Special Issue "New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set, and their Extension to Plithogenic Set/Logic/Probability, with Applications"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 31 July 2019

Special Issue Editor

Guest Editor
Prof. Dr. Florentin Smarandache

Department of Mathematics and Sciences, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
Website | E-Mail
Interests: neutrosophic triplet structures, neutrosophic duplet structures, artificial intelligence, number theory, information fusion, statistics, algebraic structures

Special Issue Information

Dear Colleagues,

The fields of neutrosophic set, logic, measure, probability, and statistics have been developed and explored very much in the last few years due to their multiple practical applications.

The neutrosophic components truth (T), indeterminacy (I), and falsehood (F) are symmetric in form, since T is symmetric to its opposite F with respect to I, which acts as an axis of symmetry:  T----------I----------F

This Special Issue invites state-of-the-art papers on new topics related of neutrosophic theories and applications, such as

- Studies of corner cases of neutrosophic sets/probabilities/statistics/logics, such as

  • neutrosophic intuitionistic sets (which are different from intuitionistic fuzzy sets), neutrosophic paraconsistent sets, neutrosophic faillibilist sets, neutrosophic paradoxist sets, neutrosophic, pseudo-paradoxist sets, neutrosophic tautological sets, neutrosophic nihilist sets, neutrosophic dialetheist sets, and neutrosophic trivialist sets;
  • neutrosophic intuitionistic probability and statistics, neutrosophic paraconsistent probability and statistics, neutrosophic faillibilist probability and statistics, neutrosophic paradoxist probability and statistics, neutrosophic pseudo-paradoxist probability and statistics, neutrosophic tautological probability and statistics, neutrosophic nihilist probability and statistics, neutrosophic dialetheist probability and statistics, and neutrosophic trivialist probability and statistics;
  • neutrosophic paradoxist logic (or paradoxism), neutrosophic pseudo-paradoxist logic (or neutrosophic pseudo-paradoxism), and neutrosophic tautological logic (or neutrosophic tautologism):

https://arxiv.org/ftp/math/papers/0301/0301340.pdf

http://fs.unm.edu/DefinitionsDerivedFromNeutrosophics.pdf

  • Refined Neutrosophic Set components (T, I, and F) of many neutrosophic sub-components:

(T1, T2, ...; I1, I2, ...; F1, F2, ...);

https://arxiv.org/ftp/arxiv/papers/1407/1407.1041.pdf

http://fs.unm.edu/n-ValuedNeutrosophicLogic-PiP.pdf

  • Degrees of Dependence and Independence between Neutrosophic Components

T, I, and F are independent components, leaving room for incomplete information (when their superior sum < 1), paraconsistent and contradictory information (when the superior sum > 1), or complete information (sum of components = 1). 

For software engineering proposals, the classical unit interval [0, 1] is used.

For single-valued neutrosophic logic, the sum of the components is:

0 ≤ t+i+f ≤ 3 when all three components are independent;

0 ≤ t+i+f ≤ 2 when two components are dependent, while the third one is independent from them;

0 ≤ t+i+f ≤ 1 when all three components are dependent.

When three or two of the components T, I, and F are independent; one leaves room for incomplete information (sum < 1), paraconsistent and contradictory information (sum > 1), or complete information (sum = 1). 

If all three components, T, I, and F are dependent; then, similarly one leaves room for incomplete information (sum < 1) or complete information (sum = 1). 

In general, the sum of two components x and y that vary in the unitary interval [0, 1] is

0 ≤ x + y ≤ 2 - d°(x, y), in which d°(x, y) is the degree of dependence between x and y, while

d°(x, y) is the degree of independence between x and y.

https://doi.org/10.5281/zenodo.571359

http://fs.unm.edu/NSS/DegreeOfDependenceAndIndependence.pdf

- Neutrosophic Overset (when some neutrosophic component is > 1), since he observed that, for example, an employee working overtime deserves a degree of membership > 1, with respect to an employee that only works regular full-time and whose degree of membership = 1;

and to Neutrosophic Underset (when some neutrosophic component is < 0), since, for example, an employee causing more damage than benefit to his company deserves a degree of membership < 0, with respect to an employee that produces benefit to the company and has the degree of membership > 0;

and to Neutrosophic Offset (when some neutrosophic components are off the interval [0, 1], i.e., some neutrosophic component > 1 and some neutrosophic component < 0).

