Single-valued Neutrosophic Numbers Research Articles

Generalised single valued neutrosophic number and its application to neutrosophic linear programming

Generalised single valued neutrosophic number and its application to neutrosophic linear programming

SINGLE VALUED (2N+1) SIDED POLYGONAL NEUTROSOPHIC NUMBERS AND SINGLE VALUED (2N) SIDED POLYGONAL NEUTROSOPHIC NUMBERS

This paper introduces a single valued (2n as well as 2n+1) sided polygonal neutrosophic numbers in continuation with other defined single valued neutrosophic numbers. The paper provides basic algebra like addition, subtraction and multiplication of a single valued (2n as well as 2n+1) sided polygonal neutrosophic numbers with examples. In addition, the paper introduces matrix for single valued (2n as well as 2n+1) sided polygonal neutrosophic matrix and its properties

Towards granular calculus of single-valued neutrosophic functions under granular computing

Neutrosophic theory studies objects whose values vary in the sets of elements and are not true or false, but in between, that can be called by neutral, indeterminate, unclear, vague, ambiguous, incomplete or contradictory quantities. In this paper, we firstly introduce preliminaries on granular calculus and analysis related to single-valued neutrosophic functions. Based on horizontal membership functions approach, we establish some basic arithmetic operations of single-valued neutrosophic numbers, that red allow us to directly introduce the terms of neutrosophic function in usual mathematical formulas. Additionally, we build metrics on the space of single-valued neutrosophic numbers induced from Hamming distance. Then, we define some backgrounds on the limit, derivative and integral of single-valued neutrosophic functions. Finally, in order to demonstrate the usable of our theoretical results, we present some applications to well-known problems arising in engineering such as logistic model, the inverted pendulum system, Mass - Spring - Damper model.

An Integrated Decision-Making Method Based on Neutrosophic Numbers for Investigating Factors of Coastal Erosion

The recent boom of various integrated decision-making methods has attracted many researchers to the field. The recent integrated Analytic Network Process and Decision Making Trial and Evaluation Laboratory (ANP–DEMATEL) methods were developed based on crisp numbers and fuzzy numbers. However, these numbers are incapable of dealing with the indeterminant and inconsistent information that exists in real-life problems. This paper proposes improvements to the integrated ANP–DEMATEL method by bringing together the neutrosophic numbers, the ANP method, and the DEMATEL method, which are later abbreviated to NS-DANP. The proposed NS-DANP method can handle the indeterminacy elements in the decision-making environment, as the single-valued neutrosophic numbers are used in the decision analysis. This proposed NS-DANP modification method includes linguistic variables representing the single-valued neutrosophic numbers (SVNNs), and also introduces the single-valued neutrosophic weighted averaging (SVNWA) aggregation operator to aggregate the decision makers’ judgments instead of the typical averaging method. The applicability of the proposed method is illustrated by a case study of the coastal erosion problem along the Peninsular Malaysia coastline, where 12 factors were considered. Three experts of coastal erosion from different organizations were invited to elicit their linguistic judgments on the cause–effect of the coastal erosion. The seven-step decision approach was developed to acquire the weightage of each coastal erosion factor. The outcome of this study reveals that coastal development is the riskiest factor toward coastal erosion. The weight of factors and the cause–effect diagram could be very helpful for government and stakeholders to project a better mitigation plan for the coastal erosion problem. Comparative analysis is also provided to check the feasibility of the proposed method.

Hybrid Weighted Arithmetic and Geometric Aggregation Operator of Neutrosophic Cubic Sets for MADM

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.

(T, S)-Based Single-Valued Neutrosophic Number Equivalence Matrix and Clustering Method

Fuzzy clustering is widely used in business, biology, geography, coding for the internet and more. A single-valued neutrosophic set is a generalized fuzzy set, and its clustering algorithm has attracted more and more attention. An equivalence matrix is a common tool in clustering algorithms. At present, there exist no results constructing a single-valued neutrosophic number equivalence matrix using t-norm and t-conorm. First, the concept of a ( T , S ) -based composition matrix is defined in this paper, where ( T , S ) is a dual pair of triangular modules. Then, a ( T , S ) -based single-valued neutrosophic number equivalence matrix is given. A λ -cutting matrix of single-valued neutrosophic number matrix is also introduced. Moreover, their related properties are studied. Finally, an example and comparison experiment are given to illustrate the effectiveness and superiority of our proposed clustering algorithm.

