Enhancing decision making with soft set theory: a novel approach to object recognition from imprecise data

Emerging trends in soft set theory and related topics.

Rough set is a very powerful invention to the whole world for dealing with uncertain, incomplete and imprecise problems. Also soft set theory and neutrosophic set theory are advance mathematical tools to handle these uncertain, incomplete, inconsistent information in a better way. The purpose of this article is to expand the scope of rough set, soft set and neutrosophic set theory. We have introduced the concept of neutrosophic soft set with roughness without using full soft set. Some definitions, properties and examples have been established on neutrosophic soft rough set. Moreover, dispensable and equalities are written on roughness with neutrosophic soft set.

Soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. The main purpose of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. This paper is devoted to the further generalization of the soft rough set theory by using the ideal notion to reduce the soft boundary region. A new soft rough set model, called soft ideal approximation space, is proposed and its properties are derived. The new soft ideal lower and upper approximation operators are presented and their related properties are surveyed.

Decision-making involves several processes such as data pre-processing, data reduction and data selection. In order to assure a valuable solution is made, each of these processes needs to be successfully conducted. When dealing with complex data, parameter reduction is one of the essential processes that the decision-makers should take into account. It helps to reduce the processing time, computational memory and data dimensionality in the decision-making process. However, some of the parameter reduction methods were unable to generate a sub-optimal value during the parameter reduction process. This problem could affect the performance of the classification process. Soft set theory is one of the parameter reduction methods that faces this kind of problem. As a result of the study, to enhance the capability of soft set parameter reduction method, an integration between soft set and rough set theories as a parameter reduction method had been proposed. It was based on the efficiency of these two theories in processing complex and uncertain data problems. These two methods were sequentially applied to simplify the initial parameters in order to improve the performance of the classification process. The experimental work had returned positive classification results and successfully assisted the standard soft set parameter reduction method in generating sub-optimal reduction set and also the classifier in the classification process.

In this particular piece of work, the new hybrid concept of vague soft rough set theory is introduced. It is a combination of rough set theory, soft set theory, and vague set theory. Based on this new concept, some definitions and operations are introduced. Furthermore, lower and upper vague soft approximations, vague soft rough positive, vague soft negative, and vague soft boundaries are discussed with examples. In addition, several theorems are presented in terms of lower and upper vague soft approximations, supported by examples for better understanding. Finally, an entirely new mathematical structure known as the vague soft rough topological structure is introduced. The related definitions of open sets, closed sets, closure, and interior, as well as their relationships in vague soft rough topological spaces, are addressed. To enhance understanding of this study, numerous examples are provided.

Multi-attribute decision-making (MADM) is a cognitive process for evaluating data under various attributes in order to find the best option in terms of the preferences of decision makers. Handling uncertainty and vagueness in most real-world decision-making problems involves the application of mathematical tools dealing with uncertainty, such as soft set (SS) theory in decision-making processes. In recent years, different approaches have been suggested for demonstrating the effectiveness and efficiency of soft set theory in MADM. To date, no systematic literature review has been presented in this area. However, a review paper on existing SS-based methods can help future researchers to find the gaps and missing MADM approaches in current studies. In this paper, we review the methodologies and applications of 71 research papers published in 30 academic journals (extracted from online databases including ScienceDirect, Scopus, Springer, World of Scientific, and Hindawi) between January 2000 and December 2017. These papers have been categorized into two main groups: single and hybrid approaches. In addition, they have been classified with respect to MADM techniques, country of origin, journal, and publication year. The statistical analysis shows that “SSAW” method which has been employed by various authors is the most common technique. Moreover, the number of research publications by academics was highest in 2017. In addition, the “Applied Mathematical Modelling” journal and China are ranked as the academic journal and country that have presented the most major contributions in this domain, respectively.