Then, similarly, neutrosophic logic/measure/probability/statistics, etc., were extended to, respectively, Neutrosophic Over-/Under-/Off- Logic, Measure, Probability, and Statistics, etc.

https://arxiv.org/ftp/arxiv/papers/1607/1607.00234.pdf

http://fs.unm.edu/NeutrosophicOversetUndersetOffset.pdf

http://fs.unm.edu/SVNeutrosophicOverset-JMI.pdf

http://fs.unm.edu/IV-Neutrosophic-Overset-Underset-Offset.pdf 

  • Neutrosophic Tripolar Set and Neutrosophic Multipolar Set

and, consequently, the Neutrosophic Tripolar Graph and Neutrosophic Multipolar Graph

http://fs.unm.edu/eBook-Neutrosophics6.pdf (p. 93)

http://fs.unm.edu/IFS-generalized.pdf

- N-norm and N-conorm

https://arxiv.org/ftp/arxiv/papers/0901/0901.1289.pdf

http://fs.unm.edu/N-normN-conorm.pdf

- Neutrosophic Measure and Neutrosophic Probability 
          (chance that an event occurs, indeterminate chance of occurrence,
           chance that the event does not occur)

https://arxiv.org/ftp/arxiv/papers/1311/1311.7139.pdf

http://fs.unm.edu/NeutrosophicMeasureIntegralProbability.pdf

  • Law of Included Multiple-Middle (as middle part of Neutrosophy)

(<A>;  <neut1A>, <neut2A>, …;  <antiA>)

http://fs.unm.edu/LawIncludedMultiple-Middle.pdf

- Neutrosophic Statistics (indeterminacy is introduced into classical statistics with respect to the sample/population, or with respect to the individuals that only partially belong to a sample/population, or with respect to the neutrosophic probability distributions)

https://arxiv.org/ftp/arxiv/papers/1406/1406.2000.pdf

http://fs.unm.edu/NeutrosophicStatistics.pdf

- Neutrosophic Precalculus and Neutrosophic Calculus

https://arxiv.org/ftp/arxiv/papers/1509/1509.07723.pdf

http://fs.unm.edu/NeutrosophicPrecalculusCalculus.pdf

Refined Neutrosophic Numbers (a+ b1I1 + b2I2 + … + bnIn), in which I1, I2, …, In are subindeterminacies of indeterminacy I;

(t,i,f)-neutrosophic graphs;

Thesis-Antithesis-Neutrothesis, and Neutrosynthesis;

Neutrosophic Axiomatic System;

Neutrosophic Dynamic Systems;

Symbolic Neutrosophic Logic;

(t, i, f)-Neutrosophic Structures;

I-Neutrosophic Structures;

Refined Literal Indeterminacy;

Quadruple Neutrosophic Algebraic Structures;

Multiplication Law of Subindeterminacies:

https://arxiv.org/ftp/arxiv/papers/1512/1512.00047.pdf   

http://fs.unm.edu/SymbolicNeutrosophicTheory.pdf

- Theory of Neutrosophic Evolution: Degrees of Evolution, Indeterminacy or Neutrality, and Involution

 http://fs.unm.edu/neutrosophic-evolution-PP-49-13.pdf

- Plithogeny as Generalization of Dialectics and Neutrosophy;

Plithogenic Set/Logic/Probability/Statistics (as generalization of fuzzy, intuitionistic fuzzy, neutrosophic set/logic/probability/statistics)

https://arxiv.org/ftp/arxiv/papers/1808/1808.03948.pdf

http://fs.unm.edu/Plithogeny.pdf

- Neutrosophic Psychology (neutropsyche, refined neutrosophic memory: conscious, aconscious, unconscious, neutropsychic personality, Eros/Aoristos/Thanatos, neutropsychic crisp personality)

http://fs.unm.edu/NeutropsychicPersonality-ed3.pdf

- Neutrosophic Applications in:

artificial intelligence, information systems, computer science, cybernetics, theory methods, mathematical algebraic structures, applied mathematics, automation, control systems, big data, engineering, electrical, electronic, philosophy, social science, psychology, biology, engineering, operational research, management science, imaging science, photographic technology, instruments, instrumentation, physics, optics, economics, mechanics, neurosciences, radiology nuclear, interdisciplinary applications, multidisciplinary sciences, etc.

