Some q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean aggregation operators and their applications to multiple-attribute group decision making

Multiple-attribute group decision making (MAGDM) under linguistic environment is an important part of modern decision sciences, and information aggregation operator plays an import role in solving MAGDM problems. In this paper, an approach for solving MAGDM problem with q-rung orthopair fuzzy 2-tuple linguistic information is developed. First, the q-rung orthopair fuzzy 2-tuple linguistic weighted averaging (q-ROFTLWA) operator and the q-rung orthopair fuzzy 2-tuple linguistic weighted geometric (q-ROFTLWG) operator are presented. Furthermore, the q-rung orthopair fuzzy 2-tuple linguistic Muirhead mean (q-ROFTLMM) operator and the q-rung orthopair fuzzy 2-tuple linguistic dual Muirhead mean (q-ROFTLDMM) operator are proposed on the basis of Muirhead mean (MM) operator and dual Muirhead mean (DMM) operator. Then, an approach is developed to deal with MAGDM problem under q-rung orthopair fuzzy 2-tuple linguistic environment based on the proposed operators. Finally, a numerical example for selecting desirable emergency alternative(s) in the process of designing emergency preplan is given to illustrate the application of the developed method and demonstrate its effectiveness.     

A novel multiple-attribute group decision-making method based on q-rung orthopair fuzzy generalized power weighted aggregation operators

In this paper, a novel approach is developed to deal with multiple-attribute group decision-making (MAGDM) problem under q-rung orthopair fuzzy environment. Firstly, some operators have been proposed to aggregate q-rung orthopair fuzzy information, such as the q-rung orthopair fuzzy generalized power averaging (q-ROFGPA) operator, the q-rung orthopair fuzzy generalized power weighted averaging (q-ROFGPWA) operator, the q-rung orthopair fuzzy generalized power geometric (q-ROFGPG) operator, and the q-rung orthopair fuzzy generalized power weighted geometric (q-ROFGPWG) operator. In addition, some desirable properties and special cases of these operators are discussed. Second, a novel approach is developed to solve MAGDM problem under the q-rung orthopair fuzzy environment based on the proposed q-ROFGPWA and q-ROFGPWG operators. Finally, a practical example is given to illustrate the application of the proposed method, and further the sensitivity analysis and comparative analysis are carried out.     

Some q‐Rung Orthopai Fuzzy Bonferroni Mean Operators and Their Application to Multi‐Attribute Group Decision Making

In the real multi‐attribute group decision making (MAGDM), there will be a mutual relationship between different attributes. As we all know, the Bonferroni mean (BM) operator has the advantage of considering interrelationships between parameters. In addition, in describing uncertain information, the eminent characteristic of q‐rung orthopair fuzzy sets (q‐ROFs) is that the sum of the qth power of the membership degree and the qth power of the degrees of non‐membership is equal to or less than 1, so the space of uncertain information they can describe is broader. In this paper, we combine the BM operator with q‐rung orthopair fuzzy numbers (q‐ROFNs) to propose the q‐rung orthopair fuzzy BM (q‐ROFBM) operator, the q‐rung orthopair fuzzy weighted BM (q‐ROFWBM) operator, the q‐rung orthopair fuzzy geometric BM (q‐ROFGBM) operator, and the q‐rung orthopair fuzzy weighted geometric BM (q‐ROFWGBM) operator, then the MAGDM methods are developed based on these operators. Finally, we use an example to illustrate the MAGDM process of the proposed methods. The proposed methods based on q‐ROFWBM and q‐ROFWGBM operators are very useful to deal with MAGDM problems.     

Some q-rung orthopair uncertain linguistic aggregation operators and their application to multiple attribute group decision making