After the paper of Molodtsov [Molodtsov D. Soft set theory — First results, Computers and Mathematics with Applications, 1999, vol. 37, no. 4-5, pp. 19-31.] first appeared, soft set theory grew at a breakneck pace. Several authors have introduced various operations, relations, results, etc. as well as other aspects in soft set theory and hybrid structures incorrectly, despite their widespread use in mathematics and allied areas. In his paper [Molodtsov D.A. Equivalence and correct operations for soft sets, International Robotics and Automation Journal, 2018, vol. 4, no. 1, pp. 18-21.], Molodtsov, the father of soft set theory, pointed out several wrong results and notions. Molodtsov [Molodtsov D.A. Structure of soft sets, Nechetkie Sistemy i Myagkie Vychisleniya, 2017, vol. 12, no. 1, pp. 5-18.] also stated that the concept of soft set had not been fully understood and used everywhere. As a result, it is important to revisit the quirks of those conceptions and provide a formal account of the notion of soft set equivalency. Molodtsov already explored many correct operations on soft sets. We use some notions and results of Molodtsov [Molodtsov D.A. Structure of soft sets, Nechetkie Sistemy i Myagkie Vychisleniya, 2017, vol. 12, no. 1, pp. 5-18.] to create matrix representations as well as related operations of soft sets, and to quantify the similarity between two soft sets.

Background Fuzzy soft sets are well-established for decision-making under uncertainty. However, integrating soft set theory with fuzzy codes remains unexplored, as fuzzy soft code sets have not addressed these concepts. This study presents novel fuzzy soft codes and discusses their application in diagnosing medical conditions and decision-making, bridging a critical gap in the literature. Methods Numerous scholars have examined fuzzy sets, fuzzy codes, and the theory of fuzzy soft sets and their applications over the years. This paper introduces the novel concept of fuzzy soft codes by combining fuzzy soft set theory with the notion of fuzzy codes, inspired by Ozkan’s 2002 definition of fuzzy codes. Results We define fuzzy soft codes as a parameterized family of fuzzy codes, detailing their matrix representation, set-theoretic operations, and algebraic properties. A new decision-making method using choice and score values from comparison tables is introduced. An example demonstrates its effectiveness for selecting optimal objects, proving its real-world potential in areas like medical diagnosis. This bridges a key theoretical gap and enables future extensions like intuitionistic fuzzy soft codes. Conclusions This study addresses a gap in the integration of soft set theory with the notion of fuzzy codes. The primary objective is to introduce the concept of fuzzy soft codes within the context of decision-making by merging fundamental ideas from soft set theory and fuzzy coding, while also examining several of their key properties. Additionally, a practical example is provided to demonstrate how this approach can be effectively utilized across a range of real world problem.

Background Fuzzy soft sets are well-established for decision-making under uncertainty. However, integrating soft set theory with fuzzy codes remains unexplored, as fuzzy soft code sets have not addressed these concepts. This study presents novel fuzzy soft codes and discusses their application in diagnosing medical conditions and decision-making, bridging a critical gap in the literature. Methods Numerous scholars have examined fuzzy sets, fuzzy codes, and the theory of fuzzy soft sets and their applications over the years. This paper introduces the novel concept of fuzzy soft codes by combining fuzzy soft set theory with the notion of fuzzy codes, inspired by Ozkan’s 2002 definition of fuzzy codes. Results We define fuzzy soft codes as a parameterized family of fuzzy codes, detailing their matrix representation, set-theoretic operations, and algebraic properties. A new decision-making method using choice and score values from comparison tables is introduced. An example demonstrates its effectiveness for selecting optimal objects, proving its real-world potential in areas like medical diagnosis. This bridges a key theoretical gap and enables future extensions like intuitionistic fuzzy soft codes. Conclusions This study addresses a gap in the integration of soft set theory with the notion of fuzzy codes. The primary objective is to introduce the concept of fuzzy soft codes within the context of decision-making by merging fundamental ideas from soft set theory and fuzzy coding, while also examining several of their key properties. Additionally, a practical example is provided to demonstrate how this approach can be effectively utilized across a range of real world problem.

