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New types of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set, Neutrosophic Refined Set,...
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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). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: closed (31 July 2019) | Viewed by 102792

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Florentin Smarandache

E-Mail Website
Guest Editor
Department of Math and Science, University of New Mexico, 705 Gurley Ave., Gallup Campus, NM 87301, USA
Interests: neutrosophic physics; neutrosophic probability; neutrosophic statistics; neutrosophic algebraic structures; plithogenic set; unmatter; superluminal physics; paradoxism; quantum paradoxes
Special Issues, Collections and Topics in MDPI journals

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 (37 papers)

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15 pages, 806 KiB  
Article
The Structure of Idempotents in Neutrosophic Rings and Neutrosophic Quadruple Rings
by Yingcang Ma, Xiaohong Zhang, Florentin Smarandache and Juanjuan Zhang
Symmetry 2019, 11(10), 1254; https://doi.org/10.3390/sym11101254 - 08 Oct 2019
Cited by 14 | Viewed by 1737
Abstract
This paper aims to reveal the structure of idempotents in neutrosophic rings and neutrosophic quadruple rings. First, all idempotents in neutrosophic rings ⟨ R ∪ I ⟩ are given when R is C , R , Q , Z or Z n . [...] Read more.
This paper aims to reveal the structure of idempotents in neutrosophic rings and neutrosophic quadruple rings. First, all idempotents in neutrosophic rings ⟨ R ∪ I ⟩ are given when R is C , R , Q , Z or Z n . Secondly, the neutrosophic quadruple ring ⟨ R ∪ T ∪ I ∪ F ⟩ is introduced and all idempotents in neutrosophic quadruple rings ⟨ C ∪ T ∪ I ∪ F ⟩ , ⟨ R ∪ T ∪ I ∪ F ⟩ , ⟨ Q ∪ T ∪ I ∪ F ⟩ , ⟨ Z ∪ T ∪ I ∪ F ⟩ and ⟨ Z n ∪ T ∪ I ∪ F ⟩ are also given. Furthermore, the algorithms for solving the idempotents in ⟨ Z n ∪ I ⟩ and ⟨ Z n ∪ T ∪ I ∪ F ⟩ for each nonnegative integer n are provided. Lastly, as a general result, if all idempotents in any ring R are known, then the structure of idempotents in neutrosophic ring ⟨ R ∪ I ⟩ and neutrosophic quadruple ring ⟨ R ∪ T ∪ I ∪ F ⟩ can be determined. Full article
(This article belongs to the 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)
18 pages, 2910 KiB  
Article
Symmetry in Hyperstructure: Neutrosophic Extended Triplet Semihypergroups and Regular Hypergroups
by Xiaohong Zhang, Florentin Smarandache and Yingcang Ma
Symmetry 2019, 11(10), 1217; https://doi.org/10.3390/sym11101217 - 01 Oct 2019
Cited by 5 | Viewed by 1688
Abstract
The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, [...] Read more.
The symmetry of hyperoperation is expressed by hypergroup, more extensive hyperalgebraic structures than hypergroups are studied in this paper. The new concepts of neutrosophic extended triplet semihypergroup (NET- semihypergroup) and neutrosophic extended triplet hypergroup (NET-hypergroup) are firstly introduced, some basic properties are obtained, and the relationships among NET- semihypergroups, regular semihypergroups, NET-hypergroups and regular hypergroups are systematically are investigated. Moreover, pure NET-semihypergroup and pure NET-hypergroup are investigated, and a strucuture theorem of commutative pure NET-semihypergroup is established. Finally, a new notion of weak commutative NET-semihypergroup is proposed, some important examples are obtained by software MATLAB, and the following important result is proved: every pure and weak commutative NET-semihypergroup is a disjoint union of some regular hypergroups which are its subhypergroups. Full article
(This article belongs to the 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)
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26 pages, 3094 KiB  
Article
On Neutrosophic Offuninorms
by Erick González Caballero, Florentin Smarandache and Maikel Leyva Vázquez
Symmetry 2019, 11(9), 1136; https://doi.org/10.3390/sym11091136 - 06 Sep 2019
Cited by 19 | Viewed by 2496
Abstract
Uninorms comprise an important kind of operator in fuzzy theory. They are obtained from the generalization of the t-norm and t-conorm axiomatic. Uninorms are theoretically remarkable, and furthermore, they have a wide range of applications. For that reason, when fuzzy sets have been [...] Read more.
