Neutrosophic Sets: An Overview

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems.

In this paper, we mainly investigate the relation theory and topological structures of rough neutrosophic sets which constructed by combining rough sets and neutrosophic sets. Firstly, we introduce the concept of algebra which is operations and identity relation. Then, discuss the properties of inverse, reflexive, symmetric as well as transitive relation of rough neutrosophic set. Further, regular sets form a Boolean algebra is introduced from the interior and closure of interior in rough neutrosophic sets. The results establish can help to gain more insight in rough neutrosophic sets.

Graph theory has numerous applications in various disciplines, including computer networks, neural networks, expert systems, cluster analysis, and image capturing. Rough neutrosophic set (NS) theory is a hybrid tool for handling uncertain information that exists in real life. In this research paper, we apply the concept of rough NS theory to graphs and present a new kind of graph structure, rough neutrosophic digraphs. We present certain operations, including lexicographic products, strong products, rejection and tensor products on rough neutrosophic digraphs. We investigate some of their properties. We also present an application of a rough neutrosophic digraph in decision-making.

Rough sets and single-valued neutrosophic sets are mathematical models to deal with incomplete and vague information. These two models can be combined into two frameworks for modelling and processing incomplete information in information systems. Thus, single-valued neutrosophic rough set model and rough single-valued neutrosophic set model are hybrid models, which give more precision, flexibility and compatibility to the system as compared to the classic and fuzzy models. In this chapter, we present rough single-valued neutrosophic digraphs (rough neutrosophic digraphs, for short) and neutrosophic rough digraphs and describe methods of their construction. We consider the concept of self-complementary rough neutrosophic digraphs. We discuss regular neutrosophic rough digraphs. We also give a comparative analysis of rough neutrosophic digraphs and neutrosophic rough digraphs.

Roughness measures for uncertainty data occur with less consideration since the data involve indeterminacy and inconsistency. The indeterminacy plus inconsistency can be solved by a rough neutrosophic set with roughness approximation. Therefore, a binary logarithm similarity measure for a rough neutrosophic set with roughness approximation was proposed in this research. A rough neutrosophic set was chosen as the uncertainty set theory information, which includes the upper and lower approximation with a boundary set approximation. The objectives of this research are to define a binary logarithm similarity measure for a rough neutrosophic set, to formulate the properties satisfied by the proposed similarity measure, and to develop a decision-making model by using a binary logarithm similarity measure for a case study (COVID-19). The roughness approximation was used in the derivation of the binary logarithm similarity measure. The proving result was finalized. Then, the derivation of binary logarithm similarity measures of a rough neutrosophic set was well defined. As a validation process, the similarity properties for identifying the most important priority group for COVID-19 vaccines were used such as age, health state, women, and job types. Following that, the decision-making for identifying the most important priority group for COVID-19 vaccines is presented.

The domain of decision‐making theory has made significant progress, especially within the industrial and management fields. Diverse decision‐making challenges such as multicriteria decision‐making (MCDM), multiattribute decision‐making (MADM), and multiattribute group decision‐making (MAGDM) have been thoroughly examined, equipping decision‐makers with effective strategies for tackling these issues. In the context of the automotive industry, a specific hurdle emerges when identifying the most suitable site for establishing a warehouse to store goods destined for multiple destinations. This article addresses a scenario amid uncertainty, leveraging substantial data. Neutrosophic sets (NSs) emerge as a comprehensive tool for effectively managing the imprecision inherent in such data. Among these sets, single‐valued neutrosophic sets (SVNSs) stand out due to their adeptness in handling inconsistent or incomplete data. The octagonal single‐valued neutrosophic numbers (OSVNNs) has a new tool for representing the uncertainty information in a simplified manner. Octagonal structure allows for representing different distinctions of truth, neutrality, and falsity of any complex situation and it is an efficient tool to compare different options based on their level of ambiguity. They have a wide range of applications in various fields, becoming increasingly important in addressing many difficult and uncertain issues. The article aims to propose a new ranking function for OSVNNs to convert the OSVNN data into precise values, drawing from the existing mean interval method (MIM). The conventional decision‐making approaches such as weighted sum model (WSM), weighted product model (WPM), technique for order of preference by similarity to ideal solution (TOPSIS), and VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) are employed to obtain the optimal warehouse location using OSVN application. Furthermore, it demonstrates the application along with the proposed ranking function using the software MATLAB to determine the most favorable alternatives. Finally, sensitivity analyses are performed to assess how different scenarios could impact the optimal location selection for automotive logistics.

A Neutrosophic differential equation approach for modelling glucose distribution in the bloodstream using neutrosophic sets

The single‐valued neutrosophic set (SVNS) is a more common platform for representing the degree of truth, indeterminacy, and falsity memberships. Indeterminacy is an important aspect of SVNS, and the score function (SF) is important in ranking alternatives in decision‐making scenarios. When the score and accuracy values for different SVNNs were equal, existing SFs were unable to rank the alternatives. To address the limitations, this paper introduces the perfect score function (PSF) as a solution for ranking single‐valued neutrosophic numbers (SVNNs) without any errors and presents the concepts of strong and weak SVNSs. This article has emphasized the drawbacks of the existing SF in SVNNs. Additionally, we propose an optimized weighted geometric operator (OWGO) for SVNNs, and its properties such as idempotency, boundedness, monotonicity, and commutativity are explored. Furthermore, the decision‐making approach has been depicted based on the proposed score function and OWGO to select the finest faculty for a university.

