On Ranking Algorithms for SV NT′ s and NV NT′ s

Ranking algorithms are very important tools in decision making. There are two ranking algorithms for nvalued neutrosophic tuplets (Single-Valued MultiNeutrosophic tuplets): The S-ranking algorithm of Single-Valued MultiNeutrosophic tuplets, which is introduced by F.Smarandache in 2023, and the N-ranking algorithm of n-valued neutrosophic tuplets, Which is introduced by V. L. Nayagam and Bharanidharan R. in 2023. In this paper we show (by examples) that these two ranking algorithms are not a total ordering for the set of n-valued neutrosophic tuplets. These algorithms do not taking into account the number of sources, which is a very important factor in neutrosophic n-valued refined sets theory. We introduce two ranking algorithms: The integrated S-ranking algorithm of Single-Valued MultiNeutrosophic tuplets, and the integrated N-ranking algorithm of n-valued neutrosophic tuplets. These algorithms are improvements of the S-ranking algorithm of Single-Valued MultiNeutrosophic tuplets, and the N-ranking algorithm of n-valued neutrosophic tuplets, respectively, and taking the number of sources into account. We construct different examples to show that each step in the integrated ranking algorithms is necessary to make them a total ordering for the set of all n-valued neutrosophic tuplets.