Big Data is a large-sized and complex dataset, which cannot be managed using traditional data processing tools. Mining process of big data is the ability to extract valuable information from these large datasets. Association rule mining is a type of data mining process, which is indented to determine interesting associations between items and to establish a set of association rules whose support is greater than a specific threshold. The classical association rules can only be extracted from binary data where an (...) item exists in a transaction, but it fails to deal effectively with quantitative attributes, through decreasing the quality of generated association rules due to sharp boundary problems. In order to overcome the drawbacks of classical association rule mining, we propose in this research a new neutrosophic association rule algorithm. The algorithm uses a new approach for generating association rules by dealing with membership, indeterminacy, and non-membership functions of items, conducting to an efficient decision-making system by considering all vague association rules. To prove the validity of the method, we compare the fuzzy mining and the neutrosophic mining. The results show that the proposed approach increases the number of generated association rules. (shrink)

To obtain effective fusion results of multi source evidences with different importance, an evidence fusion method with importance discounting factors based on neutrosopic probability analysis in DSmT framework is proposed. First, the reasonable evidence sources are selected out based on the statistical analysis of the pignistic probability functions of single focal elements. Secondly, the neutrosophic probability analysis is conducted based on the similarities of the pignistic probability functions from the prior evidence knowledge of the reasonable evidence (...) sources. Thirdly, the importance discounting factors of the reasonable evidence sources are obtained based on the neutrosophic probability analysis and the reliability discounting factors of the real-time evidences are calculated based on probabilistic-based distances. Fourthly, the real-time evidences are discounted by the importance discounting factors and then the evidences with the mass assignments of neutrosophic empty sets are discounted by the reliability discounting factors. Finally, DSmT+PCR5 of importance discounted evidences is applied. Experimental examples show that the decision results based on the proposed fusion method are different from the results based on the existed fusion methods. Simulation experiments of recognition fusion are performed and the superiority of proposed method is testified well by the simulation results. (shrink)

Bipolar neutrosophic sets are the extension of neutrosophic sets and are based on the idea of positive and negative preferences of information. Projection measure is a useful apparatus for modelling real life decision making problems. In the paper, we define projection, bidirectional projection and hybrid projection measures between bipolar neutrosophic sets. Three new methods based on the proposed projection measures are developed for solving multi-attribute decision making problems. In the solution process, the ratings of performance values of (...) the alternatives with respect to the attributes are expressed in terms of bipolar neutrosophic values. We calculate projection, bidirectional projection, and hybrid projection measures between each alternative and ideal alternative with bipolar neutrosophic information. All the alternatives are ranked to identify the best alternative. Finally, a numerical example is provided to demonstrate the applicability and effectiveness of the developed methods. Comparison analysis with the existing methods in the literature in bipolar neutrosophic environment is also performed. (shrink)

One of the most important theories in linear programming is the dualistic theory and its basic idea is that for every linear model has dual linear model, so that solving the original linear model gives a solution to the dual model. Therefore, when we solving the linear programming model, we actually obtain solutions for two linear models. In this research, we present a study of the models. The neutrosophic dual and the binary simplex algorithm, which works to find the (...) optimal solution for the original and dual models at the same time. The importance of this algorithm is evident in that it is relied upon in several operations research topics, such as integer programming algorithms, some nonlinear programming algorithms, and sensitivity analysis in linear programming. (shrink)

In this article, the main definitions and differentiation concepts of neutrosophic fuzzy environment will be reviewed. This article will introduce an analytical methodology for solving the second-order linear ordinary differential problem with neutrosophic fuzzy boundary values, this analysis will be under generalized Hukuhara differentiability to show the analytical solutions from a different point of view for the uncertain system, some of these solutions may be decreasing in uncertainty or maybe reflecting the behavior of some real-world systems better. (...) Some applications and numeral examples will be solved to show the behavior of the solution. (shrink)

