In this volume, we present a set of research that was published in cooperation with a number of researchers and those interested in keeping pace with the great scientific development that our contemporary world is witnessing, and one of its products was neutrosophic science, which was founded by the American scientist and mathematical philosopher Florentin Smarandache in 1995. Through it, we present a new vision for some research methods. Operations research to the concepts of this science.

In this paper, we define for the first time three neutrosophic actions and their properties. We then introduce the prevalence order on {T, I, F} with respect to a given neutrosophic operator “o”, which may be subjective - as defined by the neutrosophic experts; and the refinement of neutrosophic entities <A>, <neutA>, and <antiA> . Then we extend the classical logical operators to neutrosophic literal logical operators and to refined literal logical operators, (...) and we define the refinement neutrosophic literal space. (shrink)

The science of operations research is one of the modern sciences that have made a great revolution in all areas of life through the methods provided by it, suitable and appropriate to solve most of the problems that were facing researchers, scholars and those interested in the development of societies, and the most beneficiaries of this science were companies and institutions that are looking for scientific methods that help them manage their work so that they achieve the greatest profit and (...) the lowest cost, and one of the important methods that have been used in the management of companies we offer in this research two methods, Dynamic programming method. This method has been used in many practical matters and helped decision-makers in companies to achieve a maximum profit and less cost by formulating the reality of the state of the company and the data provided by decision-makers with a dynamic mathematical model that is solved using methods of solving dynamic models and we will provide in this research an example of this through the issue of choosing the optimal investment for the budget of a company so that it achieves a maximum profit, and the method of programming with integers: the method that provided these companies with solutions with integer values suitable for the nature of its work, through the use of the binary integer in the formulation of the appropriate mathematical model on the one hand, and on the other hand, the use of the binary integer variable helped to convert some nonlinear models that lead to some practical problems into linear models, and it should be noted here that in the previous two methods there is something indeterminable because we must make a decision in choosing or not choosing something, but the optimal solution that we will get remains A specific value because we are building the mathematical model for any realistic issue through the data provided by those responsible for the work and these data are calculated quantities and therefore they are uncertain values because their validity depends on the circumstances surrounding the work environment, they may be exposed to increase or decrease, and therefore the optimal solution on which the company will base its decision is suitable for specific values and any change in them can cause the company an uncalculated loss, so in this research we will use the concepts of neutrosophic science, the branch of science founded by the American scientist Florentin Smarandache in 1995 based on his belief that there is no absolute truth, a science that is interested in the study of ideas and concepts that are neither true nor false, but just in between, and we will take the data (calculated quantities) neutrosophic values that are specified or unspecified values are any set close to the calculated quantities, then the resulting mathematical model is a neutrosophic model and the optimal solution has neutrosophic values and thanks to the indefinite uncertainty that these values have, companies from the development of appropriate plans for all circumstances and thus achieve the greatest profit and the lowest cost, and we will clarify the above through two issues, the issue of optimal designation of a warehouse site, which we will formulate the mathematical model of using the neutrosophic integer programming method - and the issue of capital budget, which we will present in two different forms, we use in the first form the neutrosophic integer programming method and in the second the neutrosophic dynamic programming method. (shrink)

Enterprise resource planning (ERP) has a significant impact on modern businesses by enhancing productivity, automation, and streamlining of business processes, even accounting. Manufacturers can assure proper functioning and timely client demand using ERP software. Coordination, procurement control, inventory control, and dispatch of commodities are all features of supply chain management.

We introduce for the first time the appurtenance equation and inclusion equation, which help in understanding the operations with neutrosophic numbers within the frame of neutrosophic statistics. The way of solving them resembles the equations whose coefficients are sets (not single numbers).

We introduce now for the first time the neutrosophic modal logic. The Neutrosophic Modal Logic includes the neutrosophic operators that express the modalities. It is an extension of neutrosophic predicate logic and of neutrosophic propositional logic.

