Entropy based Single Valued Neutrosophic Digraph and its applications

119

Neutrosophic Sets and Systems, Vol. 19, 2018

University of New Mexico

Entropy based Single Valued Neutrosophic Digraph and its applications Kalyan Sinha1 , Pinaki Majumdar 2 1 Department 2 Department

of Mathematics, PSTDS Vidyapith, Chinsurah, India 712305. E-mail: kalyansinha90@gmail.com

of Mathematics, MUC Womens’ College, Burdwan, India 713104. E-mail: pmajumdar2@rediffmail.com

shown and some of the important properties of SVN digraphs are investigated. Finally SVN digraphs are applied in solving a multicriterion decision making problems.

Abstract: This paper introduces the single valued neutrosophic (i.e.

SVN) digraph. The basic terminologies and operations of SVN digraphs have been defined. Later certain types of SVN digraphs are Keywords: SVN set, SVN digraph, Entropy, Similarity, Decision making.

1

Introduction

In 1995, neutrosophic logic and set theory was introduced by Smarandache [22, 23]. The neutrosophic sets are characterized by a truth membership function(t), a falsity membership function (f) and an indeterminacy membership function(i) respectively, which lies between the nonstandard unit interval [0, 1]∗ .

Unlike intuitionistic fuzzy sets, here the uncertainties present i.e. the indeterminacy factor, is independent of truth and falsity values. Hence Neutrosophic sets are more general than intuitionistic fuzzy set [6] and draw a special attraction to the researchers.

Later on Wang et al. [25] introduced a special type of neutrosophic set say single valued neutrosophic set (SVNS). They also introduced the interval valued neutrosophic set (IVNS) in [26].

The SVN set is a generalization of classical set, fuzzy set [27], intuitionistic fuzzy set [6] etc. To see the practical application of the neutrosophic sets and SVN sets, one may see [1, 2, 3, 4, 7, 8] etc.

On the other hand, nowadays graphs and digraphs are widely used by the researchers to solve many pratical problems. The graphs are used as a tool for solving combinatorial problems in algebra, analysis, geometry etc. Many works on fuzzy graph theory, fuzzy digraph theory, intuitionistic fuzzy graphs, soft digraphs etc. are carried out by a number of researchers [12, 13, 15, 16, 17, 21]. Four main categories of neutrosophic graphs have been defined by Samarandache in the paper [24].

However the concept of single valued neutrosophic graphs was introduced by Broumi et al. [9, 10, 11].

In this paper we have introduced the notion of SVN digraphs for the first time. In section 2, some preliminaries regarding neutrosophic sets, graph theory, SVN sets etc. are discussed. In section 3, we have defined the SVN digraph and some terminologies regarding SVN digraphs with examples. We have solved a real life problem by using SVN digraph in Section 4. In Section 5, we have defined the volume of a SVN digraph and also the similarity measure between two SVN digraphs by using the volume of each SVN digraph. Finally in this section, we have computed

the similarity measure of the digraphs of Section 4 and compared the results. Section 6 concludes the paper.

2

Preliminaries

In this section, we will discuss some definitions and terminologies regarding neutrosophic sets which will be used in the rest of the paper. However, for details on the neutrosophic sets, one can see [20].

Definition 1 [20] Let X be a universal set. A neutrosophic set A on X is characterized by a truth membership function tA , an indeterminacy membership function iA and a falsity membership function fA , where tA , iA , fA : X → [0, 1], are functions and ∀ x ∈ X, x = x(tA (x), iA (x), fA (x)) ∈ A is a single valued neutrosophic element of A. A single valued neutrosophic set (SVNS) A over a finite universe X = {x1 , x2 , . . . , xn } is represented as below: A=

n X i=1

xi htA (xi ), iA (xi ), fA (xi )i

Definition 2 [20] The complement of a SVNS A is denoted by Ac and is defined by tAc (x) = fA (x), iAc (x) = 1−iA (x), fAc (x) = tA (x) ∀ x ∈ X. Definition 3 [20] A SVNS A is contained in the other SVNS B, denoted as A ⊆ B, if and only if tA (x) ≤ tB (x), iA (x) ≤ iB (x) and fA (x) ≥ fB (x) ∀x ∈ X. Two sets will be equal if A ⊆ B and B ⊆ A. Definition 4 [20] Suppose N (X) be the collection of all SVN sets on X and A, B ∈ N (X). A similarity measure between two SVN sets A and B is a function S : N (X) × N (X) → [0, 1] which satisfies the following condition: (i) 0 ≤ S(A, B) ≤ 1, (ii) S(A, B) = 1 if and only if A = B.

K. Sinha, P. Majumdar: Entropy based Single Valued Neutrosophic Digraph and its applications

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Neutrosophic Sets and Systems, Vol. 19, 2018

[25] Abdel-Basset, M., Mohamed, M., Smarandache, F., & Chang, V. (2018). Neutrosophic Association Rule Mining Algorithm for Big Data Analysis.

Symmetry, 10(4), 106.

[26] Abdel-Basset, M., & Mohamed, M. (2018). The Role of Single Valued Neutrosophic Sets and Rough Sets in Smart City: Imperfect and Incomplete Information Systems. Measurement. Volume 124, August 2018, Pages 47-55 [27] Abdel-Basset, M., Gunasekaran, M., Mohamed, M., & Smarandache, F. A novel method for solving the fully neutrosophic linear programming problems. Neural Computing and Applications, 111.

[28] Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2018). A hybrid approach of

neutrosophic sets and DEMATEL method for developing supplier selection criteria. Design Automation for Embedded Systems, 1-22.

[29] Abdel-Basset, M., Mohamed, M., & Chang, V. (2018). NMCDA: A framework for evaluating cloud computing services. Future Generation Computer Systems, 86, 12-29.

[30] Abdel-Basset, M., Mohamed, M., Zhou, Y., & Hezam, I. (2017). Multi-criteria group decision making based on neutrosophic analytic hierarchy process. Journal of Intelligent & Fuzzy Systems, 33(6), 4055-4066.

[31] Abdel-Basset, M.; Mohamed, M.; Smarandache, F. An Extension of Neutrosophic AHPâ&#x20AC;&#x201C;SWOT Analysis for Strategic Planning and DecisionMaking. Symmetry 2018, 10, 116.

Received : January 21, 2018. Accepted : March 28, 2018.

Kalyan Sinha, Pinaki Majumdar: Entropy based Single Valued Neutrosophic Digraph and its applications