In this paper, we propose a new hybrid neutrosophic multi criteria decision making (MCDM) framework that employs a collection of neutrosophic analytical network process (ANP), and order preference by similarity to ideal solution (TOPSIS) under bipolar neutrosophic numbers. The MCDM framework is applied for chief executive officer (CEO) selection in a case study at the Elsewedy Electric Group, Egypt. The proposed approach allows us to assemble individual evaluations of the decision makers and therefore perform accurate personnel selection. (...) The outcomes of the proposed method are compared with those of the related works such as weight sum model (WSM), weight product model (WPM), analytical hierarchy process (AHP), multiobjective optimization based on simple ratio analysis (MOORA) and ANP methods to prove and validate the results. (shrink)

In this paper, we propose a new technique for the enhancing images. It will work on removing the noise contained in the image as well as improving its contrast based on three different enhancing transforms, we commence by embedding the image into a neutrosophic domain; where the image will be mapped in three different levels, a level of trueness, a level of falseness and a level of indeterminacy. Hence, we act separately on each level using the enhancement transforms. Finally, (...) we introduce a new analysis in the field of analysis and processing of images using the neutrosophic crisp set theory via Mat lab program where has been obtained three images, which helps in a new analysis to improve and retrieve images. (shrink)

In the fifth version of our response-paper [26] to Imamura’s criticism, we recall that NonStandard Neutrosophic Logic was never used by neutrosophic community in no application, that the quarter of century old neutrosophic operators (1995-1998) criticized by Imamura were never utilized since they were improved shortly after but he omits to tell their development, and that in real world applications we need to convert/approximate the NonStandard Analysis hyperreals, monads and binads to tiny intervals with the desired (...) accuracy – otherwise they would be inapplicable. We point out several errors and false statements by Imamura [21] with respect to the inf/sup of nonstandard subsets, also Imamura’s “rigorous definition of neutrosophic logic” is wrong and the same for his definition of nonstandard unit interval, and we prove that there is not a total order on the set of hyperreals (because of the newly introduced Neutrosophic Hyperreals that are indeterminate), whence the Transfer Principle from R to R* is questionable. After his criticism, several response publications on theoretical nonstandard neutrosophics followed in the period 2018-2022. As such, I extended the NonStandard Analysis by adding the left monad closed to the right, right monad closed to the left, pierced binad (we introduced in 1998), and unpierced binad - all these in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of NonStandard Unit Interval and NonStandard Neutrosophic Logic, together with NonStandard Neutrosophic Operators are presented. (shrink)

Graph theory is a specific concept that has numerous applications throughout many industries. Despite the advancement of this technique, graph theory can still yield ambiguous and imprecise results. In order to cut down on these indeterminate factors, neutrosophic logic has emerged as an applicable solution that is gaining significant attention in solving many real-life decision-making problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistency, and indeterminacy. However, empirical research on this specific graph set is lacking. Neutrosophic Graph Theory and (...) Algorithms is a collection of innovative research on the methods and applications of neutrosophic sets and logic within various fields including systems analysis, economics, and transportation. While highlighting topics including linear programming, decision-making methods, and homomorphism, this book is ideally designed for programmers, researchers, data scientists, mathematicians, designers, educators, researchers, academicians, and students seeking current research on the various methods and applications of graph theory. (shrink)

In this paper, we define a new cosine similarity between two interval valued neutrosophic sets based on Bhattacharya’s distance [19]. The notions of interval valued neutrosophic sets (IVNS, for short) will be used as vector representations in 3D-vector space. Based on the comparative analysis of the existing similarity measures for IVNS, we find that our proposed similarity measure is better and more robust. An illustrative example of the pattern recognition shows that the proposed method is simple and (...) effective. (shrink)

The objective of the paper is to implement and validate diagnosis in the medical field via refined neutrosophic fuzzy logic (RNFL). As such, we have proposed a Max-Min composition (MMC) method in RNFL. This method deals with the diagnosis under certain constraints like uncertainty and indeterminacy. Further, we have considered the diagnosis problems to validate the sensitivity analysis of the novel multi attribute decision-making technique. Finally, we gave the graphical representations and compared the obtained results with other existing (...) measures in refined neutrosophic fuzzy sets. (shrink)

