Framework development of continuous non-linear Diophantine fuzzy sets and its application to renewable energy source selection

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.

Type 2 fuzzy multiset is a new concept formed by integrating type 2 fuzzy set and multiset which are very useful for modelling uncertainties. The purpose of this paper is to introduce the concept of type 2 fuzzy α multicut set and to show how α- cut in secondary level helps to relate type 2 fuzzy α- multicuts and type 2 fuzzy multisets. The unique representation of type 2 fuzzy multisets using either the family of all type 2 fuzzy α multicuts or the family of all type 2 fuzzy α strong multicuts are also showed. The construction of membership grades in different levels are explained. Furthermore, we investigate the multicriteria group decision making based on type 2 fuzzy multisets. Finally, we give an example to show the validity of type 2 fuzzy multiset in group decision making problem.

Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “(m,n)-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank (m,n)-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with (m,n)-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of (m,n)-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example.

A new fuzzy set theory, C-fuzzy set theory, is introduced in this paper. It is a particular case of the classical set theory and satisfies all formulas of the classical set theory. To add a limitation to C-fuzzy set system, in which all fuzzy sets must be “non-uniform inclusive” to each other, then it forms a family of sub-systems, the Z-fuzzy set family. It can be proved that the Z0-fuzzy set system, one of Z-fuzzy set systems, is equivalent to Zadeh’s fuzzy set system. Analysis shows that 1) Zadeh’s fuzzy set system defines the relations A = B and A ⊆ B between two fuzzy sets A and B as “∀u ∈ U, (μA ∈ (u) = μB(u))” and “∀u ∈ U, (μA(u) ≤ μB(u))” respectively is inappropriate, because it makes all fuzzy sets be “non-uniformly inclusive”; 2) it is also inappropriate to define two fuzzy sets’ union and intersection operations as the max and min of their grades of membership, because this prevents fuzzy set’s ability to correctly reflect different kinds of fuzzy phenomenon in the natural world. Then it has to work around the problem by invent unnatural functions that are hard to understand, such as augmenting max and min for union and intersection to min{a + b, 1} and max{a + b – 1, 0}, but these functions are incorrect on inclusive case. If both pairs of definitions are used together, not only are they unnatural, but also they are still unable to cover all possible set relationships in the natural world; and 3) it is incorrect to define the set complement as 1 – μA(u), because it can be proved that set complement cannot exist in Zadeh’s fuzzy set, and it causes confusion in logic and thinking. And it is seriously mistaken to believe that logics of fuzzy sets necessarily go against classical and normal thinking, logic, and conception. The C-fuzzy set theory proposed in this paper overcomes all of the above errors and shortcomings, and more reasonably reflects fuzzy phenomenon in the natural world. It satisfies all relations, formulas, and operations of the classical set theory. It is consistent with normal, natural, and classical thinking, logic, and concepts.

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.

In 1983 the Intuitionistic Fuzzy Sets (IFSs) were introduced as an extension of Zadeh's fuzzy sets. Fifteen years later the Bipolar Fuzzy Sets (BFSs) were introduced as a new extension of fuzzy sets. In the article it is proved that each BFS can be transformed injectively or bijectively to an IFSs.

Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented.

Hierarchical structures and uncertainty measures are two main aspects in granular computing, approximate reasoning and cognitive process. Typical hesitant fuzzy sets, as a prime extension of fuzzy sets, are more flexible to reflect the hesitance and ambiguity in knowledge representation and decision making. In this paper, we mainly investigate the hierarchical structures and uncertainty measures in typical hesitant fuzzy backgrounds. Firstly, we propose the parameterized scalar cardinalities of typical hesitant fuzzy elements, typical hesitant fuzzy sets and typical hesitant fuzzy relations based on a more reasonable partial orders with a disjunctive semantic meaning, respectively, where the parameters represent the decision makers' risk preferences. Secondly, we present four ordered relations for typical hesitant fuzzy space and four uncertainty measures to characterize the ambiguity in typical hesitant fuzzy approximation space and discuss their relationships. Thirdly, the hierarchical structures of a multi-granulation typical hesitant fuzzy space are analyzed by various multi-granulation typical hesitant fuzzy knowledge bases. In addition, we construct the framework of multi-granulation typical hesitant fuzzy rough sets in terms of optimistic and pessimistic attitudes. Finally, we study the uncertainty measures for the multi-granulation typical hesitant fuzzy approximation space based on the maximal and minimal knowledge bases, respectively.