Novel Open Source Python Neutrosophic Package

Neutrosophic sets has gained wide popularity and acceptance in both academia and industry. Different fields have successfully adopted and utilized neutrosophic sets. However, there is no open-source implementation that provides the basic neutrosophic concepts. Open-source is a global movement that enables developers to share their source-code, as researchers do share their research ideas and results. Presented Open-source python neutrosophic package is the first of its kind. It utilizes object oriented concepts. It is based on one of the most popular multiparadigm programming languages that is widely used in different academic and industry fields and activities; namely python. Presented package tends to dissolve the barriers and enable both researchers and developers to adopt neutrosophic sets and theory in research and applications. In this paper, the first Open-source, Object-Oriented, Python based Neutrosophic package is presented. This paper intensively scanned neutrosophic sets research attempting to reach the most widely accepted neutrosophic proofs, and then transforming them into source-code. Presented package implements most of the basic neutrosophic concepts. Presented neutrosophic package presents four different classes: Single Valued Neutrosophic Number, Single Valued Neutrosophic Sets, Interval Valued Neutrosophic Number, and Interval Valued Neutrosophic Sets. Presented package source code, test cases, usage examples, and updated documentation can be found online at https://www.github.com/helghareeb/neutrosophic. Presented neutrosophic package can be easily integrated into research and applications. This is an ongoing work and research, as neutrosophic theory is largely expanding, and there are lots of features to cover.

Gaussian single-valued neutrosophic numbers and its application in multi-attribute decision making

Gaussian single-valued neutrosophic numbers and its application in multi-attribute decision making

Multiple attribute group decision-making method with single-valued neutrosophic interval number information

ABSTRACTTo describe decision makers’ judgments for an evaluated object more comprehensively in the decision-making process, this paper firstly defines the concept of a single-valued neutrosophic interval number (SVNIN), which is composed of both its interval number (interval judgment) and its single-valued neutrosophic number (SVNN) (confident degree). Then we present the basic operational laws of SVNINs and the score function of SVNIN based on the attitude index and confident level of decision makers to compare SVNINs. Next, we propose a single-valued neutrosophic interval number weighted arithmetic averaging (SVNINWAA) operator and a single-valued neutrosophic interval number weighted geometric averaging (SVNINWGA) operator to aggregate SVNIN information. Further, a multiple attribute group decision-making (MDGDM) method using the proposed SVNINWAA or SVNINWGA operator is established under an SVNIN environment. Finally, an illustrative example regarding the selection problem of global suppliers is presented to show the application and effectiveness of the developed method.

Multiple-attribute decision making based on single-valued neutrosophic Schweizer-Sklar prioritized aggregation operator

Single-valued neutrosophic (SVN) sets can successfully describe the uncertainty problems, and Schweizer-Sklar (SS) t-norm (TN) and t-conorm (TCN) can build the information aggregation process more flexible by a parameter. To fully consider the advantages of SVNS and SS operations, in this article, we extend the SS TN and TCN to single-valued neutrosophic numbers (SVNN) and propose the SS operational laws for SVNNs. Then, we merge the prioritized aggregation (PRA) operator with SS operations, and develop the single-valued neutrosophic Schweizer-Sklar prioritized weighted averaging (SVNSSPRWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted averaging (SVNSSPROWA) operator, single-valued neutrosophic Schweizer-Sklar prioritized weighted geometric (SVNSSPRWG) operator, and single-valued neutrosophic Schweizer-Sklar prioritized ordered weighted geometric (SVNSSPROWG) operator. Moreover, we study some useful characteristics of these proposed aggregation operators (AOs) and propose two decision making models to deal with multiple-attribute decision making (MADM) problems under SVN information based on the SVNSSPRWA and SVNSSPRWG operators. Lastly, an illustrative example about talent introduction is given to testify the effectiveness of the developed methods.

