Set-theoretic Operations Research Articles

A Novel Hesitant Cubical Dombi Fuzzy Aggregation Operators for Selecting Green Supplier Chain Managements

A hesitant fuzzy (HF) set enhances the concept of fuzzy sets by addressing disagreements among decision-makers about the membership degree of an element. Similarly, the Cubical Fuzzy Set (CFS) is useful for managing uncertainty in decision-making problems. However, existing methods often lack integration of hesitation and cubical uncertainty, and there is limited exploration of their combined effects on aggregation processes. In this paper, we introduce the Hesitant Cubical Fuzzy Set (HCFS), which integrates the principles of HF sets and CFS to address these limitations. We define several set-theoretical operations for HCFSs and develop Dombi operations for them. Furthermore, we present a range of aggregation operators based on Dombi operations, including the Hesitant Cubical Dombi Fuzzy Weighted Arithmetic Averaging (HCDFWAA) Operator, the Hesitant Cubical Dombi Fuzzy Weighted Geometric Averaging (HCDFWGA) Operator, the Hesitant Cubical Dombi Fuzzy Ordered Weighted Arithmetic Averaging (HCDFOWAA) Operator, and the Hesitant Cubical Dombi Fuzzy Ordered Weighted Geometric Averaging (HCDFOWGA) Operator, and examine their properties. Additionally, we propose a multi-criteria group decision-making method and algorithm within the Hesitant Cubical Fuzzy framework. To address gaps in practical application, we provide an example of the selection of green suppliers in supply chain management. We also perform a comparative analysis with existing operators to highlight the advantages and effectiveness of our approach, emphasizing how the integration of hesitation and cubical uncertainty can enhance decision-making processes.

Selection of an Appropriate Global Partner for Companies Using the Innovative Extension of the TOPSIS Method with Intuitionistic Hesitant Fuzzy Rough Information

In this research, we introduce the intuitionistic hesitant fuzzy rough set by integrating the notions of an intuitionistic hesitant fuzzy set and rough set and present some intuitionistic hesitant fuzzy rough set theoretical operations. We compile a list of aggregation operators based on the intuitionistic hesitant fuzzy rough set, including the intuitionistic hesitant fuzzy rough Dombi weighted arithmetic averaging aggregation operator, the intuitionistic hesitant fuzzy rough Dombi ordered weighted arithmetic averaging aggregation operator, and the intuitionistic hesitant fuzzy rough Dombi hybrid weighted arithmetic averaging aggregation operator, and demonstrate several essential characteristics of the aforementioned aggregation operators. Furthermore, we provide a multi attribute decision-making approach and the technique of the suggested approach in the context of the intuitionistic hesitant fuzzy rough set. A real-world problem for selecting a suitable worldwide partner for companies is employed to demonstrate the effectiveness of the suggested approach. The sensitivity analysis of the decision-making results of the suggested aggregation operators are evaluated. The demonstrative analysis reveals that the outlined strategy has applicability and flexibility in aggregating intuitionistic hesitant fuzzy rough information and is feasible and insightful for dealing with multi attribute decision making issues based on the intuitionistic hesitant fuzzy rough set. In addition, we present a comparison study with the TOPSIS approach to illustrate the advantages and authenticity of the novel procedure. Furthermore, the characteristics and analytic comparison of the current technique to those outlined in the literature are addressed.

Novel similarity measures under complex pythagorean fuzzy soft matrices and their application in decision making problems

Complex fuzzy soft matrices play a crucial role in various applications, including decision-making, pattern recognition, signals processing, and image processing. The main objective of this study is to introduce the unique notions of complex Pythagorean fuzzy soft matrices (CPFSMs), which provide more flexibility and accuracy in modelling uncertainty. CPFSMs incorporate Pythagorean fuzzy soft matrices, allowing for more sophisticated uncertainty modeling. The key findings of CPFSMs, specific instances, and certain fundamental set-theoretic operations and principles were covered. A set of new distance metrics between two CPFSMs has been defined. In the context of complex Pythagorean fuzzy soft sets and complex Pythagorean fuzzy soft matrices, we created a CPFS decision-making technique. Moreover, the application’s numerical example and comparison analysis have been effectively demonstrated. Thus, by integrating the concepts of Pythagorean fuzzy sets, soft matrices, and complex numbers, CPFSMs provide a robust framework with membership and non-membership degrees for complex decision-making modeling and analyzing uncertain data.

