Q-rung Orthopair Fuzzy Numbers Research Articles

Integrations of q-Rung Orthopair Fuzzy Continuous Information

Yager's q-rung orthopair fuzzy sets (q-ROFSs), which extend Zadeh's fuzzy sets, use the membership and nonmembership functions to describe things’ vague characteristics, and the sum of the q th-power for the membership and nonmembership functions is less than or equal to 1. More recently, some scholars have proposed a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. However, so far, there is no research on aggregating q-rung orthopair fuzzy continuous information. Thus, we proposed q-rung orthopair fuzzy definite integrals (q-ROFDIs) to fill this vacancy. First, we further study the operations of q-rung orthopair fuzzy numbers (q-ROFNs) that are the core of q-ROFSs. We also introduce the limit of a q-ROFN sequence. Subsequently, we construct the q-ROFDIs step-by-step, give their concrete values, and discuss their integrability criteria from two perspectives. From the perspectives of modern analysis and the operational laws of q-ROFNs, we investigate the q-ROFDIs in detail, which are concise and considerably different from the investigative techniques of the previous research on aggregating continuous information. Finally, a practical example is provided to show the effectiveness, elasticity, and superiority of the q-ROFDIs via comparing them with the existing methods.

Multi-criteria group decision making based on Archimedean power partitioned Muirhead mean operators of q-rung orthopair fuzzy numbers.

Two critical tasks in multi-criteria group decision making (MCGDM) are to describe criterion values and to aggregate the described information to generate a ranking of alternatives. A flexible and superior tool for the first task is q-rung orthopair fuzzy number (qROFN) and an effective tool for the second task is aggregation operator. So far, nearly thirty different aggregation operators of qROFNs have been presented. Each operator has its distinctive characteristics and can work well for specific purpose. However, there is not yet an operator which can provide desirable generality and flexibility in aggregating criterion values, dealing with the heterogeneous interrelationships among criteria, and reducing the influence of extreme criterion values. To provide such an aggregation operator, Muirhead mean operator, power average operator, partitioned average operator, and Archimedean T-norm and T-conorm operations are concurrently introduced into q-rung orthopair fuzzy sets, and an Archimedean power partitioned Muirhead mean operator of qROFNs and its weighted form are presented and a MCGDM method based on the weighted operator is proposed in this paper. The generalised expressions of the two operators are firstly defined. Their properties are explored and proved and their specific expressions are constructed. On the basis of the specific expressions, a method for solving the MCGDM problems based on qROFNs is then designed. Finally, the feasibility and effectiveness of the method is demonstrated via a numerical example, a set of experiments, and qualitative and quantitative comparisons.

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.

A new multi-criteria group decision-making approach based on q-rung orthopair fuzzy interaction Hamy mean operators

The recently proposed q-rung orthopair fuzzy set (q-ROFS) is a powerful and effective tool to describe uncertainty and vagueness, and Hamy mean (HM) has a significant advantage of capturing the interrelationship among aggregated arguments. In order to take full advantage of q-ROFS and HM, and consider the interactions between membership and non-membership degrees at the same time, in this paper, we propose a family of q-rung orthopair fuzzy Hamy mean operators based on interaction operations. First, we define interaction operational rules for q-rung orthopair fuzzy numbers. Based on the new operational rules, q-rung orthopair fuzzy interaction HM and q-rung orthopair fuzzy interaction weighted HM operators are proposed. Further, we propose a dual Hamy mean (DHM) operator and extend it to accommodate q-rung orthopair fuzzy environment. Based on interaction operational rules and DHM, q-rung orthopair fuzzy interaction DHM operator and its weighted form are also developed. Then, a novel multi-attribute group decision-making approach based on proposed operators is introduced. Finally, a numerical instance, as well as some comparative analyses, is provided to illustrate the validity and advantages of the new approach.

Multiple‐attribute group decision‐making method of linguisticq‐rung orthopair fuzzy power Muirhead mean operators based on entropy weight

