Research

16 pages, 285 KiB

Open AccessArticle

Neutrosophic Quadruple BCK/BCI-Algebras

by Young Bae Jun, Seok-Zun Song, Florentin Smarandache and Hashem Bordbar

Axioms 2018, 7(2), 41; https://doi.org/10.3390/axioms7020041 - 18 Jun 2018

Cited by 29 | Viewed by 2980

The notion of a neutrosophic quadruple

B C K / B C I

-number is considered, and a neutrosophic quadruple

B C K / B C I

-algebra, which consists of neutrosophic quadruple

B C K / B C I

-numbers, is constructed. [...] Read more.

The notion of a neutrosophic quadruple

B C K / B C I

-number is considered, and a neutrosophic quadruple

B C K / B C I

-algebra, which consists of neutrosophic quadruple

B C K / B C I

-numbers, is constructed. Several properties are investigated, and a (positive implicative) ideal in a neutrosophic quadruple

B C K

-algebra and a closed ideal in a neutrosophic quadruple

B C I

-algebra are studied. Given subsets A and B of a

B C K / B C I

-algebra, the set

N Q (A, B)

, which consists of neutrosophic quadruple

B C K / B C I

-numbers with a condition, is established. Conditions for the set

N Q (A, B)

to be a (positive implicative) ideal of a neutrosophic quadruple

B C K

-algebra are provided, and conditions for the set

N Q (A, B)

to be a (closed) ideal of a neutrosophic quadruple

B C I

-algebra are given. An example to show that the set

{\tilde{0}}

is not a positive implicative ideal in a neutrosophic quadruple

B C K

-algebra is provided, and conditions for the set

{\tilde{0}}

to be a positive implicative ideal in a neutrosophic quadruple

B C K

-algebra are then discussed. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

17 pages, 407 KiB

Open AccessArticle

Decision-Making with Bipolar Neutrosophic TOPSIS and Bipolar Neutrosophic ELECTRE-I

by Muhammad Akram, Shumaiza and Florentin Smarandache

Axioms 2018, 7(2), 33; https://doi.org/10.3390/axioms7020033 - 15 May 2018

Cited by 61 | Viewed by 4582

Abstract

Technique for the order of preference by similarity to ideal solution (TOPSIS) and elimination and choice translating reality (ELECTRE) are widely used methods to solve multi-criteria decision making problems. In this research article, we present bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I [...] Read more.

Technique for the order of preference by similarity to ideal solution (TOPSIS) and elimination and choice translating reality (ELECTRE) are widely used methods to solve multi-criteria decision making problems. In this research article, we present bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method to solve such problems. We use the revised closeness degree to rank the alternatives in our bipolar neutrosophic TOPSIS method. We describe bipolar neutrosophic TOPSIS method and bipolar neutrosophic ELECTRE-I method by flow charts. We solve numerical examples by proposed methods. We also give a comparison of these methods. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Figure 1

13 pages, 300 KiB

Open AccessArticle

Interval Neutrosophic Sets with Applications in BCK/BCI-Algebra

by Young Bae Jun, Seon Jeong Kim and Florentin Smarandache

Axioms 2018, 7(2), 23; https://doi.org/10.3390/axioms7020023 - 09 Apr 2018

Cited by 39 | Viewed by 3319

Abstract

For

i, j, k, l, m, n \in {1, 2, 3, 4}

, the notion of [...] Read more.

For

i, j, k, l, m, n \in {1, 2, 3, 4}

, the notion of

(T (i, j), I (k, l), F (m, n))

-interval neutrosophic subalgebra in

B C K / B C I

-algebra is introduced, and their properties and relations are investigated. The notion of interval neutrosophic length of an interval neutrosophic set is also introduced, and related properties are investigated. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

25 pages, 718 KiB

Open AccessArticle

Cross Entropy Measures of Bipolar and Interval Bipolar Neutrosophic Sets and Their Application for Multi-Attribute Decision-Making

by Surapati Pramanik, Partha Pratim Dey, Florentin Smarandache and Jun Ye

Axioms 2018, 7(2), 21; https://doi.org/10.3390/axioms7020021 - 24 Mar 2018

Cited by 16 | Viewed by 4336

Abstract

The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and [...] Read more.

