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.     

Multiple-attribute group decision-making based on power Bonferroni operators of linguistic q-rung orthopair fuzzy numbers

In this paper, a new conception of linguistic q-rung orthopair fuzzy number (Lq-ROFN) is proposed where the membership and nonmembership of the q-rung orthopair fuzzy numbers ( q-ROFNs) are represented as linguistic variables. Compared with linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers, the Lq-ROFNs can more fully describe the linguistic assessment information by considering the parameter q to adjust the range of fuzzy information. To deal with the multiple-attribute group decision-making (MAGDM) problems with Lq-ROFNs, we proposed the linguistic score and accuracy functions of the Lq-ROFNs. Further, we introduce and prove the operational rules and the related properties characters of Lq-ROFNs. For aggregating the Lq-ROFN assessment information, some aggregation operators are developed, involving the linguistic q-rung orthopair fuzzy power Bonferroni mean (BM) operator, linguistic q-rung orthopair fuzzy weighted power BM operator, linguistic q-rung orthopair fuzzy power geometric BM (GBM) operator, and linguistic q-rung orthopair fuzzy weighted power GBM operator, and then presents their rational properties and particular cases, which cannot only reduce the influences of some unreasonable data caused by the biased decision-makers, but also can take the interrelationship between any two different attributes into account. Finally, we propose a method to handle the MAGDM under the environment of Lq-ROFNs by using the new proposed operators. Further, several examples are given to show the validity and superiority of the proposed method by comparing with other existing MAGDM methods.     

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.     

Differential calculus of interval-valued q-rung orthopair fuzzy functions and their applications

The interval-valued q-rung orthopair fuzzy set (IVq-ROFS) provides an extension of Yager's q-rung orthopair fuzzy set (q-ROFS), where membership and nonmembership degrees are subsets of the closed interval $\left[0,1\right]$. In such a situation, it is more superior for decision makers to provide their judgments by intervals instead of crisp numbers due to the uncertainty and vagueness in real life. In this paper, we study the calculus theories of IVq-ROFS from the microscopic. In particular, we first introduce the elementary arithmetic of interval-valued q-rung orthopair fuzzy values (IVq-ROFVs), including addition, multiplication, and their inverse. They are the basis for analysis and calculation throughout the work. In addition, we discuss and prove in detail the operation properties and aggregation operators of IVq-ROFVs. Then, we introduce the concept of interval-valued q-rung orthopair fuzzy functions (IVq-ROFFs), which is the main research object of this paper. After that, we further discuss the continuity, derivatives and differentials of IVq-ROFFs. We also find that the derivatives of IVq-ROFFs are closely related to elasticity, which is an important concept in economics. Finally, we provide some application examples to verify the feasibility and effectiveness of the derived results.     

Research on arithmetic operations over generalized orthopair fuzzy sets

The four fundamental operations of arithmetic for real (and complex) numbers are well known to everybody and quite often used in our daily life. And they have been extended to classical and generalized fuzzy environments with the demand of practical applications. In this paper, we present the arithmetic operations, including addition, subtraction, multiplication, and division operations, over -rung orthopair membership grades, where subtraction and division operations are both defined in two different ways. One is by solving the equation involving addition or multiplication operations, whereas the other is by determining the infimum or supremum of solutions of the corresponding inequality. Not all of -rung orthopairs can be performed by the former method but by the latter method, and it is proved that the former is a special case of the latter. Moreover, the elementary properties of arithmetic operations as well as mixed operations are extensively investigated. Finally, these arithmetic operations are pointwise defined on -rung orthopair fuzzy sets in which the membership degree of each element is a -rung orthopair.     

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.     