We present this research to all researchers and scholars who have realized the existence of indeterminacy in all data, through the results they obtain and the values that are not accurate enough and that may cause loss to the systems and facilities under study, and we will present through it the emergence, foundations and development of Neutrosophic theories and their applications for more than two decades (1995- 2023) since it was defined and studied, along with its applications, in order (...) to be able to present new studies and research that keep pace with the great scientific development that our contemporary world is witnessing, through the use of research that has been published by the professionals and found on the attached open links. (shrink)

In this paper, multi attribute decision making problem based on grey relational analysis in neutrosophic cubic set environment is investigated. In the decision making situation, the attribute weights are considered as single valued neutrosophic sets. The neutrosophic weights are converted into crisp weights. Both positve and negative GRA coefficients, and weighted GRA coefficients are determined. Hamming distances for weighted GRA coefficients and standard (ideal) GRA coefficients are determined. The relative closeness coefficients are derived in order to (...) rank the alternatives. The relative closeness coefficients are designed in ascending order. Finally, a numerical example is solved to demonstrate the applicability of the proposed approach. (shrink)

In this paper, we have developed an Excel package to be utilized for calculating neutrosophic data and analyze them. The use of object oriented programming techniques and concepts as they may apply to the design and development a new framework to implement neutrosophic data operations, the c# programming language, NET Framework and Microsoft Visual Studio are used to implement the neutrosophic classes. We have used Excel as it is a powerful tool that is widely accepted and used (...) for statistical analysis. Figure 1 shows Class Diagram of the implemented package. Figure 2 presents a working example of the package interface calculating the complement. Our implemented Neutrosophic package can calculate Intersection, Union, and Complement of the nuetrosophic set. Figure 3 presents our neutrosphic package capability to draw figures of presented neutrosphic set. Figure 4 presents charting of Union operation calculation, and figure 5 Intersection Operation. nuetrosophic set are characterized by its efficiency as it takes into consideration the three data items: True, Intermediate, and False. (shrink)

In the fifth version of our reply article [26] to Imamura's critique, we recall that Neutrosophic Non-Standard Logic was never used by the neutrosophic community in any application, that the quarter-century old (1995-1998) neutrosophic operators criticized by Imamura were never used as they were improved soon after, but omits to talk about their development, and that in real-world applications we need to convert/approximate the hyperreals, monads and bi-nads of Non-Standard Analysis to tiny intervals with the desired (...) precision; otherwise they would be inapplicable. We pointed out several errors and false statements by Imamura [21] regarding the inf/sup of nonstandard subsets, also Imamura's "rigorous definition of neutrosophic logic" is incorrect, as is his definition of nonstandard unit interval, and we showed that there is no total order in Neutrosophic Computing and Machine Learning , Vol. 23, 2022 Florentin Smarandache, Definición mejorada de la lógica neutrosófica no estándar e introducción a los hiperreales neutrosóficos (Quinta versión) 2 the set of hyperreals (due to the recently introduced Neutrosophic Hyperreals which are indeterminate), so the Transfer Principle from R to R* is questionable. After his critique, several reply posts on non-standard theoretical neutrosophy followed in 2018-2022. As such, I extended the Nonstandard Analysis by adding the right-closed left monad, the left-closed right monad, the punctured binad (which we introduced in 1998), and the nonpunctured binad - all in order to close the newly extended nonstandard space (R*) under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations [23, 24]. Improved definitions of the Nonstandard Unitary Interval and Nonstandard Neutrosophic Logic are presented, along with Nonstandard Neutrosophic Operators. (shrink)

