Generalized Orthopair Fuzzy Sets

<abstract> <p>Green supplier selection has been an important technique for environmental sustainability and reducing the harm of ecosystems. In the current climate, green supply chain management (GSCM) is imperative for maintaining environmental compliance and commercial growth. To handle the change related to environmental concern and how the company manages and operates, they are integrated the GSCM into traditional supplier selection process. The main aims of this study were to outline both traditional and environmental criteria for selecting suppliers, providing a comprehensive framework to assist decision-maker in prioritizing green supplier effectively. In order to address issue to simulate decision-making problems and manage inaccurate data. A useful technique of fuzzy set was proposed to handle uncertainty in various real-life problems, but all types of data could not be handled such as incomplete and indeterminate. However, several extensions of fuzzy set were considered, such as intuitionistic fuzzy set, Pythagorean fuzzy set, q-rung orthopair fuzzy set, and q-rung orthopair fuzzy soft set considering membership and nonmember ship grade to handle the uncertainty problem. However, there was a lack of information about the neutral degree and parameterization axioms lifted by existing approaches, so to fill this gap and overcome the difficulties Ali et al. proposed a generalized structure by combining the structure of picture fuzzy set and q-rung orthopair fuzzy soft set, known as q-rung orthopair picture fuzzy soft sets, characterized by positive, neutral and negative membership degree with parameterization tools and aggregation operator to solve the multi criteria group decision-making problem. Additionally, the TOPSIS method is a widely utilized to assist individuals and organizations in selecting the most appropriate option from a range of choices, taking into account various criteria. Finally, we demonstrate an illustrative example related to GSCM to enhance competitiveness, based on criteria both in general and with a focus on environmental consideration, accompanied by an algorithm and flow chart.</p> </abstract>

In recent years, q-rung orthopair fuzzy sets have been appeared to deal with an increase in the value of q > 1 , which allows obtaining membership and nonmembership grades from a larger area. Practically, it covers those membership and nonmembership grades, which are not in the range of intuitionistic fuzzy sets. The hybrid form of q-rung orthopair fuzzy sets with soft sets have emerged as a useful framework in fuzzy mathematics and decision-makings. In this paper, we presented group generalized q-rung orthopair fuzzy soft sets (GGq-ROFSSs) by using the combination of q-rung orthopair fuzzy soft sets and q-rung orthopair fuzzy sets. We investigated some basic operations on GGq-ROFSSs. Notably, we initiated new averaging and geometric aggregation operators on GGq-ROFSSs and investigated their underlying properties. A multicriteria decision-making (MCDM) framework is presented and validated through a numerical example. Finally, we showed the interconnection of our methodology with other existing methods.

Complex fuzzy set, as an extension of classical fuzzy sets, could describe the fuzzy characters of things more detail and comprehensively and is very useful in dealing with vagueness and uncertainty of problems that include the periodic or recurring phenomena. Note that a complex fuzzy set is different from the fuzzy complex set introduced and discussed by many scholars, since the membership degree of a complex fuzzy set is a complex number with length less than or equal to 1 while a fuzzy complex set is a real number with membership degree less than or equal to 1, and the universe is the complex plane. As the mathematical theoretical basis of fuzzy mathematics, fuzzy set and its mapping, corresponding fuzzy complex set and its mapping have been investigated in depth because they integrate and cross the methods and results of classical real analysis and complex analysis. However, there is no comprehensive investigation on complex fuzzy set and its corresponding mathematical theory, even include decomposition theorems, extension principles and the basic operations of the complex fuzzy set. As is well known, the cut set of fuzzy sets is the bridge between fuzzy sets and classical sets, which plays a significant role in fuzzy sets and fuzzy systems. In this paper, the concept of (r, θ)-cut sets of complex fuzzy sets is proposed and their properties are discussed. Meanwhile, the decomposition theorems and the extension principles of complex fuzzy set based on (r, θ)-cut sets are deduced and corresponding properties are investigated. All these conclusions not only deeply enrich the fundamental theory of complex fuzzy set, but also provide a powerful tool to investigate complex fuzzy set. Finally, an example application of signal detection demonstrates the utility of the (r, θ)-cut sets of complex fuzzy sets in practice.

