGDM difficulties with L^qROPF information. An illustrative numerical example is provided to showcase the efficacy of the suggested approach in identifying the optimal online teaching platform. Section 7 includes the sensitivity analysis, comparative research, and validity test for the developed approach. The main findings of the study and possible future directions are discussed in Section 8.
2. Preliminaries

In this section, we provide a concise overview of key findings about LVs, ${{}^{q}}$ ROPFSs, L^qROPFSs, and Frank operations. By reviewing these fundamental results, we aim to establish a foundation for understanding the subsequent discussions and analyses in this paper.

.

Let $\widehat{\mathbb{V}}=\{v_{g}\mid g=0,1,\ldots,r\}$ be a discrete linguistic term set (LTS) with finite and fully ordered structure. Here, $v_{g}$ represents one of the potential values associated with a linguistic variable $\widehat{\mathbb{V}}$ , and $r$ is a positive integer denoting the cardinality of the set. In order to ensure the validity and usefulness of the LTS $\widehat{\mathbb{V}}$ , it must hold the following essential characteristics as specified by [74]:

i
If $v_{i}\leqslant v_{j}\Leftrightarrow i\leqslant j$ ,
ii
$\textit{neg}(v_{g})=v_{r-g}$ ,
iii
$\max(v_{i},v_{j})=v_{i}\Leftrightarrow i\geqslant j$ ,
iv
$\min(v_{i},v_{j})=v_{i}\Leftrightarrow i\leqslant j$ .

For $t=6$ , we have

$\displaystyle\widehat{\mathbb{V}}=\left\{\begin{array}[]{l}v_{0}=\textnormal{% Neither (N)},v_{1}=\textnormal{very bad (VB)},v_{2}=\textnormal{bad (B)},v_{3}% =\textnormal{medium (M)},v_{4}=\textnormal{good (G)},\\ v_{5}=\textnormal{very good (VG)},v_{6}=\textnormal{perfect (P)}\end{array}% \right\}.$

[75] developed the concept of the extended continuous LTS in 2004. An extended continuous LTS (ECLTS) is defined by the expression $\widehat{\mathbb{V}}_{[0,r]}=\{v_{g}\mid v_{0}\leqslant v_{g}\leqslant v_{r},g% \in[0,r]\}$ . Within this framework, if $v_{g}\in\widehat{\mathbb{V}}$ , then $v_{g}$ is called the original linguistic term (OLT), otherwise, $v_{g}$ is known as the virtual linguistic term (VLT).

.

[32] A ${{}^{q}}$ ROPFS $\mathcal{P}$ in a universal set ${\mathfrak{U}}=\{u_{1},u_{2},\ldots,u_{n}\}$ mathematically defined as

$\displaystyle\mathcal{P}=\{\langle u,{{\mathrm{M}}_{\mathcal{P}}(u)},{{\mathrm% {N}}_{\mathcal{P}}(u)}\rangle|u\in{\mathfrak{U}}\},$ (1)

where ${{\mathrm{M}}_{\mathcal{P}}}:\mathfrak{U}\rightarrow[0,1]$ and ${{\mathrm{N}}_{\mathcal{P}}}:\mathfrak{U}\rightarrow[0,1]$ represent the DM and DNM for an element $u\in{\mathfrak{U}}$ , respectively. These functions satisfy the condition $0\leqslant({\mathrm{M}}_{\mathcal{P}}(u))^{q}+({\mathrm{N}}_{\mathcal{P}}(u))^% {q}\leqslant 1$ for all $u\in{\mathfrak{U}}$ , where $q\geqslant 1$ . The degree of hesitancy (DH) of $u$ to the set $\mathcal{P}$ is determined by ${{\mathrm{H}}_{\mathcal{P}}(u)}={\sqrt[q]{(1-({{\mathrm{M}}_{\mathcal{P}}(u)})% ^{q}-({{\mathrm{N}}_{\mathcal{P}}(u)})^{q})}}$ . The pair $\langle{\mathrm{M}}_{\mathcal{P}}(u),{\mathrm{N}}_{\mathcal{P}}(u)\rangle$ is referred to as a $q$ -rung orthopair fuzzy number (^qROPFN), denoted simply as $\check{\Im}=\langle\check{\mathrm{M}},\check{\mathrm{N}}\rangle$ . It is observed that the ^qROPFN $\check{\Im}$ reduces to IFN and PFN when $q=1$ and $q=2$ , respectively.

Figure 1 illustrates that the information representation space of ${{}^{q}}$ ROF-membership grade exceeds the space occupied by the IF-membership grade and PF-membership grade.

Figure 1.
Comparison of spaces of different fuzzy membership grades based on the constraints of DM and DNM.

.

[51] Consider a predetermined universal set ${\mathfrak{U}}=\{u_{1},u_{2},\ldots,u_{n}\}$ and an extended continuous LTS $\widehat{\mathbb{V}}_{[0,r]}=\{v_{g}\mid v_{0}\leqslant v_{g}\leqslant v_{r},g% \in[0,r]\}$ . A L^qROPFS $\mathcal{Q}$ in ${\mathfrak{U}}$ is defined as follows:

$\displaystyle\mathcal{Q}=\{\langle u,v_{{\mathrm{M}}_{\mathcal{Q}}(u)},v_{{% \mathrm{N}}_{\mathcal{Q}}(u)}\rangle|u\in{\mathfrak{U}}\},$ (2)

where $v_{{\mathrm{M}}_{\mathcal{Q}}(u)}$ and $v_{{\mathrm{N}}_{\mathcal{Q}}(u)}$ belong to the set $\widehat{\mathbb{V}}_{[0,r]}$ , satisfying $0\leqslant({\mathrm{M}}_{\mathcal{Q}}(u))^{q}+({\mathrm{N}}_{\mathcal{Q}}(u))^% {q}\leqslant r^{q}$ for all $u\in{\mathfrak{U}}$ , where $q\geqslant 1$ . The LTs $v_{{\mathrm{M}}_{\mathcal{Q}}(u)}$ and $v_{{\mathrm{N}}_{\mathcal{Q}}(u)}$ represent the DM and DNM of an element $u\in{\mathfrak{U}}$ with respect to the L^qROPFS $\mathcal{Q}$ . The DH of $u\in\mathfrak{U}$ to the set $\mathcal{Q}$ can be determined by $v_{{\mathrm{H}}_{\mathcal{Q}}(u)}=v_{\sqrt[q]{r^{q}-({{\mathrm{M}}_{\mathcal{Q% }}(u)})^{q}-({{\mathrm{N}}_{\mathcal{Q}}(u)})^{q}}}$ . For a given element $u\in\mathfrak{U}$ , the pair $\langle{v}_{{\mathrm{M}}_{\mathcal{Q}}(u)},{v}_{{\mathrm{N}}_{\mathcal{Q}}(u)}\rangle$ is referred to as a L^qROPFN, which can be simply denoted as $\Phi=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ . It is worth noting that the LIFN [76] and LPFN [77] are special cases of the L^qROPFN when $q=1$ and $q=2$ , respectively.

.

[51] Let ${\Phi}=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ , ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ and ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ be three L^qROPFNs. The operations defined for these L^qROPFNs are as follows:

i
${{\Phi}_{1}}\leqslant{{\Phi}_{2}}$ if ${v_{{\mathrm{M}}{1}}}\leqslant{v_{{\mathrm{M}}{2}}}$ and ${v_{{\mathrm{N}}{1}}}\geqslant{v_{{\mathrm{N}}{2}}}$ ;
ii
${{\Phi}_{1}}={{\Phi}_{2}}$ if and only if ${{\Phi}_{1}}\subseteq{{\Phi}_{2}}$ and ${{\Phi}_{2}}\subseteq{{\Phi}_{1}}$ ;
iii
${\Phi}^{C}=\langle v_{\mathrm{N}},v_{\mathrm{M}}\rangle$ ;
iv
${{\Phi}_{1}}{\overset{q}{\cup}}{{\Phi}_{2}}=\langle\max({{v}_{{\mathrm{M}}{1}}% },{{v}_{{\mathrm{M}}{2}}}),\min({v}_{{\mathrm{N}}{1}},{v}_{{{\mathrm{N}}{2}}})\rangle$ ;
v
${{\Phi}_{1}}{\overset{q}{\cap}}{{\Phi}_{2}}=\langle\min({{v}_{{\mathrm{M}}{1}}% },{{v}_{{\mathrm{M}}{2}}}),\max({v}_{{\mathrm{N}}{1}},{v}_{{{\mathrm{N}}{2}}})\rangle$ ;
vi
${{\Phi}_{1}}{\overset{q}{\oplus}}{{\Phi}_{2}}=\left\langle v_{r(\sqrt[q]{({% \frac{{{\mathrm{M}}{1}^{q}}}{r^{q}}+\frac{{{\mathrm{M}}{2}^{q}}}{r^{q}}-\frac{% {{\mathrm{M}}{1}^{q}}}{r^{q}}\frac{{{\mathrm{M}}{2}^{q}}}{r^{q}}})})},v_{r\big% {(}{{\frac{{{\mathrm{N}}{1}}{{\mathrm{N}}{2}}}{r^{2}}}}\big{)}}\right\rangle$ ;
vii
${{\Phi}_{1}}{\overset{q}{\otimes}}{{\Phi}_{2}}=\left\langle v_{r\big{(}{{\frac% {{{\mathrm{M}}{1}}{{\mathrm{M}}{2}}}{r^{2}}}}\big{)}},v_{r(\sqrt[q]{({\frac{{{% \mathrm{N}}{1}^{q}}}{r^{q}}+\frac{{{\mathrm{N}}{2}^{q}}}{r^{q}}-\frac{{{% \mathrm{N}}{1}^{q}}}{r^{q}}\frac{{{\mathrm{N}}{2}^{q}}}{r^{q}}})})}\right\rangle$ ;
viii
$\lambda{\overset{q}{\ast}}{\Phi}=\left\langle v_{r(\sqrt[q]{{(1-(1-\frac{{% \mathrm{M}}^{q}}{r^{q}})^{\lambda})}})},v_{r\big{(}{\mathrm{N}}/{r}\big{)}^{% \lambda}}\right\rangle$ ;
ix
${\Phi}^{\overset{q}{\wedge}}\lambda=\left\langle v_{r\big{(}{\mathrm{M}}/{r}% \big{)}^{\lambda}},v_{r(\sqrt[q]{{(1-(1-\frac{{\mathrm{N}}^{q}}{r^{q}})^{% \lambda})}})}\right\rangle$ .

.

[62] Consider $\Phi=\langle{v_{\mathrm{M}},v_{{\mathrm{N}}}}\rangle$ is a L^qROPFN, then

$\displaystyle\mathbscr{S}(\Phi)={{\sqrt[q]{\left(\frac{r^{q}+{\mathrm{M}}^{q}-% {\mathrm{N}}^{q}}{2}\right)}}}\quad\textnormal{and}\quad\mathbscr{A}(\Phi)={{% \sqrt[q]{({{\mathrm{M}}^{q}+{\mathrm{N}}^{q}})}}}.$ (3)

are the score and accuracy values of $\Phi$ , respectively.

For any two L^qROPFNs ${\Phi}_{1}=\langle{{v}_{{{\mathrm{M}}{1}}},{v}_{{{\mathrm{N}}{1}}}}\rangle$ and ${\Phi}_{2}=\langle{{v}_{{{\mathrm{M}}{2}}},{v}_{{{\mathrm{N}}{2}}}}\rangle$ defined on ECLTS $\widehat{\mathbb{V}}_{[0,r]}=\{v_{g}\mid v_{0}\leqslant v_{g}\leqslant v_{r},g% \in[0,r]\}$ , the comparison rules are defined as follow [62]:

(i)
If $\mathbscr{S}({\Phi_{1}})>\mathbscr{S}({{\Phi}_{2}})$ then, ${{\Phi}_{1}}>{{\Phi}_{2}}$ ;
(ii)
If $\mathbscr{S}({{\Phi}_{1}})=\mathbscr{S}({{\Phi}_{2}})$ , then

(a)
$\mathbscr{A}({{\Phi}_{1}})>\mathbscr{A}({{\Phi}_{2}})$ , then ${{\Phi}_{1}}>{{\Phi}_{2}}$ ;
(b)
$\mathbscr{A}({{\Phi}_{1}})=\mathbscr{A}({{\Phi}_{2}})$ , then ${{\Phi}_{1}}={{\Phi}_{2}}$ .

2.1 Frank operations

[78] introduced a comprehensive approach for defining a generalized union and intersection of fuzzy and intuitionistic fuzzy sets using $t$ -norms and $t$ -conorms. One of the notable families of $t$ -norms and $t$ -conorms they considered are the Frank operations, which encompass the Frank product and Frank sum, respectively. Deschrijver et al.’s method presents a formal framework for defining the generalized union and intersection and provides a theoretical foundation for handling fuzzy and intuitionistic fuzzy sets more expressively and flexibly. By leveraging the Frank operations within this approach, researchers and practitioners can effectively capture and model complex uncertainties, making it particularly valuable in domains where precise and deterministic reasoning falls short.

.

Let $a$ and $b$ be two real numbers satisfying $a,b\in[0,1]$ and let $\hslash\in(1,+\infty)$ . The Frank product ${\otimes}_{F}$ ( $t$ -norm) and Frank sum ${\oplus}_{F}$ ( $t$ -conorm) are defined as follows:

•
$a{{\otimes}_{F}}b=\log_{\hslash}\left(1+\frac{({{\hslash}^{a}-1})({{\hslash}^{% b}}-1)}{\hslash-1}\right)$ ;
•
$a{{\oplus}_{F}}b=1-\log_{\hslash}\left(1+\frac{({{\hslash}^{1-a}-1})({{\hslash% }^{1-b}}-1)}{\hslash-1}\right)$ .

It is important to note that the Frank operations satisfy the compatibility law, i.e.,

$\displaystyle a{{\oplus}_{F}}b+a{{\otimes}_{F}}b=a+b,$ (4) $\displaystyle\frac{\delta(a{{\oplus}_{F}}b)}{\delta a}+\frac{\delta(a{{\otimes% }_{F}}b)}{\delta a}=\frac{\delta(a{{\oplus}_{F}}b)}{\delta b}+\frac{\delta(a{{% \otimes}_{F}}b)}{\delta b}=1.$ (5)

In addition, Frank operations have the following special cases, which are shown as

1.
When $\hslash\rightarrow 1$ , we get $a{{\otimes}_{F}}b=a.b$ and $a{{\oplus}_{F}}b=a+b-a.b$ , which are the algebraic $t$ -norm and the algebraic $t$ -conorm, respectively,
2.
When $\hslash\rightarrow+\infty$ , then Frank $t$ -norm and Frank $t$ -conorm reduce to the Lukasiewicz $t$ -norm and the Lukasiewicz $t$ -conorm, respectively, as $a{{\otimes}_{F}}b=\max\{{a+b-1,0}\}$ and $a{{\oplus}_{F}}b=\min\{{a+b,1}\}$ .

3. An improved score function for L^qROPFNs

This section begins by examining the drawbacks of the score function $\mathcal{S}$ proposed by [62] for ranking L^qROPFNs. To illustrate these drawbacks, we provide numerical examples that highlight its inefficiencies. Subsequently, we present a novel score function that addresses these shortcomings and enables us to obtain a more efficient ranking of L^qROPFNs.

.

Let $\Phi_{1}=\langle{v_{4.50},v_{6.50}}\rangle$ and $\Phi_{2}=\langle{v_{5.00},v_{4.00}}\rangle$ be two L^qROPFNs defined on ECLTS $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . Based on the score function $\mathbscr{S}$ shown in Eq. (3) with $q=2$ , we obtain $\mathbscr{S}(\Phi_{1})=4.5826$ and $\mathbscr{S}(\Phi_{2})=6.0415$ . It indicates that $\Phi_{1}<\Phi_{2}$ . Hence, we get $\Phi_{1}<\Phi_{2}$ .

.

Consider two L^qROPFNs denoted as $\Phi_{3}=\langle{v_{5.60},v_{5.60}}\rangle$ and $\Phi_{4}=\langle{v_{6.40},v_{6.40}}\rangle$ defined on the ECLTS represented by $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . By employing the score function $\mathbscr{S}$ as presented in Eq. (3) with $q=4$ , we obtain $\mathbscr{S}(\Phi_{3})=\mathbscr{S}(\Phi_{4})=6.7272$ . It shows that $\Phi_{3}=\Phi_{4}$ . Consequently, it is not possible to differentiate the distinction between $\Phi_{3}$ and $\Phi_{4}$ using the score function proposed by [62]. The ranking order between $\Phi_{3}$ and $\Phi_{4}$ cannot be determined in this particular situation. However, if we utilize the accuracy function $\mathbscr{A}$ shown in Eq. (3), we find that $\mathbscr{A}(\Phi_{3})=6.6596$ and $\mathbscr{A}(\Phi_{4})=7.6109$ , which implies that $\Phi_{3}<\Phi_{4}$ . This demonstrates that the score and accuracy functions can enable a comparison between L^qROPFNs $\Phi_{3}$ and $\Phi_{4}$ , allowing us to determine their relative ranking.

.

Consider two L^qROPFNs denoted as $\Phi_{5}=\langle{v_{0.80},v_{4.00}}\rangle$ and $\Phi_{6}=\langle{v_{4.00},v_{5.60}}\rangle$ defined on the ECLTS represented by $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . By employing the score function $\mathbscr{S}$ as mentioned in Eq. (3) with $q=2$ , we find that $\mathbscr{S}(\Phi_{5})=\mathbscr{S}(\Phi_{6})=4.9315$ . This implies that $\Phi_{5}=\Phi_{6}$ . Consequently, it is not possible to differentiate the distinction between $\Phi_{5}$ and $\Phi_{6}$ using the score function proposed by [62]. Hence, the ranking order between the L^qROPFNs $\Phi_{5}$ and $\Phi_{6}$ cannot be determined in this particular scenario. However, if we employ the accuracy function denoted as $\mathbscr{A}$ given in Eq. (3), we obtain $\mathbscr{A}(\Phi_{5})=4.0792$ and $\mathbscr{A}(\Phi_{6})=6.8819$ . Thus, we can conclude that $\Phi_{5}<\Phi_{6}$ . This demonstrates that by utilizing both the score and accuracy functions, we can make a comparison between L^qROPFNs $\Phi_{5}$ and $\Phi_{6}$ , allowing us to establish their relative ranking.

.

Consider two L^qROPFNs denoted as $\Phi_{7}=\langle v_{7.20},v_{0.80}\rangle$ and $\Phi_{8}=\langle v_{6.40},v_{0.00}\rangle$ defined on the ECLTS represented by $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . We can evaluate the ranking order of these L^qROPFNs using different assessment measures mentioned in Definition 5. First, by employing the score function $\mathbscr{S}$ as mentioned in Eq. (3) with $q=1$ , we find that $\mathbscr{S}(\Phi_{7})=\mathbscr{S}(\Phi_{8})=7.2000$ , which implies that $\Phi_{7}=\Phi_{8}$ . It indicates that the score function fails to differentiate between $\Phi_{7}$ and $\Phi_{8}$ , implying that they are equal. Consequently, we cannot determine the difference between $\Phi_{7}$ and $\Phi_{8}$ using the score function proposed by [62] in this scenario. However, if we employ the accuracy function denoted as $\mathbscr{A}$ given in Eq. (3), we obtain $\mathbscr{A}(\Phi_{7})=8.0000$ and $\mathbscr{A}(\Phi_{8})=6.4000$ . It gives $\mathbscr{A}(\Phi_{7})>\mathbscr{A}(\Phi_{8})$ . Thus, we conclude that L^qROPFN $\Phi_{7}$ is greater than L^qROPFN $\Phi_{8}$ based on the accuracy function. In summary, while the score function fails to distinguish the ranking order between $\Phi_{7}$ and $\Phi_{8}$ , the accuracy function successfully determines that $\Phi_{7}$ is greater than $\Phi_{8}$ . This demonstrates that by utilizing both the score and accuracy functions, we can make a comparison between L^qROPFNs $\Phi_{7}$ and $\Phi_{8}$ , allowing us to establish their relative ranking.

To address the limitations mentioned earlier, we define a new score function for ranking L^qROPFN as follows:

.

Let ${\Phi}=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ be a L^qROPFN defined on ECLTS $\widehat{\mathbb{V}}_{[0,r]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{r},g\in[% 0,r]\}$ , we propose an improved score function for L^qROPFN as follows

$\displaystyle{\mathcal{S}}_{\textit{imp}}(\Phi)=r\times\left[\frac{(1+2\tilde{% \mathrm{M}}-\tilde{\mathrm{N}})\cosh(1+\tan(\tilde{\mathrm{M}}\frac{\pi}{4})-(% \tilde{\mathrm{N}})^{q})}{3\cosh(2)}\right],$ (6)

where $\tilde{\mathrm{M}}={\mathrm{M}}/{r}$ , $\tilde{\mathrm{N}}={\mathrm{N}}/{r}$ and $q\geqslant 1$ .

.

Let $\Phi=\langle v_{\mathrm{M}},v_{\mathrm{M}}\rangle$ be a L^qROPFN. The score function $\mathcal{S}_{imp}(\Phi)$ defined in Eq. (6) exhibits the following properties: (iv)]

(i)

$\mathcal{S}_{\textit{imp}}(\Phi)$ monotonically increases with respect to ${\mathrm{M}}$ and monotonically decreases with respect to ${\mathrm{N}}$ .

(ii)

$\mathcal{S}_{\textit{imp}}(\Phi)=0$ if and only if ${\mathrm{M}}=0$ and ${\mathrm{N}}=r$ .

(iii)

$\mathcal{S}_{\textit{imp}}(\Phi)=r$ if and only if ${\mathrm{M}}=r$ and ${\mathrm{N}}=0$ .

(iv)

$0\leqslant\mathcal{S}_{\textit{imp}}(\Phi)\leqslant r$ .

Proof..

According to Definition 3, it is known that ${\mathrm{M}}\in[0,r]$ , ${\mathrm{N}}\in[0,r]$ , and ${\mathrm{M}}^{q}+{\mathrm{N}}^{q}\leqslant r^{q}$ for all $q\geqslant 1$ . Then, we get $\tilde{\mathrm{M}}\in[0,1]$ , $\tilde{\mathrm{N}}\in[0,1]$ , and $\tilde{\mathrm{M}}^{q}+\tilde{\mathrm{N}}^{q}\leqslant 1$ for all $q\geqslant 1$ . We aim to prove part (i) by showing that ${\mathcal{S}}_{\textit{imp}}(\Phi)$ monotonically increases with respect to $\tilde{\mathrm{M}}$ and monotonically decreases with respect to $\tilde{\mathrm{N}}$ .