q-Rung orthopair fuzzy sets (q-ROFSs), originally presented by Yager, are a powerful fuzzy information representation model, which generalize the classical intuitionistic fuzzy sets and Pythagorean fuzzy sets and provide more freedom and choice for decision makers (DMs) by allowing the sum of the $q\mathrm{t}\mathrm{h}$ power of the membership and the $q\mathrm{t}\mathrm{h}$ power of the nonmembership to be less than or equal to 1. In this paper, a new class of fuzzy sets called q-rung orthopair uncertain linguistic sets (q-ROULSs) based on the q-ROFSs and uncertain linguistic variables (ULVs) is proposed, and this can describe the qualitative assessment of DMs and provide them more freedom in reflecting their belief about allowable membership grades. On the basis of the proposed operational rules and comparison method of q-ROULSs, several q-rung orthopair uncertain linguistic aggregation operators are developed, including the q-rung orthopair uncertain linguistic weighted arithmetic average operator, the q-rung orthopair uncertain linguistic ordered weighted average operator, the q-rung orthopair uncertain linguistic hybrid weighted average operator, the q-rung orthopair uncertain linguistic weighted geometric average operator, the q-rung orthopair uncertain linguistic ordered weighted geometric operator, and the q-rung orthopair uncertain linguistic hybrid weighted geometric operator. Then, some desirable properties and special cases of these new operators are also investigated and studied, in particular, some existing intuitionistic fuzzy aggregation operators and Pythagorean fuzzy aggregation operators are proved to be special cases of these new operators. Furthermore, based on these proposed operators, we develop an approach to solve the multiple attribute group decision making problems, in which the evaluation information is expressed as q-rung orthopair ULVs. Finally, we provide several examples to illustrate the specific decision-making steps and explain the validity and feasibility of two methods by comparing with other methods.     

q-Rung orthopair fuzzy sets-based decision-theoretic rough sets for three-way decisions under group decision making

As an extension of Pythagorean fuzzy sets, the q-rung orthopair fuzzy sets (q-ROFSs) can easily solve uncertain information in a broader perspective. Considering the fine property of q-ROFSs, we introduce q-ROFSs into decision-theoretic rough sets (DTRSs) and use it to portray the loss function. According to the Bayesian decision procedure, we further construct a basic model of q-rung orthopair fuzzy decision-theoretic rough sets (q-ROFDTRSs) under the q-rung orthopair fuzzy environment. At the same time, we design the corresponding method for the deduction of three-way decisions by utilizing projection-based distance measures and TOPSIS. Then, we extend q-ROFDTRSs to adapt the group decision-making (GDM) scenario. To fuse different experts’ evaluation results, we propose some new aggregation operators of q-ROFSs by utilizing power average (PA) and power geometric (PG) operators, that is, q-rung orthopair fuzzy power average, q-rung orthopair fuzzy power weighted average (q-ROFPWA), q-rung orthopair fuzzy power geometric, and q-rung orthopair fuzzy power weighted geometric (q-ROFPWG). In addition, with the aid of q-ROFPWA and q-ROFPWG, we investigate three-way decisions with q-ROFDTRSs under the GDM situation. Finally, we give the example of a rural e-commence GDM problem to illustrate the application of our proposed method and verify our results by conducting two comparative experiments.     

Confidence levels q-rung orthopair fuzzy aggregation operators and its applications to MCDM problems

The concept of q-rung orthopair fuzzy set (q-ROFS) is the extension of intuitionistic fuzzy set (IFS) in which the sum of the qth power of the support for and the qth power of the support against is bounded by one. Therefore, the q-ROFSs are an important way to express uncertain information in broader space, and they are superior to the IFSs and the Pythagorean fuzzy sets. In this paper, the familiarity degree of the experts with the evaluated objects is incorporated to the initial assessments under q-rung orthopair fuzzy environment. For this, some aggregation operators are proposed to combine these two types of information. Their some important properties are also well proved. Furthermore, these developed operators are utilized in a multicriteria decision-making approach and demonstrated with a real life problem of customers' choice. Then, the experimental results are compared with other existing methods to show its superiority over recent research works.     

Single variable differential calculus under q-rung orthopair fuzzy environment: Limit,derivative, chain rules,and its application

The q-rung orthopair fuzzy set ( q-ROFS) that the sum of the qth power of the membership degree and the qth power of the nonmembership degree is restricted to one is a generalization of fuzzy set (FS). Recently, many researchers have given a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. Subsequently, although some scholars have also focused on studying q-rung orthopair fuzzy continuous information and give its continuity, derivative, differential, and integral, those studies are only considered from the perspective of multivariable fuzzy functions. Thus, the main aim of the paper is to study the q-rung orthopair fuzzy continuous single variable information. In this paper, we first define the concept of q-rung orthopair single variable fuzzy function ( q-ROSVFF) to describe the fuzzy continuous information, and give its domain to make sure that this kind of function is meaningful. Afterward, we propose the limits, continuities, and infinitesimal of q-ROSVFFs, and offer the relationship between the limit of q-ROSVFF and that of q-ROSVFF infinitesimal. On the basis of the definition of derivative in mathematical analysis, we define the subtraction and division derivatives and basic operational rules, and offer the simpler proofs for the derivatives of q-ROSVFFs. What is more, we propose the subtraction and division differential invariances, and give the approximate calculation formulas of q-ROSVFFs when the value of independent variable is changed small enough. In the real situation, fundamental functions cannot be used to express more complicated functions, thus we define the compound q-ROSVFFs and give their chain rules of subtraction and division derivatives. Finally, we use numerical examples by simulation to verify the feasibility and veracity of the approximate calculation on q-ROSVFFs.     

