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The definitions provided by the authors of the current study are offered together with a discussion of the recent advances that they have contributed. We begin with an introduction of the concepts of closed and open sets. We explore characteristics and provide an idea of obtained results by adding the notion of FNbϱNOS and analyzing a few of their properties in 𝑓𝑛𝑡𝑠. We've researched the contrasts between the derived, exterior, and frontier notions that are provided.

The study introduces a hybrid weighted similarity measure (HWSM) for the analysis of symptoms and diseases in patients using a neutrosophic set (NS). NS proves valuable for modeling uncertainty by accommodating contradictory and ambiguous information. The development of a similarity measure for NS information is crucial in various applications, particularly in medical diagnostics, to quantify similarity between sets. While existing literature provides various similarity measures for NS, only a limited number incorporates hybrid techniques. This study proposes a hybrid similarity measure that combines existing measures and integrates them with an entropy weight measure. To elaborate, distance- based similarity measures for NS are initially considered. Subsequently, an entropy weight measure is employed to calculate the attributes' weight of the attributes. The work includes formulating the properties of the proposed HWSM and its practical application in medical diagnosis, focusing on assessing the possibility of medical diagnoses in a patient. The study examines five symptoms which are fever, headache, stomach pain, cough, and chest pain. The HWSM is applied to analyze these symptoms across five different diseases, resulting in consistent and reliable outcomes. This research contributes to the ongoing enhancement of diagnostic tools for medical practitioners, addressing challenges associated with uncertainty in patient information.

Rough neutrosophic multisets (RNM) is an uncertainty set theory generalized from the rough neutrosophic set. In the same equivalence relation, the universal set is a neutrosophic multisets with boundary regions involving lower and upper approximation. To date, to handle the multiplicity of information collected, the rough neutrosophic multisets geometric aggregation operator (RNMGAO) is introduced. The algebraic operations of RNM used in the derivation of RNMGAO are defined. The entropy measure of RNM is also discussed as a weighted assign for each criterion simultaneously with the geometric aggregation operator. The roughness Dice similarity measure of RNM is combined in methodology for ranking purposed. The application in medical diagnosis of three epidemic diseases Coronavirus, Influenza, and Pneumonia is implemented as a case study.

Recently, the intelligent traffic control system and its uncertainty analysis are considered one of the hot spots for utilizing the available techniques. It became more essential when the automatic car, electric vehicle, and other smart cars have introduced the transportation. To control the traffic accident and smooth road services an intelligent traffic control system required. It will be also useful in decreasing the time, reaction time, and efficiency of traffic. However, the problem arises while characterization of true, false or uncertain regions of traffic flow and its future approximation. To deal with this issue some available mathematical technique for traffic flow using rough set, fuzzy rough set, and its extension with the neutrosophic set is discussed in this paper. Some of the papers related to graphical visualization of traffic flow is also discussed for further improvement. The rough set theory can be useful for dealing the uncertain, incomplete, and indeterminate data set. Hence, the hybridization of the neutrosophic set and rough can be considered one of the efficient tools for intelligent traffic control and its approximation via automatic red, green and yellow lights. This paper tried to provide an overview of each available technique to solve the traffic problem. It is hoped that the proposed study will be helpful for several researchers working on traffic flow, traffic accident diagnosis, and its hybridization as future research.

In The traveling salesman problem (TSP) is an essential and the most popular conventional combinatorial optimization network problem in operations research, in which the TSP evaluates the shortest route or path in a network. In TSP, every node has been visited only once, excluding the starting node. In TSP, edge lengths are usually expressed to indicate journey time and expenses instead of distance from a location. The exact arc length can't be predicted because journey times and expenses vary depending on the amount of payload, climate, highway conditions, and so on. As a result, the Neutrosophic numbers introduce a new tool for dealing with unpredictability in TSP. The present article addresses TSP on a neutrosophic network where the edge weight is a neutrosophic number rather than a real number. For solving the Neutrosophic TSP, an algorithmic technique based on the genetic algorithm (GA) is proposed. We created a new mutation and crossover for our suggested GA. We used mathematical examples to show the usefulness of the algorithm that we suggested. The results of experiments suggest that the proposed GA can find the shortest path in a TSP within a neutrosophic framework. This provides valuable insights for decision-makers dealing with real-world situations characterized by imprecise and incomplete data. [ABSTRACT FROM AUTHOR]