Uninorms comprise an important kind of operator in fuzzy theory. They are obtained from the generalization of the t-norm and t-conorm axiomatic. Uninorms are theoretically remarkable, and furthermore, they have a wide range of applications. For that reason, when fuzzy sets have been generalized to others—e.g., intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets, or neutrosophic sets—then uninorm generalizations have emerged in those novel frameworks. Neutrosophic sets contain the notion of indeterminacy—which is caused by unknown, contradictory, and paradoxical information—and thus, it includes, aside from the membership and non-membership functions, an indeterminate-membership function. Also, the relationship among them does not satisfy any restriction. Along this line of generalizations, this paper aims to extend uninorms to the framework of neutrosophic offsets, which are called neutrosophic offuninorms. Offsets are neutrosophic sets such that their domains exceed the scope of the interval [0,1]. In the present paper, the definition, properties, and application areas of this new concept are provided. It is necessary to emphasize that the neutrosophic offuninorms are feasible for application in several fields, as we illustrate in this paper. Full article
(This article belongs to the 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)
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26 pages, 351 KiB  
Article
Ordinary Single Valued Neutrosophic Topological Spaces
by Junhui Kim, Florentin Smarandache, Jeong Gon Lee and Kul Hur
Symmetry 2019, 11(9), 1075; https://doi.org/10.3390/sym11091075 - 27 Aug 2019
Cited by 7 | Viewed by 1985
Abstract
We define an ordinary single valued neutrosophic topology and obtain some of its basic properties. In addition, we introduce the concept of an ordinary single valued neutrosophic subspace. Next, we define the ordinary single valued neutrosophic neighborhood system and we show that an [...] Read more.
We define an ordinary single valued neutrosophic topology and obtain some of its basic properties. In addition, we introduce the concept of an ordinary single valued neutrosophic subspace. Next, we define the ordinary single valued neutrosophic neighborhood system and we show that an ordinary single valued neutrosophic neighborhood system has the same properties in a classical neighborhood system. Finally, we introduce the concepts of an ordinary single valued neutrosophic base and an ordinary single valued neutrosophic subbase, and obtain two characterizations of an ordinary single valued neutrosophic base and one characterization of an ordinary single valued neutrosophic subbase. Full article
(This article belongs to the 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)
23 pages, 1199 KiB  
Article
A New Type of Single Valued Neutrosophic Covering Rough Set Model
by Jingqian Wang and Xiaohong Zhang
Symmetry 2019, 11(9), 1074; https://doi.org/10.3390/sym11091074 - 24 Aug 2019
Cited by 5 | Viewed by 2328
Abstract
Recently, various types of single valued neutrosophic (SVN) rough set models were presented based on the same inclusion relation. However, there is another SVN inclusion relation in SVN sets. In this paper, we propose a new type of SVN covering rough set model [...] Read more.
Recently, various types of single valued neutrosophic (SVN) rough set models were presented based on the same inclusion relation. However, there is another SVN inclusion relation in SVN sets. In this paper, we propose a new type of SVN covering rough set model based on the new inclusion relation. Furthermore, the graph and matrix representations of the new SVN covering approximation operators are presented. Firstly, the notion of SVN β 2 -covering approximation space is proposed, which is decided by the new inclusion relation. Then, a type of SVN covering rough set model under the SVN β 2 -covering approximation space is presented. Moreover, there is a corresponding SVN relation rough set model based on a SVN relation induced by the SVN β 2 -covering, and two conditions under which the SVN β 2 -covering can induce a symmetric SVN relation are presented. Thirdly, the graph and matrix representations of the new SVN covering rough set model are investigated. Finally, we propose a novel method for decision making (DM) problems in paper defect diagnosis under the new SVN covering rough set model. Full article
(This article belongs to the 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)
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26 pages, 578 KiB  
Article
Generalization of Maximizing Deviation and TOPSIS Method for MADM in Simplified Neutrosophic Hesitant Fuzzy Environment
by Muhammad Akram, Sumera Naz and Florentin Smarandache
Symmetry 2019, 11(8), 1058; https://doi.org/10.3390/sym11081058 - 17 Aug 2019
Cited by 46 | Viewed by 2937
Abstract
With the development of the social economy and enlarged volume of information, the application of multiple-attribute decision-making (MADM) has become increasingly complex, uncertain, and obscure. As a further generalization of hesitant fuzzy set (HFS), simplified neutrosophic hesitant fuzzy set (SNHFS) is an efficient [...] Read more.
With the development of the social economy and enlarged volume of information, the application of multiple-attribute decision-making (MADM) has become increasingly complex, uncertain, and obscure. As a further generalization of hesitant fuzzy set (HFS), simplified neutrosophic hesitant fuzzy set (SNHFS) is an efficient tool to process the vague information and contains the ideas of a single-valued neutrosophic hesitant fuzzy set (SVNHFS) and an interval neutrosophic hesitant fuzzy set (INHFS). In this paper, we propose a decision-making approach based on the maximizing deviation method and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) to solve the MADM problems, in which the attribute weight information is incomplete, and the decision information is expressed in simplified neutrosophic hesitant fuzzy elements. Firstly, we inaugurate an optimization model on the basis of maximizing deviation method, which is useful to determine the attribute weights. Secondly, using the idea of the TOPSIS, we determine the relative closeness coefficient of each alternative and based on which we rank the considered alternatives to select the optimal one(s). Finally, we use a numerical example to show the detailed implementation procedure and effectiveness of our method in solving MADM problems under simplified neutrosophic hesitant fuzzy environment. Full article
(This article belongs to the 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)
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23 pages, 1694 KiB  
Article
Modeling the Performance Indicators of Financial Assets with Neutrosophic Fuzzy Numbers
by Marcel-Ioan Bolos, Ioana-Alexandra Bradea and Camelia Delcea
Symmetry 2019, 11(8), 1021; https://doi.org/10.3390/sym11081021 - 07 Aug 2019
Cited by 10 | Viewed by 3194
Abstract
This research sets the basis for modeling the performance indicators of financial assets using triangular neutrosophic fuzzy numbers. This type of number allows for the modeling of financial assets performance indicators by taking into consideration all the possible scenarios of their achievement. The [...] Read more.