Fuzzy sets and fuzzy logics are used to model events with imprecise, incomplete, and uncertain information. Researchers have developed numerous methods and techniques to cope with fuzziness or uncertainty. This research intends to introduce the novel concepts of complex neutrosophic relations and its types based on the idea of complex neutrosophic sets. In addition, these concepts are supported by suitable examples. A CNR discusses the quality of a relationship using the degree of membership, the degree of abstinence, and (...) the degree of nonmembership. Each of these degrees is a complex number from the unit circle in a complex plane. The real part of complex-valued degrees represents the amplitude term, while the imaginary part represents the phase term. This property empowers CNRs to model multidimensional variables. Moreover, some interesting properties and useful results have also been proved. Furthermore, the practicality of the proposed concepts is verified by an application, which discusses the use of the proposed concepts in statistical decision-making. Additionally, a comparative analysis between the novel concepts of CNRs and the existing methods is carried out. (shrink)

In this paper, we have developed an Excel package to be utilized for calculating neutrosophic data and analyze them. The use of object oriented programming techniques and concepts as they may apply to the design and development a new framework to implement neutrosophic data operations, the c# programming language, NET Framework and Microsoft Visual Studio are used to implement the neutrosophic classes. We have used Excel as it is a powerful tool that is widely accepted and used (...) for statistical analysis. Figure 1 shows Class Diagram of the implemented package. Figure 2 presents a working example of the package interface calculating the complement. Our implemented Neutrosophic package can calculate Intersection, Union, and Complement of the neutrosophic set. Figure 3 presents our neutrosophic package capability to draw figures of presented neutrosophic set. Figure 4 presents charting of Union operation calculation, and figure 5 Intersection Operation. neutrosophic set are characterized by its efficiency as it takes into consideration the three data items: True, Intermediate, and False. -/- . (shrink)

In this book, we approach different topics related to neutrosophics, such as: Neutrosophic Set, Intuitionistic Fuzzy Set, Inconsistent Intuitionistic Fuzzy Set, Picture Fuzzy Set, Ternary Fuzzy Set, Pythagorean Fuzzy Set, Atanassov’s Intuitionistic Fuzzy Set of second type, Spherical Fuzzy Set, n-HyperSpherical Neutrosophic Set, q-Rung Orthopair Fuzzy Set, truth-membership, indeterminacy-membership, falsehood-nonmembership, Regret Theory, Grey System Theory, Three-Ways Decision, n-Ways Decision, Neutrosophy, Neutrosophication, Neutrosophic Probability, Refined Neutrosophy, Refined Neutrosophication, Nonstandard Analysis; (Theory, NeutroTheory, AntiTheory), S-denying an Axiom, Multispace with (...) Multistructure, and so on. (shrink)

The author makes an introduction to non-standard analysis, then extends the dialectics to “neutrosophy” – which became a new branch of philosophy. This new concept helps in generalizing the intuitionistic, paraconsistent, dialetheism, fuzzy logic to “neutrosophic logic” – which is the first logic that comprises paradoxes and distinguishes between relative and absolute truth. Similarly, the fuzzy set is generalized to “neutrosophic set”. Also, the classical and imprecise probabilities are generalized to “neutrosophic probability”.

A well known and referenced global result is the nilpotent characterisation of the finite p-groups. This un doubtedly transends into neutrosophy. Hence, this fact of the neutrosophic nilpotent p-groups is worth critical studying and comprehensive analysis. The nilpotent characterisation depicts that there exists a derived series (Lower Central) which must terminate at {ϵ} (an identity), after a finite number of steps. Now, Suppose that G(I) is a neutrosophic p-group of class at least m ≥ 3. We show (...) in this paper that Lm−1(G(I)) is abelian and hence G(I) possesses a characteristic abelian neutrosophic subgroup which is not supposed to be contained in Z(G(I)). Furthermore, If L3(G(I)) = 1 such that pm is the highest order of an element of G(I)/L2(G(I)) (where G(I) is any neutrosophic p-group) then no element of L2(G(I)) has an order higher than pm. (shrink)