The concepts of cubic spherical neutrosophic sets (CSNSs), introduced and investigated by Gomathi et al. [5], offer a geometric representation of collection of neutrosophic sets (NSs), enhancing their ability to capture uncertainty. The formulation characterizes information using points on a sphere with a defined center and radius, providing a more precise depiction of fuzziness inherent in uncertain data. The cubic spherical neutrosophic Archimedean triangular norms(ATN) and conorms (ATCN), expanding the model's capabilities to handle uncertainty. These algebraic operators enable the aggregation and combination of uncertain information, offering a more comprehensive approach to decision-making. The research further presents a method for solving multiple-criteria decision-making problems within the cubic spherical neutrosophic context, leveraging the newly integrated norms and conorms. The algorithm utilizes the cosine similarity measure of cubic spherical neutrosophic sets, exemplified through an application involving the selection of the most effective electric truck. This extended framework provides decision-makers with enhanced tools to navigate complex decision landscapes amidst uncertainty, facilitating more informed and robust choices across diverse domains. [ABSTRACT FROM AUTHOR]

In classical graph theory, the minimal spanning tree (MST) is a subgraph that lacks cycles and efficiently connects every vertex by utilizing edges with the minimum weights. The computation of a minimum spanning tree for a graph has been a pervasive problem over time. However, in practical scenarios, uncertainty often arises in the form of fuzzy edge weights, leading to the emergence of the Fuzzy Minimum Spanning Tree (FMST). This specialized approach is adept at managing the inherent uncertainty present in edge weights within a fuzzy graph, a situation commonly encountered in real-world applications. This study introduces the initial optimization approach for the Minimum Spanning Tree Problem within the context of interval-valued fermatean neutrosophic domain. The proposed solution involves the adaptation of the Dhouib-Matrix-MSTP (DM-MSTP) method, an innovative technique designed for optimal resolution. The DM-MSTP method operates by employing a column-row navigation strategy through the adjacency matrix. To the best of our knowledge, instances of this specific problem have not been addressed previously. To address this gap, a case study is generated, providing a comprehensive application of the novel DM-MSTP method with detailed insights into its functionality and efficacy. [ABSTRACT FROM AUTHOR]

Graph theory plays a vital role in modeling real-world scenarios like network security and expert systems. Various extensions of graph theoretical conceptions have been designed for addressing uncertainty in graphical network scenarios. The concept of Fermatean Neutrosophic Dombi Fuzzy Graphs (FNDFgraphs) represents a novel and innovative extension in graph theory, combining the principles of Fermatean Neutrosophic fuzzy graphs and the Dombi operator. FNDFGs enhances the representation and analysis of uncertain rela- tionships in a graph also it offers a more comprehensive and exible approach to modeling uncertainty in graph structures. The main objective of this present research study focused on FNDFGs and their operations. At the end, an algorithm for Fermatean Neutrosophic Dombi fuzzy multi-criteria decision-making is given, which incorporate the concepts of Fermatean Neutrosophic sets and Dombi operations. Furthermore, a numerical example based on the selection of the most suitable CIBIL score application is put forward to illuminate the aptness of the proposed research work. [ABSTRACT FROM AUTHOR]

