Q-rung Orthopair Normal Fuzzy Aggregation Operators and Their Application in Multi-Attribute Decision-Making

A q-rung orthopair fuzzy set (q-ROFS) is an effective tool for describing uncertainty and fuzziness, and is the promotion of intuitionistic fuzzy sets (IFSs) and Pythagorean fuzzy sets (PFSs). This paper extends q-rung orthopair fuzzy environment to probabilistic q-rung orthopair hesitant fuzzy environment, and proposes the concept of probabilistic q-rung orthopair hesitant fuzzy sets (P-q-ROHFSs). Some desired operational laws and properties of P-q-ROHFSs are studied. And we develop the some aggregation operators for probabilistic q-rung orthopair hesitant fuzzy information. Then the relationship among these operators is discussed by comparing the score function. In order to measure the uncertain information, this paper proposes the four distance measures between two P-q-ROHFSs. Especially, ranking and expansion of P-q-ROHFSs are discussed in detail. Furthermore, we apply them to pattern recognition and multi-attribute group decision making (MAGDM) under probabilistic q-rung orthopair hesitant fuzzy environment. Finally, examples are given to show the rationality and practicability of the proposed method.

The interval-valued Pythagorean fuzzy (IVPF) sets, describing the membership and non-membership degrees from interval values, can address uncertain information, while the normal fuzzy number (NFN) can depict normal distribution information in anthropogenic activity and natural environment. By combining the advantages of both operations, in this study, we proposed the interval-valued Pythagorean normal fuzzy (IVPNF) sets by introducing the NFN into IVPF environment. Firstly, we defined the conception, the operational laws, score function, accuracy function of IVPNF sets. Secondly, we presented four information aggregation operators to aggregate IVPNF information, including the IVPNF weighted averaging (IVPNFWA) operator, IVPNF weighted geometric (IVPNFWG) operator, the generalized IVPNFWA operator, and the generalized IVPNFWG operator. In addition, we analyzed some desirable properties of monotonicity, commutativity, and idempotency for the proposed four operators. Finally, a numerical example on multi-attribute decision-making problem is given to verify the practicality of the proposed operators, and the comparative and sensitive analysis are used to show the effectiveness and flexibility of our proposed approach.

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.

In this study, in view of expressing the uncertain information more elegantly, we shall enlighten the q-rung orthopair fuzzy sets (q-ROFSs) and the q-rung orthopair fuzzy numbers (q-ROFNs) which are considered to be superior of the intuitionistic fuzzy sets and the Pythagorean fuzzy sets, respectively. Here our aim is towards the development of some new operational laws and their corresponding weighted aggregation operators under the q-rung orthopair fuzzy environment. In this regard, at the very beginning, we define some new neutral or fair operational laws that include the concept of proportional distribution to achieve a neutral or fair treatment to the membership and non-membership functions of q-ROFN. Subsequently, with these operations, we develop q-rung orthopair fuzzy weighted fairly aggregation operator (qROFWFA) and q-rung orthopair fuzzy ordered weighted fairly aggregation operator (qROFOWFA) which can neutrally or fairly serve the membership and non-membership degrees. We observe the noteworthy features of these proposed aggregation operators. Furthermore, we exercise also an MADM (multi-attribute decision-making) approach with multiple decision makers and partial weight information in the framework of q-rung orthopair fuzzy sets. At the end of this study, we provide an illustrative example to highlight the feasibility and a practical look of the approach proposed herein.

The concept of q-rung orthopair hesitant fuzzy set represents an advancement and extension of hesitant fuzzy sets, encompassing both fuzzy sets and q-rung orthopair fuzzy sets. q-rung orthopair hesitant fuzzy set characterizes a set of membership and non-membership grades within the interval [0, 1], which enhances its adaptability compared to existing methods. This flexibility proves invaluable in providing more insightful data about various objects. The primary objective of this research is to introduce a decision-making technique in the context of q-RHF using the theory of set pair analysis (SPA). q-RHFS effectively handles ambiguous data by incorporating membership and non-membership grades, while the connection number (CN) based on SPA theory manages the intricacies of uncertainty and certainty structures by relying on "identity", "discrepancy" and "contrary" grades. Building on the relationship between q-RHFS and the connection number of set pair analysis, a comprehensive framework known as q-rung hesitant fuzzy connection number set (qHCNs) is developed. This model not only addresses uncertainty, but also offers valuable insights. Furthermore, this research introduces similarity measures derived from qHCN and examines their advantages through illustrative examples. Additionally, a novel approach to decision modeling utilizing these measures applied to medical diagnosis is also introduced. The application of this established model contributes to an effective approach and demonstrated its soundness and efficiency. In addition, a detailed comparative study is conducted with the existing models and advantages of proposed model. The research concludes with a summary of the authors' findings, highlighting the consistency and effectiveness of their work.