[See: Xindong Peng and Jingguo Dai, A bibliometric analysis of neutrosophic set: two decades review from 1998 to 2017, Artificial Intelligence Review, Springer, 18 August 2018;

http://fs.unm.edu/BibliometricNeutrosophy.pdf]

Prof. Dr. Florentin Smarandache
Guest Editor

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Published Papers (16 papers)

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Research

Open AccessArticle A Novel Approach to Neutrosophic Soft Rough Set under Uncertainty
Symmetry 2019, 11(3), 384; https://doi.org/10.3390/sym11030384
Received: 5 January 2019 / Revised: 20 February 2019 / Accepted: 25 February 2019 / Published: 15 March 2019
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Abstract
To handle indeterminate and incomplete data, neutrosophic logic/set/probability were established. The neutrosophic truth, falsehood and indeterminacy components exhibit symmetry as the truth and the falsehood look the same and behave in a symmetrical way with respect to the indeterminacy component which serves as [...] Read more.
To handle indeterminate and incomplete data, neutrosophic logic/set/probability were established. The neutrosophic truth, falsehood and indeterminacy components exhibit symmetry as the truth and the falsehood look the same and behave in a symmetrical way with respect to the indeterminacy component which serves as a line of the symmetry. Soft set is a generic mathematical tool for dealing with uncertainty. Rough set is a new mathematical tool for dealing with vague, imprecise, inconsistent and uncertain knowledge in information systems. This paper introduces a new rough set model based on neutrosophic soft set to exploit simultaneously the advantages of rough sets and neutrosophic soft sets in order to handle all types of uncertainty in data. The idea of neutrosophic right neighborhood is utilised to define the concepts of neutrosophic soft rough (NSR) lower and upper approximations. Properties of suggested approximations are proposed and subsequently proven. Some of the NSR set concepts such as NSR-definability, NSR-relations and NSR-membership functions are suggested and illustrated with examples. Further, we demonstrate the feasibility of the newly rough set model with decision making problems involving neutrosophic soft set. Finally, a discussion on the features and limitations of the proposed model is provided. Full article
Open AccessArticle Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty
Symmetry 2019, 11(3), 382; https://doi.org/10.3390/sym11030382
Received: 5 January 2019 / Revised: 20 February 2019 / Accepted: 26 February 2019 / Published: 15 March 2019
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Abstract
In the definition of the complex neutrosophic soft expert set (CNSES), parameters set is a classical set, and the parameters have the same degree of importance, which is considered as 1. This poses a limitation in modeling of some problems. This paper introduces [...] Read more.
In the definition of the complex neutrosophic soft expert set (CNSES), parameters set is a classical set, and the parameters have the same degree of importance, which is considered as 1. This poses a limitation in modeling of some problems. This paper introduces the concept of fuzzy parameterized complex neutrosophic soft expert set (FP-CNSES) to handle this issue by assigning a degree of importance to each of the problem parameters. We further develop FP-CNSES by establishing the concept of weighted fuzzy parameterized complex neutrosophic soft expert set (WFP-CNSES) based on the idea that each expert has a relative weight. These new mathematical frameworks reduce the chance of unfairness in the decision making process. Some essential operations with their properties and relevant laws related to the notion of FP-CNSES are defined and verified. The notation of mapping on fuzzy parameterized complex neutrosophic soft expert classes is defined and some properties of fuzzy parameterized complex neutrosophic soft expert images and inverse images was investigated. FP-CNSES is used to put forth an algorithm on decision-making by converting it from complex state to real state and subsequently provided the detailed decision steps. Then, we provide the comparison of FP-CNSES to the current methods to show the ascendancy of our proposed method. Full article
Open AccessArticle Logarithmic Hybrid Aggregation Operators Based on Single Valued Neutrosophic Sets and Their Applications in Decision Support Systems
Symmetry 2019, 11(3), 364; https://doi.org/10.3390/sym11030364
Received: 17 January 2019 / Revised: 6 March 2019 / Accepted: 7 March 2019 / Published: 11 March 2019
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Abstract
Recently, neutrosophic sets are found to be more general and useful to express incomplete, indeterminate and inconsistent information. The purpose of this paper is to introduce new aggregation operators based on logarithmic operations and to develop a multi-criteria decision-making approach to study the [...] Read more.