Background Fuzzy soft sets are well-established for decision-making under uncertainty. However, integrating soft set theory with fuzzy codes remains unexplored, as fuzzy soft code sets have not addressed these concepts. This study presents novel fuzzy soft codes and discusses their application in diagnosing medical conditions and decision-making, bridging a critical gap in the literature. Methods Numerous scholars have examined fuzzy sets, fuzzy codes, and the theory of fuzzy soft sets and their applications over the years. This paper introduces the novel concept of fuzzy soft codes by combining fuzzy soft set theory with the notion of fuzzy codes, inspired by Ozkan’s 2002 definition of fuzzy codes. Results We define fuzzy soft codes as a parameterized family of fuzzy codes, detailing their matrix representation, set-theoretic operations, and algebraic properties. A new decision-making method using choice and score values from comparison tables is introduced. An example demonstrates its effectiveness for selecting optimal objects, proving its real-world potential in areas like medical diagnosis. This bridges a key theoretical gap and enables future extensions like intuitionistic fuzzy soft codes. Conclusions This study addresses a gap in the integration of soft set theory with the notion of fuzzy codes. The primary objective is to introduce the concept of fuzzy soft codes within the context of decision-making by merging fundamental ideas from soft set theory and fuzzy coding, while also examining several of their key properties. Additionally, a practical example is provided to demonstrate how this approach can be effectively utilized across a range of real world problem.

Mathematical models have extensively been used in problems related to engineering, computer sciences, economics, social, natural and medical sciences etc. It has become very common to use mathematical tools to solve, study the behavior and different aspects of a system and its different subsystems. Because of various uncertainties arising in real world situations, methods of classical mathematics may not be successfully applied to solve them. Thus, new mathematical theories such as probability theory and fuzzy set theory have been introduced by mathematicians and computer scientists to handle the problems associated with the uncertainties of a model. But there are certain deficiencies pertaining to the parametrization in fuzzy set theory. Soft set theory aims to provide enough tools in the form of parameters to deal with the uncertainty in a data and to represent it in a useful way. The distinguishing attribute of soft set theory is that unlike probability theory and fuzzy set theory, it does not uphold a precise quantity. This attribute has facilitated applications in decision making, demand analysis, forecasting, information sciences, mathematics and other disciplines. medskip In this thesis we will discuss several algebraic and topological properties of soft sets and fuzzy soft sets. Since soft sets can be considered as set-valued maps, the study of fixed point theory for multivalued maps on soft topological spaces and on other related structures will be also explored. The contributions of the study carried out in this thesis can be summarized as follows: Revisit of basic operations in soft set theory and proving some new results based on these modifications which would certainly set a new dimension to explore this theory further and would help to extend its limits further in different directions. Our findings can be applied to develop and modify the existing literature on soft topological spaces. Defining some new classes of mappings and then proving the existence and uniqueness of fixed point for such mappings which can be viewed as a positive contribution towards an advancement of metric fixed point theory. Initiative of soft fixed point theory in framework of soft metric spaces and proving the results lying at the intersection of soft set theory and fixed point theory which would help in establishing a bridge between these two flourishing areas of research. Extension of Caristi-Kirk's fixed point theorem to the setting of fuzzy metric spaces, obtaining a characterization of completeness for that spaces. Our study is also a starting point for the future research in the area of fuzzy soft fixed point theory.