RETRACTED: Multiattribute decision making based on nonlinear programming methodology, score function of interval-valued intuitionistic fuzzy values, and the dispersion degree of score values

A novel approach towards multiattribute decision making using q-rung orthopair fuzzy Dombi–Archimedean aggregation operators

q-Rung orthopair fuzzy set (q-ROFS) is a powerful tool that attracts the attention of many scholars in dealing with uncertainty and vagueness. 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. Then a new exponential operational law about q-ROFNs is defined, in which the bases are positive real numbers and the exponents are q-ROFNs. Meanwhile, some properties of the operational law are investigated. Later, we apply them to derive the q-rung orthopair fuzzy weighted exponential aggregation operator. Additionally, an approach for multicriteria decision-making problems under the q-rung orthopair fuzzy data is explored by applying proposed aggregation operator. Finally, an example is 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 (1) it can obtain the optimal alternative without counterintuitive phenomena; (2) it has a great power in distinguishing the optimal alternative.

As a generalization of complex fuzzy set (CFS), complex hesitation fuzzy set (CHFS) is a powerful tool to express two-dimensional fuzzy information, but it ignores the randomness that exists widely in real life. Considering the randomness and ambiguity often simultaneously occur in real-world problems, we come up with an innovative fuzzy set called complex probabilistic hesitant fuzzy set (CPHFS) in this article, which is a joint extension of CFS and probabilistic hesitant fuzzy set (PHFS). Moreover, we establish four basic operations for complex probabilistic hesitant fuzzy element(CPHFE) and a series of useful operational laws are also developed. Secondly, several aggregation operators including complex probabilistic hesitant fuzzy weighted averaging operator(CPHFWA), complex probabilistic hesitant fuzzy weighted geometric operator(CPHFWG), complex probabilistic hesitant fuzzy Heronian Mean operator(CPHFHM) and complex probabilistic hesitant fuzzy Geometric Heronian Mean operator(CPHFGHM) as well as their weighted forms that can reflect the interrelationship among all decision attributes are proposed, which is of great use in multi-attribute decision making(MADM) or in multi-attribute group decision making(MAGDM) issues when we want to synthesize multiple evaluation values into one. Besides, we also present a number of valuable properties of these provided aggregation operators. Thirdly, a MADM process is built in the CPHF environment step by step based on the aggregation operators we have just established. Furthermore, we provide an example about purchasing a most suitable car to demonstrate the effectiveness and practicability of the proposed decision process. Finally, we point out the superiorities of MADM in working under the CPHF environment through comparing it with the existing method and also offer some possible prospects for further research. The method displayed in this work can effectively solve the MADM problems where the decision-making information is described by CPHFEs and the attributes are interactive.

This paper proposes a novel multi-attribute decision-making (MADM) method for interval rough numbers (IRNs) considering different distribution types, namely uniform, exponential, and normal distributions. Upper and lower approximate interval dominance degrees are defined and aggregated using dynamic weights to obtain pairwise comparisons of IRNs. The properties of dominance are verified, and an attribute weight determination method based on the dominance balance degree is introduced. The proposed MADM method is data-driven and does not rely on the subjective preferences of decision-makers. Case analysis demonstrates the effectiveness and rationality of the proposed method, revealing that the distribution type of IRNs significantly impacts decision results, potentially leading to reversed ranking outcomes. The proposed method offers a comprehensive framework for handling MADM problems with IRNs under different distributions.