Inequalities for totally nonnegative matrices: Gantmacher–Krein, Karlin, and Laplace

Inequalities for totally nonnegative matrices: Gantmacher–Krein, Karlin, and Laplace

N-bipolar hypersoft sets: Enhancing decision-making algorithms.

This paper introduces N-bipolar hypersoft (N-BHS) sets, a versatile extension of bipolar hypersoft (BHS) sets designed to effectively manage evaluations encompassing both binary and non-binary data, thereby exhibiting heightened versatility. The major contributions of this framework are twofold: Firstly, the N-BHS set introduces a parameterized representation of the universe, providing a nuanced and finite granularity in perceiving attributes, thereby distinguishing itself from conventional binary BHS sets and continuous fuzzy BHS sets. Secondly, this model signifies a new area of research aimed at overcoming limitations inherent in the N-bipolar soft set when handling multi-argument approximate functions. Through the strategic partitioning of attributes into distinct subattribute values using disjoint sets, the N-BHS set emerges as a powerful tool for effectively addressing uncertainty-related problems. In pursuit of these objectives, the paper outlines various algebraic definitions, including incomplete N-BHS sets, efficient N-BHS sets, normalized N-BHS sets, equivalence under normalization, N-BHS complements, and BHS sets derived from a threshold, exemplified through illustrative examples. Additionally, the article explores set-theoretic operations within the N-BHS sets framework, such as relative null/whole N-BHS sets, N-BHS subsets, and two distinct approaches to N-BHS extended/restricted union and intersection. Finally, it proposes and compares decision-making methodologies regarding N-BHS sets, including a comprehensive comparison with relevant existing models.

A multi-attribute decision making context for supply chain management using possibility single-valued neutrosophic soft settings

Argued decisions, risk estimation, empirical study, forecasts, apprehension supervision, and productive exposition depend on the degree of possibility. It enables us to assess the degree to which specific results are acceptable and to make more informed decisions by drawing on the information at our disposal and logic. Although some scholars have previously examined hybrid structures resembling fuzzy soft sets with possibility degree settings serving as fuzzy membership grades, the current study introduces an innovative structure that permits more adaptable and comprehensive settings: single-valued neutrosophic grades, to serve as possibility degrees. Thus, this study aims to introduce a new mathematical structure, i.e., possibility single-valued neutrosophic soft set (psv- NSOS), which combines three important theories (i.e., possibility theory, single-valued neutrosophic theory, and soft set theory). The basic notions, and set-theoretic operations, i.e., union and intersection, of psv-NSOS are investigated and manipulated with matrix representations. Considering evaluating suppliers for a real estate construction project as a multi-attribute decision-making issue, an algorithm that utilizes the matrix manipulations of proposed set-theoretic operations is presented. The suggested algorithm?s robustness is confirmed by comparison to evaluate its dependability and adaptability for modeling uncertainties related to the supplier selection problem.

Possibility Fermatean Neutrosophic Soft Set

In this paper, we introduce the concept of Possibility Fermatean Neutrosophic Soft Set and define some related concepts such as Possibility Fermatean Neutrosophic Soft subset, Possibility Fermatean Neutrosophic Soft null set, and Possibility Fermatean Neutrosophic Soft universal set. Then, we define set-theoretical operations of Possibility Fermatean Neutrosophic Soft Sets such as union, intersection, and complement, and investigate some properties of these operations. We also introduce AND-product and OR-product operations between two Possibility Fermatean Neutrosophic Soft Sets. We propose a decision-making method called the Possibility Fermatean Neutrosophic Soft decision-making method (PFNS-decision-making method) which can be applied to decision-making problems involving uncertainty based on AND-product operation. We finally give a numerical example to display the application of the method that can be successfully applied to the problems.

A Simple Stuck-at-faults Detection Method in Digital Combinational Circuits. II

This article proposes the improved method for detecting (diagnosing) stuck-at-faults (0/1) in PIPO-type digital combinational circuits described by a system of logical functions. Compared to already known methods and algorithms, the presented approach is characterized by a simpler implementation of the search for vectors of the test codes for detection of such malfunctions at arbitrary points of a logic circuit with many outputs due to the usage of several simple numerical set-theoretic operations and procedures. The given examples prove the advantages of the proposed method.