Recently, neutrosophic sets are found to be more general and useful to express incomplete, indeterminate and inconsistent information. The purpose of this paper is to introduce new aggregation operators based on logarithmic operations and to develop a multi-criteria decision-making approach to study the interaction between the input argument under the single valued neutrosophic (SVN) environment. The main advantage of the proposed operator is that it can deal with the situations of the positive interaction, negative interaction or non-interaction among the criteria, during decision-making process. In this paper, we also defined some logarithmic operational rules on SVN sets, then we propose the single valued neutrosophic hybrid aggregation operators as a tool for multi-criteria decision-making (MCDM) under the neutrosophic environment and discussd some properties. Finally, the detailed decision-making steps for the single valued neutrosophic MCDM problems were developed, and a practical case was given to check the created approach and to illustrate its validity and superiority. Besides this, a systematic comparison analysis with other existent methods is conducted to reveal the advantages of our proposed method. Results indicate that the proposed method is suitable and effective for decision process to evaluate their best alternative. Full article
Open AccessArticle Application of Neutrosophic Logic to Evaluate Correlation between Prostate Cancer Mortality and Dietary Fat Assumption
Symmetry 2019, 11(3), 330; https://doi.org/10.3390/sym11030330
Received: 3 January 2019 / Revised: 28 February 2019 / Accepted: 1 March 2019 / Published: 6 March 2019
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Abstract
This paper presents an epidemiological study on the dietary fat that causes prostate cancer in an uncertainty environment. To study this relationship under the indeterminate environment, data from 30 countries are selected for the prostate cancer death rate and dietary fat level in [...] Read more.
This paper presents an epidemiological study on the dietary fat that causes prostate cancer in an uncertainty environment. To study this relationship under the indeterminate environment, data from 30 countries are selected for the prostate cancer death rate and dietary fat level in the food. The neutrosophic correlation and regression line are fitted on the data. We note from the neutrosophic analysis that the prostate cancer death rate increases as the dietary fat level in the people increases. The neutrosophic regression coefficient also confirms this claim. From this study, we conclude that neutrosophic regression is a more effective model under uncertainty than the regression model under classical statistics. We also found a statistical correlation between dietary fat and prostate cancer risk. Full article
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Open AccessArticle Generalized Neutrosophic Extended Triplet Group
Symmetry 2019, 11(3), 327; https://doi.org/10.3390/sym11030327
Received: 2 February 2019 / Revised: 15 February 2019 / Accepted: 25 February 2019 / Published: 5 March 2019
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Abstract
Neutrosophic extended triplet group is a new algebra structure and is different from the classical group. In this paper, the notion of generalized neutrosophic extended triplet group is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) [...] Read more.
Neutrosophic extended triplet group is a new algebra structure and is different from the classical group. In this paper, the notion of generalized neutrosophic extended triplet group is proposed and some properties are discussed. In particular, the following conclusions are strictly proved: (1) an algebraic system is a generalized neutrosophic extended triplet group if and only if it is a quasi-completely regular semigroup; (2) an algebraic system is a weak commutative generalized neutrosophic extended triplet group if and only if it is a quasi-Clifford semigroup; (3) for each n Z + , n 2 , ( Z n , ) is a commutative generalized neutrosophic extended triplet group; (4) for each n Z + , n 2 , ( Z n , ) is a commutative neutrosophic extended triplet group if and only if n = p 1 p 2 p m , i.e., the factorization of n has only single factor. Full article
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Open AccessArticle A Generalized Approach towards Soft Expert Sets via Neutrosophic Cubic Sets with Applications in Games
Symmetry 2019, 11(2), 289; https://doi.org/10.3390/sym11020289
Received: 28 January 2019 / Revised: 18 February 2019 / Accepted: 19 February 2019 / Published: 22 February 2019
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Abstract
Games are considered to be the most attractive and healthy event between nations and peoples. Soft expert sets are helpful for capturing uncertain and vague information. By contrast, neutrosophic set is a tri-component logic set, thus it can deal with uncertain, indeterminate, and [...] Read more.