Multi-attribute group decision-making methods based on q-rung orthopair fuzzy linguistic sets

With the continuous development of the economy and society, decision-making problems and decision-making scenarios have become more complex. The q-rung orthopair fuzzy set is getting more and more attention from researchers, which is more general and flexible than Pythagorean fuzzy set and intuitionistic fuzzy set under complex vague environment. In this study, the concept of q-rung orthopair fuzzy linguistic set (q-ROFLS) is proposed and a new q-rung orthopair fuzzy linguistic method is developed to handle MAGDM problem. Firstly, the conception, operation laws, comparison methods, and distance measure methods of the q-ROFLS are proposed. Secondly, the q-ROFL weighted average operator, q-ROFL ordered weighted average operator, q-ROFL hybrid weighted average operator, q-ROFL weighted geometric operator, q-ROFL ordered weighted geometric operator, and q-ROFL hybrid weighted geometric operator are proposed, and some interesting properties, special cases of these operators are investigated. Furthermore, a new method to cope with MAGDM problem based on q-ROFL weighted average operator (q-ROFL weighted geometric operator) is developed. Finally, a practical example for suppliers selection is provided to verify the practicality of the presented method, and the effectiveness and flexibility of the presented method are illustrated by sensitive analysis and comparative analysis.     

Some q‐Rung Orthopair Fuzzy Aggregation Operators and their Applications to Multiple‐Attribute Decision Making

The q‐rung orthopair fuzzy sets (q‐ROFs) are an important way to express uncertain information, and they are superior to the intuitionistic fuzzy sets and the Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the membership degree and the qth power of the degrees of non‐membership is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we propose the q‐rung orthopair fuzzy weighted averaging operator and the q‐rung orthopair fuzzy weighted geometric operator to deal with the decision information, and their some properties are well proved. Further, based on these operators, we presented two new methods to deal with the multi‐attribute decision making problems under the fuzzy environment. Finally, we used some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.     

Additive consistency-based priority-generating method of q-rung orthopair fuzzy preference relation

The q-rung orthopair fuzzy set, whose membership function and nonmembership function belong to the interval [0,1], is more powerful than both intuitionistic fuzzy set and Pythagorean fuzzy set in expressing imprecise information of decision-makers. The aim of this paper is to investigate a method to determine the priority weights from individual or group q-rung orthopair fuzzy preference relations (q-ROFPRs). To do so, firstly, a new definition of additively consistent q-ROFPR is presented based on the preference relation of alternatives given by decision-makers. Afterward, according to individual and group q-ROFPRs, two kinds of goal programming models are proposed, respectively, to generate the q-rung orthopair fuzzy priority weight vector of the given q-ROFPR(s). Finally, two numerical examples are given to illustrate the effectiveness and superiority of the method proposed in this paper.     

Some Dombi aggregation of Q-rung orthopair fuzzy numbers in multiple-attribute decision making

The operations of $t$-norm (TN) and $t$-conorm (TCN), developed by Dombi, are generally known as Dombi operations, which have an advantage of flexibility within the working behavior of parameter. In this paper, we use Dombi operations to construct a few Q-rung orthopair fuzzy Dombi aggregation operators: Q-rung orthopair fuzzy Dombi weighted average operator, Q-rung orthopair fuzzy Dombi order weighted average operator, Q-rung orthopair fuzzy Dombi hybrid weighted average operator, Q-rung orthopair fuzzy Dombi weighted geometric operator, Q-rung orthopair fuzzy Dombi order weighted geometric operator, and Q-rung orthopair fuzzy Dombi hybrid weighted geometric operator. The different features of these proposed operators are reviewed. At that point, we have used these operators to build up a model to solve the multiple-attribute decision making issues under Q-rung orthopair fuzzy environment. Ultimately, a realistic instance is stated to substantiate the created model and to exhibit its applicability and viability.     

