Q-Fractional Fuzzy Frank Aggregation Operators and Their Application in Decision-Making

The q-rung orthopair fuzzy soft set is an advanced leeway of orthopair fuzzy sets, such as intuitionistic fuzzy soft set (IFSS) and Pythagorean fuzzy soft set (PFSS). The very lax requirement gives the evaluators excessive freedom to express their beliefs about membership degrees and non-membership degrees, making q-rung orthopair fuzzy soft set (q-ROFSS) have a broad scope of application in practical life. Considering the interaction, this paper will develop some novel operational laws for q-rung orthopair fuzzy soft numbers (q-ROFSNs). Some aggregation operators (AOs) such as q-rung orthopair fuzzy soft interactive weighted average (q-ROFSIWA) and q-rung orthopair fuzzy soft interactive weighted geometric (q-ROFSIWG) operators have been introduced. Also, we will prove some desirable properties such as idempotency, boundedness, and homogeneity for q-ROFSIWA and q-ROFSIWG operators. Multi-criteria decision-making (MCDM) plays a vital role to deals with the complications in manufacturing design for material selection. But, the prevailing MCDM methods habitually deliver incompatible outcomes. Based on the projected interactive aggregation operators, a robust MCDM method is planned for material selection in manufacturing design to accommodate this drawback. Furthermore, a comprehensive comparative analysis has been presented to confirm the pragmatism and validity of the proposed technique with some already available methods. The obtained outcomes through comparative studies show that our developed method is more efficient than existing approaches.

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.

The q-rung orthopair fuzzy sets (q-ROFSs), a novel concept for processing vague information, offer a more potent and all-encompassing method compared to traditional fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets. The inclusion of the parameter q allows for the q-rung orthopair fuzzy sets to capture a broader range of uncertainty of information. In this paper, we present two novel distance measures for q-ROFSs inspired by the Jensen–Shannon divergence, called DJS_2D and DJS_3D, and we analyze some properties they satisfy, such as non-degeneracy, symmetry, boundedness, and triangular inequality. Then, the normalized distance measures, called DJS_2D˜ and DJS_3D˜, are proposed and we verify their rationality through numerical experiments. Finally, we apply the proposed distance measures to practical scenarios, including pattern recognition and multicriteria decision-making, and the results demonstrate the effectiveness of the proposed distance measures.

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

As a generalized form of both intuitionistic fuzzy set and Pythagorean fuzzy sets, the q-rung orthopair fuzzy set (q-ROFS) has strong ability to handle uncertain or imprecision decisionmaking problems. This paper aims to introduce a new multiple criteria decision making method based on the original gain and lost dominance score (GLDS) method for investment evaluation. To do so, we first propose a new distance measure of q-rung orthopair fuzzy numbers (q-ROFNs), which takes into account the hesitancy degree of q-ROFNs. Subsequently, two methods are developed to determine the weights of DMs and criteria, respectively. Next, the original GLDS method is improved from the aspects of dominance flows and order scores of alternatives to address the multiple criteria decision making problems with q-ROFS information. Finally, a case study concerning the investment evaluation of the BE angle capital is given to illustrate the applicability and superiority of the proposed method.

Pythagorean fuzzy sets (briefly, PFSs) were created as an upgrade to intuitionistic fuzzy sets (briefly, IFSs) which helped to address some problems that IFSs couldn’t solve. The definition of q-rung orthopair fuzzy sets (briefly, q-ROFS) is then declared to generalize and solve PFS and IFS failures. Using the concept of PF beta -neighborhood, Zhan et al. defined the description of the covering through the Pythagorean fuzzy rough set (briefly, CPFRS). Hussain et al. also developed the concept of q-ROF beta -neighborhood to build the concept of covering through q-rung orthopair fuzzy rough sets (Cq-ROFRS). To enhance the results in Zhan et al.’s and Hussain et al.’s method and in a related context, the concept of PF complementary beta -neighborhood is constructed. Hence, using PF beta -neighborhood and PF complementary beta -neighborhood, three novel kinds of CPFRS are investigated and the related characteristics are analyzed. The interrelationships between Zhan et al.’s approach and our approaches are also discussed. Besides, the concept of q-ROF complementary beta -neighborhood is examined. Three new Cq-ROFRS models are differentiated using the principles of q-ROF beta -neighborhood and q-ROF complementary beta -neighborhood. As a result, the related properties and relationships between these various models and Hussain et al.’s model are established. Because of these correlations, we may consider our approach to be a generalization of Zhan et al.’s and Hussain et al’s approaches. Finally, we developed applications to solve MADM problems using CPFRS and Cq-ROFRS, as well as variances of the two methods using numerical examples are presented.