Games are considered to be the most attractive and healthy event between nations and peoples. Soft expert sets are helpful for capturing uncertain and vague information. By contrast, neutrosophic set is a tri-component logic set, thus it can deal with uncertain, indeterminate, and incompatible information where the indeterminacy is quantified explicitly and truth membership, indeterminacy membership, and falsity membership independent of each other. Subsequently, we develop a combined approach and extend this concept further to introduce the notion of the neutrosophic cubic soft expert sets (NCSESs) by using the concept of neutrosophic cubic soft sets, which is a powerful tool for handling uncertain information in many problems and especially in games. Then we define and analyze the properties of internal neutrosophic cubic soft expert sets (INCSESs) and external neutrosophic cubic soft expert sets (ENCSESs), P-order, P-union, P-intersection, P-AND, P-OR and R-order, R-union, R-intersection, R-AND, and R-OR of NCSESs. The NCSESs satisfy the laws of commutativity, associativity, De Morgan, distributivity, idempotentency, and absorption. We derive some conditions for P-union and P-intersection of two INCSESs to be an INCSES. It is shown that P-union and P-intersection of ENCSESs need not be an ENCSES. The R-union and R-intersection of the INCSESs (resp., ENCSESs) need not be an INCSES (resp. ENCSES). Necessary conditions for the P-union, R-union and R-intersection of two ENCSESs to be an ENCSES are obtained. We also study the conditions for R-intersection and P-intersection of two NCSESs to be an INCSES and ENCSES. Finally, for its applications in games, we use the developed procedure to analyze the cricket series between Pakistan and India. It is shown that the proposed method is suitable to be used for decision-making, and as good as or better when compared to existing models. Full article
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Open AccessArticle Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM
Symmetry 2019, 11(2), 278; https://doi.org/10.3390/sym11020278
Received: 4 January 2019 / Revised: 19 February 2019 / Accepted: 19 February 2019 / Published: 21 February 2019
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Abstract
Neutrosophic cubic sets (NCSs) can express complex multi-attribute decision-making (MADM) problems with its interval and single-valued neutrosophic numbers simultaneously. The weighted arithmetic average (WAA) and geometric average (WGA) operators are common aggregation operators for handling MADM problems. However, the neutrosophic cubic weighted arithmetic [...] Read more.
Neutrosophic cubic sets (NCSs) can express complex multi-attribute decision-making (MADM) problems with its interval and single-valued neutrosophic numbers simultaneously. The weighted arithmetic average (WAA) and geometric average (WGA) operators are common aggregation operators for handling MADM problems. However, the neutrosophic cubic weighted arithmetic average (NCWAA) and neutrosophic cubic geometric weighted average (NCWGA) operators may result in some unreasonable aggregated values in some cases. In order to overcome the drawbacks of the NCWAA and NCWGA, this paper developed a new neutrosophic cubic hybrid weighted arithmetic and geometric aggregation (NCHWAGA) operator and investigates its suitability and effectiveness. Then, we established a MADM method based on the NCHWAGA operator. Finally, a MADM problem with neutrosophic cubic information was provided to illustrate the application and effectiveness of the proposed method. Full article
Open AccessArticle A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making
Symmetry 2019, 11(2), 275; https://doi.org/10.3390/sym11020275
Received: 19 January 2019 / Revised: 14 February 2019 / Accepted: 15 February 2019 / Published: 21 February 2019
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Abstract
The aim of this paper is to present a multiple-attribute group decision-making (MAGDM) framework based on a new single-valued neutrosophic linguistic (SVNL) distance measure. By unifying the idea of the weighted average and ordered weighted averaging into a single-valued neutrosophic linguistic distance, we [...] Read more.
The aim of this paper is to present a multiple-attribute group decision-making (MAGDM) framework based on a new single-valued neutrosophic linguistic (SVNL) distance measure. By unifying the idea of the weighted average and ordered weighted averaging into a single-valued neutrosophic linguistic distance, we first developed a new SVNL weighted distance measure, namely a SVNL combined and weighted distance (SVNLCWD) measure. The focal characteristics of the devised SVNLCWD are its ability to combine both the decision-makers’ attitudes toward the importance, as well as the weights, of the arguments. Various desirable properties and families of the developed SVNLCWD were contemplated. Moreover, a MAGDM approach based on the SVNLCWD was formulated. Lastly, a real numerical example concerning a low-carbon supplier selection problem was used to describe the superiority and feasibility of the developed approach. Full article