(i) To begin, we calculate the first-order partial derivatives of Eq. (6) with respect to $\tilde{\mathrm{M}}$ and $\tilde{\mathrm{N}}$ . We obtain

$\displaystyle\frac{\partial{\mathcal{S}}_{\textit{imp}}(\Phi)}{\partial\hat{% \mathrm{M}}}=r\times\left[\frac{\begin{array}[]{c}2\cosh(1+\tan(\tilde{\mathrm% {M}}\frac{\pi}{4})-(\tilde{\mathrm{N}})^{q})+\frac{\pi}{4}(1+2\tilde{\mathrm{M% }}-\tilde{\mathrm{N}})\\ \sinh(1+\tan(\tilde{\mathrm{M}}\frac{\pi}{4})-(\tilde{\mathrm{N}})^{q})\text{% sec}^{2}(\tilde{\mathrm{M}}\frac{\pi}{4})\end{array}}{3\cosh(2)}\right]% \geqslant 0;$ $\displaystyle\frac{\partial{\mathcal{S}}_{\textit{imp}}(\Phi)}{\partial\tilde{% \mathrm{N}}}=-r\times\left[\frac{\begin{array}[]{c}\cosh(1+\tan(\tilde{\mathrm% {M}}\frac{\pi}{4})-(\tilde{\mathrm{N}})^{q})+q(\tilde{\mathrm{N}})^{q-1}(1+2% \tilde{\mathrm{M}}-\tilde{\mathrm{N}})\\ \sinh(1+\tan(\tilde{\mathrm{M}}\frac{\pi}{4})-(\tilde{\mathrm{N}})^{q})\end{% array}}{3\cosh(2)}\right]\leqslant 0.$

Thus, it can be deduced that ${\mathcal{S}}_{\textit{imp}}(\Phi)$ exhibits a monotonic increase in relation to $\tilde{\mathrm{M}}$ , while demonstrating a monotonic decrease in relation to $\tilde{\mathrm{N}}$ .

(ii) & (iii) Based on the information provided in part (i), the score function ${\mathcal{S}}_{\textit{imp}}(\Phi)$ reaches its minimum and maximum values when $\tilde{\mathrm{M}}=0,\tilde{\mathrm{N}}=1\rightarrow{\mathrm{M}}=0,{\mathrm{N}% }=r$ and $\tilde{\mathrm{M}}=1,\tilde{\mathrm{N}}=0\rightarrow{\mathrm{M}}=r,{\mathrm{N}% }=0$ , respectively. Furthermore, when ${\mathrm{M}}=0,{\mathrm{N}}=r\rightarrow\tilde{\mathrm{M}}=0,\tilde{\mathrm{N}% }=1$ , the corresponding value of score function ${\mathcal{S}}_{\textit{imp}}(\Phi)$ is given as

$\displaystyle{\mathcal{S}}_{\textit{imp}}(\Phi)=r\times\left[\frac{(1+2\times 0% -1)\times\cosh(1+\tan(0\times\frac{\pi}{4})-(1)^{q})}{3\cosh(2)}\right]=0.$ (7)

and when ${\mathrm{M}}=r,{\mathrm{N}}=0\rightarrow\tilde{\mathrm{M}}=1,\tilde{\mathrm{N}% }=0$ , then

$\displaystyle{\mathcal{S}}_{\textit{imp}}(\Phi)=r\times\left[\frac{(1+2\times 1% -0)\times\cosh(1+\tan(1\times\frac{\pi}{4})-(0)^{q})}{3\cosh(2)}\right]=r.$ (8)

(iv) After analyzing the preceding discussion, it is evident that the score function ${\mathcal{S}}_{imp}(\Phi)$ exhibits a minimum value of $0$ and a maximum value of $r$ . This range, specifically, implies that the value of the score function ${\mathcal{S}}_{\textit{imp}}(\Phi)$ lies within the bounds of $0$ and $r$ , thereby satisfying the inequality $0\leqslant{\mathcal{S}}_{\textit{imp}}(\Phi)\leqslant r$ .

This shows that the theorem holds true. ∎

.

Let ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ and ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ be two L^qROPFN defined on ECLTS $\widehat{\mathbb{V}}_{[0,r]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{r},g\in[% 0,r]\}$ , if $v_{{\mathrm{M}}{1}}\leqslant v_{{\mathrm{M}}{2}}$ and $v_{{\mathrm{N}}{1}}\geqslant v_{{\mathrm{N}}{2}}$ , then ${\mathcal{S}}_{\textit{imp}}({{\Phi}_{1}})<{\mathcal{S}}_{\textit{imp}}({{\Phi% }_{2}})$ .

Proof..

By applying Theorem 1, it is established that ${\mathcal{S}}_{\textit{imp}}(\Phi)$ exhibits monotonic behavior, increasing with respect to ${\mathrm{M}}$ and decreasing with respect to ${\mathrm{N}}$ . Consequently, if $v_{{\mathrm{M}}{1}}\leqslant v_{{\mathrm{M}}{2}}$ and $v_{{\mathrm{N}}{1}}\geqslant v_{{\mathrm{N}}{2}}$ hold true, it follows that ${\mathcal{S}}_{\textit{imp}}({{\Phi}_{1}})\leqslant{\mathcal{S}}_{\textit{imp}% }({{\Phi}_{2}})$ . ∎

.

The ranking rules for any two L^qROPFNs ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ and ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ defined on ECLTS $\widehat{\mathbb{V}}_{[0,r]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{r},g\in[% 0,r]\}$ , can be determined using the proposed improved score function ${\mathcal{S}}_{\textit{imp}}$ in the following manner:

•

If $\mathcal{S}_{\textit{imp}}({{\Phi}_{1}})>\mathcal{S}_{\textit{imp}}({{{\Phi}}_% {2}})$ , then ${{{\Phi}}_{1}}>{{{\Phi}}_{2}}$ ;

•

If $\mathcal{S}_{\textit{imp}}({{\Phi}_{1}})<\mathcal{S}_{\textit{imp}}({{{\Phi}}_% {2}})$ , then ${{{\Phi}}_{1}}<{{{\Phi}}_{2}}$ ;

•

If $\mathcal{S}_{\textit{imp}}({{\Phi}_{1}})=\mathcal{S}_{\textit{imp}}({{{\Phi}}_% {2}})$ , then ${{{\Phi}}_{1}}={{{\Phi}}_{2}}$ .

In order to demonstrate the benefits and practicality of the suggested score function ${\mathcal{S}}_{\textit{imp}}$ for L^qROPFNs, we apply it to solve Examples 1–4. The outcomes obtained using our proposed score function ${\mathcal{S}}_{\textit{imp}}$ are summarized in Table 1. This analysis serves to showcase the advantages and feasibility of the proposed approach.

Table 1

The comparison results

Examples

\mathbscr{S}

Ranking order

\mathbscr{A}

Ranking order

{\mathcal{S}}_{\textit{imp}}

Ranking order

\Phi_{1}=\langle{v}_{4.50},{v}_{6.50}\rangle

\Phi_{2}=\langle{v}_{5.00},{v}_{4.00}\rangle

4.5826

6.0415

\Phi_{1}<\Phi_{2}

–

1.2549

2.4124

\Phi_{1}<\Phi_{2}

\Phi_{3}=\langle{v}_{5.60},{v}_{5.60}\rangle

\Phi_{4}=\langle{v}_{6.40},{v}_{6.40}\rangle

6.7272

6.7272

\Phi_{3}=\Phi_{4}

6.6596

7.6109

\Phi_{3}<\Phi_{4}

2.5301

2.5517

\Phi_{3}<\Phi_{4}

\Phi_{5}=\langle{v}_{0.80},{v}_{4.00}\rangle

\Phi_{6}=\langle{v}_{4.00},{v}_{5.60}\rangle

4.9315

4.9315

\Phi_{5}=\Phi_{6}

4.0792

6.8819

\Phi_{5}<\Phi_{6}

0.6765

1.3438

\Phi_{5}<\Phi_{6}

\Phi_{7}=\langle{v}_{7.20},{v}_{0.80}\rangle

\Phi_{8}=\langle{v}_{6.40},{v}_{0.00}\rangle

7.2000

7.2000

\Phi_{7}=\Phi_{8}

8.0000

6.4000

\Phi_{7}>\Phi_{8}

5.6946

5.3436

\Phi_{7}>\Phi_{8}

The outcomes presented in Table 1 provide a clear demonstration of the superior ranking capabilities of the proposed score function ${\mathcal{S}}_{\textit{imp}}({\Phi})$ . This function efficiently ranks the L^qROPFNs and effectively addresses the limitations found in Liu and Liu’s score function. Consequently, based on these results, we can confidently assert that the performance of the proposed score function outperforms the existing one regarding ranking L^qROPFNs in real-world scenarios.

4. Frank operations of L^qROPFNs

In this section, we define new algebraic operational laws for L^qROPFNs based on Frank operations given in Definition 6. Further, we analyze some desirable properties of these operations in detail.

.

Let ${\Phi}=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ , ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ and ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ be three L^qROPFNs and $\hslash\in(1,+\infty)$ . Then, their Frank operations are defined as follows:

$\displaystyle(i).{{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{\Phi}_{2}}=\left% \langle v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({% {\mathrm{M}}{1}}/{r})^{q}}-1)(\hslash^{1-({{\mathrm{M}}{2}}/{r})^{q}}-1)}{% \hslash-1}\right)\right)}\right)}},\right.\left.v_{r\left(\sqrt[q]{\left(\log_% {\hslash}\left(1+\frac{(\hslash^{({{\mathrm{N}}{1}}/{r})^{q}}-1)(\hslash^{({{% \mathrm{N}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right)}\right)}\right\rangle;(% ii).{{\Phi}_{1}}{\overset{q}{\otimes}_{F}}{{\Phi}_{2}}=\left\langle v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{M}}{1}}/{r})^{% q}}-1)(\hslash^{({{\mathrm{M}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right)}% \right)},\right.\left.v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{% (\hslash^{1-({{\mathrm{N}}{1}}/{r})^{q}}-1)(\hslash^{1-({{\mathrm{N}}{2}}/{r})% ^{q}}-1)}{\hslash-1}\right)\right)}\right)}}\right\rangle;(iii).\lambda{% \overset{q}{\ast}_{F}}{{\Phi}}=\left\langle v_{{r\left(\sqrt[q]{\left(1-\log_{% \hslash}\left(1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}},v_{r\left(\sqrt[q]{\left(\log_% {\hslash}\left(1+\frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}\right\rangle;(iv).{\Phi}^{% \overset{q}{\wedge}_{F}}\lambda=\left\langle v_{r\left(\sqrt[q]{\left(\log_{% \hslash}\left(1+\frac{(\hslash^{({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash% -1)^{\lambda-1}}\right)\right)}\right)},v_{{r\left(\sqrt[q]{\left(1-\log_{% \hslash}\left(1+\frac{(\hslash^{1-({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}}\right\rangle.$

.

Let ${\Phi}=\langle v_{7.00},v_{5.65}\rangle$ , ${\Phi}_{1}=\langle v_{6.80},v_{5.92}\rangle$ and ${\Phi}_{2}=\langle{v_{7.20}},{v_{5.28}}\rangle$ be three L^qROPFNs defined on the ECLTS represented by $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . Using Definition 9 for $\hslash=3$ , $q=4$ , and $\lambda=3$ , then we get

$\displaystyle{{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{\Phi}_{2}}=\left\langle v_% {{8\times\left(\sqrt[4]{\left(1-\log_{3}\left(1+\frac{(3^{1-({6.80}/{8})^{4}}-% 1)(3^{1-({7.20}/{8})^{4}}-1)}{3-1}\right)\right)}\right)}},\right.$ $\displaystyle\quad\left.v_{8\times\left(\sqrt[4]{\left(\log_{3}\left(1+\frac{(% 3^{({5.92}/{8})^{4}}-1)(3^{({5.28}/{8})^{4}}-1)}{3-1}\right)\right)}\right)}% \right\rangle=\langle v_{7.7175},v_{3.5834}\rangle;$ $\displaystyle{{\Phi}_{1}}{\overset{q}{\otimes}_{F}}{{\Phi}_{2}}=\left\langle v% _{8\times\left(\sqrt[4]{\left(\log_{3}\left(1+\frac{(3^{({6.80}/{8})^{4}}-1)(3% ^{({7.20}/{8})^{4}}-1)}{3-1}\right)\right)}\right)},\right.$ $\displaystyle\quad\left.v_{{8\times\left(\sqrt[4]{\left(1-\log_{3}\left(1+% \frac{(3^{1-({5.92}/{8})^{4}}-1)(3^{1-({5.28}/{8})^{4}}-1)}{3-1}\right)\right)% }\right)}}\right\rangle={\langle v_{5.9793},v_{6.5499}\rangle};$ $\displaystyle 3{\overset{q}{\ast}_{F}}{{\Phi}}=\left\langle v_{{8\times\left(% \sqrt[4]{\left(1-\log_{3}\left(1+\frac{(3^{1-({7.00}/{8})^{4}}-1)^{3}}{(3-1)^{% 3-1}}\right)\right)}\right)}},v_{8\times\left(\sqrt[4]{\left(\log_{3}\left(1+% \frac{(3^{({5.65}/{8})^{4}}-1)^{3}}{(3-1)^{3-1}}\right)\right)}\right)}\right\rangle$ $\displaystyle\quad={\langle v_{7.9138},v_{2.3173}\rangle};$ $\displaystyle{\Phi}^{\overset{q}{\wedge}_{F}}3=\left\langle v_{8\times\left(% \sqrt[4]{\left(\log_{3}\left(1+\frac{(3^{({7.00}/{8})^{q}}-1)^{3}}{(3-1)^{3-1}% }\right)\right)}\right)},v_{{8\times\left(\sqrt[4]{\left(1-\log_{3}\left(1+% \frac{(3^{1-({5.65}/{8})^{4}}-1)^{3}}{(3-1)^{3-1}}\right)\right)}\right)}}\right\rangle$ $\displaystyle\quad={\langle v_{5.0139},v_{7.0829}\rangle}.$

.

Let ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ and ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ be two L^qROPFNs. If ${\Phi}_{3}={{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{\Phi}_{2}}$ and ${\Phi}_{4}={{\Phi}_{1}}{\overset{q}{\otimes}_{F}}{{\Phi}_{2}}$ , then ${\Phi}_{3}$ and ${\Phi}_{4}$ are also L^qROPFNs.

Proof..

Since ${\Phi}_{3}={{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{\Phi}_{2}}=\langle v_{% \mathrm{M}{3}},v_{{\mathrm{N}}{3}}\rangle$ . By Definition 9, we have

$\displaystyle v_{{\mathrm{M}}{3}}=v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}% \left(1+\frac{(\hslash^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)(\hslash^{1-({{% \mathrm{M}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right)}\right)}},$ (9) $\displaystyle v_{{\mathrm{N}}{3}}=v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left% (1+\frac{(\hslash^{({{\mathrm{N}}{1}}/{r})^{q}}-1)(\hslash^{({{\mathrm{N}}{2}}% /{r})^{q}}-1)}{\hslash-1}\right)\right)}\right)}.$ (10)

Since $v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\in{{\widehat{\mathbb{V}}}_{[0,r]}}$ and $({{\mathrm{M}}{i}})^{q}+({{\mathrm{N}}{i}})^{q}\leqslant r^{q}$ for $i=1,2$ , then we have

$\displaystyle 0=r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{% 1-({0}/{r})^{q}}-1)(\hslash^{1-({0}/{r})^{q}}-1)}{\hslash-1}\right)\right)}% \right)\leqslant{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash% ^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)(\hslash^{1-({{\mathrm{M}}{2}}/{r})^{q}}-1)% }{\hslash-1}\right)\right)}\right)}}\leqslant{{r\left(\sqrt[q]{\left(1-\log_{% \hslash}\left(1+\frac{(\hslash^{1-({r}/{r})^{q}}-1)(\hslash^{1-({r}/{r})^{q}}-% 1)}{\hslash-1}\right)\right)}\right)}}=r;$

and

$\displaystyle 0={r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{(% {0}/{r})^{q}}-1)(\hslash^{({0}/{r})^{q}}-1)}{\hslash-1}\right)\right)}\right)}% \leqslant{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{% \mathrm{N}}{1}}/{r})^{q}}-1)(\hslash^{({{\mathrm{N}}{2}}/{r})^{q}}-1)}{\hslash% -1}\right)\right)}\right)}\leqslant{r\left(\sqrt[q]{\left(\log_{\hslash}\left(% 1+\frac{(\hslash^{({r}/{r})^{q}}-1)(\hslash^{({r}/{r})^{q}}-1)}{\hslash-1}% \right)\right)}\right)}=r.$

Therefore

$\displaystyle v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash% ^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)(\hslash^{1-({{\mathrm{M}}{2}}/{r})^{q}}-1)% }{\hslash-1}\right)\right)}\right)}},v_{r\left(\sqrt[q]{\left(\log_{\hslash}% \left(1+\frac{(\hslash^{({{\mathrm{N}}{1}}/{r})^{q}}-1)(\hslash^{({{\mathrm{N}% }{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right)}\right)}\in{{\widehat{\mathbb{V}}% }_{[0,r]}}.$

By using Eqs (9) and (10), we have

$\displaystyle({{\mathrm{M}}{3}})^{q}+({{\mathrm{N}}{3}})^{q}=r^{q}\left(1-\log% _{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)(\hslash^{1% -({{\mathrm{M}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right){}+r^{q}\left(\log_{% \hslash}\left(1+\frac{(\hslash^{({{\mathrm{N}}{1}}/{r})^{q}}-1)(\hslash^{({{% \mathrm{N}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right)\leqslant r^{q}\left(1-% \log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)(% \hslash^{1-({{\mathrm{M}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right){}+r^{q}% \left(\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)(% \hslash^{1-({{\mathrm{M}}{2}}/{r})^{q}}-1)}{\hslash-1}\right)\right)=r^{q}.$

Hence ${\Phi}_{3}$ is a L^qROPFN. Similarly, we can prove that ${\Phi}_{4}$ is also L^qROPFN. ∎

.

Let ${\Phi}=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ be a L^qROPFN and $\lambda>0$ . If ${\Phi}_{5}=\lambda{\overset{q}{\ast}_{F}}{{\Phi}}$ and ${\Phi}_{6}={{\Phi}}^{\overset{q}{\wedge}_{F}}\lambda$ , then, both ${\Phi}_{5}$ and ${\Phi}_{6}$ are also L^qROPFNs.

Proof..

Since ${\Phi}_{5}={\lambda{\overset{q}{\ast}_{F}}{{\Phi}}}=\langle v_{{\mathrm{M}}{5}% },v_{{\mathrm{N}}{5}}\rangle$ . Using Definition 9, we get

$\displaystyle v_{{\mathrm{M}}{5}}=v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}% \left(1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{% \lambda-1}}\right)\right)}\right)}};$ (11) $\displaystyle v_{{\mathrm{N}}{5}}=v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left% (1+\frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-% 1}}\right)\right)}\right)}.$ (12)

Since $v_{\mathrm{M}},v_{\mathrm{N}}\in{{\widehat{\mathbb{V}}}_{[0,r]}}$ and $({\mathrm{M}})^{q}+({\mathrm{N}})^{q}\leqslant r^{q}$ , then we get

$\displaystyle 0={r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^% {1-({0}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)% }\leqslant{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({% \mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}% \right)}\leqslant{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash% ^{1-({r}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right% )}=r;$

and

$\displaystyle 0={r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{(% {0}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}% \leqslant{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({% \mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}% \right)}\leqslant{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{% ({r}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}=r.$

Therefore

$\displaystyle v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash% ^{1-({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)% \right)}\right)}},v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(% \hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)% \right)}\right)}\in{{\widehat{\mathbb{V}}}_{[0,r]}}.$

By using Eqs (11) and (12), we have

$\displaystyle({{\mathrm{M}}{5}})^{q}+({{\mathrm{N}}{5}})^{q}={{r^{q}{\left(1-% \log_{\hslash}\left(1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{% (\hslash-1)^{\lambda-1}}\right)\right)}}}+{r^{q}{\left(\log_{\hslash}\left(1+% \frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}% \right)\right)}}$ $\displaystyle\leqslant{{r^{q}{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-% ({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}}% }+{r^{q}{\log_{\hslash}\left(1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q}}-1)^{% \lambda}}{(\hslash-1)^{\lambda-1}}\right)}}=r^{q}.$

Hence ${\Phi}_{5}$ is a L^qROPFN. Similarly, we can prove that ${\Phi}_{6}$ is also a L^qROPFN. ∎

.

We discuss the $\lambda{\overset{q}{\ast}_{F}}{{\Phi}}$ and ${\Phi}^{\overset{q}{\wedge}_{F}}\lambda$ for some special cases of $\lambda$ and ${\Phi}$ .