The present article delves into the extension of Knuth’s fundamental Boolean logic table to accommodate the complexities of indeterminate truth values through the integration of neutrosophic logic (Smarandache & Christianto, 2008). Neutrosophic logic, rooted in Florentin Smarandache’s groundbreaking work on Neutrosophic Logic (cf. Smarandache, 2005, and his other works), introduces an additional truth value, ‘indeterminate,’ enabling a more comprehensive framework to analyze uncertainties inherent in computational systems. By bridging the gap between traditional boolean operations and the indeterminacy (...) present in various real-world scenarios, this extension redefines logic tables, introducing neutrosophic operators that capture nuances beyond the binary realm. Through a thorough exploration of neutrosophic logic's principles and its implications in computational paradigms, this study proposes a novel approach to logic design that accommodates uncertain, imprecise, and incomplete information. This paradigm shift in logic tables not only broadens the spectrum of computing methodologies but also holds promise in fields such as decision-making systems and data analytics. This article amalgamates insights from over twelve key references encompassing seminal works in boolean logic, neutrosophic logic, and their applications in diverse scientific and computational domains, aiming to pave the way for a more robust and adaptable logic framework in computation. (shrink)

As part of our small contribution in dialogue toward better peace development and reconciliation studies, and following Toffler & Toffler’s War and Antiwar (1993), the present article delves into a realm of logic beyond the traditional confines of negation and the excluded middle principle, exploring the nuances of "Otherness" that transcend classical and Nagatomo logics. Departing from the foundational premises of classical Aristotelian logic systems, this exploration ventures into alternative realms of reasoning, specifically examining Neutrosophic Logic and Klein bottle (...) logic (cf. Smarandache, 2005). The study challenges conventional boundaries and explores the implications of embracing paradoxes and self-reference in logic systems, aiming to redefine approaches to understanding truth and reasoning. The paper investigates how these alternative logics open avenues for philosophical inquiry, redefining entropy, and potentially influencing innovative perspectives in free energy systems. Through this exploration, it seeks to expand the discourse on logic, welcoming a broader spectrum of thought beyond established frameworks; and we also discuss shortly a number of possible implementations including in risk management and also Klein bottle entropy redefinition (Tang et al, 2018). (shrink)

We now found nine new topologies, such as: NonStandard Topology, Largest Extended NonStandard Real Topology, Neutrosophic Triplet Weak/Strong Topologies, Neutrosophic Extended Triplet Weak/Strong Topologies, Neutrosophic Duplet Topology, Neutrosophic Extended Duplet Topology, Neutrosophic MultiSet Topology, and recall and improve the seven previously founded topologies in the years (2019-2023), namely: NonStandard Neutrosophic Topology, NeutroTopology, AntiTopology, Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, SuperHyperTopology, and Neutrosophic SuperHyperTopology. They are called avantgarde topologies because of their (...) innovative forms. (shrink)

In this paper, we propose the introduction of a neutrosophic chi-square-test for consistency, incorporating neutrosophic statistics. Our aim is to modify the existing chi-square -test for consistency in order to analyze imprecise data. We present a novel test statistic for the neutrosophic chi-square -test for consistency, which accounts for the uncertainties inherent in the data. To evaluate the performance of the proposed test, we compare it with the traditional chi-square -test for consistency based on classical statistics. By (...) conducting a comparative analysis, we assess the efficiency and effectiveness of our proposed neutrosophic chi-square-test for consistency. Furthermore, we illustrate the application of the proposed test through a numerical example, demonstrating how it can be utilized in practical scenarios. Through this implementation, we aim to provide empirical evidence of the improved performance of our proposed test when compared to the traditional chi-square-test for consistency based on classical statistics. We anticipate that the proposed neutrosophic chi-square-test for consistency will outperform its classical counterpart, offering enhanced accuracy and reliability when dealing with imprecise data. This advancement has the potential to contribute significantly to the field of statistical analysis, particularly in situations where data uncertainty and imprecision are prevalent. (shrink)