This research sets the basis for modeling the performance indicators of financial assets using triangular neutrosophic fuzzy numbers. This type of number allows for the modeling of financial assets performance indicators by taking into consideration all the possible scenarios of their achievement. The key performance indicators (KPIs) modeled with the help of triangular fuzzy neutrosophic numbers are the return on financial assets, the financial assets risk, and the covariance between financial assets. Thus far, the return on financial assets has been studied using statistical indicators, like the arithmetic and geometric mean, or using the financial risk indicators with the help of the squared deviations from the mean and covariance. These indicators are well known as the basis of portfolio theory. This paper opens the perspective of modeling these three mentioned statistical indicators using triangular neutrosophic fuzzy numbers due to the major advantages they have. The first advantage of the neutrosophic approach is that it includes three possible symmetric scenarios of the KPIs achievement, namely the scenario of certainty, the scenario of non-realization, and the scenario of indecision, in which it cannot be appreciated whether the performance indicators are or are not achieved. The second big advantage is its data series clustering, representing the financial performance indicators by which these scenarios can be delimitated by means of neutrosophic fuzzy numbers in very good, good or weak performance indicators. This clustering is realized by means of the linguistic criteria and measuring the belonging degree to a class of indicators using fuzzy membership functions. The third major advantage is the selection of risk mitigation analysis scenarios and the formation of financial assets’ optimal portfolios. Full article
(This article belongs to the 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)
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13 pages, 300 KiB  
Article
Study on the Algebraic Structure of Refined Neutrosophic Numbers
by Qiaoyan Li, Yingcang Ma, Xiaohong Zhang and Juanjuan Zhang
Symmetry 2019, 11(8), 954; https://doi.org/10.3390/sym11080954 - 27 Jul 2019
Cited by 1 | Viewed by 1950
Abstract
This paper aims to explore the algebra structure of refined neutrosophic numbers. Firstly, the algebra structure of neutrosophic quadruple numbers on a general field is studied. Secondly, The addition operator ⊕ and multiplication operator ⊗ on refined neutrosophic numbers are proposed and the [...] Read more.
This paper aims to explore the algebra structure of refined neutrosophic numbers. Firstly, the algebra structure of neutrosophic quadruple numbers on a general field is studied. Secondly, The addition operator ⊕ and multiplication operator ⊗ on refined neutrosophic numbers are proposed and the algebra structure is discussed. We reveal that the set of neutrosophic refined numbers with an additive operation is an abelian group and the set of neutrosophic refined numbers with a multiplication operation is a neutrosophic extended triplet group. Moreover, algorithms for solving the neutral element and opposite elements of each refined neutrosophic number are given. Full article
(This article belongs to the 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)
16 pages, 744 KiB  
Article
Non-Dual Multi-Granulation Neutrosophic Rough Set with Applications
by Chunxin Bo, Xiaohong Zhang and Songtao Shao
Symmetry 2019, 11(7), 910; https://doi.org/10.3390/sym11070910 - 12 Jul 2019
Cited by 2 | Viewed by 1879
Abstract
Multi-attribute decision-making (MADM) is a part of management decision-making and an important branch of the modern decision theory and method. MADM focuses on the decision problem of discrete and finite decision schemes. Uncertain MADM is an extension and development of classical multi-attribute decision [...] Read more.
Multi-attribute decision-making (MADM) is a part of management decision-making and an important branch of the modern decision theory and method. MADM focuses on the decision problem of discrete and finite decision schemes. Uncertain MADM is an extension and development of classical multi-attribute decision making theory. When the attribute value of MADM is shown by neutrosophic number, that is, the attribute value is complex data and needs three values to express, it is called the MADM problem in which the attribute values are neutrosophic numbers. However, in practical MADM problems, to minimize errors in individual decision making, we need to consider the ideas of many people and synthesize their opinions. Therefore, it is of great significance to study the method of attribute information aggregation. In this paper, we proposed a new theory—non-dual multi-granulation neutrosophic rough set (MS)—to aggregate multiple attribute information and solve a multi-attribute group decision-making (MGDM) problem where the attribute values are neutrosophic numbers. First, we defined two kinds of non-dual MS models, intersection-type MS and union-type MS. Additionally, their properties are studied. Then the relationships between MS, non-dual MS, neutrosophic rough set (NRS) based on neutrosophic intersection (union) relationship, and NRS based on neutrosophic transitive closure relation of union relationship are outlined, and a figure is given to show them directly. Finally, the definition of non-dual MS on two universes is given and we use it to solve a MGDM problem with a neutrosophic number as the attribute value. Full article
(This article belongs to the 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)
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21 pages, 2086 KiB  
Article
A Hybrid Plithogenic Decision-Making Approach with Quality Function Deployment for Selecting Supply Chain Sustainability Metrics
by Mohamed Abdel-Basset, Rehab Mohamed, Abd El-Nasser H. Zaied and Florentin Smarandache
Symmetry 2019, 11(7), 903; https://doi.org/10.3390/sym11070903 - 11 Jul 2019
Cited by 146 | Viewed by 6331
Abstract
Supply chain sustainability has become one of the most attractive decision management topics. There are many articles that have focused on this field presenting many different points of view. This research is centred on the evaluation of supply chain sustainability based on two [...] Read more.