(a)

If ${\Phi}=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle=\langle v_{r},v_{0}\rangle$ , then

$\displaystyle\lambda{\overset{q}{\ast}_{F}}{{\Phi}}=\left\langle v_{{r\left(% \sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({r}/{r})^{q}}-1)^{% \lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}},v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{(\hslash^{({0}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}\right\rangle=\langle v_{r},v_{0% }\rangle;$ $\displaystyle{\Phi}^{\overset{q}{\wedge}_{F}}\lambda=\left\langle v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({r}/{r})^{q}}-1)^{% \lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)},v_{{r\left(\sqrt[q]{% \left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({0}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}}\right\rangle=\langle v_{r},v_{% 0}\rangle.$

(b)

If ${\Phi}=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle=\langle v_{0},v_{r}\rangle$ , then

$\displaystyle\lambda{\overset{q}{\ast}_{F}}{{\Phi}}=\left\langle v_{{r\left(% \sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({0}/{r})^{q}}-1)^{% \lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}},v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{(\hslash^{({r}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}\right\rangle=\langle v_{0},v_{r% }\rangle;$ $\displaystyle{\Phi}^{\overset{q}{\wedge}_{F}}\lambda=\left\langle v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({0}/{r})^{q}}-1)^{% \lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)},v_{{r\left(\sqrt[q]{% \left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({r}/{r})^{q}}-1)^{\lambda}}{(% \hslash-1)^{\lambda-1}}\right)\right)}\right)}}\right\rangle=\langle v_{0},v_{% r}\rangle.$

(c)

If $\lambda\rightarrow 0$ , then

$\displaystyle\lambda{\overset{q}{\ast}_{F}}{{\Phi}}=\left\langle v_{{r\left(% \sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q% }}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}},v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1% )^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}\right\rangle% \rightarrow\langle v_{0},v_{r}\rangle;$ $\displaystyle{\Phi}^{\overset{q}{\wedge}_{F}}\lambda=\left\langle v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({\mathrm{M}}/{r})^{q}}-1% )^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)},v_{{t\left(\sqrt[% q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({\mathrm{N}}/{r})^{q}}-1)^% {\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}}\right\rangle% \rightarrow\langle v_{r},v_{0}\rangle.$

(d)

If $\lambda\rightarrow+\infty$ , then

$\displaystyle\lambda{\overset{q}{\ast}_{F}}{{\Phi}}=\left\langle v_{{r\left(% \sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q% }}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}},v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1% )^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}\right\rangle% \rightarrow\langle v_{r},v_{0}\rangle;$ $\displaystyle{\Phi}^{\overset{q}{\wedge}_{F}}\lambda=\left\langle v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({\mathrm{M}}/{r})^{q}}-1% )^{\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)},v_{{r\left(\sqrt[% q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({\mathrm{N}}/{r})^{q}}-1)^% {\lambda}}{(\hslash-1)^{\lambda-1}}\right)\right)}\right)}}\right\rangle% \rightarrow\langle v_{0},v_{r}\rangle.$

(e)

If $\lambda\rightarrow 1$ , then

$\displaystyle\lim_{\hslash\to 1}\lambda{\overset{q}{\ast}_{F}}{{\Phi}}=\lim_{% \hslash\to 1}\left\langle v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \frac{(\hslash^{1-({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1% }}\right)\right)}\right)}},v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}% \right)\right)}\right)}\right\rangle.$

According to the law of equivalent infinitesimal replacement $\ln(1+a)\sim a(a>0)$ , then we have

$\displaystyle{r\lim_{\hslash\to 1}\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}}% \right)\right)}\right)}={r\lim_{\hslash\to 1}\left(\frac{\sqrt[q]{\left(\ln% \left(1+\frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{% \lambda-1}}\right)\right)}}{\ln\hslash}\right)}={r\lim_{\hslash\to 1}\left(% \sqrt[q]{\frac{(\hslash^{({\mathrm{N}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{% \lambda-1}\ln\hslash}}\right)}.$

Using Taylor expansion and $\hslash>0$ , then we obtain

$\displaystyle\hslash^{({\mathrm{N}}/{r})^{q}}=1+({\mathrm{N}}/{r})^{q}\ln{% \hslash}+\frac{({\mathrm{N}}/{r})^{q}}{2}(\ln{\hslash})^{2}+\ldots=1+({\mathrm% {N}}/{r})^{q}\ln{\hslash}+O(\ln(\hslash)).$

Thus

$\displaystyle{r\lim_{\hslash\to 1}\left(\sqrt[q]{\frac{(\hslash^{({\mathrm{N}}% /{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda-1}\ln\hslash}}\right)}={r\lim_{% \hslash\to 1}\left(\sqrt[q]{\frac{(({\mathrm{N}}/{r})\ln{\hslash})^{q}}{(% \hslash-1)^{\lambda-1}\ln\hslash}}\right)}={r\lim_{\hslash\to 1}\left(\sqrt[q]% {\frac{({\mathrm{N}}/{r})^{q}(\ln{\hslash})^{\lambda-1}}{(\hslash-1)^{\lambda-% 1}}}\right)}=r({\mathrm{N}}/{r})^{\lambda}$

Similarly, we obtain

$\displaystyle{{r\lim_{\hslash\to 1}\left(\sqrt[q]{\left(1-\log_{\hslash}\left(% 1+\frac{(\hslash^{1-({\mathrm{M}}/{r})^{q}}-1)^{\lambda}}{(\hslash-1)^{\lambda% -1}}\right)\right)}\right)}}={r\left(\sqrt[q]{{\left(1-\left(1-\frac{{\mathrm{% M}}^{q}}{r^{q}}\right)^{\lambda}\right)}}\right)}.$

Thus

$\displaystyle\lim_{\hslash\to 1}\lambda{\overset{q}{\ast}_{F}}{{\Phi}}=\left% \langle v_{r\left(\sqrt[q]{{\left(1-\left(1-\frac{{\mathrm{M}}^{q}}{r^{q}}% \right)^{\lambda}\right)}}\right)},v_{t\left({\mathrm{N}}/{r}\right)^{\lambda}% }\right\rangle.$

Similarly, we can obtain

$\displaystyle\lim_{\hslash\to 1}{\Phi}^{\overset{q}{\wedge}_{F}}\lambda=\left% \langle v_{r\left({\mathrm{M}}/{r}\right)^{\lambda}},v_{r\left(\sqrt[q]{{\left% (1-\left(1-\frac{{\mathrm{N}}^{q}}{r^{q}}\right)^{\lambda}\right)}}\right)}% \right\rangle.$

Moreover, the Frank operational laws of L^qROPFNs in Definition 9 hold the following properties listed in next theorems.

.

Let ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ , ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ and ${\Phi}_{3}=\langle v_{{\mathrm{M}}{3}},v_{{\mathrm{N}}{3}}\rangle$ be three L^qROPFNs, then the Frank operations of L^qROPFNs satisfy the following properties:

(a)

${{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{\Phi}_{2}}={{\Phi}_{2}}{\overset{q}{% \oplus}_{F}}{{\Phi}_{1}}$ ;

(b)

${{\Phi}_{1}}{\overset{q}{\otimes}_{F}}{{\Phi}_{2}}={{\Phi}_{2}}{\overset{q}{% \otimes}_{F}}{{\Phi}_{1}}$ ;

(c)

$({{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{\Phi}_{2}}){\overset{q}{\oplus}_{F}}{{% \Phi}_{3}}={{\Phi}_{1}}{\overset{q}{\oplus}_{F}}({{\Phi}_{2}}{\overset{q}{% \oplus}_{F}}{{\Phi}_{3}})$ ;

(d)

$({{\Phi}_{1}}{\overset{q}{\otimes}_{F}}{{\Phi}_{2}}){\overset{q}{\otimes}_{F}}% {{\Phi}_{3}}={{\Phi}_{1}}{\overset{q}{\otimes}_{F}}({{\Phi}_{2}}{\overset{q}{% \otimes}_{F}}{{\Phi}_{3}})$ .

Proof..

Since Definition 9 can be used to showcase the results, the proof has been left out. ∎

.

(a)

$(\lambda{\overset{q}{\ast}_{F}}{{\Phi}_{1}}){\overset{q}{\oplus}_{F}}(\lambda{% \overset{q}{\ast}_{F}}{{\Phi}_{2}})=\lambda{\overset{q}{\ast}_{F}}({{\Phi}_{1}% }{\overset{q}{\oplus}_{F}}{{\Phi}_{2}})$ ;

(b)

$({{\Phi}_{1}}^{\overset{q}{\wedge}_{F}}\lambda){\overset{q}{\otimes}_{F}}({{% \Phi}_{2}}^{\overset{q}{\wedge}_{F}}\lambda)=({{\Phi}_{1}}{\overset{q}{\otimes% }_{F}}{{\Phi}_{2}})^{\overset{q}{\wedge}_{F}}\lambda$ ;

(c)

$(\lambda_{1}{\overset{q}{\ast}_{F}}{{\Phi}}){\overset{q}{\oplus}_{F}}(\lambda_% {2}{\overset{q}{\ast}_{F}}{{\Phi}})=(\lambda_{1}+\lambda_{2}){\overset{q}{\ast% }_{F}}{{\Phi}}$ ;

(d)

$({{\Phi}}^{\overset{q}{\wedge}_{F}}\lambda_{1}){\overset{q}{\otimes}_{F}}({{% \Phi}}^{\overset{q}{\wedge}_{F}}\lambda_{2})={{\Phi}}^{\overset{q}{\wedge}_{F}% }(\lambda_{1}+\lambda_{2})$ ;

(e)

$\lambda_{1}{\overset{q}{\ast}_{F}}(\lambda_{2}{\overset{q}{\ast}_{F}}{{\Phi}})% =(\lambda_{1}\lambda_{2}){\overset{q}{\ast}_{F}}{\Phi}$ ;

(f)

$({{\Phi}}^{\overset{q}{\wedge}_{F}}\lambda_{1})^{\overset{q}{\wedge}_{F}}% \lambda_{2}={{\Phi}}^{\overset{q}{\wedge}_{F}}(\lambda_{1}\lambda_{2})$ .

Proof..

Definition 9 can be used to derive the results easily. ∎

.

Let ${\Phi}_{1}=\langle v_{{\mathrm{M}}{1}},v_{{\mathrm{N}}{1}}\rangle$ and ${\Phi}_{2}=\langle v_{{\mathrm{M}}{2}},v_{{\mathrm{N}}{2}}\rangle$ be two L^qROPFNs, then following results are true:

(a)

$({{\Phi}_{1}}{\overset{q}{\cup}}{{\Phi}_{2}}){\overset{q}{\oplus}_{F}}({{\Phi}% _{1}}{\overset{q}{\cap}}{{\Phi}_{2}})=({{\Phi}_{1}}{\overset{q}{\oplus}_{F}}{{% \Phi}_{2}})$ ;

(b)

$({{\Phi}_{1}}{\overset{q}{\cup}}{{\Phi}_{2}}){\overset{q}{\otimes}_{F}}({{\Phi% }_{1}}{\overset{q}{\cap}}{{\Phi}_{2}})=({{\Phi}_{1}}{\overset{q}{\otimes}_{F}}% {{\Phi}_{2}})$ .

Proof..

These results can be proved easily by using the Definitions 4 and 9. So, we omit the proof from here. ∎

5. AOs for L^qROPFNs based on Frank operational laws

In this section, we aim to present a set of linguistic $q$ -rung orthopair fuzzy weighted averaging and geometric AOs. These operators are derived from the Frank operational laws defined in Definition 9. The purpose of these operators is to aggregate a finite collection of L^qROPFNs denoted as ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle$ , $(i=1(1)n)$ . These AOs offer a flexible and robust approach to combine the information in the L^qROPFNs effectively. Consider that $\Omega$ represents the collection of all L^qROPFNs.

5.1 Frank weighted averaging AOs for L^qROPFNs

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs. A mapping $L^{q}\textit{ROPFFWA}:\Omega^{n}\rightarrow\Omega$ is termed as a linguistic $q$ -rung orthopair fuzzy frank weighted averaging operator, and expressed as

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=(% {w_{1}{\overset{q}{\ast}_{F}}{\Phi}_{1}}){\overset{q}{\oplus}_{F}}({w_{2}{% \overset{q}{\ast}_{F}}{\Phi}_{2}}){\overset{q}{\oplus}_{F}}\ldots{\overset{q}{% \oplus}_{F}}({w_{n}{\overset{q}{\ast}_{F}}{\Phi}_{n}}),$ (13)

where $w=(w_{1},w_{2},\ldots,w_{n})^{T}$ is the weighting vector corresponding to ${\Phi}_{i}$ with ${w}_{i}\in[0,1]$ and $\sum\limits_{i=1}^{n}{w}_{i}=1$ .

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1,(1)n)$ be a set of L^qROPFNs, then, the aggregated value by applying the L^qROPFFWA operator is also a L^qROPFN and given by

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \langle v_{r(\sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{{% \mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},v_{r(\sqrt[q]{(\log_{\hslash}(1+% \prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})}\rangle.$ (14)

Proof..

The initial result is a direct consequence of Theorem 3 and 4, whereas to establish the result described in Eq. (14), we employ the principle of mathematical induction applied to $n$ .

(i) When $n=2$ , according to Frank operational laws of L^qROPFNs, we have

$\displaystyle w_{1}{\overset{q}{\ast}_{F}}{{\Phi_{1}}}=\left\langle v_{{r\left% (\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{1}}/{% r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-1}}\right)\right)}\right)}},v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{N}}{1}}/{r})^{% q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-1}}\right)\right)}\right)}\right\rangle,$ $\displaystyle w_{2}{\overset{q}{\ast}_{F}}{{\Phi_{2}}}=\left\langle v_{{r\left% (\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{2}}/{% r})^{q}}-1)^{w_{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)}},v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{N}}{2}}/{r})^{% q}}-1)^{w_{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)}\right\rangle.$

Then

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2})=(w_{1}{\overset{q}{% \ast}_{F}}{{\Phi_{1}}}){\overset{q}{\oplus}_{F}}(w_{2}{\overset{q}{\ast}_{F}}{% {\Phi_{2}}})$ $\displaystyle=\left\langle v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \frac{(\hslash^{1-({{\mathrm{M}}{1}}/{r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-% 1}}\right)\right)}\right)}},v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{(\hslash^{({{\mathrm{N}}{1}}/{r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-1}% }\right)\right)}\right)}\right\rangle$ $\displaystyle\quad{\overset{q}{\oplus}_{F}}\left\langle v_{{r\left(\sqrt[q]{% \left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{2}}/{r})^{q}}-1% )^{w_{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)}},v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{N}}{2}}/{r})^{q}}-1)^{w% _{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)}\right\rangle$ $\displaystyle=\left\langle v_{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \frac{\prod_{i=1}^{2}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(% \hslash-1)^{\sum_{i=1}^{2}w_{i}-1}}\right)\right)}\right)},v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{\prod_{i=1}^{2}(\hslash^{1-({{{\mathrm{N}}{i% }}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{2}w_{i}-1}}\right)\right)}% \right)}\right\rangle$ $\displaystyle=\left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{2}{(% \hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},v_{r(\sqrt[q]{(\log% _{\hslash}(1+\prod_{i=1}^{2}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i% }}}))})}\right\rangle$ $\displaystyle\quad\left(\text{since }\sum_{i=1}^{2}w_{i}=1\right).$

Thus, Eq. (14) is true for $n=2$ .

(ii) Suppose that Eq. (14) holds for $n=k$ , i.e.,

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{k})=% \left\langle v_{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{\prod_{i=1% }^{k}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_% {i=1}^{k}w_{i}-1}}\right)\right)}\right)},\right.\left.v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{\prod_{i=1}^{k}(\hslash^{1-({{{\mathrm{N}}{i% }}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{k}w_{i}-1}}\right)\right)}% \right)}\right\rangle.$

When $n=k+1$ , based on the Frank operational laws, we have

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{k},{% \Phi}_{k+1})=L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{k}){% \overset{q}{\oplus}_{F}}(w_{k+1}{\overset{q}{\ast}_{F}}{\Phi}_{k+1})$ $\displaystyle=\left\langle v_{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \frac{\prod_{i=1}^{k}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(% \hslash-1)^{\sum_{i=1}^{k}w_{i}-1}}\right)\right)}\right)},v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{\prod_{i=1}^{k}(\hslash^{1-({{{\mathrm{N}}{i% }}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{k}w_{i}-1}}\right)\right)}% \right)}\right\rangle$ $\displaystyle\quad\overset{q}{\oplus}_{F}\left\langle v_{{r\left(\sqrt[q]{% \left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{k+1}}/{r})^{q}}% -1)^{w_{k+1}}}{(\hslash-1)^{w_{k+1}-1}}\right)\right)}\right)}},v_{r\left(% \sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{N}}{k+1}}/{r})% ^{q}}-1)^{w_{k+1}}}{(\hslash-1)^{w_{k+1}-1}}\right)\right)}\right)}\right\rangle.$ $\displaystyle=\left\langle v_{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \frac{\prod_{i=1}^{k+1}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{% (\hslash-1)^{\sum_{i=1}^{k+1}w_{i}-1}}\right)\right)}\right)},v_{r\left(\sqrt[% q]{\left(\log_{\hslash}\left(1+\frac{\prod_{i=1}^{k+1}(\hslash^{1-({{{\mathrm{% N}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{k+1}w_{i}-1}}\right)% \right)}\right)}\right\rangle.$ $\displaystyle=\left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{k+1}% {(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},v_{r(\sqrt[q]{(% \log_{\hslash}(1+\prod_{i=1}^{k+1}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)% ^{w_{i}}}))})}\right\rangle$ $\displaystyle\quad\left(\text{since }\sum_{i=1}^{k+1}w_{i}=1\right).$

Thus, Eq. (14) is true for $n=k+1$ . Therefore, the result given in Eq. (14) holds for all $n\in Z^{+}$ . ∎

.

Let ${\Phi}_{1}=\langle v_{4},v_{5}\rangle,{\Phi}_{2}=\langle v_{1},v_{6}\rangle,{% \Phi}_{3}=\langle v_{5},v_{7}\rangle$ , and ${\Phi}_{4}=\langle v_{3},v_{5}\rangle$ be four L^qROPFNs defined on $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . Further consider that $w=(0.30,0.10,0.20,0.40)^{T}$ denotes the weight vector corresponding to L^qROPFNs ${\Phi}_{i}(i=1(1)4)$ , $\hslash=2$ and $q=3$ . Then using Eq. (14), we have

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},{\Phi}_{3},{\Phi}_{4})$ $\displaystyle=\left\langle v_{8\times(\sqrt[3]{(1-\log_{2}(1+{(2^{1-({{4}/{8}}% )^{3}}-1)^{0.30}}\times{(2^{1-({{1}/{8}})^{3}}-1)^{0.10}}\times{(2^{1-({{5}/{8% }})^{3}}-1)^{0.20}}\times{(2^{1-({{3}/{8}})^{3}}-1)^{0.40}}))})},\right.$ $\displaystyle\quad\left.v_{8\times(\sqrt[3]{(\log_{2}(1+{(2^{({{5}/{8}})^{3}}-% 1)^{0.30}}\times{(2^{({{6}/{8}})^{3}}-1)^{0.10}}\times{(2^{({{7}/{8}})^{3}}-1)% ^{0.20}}\times{(2^{({{5}/{8}})^{3}}-1)^{0.40}}))})}\right\rangle=\langle v_{4.% 0429},v_{5.7201}\rangle.$

By using the definition of L^qROPFFWA operator, the properties can be easily derived.

If ${\Phi}_{i}=\Phi=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle\forall i$ , then

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle.$

Let ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be a collection of L^qROPFNs such that $v_{{\mathrm{M}}{i}^{\prime}}\geqslant v_{{\mathrm{M}}{i}}$ and $v_{{\mathrm{N}}{i}^{\prime}}\leqslant v_{{\mathrm{N}}{i}}\forall i$ , then

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})% \leqslant L^{q}\textit{ROPFFWA}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots% ,{\Phi}_{n}^{\prime}).$

If ${\Phi}^{\blacktriangle}=\langle\max\limits_{i}(v_{{\mathrm{M}}{i}}),\min% \limits_{i}(v_{{\mathrm{N}}{i}})\rangle$ and ${\Phi}^{\blacktriangledown}=\langle\min\limits_{i}(v_{{\mathrm{M}}{i}}),\max% \limits_{i}(v_{{\mathrm{N}}{i}})\rangle$ are two L^qROPFNs , then

$\displaystyle{\Phi}^{\blacktriangledown}\leqslant L^{q}\textit{ROPFFWA}({\Phi}% _{1},{\Phi}_{2},\ldots,{\Phi}_{n})\leqslant{\Phi}^{\blacktriangle}.$

If ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be a collection of L^qROPFNs, then

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=L% ^{q}\textit{ROPFFWA}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{n}% ^{\prime}).$

where $({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{n}^{\prime})$ is any permutation of $({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ .

Let $\Phi=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ be another L^qROPFN, then

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1}{\overset{q}{\oplus}_{F}}{\Phi},% {\Phi}_{2}{\overset{q}{\oplus}_{F}}{\Phi},\ldots,{\Phi}_{n}{\overset{q}{\oplus% }_{F}}{\Phi})=(L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n}))% {\overset{q}{\oplus}_{F}}{\Phi}.$

If $\aleph>0$ is a real number, then

$\displaystyle L^{q}\textit{ROPFFWA}(\aleph{\overset{q}{\ast}_{F}}{\Phi}_{1},% \aleph{\overset{q}{\ast}_{F}}{\Phi}_{2},\ldots,\aleph{\overset{q}{\ast}_{F}}{% \Phi}_{n})=\aleph{\overset{q}{\ast}_{F}}(L^{q}\textit{ROPFFWA}({\Phi}_{1},{% \Phi}_{2},\ldots,{\Phi}_{n})).$

Let ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be another collection of $n$ L^qROPFNs, then

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1}{\overset{q}{\oplus}_{F}}{\Phi}_% {1}^{\prime},{\Phi}_{2}{\overset{q}{\oplus}_{F}}{\Phi}_{2}^{\prime},\ldots,{% \Phi}_{n}{\overset{q}{\oplus}_{F}}{\Phi}_{n}^{\prime})$ $\displaystyle=L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n}){% \overset{q}{\oplus}_{F}}L^{q}\textit{ROPFFWA}({\Phi}_{1}^{\prime},{\Phi}_{2}^{% \prime},\ldots,{\Phi}_{n}^{\prime}).$

Let ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{M}}{i}^{% \prime}}\rangle(i=1(1)n)$ be another collection of $n$ L^qROPFNs and $\aleph_{1},\aleph_{2}>0$ be two real numbers, then

$\displaystyle L^{q}\textit{ROPFFWA}((\aleph_{1}{\overset{q}{\ast}_{F}}{\Phi}_{% 1}){\overset{q}{\oplus}_{F}}(\aleph_{2}{\overset{q}{\ast}_{F}}{\Phi}_{1}^{% \prime}),(\aleph_{1}{\overset{q}{\ast}_{F}}{\Phi}_{2}){\overset{q}{\oplus}_{F}% }(\aleph_{2}{\overset{q}{\ast}_{F}}{\Phi}_{2}^{\prime}),\ldots,$ $\displaystyle(\aleph_{1}{\overset{q}{\ast}_{F}}{\Phi}_{n}){\overset{q}{\oplus}% _{F}}(\aleph_{2}{\overset{q}{\ast}_{F}}{\Phi}_{n}^{\prime}))$ $\displaystyle=(\aleph_{1}{\overset{q}{\ast}_{F}}(L^{q}\textit{ROPFFWA}({\Phi}_% {1},{\Phi}_{2},\ldots,{\Phi}_{n}))){\overset{q}{\oplus}_{F}}(\aleph_{2}{% \overset{q}{\ast}_{F}}(L^{q}\textit{ROPFFWA}({\Phi}_{1}^{\prime},{\Phi}_{2}^{% \prime},\ldots,{\Phi}_{n}^{\prime}))).$

Next, we propose the L^qROPFFOWA operator on the basis of ordered weighted averaging (OWA) [79] operator as follows:

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs. A mapping $L^{q}\textit{ROPFFOWA}:\Omega^{n}\rightarrow\Omega$ is termed as a linguistic $q$ -rung orthopair fuzzy frank ordered weighted averaging operator, and defined as

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% ({\omega_{1}{\overset{q}{\ast}_{F}}{\Phi}_{\sigma(1)}}){\overset{q}{\oplus}_{F% }}({\omega_{2}{\overset{q}{\ast}_{F}}{\Phi}_{\sigma(2)}}){\overset{q}{\oplus}_% {F}}\ldots{\overset{q}{\oplus}_{F}}({\omega_{n}{\overset{q}{\ast}_{F}}{\Phi}_{% \sigma(n)}}),$ (15)

where $(\sigma(1),\sigma(2),\ldots,\sigma(n))$ is a permutation of $(1,2,\ldots,n)$ such that ${\Phi}_{\sigma(i-1)}\geqslant{\Phi}_{\sigma(i)}$ and $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})^{T}$ is the associated weighting vector of ${\Phi}_{\sigma(i)}$ with ${\omega}_{i}\in[0,1]$ and $\sum\limits_{i=1}^{n}{\omega}_{i}=1$ .

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs, then, the aggregated value by applying the L^qROPFFOWA operator is also a L^qROPFN and given by

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{% {\mathrm{M}}{\sigma(i)}}/{r}})^{q}}-1)^{\omega_{i}}}))})},\right.\left.v_{r(% \sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{\sigma(i)% }}/{r}})^{q}}-1)^{\omega_{i}}}))})}\right\rangle.$ (16)

Proof..

The proof can be constructed in a manner analogous to Theorem 8. ∎

.