A Fuzzy Parameterized Multiattribute Decision-Making Framework for Supplier Chain Management Based on Picture Fuzzy Soft Information

Supplier selection as a multiattribute decision-making (MADM) problem has various inherent uncertainties due to a number of symmetrical variables. In order to handle such information-based uncertainties, rational models like intuitionistic fuzzy sets have already been introduced in the literature. However, a picture fuzzy set (PiFS) with four dimensions of positive, neutral, negative, and rejection is better at capturing and interpreting such kinds of ambiguous information. Additionally, fuzzy parameterization (FPara) is helpful for evaluating the degree of uncertainty in the parameters. This study aims to develop a fuzzy parameterized picture fuzzy soft set (FpPiFSS) by integrating the ideas of PiFS and FPara. This integration is more adaptable and practical since it helps decision makers manage approximation depending on their objectivity and parameterization uncertainty. With the assistance of instructive examples, some of the set-theoretic operations are examined. A decision support framework is constructed using matrix manipulation, preferential weighting, fuzzy parameterized grades based on Pythagorean means, and the approximations of decision makers. This framework proposes a reliable algorithm to evaluate four timber suppliers (initially scrutinized by perusal process) based on eight categorical parameters for real estate projects. In order to accomplish suppliers evaluation, crucial validation outcomes are taken into account, including delivery level, purchase cost, capacity, product quality, lead time, green degree, location, and flexibility. To assess the advantages, dependability, and flexibility of the recommended strategy, comparisons in terms of computation and structure are provided. Consequently, the results are found to be reliable, analog, and consistent despite the use of fuzzy parameterization and picture fuzzy setting.

N-Hypersoft Sets: An Innovative Extension of Hypersoft Sets and Their Applications

This paper introduces N-hypersoft (N-HS) sets—an enriched and versatile extension of hypersoft (HS) sets—designed to handle evaluations involving both binary and non-binary data while embodying an inherent sense of structural symmetry. The paper presents several algebraic definitions, including incomplete N-HS sets, efficient N-HS sets, normalized N-HS sets, equivalence under normalization, N-HS complements, and HS sets derived from a threshold. These definitions are accompanied by illustrative examples. Additionally, the paper delves into various set-theoretic operations within the framework of N-HS sets, such as relative null/whole N-HS sets, N-HS subsets, and N-HS extended/restricted union and intersection, presented in two different ways. Finally, the paper presents and compares decision-making methodologies regarding N-HS sets.

An optimized multi-attribute decision-making approach to construction supply chain management by using complex picture fuzzy soft set.

Supplier selection is a critical decision-making process for any organization, as it directly impacts the quality, cost, and reliability of its products and services. However, the supplier selection problem can become highly complex due to the uncertainties and vagueness associated with it. To overcome these complexities, multi-criteria decision analysis, and fuzzy logic have been used to incorporate uncertainties and vagueness into the supplier selection process. These techniques can help organizations make informed decisions and mitigate the risks associated with supplier selection. In this article, a complex picture fuzzy soft set (cpFSS), a generalized fuzzy set-like structure, is developed to deal with information-based uncertainties involved in the supplier selection process. It can maintain the expected information-based periodicity by introducing amplitude and phase terms. The amplitude term is meant for fuzzy membership, and the phase term is for managing its periodicity within the complex plane. The cpFSS also facilitates the decision-makers by allowing them the opportunity to provide their neutral grade-based opinions for objects under observation. Firstly, the essential notions and set-theoretic operations of cpFSS are investigated and illustrated with examples. Secondly, a MADM-based algorithm is proposed by describing new matrix-based aggregations of cpFSS like the core matrix, maximum and minimum decision value matrices, and score. Lastly, the proposed algorithm is implemented in real-world applications with the aim of selecting a suitable supplier for the provision of required materials for construction projects. With the sensitivity analysis of score values through Pythagorean means, it can be concluded that the results and rankings of the suppliers are consistent. Moreover, through structural comparison, the proposed structure is proven to be more flexible and reliable as compared to existing fuzzy set-like structures.