Supply chain sustainability has become one of the most attractive decision management topics. There are many articles that have focused on this field presenting many different points of view. This research is centred on the evaluation of supply chain sustainability based on two critical dimensions. The first is the importance of evaluation metrics based on economic, environmental and social aspects, and the second is the degree of difficulty of information gathering. This paper aims to increase the accuracy of the evaluation. The proposed method is a combination of quality function deployment (QFD) with plithogenic aggregation operations. The aggregation operation is applied to aggregate: Firstly, the decision maker’s opinions of requirements that are needed to evaluate the supply chain sustainability; secondly, the evaluation metrics based on the requirements; and lastly, the evaluation of information gathering difficulty. To validate the proposed model, this study presented a real world case study of Thailand’s sugar industry. The results showed the most preferred and the lowest preferred metrics in order to evaluate the sustainability of the supply chain strategy. Full article
(This article belongs to the 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)
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24 pages, 6435 KiB  
Article
Online Visual Tracking of Weighted Multiple Instance Learning via Neutrosophic Similarity-Based Objectness Estimation
by Keli Hu, Wei He, Jun Ye, Liping Zhao, Hua Peng and Jiatian Pi
Symmetry 2019, 11(6), 832; https://doi.org/10.3390/sym11060832 - 25 Jun 2019
Cited by 31 | Viewed by 2567
Abstract
An online neutrosophic similarity-based objectness tracking with a weighted multiple instance learning algorithm (NeutWMIL) is proposed. Each training sample is extracted surrounding the object location, and the distribution of these samples is symmetric. To provide a more robust weight for each sample in [...] Read more.
An online neutrosophic similarity-based objectness tracking with a weighted multiple instance learning algorithm (NeutWMIL) is proposed. Each training sample is extracted surrounding the object location, and the distribution of these samples is symmetric. To provide a more robust weight for each sample in the positive bag, the asymmetry of the importance of the samples is considered. The neutrosophic similarity-based objectness estimation with object properties (super straddling) is applied. The neutrosophic theory is a new branch of philosophy for dealing with incomplete, indeterminate, and inconsistent information. By considering the surrounding information of the object, a single valued neutrosophic set (SVNS)-based segmentation parameter selection method is proposed, to produce a well-built set of superpixels which can better explain the object area at each frame. Then, the intersection and shape-distance criteria are proposed for weighting each superpixel in the SVNS domain, mainly via three membership functions, T (truth), I (indeterminacy), and F (falsity), for each criterion. After filtering out the superpixels with low response, the newly defined neutrosophic weights are utilized for weighting each sample. Furthermore, the objectness estimation information is also applied for estimating and alleviating the problem of tracking drift. Experimental results on challenging benchmark video sequences reveal the superior performance of our algorithm when confronting appearance changes and background clutters. Full article
(This article belongs to the 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)
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22 pages, 819 KiB  
Article
Neutrosophic Triangular Norms and Their Derived Residuated Lattices
by Qingqing Hu and Xiaohong Zhang
Symmetry 2019, 11(6), 817; https://doi.org/10.3390/sym11060817 - 20 Jun 2019
Cited by 7 | Viewed by 2473
Abstract
Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can [...] Read more.