Let ${\Phi}_{1}=\langle v_{6},v_{5}\rangle,{\Phi}_{2}=\langle v_{4},v_{7}\rangle,{% \Phi}_{3}=\langle v_{5},v_{2}\rangle$ , and ${\Phi}_{4}=\langle v_{6},v_{6}\rangle$ be four L^qROPFNs derived from $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . Further assume that $\omega=(0.1550,0.3450,\linebreak 0.3450,0.1550)^{T}$ denotes the weighting vector associated with L^qROPFNs ${\Phi}_{\sigma(i)}(i=1(1)4)$ (calculated based on normal distribution method [80]), $\hslash=2$ and $q=3$ . Then according to Eq. (6), we have

$\displaystyle{\mathcal{S}_{\textit{imp}}}(\Phi_{1})=2.9202,{\mathcal{S}_{% \textit{imp}}}(\Phi_{2})=1.0287,{\mathcal{S}_{\textit{imp}}}(\Phi_{3})=3.3924,% {\mathcal{S}_{\textit{imp}}}(\Phi_{4})=2.3351.$

Since ${\mathcal{S}_{\textit{imp}}}(\Phi_{3})>{\mathcal{S}_{\textit{imp}}}(\Phi_{1})>% {\mathcal{S}_{\textit{imp}}}(\Phi_{4})>{\mathcal{S}_{\textit{imp}}}(\Phi_{2})$ , then $\Phi_{\sigma(1)}=\Phi_{3},\Phi_{\sigma(2)}=\Phi_{1},\Phi_{\sigma(3)}=\Phi_{4},% \Phi_{\sigma(4)}=\Phi_{2}$ . Now, using Eq. (16), we have

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},{\Phi}_{3},{\Phi}_{% 4})$ $\displaystyle=\left\langle v_{8\times(\sqrt[3]{(1-\log_{2}(1+{(2^{1-({{5}/{8}}% )^{3}}-1)^{0.1550}}\times{(2^{1-({{6}/{8}})^{3}}-1)^{0.3450}}\times{(2^{1-({{6% }/{8}})^{3}}-1)^{0.3450}}\times{(2^{1-({{4}/{8}})^{3}}-1)^{0.1550}}))})},\right.$ $\displaystyle\quad\left.v_{8\times(\sqrt[3]{(\log_{2}(1+{(2^{({{2}/{8}})^{3}}-% 1)^{0.1550}}\times{(2^{({{5}/{8}})^{3}}-1)^{0.3450}}\times{(2^{({{6}/{8}})^{3}% }-1)^{0.3450}}\times{(2^{({{4}/{8}})^{3}}-1)^{0.1550}}))})}\right\rangle$ $\displaystyle=\langle v_{5.6671},v_{4.5021}\rangle.$

It is straightforward to prove that the L^qROPFFOWA operator possesses the following properties.

If ${\Phi}_{i}=\Phi=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle\forall i$ , then

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle.$

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})% \leqslant L^{q}\textit{ROPFFOWA}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},% \ldots,{\Phi}_{n}^{\prime}).$

$\displaystyle{\Phi}^{\blacktriangledown}\leqslant L^{q}\textit{ROPFFOWA}({\Phi% }_{1},{\Phi}_{2},\ldots,{\Phi}_{n})\leqslant{\Phi}^{\blacktriangle}.$

If ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be a collection of L^qROPFNs, then

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% L^{q}\textit{ROPFFOWA}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{% n}^{\prime}),$

where $({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{n}^{\prime})$ is any permutation of $({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ .

5.2 Frank weighted geometric AOs for L^qROPFNs

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs. A mapping $L^{q}\textit{ROPFFWG}:\Omega^{n}\rightarrow\Omega$ is termed as a linguistic $q$ -rung orthopair fuzzy frank weighted geometric operator, and defined as

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=(% {{{\Phi}_{1}}^{\overset{q}{\wedge}_{F}}w_{1}}){\overset{q}{\otimes}_{F}}({{{% \Phi}_{2}}^{\overset{q}{\wedge}_{F}}w_{2}}){\overset{q}{\otimes}_{F}}\ldots{% \overset{q}{\otimes}_{F}}({{{\Phi}_{n}}^{\overset{q}{\wedge}_{F}}w_{n}}),$ (17)

where $w=(w_{1},w_{2},\ldots,w_{n})^{T}$ is the weighting vector of ${\Phi}_{i}$ with ${w}_{i}\in[0,1]$ and $\sum\limits_{i=1}^{n}{w}_{i}=1$ .

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs, then the aggregated value by using the L^qROPFFWG operator is also a L^qROPFN and is given by

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \left\langle v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{% \mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},\right.\left.v_{r(\sqrt[q]{(1-\log% _{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_% {i}}}))})}\right\rangle.$ (18)

Proof..

The initial result is a direct consequence of Theorem 3 and 4, whereas to establish the result described in Eq. (18), we employ the principle of mathematical induction applied to $n$ .

(i) When $n=2$ , according to the Frank operational laws of L^qROPFNs, we obtain

$\displaystyle{{{\Phi}_{1}}^{\overset{q}{\wedge}_{F}}w_{1}}=\left\langle v_{r% \left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{M}}{1}}/% {r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-1}}\right)\right)}\right)},v_{{r\left% (\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{N}}{1}}/{% r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-1}}\right)\right)}\right)}}\right\rangle,$ $\displaystyle{{{\Phi}_{2}}^{\overset{q}{\wedge}_{F}}w_{2}}=\left\langle v_{r% \left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{M}}{2}}/% {r})^{q}}-1)^{w_{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)},v_{{r\left% (\sqrt[q]{\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{N}}{2}}/{% r})^{q}}-1)^{w_{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)}}\right\rangle.$

Then

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2})=({{{\Phi}_{1}}^{% \overset{q}{\wedge}_{F}}w_{1}}){\overset{q}{\otimes}_{F}}({{{\Phi}_{2}}^{% \overset{q}{\wedge}_{F}}w_{2}})$ $\displaystyle=\left\langle v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{(\hslash^{({{\mathrm{M}}{1}}/{r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-1}% }\right)\right)}\right)},v_{{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \frac{(\hslash^{1-({{\mathrm{N}}{1}}/{r})^{q}}-1)^{w_{1}}}{(\hslash-1)^{w_{1}-% 1}}\right)\right)}\right)}}\right\rangle$ $\displaystyle\quad{\overset{q}{\otimes}_{F}}\left\langle v_{r\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{M}}{2}}/{r})^{q}}-1)^{w% _{2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)},v_{{r\left(\sqrt[q]{\left% (1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{N}}{2}}/{r})^{q}}-1)^{w_% {2}}}{(\hslash-1)^{w_{2}-1}}\right)\right)}\right)}}\right\rangle$ $\displaystyle=\left\langle v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{\prod_{i=1}^{2}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(% \hslash-1)^{\sum_{i=1}^{2}w_{i}-1}}\right)\right)}\right)},v_{r\left(\sqrt[q]{% \left(1-\log_{\hslash}\left(1+\frac{\prod_{i=1}^{2}(\hslash^{1-({{{\mathrm{N}}% {i}}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{2}w_{i}-1}}\right)\right)% }\right)}\right\rangle$ $\displaystyle=\left\langle v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{2}{(% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},v_{r(\sqrt[q]{(1-\log% _{\hslash}(1+\prod_{i=1}^{2}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_% {i}}}))})}\right\rangle$ $\displaystyle\quad\left(\text{since }\sum_{i=1}^{2}w_{i}=1\right).$

That is, Eq. (18) is true for $n=2$ .

(ii) Suppose that Eq. (18) holds for $n=k$ , i.e.,

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{k})=% \left\langle v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\frac{\prod_{i=1}^% {k}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i% =1}^{k}w_{i}-1}}\right)\right)}\right)}\right.,\left.v_{r\left(\sqrt[q]{\left(% 1-\log_{\hslash}\left(1+\frac{\prod_{i=1}^{k}(\hslash^{1-({{{\mathrm{N}}{i}}/{% r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{k}w_{i}-1}}\right)\right)}% \right)}\right\rangle.$

When $n=k+1$ , based on the Frank operational laws, we have

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{k},{% \Phi}_{k+1})=L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{k}){% \overset{q}{\otimes}_{F}}({{{\Phi}_{k+1}}^{\overset{q}{\wedge}_{F}}w_{k+1}})$ $\displaystyle=\left\langle v_{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{\prod_{i=1}^{k}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(% \hslash-1)^{\sum_{i=1}^{k}w_{i}-1}}\right)\right)}\right)},v_{r\left(\sqrt[q]{% \left(1-\log_{\hslash}\left(1+\frac{\prod_{i=1}^{k}(\hslash^{1-({{{\mathrm{N}}% {i}}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{k}w_{i}-1}}\right)\right)% }\right)}\right\rangle$ $\displaystyle\quad{{\overset{q}{\otimes}_{F}}\left\langle v_{t\left(\sqrt[q]{% \left(\log_{\hslash}\left(1+\frac{(\hslash^{({{\mathrm{M}}{k+1}}/{r})^{q}}-1)^% {w_{k+1}}}{(\hslash-1)^{w_{k+1}-1}}\right)\right)}\right)},v_{{t\left(\sqrt[q]% {\left(1-\log_{\hslash}\left(1+\frac{(\hslash^{1-({{\mathrm{M}}{k+1}}/{r})^{q}% }-1)^{w_{k+1}}}{(\hslash-1)^{w_{k+1}-1}}\right)\right)}\right)}}\right\rangle.}$ $\displaystyle=\left\langle v_{t\left(\sqrt[q]{\left(\log_{\hslash}\left(1+% \frac{\prod_{i=1}^{k+1}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}{% (\hslash-1)^{\sum_{i=1}^{k+1}w_{i}-1}}\right)\right)}\right)},v_{t\left(\sqrt[% q]{\left(1-\log_{\hslash}\left(1+\frac{\prod_{i=1}^{k+1}(\hslash^{1-({{{% \mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}{(\hslash-1)^{\sum_{i=1}^{k+1}w_{i}-1}}% \right)\right)}\right)}\right\rangle.$ $\displaystyle=\left\langle v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{k+1}{(% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},v_{r(\sqrt[q]{(1-\log% _{\hslash}(1+\prod_{i=1}^{k+1}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{% w_{i}}}))})}\right\rangle$ $\displaystyle\quad\left(\text{since }\sum_{i=1}^{k+1}w_{i}=1\right).$

Therefore, the result given in Eq. (18) holds for all $n\in Z^{+}$ . ∎

.

Let ${\Phi}_{1}=\langle v_{3},v_{6}\rangle,{\Phi}_{2}=\langle v_{7},v_{2}\rangle,{% \Phi}_{3}=\langle v_{6},v_{1}\rangle$ , and ${\Phi}_{4}=\langle v_{5},v_{7}\rangle$ be four L^qROPFNs defined on $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . Further consider that $w=(0.20,0.15,0.35,0.35)^{T}$ denotes the weight vector corresponding to L^qROPFNs ${\Phi}_{i}(i=1,2,\ldots,4)$ , $\hslash=2$ and $q=3$ . Then using Eq. (18), we have

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},{\Phi}_{3},{\Phi}_{4})$ $\displaystyle=\langle v_{8\times(\sqrt[3]{(\log_{2}(1+{(2^{({{3}/{8}})^{3}}-1)% ^{0.20}}\times{(2^{({{7}/{8}})^{3}}-1)^{0.30}}\times{(2^{({{6}/{8}})^{3}}-1)^{% 0.35}}\times{(2^{({{5}/{8}})^{3}}-1)^{0.15}}))})},$ $\displaystyle\quad v_{8\times(\sqrt[3]{(1-\log_{2}(1+{(2^{1-({{6}/{8}})^{3}}-1% )^{0.20}}\times{(2^{1-({{2}/{8}})^{3}}-1)^{0.30}}\times{(2^{1-({{1}/{8}})^{3}}% -1)^{0.35}}\times{(2^{1-({{7}/{8}})^{3}}-1)^{0.15}}))})}\rangle$ $\displaystyle=\langle v_{5.0585},v_{5.7923}\rangle.$

The following properties of the L^qROPFFWG operator can be easily demonstrated:

If ${\Phi}_{i}=\Phi=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle\forall i$ , then

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle.$

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})% \leqslant L^{q}\textit{ROPFFWG}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots% ,{\Phi}_{n}^{\prime}).$

$\displaystyle{\Phi}^{\blacktriangledown}\leqslant L^{q}\textit{ROPFFWG}({\Phi}% _{1},{\Phi}_{2},\ldots,{\Phi}_{n})\leqslant{\Phi}^{\blacktriangle}.$

If ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be a collection of L^qROPFNs, then

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=L% ^{q}\textit{ROPFFWG}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{n}% ^{\prime}).$

where $({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{n}^{\prime})$ is any permutation of $({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ .

Let $\Phi=\langle v_{\mathrm{M}},v_{\mathrm{N}}\rangle$ be another L^qROPFN, then

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1}{\overset{q}{\otimes}_{F}}{\Phi}% ,{\Phi}_{2}{\overset{q}{\otimes}_{F}}{\Phi},\ldots,{\Phi}_{n}{\overset{q}{% \otimes}_{F}}{\Phi})=(L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi% }_{n})){\overset{q}{\otimes}_{F}}{\Phi}.$

If $\aleph>0$ is a real number, then

$\displaystyle L^{q}\textit{ROPFFWG}({{\Phi}_{1}}^{\overset{q}{\wedge}_{F}}% \aleph,{{\Phi}_{2}}^{\overset{q}{\wedge}_{F}}\aleph,\ldots,{{\Phi}_{n}}^{% \overset{q}{\wedge}_{F}}\aleph)=(L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},% \ldots,{\Phi}_{n}))^{\overset{q}{\wedge}_{F}}\aleph.$

Let ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be another collection of $n$ L^qROPFNs, then

$\displaystyle L^{q}\textit{ROPFFWG}({\Phi}_{1}{\overset{q}{\otimes}_{F}}{\Phi}% _{1}^{\prime},{\Phi}_{2}{\overset{q}{\otimes}_{F}}{\Phi}_{2}^{\prime},\ldots,{% \Phi}_{n}{\overset{q}{\otimes}_{F}}{\Phi}_{n}^{\prime})$ $\displaystyle=L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n}){% \overset{q}{\otimes}_{F}}L^{q}\textit{ROPFFWG}({\Phi}_{1}^{\prime},{\Phi}_{2}^% {\prime},\ldots,{\Phi}_{n}^{\prime}).$

Let ${\Phi}_{i}^{\prime}=\langle v_{{\mathrm{M}}{i}^{\prime}},v_{{\mathrm{N}}{i}^{% \prime}}\rangle(i=1(1)n)$ be another collection of $n$ L^qROPFNs and $\aleph_{1},\aleph_{2}>0$ be two real numbers, then

$\displaystyle L^{q}\textit{ROPFFWG}(({{\Phi}_{1}}^{\overset{q}{\wedge}_{F}}% \aleph_{1}){\overset{q}{\otimes}_{F}}({{\Phi}_{1}^{\prime}}^{\overset{q}{% \wedge}_{F}}\aleph_{2}),({{\Phi}_{2}}^{\overset{q}{\wedge}_{F}}\aleph_{1}){% \overset{q}{\otimes}_{F}}({{\Phi}_{2}^{\prime}}^{\overset{q}{\wedge}_{F}}% \aleph_{2}),\ldots,$ $\displaystyle({{\Phi}_{n}}^{\overset{q}{\wedge}_{F}}\aleph_{1}){\overset{q}{% \otimes}_{F}}({{\Phi}_{n}^{\prime}}^{\overset{q}{\wedge}_{F}}\aleph_{2}))$ $\displaystyle=((L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})% )^{\overset{q}{\wedge}_{F}}\aleph_{1}){\overset{q}{\otimes}_{F}}((L^{q}\textit% {ROPFFWG}({\Phi}_{1}^{\prime},{\Phi}_{2}^{\prime},\ldots,{\Phi}_{n}^{\prime}))% ^{\overset{q}{\wedge}_{F}}\aleph_{2}).$

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs. A mapping $L^{q}\textit{ROPFFOWG}:\Omega^{n}\rightarrow\Omega$ is termed as a linguistic $q$ -rung orthopair fuzzy frank ordered weighted geometric operator, and defined as

$\displaystyle L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% ({{{\Phi}_{\sigma(1)}}^{\overset{q}{\wedge}_{F}}\omega_{1}}){\overset{q}{% \otimes}_{F}}({{{\Phi}_{\sigma(2)}}^{\overset{q}{\wedge}_{F}}\omega_{2}}){% \overset{q}{\otimes}_{F}}\ldots{\overset{q}{\otimes}_{F}}({{{\Phi}_{\sigma(n)}% }^{\overset{q}{\wedge}_{F}}\omega_{n}}),$ (19)

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs. Then, the aggregated value by using the L^qROPFFOWG operator is also a L^qROPFN and is given by

$\displaystyle L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% \left\langle v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{% \mathrm{M}}{\sigma(i)}}/{r}})^{q}}-1)^{\omega_{i}}}))})},\right.\left.v_{r(% \sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{% \sigma(i)}}/{r}})^{q}}-1)^{\omega_{i}}}))})}\right\rangle.$ (20)

Proof..

The proof can be formulated following a similar procedure as in Theorem 10. ∎

The L^qROPFFOWG operator shares the same properties as the L^qROPFFOWA operator.

.

Let ${\Phi}_{1}=\langle v_{6},v_{4}\rangle,{\Phi}_{2}=\langle v_{5},v_{5}\rangle,{% \Phi}_{3}=\langle v_{2},v_{7}\rangle$ , and ${\Phi}_{4}=\langle v_{4},v_{6}\rangle$ be four L^qROPFNs defined on $\widehat{\mathbb{V}}_{[0,8]}=\{v_{g}|v_{0}\leqslant v_{g}\leqslant v_{8},g\in[% 0,8]\}$ . Further assume that $\omega=(0.1550,0.3450,\linebreak 0.3450,0.1550)^{T}$ denotes the weight vector corresponding to L^qROPFNs ${\Phi}_{\sigma(i)}(i=1(1)4)$ (calculated based on normal distribution method [80]), $\hslash=2$ and $q=2$ . Then using Eq. (6), we have

$\displaystyle{\mathcal{S}_{\textit{imp}}}(\Phi_{1})=3.0987,{\mathcal{S}_{% \textit{imp}}}(\Phi_{2})=1.9912,{\mathcal{S}_{\textit{imp}}}(\Phi_{3})=0.4852,% {\mathcal{S}_{\textit{imp}}}(\Phi_{4})=1.2273.$

Since ${\mathcal{S}_{\textit{imp}}}(\Phi_{1})>{\mathcal{S}_{\textit{imp}}}(\Phi_{2})>% {\mathcal{S}_{\textit{imp}}}(\Phi_{4})>{\mathcal{S}_{\textit{imp}}}(\Phi_{3})$ , then $\Phi_{\sigma(1)}=\Phi_{1},\Phi_{\sigma(2)}=\Phi_{2},\Phi_{\sigma(3)}=\Phi_{4},% \linebreak\Phi_{\sigma(4)}=\Phi_{3}$ . Now, using Eq. (20), we have

$\displaystyle L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},{\Phi}_{3},{\Phi}_{% 4})$ $\displaystyle=\langle v_{8\times(\sqrt{(\log_{2}(1+{(2^{({{6}/{8}})^{2}}-1)^{0% .1550}}\times{(2^{({{5}/{8}})^{2}}-1)^{0.3450}}\times{(2^{({{4}/{8}})^{2}}-1)^% {0.3450}}\times{(2^{({{2}/{8}})^{2}}-1)^{0.1550}}))})},$ (21) $\displaystyle\quad v_{8\times(\sqrt{(1-\log_{2}(1+{(2^{1-({{4}/{8}})^{2}}-1)^{% 0.1550}}\times{(2^{1-({{5}/{8}})^{2}}-1)^{0.3450}}\times{(2^{1-({{6}/{8}})^{2}% }-1)^{0.3450}}\times{(2^{1-({{7}/{8}})^{2}}-1)^{0.1550}}))})}\rangle$ $\displaystyle=\langle v_{4.1672},v_{5.7201}\rangle.$

.

If $\omega=(1,0,\ldots,0)^{T}$ , then

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=\max\limits_{i% }({\Phi}_{i}).$ (22)

.

If $\omega=(0,0,\ldots,1)^{T}$ , then

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=\min\limits_{i% }({\Phi}_{i}).$ (23)

.

If $\omega_{i}=1$ and $\omega_{j}=0$ for all $j\neq i$ , then

$\displaystyle L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})=% L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})={\Phi}_{\sigma% (i)}.$ (24)

where ${\Phi}_{\sigma(i)}$ is the $i^{\text{th}}$ largest of ${\Phi}_{i}(i=1,2,\ldots,n)$ .

.

[81, 82] Let $z_{i}\geqslant 0$ , $\nu_{i}>0$ $(i=1(1)n)$ and $\sum_{i=1}^{n}\nu_{i}=1$ , then

$\displaystyle\prod_{i=1}^{n}(z_{i})^{\nu_{i}}\leqslant\sum_{i=1}^{n}\nu_{i}z_{% i},$ (25)

where equality holds if only if $z_{1}=z_{2}=\ldots=z_{n}$ .

.

Let ${\Phi}_{i}=\langle v_{{\mathrm{M}}{i}},v_{{\mathrm{N}}{i}}\rangle,(i=1(1)n)$ be a set of L^qROPFNs, the following inequalities are true

(a)

$L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})\geqslant L^{q}% \textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ ,

(b)

$L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})\geqslant L^{q}% \textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ ,

with equality holds if only if ${\Phi}_{1}={\Phi}_{2}=\ldots={\Phi}_{n}$ .

Proof..

Here, we give the proof only for (a) and part (b) can be proved analogously.

