Finding the Shortest Path With Neutrosophic Theory: A Tool for Network Optimization

Finding the Shortest Path With Neutrosophic Theory: A Tool for Network Optimization

Said Broumi (Faculty of Science Ben M'Sik, University Hassan II, Morocco), Shio Gai Quek (UCSI College, Kuala Lumpur, Malaysia), Ganeshsree Selvachandran (Faculty of Business and Information Science, UCSI University, Malaysia), Florentin Smarandache (University of New Mexico, USA), Assia Bakali (Ecole Royale Navale, Morocco) and Mohamed Talea (Faculty of Science Ben M'Sik, University Hassan II, Morocco)
Copyright: © 2020 |Pages: 32
DOI: 10.4018/978-1-7998-1313-2.ch001

Abstract

In this chapter, the authors study a kind of network where the edge weights are characterized by single-valued triangular neutrosophic numbers. First, rigorous definitions of nodes, edges, paths, and cycles of such a network were proposed, which are then defined in algebraic terms. Then, characterization on the length of paths in such a network were presented. This is followed by the presentation of an algorithm for finding the shortest path length between two given nodes on the network. The proposed algorithm gives the shortest path length from source node to destination node based on a ranking method that takes both the length of edges and the number of nodes into account. Finally, a numerical example based on a real-life scenario is also presented to illustrate the efficiency and usefulness of the proposed approach.
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1. Introduction

In 1995, the concept of the neutrosophic sets and neutrosophic logic were introduced by (Smarandache, 2005;2006) in order to efficiently handle the indeterminate and inconsistent information which exists in the real world. Unlike fuzzy sets which associate to each member of a fuzzy set a degree of membership T and intuitionistic fuzzy sets which associate a degree of membership T and a degree of non-membership F, where T,F∈[0,1], NSs are characterized by a truth-membership function T(x), indeterminacy-membership function I(x) and a falsity- membership function F(x), each of which belongs to the non-standard unit interval of ]0, 1+[. Although, IFSs do have the ability to consider some indeterminacy or hesitation margin denoted by π, and can be computed via π=1‑T‑F. NSs have the ability of handling uncertainty in a better way since in the case of NSs, the indeterminacy is handled independently from the truth and falsity aspects of the information. NSs are in fact a generalization of the theory of fuzzy sets (Zadeh,1965), intuitionistic fuzzy sets (Atanassov,1986), interval-valued fuzzy sets(Turksen) and interval-valued intuitionistic fuzzy sets(Atanassov and Gargov,1989). However, neutrosophic theory is difficult to be directly applied in real life problems in the areas of engineering, science and technology.To easily use it in science and engineering areas, (Wang et al.,2010) proposed the concept of a single-valued neutrosophic set. The SVNS model is also characterized by the truth, indeterminacy and falsity membership functions, but all of these membership functions lie in [0, 1], instead of the non-standard unit interval of ]0, 1+[. Presently research on NS theory and its applications are progressing very rapidly, and some of the prominent works can be found in (http://fs.gallup.unm.edu/NSS., Liu et al. ;2016 ;2016a ; 2016b ; 2016c ; 2017 ; 2017, ŞAHİN and Liu ;2016).Recently, (Subas et al., 2015) presented the concept of triangular and trapezoidal neutrosophic numbers and applied it to multiple-attribute decision making (MADM) problems. (Biswas et al., 2014) then presented a special case of trapezoidal neutrosophic numbers which included triangular fuzzy numbers in neutrosophic sets and applied it to MADM problems by introducing a cosine similarity measure. (Deli and Subas, 2016) proposed the concept of single-valued triangular neutrosophic numbers (SVTrNNs) as a generalization of the intuitionistic triangular fuzzy numbers, and subsequently proposed a methodology for solving MADM problems using SVTrNNs.

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