Neutrosophic triangular norms (t-norms) and their residuated lattices are not only the main research object of neutrosophic set theory, but also the core content of neutrosophic logic. Neutrosophic implications are important operators of neutrosophic logic. Neutrosophic residual implications based on neutrosophic t-norms can be applied to the fields of neutrosophic inference and neutrosophic control. In this paper, neutrosophic t-norms, neutrosophic residual implications, and the residuated lattices derived from neutrosophic t-norms are investigated deeply. First of all, the lattice and its corresponding system are proved to be a complete lattice and a De Morgan algebra, respectively. Second, the notions of neutrosophic t-norms are introduced on the complete lattice discussed earlier. The basic concepts and typical examples of representable and non-representable neutrosophic t-norms are obtained. Naturally, De Morgan neutrosophic triples are defined for the duality of neutrosophic t-norms and neutrosophic t-conorms with respect to neutrosophic negators. Third, neutrosophic residual implications generated from neutrosophic t-norms and their basic properties are investigated. Furthermore, residual neutrosophic t-norms are proved to be infinitely ∨-distributive, and then some important properties possessed by neutrosophic residual implications are given. Finally, a method for producing neutrosophic t-norms from neutrosophic implications is presented, and the residuated lattices are constructed on the basis of neutrosophic t-norms and neutrosophic residual implications. Full article
(This article belongs to the 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)
14 pages, 267 KiB  
Article
Introduction to Non-Standard Neutrosophic Topology
by Mohammed A. Al Shumrani and Florentin Smarandache
Symmetry 2019, 11(5), 706; https://doi.org/10.3390/sym11050706 - 23 May 2019
Viewed by 2323
Abstract
For the first time we introduce non-standard neutrosophic topology on the extended non-standard analysis space, called non-standard real monad space, which is closed under neutrosophic non-standard infimum and supremum. Many classical topological concepts are extended to the non-standard neutrosophic topology, several theorems and [...] Read more.
For the first time we introduce non-standard neutrosophic topology on the extended non-standard analysis space, called non-standard real monad space, which is closed under neutrosophic non-standard infimum and supremum. Many classical topological concepts are extended to the non-standard neutrosophic topology, several theorems and properties about them are proven, and many examples are presented. Full article
(This article belongs to the 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)
18 pages, 341 KiB  
Article
Application of the Bipolar Neutrosophic Hamacher Averaging Aggregation Operators to Group Decision Making: An Illustrative Example
by Muhammad Jamil, Saleem Abdullah, Muhammad Yaqub Khan, Florentin Smarandache and Fazal Ghani
Symmetry 2019, 11(5), 698; https://doi.org/10.3390/sym11050698 - 21 May 2019
Cited by 12 | Viewed by 2930
Abstract
The present study aims to introduce the notion of bipolar neutrosophic Hamacher aggregation operators and to also provide its application in real life. Then neutrosophic set (NS) can elaborate the incomplete, inconsistent, and indeterminate information, Hamacher aggregation operators, and extended Einstein aggregation operators [...] Read more.
The present study aims to introduce the notion of bipolar neutrosophic Hamacher aggregation operators and to also provide its application in real life. Then neutrosophic set (NS) can elaborate the incomplete, inconsistent, and indeterminate information, Hamacher aggregation operators, and extended Einstein aggregation operators to the arithmetic and geometric aggregation operators. First, we give the fundamental definition and operations of the neutrosophic set and the bipolar neutrosophic set. Our main focus is on the Hamacher aggregation operators of bipolar neutrosophic, namely, bipolar neutrosophic Hamacher weighted averaging (BNHWA), bipolar neutrosophic Hamacher ordered weighted averaging (BNHOWA), and bipolar neutrosophic Hamacher hybrid averaging (BNHHA) along with their desirable properties. The prime gain of utilizing the suggested methods is that these operators progressively provide total perspective on the issue necessary for the decision makers. These tools provide generalized, increasingly exact, and precise outcomes when compared to the current methods. Finally, as an application, we propose new methods for the multi-criteria group decision-making issues by using the various kinds of bipolar neutrosophic operators with a numerical model. This demonstrates the usefulness and practicality of this proposed approach in real life. Full article
(This article belongs to the 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)
15 pages, 462 KiB  
Article
Neutrosophic Extended Triplet Group Based on Neutrosophic Quadruple Numbers
by Qiaoyan Li, Yingcang Ma, Xiaohong Zhang and Juanjuan Zhang
Symmetry 2019, 11(5), 696; https://doi.org/10.3390/sym11050696 - 21 May 2019
Cited by 9 | Viewed by 2138
Abstract
In this paper, we explore the algebra structure based on neutrosophic quadruple numbers. Moreover, two kinds of degradation algebra systems of neutrosophic quadruple numbers are introduced. In particular, the following results are strictly proved: (1) the set of neutrosophic quadruple numbers with a [...] Read more.
In this paper, we explore the algebra structure based on neutrosophic quadruple numbers. Moreover, two kinds of degradation algebra systems of neutrosophic quadruple numbers are introduced. In particular, the following results are strictly proved: (1) the set of neutrosophic quadruple numbers with a multiplication operation is a neutrosophic extended triplet group; (2) the neutral element of each neutrosophic quadruple number is unique and there are only sixteen different neutral elements in all of neutrosophic quadruple numbers; (3) the set which has same neutral element is closed with respect to the multiplication operator; (4) the union of the set which has same neutral element is a partition of four-dimensional space. Full article
(This article belongs to the 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)
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15 pages, 309 KiB  
Article
Multi-Attribute Decision Making Based on Probabilistic Neutrosophic Hesitant Fuzzy Choquet Aggregation Operators
by Songtao Shao, Xiaohong Zhang and Quan Zhao
Symmetry 2019, 11(5), 623; https://doi.org/10.3390/sym11050623 - 03 May 2019
Cited by 21 | Viewed by 2292
Abstract
Take the third-party logistics providers (3PLs) as an example, according to the characteristics of correlation between attributes in multi-attribute decision-making, two Choquet aggregation operators adoping probabilistic neutrosophic hesitation fuzzy elements (PNHFEs) are proposed to cope with the situations of correlation among criterions. This [...] Read more.