Extend Tversky’s Ratio Model to an Asymmetric Similarity Measurement Model with Three Conditional Parameters: 3p-ASM Model

Generally, the similarity between objects is often measured by symmetric operators, such as Cosine, Dice, and Jaccard similarity. However, the ratio model originally proposed by Tversky pointed out that the similarity using the feature matching method tends to be asymmetric. Furthermore, in many practical situations, the existing similarity measures using the feature matching method have some limitations: the calculation formulas are symmetrical, it is not intuitive based on binary features, and it is not easy to calculate based on fuzzy sets. To overcome such limitations, some other asymmetries have proposed to directly combine Tversky’s ratio model with the classical symmetric similarity metric, which in turn leads to the inability to identify different features between the compared objects and affects their similarity accuracy. Therefore, aiming to avoid these, this paper will focus on extending Tversky’s ratio model to a series of 3-parameter asymmetric similarity metrics (3p-ASM), using three conditional parameters to describe both common and different features. First, the set-based 3p-ASM is achieved due to the general and fuzzy set-theoretic operations, when estimating features in left{ 0,1right} and left[ 0,1right] , respectively. Then, considering that the estimated values of features can also be expressed as vectors, it will be extended to the vector-based 3p-ASM. Finally, a vector form of 3p-ASM is compared with existing classical methods and a comparative analysis is performed to demonstrate its effectiveness and validity. It is then applied to the KNN model in order to select the most similar items.

Refined Fuzzy Soft Sets: Properties, Set-Theoretic Operations and Axiomatic Results

This article discusses the results of an investigation into refined fuzzy soft sets, a novel variant of traditional fuzzy sets. Refined fuzzy soft sets provide a versatile method of data analysis, inspired by the need to deal with uncertainty and ambiguity in real-world data. This research expands on prior work in fuzzy set theory by investigating the nature and characteristics of refined fuzzy soft sets. They are useful in decision-making, pattern recognition, image processing, and control theory because of their capacity to deal with uncertainty, ambiguity, and the inclusion of expert information. This study analyzes these fuzzy set models and compares them to others in the field to reveal their advantages and disadvantages. The practical uses of enhanced fuzzy soft sets are also examined, along with possible future research strategies on this exciting new topic.

An extension of TOPSIS method using interval bipolar linguistic neutrosophic set and its application

TOPSIS, known as the “Technique for Order of Preference by Similarity to Ideal Solution” (TOPSIS), is one of the well-known methods for solving “multi-criteria decision-making” (MCDM) problems. Many TOPSIS methods have been developed to solve real-life problems using neutrosophic sets. However, no study has developed the TOPSIS method under interval bipolar linguistic neutrosophic environments. Therefore, this study aims to present a new concept of “interval bipolar linguistic neutrosophic set” (IBL_NS). IBL_NS is more flexible and adaptable to real-world applications than other sets. Some set-theoretic operations, such as “union”, “intersection”, and “complement”, and the “operational rules” of IBL_NS are defined. Then, a new TOPSIS procedure in IBL_NS is developed. In the proposed IBL_NS-TOPSIS method, ratings of alternatives and importance weights of criteria are expressed in IBL_NS. An application is presented demonstrating the advantages of the proposed IBL_NS-TOPSIS approach.

A Simple Stuck-at-faults Detection Method in Digital Combinational Circuits

This paper considers the new method of detection (diagnostic) stuck-at-faults (0/1) in digital combinational circuits based on a numerical set-theoretical approach. Compared to known methods and algorithms, the proposed approach differs in simpler implementation of searching for vectors of test codes at arbitrary points of the studied logic circuit. A few simple set-theoretical operations and procedures are sufficient to determine the location and the type of a stuck-at-fault (0/1). This is evidenced by the presented examples of application of the proposed method, that are borrowed from the publications of well-known authors.

Membership Functions, Set-Theoretic Operations, Distance Measurement Methods Based on Ambiguous Set Theory: A Solution to a Decision-Making Problem in Selecting the Appropriate Colleges

Membership Functions, Set-Theoretic Operations, Distance Measurement Methods Based on Ambiguous Set Theory: A Solution to a Decision-Making Problem in Selecting the Appropriate Colleges

New Possibility Soft Sets with Quality Assurance Application in Distance Education

The probability fuzzy soft set is an approach that is obtained by adding a probability degree to each approximate element of the fuzzy soft set and is formed by combining Fuzzy Set Theory and Soft Set Theory. This idea was later studied in different sets, such as intuitionistic fuzzy sets, neutrosophic fuzzy sets, Pythagorean fuzzy sets, etc. In this paper, a new Possibility Set is defined in the Fermatean Fuzzy environment and the essential features of new sets have been examined. The set-theoretic operations of new possibility sets, such as subset, soft equal, complement, union, intersection, AND, and OR, have been characterized with the help of elaborated examples. Their fundamental laws and properties are also discussed. A practical example concerning quality assurance in distance education is studied to demonstrate the applicability of new similarity measures in decision-making situations. Thus, it has been shown that the newly given similarity measure can be successfully applied to real-world decision problems. Finally, a comparison of the similarity of the proposed model is made with some existing models. Received: 13 October 2022 | Revised: 1 November 2022 | Accepted: 17 November 2022 Conflicts of Interest The author declares that he has no conflicts of interest to this work.