Linguistic q-rung orthopair fuzzy numbers (Lq-ROFNs) are a qualitative form of q-rung orthopair fuzzy numbers (q-ROFNs) where the membership and nonmembership degrees are represented by linguistic variables. The Lq-ROFNs can describe a broader range of linguistic assessment information flexibly by adjusting the parameter q based on different situations, so they are more superior to the linguistic intuitionistic fuzzy numbers and linguistic Pythagorean fuzzy numbers in real application. Based on the Lq-ROFNs, we introduce the entropy measure which can be used to determine the indefiniteness of the assessment information. Then, based on the linguistic entropy measure, we further propose a method to obtain the attribute weights when the weight information is incomplete known. For aggregating the assessment information, the power average (PA) operator can reduce the influence of extreme data caused by the biased decision-makers by considering the support degree of different evaluation individuals, and the Muirhead mean (MM) operator can take the interrelationship of different numbers of attributes into account by adjusting the parameter vector based on the real situations. In this paper, based on these two operators, we firstly propose the linguistic q-rung orthopair fuzzy PA operator and linguistic q-rung orthopair fuzzy weighted PA operator. Further, for combing the advantages of the MM operator and PA operator, we propose the linguistic q-rung orthopair fuzzy power MM (PMM) operator and linguistic q-rung orthopair fuzzy weighted PMM operator, and then investigate some properties of them. Finally, a new multiple-attribute group decision-making (MAGDM) method is proposed to process the Lq-ROFNs, and some practical examples are given to illustrate the effectiveness and superiority of this new method in comparison with other existing MAGDM methods.

Some q-rung orthopair fuzzy point weighted aggregation operators for multi-attribute decision making

q-Rung orthopair fuzzy sets, originally proposed by Yager, can dynamically adjust the range of indication of decision information by changing a parameter q based on the different hesitation degree, and point operator is a useful aggregation technology that can control the uncertainty of valuating data from some experts and thus get intensive information in the process of decision making. However, the existing point operators are not available for decision-making problems under q-Rung orthopair fuzzy environment. Thus, in this paper, we firstly propose some new point operators to make it conform to q-rung orthopair fuzzy numbers (q-ROFNs). Then, associated with classic arithmetic and geometric operators, we propose a new class of point weighted aggregation operators to aggregate q-rung orthopair fuzzy information. These proposed operators can redistribute the membership and non-membership in q-ROFNs according to different principle. Furthermore, based on these operators, a novel approach to multi-attribute decision making (MADM) in q-rung orthopair fuzzy context is introduced. Finally, we give a practical example to illustrate the applicability of the new approach. The experimental results show that the novel MADM method outperforms the existing MADM methods for dealing with MADM problems.

Multiple-Attribute Group Decision-Making Based on q-Rung Orthopair Fuzzy Power Maclaurin Symmetric Mean Operators

To be able to describe more complex fuzzy uncertainty information effectively, the concept of q-rung orthopair fuzzy sets (q-ROFSs) was first proposed by Yager. The q-ROFSs can dynamically adjust the range of indication of decision information by changing a parameter q based on the different hesitation degree from the decision-makers, where q ≥ 1, so they outperform the traditional intuitionistic fuzzy sets and Pythagorean fuzzy sets. In real decision-making problems, there is often an interaction phenomenon between attributes. For aggregating these complex fuzzy information, the Maclaurin symmetric mean (MSM) operator is more superior by considering interrelationships among attributes. In addition, the power average (PA) operator can reduce the effects of extreme evaluating data from some experts with prejudice. In this paper, we introduce the PA operator and the MSM operator based on q-rung orthopair fuzzy numbers (q-ROFNs). Then, we put forward the q-rung orthopair fuzzy power MSM (q-ROFPMSM) operator and the q-rung orthopair fuzzy power weighed MSM (q-ROFPWMSM) operator of q-ROFNs and present some of their properties. Finally, we present a novel multiple-attribute group decision-making (MAGDM) method based on the q-ROFPWA and the q-ROFPWMSM operators. The experimental results show that the novel MAGDM method outperforms the existing MAGDM methods for dealing with MAGDM problems.

Dual Hesitant q-Rung Orthopair Fuzzy Muirhead Mean Operators in Multiple Attribute Decision Making

On account of the indeterminacy and subjectivity of decision makers (DMs) in complexity decision-making environments, the evaluation information over alternatives presented by DMs is usually fuzzy and ambiguous. As the generalization of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs), the q-rung orthopair fuzzy sets (q-ROFSs) are more useful to express more fuzzy and ambiguous information. Meanwhile, to consider human's hesitance, the dual hesitant q-rung orthopair fuzzy sets (DHq-ROFSs) are presented which can be more valid of handling real MADM problems. To fuse the information in DHq-ROFSs more effectively, in this article, some Muirhead mean (MM) operators based on DHq-ROFSs environment, which consider any number of being fused arguments, are defined and studied. Evidently, the new proposed operators can obtain more exact results than other existing methods. In addition, some precious properties of these MM operators are discussed and all the special cases of them are investigated which indicates MM operator is more powerful than others. Afterward, the defined aggregation operators are used to solve the MADM with dual hesitant q-rung orthopair fuzzy numbers (DHq-ROFNs) and the MADM decision-making model is developed. In accordance of the defined operators and built model, two operators are applied to deal with the MADM problems for supplier selection with the DHPFNs information and the availability and superiority of the proposed operators are analyzed by comparing with some existing approaches. The method presented in this paper can effectually solve the MADM problems in which the decision-making information is expressed by the DHq-ROFNs and the attributes are interactive.