Take the third-party logistics providers (3PLs) as an example, according to the characteristics of correlation between attributes in multi-attribute decision-making, two Choquet aggregation operators adoping probabilistic neutrosophic hesitation fuzzy elements (PNHFEs) are proposed to cope with the situations of correlation among criterions. This measure not only provides support for the correlation phenomenon between internal attributes, but also fully concerns the incidental uncertainty of the external space. Our goal is to make it easier for decision makers to cope with this uncertainty, thus we establish the notion of probabilistic neutrosophic hesitant fuzzy Choquet averaging (geometric) (PNHFCOA, PNHFCOG) operator. Based on this foundation, a method for aggregating decision makers’ information is proposed, and then the optimal decision scheme is obtained. Finally, an example of selecting optimal 3PL is given to demonstrate the objectivity of the above-mentioned standpoint. Full article
(This article belongs to the 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)
21 pages, 1745 KiB  
Article
IoT and Its Impact on the Electronics Market: A Powerful Decision Support System for Helping Customers in Choosing the Best Product
by Mohamed Abdel-Basset, Mai Mohamed, Victor Chang and Florentin Smarandache
Symmetry 2019, 11(5), 611; https://doi.org/10.3390/sym11050611 - 01 May 2019
Cited by 15 | Viewed by 3908
Abstract
Many companies have observed the significant benefits they can get via using internet. Since then, large companies have been able to develop business transactions with customers at anytime, anywhere, and in relation to anything, so that we now need a more comprehensive concept [...] Read more.
Many companies have observed the significant benefits they can get via using internet. Since then, large companies have been able to develop business transactions with customers at anytime, anywhere, and in relation to anything, so that we now need a more comprehensive concept than the internet. This concept is the Internet of Things (IoT). IoT will influence decision making style in various phases of selling, buying and marketing process. Therefore, every individual and company should know precisely what IoT is, and how and why they should incorporate it in their operations. This motivated us to propose a smart system based on IoT to help companies and marketers make a powerful marketing strategy via utilizing obtained data from IoT devices. Not only this, but the proposed system can also solve the problems which face companies and customers in online shopping. Since there are different types of the same product, and also different criteria for purchasing which can be different between individuals, customers will need a decision support system to recommend them with the best selection. This motivates us to also propose a neutrsophic technique to deal with unclear and conflicting information which exists usually in the purchasing process. Therefore, the smart system and neutrosophic technique is considered as a comprehensive system which links between customers, companies, marketers to achieve satisfaction for each of them. Full article
(This article belongs to the 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)
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34 pages, 509 KiB  
Article
Neutrosophic Optimization Model and Computational Algorithm for Optimal Shale Gas Water Management under Uncertainty
by Firoz Ahmad, Ahmad Yusuf Adhami and Florentin Smarandache
Symmetry 2019, 11(4), 544; https://doi.org/10.3390/sym11040544 - 15 Apr 2019
Cited by 42 | Viewed by 3235
Abstract
Shale gas energy is the most prominent and dominating source of power across the globe. The processes for the extraction of shale gas from shale rocks are very complex. In this study, a multiobjective optimization framework is presented for an overall water management [...] Read more.
Shale gas energy is the most prominent and dominating source of power across the globe. The processes for the extraction of shale gas from shale rocks are very complex. In this study, a multiobjective optimization framework is presented for an overall water management system that includes the allocation of freshwater for hydraulic fracturing and optimal management of the resulting wastewater with different techniques. The generated wastewater from the shale fracking process contains highly toxic chemicals. The optimal control of a massive amount of contaminated water is quite a challenging task. Therefore, an on-site treatment plant, underground disposal facility, and treatment plant with expansion capacity were designed to overcome environmental issues. A multiobjective trade-off between socio-economic and environmental concerns was established under a set of conflicting constraints. A solution method—the neutrosophic goal programming approach—is suggested, inspired by independent, neutral/indeterminacy thoughts of the decision-maker(s). A theoretical computational study is presented to show the validity and applicability of the proposed multiobjective shale gas water management optimization model and solution procedure. The obtained results and conclusions, along with the significant contributions, are discussed in the context of shale gas supply chain planning policies over different time horizons. Full article
(This article belongs to the 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)
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25 pages, 300 KiB  
Article
Extended Nonstandard Neutrosophic Logic, Set, and Probability Based on Extended Nonstandard Analysis
by Florentin Smarandache
Symmetry 2019, 11(4), 515; https://doi.org/10.3390/sym11040515 - 10 Apr 2019
Cited by 3 | Viewed by 1805
Abstract
We extend for the second time the nonstandard analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad—all these in order [...] Read more.