Multisets alias bags are similar to fuzzy sets but essentially different in basic concepts and operations. We overview multisets together with basics of fuzzy sets in order to observe differences between the two. We then introduce fuzzy multisets and the combination of the both concepts. There is another concept of real-valued multisets as a generalization of multisets. Rough approximations of multisets and fuzzy multisets are discussed which uses a natural projection of the universal set onto the set of equivalence classes.

The existing concepts of picture fuzzy sets (PFS), spherical fuzzy sets (SFSs), T-spherical fuzzy sets (T-SFSs) and neutrosophic sets (NSs) have numerous applications in decision-making problems, but they have various strict limitations for their satisfaction, dissatisfaction, abstain or refusal grades. To relax these strict constraints, we introduce the concept of spherical linear Diophantine fuzzy sets (SLDFSs) with the inclusion of reference or control parameters. A SLDFS with parameterizations process is very helpful for modeling uncertainties in the multi-criteria decision making (MCDM) process. SLDFSs can classify a physical system with the help of reference parameters. We discuss various real-life applications of SLDFSs towards digital image processing, network systems, vote casting, electrical engineering, medication, and selection of optimal choice. We show some drawbacks of operations of picture fuzzy sets and their corresponding aggregation operators. Some new operations on picture fuzzy sets are also introduced. Some fundamental operations on SLDFSs and different types of score functions of spherical linear Diophantine fuzzy numbers (SLDFNs) are proposed. New aggregation operators named spherical linear Diophantine fuzzy weighted geometric aggregation (SLDFWGA) and spherical linear Diophantine fuzzy weighted average aggregation (SLDFWAA) operators are developed for a robust MCDM approach. An application of the proposed methodology with SLDF information is illustrated. The comparison analysis of the final ranking is also given to demonstrate the validity, feasibility, and efficiency of the proposed MCDM approach.

A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

THE CONCEPT OF GRADE OF MEMBERSHIP

This paper describes a generalized Diophantine fuzzy sets, which can be seen as a generalization of both Diophantine fuzzy sets and Pythagorean fuzzy sets. We define the basic properties of generalized Diophantine fuzzy set, as well as their relationships and distances. We compare Diophantine fuzzy sets with other Diophantine Pythagorean fuzzy sets to demonstrate their importance in the literature. We introduce new operators including necessity, possibility, accuracy function and score function. Furthermore, we discuss the new distance between normalized Euclidean distance and normalized Hamming distance. For a generalized Diophantine fuzzy relation, image and inverse image functions are defined. Numerous real-world applications can be found for the prevalent ideas of intuitionistic fuzzy sets, Pythagorean fuzzy sets, Diophantine fuzzy sets and q-rung orthopair fuzzy sets. Regretfully, these theories about the membership and non-membership grades have their own limits. We provide a new idea the generalized Diophantine fuzzy set that eliminates these limitations by including reference parameters. Compared to other kinds of fuzzy sets, there are more applications for generalized Diophantine fuzzy sets. We offer practical examples that show how different enhanced distances might be used in everyday situations. Additionally, to demonstrate the effectiveness of the suggested approach, flowchart based multi-criteria decision-making is provided and used to a numerical example. The outcomes are assessed for various parameter values. Furthermore, a comparative analysis developed to demonstrate the superiority of the suggested technique over current methodologies.

A systematic clinical healthcare model for assessing post-pandemic recovery in heart diagnosis