Let us assume

$\displaystyle{{v}_{\mathrm{M}{{}^{\prime}}}}=v_{r(\sqrt[q]{(1-\log_{\hslash}(1% +\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})};% \;\;{{v}_{\mathrm{N}{{}^{\prime}}}}=v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1% }^{n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})};$ $\displaystyle{{v}_{\mathrm{M}{{}^{\prime\prime}}}}=v_{r(\sqrt[q]{(\log_{% \hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}% }))})};\;\;{{v}_{\mathrm{N}{{}^{\prime\prime}}}}=v_{r(\sqrt[q]{(1-\log_{% \hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i% }}}))})}.$

In order to prove (a), we first prove

$\displaystyle{v_{\mathrm{M}{{}^{\prime}}}}\geqslant{v_{\mathrm{M}{{}^{\prime% \prime}}}}\;\;\text{and}\;\;{v_{\mathrm{N}{{}^{\prime}}}}\leqslant{v_{\mathrm{% N}{{}^{\prime\prime}}}}.$ (26)

which implies

$\displaystyle{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\prod_{i=1}^{n}{(% \hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\right)}\right)}$ $\displaystyle\geqslant{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\prod_{i=1}% ^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\right)}\right)}$ $\displaystyle\text{and}\;\;{r\left(\sqrt[q]{\left(\log_{\hslash}\left(1+\prod_% {i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\right)}% \right)}$ $\displaystyle\leqslant{r\left(\sqrt[q]{\left(1-\log_{\hslash}\left(1+\prod_{i=% 1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\right)}% \right)}.$

$\displaystyle 1-\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{% M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\geqslant\log_{\hslash}\left(1+\prod_{i=1% }^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)$ (27) $\displaystyle\text{and}\;\;\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash^{({{% {\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\leqslant 1-\log_{\hslash}\left(% 1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right).$ (28)

As $\hslash>1\rightarrow\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}>1$ , so using Lemma 12, we obtain

$\displaystyle\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_% {i}}}\leqslant\sum_{i=1}^{n}{{w_{i}}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}% -1)}.$ (29)

Then

$\displaystyle 1-\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{% M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)-\log_{\hslash}\left(1+\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)$ $\displaystyle={{\log_{\hslash}\left(\frac{\hslash}{(1+\prod_{i=1}^{n}{(\hslash% ^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}){{(1+\prod_{i=1}^{n}{(\hslash^{% ({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}}}\right)}}$ (30) $\displaystyle\geqslant{{\log_{\hslash}\left(\frac{\hslash}{(1+\sum_{i=1}^{n}{{% w_{i}}(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)}){{(1+\sum_{i=1}^{n}{{w_{i% }}(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)})}}}\right)}}$ $\displaystyle={{\log_{\hslash}\left(\frac{\hslash}{\left(\sum_{i=1}^{n}{{w_{i}% }\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}}\right)\left(\sum_{i=1}^{n}{{w_{i}}% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}}\right)}\right)}}\geqslant 0,$

with equality if and only if ${{\mathrm{M}}{1}}={{\mathrm{M}}{2}}=\ldots={{\mathrm{M}}{n}}$ .

Similarly, we can obtain

$\displaystyle\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i% }}/{r}})^{q}}-1)^{w_{i}}}\right)\leqslant 1-\log_{\hslash}\left(1+\prod_{i=1}^% {n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right).$ (31)

Thus, when $\hslash>1$ , Eqs (5.2) and (31) give

$\displaystyle L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})% \geqslant L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n}),$

with equality if and only ${\Phi}_{1}={\Phi}_{2}=\ldots={\Phi}_{n}$ . ∎

The subsequent analysis will focus on investigating special cases of the proposed AOs.

If ${\hslash}\rightarrow 1$ , then L^qROPFFWA and L^qROPFFOWA operators reduce to the L^qROPF averaging AOs based on the Algebraic $t$ -conorm and $t$ -norm:

$\displaystyle(a)\;L^{q}\textit{ROPFFWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ $\displaystyle=\left\langle r\left({{\sqrt[q]{\left(1-\prod_{i=1}^{n}{({1-{({{{% \mathrm{M}}{i}}/{r}})^{q}}})^{w_{i}}}\right)}}}\right).r\left(\prod_{i=1}^{n}{% {({{{\mathrm{N}}{i}}/{r}})}^{w_{i}}}\right)\right\rangle,$ $\displaystyle(b)\;L^{q}\textit{ROPFFOWA}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{% n})$ $\displaystyle=\left\langle r\left({{\sqrt[q]{\left(1-\prod_{i=1}^{n}{({1-{({{{% \mathrm{M}}{\sigma(i)}}/{r}})^{q}}})^{w_{i}}}\right)}}}\right).r\left(\prod_{i% =1}^{n}{{({{{\mathrm{N}}{\sigma(i)}}/{r}})}^{w_{i}}}\right)\right\rangle.$

Proof. We give the proof of part (a) only, part (b) can be prove similarly.

(a) Because $\hslash\rightarrow 1$ , $\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\rightarrow 0$ . Therefore, we get

$\displaystyle r\sqrt[q]{\left(1-\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash% ^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\right)}$ $\displaystyle=r\sqrt[q]{\left(1-\frac{\ln(1+\prod_{i=1}^{n}{(\hslash^{1-({{{% \mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}$ $\displaystyle=r\sqrt[q]{\left(1-\frac{(\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm% {M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}.$

According to Taylor’s expansion and $\hslash>0$ , we can get

$\displaystyle\hslash^{1-({\mathrm{M}{i}}/{r})^{q}}=1+(1-({\mathrm{M}{i}}/{r})^% {q})\ln{\hslash}+\frac{(1-({\mathrm{M}{i}}/{r})^{q})}{2}(\ln{\hslash})^{2}+\ldots.$

Then we have

$\displaystyle\hslash^{1-({\mathrm{M}{i}}/{r})^{q}}-1=(1-({\mathrm{M}{i}}/{r})^% {q})\ln{\hslash}+O(\ln(\hslash)).$ $\displaystyle r\sqrt[q]{\left(1-\frac{(\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm% {M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}=r\sqrt[q]{\left(1-\frac{% (\prod_{i=1}^{n}{((1-({\mathrm{M}{i}}/{r})^{q})\ln\hslash)^{w_{i}}})}{\ln% \hslash}\right)}$ $\displaystyle=r\sqrt[q]{\left(1-{\left(\prod_{i=1}^{n}{(1-({\mathrm{M}{i}}/{r}% )^{q})^{w_{i}}}\right)}\right)}.$

Similarly, we have $\hslash\rightarrow 1$ , $\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\rightarrow 0$ . Therefore, we get

$\displaystyle r\sqrt[q]{\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash^{({{{% \mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)}=r\sqrt[q]{\left(\frac{\ln(1+% \prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln% \hslash}\right)}$ $\displaystyle=r\sqrt[q]{\left(\frac{(\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}% {i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}.$

Using Taylor’s expansion and $\hslash>0$ , we can get

$\displaystyle\hslash^{({\mathrm{N}{i}}/{r})^{q}}=1+({\mathrm{N}{i}}/{r})^{q}% \ln{\hslash}+\frac{({\mathrm{N}{i}}/{r})^{q}}{2}(\ln{\hslash})^{2}+\ldots.$

Then, we obtain

$\displaystyle\hslash^{({\mathrm{N}{i}}/{r})^{q}}-1=({\mathrm{N}{i}}/{r})^{q}% \ln{\hslash}+O(\ln(\hslash)).$ $\displaystyle r\sqrt[q]{\left(\frac{(\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}% {i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}=r\sqrt[q]{\left(\frac{(\prod% _{i=1}^{n}{(({\mathrm{N}{i}}/{r})^{q}\ln\hslash)^{w_{i}}})}{\ln\hslash}\right)% }=r{{\left(\prod_{i=1}^{n}{(({\mathrm{N}{i}}/{r})^{q})^{w_{i}}}\right)}}.$

Therefore, we obtain

$\displaystyle\lim_{\hslash\to 1}L^{q}ROPFFWA({\Phi}_{1},{\Phi}_{2},\ldots,{% \Phi}_{n})$ $\displaystyle=\lim_{\hslash\to 1}\left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(% 1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},% v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/% {r}})^{q}}-1)^{w_{i}}}))})}\right\rangle$ $\displaystyle=\left\langle r\left({{\sqrt[q]{\left(1-\prod_{i=1}^{n}{({1-{({{{% \mathrm{M}}{i}}/{r}})^{q}}})^{w_{i}}}\right)}}}\right),r\left(\prod_{i=1}^{n}{% {({{{\mathrm{N}}{i}}/{r}})}^{w_{i}}}\right)\right\rangle.$

When ${\hslash}\rightarrow+\infty$ , then L^qROPFFWA and L^qROPFFOWA operators become the traditional weighted average operator and ordered weighted average operator, respectively.

$\displaystyle(c)\;L^{q}ROPFFWA({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ $\displaystyle=\left\langle{r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{\mathrm{M}% }{i}}/{r}})^{q}}\right)},r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{\mathrm{N}}{% i}}/{r}})^{q}}\right)\right\rangle,$ $\displaystyle(d)\;L^{q}ROPFFOWA({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ $\displaystyle=\left\langle{r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{\mathrm{M}% }{\sigma(i)}}/{r}})^{q}}\right)},r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{% \mathrm{N}}{\sigma(i)}}/{r}})^{q}}\right)\right\rangle.$

Proof. We give the proof of part (c) only, part (d) can be prove similarly.

$\displaystyle\lim_{\hslash\to+\infty}r\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)% \right)}$ $\displaystyle=r\sqrt[q]{\left(1-\lim_{\hslash\to+\infty}\frac{\ln(1+\prod_{i=1% }^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}% \right)}.$

Applying the L’Hospital’s rule, we have

$\displaystyle r\sqrt[q]{\left(1-\lim_{\hslash\to+\infty}\frac{\frac{\prod_{i=1% }^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n% }{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\left(\sum_{i=1}^{n}w% _{i}{(1-({{{\mathrm{M}}{i}}/{r}})^{q})\frac{\hslash^{-({{{\mathrm{M}}{i}}/{r}}% )^{q}}}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)}}\right)}{{1}/{\hslash}}% \right)}$ $\displaystyle=r\sqrt[q]{\!\left(\!1-\lim_{\hslash\to+\infty}{\frac{\prod_{i=1}% ^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n}% {(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\!\left(\!\sum_{i=1}^{% n}w_{i}{(1-({{{\mathrm{M}}{i}}/{r}})^{q})\frac{\hslash^{1-({{{\mathrm{M}}{i}}/% {r}})^{q}}}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)}}\!\right)}\!\right)}$ $\displaystyle=r\sqrt[q]{\left(1-\sum_{i=1}^{n}w_{i}{(1-({{{\mathrm{M}}{i}}/{r}% })^{q})}\right)}{=r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{M}}{i}}/{r}})^{q}}% }},$

and

$\displaystyle\lim_{\hslash\to+\infty}r\sqrt[q]{\left(\log_{\hslash}\left(1+% \prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)% \right)}$ $\displaystyle=r\sqrt[q]{\left(\lim_{\hslash\to\infty}\frac{\ln(1+\prod_{i=1}^{% n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}.$

Applying the L’Hospital’s rule, we have

$\displaystyle r\sqrt[q]{\left(\lim_{\hslash\to+\infty}\frac{\frac{\prod_{i=1}^% {n}{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\left(\sum_{i=1}^{n}w_{i}{% ({{{\mathrm{N}}{i}}/{r}})^{q}\frac{\hslash^{{({{{\mathrm{N}}{i}}/{r}})^{q}}-1}% }{(\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)}}\right)}{{1}/{\hslash}}\right)}$ $\displaystyle=r\sqrt[q]{\lim_{\hslash\to+\infty}{\frac{\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\left(\sum_{i=1}^{n}w_{i}{% ({{{\mathrm{N}}{i}}/{r}})^{q}\frac{\hslash^{({{{\mathrm{N}}{i}}/{r}})^{q}}}{(% \hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)}}\right)}}$ $\displaystyle=r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{N}}{i}}/{r}})^{q}}}.$

Therefore, we obtain

$\displaystyle\lim_{\hslash\to+\infty}L^{q}ROPFFWA({\Phi}_{1},{\Phi}_{2},\ldots% ,{\Phi}_{n})$ $\displaystyle=\lim_{\hslash\to 1}\left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(% 1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},% v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{N}}{i}}/% {r}})^{q}}-1)^{w_{i}}}))})}\right\rangle$ $\displaystyle=\left\langle r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{M}}{i}}/{% r}})^{q}}},r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{N}}{i}}/{r}})^{q}}}\right\rangle.$

If ${\hslash}\rightarrow 1$ , then L^qROPFFWG and L^qROPFFOWG operators reduce to the L^qROPF geometric AOs based on the Algebraic $t$ -conorm and $t$ -norm:

$\displaystyle(e)\;L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ $\displaystyle=\left\langle{{r\left(\prod_{i=1}^{n}{{({{{\mathrm{M}}{i}}/{r}})}% ^{w_{i}}}\right)}},r\left({{\sqrt[q]{\left(1-\prod_{i=1}^{n}{({1-{({{{\mathrm{% N}}{i}}/{r}})^{q}}})^{w_{i}}}\right)}}}\right)\right\rangle,$ $\displaystyle(f)\;L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{% n})$ $\displaystyle=\left\langle{{r\left(\prod_{i=1}^{n}{{({{{\mathrm{M}}{\sigma(i)}% }/{r}})}^{w_{i}}}\right)}},r\left({{\sqrt[q]{\left(1-\prod_{i=1}^{n}{({1-{({{{% \mathrm{N}}{\sigma(i)}}/{r}})^{q}}})^{w_{i}}}\right)}}}\right)\right\rangle.$

Proof. We give the proof of part (e) only, part (f) can be prove similarly.

(e) Since $\hslash\rightarrow 1$ , $\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\rightarrow 0$ . Therefore, we get

$\displaystyle r\sqrt[q]{\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash^{({{{% \mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)}=r\sqrt[q]{\left(\frac{\ln(1+% \prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln% \hslash}\right)}$ $\displaystyle\rightarrow r\sqrt[q]{\left(\frac{(\prod_{i=1}^{n}{(\hslash^{({{{% \mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}.$

Using Taylor’s expansion and $\hslash>0$ , we can get

$\displaystyle\hslash^{({\mathrm{M}{i}}/{r})^{q}}=1+({\mathrm{M}{i}}/{r})^{q}% \ln{\hslash}+\frac{({\mathrm{M}{i}}/{r})^{q}}{2}(\ln{\hslash})^{2}+\ldots.$

Then, we get

$\displaystyle\hslash^{({\mathrm{M}{i}}/{r})^{q}}-1=({\mathrm{M}{i}}/{r})^{q}% \ln{\hslash}+O(\ln(\hslash)).$ $\displaystyle r\sqrt[q]{\left(\frac{(\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}% {i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}=r\sqrt[q]{\left(\frac{(\prod% _{i=1}^{n}{(({\mathcal{M}{i}}/{r})^{q}\ln\hslash)^{w_{i}}})}{\ln\hslash}\right)}$ $\displaystyle=r{{\left(\prod_{i=1}^{n}{(({\mathcal{M}{i}}/{r})^{q})^{w_{i}}}% \right)}}.$

Because $\hslash\rightarrow 1$ , $\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\rightarrow 0$ . Therefore, we obtain

$\displaystyle r\sqrt[q]{\left(1-\log_{\hslash}\left(1+\prod_{i=1}^{n}{(\hslash% ^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)\right)}$ $\displaystyle=r\sqrt[q]{\left(1-\frac{\ln(1+\prod_{i=1}^{n}{(\hslash^{1-({{{% \mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}=r\sqrt[q]{\left(1% -\frac{(\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}% )}{\ln\hslash}\right)}.$

According to Taylor’s expansion and $\hslash>0$ , we can get

$\displaystyle\hslash^{1-({\mathrm{N}{i}}/{r})^{q}}=1+(1-({\mathscr{N}{i}}/{r})% ^{q})\ln{\hslash}+\frac{(1-({\mathbscr{N}{i}}/{r})^{q})}{2}(\ln{\hslash})^{2}+\ldots.$

Then we have

$\displaystyle\hslash^{1-({\mathrm{N}{i}}/{r})^{q}}-1=(1-({\mathrm{N}{i}}/{r})^% {q})\ln{\hslash}+O(\ln(\hslash)).$ $\displaystyle r\sqrt[q]{\left(1-\frac{(\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm% {N}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}=r\sqrt[q]{\left(1-\frac{% (\prod_{i=1}^{n}{((1-({\mathrm{N}{i}}/{r})^{q})\ln\hslash)^{w_{i}}})}{\ln% \hslash}\right)}$ $\displaystyle=r\sqrt[q]{\left(1-{\left(\prod_{i=1}^{n}{(1-({\mathrm{N}{i}}/{r}% )^{q})^{w_{i}}}\right)}\right)}.$

Therefore, we obtain

$\displaystyle\lim_{\hslash\to 1}L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},% \ldots,{\Phi}_{n})$ $\displaystyle=\lim_{\hslash\to 1}\left\langle v_{r(\sqrt[q]{(\log_{\hslash}(1+% \prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})},v_{r% (\sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/% {r}})^{q}}-1)^{w_{i}}}))})}\right\rangle$ $\displaystyle=\left\langle{{r\left(\prod_{i=1}^{n}{{({{{\mathrm{M}}{i}}/{r}})}% ^{w_{i}}}\right)}},r\left({{\sqrt[q]{\left(1-\prod_{i=1}^{n}{({1-{({{{\mathrm{% N}}{i}}/{r}})^{q}}})^{w_{i}}}\right)}}}\right)\right\rangle.$

When ${\hslash}\rightarrow+\infty$ , then L^qROPFFWG and L^qROPFFOWG operators become the traditional weighted average operator and ordered weighted average operator, respectively.

$\displaystyle(g)\;L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{n})$ $\displaystyle=\left\langle{r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{\mathrm{M}% }{i}}/{r}})^{q}}\right)},{r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{\mathrm{N}}% {i}}/{r}})^{q}}\right)}\right\rangle,$ $\displaystyle(h)\;L^{q}\textit{ROPFFOWG}({\Phi}_{1},{\Phi}_{2},\ldots,{\Phi}_{% n})$ $\displaystyle=\left\langle{r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{\mathrm{M}% }{\sigma(i)}}/{r}})^{q}}\right)},{r\left(\sqrt[q]{\sum_{i=1}^{n}{w_{i}}({{{% \mathrm{N}}{\sigma(i)}}/{r}})^{q}}\right)}\right\rangle.$

Proof. We give the proof of part (g) only, part (h) can be prove similarly.

(g) Let us consider

$\displaystyle\lim_{\hslash\to+\infty}r\sqrt[q]{\left(\log_{\hslash}\left(1+% \prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)% \right)}$ $\displaystyle=r\sqrt[q]{\left(\lim_{\hslash\to+\infty}\frac{\ln(1+\prod_{i=1}^% {n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}\right)}.$

Applying the L’Hospital’s rule, we have

$\displaystyle r\sqrt[q]{\left(\lim_{\hslash\to+\infty}\frac{\frac{\prod_{i=1}^% {n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\left(\sum_{i=1}^{n}w_{i}{% ({{{\mathrm{M}}{i}}/{r}})^{q}\frac{\hslash^{{({{{\mathrm{M}}{i}}/{r}})^{q}}-1}% }{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)}}\right)}{{1}/{\hslash}}\right)}$ $\displaystyle=r\sqrt[q]{\lim_{\hslash\to+\infty}{\frac{\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n}{(% \hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\left(\sum_{i=1}^{n}w_{i}{% ({{{\mathrm{M}}{i}}/{r}})^{q}\frac{\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}}{(% \hslash^{1-({{{\mathrm{M}}{i}}/{r}})^{q}}-1)}}\right)}}$ $\displaystyle=r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{M}}{i}}/{r}})^{q}}},$

and

$\displaystyle\lim_{\hslash\to+\infty}r\sqrt[q]{\left(1-\log_{\hslash}\left(1+% \prod_{i=1}^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}\right)% \right)}$ $\displaystyle=r\sqrt[q]{\left(1-\lim_{\hslash\to+\infty}\frac{\ln(1+\prod_{i=1% }^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}})}{\ln\hslash}% \right)}.$

Applying the L’Hospital’s rule, we have

$\displaystyle r\sqrt[q]{\left(1-\lim_{\hslash\to+\infty}\frac{\frac{\prod_{i=1% }^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n% }{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\left(\sum_{i=1}^{n}w% _{i}{(1-({{{\mathrm{N}}{i}}/{r}})^{q})\frac{\hslash^{-({{{\mathrm{N}}{i}}/{r}}% )^{q}}}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)}}\right)}{{1}/{\hslash}}% \right)}$ $\displaystyle=r\sqrt[q]{\!\left(\!1-\lim_{\hslash\to+\infty}{\frac{\prod_{i=1}% ^{n}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}{1+\prod_{i=1}^{n}% {(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}}\!\left(\!\sum_{i=1}^{% n}w_{i}{(1-({{{\mathrm{N}}{i}}/{r}})^{q})\frac{\hslash^{1-({{{\mathrm{N}}{i}}/% {r}})^{q}}}{(\hslash^{1-({{{\mathrm{N}}{i}}/{r}})^{q}}-1)}}\right)}\right)}$ $\displaystyle=r\sqrt[q]{\left(1-\sum_{i=1}^{n}w_{i}{(1-({{{\mathrm{N}}{i}}/{r}% })^{q})}\right)}=r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{N}}{i}}/{r}})^{q}}}.$

Therefore, we obtain

$\displaystyle\lim_{\hslash\to+\infty}L^{q}\textit{ROPFFWG}({\Phi}_{1},{\Phi}_{% 2},\ldots,{\Phi}_{n})$ $\displaystyle=\lim_{\hslash\to+\infty}\left\langle v_{r(\sqrt[q]{(\log_{% \hslash}(1+\prod_{i=1}^{n}{(\hslash^{({{{\mathrm{M}}{i}}/{r}})^{q}}-1)^{w_{i}}% }))})},v_{r(\sqrt[q]{(1-\log_{\hslash}(1+\prod_{i=1}^{n}{(\hslash^{1-({{{% \mathrm{N}}{i}}/{r}})^{q}}-1)^{w_{i}}}))})}\right\rangle$ $\displaystyle=\left\langle r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{M}}{i}}/{% r}})^{q}}},r\sqrt[q]{\sum_{i=1}^{n}w_{i}{({{{\mathrm{N}}{i}}/{r}})^{q}}}\right\rangle.$

6. L^qROPF-Entropy-WASPAS method for MAGDM with L^qROPF information

In this section, by using proposed Frank AOs and improved score function, we formulate the L^qROPF-Entropy-WASPAS method to solve MAGDM problems in which the information related to the weight vector of the attributes is uknown and the decision information given in the form of L^qROPFNs. We also consider a real-life numerical example to demonstrate the decision steps and flexibility of the developed approach. The procedural steps of the formulated L^qROPF-Entropy-WASPAS method are discussed as follows (see Fig. 2).

Figure 2.

Proposed L^qROPF-Entropy-WASPAS model for ranking online platform for teaching.