We extend for the second time the nonstandard analysis by adding the left monad closed to the right, and right monad closed to the left, while besides the pierced binad (we introduced in 1998) we add now the unpierced binad—all these in order to close the newly extended nonstandard space under nonstandard addition, nonstandard subtraction, nonstandard multiplication, nonstandard division, and nonstandard power operations. Then, we extend the Nonstandard Neutrosophic Logic, Nonstandard Neutrosophic Set, and Nonstandard Probability on this Extended Nonstandard Analysis space, and we prove that it is a nonstandard neutrosophic lattice of first type (endowed with a nonstandard neutrosophic partial order) as well as a nonstandard neutrosophic lattice of second type (as algebraic structure, endowed with two binary neutrosophic laws: infN and supN). Many theorems, new terms introduced, better notations for monads and binads, and examples of nonstandard neutrosophic operations are given. Full article
(This article belongs to the 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)
12 pages, 276 KiB  
Article
Simplified Neutrosophic Exponential Similarity Measures for Evaluation of Smart Port Development
by Jihong Chen, Kai Xue, Jun Ye, Tiancun Huang, Yan Tian, Chengying Hua and Yuhua Zhu
Symmetry 2019, 11(4), 485; https://doi.org/10.3390/sym11040485 - 03 Apr 2019
Cited by 18 | Viewed by 3055
Abstract
Smart ports represent the current trend of port development. Intelligent operations reduce the daily production cost of ports, facilitate efficient production, strengthen the risk mitigation ability and comply with the requirements for long-term development. However, a systematic and scientific smart port evaluation method [...] Read more.
Smart ports represent the current trend of port development. Intelligent operations reduce the daily production cost of ports, facilitate efficient production, strengthen the risk mitigation ability and comply with the requirements for long-term development. However, a systematic and scientific smart port evaluation method is missing to nail down the evaluation indicators of a smart port and enable accurate evaluation of a port’s degree of intelligence. This paper analyzes the concept of the smart port, establishes a set of smart port evaluation indicator systems, and applies a single-valued neutrosophic exponential similarity measure to port evaluation to enable quantitative evaluation of port integrity. This evaluation method is capable of decision-making in the event of incomplete, uncertain, and inconsistent information during general evaluation, opening up a new method for smart port evaluation, and acting as a helpful tool for ports to carry out improvements during actual application. Full article
(This article belongs to the 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)
23 pages, 850 KiB  
Article
A Refined Approach for Forecasting Based on Neutrosophic Time Series
by Mohamed Abdel-Basset, Victor Chang, Mai Mohamed and Florentin Smarandache
Symmetry 2019, 11(4), 457; https://doi.org/10.3390/sym11040457 - 01 Apr 2019
Cited by 48 | Viewed by 3192
Abstract
This research introduces a neutrosophic forecasting approach based on neutrosophic time series (NTS). Historical data can be transformed into neutrosophic time series data to determine their truth, indeterminacy and falsity functions. The basis for the neutrosophication process is the score and accuracy functions [...] Read more.
This research introduces a neutrosophic forecasting approach based on neutrosophic time series (NTS). Historical data can be transformed into neutrosophic time series data to determine their truth, indeterminacy and falsity functions. The basis for the neutrosophication process is the score and accuracy functions of historical data. In addition, neutrosophic logical relationship groups (NLRGs) are determined and a deneutrosophication method for NTS is presented. The objective of this research is to suggest an idea of first-and high-order NTS. By comparing our approach with other approaches, we conclude that the suggested approach of forecasting gets better results compared to the other existing approaches of fuzzy, intuitionistic fuzzy, and neutrosophic time series. Full article
(This article belongs to the 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)
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16 pages, 295 KiB  
Article
A Novel Approach to Neutrosophic Soft Rough Set under Uncertainty
by Ashraf Al-Quran, Nasruddin Hassan and Emad Marei
Symmetry 2019, 11(3), 384; https://doi.org/10.3390/sym11030384 - 15 Mar 2019
Cited by 20 | Viewed by 2445
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
(This article belongs to the 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)
19 pages, 320 KiB  
Article
Fuzzy Parameterized Complex Neutrosophic Soft Expert Set for Decision under Uncertainty
by Ashraf Al-Quran, Nasruddin Hassan and Shawkat Alkhazaleh
Symmetry 2019, 11(3), 382; https://doi.org/10.3390/sym11030382 - 15 Mar 2019
Cited by 17 | Viewed by 2315
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
(This article belongs to the 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)
23 pages, 316 KiB  
Article
Logarithmic Hybrid Aggregation Operators Based on Single Valued Neutrosophic Sets and Their Applications in Decision Support Systems
by Shahzaib Ashraf, Saleem Abdullah, Florentin Smarandache and Noor ul Amin
Symmetry 2019, 11(3), 364; https://doi.org/10.3390/sym11030364 - 11 Mar 2019
Cited by 48 | Viewed by 2897
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
(This article belongs to the 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)
