A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems
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ORIGINAL ARTICLE
A ranking method of single valued neutrosophic numbers and its applications to multi-attribute decision making problems I. Deli1 · Y. Şubaş1
Received: 2 June 2015 / Accepted: 25 January 2016 © Springer-Verlag Berlin Heidelberg 2016
Abstract The concept of a single valued neutrosophic number (SVN-number) is of importance for quantifying an ill-known quantity and the ranking of SVN-numbers is a very difficult problem in multi-attribute decision making problems. The aim of this paper is to present a methodology for solving multi-attribute decision making problems with SVN-numbers. Therefore, we firstly defined the concepts of cut sets of SVN-numbers and then applied to single valued trapezoidal neutrosophic numbers (SVTNnumbers) and triangular neutrosophic numbers (SVTrNnumbers). Then, we proposed the values and ambiguities of the truth-membership function, indeterminacy-membership function and falsity-membership function for a SVNnumbers and studied some desired properties. Also, we developed a ranking method by using the concept of values and ambiguities, and applied to multi-attribute decision making problems in which the ratings of alternatives on attributes are expressed with SVTN-numbers. Keywords Neutrosophic set · Single valued neutrosophic numbers · Trapezoidal neutrosophic numbers · Triangular neutrosophic numbers · Decision making
& I. Deli [email protected] Y. S¸ubas¸ [email protected] 1
Muallim Rıfat Faculty of Education, 7 Aralık University, 79000 Kilis, Turkey
1 Introduction Smarandache [32] proposed concept of neutrosophic set which is generalization of classical set, fuzzy set [50], intuitionistic fuzzy set [3], and so on. In the neutrosophic set, for an element x of the universe, the membership functions independently indicates the truth-membership degree, indeterminacy-membership degree, and falsemembership degree of the element x belonging to the neutrosophic set. Also, fuzzy, intuitionistic and neutrosophic models have been studied by many authors (e.g. [1, 2, 5, 7–9, 14, 15, 19, 31, 33, 34, 36, 41, 42]). Multi-attribute decision making (MADM) which is an important part of decision science is to find an optimal alternative, which are characterized in terms of multiple attributes, from alternative sets. In some practical applications, the decision makers may be not able to evaluate exactly the values of the MADM problems due to uncertain and asymmetric information between decision makers. As a result, values of the MADM problems are not measured by accurate numbers. It is feasible for some sets which contain uncertainty such as; a fuzzy set, intuitionistic set and neutrosophic set to represent an uncertainty of values of the MADM problems. Intuitionistic fuzzy numbers, intuitionistic triangular fuzzy numbers and intuitionistic trapezoidal fuzzy numbers is introduced by Mahapatra and Roy [27], Liang [26] and Jianqiang [20], respectively. Li [23] gave a ranking method of intuitionistic fuzzy numbers and application to multiattribute decision-making problems in which the
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