Cloud computing (ClC) has become a more popular computer paradigm in the preceding few years. Quality of Service (QoS) is becoming a crucial issue in service alteration because of the rapid growth in the number of cloud services. When evaluating cloud service functioning using several performance measures, the issue becomes more complex and non-trivial. It is therefore quite difficult and crucial for consumers to choose the best cloud service. The user's choices are provided in a quantifiable manner in the current (...) methods for choosing cloud services. Hence, this study attempts to achieve this objective through construction. decision-making framework so-called cloud services evaluator (CSsEv). The main indicator and its sub-indicators are formed as nodes at levels(n) in tree soft sets (TSSs). Thereafter Single Value Neutrosophic Sets (SVNSs) as branch of neutrosophic sets which conjunction with the Multi-Criteria Decision Making (MCDM) technique to facilitate analysis and evaluation process for the available Cloud services providers. Hence, entropy is employed to obtain indicators and sub_indicators weights and Complex Proportional Assessment utilizes these weights to facilitate the decision process of selecting optimal ClSPs. (shrink)

In this paper, the cosine similarity measure of neutrosophic refined (multi-) sets is proposed and its properties are studied. The concept of this cosine similarity measure of neutrosophic refined sets is the extension of improved cosine similarity measure of single valued neutrosophic. Finally, using this cosine similarity measure of neutrosophic refined set, the application of medical diagnosis is presented.

Impostor syndrome or Impostor phenomenon is a belief that a person thinks their success is due to luck or external factors, not their abilities. This psychological trait is present in certain groups like women. In this paper, we propose a neutrosophic trait measure to represent the psychological concept of the trait-anti trait using refined neutrosophic sets. This study analysed a group of 200 undergraduate students for impostor syndrome, perfectionism, introversion and self-esteem: after the COVID pandemic break in 2021. (...) Data labelling was carried out using these neutrosophic trait measures. Machine learning models like Support Vector Machine(SVM), K-nearest neighbour (K-NN), and random forest were used to model the data; SVM provided the best accuracy of 92.15%. (shrink)

We use a neutrosophic set, instead of an intuitionistic fuzzy because the neutrosophic set is more general, and it allows for independent and partial independent components, while in an intuitionistic fuzzy set, all components are totally dependent. In this article, we present and demonstrate the concept of neutrosophic invariant subgroups. We delve into the exploration of this notion to establish and study the neutrosophic quotient group. Further, we give the concept of a neutrosophic normal subgroup (...) as a novel concept. (shrink)

This paper is devoted to present Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method for multi-attribute group decision making under rough neutrosophic environment. The concept of rough neutrosophic set is a powerful mathematical tool to deal with uncertainty, indeterminacy and inconsistency. In this paper, a new approach for multi-attribute group decision making problems is proposed by extending the TOPSIS method under rough neutrosophic environment. Rough neutrosophic set is characterized by the upper and lower (...) approximation operators and the pair of neutrosophic sets that are characterized by truth-membership degree, indeterminacy membership degree, and falsity membership degree. In the decision situation, ratings of alternatives with respect to each attribute are characterized by rough neutrosophic sets that reflect the decision makers’ opinion. Rough neutrosophic weighted averaging operator has been used to aggregate the individual decision maker’s opinion into group opinion for rating the importance of attributes and alternatives. Finally, a numerical example has been provided to demonstrate the applicability and effectiveness of the proposed approach. (shrink)

The notion of neutrosophic cubic set is originated from the hybridization of the concept of neutrosophic set and interval valued neutrosophic set. We define similarity measure for neutrosophic cubic sets and prove some of its basic properties. We present a new multi criteria group decision making method with linguistic variables in neutrosophic cubic set environment. Finally, we present a numerical example to demonstrate the usefulness and applicability of the proposed method.

The objective of this paper is to develop neutrosophic quadruple algebraic hyperstructures. Specifically, we develop neutrosophic quadruple semihypergroups, neutrosophic quadruple canonical hypergroups and neutrosophic quadruple hyperrings and we present elementary properties which characterize them.

Both neutrosophic sets theory and rough sets theory are emerging as powerful tool for managing uncertainty, indeterminate, incomplete and imprecise information .In this paper we develop an hybrid structure called “ rough neutrosophic sets” and studied their properties.