New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making

The q-rung orthopair fuzzy sets are superior to intuitionistic fuzzy sets or Pythagorean fuzzy sets in expressing fuzzy and uncertain information. In this paper, some partitioned Bonferroni means (BMs) for q-rung orthopair fuzzy values have been developed. First, the q-rung orthopair fuzzy partitioned BM (q-ROFPBM) operator and the q-rung orthopair fuzzy partitioned geometric BM (q-ROFPGBM) operator are developed. Some desirable properties and some special cases of the new aggregation operators have been studied. The q-rung orthopair fuzzy weighted partitioned BM (q-ROFWPBM) operator and the q-rung orthopair fuzzy partitioned geometric weighted BM (q-ROFPGWBM) operator are also developed. Then, a new multiple-attribute decision-making method based on the q-ROFWPBM (q-ROFPGWBM) operator is proposed. Finally, a numerical example of investment company selection problem is given to illustrate feasibility and practical advantages of the new method.     

Complex q-rung orthopair fuzzy 2-tuple linguistic Maclaurin symmetric mean operators and its application to emergency program selection

This essay designs an innovate approach to work out linguistic multiattribute group decision-making (MAGDM) issues with complex q-rung orthopair fuzzy 2-tuple linguistic (Cq-ROF2TL) evaluation information. To begin with, the conception of Cq-ROF2TL set is propounded to express uncertain and fuzzy assessment information. Meanwhile, the score and accuracy function, a comparison approach, Cq-ROF2TL weighted averaging, and Cq-ROF2TL weighted geometric operator are put forward. Furthermore, to take into consideration the correlation among multiple input data, the Cq-ROF2TL Maclaurin symmetric mean (MSM) operator, the Cq-ROF2TL dual MSM operator and their weighted forms are presented. Several attractive characteristics and particular instances of the developed operators are also explored at length. Later, an innovative MAGDM methodology is designed based upon the propounded operators to settle the emergency program evaluation issue under the Cq-ROF2TL circumstance. Consequently, the efficiency and outstanding superiority of the created approach are severally substantiated by parameter exploration and detailed comparative analysis.     

Some power Maclaurin symmetric mean aggregation operators based on Pythagorean fuzzy linguistic numbers and their application to group decision making

The power average (PA) operator and Maclaurin symmetric mean (MSM) operator are two important tools to handle the multiple attribute group decision‐making (MAGDM) problems, and the combination of two operators can eliminate the influence of unreasonable information from biased decision makers (DMs) and can capture the interrelationship among any number of arguments. The Pythagorean fuzzy linguistic set (PFLS) is parallel to the intuitionistic linguistic set (ILS), which is more powerful to convey the uncertainty and ambiguity of the DMs than ILS. In this paper, we propose some power MSM aggregation operators for Pythagorean fuzzy linguistic information, such as Pythagorean fuzzy linguistic power MSM operator and Pythagorean fuzzy linguistic power weighted MSM (PFLPWMSM) operator. At the same time, we further discuss the properties and special cases of these operators. Then, we propose a new method to solve the MAGDM problems with Pythagorean fuzzy linguistic information based on the PFLPWMSM operator. Finally, some illustrative examples are utilized to show the effectiveness of the proposed method.     

A decision-making algorithm for online shopping using deep-learning–based opinion pairs mining and q-rung orthopair fuzzy interaction Heronian mean operators

In the process of online shopping, consumers usually compare the review information of the same product in different e-commerce platforms. The sentiment orientation of online reviews from different platforms interactively influences on consumers’ purchase decision. However, due to the limitation of the ability to process information manually, it is difficult for a consumer to accurately identify the sentiment orientation of all reviews one by one and describe the process of their interactive influence. To this end, we proposed an online shopping support model using deep-learning–based opinion mining and q-rung orthopair fuzzy interaction weighted Heronian mean (q-ROFIWHM) operators. First, in the proposed method, the deep-learning model is used to automatically extract different product attribute words and opinion words from online reviews, and match the corresponding attribute-opinion pairs; meanwhile, the sentiment dictionary is used to calculate sentiment orientation, including positive, negative, and neutral sentiments. Second, the proportions of the three kinds of sentiments about each attribute of the same product are calculated. According to the proportion value of attribute sentiment from different platforms, the sentiment information is converted into multiple cross-decision matrices, which are represented by the q-rung orthopair fuzzy set. Third, considering the interactive characteristics of decision matrix, the q-ROFIWHM operators are proposed to aggregate this cross-decision information, and then the ranking result was determined by score function to support consumers' purchase decisions. Finally, an actual example of mobile phone purchase is given to verify the rationality of the proposed method, and the sensitivity and the comparison analysis are used to show its effectiveness and superiority.     