Correlation and correlation coefficient of generalized orthopair fuzzy sets

Generalized orthopair fuzzy sets are extensions of ordinary fuzzy sets by relaxing restrictions on the degrees of support for and support against. Correlation analysis is to measure the statistical relationships between two samples or variables. In this paper, we propose a function measuring the interrelation of two -rung orthopair fuzzy sets, whose range is the unit interval . First, the correlation and correlation coefficient of -rung orthopair membership grades are presented, and their basic properties are investigated. Second, these concepts are extended to -rung orthopair fuzzy sets on discrete universes. Then, we discuss their applications in cluster analysis under generalized orthopair fuzzy environments. And, a real-world problem involving the evaluation of companies is used to illustrate the detailed processes of the clustering algorithm. Finally, we introduce the correlation and correlation coefficient of -rung orthopair fuzzy sets on both bounded and unbounded continuous universes and provide some numerical examples to substantiate such arguments.     

Some preference relations based on q-rung orthopair fuzzy sets

As a generalization of intuitionistic fuzzy sets and Pythagorean fuzzy sets, $q$-rung orthopair fuzzy sets provide decision makers more flexible space in expressing their opinions. Preference relations have received widespread acceptance as an efficient tool in representing decision makers’ preference over alternatives in the decision-making process. In this paper, some new preference relations are investigated based on the $q$-rung orthopair fuzzy sets. First, a novel score function is presented for ranking $q$-rung orthopair fuzzy numbers. Second, $q$-rung orthopair fuzzy preference relation, consistent $q$-rung orthopair fuzzy preference relation, incomplete $q$-rung orthopair fuzzy preference relation, consistent incomplete $q$-rung orthopair fuzzy preference relation, and acceptable incomplete $q$-rung orthopair fuzzy preference relation are defined. In the end, based on the new score function and these preference relations, some algorithms are constructed for ranking and selection of the decision-making alternatives.     

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.     

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.     

q-Rung orthopair fuzzy Choquet integral aggregation and its application in heterogeneous multicriteria two-sided matching decision making

In the real decision making, $q$-rung orthopair fuzzy sets ($q$-ROFSs) as a novel effective tool can depict and handle uncertain information in a broader perspective. Considering the interrelationships among the criteria, this paper extends Choquet integral to the $q$-rung orthopair fuzzy environment and further investigates its application in multicriteria two-sided matching decision making. To determine the fuzzy measures used in Choquet integral, we first define a pair of $q$-rung orthopair fuzzy entropy and cross-entropy. Then, by utilizing $\lambda$-fuzzy measure theory, we propose an entropy-based method to calculate the fuzzy measures upon criteria. Furthermore, we discuss $q$-rung orthopair fuzzy Choquet integral operator and its properties. Thus, with the aid of $q$-rung orthopair fuzzy Choquet integral, we consider the preference heterogeneity of the matching subjects and further explore the corresponding generalized model and approach for the two-sided matching. Finally, a simulated example of loan market matching is given to illustrate the validity and applicability of our proposed approach.     

Neutrality aggregation operators based on complex q-rung orthopair fuzzy sets and their applications in multiattribute decision-making problems

Complex q-rung orthopair fuzzy sets (CQROFSs) are proposed to convey vague material in decision-making problems. The CQROFSs can enthusiastically modify the region of proof by altering the factor $q\ge 1$ for real and imaginary parts based on the variation degree and, therefore, favor further uncountable options. Consequently, this set reverses over the existing theories, such as complex intuitionistic fuzzy sets (CIFSs) and complex Pythagorean fuzzy sets (CPFS). In everyday life, there are repeated situations that can occur, which involve an impartial assertiveness of the decision-makers. To determine the best decision to handle such situations, in this study, we propose modern operational laws by joining the characteristics of the truth factor sum and collaboration between the truth degrees into the analysis for CQROFSs. Based on these principles, we determined several weighted averaging neutral aggregation operators (AOs) to collect the CQROF knowledge. Subsequently, we established an original multiattribute decision-making (MADM) procedure by using the demonstrated AOs based on CQROFS. To evaluate the effectiveness, in terms of reliability and consistency, of the proposed operators, they were applied to some numerical examples. A comparative analysis of the investigated operators and other existing operators was also performed to find the dominance and validity of the introduced MADM method.     