A neutrosophic set is a mathematical framework that extends fuzzy and intuitionistic sets to handle indeterminate or contradictory information using three components: truth, falsity, and indeterminacy -membership degrees which deal with handling indeterminate, imprecise, and uncertain data, while Extended Fuzzy Theory extends the standard fuzzy set theory to manage more intricate membership degrees. The primary objective of this review is to thoroughly investigate and summarize the existing literature on trapezoidal neutrosophic environments, particularly focusing on aspects such as the NFMOLP Problems in SVTpN environments. Ultimately, this comprehensive review article aims to enhance the understanding of the potential of these integrated methodologies for effectively presenting decision-making amidst complex and uncertain conditions. [ABSTRACT FROM AUTHOR]

Distance measure is a numerical measurement of the distance between any two objects. The aim of this paper is to propose a new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids with graphical representation. In addition, the metric properties of the proposed measure are examined in detail. A decision making problem also has been solved using the proposed distance measure for a software selection process. comparative analysis has been done with the existing methods to show the potential of the proposed distance measure and various forms of trapezoidal fuzzy neutrosophic number have been listed out to show the uniqueness of the proposed graphical representation. Further, advantages of the proposed distance measure have been given.

The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center. [ABSTRACT FROM AUTHOR]

The shortest path problem is a classic optimization problem in graph theory and computer technology. It involves identifying the shortest path between two nodes in a graph, where each edge has a numerical weight. In this paper, we put our effort into examining the use of the dynamic programming method to evaluate the shortest path (SP) between the two specified nodes in a multistage network where the parameter is a multi-value neutrosophic number (MVNN). Firstly, we propose an algorithm based on the forward and backward approach in an uncertain environment and also implement our approach in the Python-3 programming language. Furthermore, a numerical illustration has been provided to showcase the effectiveness and robustness of the novel model. [ABSTRACT FROM AUTHOR]

The goal of this article is to establish a methodology for ordering of single-valued neutrosophic numbers (SVN-numbers) on the basis of values and ambiguities. First of all, the idea of neutrosophic numbers is discussed, and (α, β, γ)-cut and arithmetic oprations definecd over SVN-numbers are examined. Thereafter, corresponding to each components, the values and ambiguities are defined and using these definitions, the ratio ranking function is constructed. Then, for the stability of the ratio ranking function, some examples are provided for comparing this method with other approaches. Applying this ratio ranking function, neutrosophic linear programming problem(Neu-LPP) converts to the crisp linear programmning problems (CLP-Problems) and solved it by computational lingo method. At last, Neu-LPP is illustrated by two numerical real-life examples. [ABSTRACT FROM AUTHOR]

Business has a massive effect on both people and society as a whole in today's culture. Businesses make significant contributions to humanity's general well-being and progress by promoting economic growth, supporting innovation, producing wealth, and addressing social challenges. However, choosing the right location for a business is a complex task, involving multiple criteria and qualitative and quantitative factors that heavily rely on expert judgement. The researchers propose a unique approach called the SVNZN multi-attributed decision-making method to aid decision-makers in this process. They introduce new operating laws for SVNZNs based on the sine trigonometric (ST) function, known for its periodicity and symmetry over the origin, making it favorable for decision-makers over multi-time phase parameters. Additionally, novel AOs such as SVNZN weighted averaging and geometric operators are defined to combine SVNZNs effectively. Based on these AOs, a decision-making technique for MADM problems is presented, and its applicability is demonstrated through a numerical example of selecting the best location for a business. To validate and enhance the understanding of these proposed techniques, the researchers conducted a comprehensive comparison analysis, including a sensitivity analysis, considering existing literature on MADM difficulties. Figure 1 provides a thorough graphical summary by visually representing all of the contributions and outcomes. [ABSTRACT FROM AUTHOR]