区间值广义正交模糊Frank算子及群决策方法

基于区间值广义正交模糊环境和Frank算子,定义了区间值广义正交模糊Frank算子的运算法则,提出了区间值广义正交模糊Frank加权平均算子(IVq-ROFFWA)和加权几何算子(IVq-ROFFWG),并研究了它们的幂等性、有界性和单调性。然后提出了基于IVq-ROFFWA算子的多属性群决策方法(MAGDM),该方法通过选取满足条件的q值,使用IVq-ROFFWA算子集结得到目标区间值模糊数,比较它们的得分得到最优方案,还得出了不同q值不影响最优方案排序的结论。最后通过实际案例验证了基于IVq-ROFFWA算子的多属性群决策方法的可行性和有效性,验证了不同q值不影响最优方案排序的结论。经过比较分析,基于IVq-ROFFWA算子和IVq-ROFFWG算子的群决策方法与基于其他算子的群决策方法运算结果一致。     

Undergraduate teaching audit and evaluation using an extended MABAC method under q-rung orthopair fuzzy environment

Undergraduate teaching audit and evaluation (UTAE) is a new type of evaluation pattern, which is extremely important for a university to improve its quality assurance system and enhance teaching quality. Selecting an optimal university for benchmarking through UTAE to promote the quality of teaching can be regarded as a complex multicriteria decision making (MCDM) problem. Furthermore, in the process of UTAE, experts' evaluations over the teaching quality of universities are often imprecise and fuzzy due to the subjective nature of human thinking. In this paper, we propose a new UTAE approach based on q-rung orthopair fuzzy sets and the multiattribute border approximation area comparison (MABAC) method for evaluating and selecting the best university for benchmarking. The introduced method deals with the linguistic assessments given by experts by using q-ROFSs, assigns the weights of audit elements based on the indifference threshold-based attribute ratio analysis method, and acquires the ranking of universities with an extended MABAC method. The feasibility and effectiveness of the proposed q-rung orthopair fuzzy MABAC method is demonstrated through a realistic UTAE example. Results show that the UTAE method being proposed is valid and practical for UTAE.     

Research on the assessment of classroom teaching quality with q-rung orthopair fuzzy information based on multiparametric similarity measure and combinative distance-based assessment

The assessment of classroom teaching quality is critically important for producing a positive incentive and guidance role to improve service and management of universities, stimulating the enthusiasm of teachers, enhancing the teacher’s teaching ability, and improving the quality of talent training. In considering the case of teaching quality evaluation, the essential question that arises concerns strong ambiguity, fuzziness, and inexactness. The q-rung orthopair fuzzy sets ( q-ROFSs) dealing the indeterminacy characterized by membership degrees and nonmembership degrees are a more flexible and effective way to capture indeterminacy. In this paper, firstly, the new score function for q-rung orthopair fuzzy number is initiated for tackling the comparison problem. Subsequently, a new distance measure for q-ROFSs with multiple parameters is studied along with their detailed proofs. The various desirable properties among the developed similarity measures and distance measures have also been derived. Then, the objective weights of various attributes are determined via antientropy weighting method. Also, we develop the combined weights, which can reveal both the subjective information and the objective information. Moreover, two algorithms to solve q-rung orthopair fuzzy decision-making problem by combinative distance-based assessment and multiparametric similarity measure are presented. Later, the feasibility of approaches is demonstrated by a classroom teaching quality problem, along with the effect of the different parameters on the ordering. Finally, a comparison between the proposed and the existing decision-making methods has been performed for showing their effectiveness. The salient features of the proposed methods, compared to the existing q-ROFS decision-making methods, are as follows: (a) it can obtain the optimal alternative without counterintuitive phenomena and (b) it has a lower computational complexity.     

Some q‐rung orthopair fuzzy Heronian mean operators in multiple attribute decision making

The generalized Heronian mean and geometric Heronian mean operators provide two aggregation operators that consider the interdependent phenomena among the aggregated arguments. In this paper, the generalized Heronian mean operator and geometric Heronian mean operator under the q‐rung orthopair fuzzy sets is studied. First, the q‐rung orthopair fuzzy generalized Heronian mean (q‐ROFGHM) operator, q‐rung orthopair fuzzy geometric Heronian mean (q‐ROFGHM) operator, q‐rung orthopair fuzzy generalized weighted Heronian mean (q‐ROFGWHM) operator, and q‐rung orthopair fuzzy weighted geometric Heronian mean (q‐ROFWGHM) operator are proposed, and some of their desirable properties are investigated in detail. Furthermore, we extend these operators to q‐rung orthopair 2‐tuple linguistic sets (q‐RO2TLSs). Then, an approach to multiple attribute decision making based on q‐ROFGWHM (q‐ROFWGHM) operator is proposed. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.