7 pages, 639 KiB  
Article
Application of Neutrosophic Logic to Evaluate Correlation between Prostate Cancer Mortality and Dietary Fat Assumption
by Muhammad Aslam and Mohammed Albassam
Symmetry 2019, 11(3), 330; https://doi.org/10.3390/sym11030330 - 06 Mar 2019
Cited by 39 | Viewed by 2780
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
(This article belongs to the 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)
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16 pages, 256 KiB  
Article
Generalized Neutrosophic Extended Triplet Group
by Yingcang Ma, Xiaohong Zhang, Xiaofei Yang and Xin Zhou
Symmetry 2019, 11(3), 327; https://doi.org/10.3390/sym11030327 - 05 Mar 2019
Cited by 29 | Viewed by 2402
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
(This article belongs to the 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)
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26 pages, 404 KiB  
Article
A Generalized Approach towards Soft Expert Sets via Neutrosophic Cubic Sets with Applications in Games
by Muhammad Gulistan and Nasruddin Hassan
Symmetry 2019, 11(2), 289; https://doi.org/10.3390/sym11020289 - 22 Feb 2019
Cited by 9 | Viewed by 2780
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
(This article belongs to the 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)
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10 pages, 480 KiB  
Article
Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM
by Lilian Shi and Yue Yuan
Symmetry 2019, 11(2), 278; https://doi.org/10.3390/sym11020278 - 21 Feb 2019
Cited by 6 | Viewed by 2607
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
(This article belongs to the 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)
11 pages, 256 KiB  
Article
A Single-Valued Neutrosophic Linguistic Combined Weighted Distance Measure and Its Application in Multiple-Attribute Group Decision-Making
by Chengdong Cao, Shouzhen Zeng and Dandan Luo
Symmetry 2019, 11(2), 275; https://doi.org/10.3390/sym11020275 - 21 Feb 2019
Cited by 13 | Viewed by 2597
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
(This article belongs to the 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)
13 pages, 257 KiB  
Article
Multi-Attribute Decision Making Method Based on Aggregated Neutrosophic Set
by Wen Jiang, Zihan Zhang and Xinyang Deng
Symmetry 2019, 11(2), 267; https://doi.org/10.3390/sym11020267 - 20 Feb 2019
Cited by 9 | Viewed by 2834
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
(This article belongs to the 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)
24 pages, 340 KiB  
Article
Neutrosophic Cubic Einstein Geometric Aggregation Operators with Application to Multi-Criteria Decision Making Method
by Majid Khan, Muhammad Gulistan, Naveed Yaqoob, Madad Khan and Florentin Smarandache
Symmetry 2019, 11(2), 247; https://doi.org/10.3390/sym11020247 - 16 Feb 2019
Cited by 23 | Viewed by 2705
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
(This article belongs to the 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)
8 pages, 259 KiB  
Article
Inspection Strategy under Indeterminacy Based on Neutrosophic Coefficient of Variation
by Muhammad Aslam and Mansour Sattam Aldosari
Symmetry 2019, 11(2), 193; https://doi.org/10.3390/sym11020193 - 09 Feb 2019
Cited by 5 | Viewed by 2418
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
(This article belongs to the 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)
29 pages, 4513 KiB  
Article
Group Decision Making Based on Triangular Neutrosophic Cubic Fuzzy Einstein Hybrid Weighted Averaging Operators
by Aliya Fahmi, Fazli Amin, Madad Khan and Florentin Smarandache
Symmetry 2019, 11(2), 180; https://doi.org/10.3390/sym11020180 - 02 Feb 2019
Cited by 23 | Viewed by 2801
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
(This article belongs to the 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)
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9 pages, 1108 KiB  
Article
Neutrosophic Compound Orthogonal Neural Network and Its Applications in Neutrosophic Function Approximation
by Jun Ye and Wenhua Cui
Symmetry 2019, 11(2), 147; https://doi.org/10.3390/sym11020147 - 29 Jan 2019
Cited by 6 | Viewed by 2748
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
(This article belongs to the 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)
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18 pages, 293 KiB  
Article
An Approach toward a Q-Neutrosophic Soft Set and Its Application in Decision Making
by Majdoleen Abu Qamar and Nasruddin Hassan
Symmetry 2019, 11(2), 139; https://doi.org/10.3390/sym11020139 - 27 Jan 2019
Cited by 41 | Viewed by 3052
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
(This article belongs to the 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)
7 pages, 234 KiB  
Article
A Variable Acceptance Sampling Plan under Neutrosophic Statistical Interval Method
by Muhammad Aslam
Symmetry 2019, 11(1), 114; https://doi.org/10.3390/sym11010114 - 19 Jan 2019
Cited by 29 | Viewed by 3200
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
(This article belongs to the 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)
19 pages, 826 KiB  
Article
A Robust Single-Valued Neutrosophic Soft Aggregation Operators in Multi-Criteria Decision Making
by Chiranjibe Jana and Madhumangal Pal
Symmetry 2019, 11(1), 110; https://doi.org/10.3390/sym11010110 - 18 Jan 2019
Cited by 77 | Viewed by 3454
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
(This article belongs to the 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)
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