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 neutrosophic cubic hesitant fuzzy set can efficiently handle the complex information in a decision-making problem because it combines the advantages of the neutrosophic cubic set and the hesitant fuzzy set. The algebraic operations based on Dombi norms and co-norms are more flexible than the usual algebraic operations as they involve an operational parameter. First, this paper establishes Dombi algebraic operational laws, score functions, and similarity measures in neutrosophic cubic hesitant fuzzy sets. Then, we proposed Dombi exponential (...) operational laws in which the exponents are neutrosophic cubic hesitant fuzzy values and bases are positive real numbers. To use neutrosophic cubic hesitant fuzzy sets in decision-making, we are developing Dombi exponential aggregation operators in the current study. In the end, we present applications of exponential aggregation operators in multiattribute decision-making problems. (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)

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.

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)

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)

As a generalization of both single-valued neutrosophic element and hesitant fuzzy element, single-valued neutrosophic hesitant fuzzy element is an efficient tool for describing uncertain and imprecise information. Thus, it is of great significance to deal with single-valued neutrosophic hesitant fuzzy information for many practical problems. In this paper, we study the aggregation of SVNHFEs based on some normalized operations from geometric viewpoint. Firstly, two normalized operations are defined for processing SVNHFEs. Then, a series of normalized aggregation (...) class='Hi'>operators which fulfill some basic conditions of a valid aggregation operator are proposed. Additionally, a decision-making method is developed for resolving multiattribute decision-making problems based on the proposed operators. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the method. (shrink)

We introduce Pura Vida Neutrosophic Algebra, an algebraic structure consisting of neutrosophic numbers equipped with two binary operations namely addition and multiplication. The addition can be calculated sometimes with the function min and other times with the max function. The multiplication operation is the usual sum between numbers. Pura Vida Neutrosophic Algebra is an extension of both Tropical Algebra (also known as Min-Plus, or Min-Algebra) and Max-Plus Algebra (also known as Max-algebra). Tropical and Max-Plus algebras are algebraic (...) structures included in semirings and their operations can be used in matrices and vectors. Pura Vida Neutrosophic Algebra is included in Neutrosophic semirings and can be used in Neutrosophic matrices and vectors. (shrink)

In this paper, we introduce the notion of single-valued neutrosophic soft uniform spaces as a view point of the entourage approach. We investigate the relationship among single-valued neutrosophic soft uniformities, single-valued neutrosophic soft topologies and single-valued neutrosophic soft interior operators. Also, we study several single-valued neutrosophic soft topologies induced by a single-valued neutrosophic soft uniform space.

The traditional soft set is a mapping from a parameter set to family of all crisp subsets of a universe. Molodtsov introduced the soft set as a generalized tool for modelling complex systems involving uncertain or not clearly defined objects. In this paper, the notion of neutrosophic soft set is reanalysed. The novel theory is a combination of neutrosophic set theory and soft set theory. The complement, “and”, “or”, intersection and union operations are defined on the neutrosophic (...) soft sets. The neutrosophic soft relations accompanied with their compositions are also defined. The basic properties of the neutrosophic soft sets, neutrosophic soft relations and neutrosophic soft compositions are also discussed. (shrink)

In this paper, we propose a new concept named the uniform single valued neutrosophic graph. An illustrative example and some properties are examined. Next, we develop an algorithmic approach for computing the complement of the single valued neutrosophic graph. A numerical example is demonstrated for computing the complement of single valued neutrosophic graphs and uniform single valued neutrosophic graph.