Open AccessArticle Multi-Attribute Decision Making Method Based on Aggregated Neutrosophic Set
Symmetry 2019, 11(2), 267; https://doi.org/10.3390/sym11020267
Received: 30 December 2018 / Revised: 6 February 2019 / Accepted: 17 February 2019 / Published: 20 February 2019
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Abstract
Multi-attribute decision-making refers to the decision-making problem of selecting the optimal alternative or sorting the scheme when considering multiple attributes, which is widely used in engineering design, economy, management and military, etc. But in real application, the attribute information of many objects is [...] Read more.
Multi-attribute decision-making refers to the decision-making problem of selecting the optimal alternative or sorting the scheme when considering multiple attributes, which is widely used in engineering design, economy, management and military, etc. But in real application, the attribute information of many objects is often inaccurate or uncertain, so it is very important for us to find a useful and efficient method to solve the problem. Neutrosophic set is proposed from philosophical point of view to handle inaccurate information efficiently, and a single-valued neutrosophic set (SVNS) is a special case of neutrosophic set, which is widely used in actual application fields. In this paper, a new method based on single-valued neutrosophic sets aggregation to solve multi-attribute decision making problem is proposed. Firstly, the neutrosophic decision matrix is obtained by expert assessment, a score function of single-valued neutrosophic sets (SVNSs) is defined to obtain the positive ideal solution (PIS) and the negative ideal solution (NIS). Then all alternatives are aggregated based on TOPSIS method to make decision. Finally numerical examples are given to verify the feasibility and rationality of the method. Full article
Open AccessArticle Neutrosophic Cubic Einstein Geometric Aggregation Operators with Application to Multi-Criteria Decision Making Method
Symmetry 2019, 11(2), 247; https://doi.org/10.3390/sym11020247
Received: 26 December 2018 / Revised: 28 January 2019 / Accepted: 31 January 2019 / Published: 16 February 2019
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Abstract
Neutrosophic cubic sets (NCs) are amore generalized version of neutrosophic sets(Ns) and interval neutrosophic sets (INs). Neutrosophic cubic setsare better placed to express consistent, indeterminate and inconsistent information, which provides a better platform to deal with incomplete, inconsistent and vague data. Aggregation operators [...] Read more.
Neutrosophic cubic sets (NCs) are amore generalized version of neutrosophic sets(Ns) and interval neutrosophic sets (INs). Neutrosophic cubic setsare better placed to express consistent, indeterminate and inconsistent information, which provides a better platform to deal with incomplete, inconsistent and vague data. Aggregation operators play a key role in daily life, and in relation to science and engineering problems. In this paper we defined the algebraic and Einstein sum, multiplication and scalar multiplication, score and accuracy functions. Using these operations we defined geometric aggregation operators and Einstein geometric aggregation operators. First, we defined the algebraic and Einstein operators of addition, multiplication and scalar multiplication. We defined score and accuracy function to compare neutrosophic cubic values. Then we definedthe neutrosophic cubic weighted geometric operator (NCWG), neutrosophic cubic ordered weighted geometric operator (NCOWG), neutrosophic cubic Einstein weighted geometric operator (NCEWG), and neutrosophic cubic Einstein ordered weighted geometric operator (NCEOWG) over neutrosophic cubic sets. A multi-criteria decision making method is developed as an application to these operators. This method is then applied to a daily life problem. Full article
Open AccessArticle Inspection Strategy under Indeterminacy Based on Neutrosophic Coefficient of Variation
Symmetry 2019, 11(2), 193; https://doi.org/10.3390/sym11020193
Received: 4 January 2019 / Revised: 22 January 2019 / Accepted: 30 January 2019 / Published: 9 February 2019
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Abstract
The existing sampling plans which use the coefficient of variation (CV) are designed under classical statistics. These available sampling plans cannot be used for sentencing if the sample or the population has indeterminate, imprecise, unknown, incomplete or uncertain data. In this paper, we [...] Read more.