Characterizations of an (∈, ∈)-neutrosophic ideal are considered. Any ideal in a BCK/BCI-algebra will be realized as level neutrosophic ideals of some (∈, ∈)-neutrosophic ideal. The relation between (∈, ∈)-neutrosophic ideal and (∈, ∈)-neutrosophic subalgebra in a BCK-algebra is discussed. Conditions for an (∈, ∈)-neutrosophic subalgebra to be a (∈, ∈)-neutrosophic ideal are provided. Using a collection of ideals in a BCK/BCI-algebra, an (∈, ∈)-neutrosophic ideal is established. Equivalence relations on the family (...) of all (∈, ∈)-neutrosophic ideals are introduced, and related properties are investigated. (shrink)

In this paper, we firstly review the neutrosophic set, and then construct two new concepts called neutrosophic implication of type 1 and of type 2 for neutrosophic sets. Furthermore, some of their basic properties and some results associated with the two neutrosophic implications are proven.

The Vehicle Routing Problem (VRP) has been extensively studied by different researchers from all over the world in recent years. Multiple solutions have been proposed for different variations of the problem, such as Capacitive Vehicle Routing Problem (CVRP), Vehicle Routing Problem with Time Windows (VRP-TW), Vehicle Routing Problem with Pickup and Delivery (VRPPD), among others, all of them with deterministic times. In the last years, researchers have been interested in including in their different models the variations that travel times may (...) experience when exposed to all kind of phenomena, mainly vehicle traffic. This article addresses the VRP from this perspective, proposing the design and implementation of a genetic algorithm based on neutrosophic theory for calculating the fitness function of each route, considering the variability and uncertainty present in travel times. A deterministic genetic algorithm is also implemented with the average travel times to compare it with the neutrosophic algorithm using simulation. As conclusion, a deterministic algorithm does not necessarily generate the best solution in the real world, full of uncertainty. Also, the quantification of uncertainty using neutrosophic theory can be used in route planning, opening a broad and interesting field of research for future investigations. (shrink)

The Neutrosophic Precalculus and the Neutrosophic Calculus can be developed in many ways, depending on the types of indeterminacy one has and on the method used to deal with such indeterminacy. This article is innovative since the form of neutrosophic binomial factorial theorem was constructed in addition to its refrains. Two other important theorems were proven with their corollaries, and numerical examples as well. As a conjecture, we use ten (indeterminate) forms in neutrosophic calculus taking an (...) important role in limits. To serve article's aim, some important questions had been answered. (shrink)

Today, soft computing is a field that is used a lot in solving real-world problems, such as problems in economics, finance, banking... With the aim to serve for solving the real problem, many new theories and/or tools which were proposed, improved to help soft computing used more efficiently. We can mention some theories as fuzzy sets theory (L. Zadeh, 1965), intuitionistic fuzzy set (K Atanasov, 1986), neutrosophic set (F. Smarandache 1999). In this paper, we introduce a new notion of (...) sup port-neutrosophic set (SNS), which is the combination a neutrosophic set with a fuzzy set. So, SNS set is a direct extension of fuzzy set and neutrosophic sets (F. Smarandache). Then, we define some operators on the support-neutrosophic sets, and investigate some properties of these operators. (shrink)

Neutrosophy has been introduced some years ago by Florentin Smarandache as a new branch of philosophy dealing with “the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra”. A variety of new theories have been developed on the basic principles of neutrosophy: among them is neutrosophic logics, a family of many-valued systems that can be regarded as a generalization of fuzzy logics. In this paper we present a critical introduction to neutrosophic logics, (...) focusing on the problem of defining suitable neutrosophic propositional connectives and discussing the relationship between neutrosophic logics and other well-known frameworks for reasoning with uncertainty and vagueness, such as (intuitionistic and interval-valued) fuzzy systems and Belnap’s logic. (shrink)

In this paper , the authors explore neutrosophic statistics, that was initiated by Florentin Smarandache in 1998 and developed in 2014, by presenting various examples of several statistical distributions, from the work [1]. The paper is presented with more case studies, by means of which this neutrosophic version of statistical distribution becomes more pronounced.