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)

Product acceptance determination using similarity measure index by neutrosophic statistics / Muhammad Aslam and Rehan Ahmed Khan Sherwani -- New concepts of strongly edge irregular interval-valued neutrosophic graphs / A.A.Talebi, Hossein Rashmanlou and Masoomeh Ghasemi -- The link between neutrosophy and learning : through the related concepts of representation and compression / Philippe Schweizer -- Neutrosophic soft cubic M-subalgebras of B-algebras / Mohsin Khalid, Neha Andaleeb Khalid and Hasan Khalid -- Alpha, beta and gamma product of (...) class='Hi'>neutrosophic graphs / Nasir Shah, Said Broumi, Abdul Raheem and Tahir Imran -- Neutrosophic cubic normal ideal and neutrosophic cubic closed normal ideal of PS-algebra / Mohsin Khalid, Hasan khalid and Neha Andaleeb Khalid -- Neutrosophic multisets : an overview / Vakkas Uluçay -- Neutrosophy as a model for knowledge : the influence of representative models on thinking / Philippe Schweizer -- When neutrosophic theory meets three-way decisions / Rui Ren, Chao Zhang and Deyu Li -- Neutrosophic economic order quantity model with more than one price breaks / R. Surya, M. Mullai and G. Madhan Kumar -- General non-square systems of linear equations in single-valued triangular neutrosophic number environment / S.A. Edalatpanah -- Hesitant neutrosophic soft set relations / Abhijit Saha -- Computer application of neutrosophic set operations / S. Saranya, M. Vigneshwaran, Said Broumi -- Triangular bipolar neutrosophic based transportation problem / Avishek Chakraborty -- Studying neutrosophic variables / Rafif Alhabib and Ahmad Salama -- Decision making for logistics center location selection in trapezoidal neutrosophic environment / Kalyan Mondal and Surapati Pramanik -- Weighted similarity measure and decision making model for clinical application of single valued neutrosophic set / R.Binu and P. Isaac -- Rough neutrosophic set : an overview / Surapati Pramanik -- Some new concepts on normalized SVTN-number and multiple criteria decision making / İrfan Deli and Emel Kırmızı Öztürk. (shrink)

In this book, we present a study of linear models and algorithms to find the optimal solution for them using the concepts of neuroscientific science. We know that the linear programming method is one of the important methods of operations research, the science that was the product of the great scientific development that our contemporary world is witnessing. The name operations research is given to the group of scientific methods used. In analyzing problems and searching for optimal solutions, it is (...) a science whose applications have achieved widespread success in various fields of life. What is meant by neutrosophic models are models in which the data are neutrosophic values, that is, variables such as in the objective function, which expresses profit if the model is a maximization model, and expresses a cost if the model is a minimization model, which in turn is affected by environmental conditions. (shrink)

Operations research science is defined as the science that is concerned with applying scientific methods to complex problems in managing and directing large systems of people, including resources and tools in various fields, private and governmental work, peace and war, politics, administration, economics, planning and implementation in various domains. It uses scientific methods that take the language of mathematics as a basis for it and uses computer, without which it would not have been possible to achieve numerical solutions to the (...) raised problems, those that need correct solutions, when the solutions abound and the options are multiple, so we need a decision based on correct scientific foundations and takes into account all the circumstances and changes that you can encounter the decision-maker during the course of work, and nothing is left to chance or luck, but rather everything that enters into the account and plays its role in decision-making, and we get that when we use the concepts of neutrosophic science to reformulate what the science of operations research presented in terms of methods and methods to solve many practical problems, so we will present in this research a study aimed at shedding light on the most important methods used to solve nonlinear models, which is the Lagrangian multiplier method for nonlinear models constrained by equality and then reformulated using the concepts of neutrosophic science. (shrink)

The paper proposes a new technique for dealing with multi-attribute decision making problems through an extended TOPSIS method under neutrosophic cubic environment. Neutrosophic cubic set is the generalized form of cubic set and is the hybridization of a neutrosophic set with an interval neutrosophic set. In this study, we have defined some operation rules for neutrosophic cubic sets and proposed the Euclidean distance between neutrosophic cubic sets. In the decision making situation, the rating of (...) alternatives with respect to some predefined attributes are presented in terms of neutrosophic cubic information where weights of the attributes are completely unknown. In the selection process, neutrosophic cubic positive and negative ideal solutions have been defined. An extended TOPSIS method is then proposed for ranking the alternatives and finally choosing the best one. Lastly, an illustrative example is solved to demonstrate the decision making procedure and effectiveness of the developed approach. (shrink)

