Intuitionistic Fuzzy Soft Sets Research Articles

An intelligent fuzzy parameterized multi-criteria decision-support system based on intuitionistic fuzzy hypersoft expert set for automobile evaluation

In every decision-making scenario, the role of parameters is pertinent but it becomes critical for the situations when parameters are ambiguous to be assessed by the experts so this sort of uncertainty is judged by assigning a fuzzy membership-grade to each parameter. The existing literature on fuzzy parameterization is unable to provide an appropriate model which may cope with hypersoft setting, multi-decisive opinions of experts and intuitionistic setting collectively. This shortcoming leads to the motivation of this study. In this paper, fuzzy parameterized intuitionistic fuzzy hypersoft expert set (FPIFHsES) is characterized which is capable to address the insufficiencies of existing models like fuzzy parameterized intuitionistic fuzzy soft expert set (FPIFSES) for the consideration of multi-argument approximate function. With the entitlement of this function, FPIFHsES tackles the real-life scenario where each attribute is meant to be further classified into its respective sub-attribute valued disjoint set. The FPIFHsES is more flexible and the reliable with the deep analysis of attributes in the decision-support system. The characterization of FPIFHsES is accomplished by employing theoretic, axiomatic, and algorithmic approaches. In order to validate the proposed model, an algorithm is proposed to study its role in decision-making while dealing with a real-world scenario.

Multi-attribute decision-making based on aggregations and similarity measures of neutrosophic hypersoft sets with possibility setting

ABSTRACT Possibility neutrosophic hypersoft set is the generalisation of possibility neutrosophic soft set, and it is the blend of possibility neutrosophic set and hypersoft set. It tackles the insufficiencies of possibility intuitionistic fuzzy set and soft set for the entitlement of degree of indeterminacy and multi-argument approximate function, respectively. This kind of function maps the sub-parametric tuples to the power set of universe. It emphasises the partitioning of each attribute into its attribute-valued set that is missing in the existing fuzzy soft set-like structures. These features make it a completely new mathematical tool for solving problems dealing with uncertainties. This makes the decision-making process more flexible and reliable. This research aims to initiate a novel structure that hybridises the possibility neutrosophic set with the hypersoft set for dealing with uncertainties efficiently. Therefore, the possibility neutrosophic hypersoft set is developed by employing theoretic, axiomatic, graphical and algorithmic approaches. After conceptual characterisation of its essential elementary notions, decision-support systems are presented based on AND and OR operations with the proposal of algorithms to assist the decision-making process. Similarity measures between possibility neutrosophic hypersoft sets are elaborated with the support of daily life applications. The comparison of the proposed work is discussed with the existing relevant models under certain evaluating features.

Cubic Intuitionistic Fuzzy Soft Set and it's Distance Measures

To deal with vagueness, falsity, attributive values and inconsistency, we introduce the cubic intuitionistic fuzzy soft set (CIFS-set) which is the extension of the cubic intuitionistic fuzzy set and propose distance measure, Hamming distance, Euclidean distance and separation measures of CIFS-set. Moreover, we present aggregate operator (P-union, R-intersection) of CIFS-sets. The proposed CIFS-set is more reliable, efficient and accurate. In future MCDM and MCGDM techniques can be propoed to deal with real life issues, and also this CIFS-set can be extended for its hybrids.

Recognition of cancer mediating biomarkers using rough approximations enabled intuitionistic fuzzy soft sets based similarity measure

In recent years, gene expression analysis has become crucial in studying microarray and provides several techniques related to computational biology and bioinformatics. This article introduces a novel Intuitionistic fuzzy set under rough multigranulation approximation based feature extraction from microarray gene expression dataset. The proposed model identifies cancer mediating human biomarkers using an Intuitionistic fuzzy soft set based similarity measure. Firstly, Intuitionistic fuzzy rough reduct is devised in the microarray datasets preceded by an entropy based pre-processing step. Intuitionistic fuzzy rough reduct produces the feature sets from the microarray datasets through roughness and accuracy measures that are differentially expressed in a cancerous state. Rough multigranulation approximations reduce the dimensions of the microarray datasets. Thereafter, a weighted similarity measure is designed using the intuitionistic fuzzy soft set for the classification of biomarkers having significant expression patterns from the normal state to the carcinogenic state. The proposed method is demonstrated on six microarray datasets and is validated with different performance metrics resulting in better effectiveness than the state-of-the-art methods.

