Shortest Path Problem under Bipolar Neutrosphic Setting

This main purpose of this paper is to develop an algorithm to find the shortest path on a network in which the weights of the edges are represented by bipolar neutrosophic numbers. Finally, a numerical example has been provided for illustrating the proposed approach. Introduction Smarandache [1, 2] introduced neutrosophic set and neutrosophic logic by considering the nonstandard analysis. The concept of neutrosophic sets generalized the concepts of fuzzy sets [3] and intuitionistic fuzzy set [4] by adding an independent indeterminacy-membership. Neutrosophic set is a powerful tool to deal with incomplete, indeterminate and inconsistent information in real world, which have attracted the widespread concerns for researchers. The concept of neutrosophic set is characterized by three independent degrees namely truth-membership degree (T), indeterminacymembership degree (I), and falsity-membership degree (F). From scientific or engineering point of view, the neutrosophic set and settheoretic operator will be difficult to apply in the real application. The subclass of the neutrosophic sets called single-valued neutrosophic sets [5] (SVNS for short) was studied deeply by many researchers. The concept of single valued neutrosophic theory has proven to be useful in many different field such as the decision making problem, medical diagnosis and so on. Additional literature on neutrosophic sets can be found in [6]. Recently, Deli et al. [7] introduced the concept of bipolar neutrosophic sets which is an extension of the fuzzy sets, bipolar fuzzy sets, intuitionistic fuzzy sets and neutrosophic sets. The bipolar neutrosophic set (BNS) is an important concept to handle uncertain and vague information exciting in real life, which consists of three membership functions including bipolarity. Also, they give some operations including the score, certainty and accuracy functions to compare the bipolar neutrosophic sets and operators on the bipolar neutrosophic sets. The shortest path problem is a fundamental algorithmic problem, in which a minimum weight path is computed between two nodes of a weighted, directed graph. The shortest path problem has been widely studied in the fields of operations research, computer science, and transportation engineering. In literature, there are many publications which deal with shortest path problems [8-13] that have been studied with different types of input data, including fuzzy set, intuitionstic fuzzy sets, trapezoidal intuitionistic fuzzy sets and vague set. Recently, Broumi et al. [14-17] presented the concept of neutrosophic graphs, interval valued neutrosophic graphs and bipolar single valued neutrosophic graphs. Smarandache [18-19] proposed another variant of neutrosophic graphs based on literal indeterminacy component (I). Also Kandasamy et al. [20] studied the concept of neutrosophic graphs, To do best of our knowledge, few research papers deal with shortest path in neutrosophic environment. Broumi et al. [21] proposed an algorithm for solving neutrosophic shortest path problem based on score function. The same authors in [22] proposed a study of neutrosphic shortest path with interval valued Applied Mechanics and Materials Submitted: 2016-07-25 ISSN: 1662-7482, Vol. 859, pp 59-66 Revised: 2016-08-13 doi:10.4028/www.scientific.net/AMM.859.59 Accepted: 2016-09-29 © 2017 Trans Tech Publications, Switzerland Online: 2016-12-01 All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (#71046158-04/11/16,08:42:32) neutrosophic number on a network. Till now, there is no study in the literature for computing shortest path problem in bipolar neutrosophic environment. The structure of the paper is as follows. In Section 2, we review some basic concepts about neutrosophic sets, single valued neutrosophic sets and bipolar neutrosophic sets. In section 3, we give the network terminology. In Section 4, an algorithm is proposed for finding the shortest path and shortest distance in bipolar neutrosophic graph. In section 5 an illustrative example is provided to find the shortest path and shortest distance between the source node and destination node. Finally, in Section 6 we provide conclusion and proposal for further research. Preliminaries In this section, some basic concepts and definitions on neutrosophic sets, single valued neutrosophic sets and bipolar neutrosophic sets are reviewed from the literature. Definition 2.1 [1-2]. Let X be a space of points (objects) with generic elements in X denoted by x; then the neutrosophic set A (NS A) is an object having the form A = {< x: ( ) A T x , ( ) A I x , ( ) A F x >, x ∈ X}, where the functions T, I, F: X→]0,1[define respectively the truth-membership function, an indeterminacy-membership function, and a falsity-membership function of the element x ∈ X to the set A with the condition: − 0 ≤ ( ) A T x + ( ) A I x + ( ) A F x ≤ 3 + . (1) The functions ( ) A T x , ( ) A I x and ( ) A F x are real standard or nonstandard subsets of ] − 0,1 + [. Since it is difficult to apply NSs to practical problems, Wang et al. [14] introduced the concept of a SVNS, which is an instance of a NS and can be used in real scientific and engineering applications. Definition 2.2 [3]. Let X be a space of points (objects) with generic elements in X denoted by x. A single valued neutrosophic set A (SVNS A) is characterized by truth-membership function ( ) A T x , an indeterminacy-membership function ( ) A I x , and a falsity-membership function ( ) A F x . For each point x in X ( ) A T x , ( ) A I x , ( ) A F x ∈ [0, 1]. A SVNS A can be written as A = {< x: ( ) A T x , ( ) A I x , ( ) A F x >, x ∈X} (2) Deli et al. [15] proposed the concept of bipolar neutrosophic set, which is an instance of a neutrosophic set, and introduced the definition of an BNS. Definition 2.3 [4]. A bipolar neutrosophic set A in X is defined as an object of the form A={: x ∈ X}, where p T , p I , p F :X→ [1, 0] and n T , n I , n F : X → [-1, 0] .The positive membership degree ( ) p T x , ( ) p I x , ( ) p F x denotes the truth membership, indeterminate membership and false membership of an element ∈ X corresponding to a bipolar neutrosophic set A and the negative membership degree ( ) n T x , ( ) n I x , ( ) n F x denotes the truth membership, indeterminate membership and false membership of an element ∈ X to some implicit counter-property corresponding to a bipolar neutrosophic set A. Definition 2.4 [4]. An empty bipolar neutrosophic set 1 1 1 1 1 1 1 , I , F , , I , F p p p n n n A T T =< > is defined as 1 1 1 0, I 0, F 1 p p p T = = = and 1 1 1 1, I 0, F 0 n n n T = − = = (4) 60 Advanced Research in Area of Materials, Aerospace, Robotics and Modern Manufacturing Systems