Generalized orthopair fuzzy weighted distance-based approximation (WDBA) algorithm in emergency decision-making

With the intensification of global warming trends, the frequent occurrence of natural disasters has brought severe challenges to the sustainable development of society. Emergency decision-making (EDM) in natural disasters is playing an increasingly important role in improving disaster response capacity. In the case of EDM evaluation, the essential problem arises serious incompleteness, impreciseness, subjectivity, and incertitude. The q-rung orthopair fuzzy set (q-ROFS), disposing the indeterminacy portrayed by membership and nonmembership with the sum of $q$th power of them, is a more viable and effective means to seize indeterminacy. The aim of paper is to present a new score function of q-rung orthopair fuzzy number (q-ROFN) for solving the failure problems when comparing two q-ROFNs. Firstly, we introduce some basic set operations for q-ROFS. The properties of these operations are also discussed in detail. Later, we propose a q-rung orthopair fuzzy decision-making method based on weighted distance-based approximation (WDBA), in which the weights of decision-makers are obtained from a nonliner optimization model according to the deviation-based method. Finally, some examples are investigated to illustrate the feasibility and validity of the proposed approach. The salient features of the proposed method, compared to the existing q-rung orthopair fuzzy decision-making methods, are as follows: (a) it can obtain the optimal alternative without counterintuitive phenomena and (b) it has a great power in distinguishing the optimal alternative.     

Weighted power means of q-rung orthopair fuzzy information and their applications in multiattribute decision making

Weighted power means with weights and exponents serving as their parameters are generalizations of arithmetic means. Taking into account decision makers' flexibility in decision making, each attribute value is usually expressed by a $q$-rung orthopair fuzzy value ($q$-ROFV, $q\ge 1$), where the former indicates the support for membership, the latter support against membership, and the sum of their $q$th powers is bounded by one. In this paper, we propose the weighted power means of $q$-rung orthopair fuzzy values to enrich and flourish aggregations on $q$-ROFVs. First, the $q$-rung orthopair fuzzy weighted power mean operator is presented, and its boundedness is precisely characterized in terms of the power exponent. Then, the $q$-rung orthopair fuzzy ordered weighted power mean operator is introduced, and some of its fundamental properties are investigated in detail. Finally, a novel multiattribute decision making method is explored based on developed operators under the $q$-rung orthopair fuzzy environment. A numerical example is given to illustrate the feasibility and validity of the proposed approach, and it is shown that the power exponent is an index suggesting the degree of the optimism of decision makers.     

Some q-rung orthopair fuzzy Hamy mean operators in multiple attribute decision-making and their application to enterprise resource planning systems selection

In this paper, the Hamy mean (HM) operator, weighted HM (WHM), dual HM (DHM) operator, and dual WHM (WDHM) operator under the q-rung orthopair fuzzy sets (q-ROFSs) is studied to propose the q-rung orthopair fuzzy HM (q-ROFHM) operator, q-rung orthopair fuzzy WHM (q-ROFWHM) operator, q-rung orthopair fuzzy DHM (q-ROFDHM) operator, and q-rung orthopair fuzzy weighted DHM (q-ROFWDHM) operator and some of their desirable properties are investigated in detail. Then, we apply these operators to multiple attribute decision-making problems. Finally, a practical example for enterprise resource planning system selection is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

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.     

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.     

The distance measures between q-rung orthopair hesitant fuzzy sets and their application in multiple criteria decision making

In this paper, we first introduce the concept of q-rung orthopair hesitant fuzzy set ($q$-ROHFS) and discuss the operational laws between any two $q$-ROHFSs. Then the distance measures between $q$-ROHFSs are proposed based on the concept of “multiple fuzzy sets”, and we develop the TOPSIS method to the proposed distance measures. The proposed distance measures not only retain the preference information expressed by $q$-ROHFSs, but also deal with the $q$-rung orthopair hesitant fuzzy decision information more objectively, In fact, the method can avoid the loss and distortion of the information in actual decision-making process. Furthermore, we give an illustrative example about the selection of energy projects to illustrate the reasonableness and effectiveness of the proposed method, which is also compared with other existing methods. Finally, we make the sensitivity analysis of the parameters in proposed distance measures about the selection of energy projects.