(<img src=image/13426382_01.gif>)-Anti-Intuitionistic Fuzzy Soft b-Ideals in BCK/BCI-Algebras

Among many algebraic structures, algebras of logic form an essential class of algebras. BCK and BCI-algebras are two classes of logical algebras. They were introduced by Imai and Iséki [6, 7] in 1966 and have been extensively investigated by many researchers. The concept of fuzzy soft sets is introduced in [17] to generalize standard soft sets [21]. The concept of intuitionistic fuzzy soft sets is introduced by Maji et al. [18], which is based on a combination of the intuitionistic fuzzy set [2] and soft set models. The first section will discuss the origins and importance of studies in this article. Section 2 will review the definitions of a BCK/BCI-algebra, a soft set, a fuzzy soft set, and an intuitionistic fuzzy soft set and show the essential properties of BCK/BCI-algebras to be applied in the next section. In Section 3, the concept of an anti-intuitionistic fuzzy soft b-ideal (AIFSBI) is discussed in BCK/BCI-algebras, and essential properties are provided. A set of conditions is provided for an AIFSBI to be an anti-intuitionistic fuzzy soft ideal (AIFSI). The definition of quasi-coincidence of an intuitionistic fuzzy soft point with an intuitionistic fuzzy soft set (IFSS) is considered in a more general form. In Section 4, the concepts of an (<img src=image/13426382_01.gif>)-AFSBI and an (<img src=image/13426382_01.gif>)-AIFSBI of <img src=image/13426382_02.gif> are introduced, and some characterizations of (<img src=image/13426382_01.gif>)-AIFSBI are discussed using the concept of an AIFSBI with thresholds. Finally, conditions are given for a (<img src=image/13426382_01.gif>)-AIFSBI to be a (∈,∈)-AIFSBI.

Sanal sezgisel bulanık parametreli sezgisel bulanık esnek kümeler ve karar verme üzerine bir uygulaması

In this paper, it is aimed to express more accurately the intuitionistic fuzzy values expressed by the decision-makers for intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets. For this purpose, virtual intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets are proposed by defining the concepts of virtual lower and upper intuitionistic fuzzy approaches. Moreover, the basic set operations for this hybrid set type are examined together with their associated properties. In addition, a decision-making algorithm focusing on virtual intuitionistic fuzzy parameterized intuitionistic fuzzy soft sets is constructed. Finally, an analysis of the obtained results was made.

Einstein Aggregation Operators for Pythagorean Fuzzy Soft Sets with Their Application in Multiattribute Group Decision-Making

The Pythagorean fuzzy soft set (PFSS) is the most proficient and manipulative leeway of the Pythagorean fuzzy set (PFS), which contracts with parameterized values of the alternatives. It is a generalized form of the intuitionistic fuzzy soft set (IFSS), which provides healthier and more accurate evaluations through decision-making (DM). The main determination of this research is to prolong the idea of Einstein’s aggregation operators for PFSS. We introduce the Einstein operational laws for Pythagorean fuzzy soft numbers (PFSNs). Based on Einstein operational laws, we construct two novel aggregation operators (AOs) such as Pythagorean fuzzy soft Einstein-weighted averaging (PFSEWA) and Pythagorean fuzzy soft Einstein-weighted geometric (PFSEWG) operators. In addition, important possessions of proposed operators, such as idempotency, boundedness, and homogeneity, are discussed. Furthermore, to validate the practicability of the anticipated operators, a multiple attribute group decision-making (MAGDM) method is developed. We intend innovative AOs considering the Einstein norms for PFSS to elect the most subtle business. Pythagorean fuzzy soft numbers (PFSNs) support us to signify unclear data in real-world perception. Furthermore, a numerical description is planned to certify the efficacy and usability of the projected method in the DM practice. The recent approach’s pragmatism, usefulness, and tractability are validated through comparative exploration with the support of some prevalent studies.

A q-rung orthopair basic probability assignment and its application in medical diagnosis

Dempster-Shafer theory is widely used in decision-making and considered as one of the potential mathematical tools in order to fuse the evidence. However, existing studies in this theory show disadvantage due to conflicting nature of standard evidence set and the combination rule of evidence. In this paper, we have constructed the framework of q-rung evidence set to address the issue of conflicts based on the q-rung fuzzy number due to its more comprehensive range of advantage compared to the other fuzzy or discrete numbers. The proposed q-rung evidence set has the flexibility in assessing a parameter through the q-rung orthopair basic probability assignment consisting of membership and non-membership belief degree. Moreover, as the proposed q-rung orthopair basic probability assignment consists of pair of belief degrees, the possibility of conflicts cannot be ignored entirely. In this regard, a new association coefficient measure is introduced where each component of the belief degrees is modified through the weighted average mass technique. This paper uses various concept such as fuzzy soft sets, Deng entropy, association coefficient measure and score function for decision-making problem. Firstly, to obtain the initial q-rung belief function, we have implemented the Intuitionistic fuzzy soft set to assess the parameter of the alternatives and Deng entropy to find the uncertainty of the parameters. Secondly, the association coefficient measure is used to avoid the conflict through the modified form of evidence. Finally, we combined the evidence and found the score value of the Intuitionistic fuzzy numbers for the ranking of the alternatives based on the score values of alternatives. This study is validated with the case study in the medical diagnosis problem from the existing paper and compared the ranking of alternatives based on the score function of belief measures of the alternatives.

Application of lattice ordered interval valued intutitionistic fuzzy soft sets in decision-making

Motivated by intuitionistic fuzzy sets and soft sets, a novel concept of lattice ordered interval-valued intuitionistic fuzzy soft sets (LOIVIFSSs) is introduced in this article. Operational rules like union, intersection, complement and some properties of LOIVIFSSs are demonstrated with examples. In this regard, an algorithm is developed to solve the multiple criteria decision-making (MCDM) problems based on LOIVIFSSs. Further, a benchmark problem concerning medical diagnosis have been investigated and a comparative analysis with existing technique is furnished to strengthen our approach.

On m-polar Diophantine Fuzzy N-soft Set with Applications.

In this paper, we present a novel hybrid model m-polar Diophantine fuzzy N-soft set and define its operations. We generalize the concepts of fuzzy sets, soft sets, N-soft sets, fuzzy soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets and Pythagorean fuzzy N-soft sets by incorporating our proposed model. Additionally, we define three different sorts of complements for Pythagorean fuzzy N-soft sets and examine few outcomes, which do not hold in Pythagorean fuzzy N-soft sets complements unlike to crisp set. We further discuss (α, β, γ) -cut of m-polar Diophantine fuzzy N-soft sets and their properties. Lastly, we prove our claim that the defined model is a generalization of the soft set, N-soft set, fuzzy Nsoft set, intuitionistic fuzzy N soft set, and Pythagorean fuzzy N-soft set. m-polar Diophantine fuzzy N-soft set is more efficient and an adaptable model to manage uncertainties as it also overcomes drawbacks of existing models, which are to be generalized. We introduced the novel concept of m-polar Diophantine fuzzy N-soft sets (MPDFNS sets).

Interval-Valued Intuitionistic Fuzzy Soft Rough Approximation Operators and Their Applications in Decision Making Problem.

It has been found that fuzzy sets, rough sets and soft sets are closely related concepts. Many complicated problems in economics, engineering, social sciences, medical science and many other fields involve uncertain data. These problems, which one comes in real life, cannot be solved using classical mathematical methods. There are several well-known theories to describe uncertainty, for instance, fuzzy set theory, rough set theory, and other mathematical tools. But all of these theories have their inherit difficulties as pointed out by D. Molodtsov. In 1999, D. Molodtsov introduced the concept of soft sets, which can be seen as a new mathematical tool for dealing with uncertainties. The concept of rough sets, proposed by Z. Pawlak as a framework for the construction of approximations of concepts. It is a formal tool for modeling and processing insufficient and incomplete information. Zhou and Wu first proposed the concept of intuitionistic fuzzy rough sets (IFrough sets). The aim of this paper is to introduce the concept of interval-valued intuitionistic fuzzy soft rough sets (IVIFS rough sets). We also investigate some properties of IVIFS rough approximation operators. Some basic operations and properties are studied. Lastly applications have been shown in decision making problems.

Einstein-Ordered Weighted Geometric Operator for Pythagorean Fuzzy Soft Set with Its Application to Solve MAGDM Problem

The Pythagorean fuzzy soft set (PFSS) is the most influential and operative tool for maneuvering compared to the Pythagorean fuzzy set (PFS), which can accommodate the parameterization of alternatives. It is also a generalized form of intuitionistic fuzzy soft sets (IFSS), which delivers healthier and more exact valuations in the decision-making (DM) procedure. The primary purpose is to extend and propose ideas related to Einstein’s ordered weighted geometric aggregation operator from fuzzy structure to PFSS structure. The core objective of this work is to present a PFSS aggregation operator, such as the Pythagorean fuzzy soft Einstein-ordered weighted geometric (PFSEOWG) operator. In addition, the basic properties of the proposed operator are introduced, such as idempotency, boundedness, and homogeneity. Moreover, a DM method based on a developed operator has been presented to solve the multiattribute group decision-making (MAGDM) problem. A real-life application of the anticipated method has been offered for a capitalist to choose the most delicate business to finance his money. Finally, a brief comparative analysis with some current methods demonstrates the proposed approach’s effectiveness and reliability.

Group decision-making with Fermatean fuzzy soft expert knowledge

With the rapid growth of population, the global impact of solar technology is increasing by the day due to its advantages over other power production technologies. Demand for solar panel systems is soaring, thus provoking the arrival of many new manufacturers. Sale dealers or suppliers face an uncertain problem to choose the most adequate technological solution. To effectively address such kind of issues, in this paper we propose the Fermatean fuzzy soft expert set model by combining Fermatean fuzzy sets and soft expert sets. We describe this hybrid model with numerical examples. From a theoretical standpoint, we demonstrate some essential properties and define operations for this setting. They comprise the definitions of complement, union and intersection, the OR operation and the AND operation. Concerning practice in this new environment, we provide an algorithm for multi-criteria group decision making whose productiveness and authenticity is dutifully tested. We explore a practical application of this approach (that is, the selection of a suitable brand of solar panel system). Lastly, we give a comparison of our model with certain related mathematical tools, including fuzzy and intuitionistic fuzzy soft expert set models.

Novel Multicriteria Decision Making Approach for Interactive Aggregation Operators of q-Rung Orthopair Fuzzy Soft Set

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.

Decision making algorithmic techniques based on aggregation operations and similarity measures of possibility intuitionistic fuzzy hypersoft sets

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

Spherical Fuzzy Soft Rough Average Aggregation Operators and Their Applications to Multi-Criteria Decision Making

The objective of this article is to explore and generalize the notions of soft set and rough set along with spherical fuzzy set and to introduce the novel concept called spherical fuzzy soft rough set that is free from all those complications faced by many modern concepts like intuitionistic fuzzy soft rough set, Pythagorean fuzzy soft rough set, and q-rung orthopair fuzzy soft rough set. Since aggregation operators are the fundamental tools to translate the complete information into a distinct number, so some spherical fuzzy soft rough new average aggregation operators are introduced, such as spherical fuzzy soft rough weighted average, spherical fuzzy soft rough ordered weighted average, and spherical fuzzy soft rough hybrid average aggregation operators. Also, the basic characteristics of these introduced operators have been elaborated in detail. Furthermore, a multi-criteria decision-making (MCDM) technique has been developed and a descriptive example is given to support newly presented work. At the end of this article, a comparative study of the introduced technique has been established that shows how our work is more superior and efficient compared to the picture fuzzy soft set.