T-spherical fuzzy set is a recently developed model that copes with imprecise and uncertain events of real-life with the help of four functions having no restrictions. This article’s aim is to define some improved algebraic operations for T-SFSs known as Einstein sum, Einstein product and Einstein scalar multiplication based on Einstein t-norms and t-conorms. Then some geometric and averaging aggregation operators have been established based on defined Einstein operations. The validity of the defined aggregation operators has been investigated thoroughly. The multi-attribute decision-making method is described in the environment of T-SFSs and is supported by a comprehensive numerical example using the proposed Einstein aggregation tools. As consequences of the defined aggregation operators, the same concept of Einstein aggregation operators has been proposed for q-rung orthopair fuzzy sets, spherical fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. To signify the importance of proposed operators, a comparative analysis of proposed and existing studies is developed, and the results are analyzed numerically. The advantages of the proposed study are demonstrated numerically over the existing literature with the help of examples.

Picture fuzzy set (PFS) is a direct generalization of the fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs) and is quite powerful in modelling the situations that involve more answers of the type yes, no, abstain, and refuse. In this study, we introduce a novel picture fuzzy (PF) similarity measure on the basis of direct operation on the function of membership, the function of non-membership, the function of neutrality, the function of refusal, and the upper bound of the function of membership of two PFSs instead on the basis of distance measure or the association between the functions of membership, non-membership, and neutrality. The comparison of the proposed PF similarity measure with the existing PF similarity measures reveals that it does not give unreasonable results and also overcomes the drawbacks of the existing PF similarity measures. The application of the proposed measure in pattern recognition is also discussed. Moreover, we also introduce a new multi-attribute decision-making (MADM) method using the proposed PF similarity measure that overcomes a major drawback of the technique for order preference by similarity to ideal solution (TOPSIS). Finally, we contrast the performance of the proposed MADM method with several existing MADM methods in the PF environment.

In several practical decision procedures, it is not accessible to denote assessments by a single crisp number due to a lack of information. However, representing information by an interval number within [0, 1] is a more credible approach. In multi-criteria decision making (MCDM) such an interval number can significantly catch information. In addition, the combination of soft sets with interval-valued q-rung orthopair fuzzy sets can be viewed as interval-valued q-rung orthopair fuzzy soft sets (IVq-ROFSs). It can be a reliable tool to cope with uncertainties. Usually, aggregation operators are functional in MCDM techniques; therefore, aggregation operators on IVq-ROFSs can significantly aggregate pieces of information in intervals with IVq-ROFSs. In this paper, we investigated some crucial properties of interval valued q-rung orthopair fuzzy soft sets (IVq-ROFSSs) and expressed a different representation of IVq-ROFSS in the form of IVq-ROFS number. Based on this representation, we investigated IVq-ROF weighted averaging, IVq-ROF weighted geometric operators and given their basic properties. Moreover, we consider interactions between non-memberships and memberships of different interval-valued q-rung orthopair fuzzy values and defined IVq-ROF weighted interaction averaging, IVq-ROF weighted interaction geometric aggregation operators in IVq-ROFS environments. A decision-making process is given, and an illustration is provided by tackling application in automation company evaluation.

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can hold much more information and can be used to describe both an interval-valued Pythagorean fuzzy set (IVPFS) and Pythagorean fuzzy set (PFS) at the same time to handle data uncertainties. Based on it, the present study is classified into three phases. The first phase is to modify the existing operational laws and aggregation operators (AOs) in the article presented by Abbas et al. (Journal of Intelligent & Fuzzy Systems, vol. 37, no. 1, pp. 1529–1544, (2019)). The main objective of improved operational laws is to eliminate the flows and ambiguities in existing AOs. Secondly, based on these laws, various AOs to aggregate the information are acquired along with their requisite properties and relations. Lastly, an approach for interpreting the multi-attribute decision-making (MCDM) problem based on the stated operators is given and illustrated with an example. Some of the existing models are used to perform a comprehensive comparative analysis to demonstrate their impacts.

In this study, the authors explore the complex q-rung orthopair fuzzy sets, which are preferred to be enhanced of the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets, individually. The intention is regarding the buildup of certain innovative operational laws and their related weighted aggregation operators based on the complex q-rung orthopair fuzzy (CQROF) information. In this regard, they characterize certain original neutral or fair operational laws that involve the model of proportional distribution to accomplish a neutral or fair usage to the truth and falsity functions of CQROFSs. Consequently, with these operations, they acquire CQROF weighted fairly aggregation (CQROFWFA) and CQROF ordered weighted fairly aggregation (CQROFOWFA) operators which can neutrally or fairly provide the truth and falsity degrees. They implement an MADM (multi-attribute decision-making) methodology with multiple decision makers and partial weight knowledge in the structure of CQROFSs.