Efficiently determining optimal paths and calculating the least travel time within complex networks is of utmost importance in addressing transportation challenges. Several techniques have been developed to identify the most effective routes within graphs, with the Reversal Dijkstra algorithm serving as a notable variant of the classical Dijkstra's algorithm. To accommodate uncertainty within the Reversal Dijkstra algorithm, Fermatean neutrosophic numbers are harnessed. The travel time associated with the edges, which represents the connection between two nodes, can be described using fermatean neutrosophic numbers. Furthermore, the edge weights in fermatean neutrosophic graphs can be subject to temporal variations, meaning they can change over time. In this study, an extended version of the Reversal Dijkstra algorithm is employed to discover the shortest path and compute the minimum travel time within a single-source time-dependent network, where the edges are weighted using fermatean neutrosophic representations. The proposed method is exemplified, and the outcomes affirm the effectiveness of the expanded algorithm. The primary aim of this article is to serve as a reference for forthcoming shortest path algorithms designed for time-dependent fuzzy graphs. [ABSTRACT FROM AUTHOR]

Fractional transportation problem that includes source and destination may have fractional objective functions in real-world applications to maximize the profitability ratio like profit/cost or profit/time. We refere to such transportation problems as fractional transportation problem. The paper considers the interval-valued neutrosophic numbers and its aritemematic operations. This paper deals with fractional transportation problem having discounting cost in neutrosophic environment, where the supply, demand and transportation costs are uncertain. The problem is considered by introducing all the parameters as neutrosophic numbers. Using the benefits of the score function definition, the problem is transformed into the corresponding deterministic form which can be illustrated by any method. and hence by applying of least cost method with the help of Kuhn-Tucker' optimality conditions, the optimal solution is resulted. Our strategy is to assess the issue and can rank different sort of neutrosophic numbers. To claify the proposed technique, a numerical example is given to show the adequacy of the new model. [ABSTRACT FROM AUTHOR]

Municipal solid waste management (MSWM) has emerged as a major issue in India due to the massive amounts of waste generated on a daily basis. Governments are focusing on establishing a proper waste management system, including a timeline for the installation of waste processing and disposal facilities to reduce waste. For this problem, waste to energy (WtE) technologies have been identified because of their ability to convert waste into green energy while minimizing associated issues. This research investigates the different kinds of WtE treatments based on various factors, namely eco-friendly, budget friendly, technical, and social aspects. The findings of this study revealed which WtE treatments are best suited for waste management systems that convert green energy. In this paper, we propose the best WtE treatment for India using the interval-valued pythagorean neutrosophic fuzzy set (PNFS). We employ the WASPAS-CRITIC method in the interval-valued pythagorean neutrosophic fuzzy environment for this WtE treatment problem. Finally, a numerical example and a comparison are provided to illustrate the reliability and efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]

Uncertain and ambiguous data is usually presented as a data point. When dealing with uncertain data, it is challenging to deal with incompleteness, imprecision, and incertitude; therefore, various mathematical models were developed to resolve issues concerning uncertainty points. To overcome this difficulty, a mathematical model with redefined control points characterised by three important components of the neutrosophic set was produced. The neutrosophic set can then be translated by creating models based on the neutrosophic theory and its relationships to produce control points. Hence, this paper represents a curve generated by combining neutrosophic control points with the B'ezier basic functions using their relation as a Neutrosophic B'ezier Curve. With an illustration example, we show how to visualise neutrosophic data sets into Neutrosophic B'ezier Curve and their relationship. [ABSTRACT FROM AUTHOR]

The characterization of interval valued (IV) secondary k- range symmetric (RS) Neutrosophic fuzzy matrices have been examined in this study with an example. It is discussed how IV s-k RS, s- RS, IV k- RS, and IV RS matrices relate to one another. We establish the necessary and sufficient criteria for IV s-k RS Neutrosophic fuzzy matrices. The existence of several generalized inverses of a matrix in IV Neutrosophic fuzzy matrices. It is also established what are the equivalent criteria for various g-inverses of an IV s - κ RS fuzzy matrix to be an IV s - κ RS. The generalized inverses of an IV s - κ RS P corresponding to the sets P{1, 2}, P{1, 2, 3} and P{1, 2, 4} are characterized. [ABSTRACT FROM AUTHOR]