Abstract. Molodtsov introduced the concept of soft sets, which can be seen as a newmathematical tool for dealing with uncertainty. In this paper, we initiate the study ofsoft BL-algebras by using the soft set theory. The notion of lteristic soft BL-algebras isintroduced and some related properties are investigated. 1. IntroductionTo solve complicated problems in economics, engineering, and environment, wecannot successfully use classical methods because of various uncertainties typicalfor those problems. There are three theories: theory of probability, theory of fuzzysets, and the interval mathematics which we can consider as mathematical toolsfor dealing with uncertainties. But all these theories have their own difficulties.Uncertainties cannot be handled using traditional mathematical tools but may bedealt with using a wide range of existing theories such as probability theory, theoryof (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics,and theory of rough sets. However, all of these theories have their own difficultieswhich have been pointed out in [12]. Maji et al. [10] and Molodtsov [12] sug-gested that one reason for these difficulties may be due to the inadequacy of theparametrization tool of the theory. To overcome these difficulties, Molodtsov [12]introduced the concept of soft set as a new mathematical tool for dealing with un-certainties that is free from the difficulties that have troubled the usual theoreticalapproaches. Molodtsov pointed out several directions for the applications of softsets. At present, research on the soft set theory is progressing rapidly. Maji etal. [11] described the application of soft set theory to a decision making problem.They also studied several operations on the theory of soft sets. The most appro-

Dimensionality reduction is the most popular technique, which is used in data analytics for the optimal solution over high-dimensional data. Many times some inconsistency and uncertainty are presented in huge data so to deal with such a type of data soft set theory is being developed. Soft set theory handles the difficulties of older theories like fuzzy set, rough set using new property that is parameterization reduction. This paper presents different soft set based NPR algorithms for information systems that give optimal solutions for big data. The goal of this article is to focus on the dimensionality reduction approach of data mining using soft set theory which can be applied to machine learning field. Thus normal dimensionality reduction of soft set (NDRSS) algorithm is proposed for real-time data to give a better optimal solution than existing soft set algorithms. The result shows that how-high dimensional data is to be reduced using soft set theory in machine learning.KeywordsSoft set theoryNormal parameter reductionDimensionality reductionBig data analysis

Purpose – The purpose of this paper is to present a new method for evaluation of emergency plans for unconventional emergency events by using the soft fuzzy rough set theory and methodology. Design/methodology/approach – In response to the problems of insufficient risk identification, incomplete and inaccurate data and different preference of decision makers, a new model for emergency plan evaluation is established by combining soft set theory with classical fuzzy rough set theory. Moreover, by combining the TOPSIS method with soft fuzzy rough set theory, the score value of the soft fuzzy lower and upper approximation is defined for the optimal object and the worst object. Finally, emergency plans are comprehensively evaluated according to the soft close degree of the soft fuzzy rough set theory. Findings – This paper presents a new perspective on emergency management decision making in unconventional emergency events. Also, the paper provides an effective model for evaluating emergency plans for unconventional events. Originality/value – The paper contributes to decision making in emergency management of unconventional emergency events. The model is useful for dealing with decision making with uncertain information.

Soft set theory, introduced by Molodtsov [D. Molodtsov, Soft set theory–First results, Computers & Mathematics with Applications 37(4–5) (1999) 19–31], provides a flexible framework for managing uncertainty and vagueness, addressing limitations in traditional approaches such as fuzzy set theory, rough set theory, and probability theory. Over time, fuzzy soft set theory has emerged as a significant extension, blending the principles of fuzzy set theory and soft set theory to support applications in various decision-making processes. This study revisits fuzzy soft set theory, addressing conceptual errors and inaccuracies in the definitions of t-norm, t-conorm, strong negation, and implication that deviated from Molodtsov’s foundational principles. Corrected definitions, viz., fuzzy soft t-norm, fuzzy soft t-conorm, fuzzy soft negation, and fuzzy soft implication are proposed to ensure theoretical rigor. The paper rectifies conceptual errors in prior work by Ali and Shabir [M. I. Ali and M. Shabir, Logic connectives for soft sets and fuzzy soft sets, IEEE Transactions on Fuzzy Systems 22(6) (2013) 1431–1442] and introduces refined results to strengthen the logical framework, providing a consistent foundation for future research and hybrid model development in this domain.