In this paper, we introduce the concepts of Pythagorean fuzzy valued neutrosophic set (PFVNS) and Pythagorean fuzzy valued neutrosophic (PFVNV) constructed by considering Pythagorean fuzzy values (PFVs) instead of numbers for the degrees of the truth, the indeterminacy and the falsity, which is a new extension of intuitionistic fuzzy valued neutrosophic set (IFVNS). By means of PFVNSs, the degrees of the truth, the indeterminacy and the falsity can be given in Pythagorean fuzzy environment and more sensitive evaluations (...) are made by a decision maker in decision making problems compared to IFVNSs. In other words, such sets enable a decision maker to evaluate the degrees of the truth, the indeterminacy and the falsity as PFVs to model the uncertainty in the evaluations. (shrink)

The interval neutrosophic uncertain linguistic variables can easily express the indeterminate and inconsistent information in real world, and TOPSIS is a very effective decision making method more and more extensive applications. In this paper, we will extend the TOPSIS method to deal with the interval neutrosophic uncertain linguistic information, and propose an extended TOPSIS method to solve the multiple attribute decision making problems in which the attribute value takes the form of the interval neutrosophic uncertain linguistic variables (...) and attribute weight is unknown. Firstly, the operational rules and properties for the interval neutrosophic variables are introduced. Then the distance between two interval neutrosophic uncertain linguistic variables is proposed and the attribute weight is calculated by the maximizing deviation method, and the closeness coefficients to the ideal solution for each alternatives. Finally, an illustrative example is given to illustrate the decision making steps and the effectiveness of the proposed method. (shrink)

This work introduces the framework for selecting architecture in 5G networks, considering various technological, performance, economic, and operational factors. With the emergence of 5G technology, the architecture selection process has become pivotal in meeting diverse requirements for ultra-high-speed connectivity, low latency, scalability, and diverse service demands. The evaluation comprehensively analyses different architecture options, including centralized, distributed, cloud-based, and virtualized architectures. Factors such as network performance, scalability, cost-effectiveness, security, and compatibility are considered within a multi-criteria decision-making framework. Findings reveal each architecture (...) option's diverse strengths and limitations, emphasizing the importance of aligning architectural choices with specific use cases, deployment scenarios, and business objectives. We used the concept of multi-criteria decision-making (MCDM) methodology to control the various criteria. We used the single-valued neutrosophic set (SVNS) to deal with vague and inconsistent data. The SVNS is integrated with the hyperSoft set. The proposed methodology uses a single-valued neutrosophic hyperSoft set (SVNHSS) to select the best 5G architecture. We used the VIKOR method as an MCDM method to rank the alternatives. The proposed framework provides stakeholders with a structured methodology to evaluate and prioritize architecture options, facilitating informed decision-making in the complex landscape of 5G network deployments. (shrink)

The linear programming method is one of the important methods of operations research that has been used to address many practical issues and provided optimal solutions for many institutions and companies, which helped decision makers make ideal decisions through which companies and institutions achieved maximum profit, but these solutions remain ideal and appropriate in If the conditions surrounding the work environment are stable, because any change in the data provided will affect the optimal solution and to avoid losses and achieve (...) maximum profit, we have, in previous research, reformulated the linear models using the concepts of neutrosophic science, the science that takes into account the instability of conditions and fluctuations in the work environment and leaves nothing to chance. While taking data, neutrosophic values carry some indeterminacy, giving a margin of freedom to decision makers. In another research, we reformulated one of the most important methods used to solve linear models, which is the simplex method, using the concepts of this science, and as a continuation of what we did in the previous two researches, we will reformulate in this research. The graphical method for solving linear models using the concepts of neutrosophics. We will also shed light on a case that is rarely mentioned in most operations research references, which is that when the difference between the number of unknowns and the number of constraints is equal to one, two, or three, we can also find the optimal solution graphically for some linear models. This is done by taking advantage of the conditions of non-negativity that linear models have, and we will explain this through an example in which the difference is equal to two. Also, through examples, we will explain the difference between using classical values and neutrosophic values and the extent of this’s impact on the optimal solution. (shrink)