Multiple-attribute group decision-making based on power Bonferroni operators of linguisticq-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.

A Novel Approach for Green Supplier Selection under a q-Rung Orthopair Fuzzy Environment

With environmental issues becoming increasingly important worldwide, plenty of enterprises have applied the green supply chain management (GSCM) mode to achieve economic benefits while ensuring environmental sustainable development. As an important part of GSCM, green supplier selection has been researched in many literatures, which is regarded as a multiple criteria group decision making (MCGDM) problem. However, these existing approaches present several shortcomings, including determining the weights of decision makers subjectively, ignoring the consensus level of decision makers, and that the complexity and uncertainty of evaluation information cannot be adequately expressed. To overcome these drawbacks, a new method for green supplier selection based on the q-rung orthopair fuzzy set is proposed, in which the evaluation information of decision makers is represented by the q-rung orthopair fuzzy numbers. Combined with an iteration-based consensus model and the q-rung orthopair fuzzy power weighted average (q-ROFPWA) operator, an evaluation matrix that is accepted by decision makers or an enterprise is obtained. Then, a comprehensive weighting method can be developed to compute the weights of criteria, which is composed of the subjective weighting method and a deviation maximization model. Finally, the TODIM (TOmada de Decisao Interativa e Multicritevio) method, based on the prospect theory, can be extended into the q-rung orthopair fuzzy environment to obtain the ranking result. A numerical example of green supplier selection in an electric automobile company was implemented to illustrate the practicability and advantages of the proposed approach.

Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making

The q-rung orthopair fuzzy sets (q-ROFSs), originated by Yager, are good tools to describe fuzziness in human cognitive processes. The basic elements of q-ROFSs are q-rung orthopair fuzzy numbers (q-ROFNs), which are constructed by membership and nonmembership degrees. As realistic decision-making is very complicated, decision makers (DMs) may be hesitant among several values when determining membership and nonmembership degrees. By incorporating dual hesitant fuzzy sets (DHFSs) into q-ROFSs, we propose a new technique to deal with uncertainty, called q-rung dual hesitant fuzzy sets (q-RDHFSs). Subsequently, we propose a family of q-rung dual hesitant fuzzy Heronian mean operators for q-RDHFSs. Further, the newly developed aggregation operators are utilized in multiple attribute group decision-making (MAGDM). We used the proposed method to solve a most suitable supplier selection problem to demonstrate its effectiveness and usefulness. The merits and advantages of the proposed method are highlighted via comparison with existing MAGDM methods. The main contribution of this paper is that a new method for MAGDM is proposed.

Some Partitioned Maclaurin Symmetric Mean Based on q-Rung Orthopair Fuzzy Information for Dealing with Multi-Attribute Group Decision Making

In respect to the multi-attribute group decision making (MAGDM) problems in which the evaluated value of each attribute is in the form of q-rung orthopair fuzzy numbers (q-ROFNs), a new approach of MAGDM is developed. Firstly, a new aggregation operator, called the partitioned Maclaurin symmetric mean (PMSM) operator, is proposed to deal with the situations where the attributes are partitioned into different parts and there are interrelationships among multiple attributes in same part whereas the attributes in different parts are not related. Some desirable properties of PMSM are investigated. Then, in order to aggregate the q-rung orthopair fuzzy information, the PMSM is extended to q-rung orthopair fuzzy sets (q-ROFSs) and two q-rung orthopair fuzzy partitioned Maclaurin symmetric mean (q-ROFPMSM) operators are developed. To eliminate the negative influence of unreasonable evaluation values of attributes on aggregated result, we further propose two q-rung orthopair fuzzy power partitioned Maclaurin symmetric mean (q-ROFPPMSM) operators, which combine the PMSM with the power average (PA) operator within q-ROFSs. Finally, a numerical instance is provided to illustrate the proposed approach and a comparative analysis is conducted to demonstrate the advantage of the proposed approach.

Someq-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.