Neutrosophic sets play a crucial role in handling uncertainty, ambiguity, and indeterminacy in numerous theories. They are a kind of extension of the two types of fuzzy sets and intuitionistic fuzzy sets. In the context of modeling drug diffusion within human connective tissues, a differential equation is employed within the neutrosophic framework, utilizing Hukuhara differentiability. We establish the initial conditions and parameters in the form of Type 2 triangular single-valued neutrosophic numbers. This study explores the stability and existence of equilibrium points, providing precise solutions. To conduct numerical simulations across various values of the (α, β, γ)-cut of the triangular neutrosophic number, we utilize MATLAB 2018a. [ABSTRACT FROM AUTHOR]

In this paper, a maple code is presented to do many operations on interval valued neutrosophic matrices including entering the elements of the matrix, checking whether a given matrix is an interval valued neutrosophic matrix or not, finding complement of an interval valued neutrosophic matrix, finding score, accuracy, and certainty measures, union and intersection of two interval valued neutrosophic matrices, sum and product of two interval valued neutrosophic matrices and finding the transpose of a given interval valued of neutrosophic matrix. What distinguishes this code is its simplicity to understand and to call the functions. Many examples are presented and solved successfully. [ABSTRACT FROM AUTHOR]

The discovery of soft sets is accredited to Molodtsov. This theory can cope with difficult circumstances with a lot of ambiguity, like those where deciding is hard. The bipolar soft set (BSS) and neutrosophic soft set (NSS) are algebraic models that can be viewed as soft set expansions. The BSS theory states that we weigh the pros and cons when deciding and NSS theory can handle belief system ambiguity, contradiction, and lack of knowledge due to its truth and falsity membership values. The concept of BSS and NSS are explained in comprehensive detail in this article. This article examined the weighted bipolar soft set (WBSS) and the weighted neutrosophic soft set (WNSS), as well as how to make accurate decisions under uncertain or inadequate information. A detailed comparison of information extraction approaches using weighted bipolar and neutrosophic soft sets may be lacking in the literature. These strategies may have been studied separately, but there may be little research comparing their performance under different settings and with diverse data. Filling this gap with a thorough and rigorous comparison study would help comprehend these techniques' practical benefits and drawbacks. [ABSTRACT FROM AUTHOR]

This paper aims to define a special case of neutrosophic matrices referred to as Fermatean neutrosophic matrices (FNMs). FNMs were introduced as generalization of Fermatean fuzzy matrices, intuitionistic fuzzy matrices, and Pythagorean neutrosophic matrices. In FNMs, some properties were discussed in connection with the well-known standard operations (⊕, ⊗, ∧ and ∨). In addition, the scalar multiplication (nA) and exponentiation ( An ) of a Fermatean neutrosophic matrix A were proposed, and their basic properties were investigated. Lastly, a new operation denoted by @ was defined on Fermatean neutrosophic matrices, and some of the properties of these matrices were examined. [ABSTRACT FROM AUTHOR]

A new growing area of the neutrosophic set theory called complex neutrosophic sets (CNS) provides useful tools for dealing with uncertainty in complex valued physical variables that are observed in the actual world. A CNS takes values for the truth, indeterminacy and falsity membership functions in the complex plane's unit circle. In this research, a novel concept of complex fermatean neutrosophic graph (CFNG) is established and various basic graphical ideas such as the order, size, degree and total degree of a vertex of it are introduced. Also, set theoretical operations such as complement, union, join, ring-sum and cartesian product of CFNG are studied. Further, the concept of a regular graph under a complex fermatean neutrosophic environment is introduced. Finally, we make use of the proposed CFNG in solving a multi-criteria decision-making problem in which the graphical structure of attributes is uncertain. This study also demonstrates the application of a CFNG in the educational system to evaluate a lecturer's research productivity. [ABSTRACT FROM AUTHOR]

The Intuitionistic Fuzzy Sets (IFSs) are generalizations of Zadeh's fuzzy sets, in which the elements of the universe have apart from Zadeh's membership and the degree of non-membership in [0, 1]. This paper studies applications of intuitionistic Fuzzy Sets (FS) to assessment and multi-criteria decision making, which are very useful when uncertainty characterizes the grades or parameters respectively assigned to the elements of the universal set. Applications to everyday life situations are also presented illustrating our results. [ABSTRACT FROM AUTHOR]

The shortest path (SP) problem (SPP) has several applications in graph theory. It can be used to calculate the distance between the provided initial and final vertex in a network. In this paper, we employed the Fermatean neutrosophic number as the appropriate edge weight of the network to estimate the SP connecting the start and end vertex. This technique is highly useful in establishing the shortest path for the decision-maker under uncertainty. We also investigated its effectiveness in comparison to several existing methods. Finally, a few numerical tests were performed to demonstrate the validity and stability of this new technique, as well as to compare different types of shortest paths with different networks. [ABSTRACT FROM AUTHOR]

The most well-known subject in graph theory is the shortest path problem (SPP), which has real-world applications in several different fields of study, including transportation, emergency services, network communications, fire station services, etc. The arc weights of the applicable SP problems are typically represented by fuzzy numbers in real-world applications. In this paper, we discussed the process of finding the shortest distance in a connected graph network in which the arc weights are multi-valued neutrosophic numbers (MNNs). Moreover, here we compare our method with some of the existing results and illustrate one implementation of our method with the help of one numerical example. [ABSTRACT FROM AUTHOR]

Multicriteria group decision-making scenarios with a large number of criteria values may be challenging for experts to control. This is a result of the specialists' need to consider an excessive amount of data. They find it difficult to make the optimal decision since the possibilities overwhelm them. We propose a novel multicriteria group decisionmaking method that methodically eliminates the initial set of criterion values in order to address this issue. One of the most promising emerging technologies currently in development is an additive manufacturing (AM), which includes 3D printing. It has been hypothesized that 3D printing technology could eventually replace the conventional production machinery that is commonly used in the industrial sector. Making conclusions through accurate figures is difficult for decision-makers due to the complexity and ambiguity of reality. Neutrosophic ensembles are used to tackle uncertainty and indeterminacy in a practical environment. By concentrating on ranking the smaller set of criterion values, the proposed method enables the experts to carry out the group decision-making process. As a result, a relaxed decision-making environment is created, allowing the experts to handle a reasonable amount of information while still making decisions. To demonstrate the decision process of a 3D printer, we combine a single valued Neutrosophic with hybrid score and accuracy function, the single valued Neutrosophic number ranking approach, and the single valued Neutrosophic score and accuracy function. To determine the best attributes, the score function was used to rank the total values of each possibility. Concrete examples have been given to support the suggested solution to the multi-attribute decision-making problem (MADM). [ABSTRACT FROM AUTHOR]

We develop the concept of range symmetric Neutrosophic Fuzzy Matrix and Kernel symmetric Neutrosophic Fuzzy Matrix analogous to that of an EP -matrix in the complex field. First we present equivalent characterizations of a range symmetric matrix and then derive equivalent conditions for a Neutrosophic Fuzzy Matrix to be kernel symmetric matrix and study the relation between range symmetric and kernel symmetric Neutrosophic Fuzzy Matrices. The idea of Kernel and k-Kernel Symmetric (k-KS) Neutrosophic Fuzzy Matrices (NFM) are introduced with an example. We present some basic results of kernel symmetric matrices. We show that k-symmetric implies k-Kernel symmetric but the converse need not be true. The equivalent relations between kernel symmetric, k-kernel symmetric and Moore-Penrose inverse of NFM are explained with numerical results. [ABSTRACT FROM AUTHOR]