An important method for finding the optimal solution for linear and nonlinear models is the graphical method, which is used if the linear or nonlinear mathematical model contains one, two, or three variables. The models that contain only two variables are among the most models for which the optimal solution has been obtained graphically, whether these models are linear or non-linear in references and research that are concerned with the science of operations research, when the data of the issue under (...) study is classical data. In this research, we will present a study through, which we present the graphical method for solving Neutrosophical nonlinear models in the following case: A nonlinear programming issue, the objective function is a nonlinear function, and the constraints are linear functions. Note that we can use the same method if (i) the objective function follower is a linear follower and the constraints are nonlinear; (ii) the objective function is a non-linear follower and the constraints are non-linear. In the three cases, the nonlinear models are neutrosophic, and as we know, the mathematical model is a nonlinear model if any of the components of the objective function or the constraints are nonlinear expressions, and the nonlinear expressions may be in both. At the left end of the constraints are neutrosophic values, at least one or all of them. Then, the possible solutions to the neutrosophic nonlinear programming problem are the set of rays that fulfills all the constraints. As for the region of possible solutions, it is the region that contains all the rays that fulfill the constraints. The optimal solution is the beam that fulfills all constraints and at which the function reaches a maximum or minimum value, depending on the nature of the issue under study (noting that it is not necessary to be alone). (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)

On the basis of the combination of single-valued neutrosophic sets and hesitant fuzzy sets, this article proposes a single-valued neutrosophic hesitant fuzzy set as a further generalization of the concepts of fuzzy set, intuitionistic fuzzy set, single-valued neutrosophic set, hesitant fuzzy set, and dual hesitant fuzzy set. Then, we introduce the basic operational relations and cosine measure function of SVNHFSs. Also, we develop a single-valued neutrosophic hesitant fuzzy weighted averaging operator and a single-valued neutrosophic hesitant (...) fuzzy weighted geometric operator and investigate their properties. Furthermore, a multiple-attribute decision-making method is established on the basis of the SVNHFWA and SVNHFWG operators and the cosine measure under a single-valued neutrosophic hesitant fuzzy environment. Finally, an illustrative example of investment alternatives is given to demonstrate the application and effectiveness of the developed 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)

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; (...) multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more. (shrink)

The purpose of this paper is to introduce and study the characteristic function of a neutrosophic set. After given the fundamental definitions of neutrosophic set operations generated by the characteristic function of a neutrosophic set ( Ng for short), we obtain several properties, and discussed the relationship between neutrosophic sets generated by Ng and others. Finally, we introduce the neutrosophic topological spaces generated by Ng . Possible application to GIS topology rules are touched upon.

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)

The topics approached in this collection of papers are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (...) (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; fuzzy measure; clustering algorithm; and many more. (shrink)

This paper is an attempt of proposing the processing approach of neutrosophic technique in image processing. As neutrosophic sets is a suitable tool to cope with imperfectly defined images, the properties, basic operations distance measure, entropy measures, of the neutrosophic sets method are presented here. İn this paper we, introduce the distances between neutrosophic sets: the Hamming distance, the normalized Hamming distance, the Euclidean distance and normalized Euclidean distance. We will extend the concepts of distances to (...) the case of neutrosophic hesitancy degree. Entropy plays an important role in image processing. In our further considertions on entropy for neutrosophic sets the concept of cardinality of a neutrosophic set will also be useful. Possible applications to image processing are touched upon. (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 paper, we introduce the plithogenic set (as generalization of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets), which is a set whose elements are characterized by many attributes (parameters)’ values. An attribute value v has a corresponding (fuzzy, intuitionistic fuzzy, or neutrosophic) degree of appurtenance d(x,v) of the element x, to the set P, with respect to some given criteria. In order to obtain a better accuracy for the plithogenic aggregation operators in the plithogenic set, and (...) for a more exact inclusion (partial order), a (fuzzy, intuitionistic fuzzy, or neutrosophic) contradiction (dissimilarity) degree is defined be-tween each attribute value and the dominant (most important) attribute value. The plithogenic intersection and union are linear combinations of the fuzzy operators tnorm and tconorm, while the plithogenic complement, inclusion (inequality), equality are influenced by the attribute values contradiction (dissimilarity) degrees. This article offers some examples and applications of these new concepts in our everyday life. (shrink)

