A Study of Neutrosophic Shortest Path Problem

A Study of Neutrosophic Shortest Path Problem

Ranjan Kumar (Jain Deemed to be University, Jayanagar, Bengaluru, India), Arindam Dey (Saroj Mohan Institute of Technology, India), Said Broumi (Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Casablanca, Morocco & Regional Center for the Professions of Education and Training (CRMEF), Casablanca-Settat, Morocco) and Florentin Smarandache (University of New Mexico, USA)
Copyright: © 2020 |Pages: 32
DOI: 10.4018/978-1-7998-1313-2.ch006
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Abstract

Shortest path problem (SPP) is an important and well-known combinatorial optimization problem in graph theory. Uncertainty exists almost in every real-life application of SPP. The neutrosophic set is one of the popular tools to represent and handle uncertainty in information due to imprecise, incomplete, inconsistent, and indeterminate circumstances. This chapter introduces a mathematical model of SPP in neutrosophic environment. This problem is called as neutrosophic shortest path problem (NSPP). The utility of neutrosophic set as arc lengths and its real-life applications are described in this chapter. Further, the chapter also includes the different operators to handle multi-criteria decision-making problem. This chapter describes three different approaches for solving the neutrosophic shortest path problem. Finally, the numerical examples are illustrated to understand the above discussed algorithms.
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1. Introduction

Neutrosophic sets (NS) theory is proposed by Smarandache [ (1999); (2005)], and this is generalised from the intuitionistic fuzzy set (Atanassov K. T., 1986) and fuzzy set (Zadeh, 1965). NS deals with indeterminate, uncertain and incongruous data where the indeterminacy is calculated explicitly. Moreover, indeterminacy, falsity and truth membership are fully independent. It overcomes some restrictions of the current methods in portraying uncertain decision. Samarandche’s contribution towards neutrosophic theory is summarized in Table 1.

Table 1.
Significance influences of neutrosophic set in different area of mathematics
Author and YearContribution in Different Area of Mathematics
Kandasamy &
Smarandache (2003)		Introduced neutrosophic number.
Wang et al. (2005) Introduced of interval neutrosophic set/logic.
Smarandache (2005) Introduced the neutrosophic tri-polar set and neutrosophic multipolar set.
Smarandache (2009) Introduced of n-norm and n-conorm.
Smarandache (2013a) Development of neutrosophic measure and neutrosophic probability.
Smarandache (2013b) Extended or refined the neutrosophic components.
Smarandache (2014a) Introduced the law of included multiple middle.
Smarandache (2014b) Suggested some of the development of neutrosophic statistics.
Smarandache (2014c)Introduced neutrosophic crisp set and topology.
Smarandache (2015a) Introduced neutrosophic pre-calculus and neutrosophic calculus.
Smarandache (2015b) Extended or refined neutrosophic numbers.
Smarandache (2016) Introduced degree of dependence and degree of independence between the neutrosophic components t, i, f
Smarandache (2017a) In biology, Author introduced the theory of neutrosophic evolution: degrees of evolution, indeterminacy or neutrality, and involution.
Salama &
Smarandache (2017b)		Introduced plithogeny (as generalization of dialectics and neutrosophy), and Plithogenic set/logic/probability/statistics.
Smarandache (2018) Introduced neutrosophic psychology.

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