The bipolar neutrosophic set is an important extension of the bipolar fuzzy set. The bipolar neutrosophic set is a hybridization of the bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic set consists of three independent positive membership functions and three independent negative membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic sets and prove their basic properties. We also define cross entropy measures of interval bipolar neutrosophic sets and prove their basic properties. Thereafter, we develop two novel multi-attribute decision-making strategies based on the proposed cross entropy measures. In the decision-making framework, we calculate the weighted cross entropy measures between each alternative and the ideal alternative to rank the alternatives and choose the best one. We solve two illustrative examples of multi-attribute decision-making problems and compare the obtained result with the results of other existing strategies to show the applicability and effectiveness of the developed strategies. At the end, the main conclusion and future scope of research are summarized. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Graphical abstract

24 pages, 387 KiB

Open AccessArticle

Multi-Attribute Decision-Making Method Based on Neutrosophic Soft Rough Information

by Muhammad Akram, Sundas Shahzadi and Florentin Smarandache

Axioms 2018, 7(1), 19; https://doi.org/10.3390/axioms7010019 - 20 Mar 2018

Cited by 17 | Viewed by 3850

Abstract

Soft sets (SSs), neutrosophic sets (NSs), and rough sets (RSs) are different mathematical models for handling uncertainties, but they are mutually related. In this research paper, we introduce the notions of soft rough neutrosophic sets (SRNSs) and neutrosophic soft rough sets (NSRSs) as [...] Read more.

Soft sets (SSs), neutrosophic sets (NSs), and rough sets (RSs) are different mathematical models for handling uncertainties, but they are mutually related. In this research paper, we introduce the notions of soft rough neutrosophic sets (SRNSs) and neutrosophic soft rough sets (NSRSs) as hybrid models for soft computing. We describe a mathematical approach to handle decision-making problems in view of NSRSs. We also present an efficient algorithm of our proposed hybrid model to solve decision-making problems. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Figure 1

27 pages, 603 KiB

Open AccessArticle

Neutrosophic Soft Rough Graphs with Application

by Muhammad Akram, Hafsa M. Malik, Sundas Shahzadi and Florentin Smarandache

Axioms 2018, 7(1), 14; https://doi.org/10.3390/axioms7010014 - 26 Feb 2018

Cited by 17 | Viewed by 4022

Abstract

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The [...] Read more.

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Figure 1

9 pages, 246 KiB

Open AccessArticle

Neutrosophic Number Nonlinear Programming Problems and Their General Solution Methods under Neutrosophic Number Environments

by Jun Ye, Wenhua Cui and Zhikang Lu

Axioms 2018, 7(1), 13; https://doi.org/10.3390/axioms7010013 - 24 Feb 2018

Cited by 21 | Viewed by 4151

Abstract

In practical situations, we often have to handle programming problems involving indeterminate information. Building on the concepts of indeterminacy I and neutrosophic number (NN) (z = p + qI for p, q ∈ ℝ), this paper introduces some basic operations [...] Read more.

In practical situations, we often have to handle programming problems involving indeterminate information. Building on the concepts of indeterminacy I and neutrosophic number (NN) (z = p + qI for p, q ∈ ℝ), this paper introduces some basic operations of NNs and concepts of NN nonlinear functions and inequalities. These functions and/or inequalities contain indeterminacy I and naturally lead to a formulation of NN nonlinear programming (NN-NP). These techniques include NN nonlinear optimization models for unconstrained and constrained problems and their general solution methods. Additionally, numerical examples are provided to show the effectiveness of the proposed NN-NP methods. It is obvious that the NN-NP problems usually yield NN optimal solutions, but not always. The possible optimal ranges of the decision variables and NN objective function are indicated when the indeterminacy I is considered for possible interval ranges in real situations. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Graphical abstract

16 pages, 1115 KiB

Open AccessArticle

NN-Harmonic Mean Aggregation Operators-Based MCGDM Strategy in a Neutrosophic Number Environment

by Kalyan Mondal, Surapati Pramanik, Bibhas C. Giri and Florentin Smarandache

Axioms 2018, 7(1), 12; https://doi.org/10.3390/axioms7010012 - 23 Feb 2018

Cited by 22 | Viewed by 4595

Abstract

A neutrosophic number (a + bI) is a significant mathematical tool to deal with indeterminate and incomplete information which exists generally in real-world problems, where a and bI denote the determinate component and indeterminate component, respectively. We define score functions and [...] Read more.