The existing sampling plans which use the coefficient of variation (CV) are designed under classical statistics. These available sampling plans cannot be used for sentencing if the sample or the population has indeterminate, imprecise, unknown, incomplete or uncertain data. In this paper, we introduce the neutrosophic coefficient of variation (NCV) first. We design a sampling plan based on the NCV. The neutrosophic operating characteristic (NOC) function is then given and used to determine the neutrosophic plan parameters under some constraints. The neutrosophic plan parameters such as neutrosophic sample size and neutrosophic acceptance number are determined through the neutrosophic optimization solution. We compare the efficiency of the proposed plan under the neutrosophic statistical interval method with the sampling plan under classical statistics. A real example which has indeterminate data is given to illustrate the proposed plan. Full article
Open AccessArticle Group Decision Making Based on Triangular Neutrosophic Cubic Fuzzy Einstein Hybrid Weighted Averaging Operators
Symmetry 2019, 11(2), 180; https://doi.org/10.3390/sym11020180
Received: 17 December 2018 / Revised: 22 January 2019 / Accepted: 28 January 2019 / Published: 2 February 2019
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Abstract
In this paper, a new concept of the triangular neutrosophic cubic fuzzy numbers (TNCFNs), their score and accuracy functions are introduced. Based on TNCFNs, some new Einstein aggregation operators, such as the triangular neutrosophic cubic fuzzy Einstein weighted averaging (TNCFEWA), triangular neutrosophic cubic [...] Read more.
In this paper, a new concept of the triangular neutrosophic cubic fuzzy numbers (TNCFNs), their score and accuracy functions are introduced. Based on TNCFNs, some new Einstein aggregation operators, such as the triangular neutrosophic cubic fuzzy Einstein weighted averaging (TNCFEWA), triangular neutrosophic cubic fuzzy Einstein ordered weighted averaging (TNCFEOWA) and triangular neutrosophic cubic fuzzy Einstein hybrid weighted averaging (TNCFEHWA) operators are developed. Furthermore, their application to multiple-attribute decision-making with triangular neutrosophic cubic fuzzy (TNCF) information is discussed. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness. Full article
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Open AccessArticle Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation
Symmetry 2019, 11(2), 147; https://doi.org/10.3390/sym11020147
Received: 16 January 2019 / Revised: 25 January 2019 / Accepted: 28 January 2019 / Published: 29 January 2019
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Abstract
Neural networks are powerful universal approximation tools. They have been utilized for functions/data approximation, classification, pattern recognition, as well as their various applications. Uncertain or interval values result from the incompleteness of measurements, human observation and estimations in the real world. Thus, a [...] Read more.
Neural networks are powerful universal approximation tools. They have been utilized for functions/data approximation, classification, pattern recognition, as well as their various applications. Uncertain or interval values result from the incompleteness of measurements, human observation and estimations in the real world. Thus, a neutrosophic number (NsN) can represent both certain and uncertain information in an indeterminate setting and imply a changeable interval depending on its indeterminate ranges. In NsN settings, however, existing interval neural networks cannot deal with uncertain problems with NsNs. Therefore, this original study proposes a neutrosophic compound orthogonal neural network (NCONN) for the first time, containing the NsN weight values, NsN input and output, and hidden layer neutrosophic neuron functions, to approximate neutrosophic functions/NsN data. In the proposed NCONN model, single input and single output neurons are the transmission notes of NsN data and hidden layer neutrosophic neurons are constructed by the compound functions of both the Chebyshev neutrosophic orthogonal polynomial and the neutrosophic sigmoid function. In addition, illustrative and actual examples are provided to verify the effectiveness and learning performance of the proposed NCONN model for approximating neutrosophic nonlinear functions and NsN data. The contribution of this study is that the proposed NCONN can handle the approximation problems of neutrosophic nonlinear functions and NsN data. However, the main advantage is that the proposed NCONN implies a simple learning algorithm, higher speed learning convergence, and higher learning accuracy in indeterminate/NsN environments. Full article
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Open AccessArticle An Approach toward a Q-Neutrosophic Soft Set and Its Application in Decision Making
Symmetry 2019, 11(2), 139; https://doi.org/10.3390/sym11020139
Received: 20 December 2018 / Revised: 20 January 2019 / Accepted: 23 January 2019 / Published: 27 January 2019
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Abstract
A neutrosophic set was proposed as an approach to study neutral uncertain information. It is characterized through three memberships, T,I and F, such that these independent functions stand for the truth, indeterminate, and false-membership degrees of an object. The neutrosophic [...] Read more.