Green Supply Chain Management (GSCM) is essential to ensure environmental compliance and commercial growth in the current climate. Businesses constantly look for fresh concepts and techniques for ensuring environmental sustainability. To keep up with the new trends in environmental concerns related to company management and procedures, Green Supplier Selection (GSS) criteria are added to the traditional supplier selection processes. This study aims to identify general and environmental supplier selection criteria to provide a framework that can assist decision-makers in choosing and prioritizing appropriate green supplier selection. The development and implementation of decision support systems aimed to solve these difficulties at a rapid rate. In order to manage inaccurate data and simulate decision-making problems. Fuzzy sets introduced by Zadeh, are a useful technique to handle the imperfectness and uncertainty in different problems. Although fuzzy sets can handle incomplete information in different real worlds problems, but its cannot handle all type of uncertainty such as incomplete and indeterminate data. Therefore different extensions of fuzzy sets such as intuitionistic fuzzy, pythagorean fuzzy and q-rung orthopair fuzzy sets introduced to address the problems of uncertainty by considering the membership and non-membership grade. However, these concepts have some shortcomings in the handling uncertainty with sub-attributes. To overcome this difficulties Khan et al. developed the structure of q-rung orthopair fuzzy hypersoft sets by combining q-rung orthopair fuzzy sets with hypersoft sets. A remarkable and beneficial research work is done in the field of q-rung orthopair fuzzy hypersoft sets, and then we think about the application. In this paper, we use the structure of q-rung orthopair fuzzy hypersoft in multi-criteria supplier selection problems. For this, we present aggregation operator to solve multi-criteria decision-making (MCDM) problems with q-rung orthopair fuzzy hypersoft (q-ROFH) information, known as ordered weighted geometric aggregation operator. Since the uncertainty and vagueness is an unavoidable feature of multi-criteria decision-making problems, the proposed structure can be a useful tool for decision making in an uncertain environment. Further, the expert opinions were investigated using the multi-criteria decision-making (MCDM) technique, which helped identify interrelationship and causal preference of green supplier evaluation aspects that used aggregation operators. Finally, a numerical example of the proposed method for the task of Green Supplier Selection is presented.

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.

THE CONCEPT OF GRADE OF MEMBERSHIP

The concept of grade of membership

Modeling uncertainties with multipolar information is an important tool in computational intelligence to address complexities in real-world circumstances. An m-polar fuzzy set (mPFS) is the strong model to express multipolarity with m $m$ membership grades (MGs) in the unit closed interval [ 0 , 1 ] $[0,1]$ . A q-rung orthopair fuzzy set (qROFS) is the strong model to express vague and uncertain information with MGs and nonmembership grades (NMGs). The notion of q-rung orthopair m-polar fuzzy set is a new hybrid extension of both mPFS and qROFS. An ROmPFS is a generalized concept that has the ability to deal with multipolarity with m $m$ ordered pairs of MGs and NMGs. Motivated by these robust concepts, in this article, various aggregation operators (AOs) for the aggregation of q-rung orthopair m-polar fuzzy numbers are proposed, including q-rung orthopair m-polar fuzzy weighted averaging operator, symmetric q-rung orthopair m-polar fuzzy weighted averaging operator, q-rung orthopair m-polar fuzzy weighted geometric operator, symmetric q-rung orthopair m-polar fuzzy weighted geometric operator, and q $q$ -rung orthopair m-polar fuzzy Maclaurin symmetric mean operator. On the basis of proposed AOs, a robust multicriteria decision-making approach is proposed. An application of proposed AOs is presented to address economic crises during COVID-19. Furthermore, the comparison analysis is designed to discuss the validity and rationality of proposed AOs.

In this chapter, we provide an introduction to more advanced, hierarchically structured information granules such as those of higher type and higher order. In general, when talking about information granules of higher type, say type-2, we mean information granules whose elements are characterized by membership grades, which themselves are information granules (instead of being plain numeric values). For instance, in type-2 fuzzy sets, membership grades are quantified as fuzzy sets in [0,1] or intervals in the unit interval. Of course, one could envision a plethora of the constructs along this line; for instance, the membership grades could be rough sets or probability density functions (as this is the case in probabilistic sets). When talking about higher order information granules, we mean constructs for which the universe of discourse comprises a family of information granules instead of single elements. Hybridization, on the other hand, is about bringing several formalisms of information granules and using them in an orthogonal setting. This situation is visible in fuzzy probabilities.