A neutrosophic number (a + bI) is a significant mathematical tool to deal with indeterminate and incomplete information which exists generally in real-world problems, where a and bI denote the determinate component and indeterminate component, respectively. We define score functions and accuracy functions for ranking neutrosophic numbers. We then define a cosine function to determine the unknown weight of the criteria. We define the neutrosophic number harmonic mean operators and prove their basic properties. Then, we develop two novel multi-criteria group decision-making (MCGDM) strategies using the proposed aggregation operators. We solve a numerical example to demonstrate the feasibility, applicability, and effectiveness of the two proposed strategies. Sensitivity analysis with the variation of “I” on neutrosophic numbers is performed to demonstrate how the preference ranking order of alternatives is sensitive to the change of “I”. The efficiency of the developed strategies is ascertained by comparing the results obtained from the proposed strategies with the results obtained from the existing strategies in the literature. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Graphical abstract

20 pages, 2347 KiB

Open AccessArticle

Rough Neutrosophic Digraphs with Application

by Sidra Sayed, Nabeela Ishfaq, Muhammad Akram and Florentin Smarandache

Axioms 2018, 7(1), 5; https://doi.org/10.3390/axioms7010005 - 18 Jan 2018

Cited by 9 | Viewed by 4086

Abstract

A rough neutrosophic set model is a hybrid model which deals with vagueness by using the lower and upper approximation spaces. In this research paper, we apply the concept of rough neutrosophic sets to graphs. We introduce rough neutrosophic digraphs and describe methods [...] Read more.

A rough neutrosophic set model is a hybrid model which deals with vagueness by using the lower and upper approximation spaces. In this research paper, we apply the concept of rough neutrosophic sets to graphs. We introduce rough neutrosophic digraphs and describe methods of their construction. Moreover, we present the concept of self complementary rough neutrosophic digraphs. Finally, we consider an application of rough neutrosophic digraphs in decision-making. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Figure 1

13 pages, 263 KiB

Open AccessArticle

Neutrosophic Positive Implicative N -Ideals in BCK-Algebras

by Young Bae Jun, Florentin Smarandache, Seok-Zun Song and Madad Khan

Axioms 2018, 7(1), 3; https://doi.org/10.3390/axioms7010003 - 15 Jan 2018

Cited by 39 | Viewed by 4336

Abstract

The notion of a neutrosophic positive implicative

N

-ideal in

B C K

-algebras is introduced, and several properties are investigated. Relations between a neutrosophic

N

-ideal and a neutrosophic positive implicative

N

-ideal are discussed. Characterizations of a neutrosophic positive implicative [...] Read more.

The notion of a neutrosophic positive implicative

N

-ideal in

B C K

-algebras is introduced, and several properties are investigated. Relations between a neutrosophic

N

-ideal and a neutrosophic positive implicative

N

-ideal are discussed. Characterizations of a neutrosophic positive implicative

N

-ideal are considered. Conditions for a neutrosophic

N

-ideal to be a neutrosophic positive implicative

N

-ideal are provided. An extension property of a neutrosophic positive implicative

N

-ideal based on the negative indeterminacy membership function is discussed. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

3630 KiB

Open AccessArticle

Neutrosophic Hough Transform

by Ümit Budak, Yanhui Guo, Abdulkadir Şengür and Florentin Smarandache

Axioms 2017, 6(4), 35; https://doi.org/10.3390/axioms6040035 - 18 Dec 2017

Cited by 3 | Viewed by 5332

Abstract

Hough transform (HT) is a useful tool for both pattern recognition and image processing communities. In the view of pattern recognition, it can extract unique features for description of various shapes, such as lines, circles, ellipses, and etc. In the view of image [...] Read more.

Hough transform (HT) is a useful tool for both pattern recognition and image processing communities. In the view of pattern recognition, it can extract unique features for description of various shapes, such as lines, circles, ellipses, and etc. In the view of image processing, a dozen of applications can be handled with HT, such as lane detection for autonomous cars, blood cell detection in microscope images, and so on. As HT is a straight forward shape detector in a given image, its shape detection ability is low in noisy images. To alleviate its weakness on noisy images and improve its shape detection performance, in this paper, we proposed neutrosophic Hough transform (NHT). As it was proved earlier, neutrosophy theory based image processing applications were successful in noisy environments. To this end, the Hough space is initially transferred into the NS domain by calculating the NS membership triples (T, I, and F). An indeterminacy filtering is constructed where the neighborhood information is used in order to remove the indeterminacy in the spatial neighborhood of neutrosophic Hough space. The potential peaks are detected based on thresholding on the neutrosophic Hough space, and these peak locations are then used to detect the lines in the image domain. Extensive experiments on noisy and noise-free images are performed in order to show the efficiency of the proposed NHT algorithm. We also compared our proposed NHT with traditional HT and fuzzy HT methods on variety of images. The obtained results showed the efficiency of the proposed NHT on noisy images. Full article

(This article belongs to the Special Issue Neutrosophic Multi-Criteria Decision Making)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Neutrosophic Multi-Criteria Decision Making

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Published Papers (11 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI