Exploring n‐Dimensional Fuzzy Soft Sets: A Framework for Multicriteria Decision‐Making Problems

In general most of real life problem of decision making involve imprecise parameters. In recent past the major emphasis of research workers in this area have been to develop the reliable models to deal with such imprecision and vagueness effectively. Several theories have been developed such as fuzzy set theory, interval valued fuzzy set, intuitionistic fuzzy set, and interval valued intuitionistic fuzzy set, rough set and soft set. The primary objectives of all the above developed theories are to deal with various kinds of uncertainty, imprecision and vagueness but unfortunately every theory has certain limitations. In the present paper we briefly introduced the notion of soft set, fuzzy soft set and intuitionistic fuzzy soft set. We extend the Jurio et al construction method of converting fuzzy set into intuitionistic fuzzy set to fuzzy soft set into intuitionistic fuzzy soft set. Here we consider a problem of decision making in fuzzy soft set and presented a method to generalize it into intuitionistic fuzzy soft set based decision making problem for modelling the problem in a better way. In the process we used the construction method and score function of intuitionistic fuzzy number.

In our daily life, we often come across various problems related to the high dimensionality of data. In such type of problems irrelevant and superfluous data along with useful data is also present. Thus, dimensionality reduction has found wide applications in data analysis and management. In recent years the issue of dimensionality reduction in a fuzzy situation has also gained importance and has invited attention of researchers. The various techniques and theories have been developed by them to solve these types of problems. Some of these techniques based on probabilistic approach and others are non-probabilistic approach. For finding coherent and logical solution to various real life problems containing uncertainty, impreciseness and vagueness, fuzzy soft set theory is gaining significance. Recently, on a theoretical study of intuitionistic fuzzy soft has also been developed. The combination of intuitionistic fuzzy set and intuitionistic fuzzy soft set are more useful for application point of view in the environment when uncertainty due to vagueness is more complex. In this chapter the concept of fuzzy soft set is defined as hybridization of fuzzy set and soft set theory. A new technique is proposed to convert the soft set table into fuzzy soft set table, which is applied in dimension reduction of big data. The concepts of fuzzy soft set and intuitionistic fuzzy soft set are defined as hybridization of fuzzy set and soft set theory. New methods of applications of fuzzy soft and intuitionistic soft sets are also described in Medical Diagnosis following Sanchez’s approach.

In recent years, the fuzzy sets, interval-valued fuzzy sets, intuitionistic fuzzy sets and soft sets, which offer different perspectives for the structures containing the uncertainties, have attracted the interest of many researchers. Also, the intuitionistic fuzzy soft sets produced by combining the intuitionistic fuzzy sets with the soft sets have been widely studied. In this work, the concept of interval-valued fuzzy parameterized intuitionistic fuzzy soft set (IVFPIFS set) is introduced. This set is the generalization of soft sets, fuzzy soft sets, fuzzy parameterized (fuzzy) soft sets, interval-valued fuzzy parameterized (fuzzy) soft sets, intuitionistic fuzzy soft sets and fuzzy parameterized intuitionistic fuzzy soft sets. For the IVFPIFS sets, basic operations such as complement, union and intersection are defined. Also, the properties of these operations are investigated in detail. Lastly, an algorithm by using the aggregation operators based on the IVFPIFS sets is constructed. The examples are given to verify the feasibility and validity of the proposed algorithm.

Emerging trends in soft set theory and related topics.

Objective : The objective of Intuitionistic fuzzy binary soft sets is to extend the existing frameworks of binary soft sets and Intuitionistic fuzzy sets to accommodate both uncertainty and ambiguity in decision-making processes. This extension aims to provide a more flexible, representation of real-world data and phenomena, allowing for analysis and reasoning. Specifically, the objective involves defining the fundamental structures of Intuitionistic fuzzy binary soft sets over two initial universal sets, U1 and U2. Method: Reviewing the existing literature on binary soft sets, Intuitionistic fuzzy sets, and their extensions. Defining the fundamental structures of Intuitionistic fuzzy binary soft sets over two initial universal sets, U1 and U2. The next step is to propose the operations on Intuitionistic fuzzy binary soft sets and analyze their characteristics. Findings: The extension of binary soft sets and Intuitionistic fuzzy sets to Intuitionistic fuzzy binary soft sets is feasible. Intuitionistic fuzzy binary soft sets possess distinct properties worthy of exploration. Operations such as union, intersection, difference, AND, Or of Intuitionistic fuzzy binary soft sets exhibit specific behaviors elucidated through our findings. Novelty: The development of operations on Intuitionistic fuzzy binary soft sets provides a comprehensive framework and analysis. Exploration of properties unique to Intuitionistic fuzzy binary soft sets is contributing to the advancement of soft computing and set theory. Mathematics Classification code: 03D45, 03F55, 03E72 Keywords: Fuzzy, Soft, Intuitionistic, Binary soft, Fuzzy binary soft

For finding coherent and logical solution to various real life problems containing uncertainty, impreciseness and vagueness, fuzzy soft set theory is gaining importance. Later on a theoretical study of the intuitionistic fuzzy soft set was developed. The combination of intuitionistic fuzzy set and intuitionistic fuzzy soft set are more useful for application point of view in the field wherever uncertainty due to vagueness appear in more complex form. In the present communication the concepts of fuzzy soft set and Intuitionistic fuzzy soft Setare defined as hybridization of fuzzy set and soft set theory. A new method of application of intuitionistic fuzzy soft set is studied in Medical Diagnosis following Sanchez’s approach. A hypothetical case study is also discussed in brief using the proposed method.