Rough set concept is a methodology of information processing for relational databases. It is a unique uncertainty mathematics topic closely connected to fuzzy set theory. When the rough set is combined with neutrosophic set theory, an effective tool for working with indeterminacy arises. In this study, we defined a multi-valued rough neutrosophic set and a multi-valued rough neutrosophic matrix. Using separation measures, we introduced a new approach for a multi-valued neutrosophic with a rough structure. We consider the problem of determining the condition of dengue-affected patients in a specific hospital. Using this method, we create a multi-valued rough neutrosophic decision matrix that clearly displays the relationship between patient conditions and symptoms. We can determine which one has a serious condition by solving this problem and presenting it on the graph.

In recent years, hesitant fuzzy sets (HFSs) and neutrosophic sets (NSs) have become a subject of great interest for researchers and have been widely applied to multi-criteria group decision-making (MCGDM) problems. In this paper, multi-valued neutrosophic sets (MVNSs) are introduced, which allow the truth-membership, indeterminacy membership and falsity-membership degree have a set of crisp values between zero and one, respectively. Then the operations of multi-valued neutrosophic numbers (MVNNs) based on Einstein operations are defined, and a comparison method for MVNNs is developed depending on the related research of HFSs and Atanassov’s intuitionistic fuzzy sets (IFSs). Furthermore, the multi-valued neutrosophic power weighted average (MVNPWA) operator and the multi-valued neutrosophic power weighted geometric (MVNPWG) operator are proposed and the desirable properties of two operators are also discussed. Finally, an approach for solving MCGDM problems is explored by applying the power aggregation operators, and an example is provided to illustrate the application of the proposed method, together with a comparison analysis.

The paper presents two new correlation coefficients of simplified neutrosophic sets (SNSs) as the further extension of the correlation coefficient of single valued neutrosophic sets (SVNSs) and investigates their properties.

This papers proposes a reflexion on the concept of vagueness. We relate the different kinds of vagueness to relevant mathematical frameworks with particular emphasis on recent works on neutrosophic logic, Dezert-Smarandache theory, and rough sets. The different facets of vagueness are illustrated through a case study on Ambrosia artemisiifolia predictive cartography

During the recent years the smart cities knows a great extension as a modern shape of sustainable expansion. It's a urban area that utilize various devices connected with internet and integrates them with ICTs to promote goodness and execution of services for the best interaction among citizens and city's government. The basic for smart cities is distributed and independent information infrastructure. Using information effectively is going to be a main factor for success in the smart cities. The sources of information's (models, experts, and sensors) must be reason, perfect and complete. The generated information from independent and distributed sources can be imprecise, uncertain, and/or incomplete in real life. Any deficiency in gathered information will have a negative effect on the performance of services and decision making process within smart cities. So, we need a general framework to represent all types of imperfect and incomplete information. Since the classical methods fails to deal with vague, inconsistent and incomplete information, the fuzzy set was introduced to solve this drawback. The fuzzy set was not the perfect method for dealing with these drawbacks because it considers only truthiness and fails to deal with indeterminacy. The efficient mathematical tool for dealing with uncertain, vague and inconsistent objects is rough sets theory which introduced by Pawlak. The theory of neutrosophic rough sets is powerful for dealing with incompleteness and neutrosophic set deals with indeterminate and inconsistent data efficiently through considering truthiness, indeterminacy and falsity degrees. So, in this research we will propose a general framework for dealing with imperfect and incomplete information through using single valued neutrosophic and rough set theories. The combination of two sets will deal with all aspects of vagueness, inconsistency and incompleteness of data and information, and then will enhance the quality of introduced services and decisions from smart cities to their citizens. As experimentation, we applied the proposed framework for modeling imperfect and incomplete data in healthcare field.

The neutrosophic cubic set (NCS) is a hybrid structure, which consists of interval neutrosophic sets (INS) (associated with the undetermined part of information associated with entropy) and single-valued neutrosophic set (SVNS) (associated with the determined part of information). NCS is a better tool to handle complex decision-making (DM) problems with INS and SVNS. The main purpose of this article is to develop some new aggregation operators for cubic neutrosophic numbers (NCNs), which is a basic member of NCS. Taking the advantages of Muirhead mean (MM) operator and power average (PA) operator, the power Muirhead mean (PMM) operator is developed and is scrutinized under NC information. To manage the problems upstretched, some new NC aggregation operators, such as the NC power Muirhead mean (NCPMM) operator, weighted NC power Muirhead mean (WNCPMM) operator, NC power dual Muirhead mean (NCPMM) operator and weighted NC power dual Muirhead mean (WNCPDMM) operator are proposed and related properties of these proposed aggregation operators are conferred. The important advantage of the developed aggregation operator is that it can remove the effect of awkward data and it considers the interrelationship among aggregated values at the same time. Furthermore, a novel multi-attribute decision-making (MADM) method is established over the proposed new aggregation operators to confer the usefulness of these operators. Finally, a numerical example is given to show the effectiveness of the developed approach.

Uncertainty assessments of linear time-cost tradeoffs using neutrosophic set