In the real word, in most cases, everything (an attribute, event, proposition, theory, idea, person, object, action, etc.) is evaluated in general by many sources (called experts), not only one. The more sources evaluate a subject, the better accurate result (after fusioning all evaluations). That’s why, in this paper, we straightforwardly extend the Refined Neutrosophic Set to the MultiNeutrosophic Set, and we show that the last two are isomorphic. A MultiNeutrosophic Set is a Neutrosophic Set whose all elements’ (...) degrees of truth/indeterminacy/falsehood are evaluated by many (Multi) sources. Afterwards, we introduce a total order on the set of n-plets of the form (p, r, s), we build the operators on the (p, r, s)-plets, and show several applications of the MultiNeutrosophic Sets. Several particular cases of the MultiNeutrosophic Sets are presented: such as MultiFuzzy Set, MultiIntuitionistic Fuzzy Set, MultiPicture Fuzzy Set, and other Multi(Fuzzy Extension) Set. (shrink)

In this article, the concepts of Nth Power Set of a Set, Super-Hyper-Oper-Operation, Super-Hyper-Axiom, SuperHyper-Algebra, and their corresponding Neutrosophic Super-Hyper-Oper-Operation, Neutrosophic Super-Hyper-Axiom and Neutrosophic Super-Hyper-Algebra are reviewed. In general, in any field of knowledge, really what are found are Super-HyperStructures (or more specifically Super-Hyper-Structures (m, n)).

We want to highlight the strong link between learning systems such as deep learning neural networks and neutrosophy. The latter is above all a representation considering a neutral state which is at the heart of many phenomena of reality as well as mathematical and information theories. Here, we start from the recent understanding of neural networks, which considers their internal functioning and the learning that characterizes them as based on adapted representations (both of the information to be processed and of (...) the task to be performed) that are in fact optimal compressions. Compression means discriminating the similar and the different between cases by identifying representative characteristics through focusing on the most significant and rejecting of the rest. These three operations of neural processing, learning and compression owe their existence to non-linear treatments which essentially amount to distinguishing a neutral zone between two filtered zones, corresponding to the neutrosophical approach. (shrink)

This paper presents for the first time the IndetermSoft Set, as an extension of the classical (determinate) Soft Set, which operates on indeterminate data, and similarly the HyperSoft Set extended to the IndetermHyperSoft Set, where 'Indeterm' means 'Indeterminate' (uncertain, conflicting, non-unique result). They are built on an IndetermSoft Algebra which is an algebra dealing with IndetermSoft Operators resulting from our real world. Subsequently, the IndetermSoft and IndetermHyperSoft Sets and their Fuzzy/Fuzzy Intuitionistic/Neutrosophic and other fuzzy extensions and their applications (...) are presented. (shrink)

In this paper one introduces for the first time the IndetermSoft Set, as extension of the classical (determinate) Soft Set, that deals with indeterminate data, and similarly the HyperSoft Set extended to IndetermHyperSoft Set, where ‘Indeterm’ stands for ‘Indeterminate’ (uncertain, conflicting, not unique outcome). They are built on an IndetermSoft Algebra that is an algebra dealing with IndetermSoft Operators resulted from our real world. Afterwards, the corresponding Fuzzy / Intuitionistic Fuzzy / Neutrosophic / and other fuzzy-extension IndetermSoft Set (...) & IndetermHyperSoft Set are presented together with their applications. (shrink)

In this paper, we recall and study the new type of algebraic structures called Symbolic Plithogenic Algebraic Structures. Their operations are given under the Absorbance Law and the Prevalence Order.