A study of quadratic Diophantine fuzzy sets with structural properties and their application in face mask detection during COVID-19

<abstract><p>During the COVID-19 pandemic, identifying face masks with artificial intelligence was a crucial challenge for decision support systems. To address this challenge, we propose a quadratic Diophantine fuzzy decision-making model to rank artificial intelligence techniques for detecting masks, aiming to prevent the global spread of the disease. Our paper introduces the innovative concept of quadratic Diophantine fuzzy sets (QDFSs), which are advanced tools for modeling the uncertainty inherent in a given phenomenon. We investigate the structural properties of QDFSs and demonstrate that they generalize various fuzzy sets. In addition, we introduce essential algebraic operations, set-theoretical operations, and aggregation operators. Finally, we present a numerical case study that applies our proposed algorithms to select a unique face mask detection method and evaluate the effectiveness of our techniques. Our findings demonstrate the viability of our mask identification methodology during the COVID-19 outbreak.</p></abstract>

Multi-Attribute Decision Making with Einstein Aggregation Operators in Complex Q-Rung Orthopair Fuzzy Hypersoft Environments

The purpose of our research is to extend the formal representation of the human mind to the concept of the complex q-rung orthopair fuzzy hypersoft set (Cq-ROFHSS), a more general hybrid theory. A great deal of imprecision and ambiguity can be captured by it, which is common in human interpretations. It provides a multiparameterized mathematical tool for the order-based fuzzy modeling of contradictory two-dimensional data, which provides a more effective way of expressing time-period problems as well as two-dimensional information within a dataset. Thus, the proposed theory combines the parametric structure of complex q-rung orthopair fuzzy sets and hypersoft sets. Through the use of the parameter q, the framework captures information beyond the limited space of complex intuitionistic fuzzy hypersoft sets and complex Pythagorean fuzzy hypersoft sets. By establishing basic set-theoretic operations, we demonstrate some of the fundamental properties of the model. To expand the mathematical toolbox in this field, Einstein and other basic operations will be introduced to complex q-rung orthopair fuzzy hypersoft values. The relationship between it and existing methods demonstrates its exceptional flexibility. The Einstein aggregation operator, score function, and accuracy function are used to develop two multi-attribute decision-making algorithms, which prioritize based on the score function and accuracy function to ideal schemes under Cq-ROFHSS, which captures subtle differences in periodically inconsistent data sets. The feasibility of the approach will be demonstrated through a case study of selected distributed control systems. The rationality of these strategies has been confirmed by comparison with mainstream technologies. Additionally, we demonstrate that these results are compatible with explicit histograms and Spearman correlation analyses. The strengths of each approach are analyzed in a comparative manner. The proposed model is then examined and compared with other theories, demonstrating its strength, validity, and flexibility.

An innovative decisive framework for optimized agri-automobile evaluation and HRM pattern recognition via possibility fuzzy hypersoft setting

In multi-attribute decision-making system, decision-makers have to face two kinds of situations: (i) the opted parameters are likely to be classified into their respective parametric-valued sub-collections and (ii) the acceptance degree for approximate opinions of decision-makers is required to be assessed by possibility setting. The literature related to fuzzy soft sets is unable to provide any model which can tackle such situations collectively. Therefore, this study aims to address this scarcity through the development of a novel structure, that is, possibility fuzzy hypersoft set ([Formula: see text]-set). Firstly, the algebraic properties and set-theoretic operations of [Formula: see text]-set are characterized by mathematical illustration. Secondly, two algorithms based on AND and OR-operations of [Formula: see text]-set are proposed and authenticated through application in multi-attribute decision-making real-world problems for the evaluation of agri-automobile and then their suitability is judged through vivid comparison. Thirdly, similarity measures between [Formula: see text]-sets are formulated and validated with the help of application in recruitment-based pattern recognition and its significance is assessed through comparison with most relevant models. Lastly, the advantageous aspects of the proposed structure are analyzed by its comparison with possibility fuzzy soft set-like models by observing some important evaluating features.