Statistically grounded logic operators in fuzzy sets

The notion of linear Diophantine fuzzy sets (LD-FSs) is a novel mechanism to combat uncertainties in decision analysis. Due to reference parameters associated with membership grade (MG) and non-membership grade (NMG), LD-FS is more efficient and reliable than ideas of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS). The main goal of this article is to present an innovative roughness strategy for LD-FSs using a fuzzy relation (FR) over dual universes, known as a linear Diophantine fuzzy rough set (LD-FRS). The lower and upper approximations of an LD-FS are formulated using fuzzy relation (FR) over dual universes, and several axiomatic systems are investigated. The suggested model of LD-FRS is more flexible to address fuzziness and roughness. Meanwhile, a relationship is made between LD-FRSs and linear Diophantine fuzzy topologies (LDF-topologies). It is shown that the collection of all lower approximations based on a reflexive FR leads to an LDF-topology. Moreover, several similarity relations among LD-FSs are also examined based on their lower and upper approximations. An application of multi-criteria group decision-making (MCGDM) is demonstrated by a supplier selection problem. Finally, a detailed comparative analysis with certain existing methods is given to verify the feasibility and superiority of the suggested model.

In Fuzzy Inference Systems (FIS) the rule base consists of fuzzy relations between antecedents and consequents represented by classical fuzzy sets. Because their membership grades are exact real numbers in the unit interval [0, 1], there is no uncertainty in this sort of specification. In many applications there is some uncertainty as to the memberships, hence they can be stated as ordinary fuzzy sets of type 1 and can constitute type 2 fuzzy sets.In the world literature exists a global model of type 2 FIS. However it consists of an enormous number of embedded subsystems of type 1 and with regard to this model it has not found any use in connectionist realizations. In this paper we derive connectionist structures of type 2 FIS.

Orthopair fuzzy sets are fuzzy sets in which every element is represented by a pair of values in the unit interval, one of which refers to membership and the other refers to non-membership. The different types of orthopair fuzzy sets given in the literature are distinguished according to the proposed constrain for membership and non-membership grades. The aim of writing this manuscript is to familiarize a new class of orthopair fuzzy sets called “(2,1)-Fuzzy sets” which are good enough to control some real-life situations. We compare (2,1)-Fuzzy sets with IFSs and some of their celebrated extensions. Then, we put forward the fundamental set of operations for (2,1)-Fuzzy sets and investigate main properties. Also, we define score and accuracy functions which we apply to rank (2,1)-Fuzzy sets. Moreover, we reformulate aggregation operators to be used with (2,1)-Fuzzy sets. Finally, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making (MCDM) problems in the environment of (2,1)-Fuzzy sets. To show the effectiveness and usability of the proposed technique in MCDM problems, an illustrative example is provided.

Constructing shadowed sets and three-way approximations of fuzzy sets

Intuitionistic fuzzy sets, Pythagorean fuzzy sets, and q-rung orthopair fuzzy sets are rudimentary concepts in computational intelligence, which have a myriad of applications in fuzzy system modeling and decision-making under uncertainty. Nevertheless, all these notions have some strict restrictions imposed on the membership and non-membership grades (e.g., the sum of the grades or the sum of the squares of the grades or the sum of the qth power of the grades is less than or equal to 1). To relax these restrictions, linear Diophantine fuzzy set is a new extension of fuzzy sets, by additionally considering reference/control parameters. Thereby, the sum of membership grade and non-membership grade can be greater than 1, and even both of these grades can be 1. By selecting different pairs of reference parameters, linear Diophantine fuzzy sets can naturally categorize concerned problems and produce appropriate solutions accordingly. In this paper, the interval-valued linear Diophantine fuzzy set, which is a generalization of linear Diophantine fuzzy set, is studied. The interval-valued linear Diophantine fuzzy set is more efficient to deal with uncertain and vague information due to its flexible intervals of membership grades, non-membership grades, and reference parameters. Some basic operations on interval-valued linear Diophantine fuzzy sets are presented. We define interval-valued linear Diophantine fuzzy weighted average and interval-valued linear Diophantine fuzzy weighted geometric aggregation operators. Based on these new aggregation operators, we propose a method for multi-criteria decision-making based on supplier selection under the interval-valued linear Diophantine fuzzy environment. Besides, a real-life example, comparison study, and advantages of proposed aggregation operators are presented. We describe some correlation coefficient measures (type-1 and type-2) for the interval-valued linear Diophantine fuzzy sets and they are applied in medical diagnosis for Coronavirus Disease 2019 (COVID-19). Lastly, a comparative examination and the benefits of proposed correlation coefficient measures are also discussed.