In this paper, we flrstly deflned neutrosophic parameterized neutrosophic soft sets(npnisoft sets) which is combination of a neutro- sophic sets and a soft sets. Our npnisoft sets generalizes the concept of the other soft sets such as; fuzzy soft sets, intuitionistic fuzzy soft sets, neutrosophic soft sets, fuzzy parameterized soft sets, intuitionistic fuzzy parameterized soft sets, neutrosophic parameterized soft sets and so on. Then, we introduce some deflnitions and operations on npnisoft sets and some properties of the sets which are connected to operations have been established. Also, we have introduced the concept of npnisoft matrix and their operators which are more functional to make theoretical studies in the npnisoft set theory. Finally, we proposed the decision making method on the npnisoft set theory which can be applied to problems of many flelds that contain uncertainty and provided an example that demonstrated that this method can be successfully worked. 2010 AMS Classiflcation: 03E72, 08A72

Entropy on intuitionistic fuzzy soft sets and on interval-valued fuzzy soft sets

In recent years, the application of fuzzy sets has gained significant attraction in various fields, including medical diagnosis, due to their ability to manage uncertainties and imprecise information. This paper focuses on the comparative analysis of similarity measures within the realm of Generalized Interval-Valued Intuitionistic Fuzzy Soft Expert Sets (GIVIFSESs) and explores their application in the domain of medical diagnosis. Most of the important topics in fuzzy set theory are the similarity measures between the generalizations of fuzzy set theory. Similarity measures are a crucial tool which was used in data science. In this process, we measure how much the data sets are related and comparable. Measures of similarity give a numerical value that reveals the strength of associations between sets or sets of variables. In this paper, we initiate a new concept of generalized interval-valued intuitionistic fuzzy soft expert sets and their fundamental operations. This new concept is more flexible than existing concepts based on their algebraic definition. Unlike fuzzy sets, the concept of generalized interval-valued intuitionistic fuzzy soft expert sets is characterized by a degree of membership and degree of non-membership along with fuzzy set theory. The proposed methodology is validated through an empirical application in medical diagnosis, where (GIV-IFSESs) are employed to model the uncertainty and imprecision inherent in expert assessments. The selected similarity measures are then applied to quantify the degree of resemblance between different medical cases, facilitating a more informed decision-making process. We introduce several types of similarity measures on generalized interval-valued intuitionistic fuzzy soft expert sets. We also discuss a similarity measure of Type-I, Type-II, and Type-III for two (GIVIFSESs) and its application in medical diagnosis problems.

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).

<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>

In this paper, the notion of the interval valued neutrosophic soft sets (ivn soft sets) is defined which is a combination of an interval valued neutrosophic sets [36] and a soft sets [30]. Our ivn soft sets generalizes the concept of the soft set, fuzzy soft set, interval valued fuzzy soft set, intuitionistic fuzzy soft set, interval valued intuitionistic fuzzy soft set and neutrosophic soft set. Then, we introduce some definitions and operations on ivn soft sets sets. Some properties of ivn soft sets which are connected to operations have been established. Also, the aim of this paper is to investigate the decision making based on ivn soft sets by level soft sets. Therefore, we develop a decision making methods and then give a example to illustrate the developed approach. Keyword: Interval sets, soft sets, fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, level soft set.

In recent years, soft sets and neutrosophic sets have become a subject of great interest for researchers and have been widely studied based on decision-making problems. In this paper, we propose a new concept of the soft sets that is called interval-valued neutrosophic parameterized interval-valued neutrosophic soft sets (ivnpivn-soft sets). It is a generalization of the other soft sets such as fuzzy soft sets, intuitionistic fuzzy soft sets, neutrosophic soft sets, fuzzy parameterized soft sets, intuitionistic fuzzy parameterized soft sets, neutrosophic parameterized neutrosophic soft sets. Also, we proposed ivnpivn-soft matrices which are representative of the ivnpivn-soft sets. We then developed a decision-making method on the ivnpivn-soft sets and ivnpivn-soft matrices. Then, we proposed a numerical example to verify validity and feasibility of the developed method. Finally, the proposed method is compared with several different methods to verify its feasibility.

Decision Making | Soft Set | Level Soft Set | Intuitionistic Fuzzy Soft Sets | Weight Function | Opinion Weighted Vector | Weighted Interval Valued Intuitionistic Fuzzy Soft Multi Set | Reduct Weighted Intuitionistic Fuzzy Soft Multi Set

Earlier fuzzy set, vague set, intuitionistic fuzzy set, L fuzzy set etc are used as a mathematical tools for solving problems based on uncertainties or ambiguous in nature. But due to more complexity involves in problems exist in nature, traditional tools are unable to handle those in a systematic manner. So we need a tool which is more flexible to handle those problems. Which leads to the invention of soft set which was introduced by Molodtsov in 1999. Soft set (SS) theory is a mathematical tool deals with parametric data which are imprecise in nature. Ithis a generalization of fuzzy set theory. On the other hand Rough set (RS) theory and Neutrosophic set (NS) theory both rising as a powerful tool to handle these uncertain, incomplete, inconsistent and imprecise information in an effective manner. Actually Neutrosophic set is a generalization of intuitionistic fuzzy set. Sometimes it is not possible to handle all sorts of uncertain problems with a single mathematical tool. Fusion of two or more mathematical tools give rise to a new mathematical concept which gives an idea how to solve such type of problems in a more sophisticated ways. Which leads to the introduction of fuzzy soft set, rough soft set, intuitionistic fuzzy soft set, soft rough set etc. Neutrosophic soft set (NSS) was established by combining the concept of Soft set and Neutrosophic set. In this paper, using the concept of Rough set and Neutrosophic soft set a new concept known as Rough neutrosophic soft set (RNSS) is developed. Some properties and operations on them are introduced.