A multicriteria model for the selection of the transport service provider: A single valued neutrosophic DEMATEL multicriteria model

The decision-making process requires, a priori, defining and considering certain factors, especially when it comes to complex areas such as transport management in companies. One of the most important items in the initial phase of the transport process that significantly influences its further flow is decision-making about the choice of the most favorable transport provider. In this paper a model for evaluating and selecting a transport service provider based on a single valued neutrosophic number (SVNN) is presented. The neutrosophic set concept represents a general platform that extends the concepts of classical sets, fuzzy sets, intuitionistic fuzzy sets, and an interval valued intuitionistic fuzzy sets. The application of the SVNN concept made a modification of the DEMATEL method (Decision-making Trial and Evaluation Laboratory Method) and proposed a model for ranking alternative solutions. The SVNN-DEMATEL model defines the mutual effects of the provider's evaluation criteria, while, in the second phase of the model, alternative providers are evaluated and ranked. The SVNN-DEMATEL model was tested on a hypothetical example of evaluation of five providers of transport services.

New logarithmic operational laws and their applications to multiattribute decision making for single-valued neutrosophic numbers

Neutrosophic set, initiated by Smarandache, is a novel tool to deal with vagueness considering the truth, indeterminacy and falsity memberships satisfying the condition that their sum is less than 3. This set can be used to characterize the information more accurately than the intuitionistic fuzzy set. Under this set, the objective of this manuscript is to present some new operational laws called as logarithm operational laws with real number base λ for the single-valued neutrosophic (SVN) numbers. Various desirable properties of the proposed operational laws are contemplated. Further, based on these laws, different weighted averaging and geometric aggregation operators are developed. The properties such as idempotency, monotonicity, boundedness are provided to support the proposed operators. Then, we utilized these operations and operators to present a multiattribute decision making method to solve the decision-making problems. A real numerical example is given to demonstrate the approach under SVN environment. The legitimacy of the proposed strategy is exhibited with a numerical illustration and compared the results with the several existing approaches result.

Operations and Aggregation Methods of Single-Valued Linguistic Neutrosophic Interval Linguistic Numbers and Their Decision Making Method

To comprehensively describe uncertain/interval linguistic arguments and confident linguistic arguments in the decision making process by a linguistic form, this study first presents the concept of a single-valued linguistic neutrosophic interval linguistic number (SVLN-ILN), which is comprehensively composed of its uncertain/interval linguistic number (determinate linguistic argument part) and its single-valued linguistic neutrosophic number (confident linguistic argument part), and its basic operations. Then, the score function of SVLN-ILN based on the attitude index and confident degree/level is presented for ranking SVLN-ILNs. After that, SVLN-ILN weighted arithmetic averaging (SVLN-ILNWAA) and SVLN-ILN weighted geometric averaging (SVLN-ILNWGA) operators are proposed to aggregate SVLN-ILN information and their properties are investigated. Further, a multi-attribute decision-making (MADM) method based on the proposed SVLN-ILNWAA or SVLN-ILNWGA operator and the score function is established under consideration of decision makers’ preference attitudes (pessimist, moderate, and optimist). Lastly, an actual example is given to show the applicability of the established MADM approach with decision makers’ attitudes.

A new algorithm for finding minimum spanning trees with undirected neutrosophic graphs

In this paper, we discuss the minimum spanning tree (MST) problem of an undirected neutrosophic weighted connected graph in which a single-valued neutrosophic number, instead of a real number/fuzzy number, is assigned to each arc as its arc length.We define this type of MST as neutrosophic minimum spanning tree (NMST).

Multiple Attribute Decision-Making Method Using Similarity Measures of Neutrosophic Cubic Sets

In inconsistent and indeterminate settings, as a usual tool, the neutrosophic cubic set (NCS) containing single-valued neutrosophic numbers and interval neutrosophic numbers can be applied in decision-making to present its partial indeterminate and partial determinate information.

Multiple Attribute Decision-Making Method Using Similarity Measures of Neutrosophic Cubic Sets

In inconsistent and indeterminate settings, as a usual tool, the neutrosophic cubic set (NCS) containing single-valued neutrosophic numbers and interval neutrosophic numbers can be applied in decision-making to present its partial indeterminate and partial determinate information. However, a few researchers have studied neutrosophic cubic decision-making problems, where the similarity measure of NCSs is one of the useful measure methods. For this work, we propose the Dice, cotangent, and Jaccard measures between NCSs, and indicate their properties. Then, under an NCS environment, the similarity measures-based decision-making method of multiple attributes is developed. In the decision-making process, all the alternatives are ranked by the similarity measure of each alternative and the ideal solution to obtain the best one. Finally, two practical examples are applied to indicate the feasibility and effectiveness of the developed method.

Analysis of Faculty Teaching using Multi-criteria Decision Making Approach

In order to maintain a good standardof any college, the institution should follow paper based approach or web-based approach to gather student feedback on faculty teaching to maintain the qualities of the faculty. Students have very much concerned about their teachers who have played a vital role in their life in both outside or inside of the institutions which helps them to justify a teacher and it comes in form of their feedback. In this paper, we introduce a ranking method of the given feedback's factors of teacher's performances with the help of multi-criteria decision making approach using neutrosophic logic. We use the score and accuracy functions and the hybrid score-accuracy functions of single valued neutrosophic numbers (SVNNs) and ranking method for SVNNs.

A new algorithm for finding minimum spanning trees with undirected neutrosophic graphs

In this paper, we discuss the minimum spanning tree (MST) problem of an undirected neutrosophic weighted connected graph in which a single-valued neutrosophic number, instead of a real number/fuzzy number, is assigned to each arc as its arc length. We define this type of MST as neutrosophic minimum spanning tree (NMST). We describe the utility of neutrosophic numbers as arc lengths and its application in different real world MST problems. Here, a new algorithm for designing the MST of a neutrosophic graph is introduced. In the proposed algorithm, we incorporate the uncertainty in Kruskal algorithm for designing MST using neutrosophic number as arc length. A score function is used to compare different NMSTs whose weights are computed using the addition operation of neutrosophic numbers. We compare this weight of the NMST with that of an equivalent classical MST with real numbers as arc lengths. Compared with the existing algorithms for NMST, the proposed algorithm is more efficient due to the fact that the addition operation and the ranking of neutrosophic number can be done in straightforward manners. The proposed algorithm is illustrated by numerical examples.

Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making

The neutrosophic cubic set can describe complex decision-making problems with its single-valued neutrosophic numbers and interval neutrosophic numbers simultaneously. The Dombi operations have the advantage of good flexibility with the operational parameter. In order to solve decision-making problems with flexible operational parameter under neutrosophic cubic environments, the paper extends the Dombi operations to neutrosophic cubic sets and proposes a neutrosophic cubic Dombi weighted arithmetic average (NCDWAA) operator and a neutrosophic cubic Dombi weighted geometric average (NCDWGA) operator. Then, we propose a multiple attribute decision-making (MADM) method based on the NCDWAA and NCDWGA operators. Finally, we provide two illustrative examples of MADM to demonstrate the application and effectiveness of the established method.

Linguistic single-valued neutrosophic prioritized aggregation operators and their applications to multiple-attribute group decision-making

In this paper, we introduce some new linguistic prioritized aggregation operators which simultaneously considers the priority among the attributes and the uncertainty in linguistic terms under the linguistic single-valued neutrosophic set (LSVNS). For this, firstly the operational laws for LSVNSs are introduced along with their properties. Based on these operations, we proposes some prioritized weighted and ordered weighted averaging as well as geometric aggregation operators for a collection of linguistic single-valued neutrosophic numbers. Further, the desirable properties of these operators are studied. Finally, a decision-making approach presents and illustrate with a numerical example.