6.1 Problem formulation

A general MAGDM problem incorporating the L^qROPF information can be defined as follows:

Consider a set of $m$ alternatives to be evaluated, denoted by $\mathcal{Q}=\{\mathcal{Q}_{1},\mathcal{Q}_{2},\ldots,\mathcal{Q}_{m}\}$ . Additionally, let $\mathfrak{Y}=\{\mathfrak{Y}_{1},\mathfrak{Y}_{2},\ldots,\mathfrak{Y}_{n}\}$ represent a set of $n$ attributes associated with these alternatives. Each attribute $\mathfrak{Y}_{j}$ is assigned a weight $w_{j}\in[0,1]$ $(j=1(1)n)$ in the weight vector $w=(w_{1},w_{2},\ldots,w_{n})^{T}$ , satisfying the constraint $\sum_{j=1}^{n}w_{j}=1$ . Furthermore, let ${DE}=\{{DE}_{1},{DE}_{2},\ldots,{DE}_{l}\}$ represent a set of $l$ decision experts (DEs). Each ${DE}_{k}$ is assigned a weight $z_{k}\in[0,1]$ $(k=1(1)l)$ in the weight vector $z=(z_{1},z_{2},\ldots,z_{l})^{T}$ , satisfying the constraint $\sum_{k=1}^{l}z_{k}=1$ . To facilitate the decision-making process, we introduce L^qROPF decision matrices $A^{(k)}=(\Phi_{ij}^{(k)})_{m\times n}$ for each ${DE}_{k}$ $(k=1(1)l)$ . These matrices incorporate L^qROPF information and can be represented as follows:

$\displaystyle A^{(k)}=(\Phi_{ij}^{(k)})_{m\times n}=\bordermatrix{&\mathfrak{Y% }_{1}&\mathfrak{Y}_{2}&\cdots&\mathfrak{Y}_{n}\cr\mathcal{Q}_{1}&\Phi_{11}^{(k% )}&\Phi_{12}^{(k)}&\cdots&\Phi_{1n}^{(k)}\cr\mathcal{Q}_{2}&\Phi_{21}^{(k)}&% \Phi_{22}^{(k)}&\cdots&\Phi_{2n}^{(k)}\cr\vdots&\vdots&\vdots&\vdots&\vdots\cr% \mathcal{Q}_{m}&\Phi_{m1}^{(k)}&\Phi_{m2}^{(k)}&\cdots&\Phi_{mn}^{(k)}}$

where $\Phi_{ij}^{(k)}=\langle v_{{\mathrm{M}}{ij}^{(k)}},v_{{\mathrm{N}}{ij}^{(k)}}\rangle$ represents the attribute evaluation value provided by the decision expert ${DE}_{k}$ in terms of L^qROPFN for a specific alternative $\mathcal{Q}_{i}$ corresponding to the attribute $\mathfrak{Y}_{j}$ and satisfy $v_{{\mathrm{M}}{ij}^{(k)}},v_{{\mathrm{N}}{ij}^{(k)}}\in\widehat{\mathbb{V}}_{% [0,t]}$ and ${{\mathrm{M}}{ij}^{(k)}}+{{\mathrm{N}}{ij}^{(k)}}\leqslant(r)^{q}$ for all DEs. The DEs’ objective in this MAGDM process is to determine the best alternative among a set of available alternatives, denoted as $\mathcal{Q}=\{\mathcal{Q}_{1},\mathcal{Q}_{2},\ldots,\mathcal{Q}_{m}\}$ . The decision-making steps of the proposed MAGDM method can be outlined as follows:

Step 1: Calculate the DEs’ weights

Let us assume that the importance levels $\phi_{k}$ ’s $(k=1(1)l)$ of ${{DE}}_{k}$ ’s $(k=1(l)l)$ are given in the form of L^qROPFNs such as $\phi_{k}=\langle{\breve{v}}_{{\mathrm{M}}{k}},{\breve{v}}_{{\mathrm{N}}{k}}\rangle$ , ${\breve{v}}_{{\mathrm{M}}{k}},{\breve{v}}_{{\mathrm{N}}{k}}\in\breve{\mathbb{V% }}_{[0,{r}^{\prime}]}\forall k$ such that ${({\mathrm{M}}{k})}^{q}+{({\mathrm{N}}{k})}^{q}\leqslant(r^{\prime})^{q}\forall k$ . The weight of the $k^{\text{th}}$ DE is calculated as follows:

$\displaystyle z_{k}=\frac{{r}^{\prime}\times\left[\frac{(1+2\frac{{\mathrm{M}}% {k}}{{r}^{\prime}}-\frac{{\mathrm{N}}{k}}{{r}^{\prime}})\cosh(1+\tan(\frac{{% \mathrm{M}}{k}}{{r}^{\prime}}\frac{\pi}{4})-(\frac{{\mathrm{N}}{k}}{{r}^{% \prime}})^{q})}{3\cosh(2)}\right]}{\sum_{k=1}^{l}{r}^{\prime}\times\left[\frac% {(1+2\frac{{\mathrm{M}}{k}}{{r}^{\prime}}-\frac{{\mathrm{N}}{k}}{{r}^{\prime}}% )\cosh(1+\tan(\frac{{\mathrm{M}}{k}}{{r}^{\prime}}\frac{\pi}{4})-(\frac{{% \mathrm{N}}{k}}{{r}^{\prime}})^{q})}{3\cosh(2)}\right]},\;k=1(1)l;\;q\geqslant 1.$ (32)

Clearly, $z_{k}\in[0,1]$ and $\sum_{k=1}^{l}z_{k}=1$ .

Step 2: Generate the collective linguistic $q$ -rung orthopair fuzzy decision matrix ${A}=({\Phi}_{ij})_{m\times n}$

In this step, we employ the L^qROPFFWA operator to aggregate the individual decision matrices, denoted as $A^{(k)}=(\Phi_{ij}^{(k)})_{m\times n}$ , into a single comprehensive matrix $A=(\Phi_{ij})_{m\times n}$ . By applying the L^qROPFFWA operator, we combine the attribute evaluation values $\Phi_{ij}^{(k)}$ from all decision matrices to obtain the aggregated matrix ${A}$ , with entries denoted as ${\Phi}_{ij}$ . The aggregation process considers the preferences and weights assigned by the DEs, allowing for a comprehensive representation of the collective evaluations for each alternative across all attributes. This aggregated matrix $A=(\Phi_{ij})_{m\times n}$ serves as a foundation for subsequent analyses and decision-making tasks, where

$\displaystyle{\Phi}_{ij}=L^{q}\textit{ROPFFWA}({\Phi}_{ij}^{(1)},{\Phi}_{ij}^{% (2)},\ldots,{\Phi}_{ij}^{(l)})=\left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(1+% \prod_{k=1}^{l}{(\hslash^{1-({{{{\mathrm{M}}{ij}^{(k)}}}/{r}})^{q}}-1)^{z_{k}}% }))})},v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{k=1}^{l}{(\hslash^{({{{{\mathrm{% N}}{ij}^{(k)}}}/{r}})^{q}}-1)^{z_{k}}}))})}\right\rangle.$ (33)

Step 3: Construct the normalized collective decision matrix $\hat{A}=(\hat{\Phi}_{ij})_{m\times n}$

In attribute-based decision-making, we often encounter two types of attributes: benefit-type and cost-type. When all attributes $\mathfrak{Y}_{j}$ $(j=1(1)n)$ belong to the same type, there is no need to perform a normalization process. However, if the attributes involve different types, we need to construct a normalized decision matrix $\hat{A}=(\hat{\Phi}{ij})_{m\times n}$ . We establish the normalized decision matrix $\hat{A}$ by using mathematical expression given Eq. (34) as follows

$\displaystyle{\hat{\Phi}}_{ij}=\langle{v}_{{\hat{\mathrm{M}}_{ij}}},{v}_{{\hat% {\mathrm{N}}_{ij}}}\rangle=\left\{\begin{array}[]{ll}\langle{v}_{{{\mathrm{M}}% _{ij}}},{v}_{{{\mathrm{N}}_{ij}}}\rangle,&\mbox{for benefit attribute}\,% \mathfrak{Y}_{j},\\ \langle{v}_{{{\mathrm{N}}_{ij}}},{v}_{{{\mathrm{M}}_{ij}}}\rangle,&\mbox{for % cost attribute}\,\mathfrak{Y}_{j}.\end{array}\right.$ (34)

The normalization process ensures that attributes of different types are placed on a comparable scale, enabling fair and accurate comparisons during the decision-making process. This step becomes necessary when there is a mix of benefit-type and cost-type attributes, as their measurement units or scales might differ. The normalized decision matrix is a valuable input for subsequent analysis and decision-making processes.

Step 4: Determination of attribute weights by employing entropy method

In the context of MAGDM problems, assigning appropriate weights to attributes is crucial for achieving a meaningful ranking order among the alternatives. It is important to note that not all attributes are assumed to have equal importance in the decision-making process. To address this, we introduce a weight vector denoted as $w=(w_{1},w_{2},\ldots,w_{n})$ , where $w_{j}\in[0,1]$ and $\sum_{j=1}^{n}w_{j}=1$ . This weight vector represents the relative importance or significance assigned to each attribute. Entropy, a concept derived from information theory, measures uncertainty or randomness within a system. In the context of MAGDM, the entropy method [71] is utilized to calculate the weights or importance of attributes in a data set. The fundamental principle behind the entropy method is to quantify the disorder or uncertainty present in a dataset before and after splitting the data based on a particular attribute. The attribute that leads to the greatest reduction in disorder is considered to have the highest weight or importance. The entropy method has been extensively applied in decision-making problems involving multiple attributes [83, 84, 85]. In this specific case, we present the application of the entropy method within the L^qROPF context. This approach allows for a comprehensive and effective determination of attribute weights, considering linguistic assessments and the inherent uncertainty associated with decision-making processes. Here, we outline the working process of the entropy method within the L^qROPF framework.

Step 4a: Construct the score matrix $\mathfrak{Q}=({\varphi}_{ij})_{m\times n}(i=1(1)m;j=1(1)n)$ , where

$\displaystyle{\varphi}_{ij}={\mathcal{S}}_{\textit{imp}}(\Phi_{ij})=r\times% \left[\frac{\left(1+2\frac{\hat{\mathrm{M}}_{ij}}{r}-\frac{\hat{\mathrm{N}}_{% ij}}{r}\right)\cosh\left(1+\tan\left(\frac{\hat{\mathrm{M}}_{ij}}{r}\frac{\pi}% {4}\right)-\left(\frac{\hat{\mathrm{N}}_{ij}}{r}\right)^{q}\right)}{3\cosh(2)}% \right]\forall i,j.$ (35)

Step 4b: Compute the normalize score matrix $\hat{\mathfrak{Q}}=(\hat{\varphi}_{ij})_{m\times n}(i=1(1)m;j=1,(1)n)$ by using Eq. (36) as given below

$\displaystyle{\hat{\varphi}}_{ij}=\frac{{\varphi}_{ij}}{\sum_{i=1}^{m}{\varphi% }_{ij}}.$ (36)

Step 4c: Evaluate the entropy value corresponding to each attribute based on the following expression:

$\displaystyle{\mathcal{E}}_{j}=\frac{1}{(e^{1-{1}/{m}}-1)}\sum_{i=1}^{m}{\hat{% \varphi}}_{ij}(e^{1-{\hat{\varphi}}_{ij}}-1).$ (37)

Step 4d: Calculate the weight corresponding to each attribute $\mathfrak{Y}_{j}(j=1(1),n)$ as follows

$\displaystyle w_{j}=\frac{{1-\mathcal{E}}_{j}}{\sum_{j=1}^{n}{1-\mathcal{E}}_{% j}}.$ (38)

where $w_{j}\in[0,1]$ and $\sum_{j=1}^{n}w_{j}=1$ .

Step 5: Computation of utility measures of weighted sum and weighted power models for alternatives

In this step, first we employ the L^qROPFFWA operator to determine the utility measure ${\hat{\Phi}}_{i}^{(k1)}(i=1,2,\ldots,n)$ of weighted sum model (WSM) as follows:

$\displaystyle{{\hat{\Phi}}_{i}}^{(k1)}=L^{q}\textit{ROPFFWA}(\hat{\Phi}_{i1},% \hat{\Phi}_{i2},\ldots,\hat{\Phi}_{in})$ (39) $\displaystyle=\left\langle v_{r(\sqrt[q]{(1-\log_{\hslash}(1+\prod_{j=1}^{n}{(% \hslash^{1-({{{\hat{\mathrm{M}}_{ij}}}/{r}})^{q}}-1)^{w_{j}}}))})},v_{r(\sqrt[% q]{(\log_{\hslash}(1+\prod_{j=1}^{n}{(\hslash^{({{{\hat{\mathrm{N}}_{ij}}}/{r}% })^{q}}-1)^{w_{j}}}))})}\right\rangle,$

Next, we use the L^qROPFFWG operator to evaluate the utility measure ${\hat{\Phi}}_{i}^{(k2)}(i=1,2,\ldots,n)$ of weighted product model (WPM) as follows:

$\displaystyle{{\hat{\Phi}}_{i}}^{(k2)}=L^{q}\textit{ROPFFWG}(\hat{\Phi}_{i1},% \hat{\Phi}_{i2},\ldots,\hat{\Phi}_{in})$ (40) $\displaystyle=\left\langle v_{r(\sqrt[q]{(\log_{\hslash}(1+\prod_{j=1}^{n}{(% \hslash^{({{{\hat{\mathrm{M}}_{ij}}}/{r}})^{q}}-1)^{w_{j}}}))})},v_{r(\sqrt[q]% {(1-\log_{\hslash}(1+\prod_{j=1}^{n}{(\hslash^{1-({{{\hat{\mathrm{N}}_{ij}}}/{% r}})^{q}}-1)^{w_{j}}}))})}\right\rangle.$

Step 6: Determination of the WASPAS measure for each alternative

The WASPAS measure corresponding to each alternative $\mathcal{Q}_{i}(i=1(1)m)$ is computed according to the following formula:

$\displaystyle\mathbb{K}_{i}=(\vartheta{\overset{q}{\otimes}_{F}}{{\hat{\Phi}}_% {i}}^{(k1)}){\overset{q}{\oplus}_{F}}((1-\vartheta){\overset{q}{\otimes}_{F}}{% {\hat{\Phi}}_{i}}^{(k2)})$ (41)

where $\vartheta$ denotes the decision precision parameter and $\gamma\in[0,1]$ .

Step 7: Rank the alternatives

Once the WASPAS measures, denoted as $\mathbb{K}_{i}(i=1(1)m)$ , have been obtained, the subsequent task involves ranking all the alternatives $\mathcal{Q}_{i}(i=1(1)m)$ based on Definition 8. This ranking process allows us to establish a clear order of preference among the alternatives based on their overall performance. Utilizing the ranking established through this evaluation, we identify the most desirable alternative(s) among the set of alternatives.The alternative possess the highest value of WASPAS measure $\mathbb{K}_{i}$ is considered to be the most suitable alternative based on the decision criteria and preferences considered in the MAGDM problem.

6.2 Numerical illustration

In this subsection, the developed L^qROPF-Entropy-WASPAS method has been demonstrated by a real-life decision-making problem of online teaching platform selection in the L^qROPF information context.

Problem description

The education sector has witnessed a significant transformation in recent years, driven by technological advancements and changing learning preferences [86]. Online teaching platforms have emerged as indispensable tools in modern education, offering numerous benefits and transforming knowledge acquisition. These platforms offer a wide range of features and capabilities that enhance the educational experience for both educators and learners. Students now seek access to education beyond the confines of physical classrooms, allowing them to learn at their own pace, from anywhere in the world. With the increasing demand for remote and flexible learning, online teaching platforms have gained significant importance and have become essential for educational institutions worldwide [87]. It is worth noting that these platforms also break down barriers by enabling students to learn from reputable institutions and expert instructors, even if they cannot physically attend traditional classrooms.

Furthermore, unforeseen circumstances, such as the COVID-19 pandemic, have highlighted the urgent need for robust online education systems that can ensure uninterrupted learning even in times of crisis. Compared to traditional teaching methods, online teaching platforms offer greater flexibility, personalized learning experiences, diverse resources, interactivity, global connectivity, enhanced assessment and feedback, cost-effectiveness, and professional development opportunities [88]. In addition, online teaching platforms offer a wide range of multimedia resources, including videos, simulations, and interactive content, which enhance the learning experience and make complex concepts more accessible and engaging. By harnessing the power of technology, online teaching platforms have the potential to revolutionize education, empower learners, and pave the way for a more inclusive and effective educational system.

.

A Chilean university is planning to implement an online teaching platform to enhance its educational offerings and accommodate remote learning. The university has shortlisted the seven online education platforms as potential candidates for further evaluation, represented by $\mathcal{Q}=({\mathcal{Q}}_{1},{\mathcal{Q}}_{2},{\mathcal{Q}}_{3},{\mathcal{Q% }}_{4},{\mathcal{Q}}_{5},{\mathcal{Q}}_{6},{\mathcal{Q}}_{7})$ . The university has formed a committee of four experts consisting of faculty members, administrators, and IT professionals, denoted by ${DE}=({{DE}}_{1},{{DE}}_{2},{{DE}}_{3},{{DE}}_{4})$ , to evaluate the shortlisted online education platforms and make well-informed decisions. The decision-making process involves multiple attributes that must be considered to ensure the platform meets the university’s requirements. The expert committee considers the following attributes to evaluate the potential online teaching platforms: (i) Cost-effectiveness $({{\mathfrak{Y}}_{1}})$ (ii) Audio and video quality $({{\mathfrak{Y}}_{2}})$ (iii) Security and privacy $({{\mathfrak{Y}}_{3}})$ (iv) Mobile accessibility $({{\mathfrak{Y}}_{4}})$ (v) Technical support and training $({{\mathfrak{Y}}_{5}})$ (vi) User friendly $({{\mathfrak{Y}}_{6}})$ (vii) Attractive interface $({{\mathfrak{Y}}_{7}})$ (viii) Tracking and reporting $({{\mathfrak{Y}}_{8}})$ (ix) Automated evaluation $({{\mathfrak{Y}}_{9}})$ (x) Collaboration and social learning tools $({{\mathfrak{Y}}_{10}})$ . The hierarchical framework structure for selecting the best platform for online teaching is illustrated in Fig. 3.

Figure 3.

The hierarchical framework to select online platform for teaching.

Table 2

The importance levels of experts in assessing alternatives

Experts	$DE_{1}$	$DE_{2}$	$DE_{3}$	$DE_{4}$
Importance levels	$\langle\breve{v}_{3},\breve{v}_{4}\rangle$	$\langle\breve{v}_{4},\breve{v}_{2}\rangle$	$\langle\breve{v}_{2},\breve{v}_{1}\rangle$	$\langle\breve{v}_{4},\breve{v}_{3}\rangle$

Note that the importance levels $\phi_{k}$ ’s $(k=1(1)4)$ of ${{DE}}_{k}$ ’s $(k=1(1)4)$ are represented by using L^qROPFNs according to the following LTSs: ${\breve{\mathbb{V}}}=\{\breve{v}_{0},\breve{v}_{1},\breve{v}_{2},\breve{v}_{3}% ,\breve{v}_{4},\breve{v}_{5},\breve{v}_{6}\}$ and summarized in Table 2. The committee members evaluate the available alternatives according to the attributes ${{\mathfrak{Y}}_{j}}(j=1(1)10)$ and express their preference information based on the LTS $\widehat{\mathbb{V}}=\{{v}_{0},v_{1},v_{2},v_{3},v_{4},v_{5},v_{6},v_{7},v_{8}\}$ . The evaluation information provided by the DEs are summarized in the following L^qROPF decision matrices $A^{(k)}=(\Phi_{ij}^{(k)})_{m\times n}(k=1(1)4)$ , which is shown in Table 3, respectively.

Table 3

L^qROPF decision matrices $A^{(k)}(k=1,2,3,4)$ suggested by DEs

$A^{(1)}$ suggested by ${DE}_{1}$
	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
${{\mathcal{Q}}_{1}}$	$\langle v_{2},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{1}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{7},v_{3}\rangle$
${{\mathcal{Q}}_{2}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{7},v_{1}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{5},v_{3}\rangle$
${{\mathcal{Q}}_{3}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{4}}$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{2},v_{2}\rangle$	$\langle v_{6},v_{1}\rangle$	$\langle v_{3},v_{5}\rangle$
${{\mathcal{Q}}_{5}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{6}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$
${{\mathcal{Q}}_{6}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{7},v_{2}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{3},v_{4}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{3}\rangle$
${{\mathcal{Q}}_{7}}$	$\langle v_{2},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{7},v_{2}\rangle$	$\langle v_{1},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{1}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$
$A^{(2)}$ suggested by ${DE}_{2}$
${{\mathcal{Q}}_{1}}$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{7},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$
${{\mathcal{Q}}_{2}}$	$\langle v_{2},v_{6}\rangle$	$\langle v_{7},v_{1}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{4},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{6},v_{2}\rangle$
${{\mathcal{Q}}_{3}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{7},v_{2}\rangle$	$\langle v_{3},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$
${{\mathcal{Q}}_{4}}$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{5},v_{1}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{5}}$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{7}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{7},v_{1}\rangle$
${{\mathcal{Q}}_{6}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{4}\rangle$	$\langle v_{3},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{3}\rangle$
${{\mathcal{Q}}_{7}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{3},v_{7}\rangle$	$\langle v_{4},v_{3}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$
$A^{(3)}$ suggested by ${DE}_{3}$
${{\mathcal{Q}}_{1}}$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{2}}$	$\langle v_{2},v_{6}\rangle$	$\langle v_{7},v_{3}\rangle$	$\langle v_{5},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{7},v_{1}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{7},v_{1}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{6},v_{2}\rangle$
${{\mathcal{Q}}_{3}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{4}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{7},v_{2}\rangle$	$\langle v_{3},v_{3}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{6},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$
${{\mathcal{Q}}_{5}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{7}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{1},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{6}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{7},v_{2}\rangle$
${{\mathcal{Q}}_{7}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{7},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{6}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$
$A^{(4)}$ suggested by ${DE}_{4}$
${{\mathcal{Q}}_{1}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{2}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{7},v_{1}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{7},v_{1}\rangle$
${{\mathcal{Q}}_{3}}$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{5},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{5},v_{3}\rangle$
${{\mathcal{Q}}_{4}}$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{2},v_{5}\rangle$	$\langle v_{3},v_{6}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$
${{\mathcal{Q}}_{5}}$	$\langle v_{3},v_{5}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{7}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{3},v_{5}\rangle$
${{\mathcal{Q}}_{6}}$	$\langle v_{2},v_{6}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{7},v_{1}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{6},v_{2}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$
${{\mathcal{Q}}_{7}}$	$\langle v_{2},v_{6}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{4}\rangle$	$\langle v_{4},v_{5}\rangle$	$\langle v_{5},v_{4}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{3}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{3},v_{5}\rangle$

According to available data, the decision steps of developed algorithm are performed as follows:

Step 1: The weight assigned to each DE is determined by utilizing the information presented in Table 2 as follows:

$\displaystyle{\mathcal{S}}_{\textit{imp}}(\phi_{1})=6\times\left[\frac{(1+2% \times\frac{3}{6}-\frac{4}{6})\times\cosh(1+\tan(\frac{3}{6}\frac{\pi}{4})-(% \frac{4}{6})^{2})}{3\cosh(2)}\right]=1.4254,$ $\displaystyle{\mathcal{S}}_{\textit{imp}}(\phi_{2})=6\times\left[\frac{(1+2% \times\frac{4}{6}-\frac{2}{6})\times\cosh(1+\tan(\frac{4}{6}\frac{\pi}{4})-(% \frac{2}{6})^{2})}{3\cosh(2)}\right]=3.2348,$ $\displaystyle{\mathcal{S}}_{\textit{imp}}(\phi_{3})=6\times\left[\frac{(1+2% \times\frac{2}{6}-\frac{1}{6})\times\cosh(1+\tan(\frac{2}{6}\frac{\pi}{4})-(% \frac{1}{6})^{2})}{3\cosh(2)}\right]=1.9911,$ $\displaystyle{\mathcal{S}}_{\textit{imp}}(\phi_{4})=6\times\left[\frac{(1+2% \times\frac{4}{6}-\frac{3}{6})\times\cosh(1+\tan(\frac{4}{6}\frac{\pi}{4})-(% \frac{3}{6})^{2})}{3\cosh(2)}\right]=2.6225.$

Using Eq. (32), we calculate the following values:

$\displaystyle z_{1}=0.1536,z_{2}=0.3490,z_{3}=0.2146,z_{4}=0.2827.$

Step 2: By using L^qROPFFWA operator given in Eq. (33), we aggregate all the individual decision matrix $A^{(k)}=({\Phi}_{ij}^{(k)})_{7\times 10}(k=1(1)4)$ to obtain a collective decision matrix $A=({\Phi}_{ij})_{7\times 10}$ . The obtained results are summarized in Table 4.

Table 4

Collective decision matrix $A$ based on L^qROPFFWA operator with $\hslash=2$

	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$
${\mathcal{Q}_{1}}$	$\langle v_{3.5107},v_{4.0858}\rangle$	$\langle v_{4.0780},v_{5.0194}\rangle$	$\langle v_{4.6879},v_{3.7639}\rangle$	$\langle v_{5.4965},v_{3.3202}\rangle$	$\langle v_{5.0649},v_{2.7908}\rangle$
${\mathcal{Q}_{2}}$	$\langle v_{2.4939},v_{5.5484}\rangle$	$\langle v_{6.5646},v_{1.8306}\rangle$	$\langle v_{5.1853},v_{3.0047}\rangle$	$\langle v_{1.8830},v_{6.1479}\rangle$	$\langle v_{6.2235},v_{1.6395}\rangle$
${\mathcal{Q}_{3}}$	$\langle v_{3.3242},v_{4.6997}\rangle$	$\langle v_{5.5925},v_{3.1571}\rangle$	$\langle v_{3.8728},v_{3.1379}\rangle$	$\langle v_{4.0378},v_{3.6002}\rangle$	$\langle v_{4.5787},v_{3.6262}\rangle$
${\mathcal{Q}_{4}}$	$\langle v_{4.2356},v_{3.8066}\rangle$	$\langle v_{4.0566},v_{4.2890}\rangle$	$\langle v_{3.2136},v_{4.6115}\rangle$	$\langle v_{4.8439},v_{4.0481}\rangle$	$\langle v_{2.5545},v_{4.9489}\rangle$
${\mathcal{Q}_{5}}$	$\langle v_{3.3946},v_{4.6312}\rangle$	$\langle v_{5.0493},v_{2.9848}\rangle$	$\langle v_{4.0015},v_{3.4472}\rangle$	$\langle v_{3.6100},v_{4.4150}\rangle$	$\langle v_{4.5033},v_{4.4424}\rangle$
${\mathcal{Q}_{6}}$	$\langle v_{2.7600},v_{5.2701}\rangle$	$\langle v_{5.6838},v_{3.0047}\rangle$	$\langle v_{5.6192},v_{2.4566}\rangle$	$\langle v_{3.6277},v_{3.5873}\rangle$	$\langle v_{4.5276},v_{3.1895}\rangle$
${\mathcal{Q}_{7}}$	$\langle v_{2.6184},v_{5.2701}\rangle$	$\langle v_{3.8948},v_{4.7122}\rangle$	$\langle v_{5.8244},v_{3.0650}\rangle$	$\langle v_{4.1601},v_{4.0048}\rangle$	$\langle v_{5.6186},v_{3.3162}\rangle$
	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
${\mathcal{Q}_{1}}$	$\langle v_{5.0649},v_{2.9681}\rangle$	$\langle v_{4.0535},v_{3.9999}\rangle$	$\langle v_{4.1805},v_{2.8113}\rangle$	$\langle v_{4.2979},v_{3.4717}\rangle$	$\langle v_{4.4502},v_{4.4379}\rangle$
${\mathcal{Q}_{2}}$	$\langle v_{3.3946},v_{4.6312}\rangle$	$\langle v_{5.8213},v_{2.1840}\rangle$	$\langle v_{6.0149},v_{1.7440}\rangle$	$\langle v_{1.8343},v_{6.2072}\rangle$	$\langle v_{6.2556},v_{1.7541}\rangle$
${\mathcal{Q}_{3}}$	$\langle v_{5.2923},v_{2.7288}\rangle$	$\langle v_{4.1834},v_{3.4854}\rangle$	$\langle v_{4.9998},v_{3.0003}\rangle$	$\langle v_{3.0262},v_{5.0866}\rangle$	$\langle v_{4.0535},v_{3.9999}\rangle$
${\mathcal{Q}_{4}}$	$\langle v_{4.9378},v_{3.1044}\rangle$	$\langle v_{4.3016},v_{4.1699}\rangle$	$\langle v_{3.9782},v_{3.4854}\rangle$	$\langle v_{4.6343},v_{1.8912}\rangle$	$\langle v_{3.1919},v_{4.8862}\rangle$
${\mathcal{Q}_{5}}$	$\langle v_{3.1674},v_{5.0375}\rangle$	$\langle v_{4.6928},v_{3.3211}\rangle$	$\langle v_{3.8212},v_{3.9448}\rangle$	$\langle v_{4.7099},v_{3.3258}\rangle$	$\langle v_{5.5671},v_{2.5534}\rangle$
${\mathcal{Q}_{6}}$	$\langle v_{2.7161},v_{5.5610}\rangle$	$\langle v_{5.2174},v_{2.8026}\rangle$	$\langle v_{5.1183},v_{2.9550}\rangle$	$\langle v_{4.1805},v_{3.8302}\rangle$	$\langle v_{5.7047},v_{2.9913}\rangle$
${\mathcal{Q}_{7}}$	$\langle v_{3.8690},v_{4.1425}\rangle$	$\langle v_{5.4149},v_{2.7490}\rangle$	$\langle v_{4.7541},v_{2.7653}\rangle$	$\langle v_{3.3603},v_{4.7240}\rangle$	$\langle v_{4.3523},v_{3.7013}\rangle$

Step 3: In this case, ${{\mathfrak{Y}}_{1}}$ represents an attribute of cost type, while ${{\mathfrak{Y}}_{j}}$ for $(j=2(1)10)$ represent attributes of benefit type. Therefore, it is necessary to convert the attribute values of the cost type into the attribute values of the benefit type using Eq. (34). After transformation, the resulting normalized collective decision matrix $\hat{A}=(\hat{\Phi}_{ij})_{7\times 10}$ is displayed in Table 5.

Table 5

Normalized decision matrix $\hat{A}$

	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$
${\mathcal{Q}_{1}}$	$\langle v_{4.0858},v_{3.5107}\rangle$	$\langle v_{4.0780},v_{5.0194}\rangle$	$\langle v_{4.6879},v_{3.7639}\rangle$	$\langle v_{5.4965},v_{3.3202}\rangle$	$\langle v_{5.0649},v_{2.7908}\rangle$
${\mathcal{Q}_{2}}$	$\langle v_{5.5484},v_{2.4939}\rangle$	$\langle v_{6.5646},v_{1.8306}\rangle$	$\langle v_{5.1853},v_{3.0047}\rangle$	$\langle v_{1.8830},v_{6.1479}\rangle$	$\langle v_{6.2235},v_{1.6395}\rangle$
${\mathcal{Q}_{3}}$	$\langle v_{4.6997},v_{3.3242}\rangle$	$\langle v_{5.5925},v_{3.1571}\rangle$	$\langle v_{3.8728},v_{3.1379}\rangle$	$\langle v_{4.0378},v_{3.6002}\rangle$	$\langle v_{4.5787},v_{3.6262}\rangle$
${\mathcal{Q}_{4}}$	$\langle v_{3.8066},v_{4.2356}\rangle$	$\langle v_{4.0566},v_{4.2890}\rangle$	$\langle v_{3.2136},v_{4.6115}\rangle$	$\langle v_{4.8439},v_{4.0481}\rangle$	$\langle v_{2.5545},v_{4.9489}\rangle$
${\mathcal{Q}_{5}}$	$\langle v_{4.6312},v_{3.3946}\rangle$	$\langle v_{5.0493},v_{2.9848}\rangle$	$\langle v_{4.0015},v_{3.4472}\rangle$	$\langle v_{3.6100},v_{4.4150}\rangle$	$\langle v_{4.5033},v_{4.4424}\rangle$
${\mathcal{Q}_{6}}$	$\langle v_{5.2701},v_{2.7600}\rangle$	$\langle v_{5.6838},v_{3.0047}\rangle$	$\langle v_{5.6192},v_{2.4566}\rangle$	$\langle v_{3.6277},v_{3.5873}\rangle$	$\langle v_{4.5276},v_{3.1895}\rangle$
${\mathcal{Q}_{7}}$	$\langle v_{5.2701},v_{2.6184}\rangle$	$\langle v_{3.8948},v_{4.7122}\rangle$	$\langle v_{5.8244},v_{3.0650}\rangle$	$\langle v_{4.1601},v_{4.0048}\rangle$	$\langle v_{5.6186},v_{3.3162}\rangle$
	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
${\mathcal{Q}_{1}}$	$\langle v_{5.0649},v_{2.9681}\rangle$	$\langle v_{4.0535},v_{3.9999}\rangle$	$\langle v_{4.1805},v_{2.8113}\rangle$	$\langle v_{4.2979},v_{3.4717}\rangle$	$\langle v_{4.4502},v_{4.4379}\rangle$
${\mathcal{Q}_{2}}$	$\langle v_{3.3946},v_{4.6312}\rangle$	$\langle v_{5.8213},v_{2.1840}\rangle$	$\langle v_{6.0149},v_{1.7440}\rangle$	$\langle v_{1.8343},v_{6.2072}\rangle$	$\langle v_{6.2556},v_{1.7541}\rangle$
${\mathcal{Q}_{3}}$	$\langle v_{5.2923},v_{2.7288}\rangle$	$\langle v_{4.1834},v_{3.4854}\rangle$	$\langle v_{4.9998},v_{3.0003}\rangle$	$\langle v_{3.0262},v_{5.0866}\rangle$	$\langle v_{4.0535},v_{3.9999}\rangle$
${\mathcal{Q}_{4}}$	$\langle v_{4.9378},v_{3.1044}\rangle$	$\langle v_{4.3016},v_{4.1699}\rangle$	$\langle v_{3.9782},v_{3.4854}\rangle$	$\langle v_{4.6343},v_{1.8912}\rangle$	$\langle v_{3.1919},v_{4.8862}\rangle$
${\mathcal{Q}_{5}}$	$\langle v_{3.1674},v_{5.0375}\rangle$	$\langle v_{4.6928},v_{3.3211}\rangle$	$\langle v_{3.8212},v_{3.9448}\rangle$	$\langle v_{4.7099},v_{3.3258}\rangle$	$\langle v_{5.5671},v_{2.5534}\rangle$
${\mathcal{Q}_{6}}$	$\langle v_{2.7161},v_{5.5610}\rangle$	$\langle v_{5.2174},v_{2.8026}\rangle$	$\langle v_{5.1183},v_{2.9550}\rangle$	$\langle v_{4.1805},v_{3.8302}\rangle$	$\langle v_{5.7047},v_{2.9913}\rangle$
${\mathcal{Q}_{7}}$	$\langle v_{3.8690},v_{4.1425}\rangle$	$\langle v_{5.4149},v_{2.7490}\rangle$	$\langle v_{4.7541},v_{2.7653}\rangle$	$\langle v_{3.3603},v_{4.7240}\rangle$	$\langle v_{4.3523},v_{3.7013}\rangle$

Step 4: In this step, we use the entropy method to determine the importance weights of the attributes. The specific procedure is described as follows:

Step 4a: Firstly, the score matrix $\mathfrak{Q}=({\varphi}_{ij})_{7\times 10}$ is calculated with the use of Eq. (35) and Table 5. The obtained score matrix $\mathfrak{Q}=({\varphi}_{ij})_{7\times 10}$ is shown in Table 6.

Table 6

Score matrix $\mathfrak{Q}$ corresponding to normalized collective decision matrix $\hat{A}$

	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
${{\mathcal{Q}}_{1}}$	2.0856	1.5575	2.3253	3.0584	2.9783	2.9024	1.8952	2.3900	2.2157	1.9330
${{\mathcal{Q}}_{2}}$	3.4878	4.8323	2.9730	0.5890	4.5336	1.4006	3.8734	4.2646	0.5676	4.5157
${{\mathcal{Q}}_{3}}$	2.5106	3.2109	2.1020	2.0284	2.3167	3.1753	2.1473	2.8429	1.1399	1.8952
${{\mathcal{Q}}_{4}}$	1.6980	1.7956	1.3360	2.2995	1.0168	2.7562	1.9580	2.0377	3.0186	1.2516
${{\mathcal{Q}}_{5}}$	2.4401	2.8842	2.0633	1.5541	1.9580	1.2019	2.5075	1.8016	2.5163	3.4762
${{\mathcal{Q}}_{6}}$	3.1444	3.3588	3.5653	1.8273	2.4578	0.9265	3.0857	2.9463	2.0202	3.3826
${{\mathcal{Q}}_{7}}$	3.2067	1.5805	3.4463	1.9466	3.1538	1.7577	3.2621	2.7711	1.3602	2.1597

Step 4b: Covert the score matrix $\mathfrak{Q}=({\varphi}_{ij})_{7\times 10}$ into normalized score matrix $\hat{\mathfrak{Q}}=(\hat{\varphi}_{ij})_{7\times 10}$ by using the Eq. (36). Table 7 shows the normalized score matrix $\hat{\mathfrak{Q}}=(\hat{\varphi}_{ij})_{7\times 10}$ .

Table 7

Normalized score matrix $\hat{\mathfrak{Q}}$

	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
${{\mathcal{Q}}_{1}}$	0.1123	0.0810	0.1306	0.2299	0.1617	0.2055	0.1012	0.1254	0.1726	0.1038
${{\mathcal{Q}}_{2}}$	0.1878	0.2514	0.1669	0.0443	0.2462	0.0992	0.2068	0.2238	0.0442	0.2426
${{\mathcal{Q}}_{3}}$	0.1352	0.1671	0.1180	0.1525	0.1258	0.2249	0.1146	0.1492	0.0888	0.1018
${{\mathcal{Q}}_{4}}$	0.0914	0.0934	0.0750	0.1729	0.0552	0.1952	0.1045	0.1069	0.2351	0.0672
${{\mathcal{Q}}_{5}}$	0.1314	0.1501	0.1158	0.1168	0.1063	0.0851	0.1339	0.0946	0.1960	0.1868
${{\mathcal{Q}}_{6}}$	0.1693	0.1748	0.2002	0.1374	0.1335	0.0656	0.1648	0.1546	0.1574	0.1817
${{\mathcal{Q}}_{7}}$	0.1727	0.0822	0.1935	0.1463	0.1713	0.1245	0.1742	0.1454	0.1059	0.1160

Steps 4c: Utilizing the Eq. (37), we obtain the entropy measure $\mathcal{E}_{i}(i=1(1)7)$ corresponding to each attribute $\mathfrak{Y}_{j}(j=1(1)10)$ as follows

$\displaystyle\mathcal{E}_{1}=0.9881,\mathcal{E}_{2}=0.9628,\mathcal{E}_{3}=0.9% 798,\mathcal{E}_{4}=0.9691,\mathcal{E}_{5}=0.9660,$ $\displaystyle\mathcal{E}_{6}=0.9599,\mathcal{E}_{7}=0.9846,\mathcal{E}_{8}=0.9% 830,\mathcal{E}_{9}=0.9571,\mathcal{E}_{10}=0.9632.$

Steps 4d: We employ Eq. (38) to determine the weight $w_{j}$ of each attribute $\mathfrak{Y}_{j}(j=1(1)10)$ . The calculated weights are presented in Table 8 for reference. (graphically shown in Fig. 4)

Table 8

The weights $(w_{j})$ corresponding to attributes $(\mathfrak{Y}_{j})$

	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
$w_{j}$	0.0414	0.1298	0.0707	0.1078	0.1187	0.1400	0.0539	0.0593	0.1498	0.1286

Figure 4.

Criteria weights for the evaluation of platform for online teaching.

Steps 5: (i) By using the Eq. (6.1), the utility measures ${\hat{\Phi}}_{i}^{(k1)}(i=1(1)n)$ of the WSM are obtained as follows:

$\displaystyle{\hat{\Phi}}_{1}^{(k1)}=\langle v_{4.6621},v_{3.5805}\rangle,{% \hat{\Phi}}_{2}^{(k1)}=\langle v_{5.2350},v_{3.0098}\rangle,{\hat{\Phi}}_{3}^{% (k1)}=\langle v_{4.5357},v_{3.5551}\rangle,$ $\displaystyle{\hat{\Phi}}_{4}^{(k1)}=\langle v_{4.1167},v_{3.7153}\rangle,{% \hat{\Phi}}_{5}^{(k1)}=\langle v_{4.5048},v_{3.6457}\rangle,{\hat{\Phi}}_{6}^{% (k1)}=\langle v_{4.7992},v_{3.210}\rangle,$ $\displaystyle{\hat{\Phi}}_{7}^{(k1)}=\langle v_{4.5634},v_{3.7792}\rangle;$

(ii) Utilizing the Eq. (6.1), the utility measures ${\hat{\Phi}}_{i}^{(k2)}(i=1(1)n)$ of the WPM are obtained as follows:

$\displaystyle{\hat{\Phi}}_{1}^{(k2)}=\langle v_{4.6621},v_{3.5805}\rangle,{% \hat{\Phi}}_{2}^{(k2)}=\langle v_{5.2350},v_{3.0098}\rangle,{\hat{\Phi}}_{3}^{% (k2)}=\langle v_{4.5357},v_{3.5551}\rangle,$ $\displaystyle{\hat{\Phi}}_{4}^{(k2)}=\langle v_{4.1167},v_{3.7153}\rangle,{% \hat{\Phi}}_{5}^{(k2)}=\langle v_{4.5048},v_{3.6457}\rangle,{\hat{\Phi}}_{6}^{% (k2)}=\langle v_{4.7992},v_{3.210}\rangle,$ $\displaystyle{\hat{\Phi}}_{7}^{(k2)}=\langle v_{4.5634},v_{3.7792}\rangle;$

Steps 6: The WASPAS measure for each alternative $\mathcal{Q}_{i}(i=1(1)7)$ is calculated and summarized as (without any loss of generality, we take $\vartheta=0.5$ )

$\displaystyle\mathbb{K}_{1}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{1}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{1}^{(k2)})=% \langle v_{4.6294},v_{3.6645}\rangle,$ $\displaystyle\mathbb{K}_{2}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{2}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{2}^{(k2)})=% \langle v_{4.7725},v_{3.5523}\rangle,$ $\displaystyle\mathbb{K}_{3}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{3}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{3}^{(k2)})=% \langle v_{4.4431},v_{3.6372}\rangle,$ $\displaystyle\mathbb{K}_{4}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{4}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{4}^{(k2)})=% \langle v_{4.0231},v_{3.8734}\rangle,$ $\displaystyle\mathbb{K}_{5}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{5}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{5}^{(k2)})=% \langle v_{4.4257},v_{3.7446}\rangle,$ $\displaystyle\mathbb{K}_{6}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{6}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{6}^{(k2)})=% \langle v_{4.6546},v_{3.5556}\rangle,$ $\displaystyle\mathbb{K}_{7}=(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{7}^{(k1)}% ){\overset{q}{\oplus}_{F}}(0.5{\overset{q}{\ast}_{F}}{\hat{\Phi}}_{7}^{(k2)})=% \langle v_{4.4698},v_{3.8548}\rangle.$

Steps 6: Finally, we calculate the score values of the computed WASPAS measures $\mathbb{K}_{i}(i=1(1)7)$ using Eq. (6) as

$\displaystyle{\mathcal{S}}_{\textit{imp}}(\mathbb{K}_{1})=2.3310,{\mathcal{S}}% _{\textit{imp}}(\mathbb{K}_{2})=2.4624,{\mathcal{S}}_{\textit{imp}}(\mathbb{K}% _{3})=2.2346,{\mathcal{S}}_{\textit{imp}}(\mathbb{K}_{4})=1.9246,$ $\displaystyle{\mathcal{S}}_{\textit{imp}}(\mathbb{K}_{5})=2.1837,{\mathcal{S}}% _{\textit{imp}}(\mathbb{K}_{6})=2.3896,{\mathcal{S}}_{\textit{imp}}(\mathbb{K}% _{7})=2.1656.$

Step 7: According to the score values ${\mathcal{S}}_{\textit{imp}}(\mathbb{K}_{i})$ , the ranking order of the alternatives ${\mathcal{Q}_{i}}(i=1(1)7)$ is obtained as ${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$ . Hence, $\mathcal{Q}_{2}$ is the best online platform for teaching.

7. Sensitivity analysis, validity test and comparative study

7.1 Sensitivity analysis

It is important to note that the analysis conducted above was based on the assumption of $\hslash=2$ . However, in real-world scenarios, the parameter $\hslash$ can vary depending on the DE’s preference and the specific circumstances involved. A sensitivity analysis has been conducted to assess the influence of the parameter $\hslash$ on the ranking of the alternatives in the decision-making process. This analysis involved considering different values of the parameter $\hslash$ for the problem under consideration.

To begin, the weights assigned to the relevant attributes are determined for various values of the parameter $\hslash$ , specifically $\hslash=$ 3, 5, 7, 10, 12, 15, and 20. The weight calculation is performed using Eqs (35) to (38). The resulting weights are then compiled and presented in Table 9 and graphically shown in Fig. 5. This comprehensive table provides insights into the attribute weights obtained for each $\hslash$ value, thereby enabling a thorough examination of the impact of varying $\hslash$ on the decision-making process.

Table 9
The weights $(w_{j})$ corresponding to attributes $(\mathfrak{Y}_{j})$ for different values of $\hslash$

	$\hslash$	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
$w_{j}$	$\hslash=2$	0.0414	0.1298	0.0707	0.1078	0.1187	0.1400	0.0539	0.0593	0.1498	0.1286
	$\hslash=3$	0.0412	0.1313	0.0702	0.1057	0.1191	0.1405	0.0544	0.0597	0.1495	0.1284
	$\hslash=5$	0.0412	0.1321	0.0694	0.1035	0.1199	0.1407	0.0552	0.0595	0.1491	0.1293
	$\hslash=7$	0.0411	0.1326	0.0689	0.1024	0.1207	0.1406	0.0553	0.0601	0.1488	0.1294
	$\hslash=10$	0.0410	0.1336	0.0682	0.1008	0.1210	0.1404	0.0560	0.0608	0.1490	0.1293
	$\hslash=12$	0.0409	0.1339	0.0682	0.1004	0.1213	0.1406	0.0559	0.0609	0.1485	0.1294
	$\hslash=15$	0.0408	0.1343	0.0678	0.0994	0.1213	0.1407	0.0563	0.0609	0.1485	0.1300
	$\hslash=20$	0.0397	0.1350	0.0674	0.0989	0.1218	0.1410	0.0564	0.0612	0.1483	0.1302

In addition, the process continues with Steps 5 to 7, which are employed to determine the WASPAS denoted as $\mathbb{K}_{i}$ , $i=(1(1)7)$ , the score values ${\mathcal{S}}_{\textit{imp}}(\mathbb{K}_{i})$ and the ranking order of the alternatives based on various values of the parameter $\hslash$ . For a comprehensive overview of the obtained results, see Tables 10 and 11. These tables present the WASPAS measures $\mathbb{K}_{i}$ , the corresponding score values ${\mathcal{S}}_{\textit{imp}}(\mathbb{K}_{i})$ , and the ranking of alternatives for different values of the parameter $\hslash$ .

The findings presented in Table 11 demonstrate that ${\mathcal{Q}_{2}}$ emerges as the most favorable alternative across all values of the parameter $\hslash$ , while ${\mathcal{Q}_{4}}$ consistently ranks as the least favorable alternative. It is also worth noting that the relative ordering of the alternatives remains consistent for all values of the parameter $\hslash$ . A graphical representation of the sensitivity results is offered in Fig. 5 for better understanding of readers. Furthermore, Fig. 6 is also drawn to illustrate the linear ranking orders of the online education platforms, with ascending score values $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{i})(i=1(1)7)$ corresponding to each value of the parameter $\hslash$ .

Table 10

The WASPAS measures $\mathbb{V}_{i}$ for different values of the parameter $\hslash$

$\hslash$	WASPAS measures
	$\mathbb{K}_{1}$	$\mathbb{K}_{2}$	$\mathbb{K}_{3}$	$\mathbb{K}_{4}$	$\mathbb{K}_{5}$	$\mathbb{K}_{6}$	$\mathbb{K}_{7}$
2	$\langle v_{4.6294}$ , $v_{3.6645}\rangle$	$\langle v_{4.7725}$ , $v_{3.5523}\rangle$	$\langle v_{4.4431}$ , $v_{3.6372}\rangle$	$\langle v_{4.0231}$ , $v_{3.8734}\rangle$	$\langle v_{4.4257}$ , $v_{3.7446}\rangle$	$\langle v_{4.6546}$ , $v_{3.5556}\rangle$	$\langle v_{4.4698}$ , $v_{3.8548}\rangle$
3	$\langle v_{4.6079}$ , $v_{3.6752}\rangle$	$\langle v_{4.7739}$ , $v_{3.5509}\rangle$	$\langle v_{4.4350}$ , $v_{3.6464}\rangle$	$\langle v_{4.0098}$ , $v_{3.8885}\rangle$	$\langle v_{4.4157}$ , $v_{3.7606}\rangle$	$\langle v_{4.6445}$ , $v_{3.5646}\rangle$	$\langle v_{4.4633}$ , $v_{3.8682}\rangle$
5	$\langle v_{4.6140}$ , $v_{3.6611}\rangle$	$\langle v_{4.6547}$ , $v_{3.6809}\rangle$	$\langle v_{4.3962}$ , $v_{3.6698}\rangle$	$\langle v_{4.0248}$ , $v_{3.8829}\rangle$	$\langle v_{4.3688}$ , $v_{3.8103}\rangle$	$\langle v_{4.5844}$ , $v_{3.5967}\rangle$	$\langle v_{4.4424}$ , $v_{3.8957}\rangle$
7	$\langle v_{4.5715}$ , $v_{3.6963}\rangle$	$\langle v_{4.7831}$ , $v_{3.5518}\rangle$	$\langle v_{4.4187}$ , $v_{3.6683}\rangle$	$\langle v_{3.9854}$ , $v_{3.9220}\rangle$	$\langle v_{4.4001}$ , $v_{3.7939}\rangle$	$\langle v_{4.6284}$ , $v_{3.5820}\rangle$	$\langle v_{4.4544}$ , $v_{3.8942}\rangle$
10	$\langle v_{4.5601}$ , $v_{3.7040}\rangle$	$\langle v_{4.7824}$ , $v_{3.5608}\rangle$	$\langle v_{4.4117}$ , $v_{3.6777}\rangle$	$\langle v_{3.9789}$ , $v_{3.9343}\rangle$	$\langle v_{4.3931}$ , $v_{3.8088}\rangle$	$\langle v_{4.6214}$ , $v_{3.5898}\rangle$	$\langle v_{4.4510}$ , $v_{3.9052}\rangle$
12	$\langle v_{4.5520}$ , $v_{3.7093}\rangle$	$\langle v_{4.7929}$ , $v_{3.5543}\rangle$	$\langle v_{4.4116}$ , $v_{3.6811}\rangle$	$\langle v_{3.9740}$ , $v_{3.9409}\rangle$	$\langle v_{4.3919}$ , $v_{3.8146}\rangle$	$\langle v_{4.6213}$ , $v_{3.5924}\rangle$	$\langle v_{4.4500}$ , $v_{3.9102}\rangle$
15	$\langle v_{4.5445}$ , $v_{3.7150}\rangle$	$\langle v_{4.7981}$ , $v_{3.5552}\rangle$	$\langle v_{4.4086}$ , $v_{3.6871}\rangle$	$\langle v_{3.9697}$ , $v_{3.9486}\rangle$	$\langle v_{4.3901}$ , $v_{3.8222}\rangle$	$\langle v_{4.6195}$ , $v_{3.5969}\rangle$	$\langle v_{4.4488}$ , $v_{3.9163}\rangle$
20	$\langle v_{4.5375}$ , $v_{3.7227}\rangle$	$\langle v_{4.8023}$ , $v_{3.5607}\rangle$	$\langle v_{4.4062}$ , $v_{3.6945}\rangle$	$\langle v_{3.9661}$ , $v_{3.9586}\rangle$	$\langle v_{4.3863}$ , $v_{3.8344}\rangle$	$\langle v_{4.6154}$ , $v_{3.6047}\rangle$	$\langle v_{4.4455}$ , $v_{3.9104}\rangle$

Figure 5.

Attributes weight according to different values of $\hslash$ .

Table 11

The score values $\mathcal{S}_{\textit{imp}}({\hat{\Phi}}_{i})$ and ranking of alternatives for various values of the parameter $\hslash$

$\hslash$	Score values							Ranking results
	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{1})$	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{2})$	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{3})$	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{4})$	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{5})$	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{6})$	$\mathcal{S}_{\textit{imp}}(\mathbb{K}_{7})$
2	2.3310	2.4624	2.2346	1.9246	2.1837	2.3896	2.1656	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
3	2.3142	2.4639	2.2265	1.9126	2.1720	2.3800	2.1569	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
5	2.3234	2.3393	2.1959	1.9221	2.1275	2.3317	2.1349	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
7	2.2848	2.4692	2.2090	1.8889	2.1507	2.3634	2.1420	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
10	2.2752	2.4650	2.2015	1.8814	2.1413	2.3562	2.1360	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
12	2.2685	2.4742	2.2001	1.8767	2.1384	2.3551	2.1335	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
15	2.2619	2.4770	2.1962	1.8719	2.1345	2.3522	2.1305	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$
20	2.2549	2.4773	2.1920	1.8667	2.1278	2.3467	2.1310	${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{4}}$

By examining these results, we can conclude that the proposed L^qROPF-Entropy-WASPAS decision methodology exhibits robustness across various values of the parameter $\hslash$ , indicating its broad applicability in addressing real-world decision problems within a qualitative information environment. This analysis underscores the stability and versatility of the proposed approach, thereby reinforcing its efficacy as a practical decision-making tool.

Figure 6.

Sensitivity analysis on the alternatives using varying values of $\hslash$ .

Figure 7.

Linear ranking orders for the alternatives based on the different values of $\hslash$ .

7.2 Validity test

In real-world complex and uncertain scenarios, identifying the optimal alternative for a given decision problem is a challenging task. Various decision-making approaches may yield distinct ranking orders of alternatives when presented with the same information data. Moreover, even slight variations in the available data can lead to changes in the ranking results. Consequently, decision-making approaches do not consistently provide reliable outcomes. Thus, it becomes imperative to assess the validity and reliability of any developed decision-making approach. To address this issue, [89] proposed a set of testing criteria to evaluate the validity of MCDM approaches. These criteria serve as a valuable framework for scrutinizing the efficacy of the decision-making methodology. The testing criteria are presented as follows:

•
Test criterion 1: “The optimal alternative should always remain unchanged when replacing a non-optimal alternative with a worse one without altering the relative importance of each decision criterion.”
•
Test criterion 2: “An effective decision-making approach should follow transitive property.”
•
Test criterion 3: “The final ranking order, after splitting the decision-making problems into the smaller ones, remains identical to the original ones.”

The efficiency and reliability of the formulated MAGDM approach are tested and discussed based on these test criteria as follows:
7.2.1 Validity test of the formulated approach with test criterion 1

To evaluate criterion 1 in our developed L^qROPF-Entropy-WASPAS approach, we will replace the non-optimal alternative $\mathcal{Q}_{5}$ with an arbitrary worst alternative $\widetilde{\mathcal{Q}}_{5}$ in the original decision matrices $A^{(k)}(k=1,2,3,4)$ for each expert. The assessment values of the alternative $\widetilde{\mathcal{Q}}_{5}$ are selected arbitrarily and recorded in Table 12.

Table 12
Assesment values of $\widetilde{\mathcal{Q}}_{5}$ for each expert

	${{\mathfrak{Y}}_{1}}$	${{\mathfrak{Y}}_{2}}$	${{\mathfrak{Y}}_{3}}$	${{\mathfrak{Y}}_{4}}$	${{\mathfrak{Y}}_{5}}$	${{\mathfrak{Y}}_{6}}$	${{\mathfrak{Y}}_{7}}$	${{\mathfrak{Y}}_{8}}$	${{\mathfrak{Y}}_{9}}$	${{\mathfrak{Y}}_{10}}$
${{{DE}}_{1}}$	$\langle v_{2},v_{6}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{1},v_{6}\rangle$	$\langle v_{2},v_{7}\rangle$	$\langle v_{4},v_{5}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{3},v_{4}\rangle$	$\langle v_{2},v_{5}\rangle$	$\langle v_{4},v_{6}\rangle$	$\langle v_{1},v_{4}\rangle$
${{{DE}}_{2}}$	$\langle v_{2},v_{5}\rangle$	$\langle v_{3},v_{6}\rangle$	$\langle v_{1},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{5},v_{6}\rangle$	$\langle v_{5},v_{3}\rangle$
${{{DE}}_{3}}$	$\langle v_{1},v_{6}\rangle$	$\langle v_{2},v_{5}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{2},v_{5}\rangle$	$\langle v_{4},v_{6}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{1},v_{6}\rangle$	$\langle v_{2},v_{6}\rangle$	$\langle v_{1},v_{5}\rangle$
${{{DE}}_{4}}$	$\langle v_{1},v_{6}\rangle$	$\langle v_{4},v_{4}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{3},v_{6}\rangle$	$\langle v_{3},v_{6}\rangle$	$\langle v_{4},v_{5}\rangle$	$\langle v_{1},v_{7}\rangle$	$\langle v_{3},v_{5}\rangle$	$\langle v_{2},v_{6}\rangle$

After applying our proposed L^qROPF-Entropy-WASPAS method with modified data, where $\hslash=2$ and $q=2$ , we obtain the final score values for the alternatives. The scores are as follows: $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{1})=2.3431$ , $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{2})=2.5686$ , $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{3})=2.285$ , $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{4})=1.8786$ , $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{5})=1.0541$ , $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{6})=2.4346$ , and $\mathcal{S}_{\textit{imp}}({\mathbb{K}}_{7})=2.2464$ . Based on these score values, the ranking order of the alternatives is determined as follows: ${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}\succ{\tilde{\mathcal{Q}}_{5}}$ . This ranking order indicates that the best alternative remains ${\mathcal{Q}_{2}}$ . Therefore, the results validate that our proposed approach satisfies test criterion 1.

7.2.2 Validity test of the formulated approach with test criteria 2 and 3

In order to evaluate the validity of the L^qROPF-Entropy-WASPAS approach based on test criteria 2 and 3, we employ a decomposition strategy to break down the original MAGDM problem into four sub-problems. Each sub-problem consists of a specific set of alternatives. These sets are denoted as $\{\mathcal{Q}_{1},\mathcal{Q}_{2},\mathcal{Q}_{3},\mathcal{Q}_{4}\}$ , $\{\mathcal{Q}_{3},\mathcal{Q}_{5},\mathcal{Q}_{6},\mathcal{Q}_{7}\}$ , $\{\mathcal{Q}_{4},\mathcal{Q}_{6},\mathcal{Q}_{7},\mathcal{Q}_{1}\}$ , and $\{\mathcal{Q}_{5},\mathcal{Q}_{7},\mathcal{Q}_{2},\mathcal{Q}_{3}\}$ . By applying the developed approach to each of these sub-problems, we obtain their respective ranking orders as follows: Sub-problem 1: $\mathcal{Q}_{2}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{3}\succ\mathcal{Q}_{4}$ Sub-problem 2: $\mathcal{Q}_{6}\succ\mathcal{Q}_{3}\succ\mathcal{Q}_{5}\succ\mathcal{Q}_{7}$ Sub-problem 3: $\mathcal{Q}_{6}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{7}\succ\mathcal{Q}_{4}$ Sub-problem 4: $\mathcal{Q}_{2}\succ\mathcal{Q}_{3}\succ\mathcal{Q}_{5}\succ\mathcal{Q}_{7}$ It is important to note that we have chosen the parameter values $\hslash=2$ and $q=2$ in this context. To obtain the overall ranking order of the alternatives, we combine all these individual rankings. The final ranking order is as follows: ${\mathcal{Q}_{2}}\succ{\mathcal{Q}_{6}}\succ{\mathcal{Q}_{1}}\succ{\mathcal{Q}% _{3}}\succ{\mathcal{Q}_{5}}\succ{\mathcal{Q}_{7}}\succ{\mathcal{Q}_{4}}$

Table 13
The score values/relative closeness coefficients and ranking results of alternatives based on existing methods

Methods	$\mathcal{Q}_{1}$	$\mathcal{Q}_{2}$	$\mathcal{Q}_{3}$	$\mathcal{Q}_{4}$	$\mathcal{Q}_{5}$	$\mathcal{Q}_{6}$	$\mathcal{Q}_{7}$	Ranking results
	Score values/relative closeness coefficients
Lin et al.’s method with L^qROFWA operator [33]	${6.0668}$	${6.4630}$	${6.0231}$	${5.8269}$	${5.9155}$	${6.1669}$	${5.9639}$	$\mathcal{Q}_{2}\succ\mathcal{Q}_{6}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{3}% \succ\mathcal{Q}_{7}\succ\mathcal{Q}_{5}\succ\mathcal{Q}_{4}$
Liu et al.’s TOPSIS method [53]	${0.5517}$	${0.6011}$	${0.4116}$	${0.2298}$	${0.4425}$	${0.5719}$	${0.5021}$	$\mathcal{Q}_{2}\succ\mathcal{Q}_{6}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{7}% \succ\mathcal{Q}_{5}\succ\mathcal{Q}_{3}\succ\mathcal{Q}_{4}$
Liu et al.’s method with L^qROPFWAG operator [56]	${5.7470}$	${5.8323}$	${5.7366}$	${5.6883}$	${5.7089}$	${5.7697}$	${5.7242}$	$\mathcal{Q}_{2}\succ\mathcal{Q}_{6}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{3}% \succ\mathcal{Q}_{7}\succ\mathcal{Q}_{5}\succ\mathcal{Q}_{4}$
Garg’s method with LPFWA operator [77]	${6.0668}$	${6.4630}$	${6.0231}$	${5.8269}$	${5.9155}$	${6.1669}$	${5.9639}$	$\mathcal{Q}_{2}\succ\mathcal{Q}_{6}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{3}% \succ\mathcal{Q}_{7}\succ\mathcal{Q}_{5}\succ\mathcal{Q}_{4}$
Zhao’s method with L^qROPFHWA operator [57]	${5.9885}$	${6.3210}$	${5.9468}$	${5.7402}$	${5.8224}$	${6.0777}$	${5.8957}$	$\mathcal{Q}_{2}\succ\mathcal{Q}_{6}\succ\mathcal{Q}_{1}\succ\mathcal{Q}_{3}% \succ\mathcal{Q}_{7}\succ\mathcal{Q}_{5}\succ\mathcal{Q}_{4}$

Based on this combined ranking, we conclude that the proposed approach is valid according to test criteria 2 and 3. In summary, the developed approach successfully addresses test criteria 2 and 3 by decomposing the MAGDM problem into subproblems, solving them individually using the approach, and combining the results to obtain the overall ranking order of the alternatives. The resulting ranking confirms the validity of the proposed L^qROPF-Entropy-WASPAS approach.

7.3 Comparative analysis

This section compares the proposed L^qROPF-Entropy-WASPAS methodology with some existing decision-making methods with linguistic information. We chose approaches with high efficacy for resolving MADM issues involving multiple experts proposed in the following references: [49], [53], [56], [77], and [57]. We apply all these methods to solve the MAGDM problem given in Example 10. Table 13 presents the obtained score values/relative closeness coefficients with ranking results of the alternatives.

Based on the findings presented in Table 13, it is evident that alternative $\mathcal{Q}_{2}$ obtains as the best alternative according to all the methods employed, which aligns perfectly with the best alternative identified by our suggested L^qROPF-Entropy-WASPAS approach.However, it should be acknowledged that there is a slight variation in the ranking order of the alternatives $\mathcal{Q}_{3}$ , $\mathcal{Q}_{5}$ , and $\mathcal{Q}_{7}$ when compared to our L^qROPF-Entropy-WASPAS technique.This variation in ranking can be attributed to the utilization of diverse AOs and ranking strategies by the existing approaches during the decision-making process. Additionally, we calculate the Wojciech Salabun (WS)-coefficient [90] to assess the concordance between ranking orders derived from various approaches. Notably, the WS coefficient is responsive to substantial changes in rankings. The WS-coefficient values for Lin et al.’s method with L^qROFWA operator [33], Liu et al.’s TOPSIS method [53], Liu et al.’s method with L^qROPFWAG operator [56], Garg’s method with LPFWA operator [77], and Zhao’s method with L^qROPFHWA operator [57] models with the proposed approach are determined as 0.9891, 0.9625, 0.9891, 0.9891, and 0.9891, respectively. It’s worth noting that the WS-coefficient exceeds 0.98 for all existing methods except for Liu et al.’s TOPSIS method [53]. This underscores the substantial ranking similarity between the devised L^qROPF-Entropy-WASPAS method and other approaches, validating its efficiency in addressing MAGDM problems with linguistic $q$ -rung orthopair fuzzy information through the introduced score function and AOs. As a result, it can be inferred that a highly robust correlation exists among preference outcomes.

Consequently, we can deduce that our suggested L^qROPF-Entropy-WASPAS approach yields consistent results and demonstrates its validity and practicality in addressing MAGDM problems within real-world scenarios. By utilizing the L^qROPF-Entropy-WASPAS method, decision-makers can effectively navigate the complexities associated with MAGDM and arrive at reasonable and practical solutions that align with their objectives and constraints. The robustness of our L^qROPF-Entropy-WASPAS approach in identifying the best alternative, as corroborated by the unanimous selection of $\mathcal{Q}_{2}$ across all methods, reinforces its efficacy and highlights its suitability for real-world decision-making contexts.

8. Conclusion

This paper primarily aims to investigate decision-making within the L^qROPF setting thoroughly. In order to achieve this objective, we have introduced an improved score function to rank L^qROPFNs more efficiently. Furthermore, we have established the Frank operational rules for L^qROPFNs and examined their characteristics. The study has utilized the Frank operations to introduce various arithmetic and geometric AOs for combining different L^qROPFNs. These operators include the L^qROPFFWA operator, L^qROPFFWG operator, L^qROPFFOWA operator, and L^qROPFFOWG operator, each possessing distinct characteristics. It is crucial to acknowledge that these AOs provide a more adaptable and comprehensive method for combining L^qROPFNs by adjusting the parameter $\hslash$ . In addition, the L^qROPF-Entropy-WASPAS methodology has been designed to address MAGDM problems within the L^qROPF framework. Ultimately, we have implemented the recommended methodology to address a practical decision-making issue regarding choosing an online educational platform to showcase the decision-making process and adaptability of the proposed approach. An analysis has also been conducted on how the decision parameter $\hslash$ affects the ranking outcome. To summarize, the suggested score function, AOs, and decision-making technique provide several significant benefits compared to current methodologies, which can be described as follows:

•

The given score function can establish a ranking order among the L^qROPFNs more efficiently without using the accuracy function.

•

The proposed AOs can fuse a wide range of linguistic information by taking an appropriate value of parameter $q$ .

•

We can consider several aggregation approaches by assigning different values of the parameter $\hslash$ according to the performance of the experts.

•

The designed AOs can also be utilized to aggregate the linguistic intuitionistic fuzzy information (LIFI) and the linguistic Pythagorean fuzzy information (LPFI) in a single formation.

•

The developed L^qROPF-Entropy-WASPAS decision-making model provides more flexibility to the experts for analyzing the decision result under different situations.

In future research, we will utilize the proposed AOs across various fields, encompassing game theory, clustering analysis, image processing, and pattern recognition. Our objective is to assess the adaptability and reliability of the AOs by evaluating their performance in these domains. Additionally, we aim to augment the existing AOs by integrating additional capabilities and features, such as confidence degree, probability, and interaction information. Moreover, our research will explore the potential applications of the developed MAGDM approach in various areas, including environmental management, transportation and logistics, urban planning, renewable energy management, and the agriculture industry.

Data availability statement

The author verify that the data supporting the findings of this study are accessible within the article.

Ethical statement

This article does not contain any studies with human participants or animals performed by the author.

Footnotes

Acknowledgments

Rajkumar Verma expresses gratitude for the financial support received from the Universidad de Talca through FONDO PARA ATRACCIÓN DE INVESTIGADORES POSTDOCTORALES.

Conflict of interest

The author confirm that there are no conflicts of interest to declare.

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A novel L q ROPF-Entropy-WASPAS group model based on Frank aggregation operators and improved score function in linguistic q -rung orthopair fuzzy framework

Abstract

Keywords

1. Introduction

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Proof..

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Proof..

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Proof..

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Proof..

5.1 Frank weighted averaging AOs for LqROPFNs

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Proof..

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Proof..

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Proof..

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Proof..

Problem description

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7.1 Sensitivity analysis

Table 9 The weights ( w j ) corresponding to attributes ( 𝔜 j ) for different values of ℏ

Table 12 Assesment values of 𝒬 ∼ 5 for each expert

Table 13 The score values/relative closeness coefficients and ranking results of alternatives based on existing methods

8. Conclusion

Data availability statement

Ethical statement

Footnotes

Acknowledgments

Conflict of interest

References

5.1 Frank weighted averaging AOs for L^qROPFNs

Table 9
The weights $(w_{j})$ corresponding to attributes $(\mathfrak{Y}_{j})$ for different values of $\hslash$

Table 12
Assesment values of $\widetilde{\mathcal{Q}}_{5}$ for each expert

Table 13
The score values/relative closeness coefficients and ranking results of alternatives based on existing methods