In this paper, we introduce the integer programming in neutrosophic environment, by considering coffecients of problem as a triangulare neutrosophic numbers. The degrees of acceptance, indeterminacy and rejection of objectives are simultaneously considered. The Neutrosophic Integer Programming Problem (NIP) is transformed into a crisp programming model, using truth membership (T), indeterminacy membership (I), and falsity membership (F) functions as well as single valued triangular neutrosophic numbers. To measure the efficiency of the model, we solved several numerical (...) examples. (shrink)

The purpose of this paper is to introduce new types of neutrosophic crisp sets with three types 1, 2, 3. After given the fundamental definitions and operations, we obtain several properties, and discussed the relationship between neutrosophic crisp sets and others. Also, we introduce and study the neutrosophic crisp point and neutrosophic crisp relations. Possible applications to database are touched upon.

This Paper combines interval- valued neutrouphic sets and rough sets. It studies roughness in interval- valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower and upper approximations of interval valued neutrosophic sets.

Science is the basis for managing the affairs of life and human activities, and living without knowledge is a form of wandering and a kind of loss. Using scientific methods helps us understand the foundations of choice, decision-making, and adopting the right solutions when solutions abound and options are numerous. Operational research is considered the best that scientific development has provided because its methods depend on the application of scientific methods in solving complex issues and the optimal use of available (...) resources in various fields, private and governmental work in peace and war, in politics and economics, in planning and implementation, and in various aspects of life. Its basic essence is to use the data provided for the issue under study to build a mathematical model that is the optimal solution. It is the basis on which decision makers rely in managing institutions and companies, and when operations research methods meet with the neutrosophic teacher, we get ideal solutions that take into account all the circumstances and fluctuations that may occur in the work environment over time. One of the most important operations research methods is the linear programming method. Which prompted us to reformulate the linear models, the graphical method, and the simplex method, which are used to obtain the optimal solution for linear models using the concepts of neutrosophic science. In this research, and as a continuation of what we presented previously, we will reformulate the modified simplex algorithm that was presented to address the difficulty that we were facing when applying the direct simplex algorithm. It is the large number of calculations required to be performed in each step of the solution, which requires a lot of time and effort. (shrink)

This book introduces readers to the concept of the neutrosophic set which can deal with dynamic and complex decision-making problems. With the complexity of the socio-economic environment, today's decision-making is one of the most notable ventures, whose mission is to decide the best alternative under numerous known or unknown criteria. This book provides a large amount of theoretical and practical information about the latest research in the field, allowing readers to gain an extensive understanding of both the fundamentals and (...) applications of neutrosophic sets to solve different kinds of decision-making problems and mathematical programming such as medical diagnosis, pattern recognition, construction problems, technology selection etc. (shrink)

In this paper, we introduce the concept of ""neutrosophic crisp neighborhoods system for the neutrosophic crisp point ". Added to, we introduce and study the concept of neutrosophic crisp local function, and construct a new type of neutrosophic crisp topological space via neutrosophic crisp ideals. Possible application to GIS topology rules are touched upon.

In this paper, we introduce and study the concept of" neutrosophic closed set" and" neutrosophic continuous function". Possible application to GIS topology rules are touched upon.

Neutrosophic set is a new mathematical tool for handling problems involving imprecise, indetermi nacy and inconsistent data. Pseudo-BCI algebra is a kind of non-classical logic algebra in close connection with various non-commutative fuzzy logics. Recently, we applied neutrosophic set theory to pseudo-BCI al gebras. In this paper, we study neutrosophic filters in pseudo-BCI algebras. The concepts of neutrosophic regular filter, neutrosophic closed filter and fuzzy regular filter in pseudo-BCI algebras are introduced, and some basic properties (...) are discussed. Moreover, the relationships among neutrosophic regular filter, fuzzy filters and anti-grouped neutrosophic filters are prese nted, and the results are proved: a neutrosophic filter (fuzzy filter) is a neutrosophic regular filter (fuzzy regular filter), if and only if it is both a neutrosophic closed filter (fuzzy closed filter) and an anti-grouped neutrosophic filter (fuzzy anti-grouped filter). (shrink)