A neutrosophic set was proposed as an approach to study neutral uncertain information. It is characterized through three memberships, T , I and F, such that these independent functions stand for the truth, indeterminate, and false-membership degrees of an object. The neutrosophic set presents a symmetric form since truth enrolment T is symmetric to its opposite false enrolment F with respect to indeterminacy enrolment I that acts as an axis of symmetry. The neutrosophic set was further extended to a Q-neutrosophic soft set, which is a hybrid model that keeps the features of the neutrosophic soft set in dealing with uncertainty, and the features of a Q-fuzzy soft set that handles two-dimensional information. In this study, we discuss some operations of Q-neutrosophic soft sets, such as subset, equality, complement, intersection, union, AND operation, and OR operation. We also define the necessity and possibility operations of a Q-neutrosophic soft set. Several properties and illustrative examples are discussed. Then, we define the Q-neutrosophic-set aggregation operator and use it to develop an algorithm for using a Q-neutrosophic soft set in decision-making issues that have indeterminate and uncertain data, followed by an illustrative real-life example. Full article
Open AccessFeature PaperArticle A Variable Acceptance Sampling Plan under Neutrosophic Statistical Interval Method
Symmetry 2019, 11(1), 114; https://doi.org/10.3390/sym11010114
Received: 3 January 2019 / Revised: 13 January 2019 / Accepted: 15 January 2019 / Published: 19 January 2019
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Abstract
The acceptance sampling plan plays an important role in maintaining the high quality of a product. The variable control chart, using classical statistics, helps in making acceptance or rejection decisions about the submitted lot of the product. Furthermore, the sampling plan, using classical [...] Read more.
The acceptance sampling plan plays an important role in maintaining the high quality of a product. The variable control chart, using classical statistics, helps in making acceptance or rejection decisions about the submitted lot of the product. Furthermore, the sampling plan, using classical statistics, assumes the complete or determinate information available about a lot of product. However, in some situations, data may be ambiguous, vague, imprecise, and incomplete or indeterminate. In this case, the use of neutrosophic statistics can be applied to guide the experimenters. In this paper, we originally proposed a new variable sampling plan using the neutrosophic interval statistical method. The neutrosophic operating characteristic (NOC) is derived using the neutrosophic normal distribution. The optimization solution is also presented for the proposed plan under the neutrosophic interval method. The effectiveness of the proposed plan is compared with the plan under classical statistics. The tables are presented for practical use and a real example is given to explain the neutrosophic fuzzy variable sampling plan in the industry. Full article
Open AccessArticle A Robust Single-Valued Neutrosophic Soft Aggregation Operators in Multi-Criteria Decision Making
Symmetry 2019, 11(1), 110; https://doi.org/10.3390/sym11010110
Received: 14 December 2018 / Revised: 9 January 2019 / Accepted: 17 January 2019 / Published: 18 January 2019
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Abstract
Molodtsov originated soft set theory that was provided a general mathematical framework for handling with uncertainties in which we meet the data by affix parameterized factor during the information analysis as differentiated to fuzzy as well as neutrosophic set theory. The main object [...] Read more.
Molodtsov originated soft set theory that was provided a general mathematical framework for handling with uncertainties in which we meet the data by affix parameterized factor during the information analysis as differentiated to fuzzy as well as neutrosophic set theory. The main object of this paper is to lay a foundation for providing a new approach of single-valued neutrosophic soft tool which is considering many problems that contain uncertainties. In present study, a new aggregation operators of single-valued neutrosophic soft numbers have so far not yet been applied for ranking of the alternatives in decision-making problems. To this propose work, single-valued neutrosophic soft weighted arithmetic averaging (SVNSWA) operator, single-valued neutrosophic soft weighted geometric averaging (SVNSWGA) operator have been used to compare two single-valued neutrosophic soft numbers (SVNSNs) for aggregating different single-valued neutrosophic soft input arguments in neutrosophic soft environment. Then, its related properties have been investigated. Finally, a practical example for Medical diagnosis problems provided to test the feasibility and applicability of the proposed work. Full article
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