<abstract><p>The study on N-soft sets (NSSs) has been significantly developed recently. Hybrid models such as fuzzy N-soft sets, Intuitionistic fuzzy N-soft sets, and hesitant fuzzy N-soft sets were introduced to combine fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets with NSSs. Related to the hybrid models, it was also constructed some complements, operations and related properties. This article aims to construct a new hybrid model called hesitant intuitionistic fuzzy N-soft sets (HIFNSSs) to combine intuitionistic fuzzy N-soft sets and hesitant fuzzy N-soft sets. Moreover, we generalise HIFNSSs to generalized hesitant intuitionistic fuzzy N-soft sets (GHIFNSSs) as a hybrid model between generalized hesitant intuitionistic fuzzy sets and N-soft sets. It was also defined some complements of GHIFNSSs, intersection and union operations between GHIFNSSs, and proved that the operations between some particular complements hold De Morgan Law. In applying a GHIFNSS, we provide an algorithm for decision-making problems and its numerical illustration.</p></abstract>

Every problem in decision-making has a solution when the information that is available is properly and precisely modeled. This study focuses on non-binary data from N-soft sets and q-rung orthopair fuzzy values, referred to as group-based generalized q-rung orthopair fuzzy N-soft sets (GGq-ROFNSSs). The GGq-ROFNSSs model provides information simultaneously on numerous competing criteria, alternatives, sub-alternatives, and data summarization. We introduce properties of GGq-ROFNSSs such as distinct inclusion features of GGq-ROFNSSs, weak complements of the GGq-ROFNSS, top weak complements the GGq-ROFNSS, bottom weak complements the GGq-ROFNSS. We provide the notion of GGq-ROFNSWA and GGq-ROFNSWG operators as well as their idempotency, monotonicity, and boundedness features. The notion of GGq-ROFNSSs requires a sound methodology of multiple criteria decision making (MCDM) since GGq-ROFNSS combines numerous elements of complex decision-making. We provide a MCDM methodology for the GGq-ROFNSWA and GGq-ROFNSWG operators and depict it in a flowchart. The selection of solar panels for a city is a difficult procedure because it depends on several components such as environment, where the area is located, what kinds of needs are being met, etc. We find a solution to the problem of selecting a suitable solar panel for a city with their underlying characteristics. Finally, we provide a comparison of the suggested method with other techniques to demonstrate its advantages.

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.

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.

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 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.

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.

In this paper we introduce the concentration and dilation operators on interval valued intuitionistic fuzzy soft sets of root type which is a generalization of fuzzy set, intuitionistic fuzzy set, interval valued intuitionistic fuzzy set and intuitionistic fuzzy set of root type and establish some properties of these operators. We define Hamming distance between two interval valued intuitionistic fuzzy soft sets of root type and it is shown that it is a metric. A similarity measure based on this Hamming distance is defined and some properties are established. We also develop a decision making method based on the similarity measure of Hamming distance between interval valued intuitionistic fuzzy soft sets of root type. We develop an algorithm for the decision making problem and illustrate its working by means of examples.

As an attractive generalization of the intuitionistic fuzzy set (IFS), q-rung orthopair fuzzy set (q-ROFS) provides the decision makers (DMs) with a wide window for preference elicitation. Previous studies on q-ROFS indicate that there is an urge for a decision framework which can make use of the available information in a proper manner for making rational decisions. Motivated by the superiority of q-ROFS, in this paper, a new decision framework is proposed, which provides scientific methods for multi-attribute group decision-making (MAGDM). Initially, a programming model is developed for calculating weights of attributes with the help of partially known information. Later, another programming model is developed for determining the weights of DMs with the help of partially known information. Preferences from different DMs are aggregated rationally by using the weights of DMs and extending generalized Maclaurin symmetric mean (GMSM) operator to q-ROFS, which can properly capture the interrelationship among attributes. Further, complex proportional assessment (COPRAS) method is extended to q-ROFS for prioritization of objects by using attributes’ weight vector and aggregated preference matrix. The applicability of the proposed framework is demonstrated by using a renewable energy source prioritization problem from an Indian perspective. Finally, the superiorities and weaknesses of the framework are discussed in comparison with state-of-the-art methods.