Neutrosophic κ-Structures in Ordered Semigroups

In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic κ-ideals and neutrosophic κ-interior ideals. We also use neutrosophic κ-ideals and neutrosophic κ-interior ideals to characterize ordered semigroups.

Retrieving the Missing Data From Incomplete Soft Set, Incomplete Fuzzy Soft Set and Incomplete Intuitionistic Fuzzy Soft Set

Dealing with an incomplete information has been a major issue in the theory of soft sets. In this paper, we have presented an approach to deal with incomplete soft set, incomplete fuzzy soft set and incomplete intuitionistic fuzzy soft set. For this purpose, we have introduced the notion of distance between two objects (parameters) which will be used to compute the degree of interdependence between them. This approach will use the full information of known data and the relationships between them. Data filling converts an incomplete soft set into a complete one which makes the soft sets applicable not only to decision making but also to other fields.

Pythagorean Fuzzy Soft Einstein Ordered Weighted Average Operator in Sustainable Supplier Selection Problem

Pythagorean fuzzy soft set (PFSS) is the most influential and operative extension of the Pythagorean fuzzy set (PFS), which contracts with the parametrized standards of the substitutes. It is also a generalized form of the intuitionistic fuzzy soft set (IFSS) and delivers a well and accurate estimation in the decision-making (DM) procedure. The primary purpose is to prolong and propose ideas related to Einstein’s ordered weighted aggregation operator from fuzzy to PFSS, comforting the condition that the sum of the degrees of membership function and nonmembership function is less than one and the sum of the squares of the degree of membership function and nonmembership function is less than one. We present a novel Pythagorean fuzzy soft Einstein ordered weighted averaging (PFSEOWA) operator based on operational laws for Pythagorean fuzzy soft numbers. Furthermore, some essential properties such as idempotency, boundedness, and homogeneity for the proposed operator have been presented in detail. The choice of a sustainable supplier is also examined as an essential part of sustainable supply chain management (SSCM) and is considered a crucial multiattribute group decision-making (MAGDM) issue. In some MAGDM problems, the relationship between alternatives and uncertain environments will be the main reason for deficient consequences. We have presented a novel aggregation operator for PFSS information to choose sustainable suppliers to cope with those complex issues. The Pythagorean fuzzy soft number (PFSN) helps to represent the obscure information in such real-world perspectives. The priority relationship of PFSS details is beneficial in coping with SSCM. The proposed method’s effectiveness is proved by comparing advantages, effectiveness, and flexibility among the existing studies.

Novel Technique for Group Decision-Making under Fuzzy Parameterized q -Rung Orthopair Fuzzy Soft Expert Framework

The q -rung orthopair fuzzy sets and their hybrid models are capable of dealing with uncertain situations very effectively than the theories of intuitionistic and Pythagorean fuzzy sets and thus have numerous decision-making applications in daily life, while the fuzzy parameterized soft set theory has its impact on different decision-making scenarios. Motivated by these facts, in this research article, these theories are combined to form a new structure named fuzzy parameterized q -rung orthopair fuzzy soft expert sets ( FP q ROFSESs) for dealing with more generalized information. The developed model is an efficient extension of fuzzy parameterized intuitionistic fuzzy soft expert sets. Some of its basic notions, including subset, complement, OR operation, AND operation, intersection, and union are studied and illustrated via examples. Moreover, to show the applicability and efficiency of the developed model, two real-life applications are solved under the FP q ROFSES approach, which is supported by an algorithm, the first application is about selecting an appropriate site for a cafe outlet, and the second application is about selecting the Best News Channel for an award. At last, a comparison of the initiated model with some existing approaches is presented to verify its advantages over them.