The single-valued neutrosophic set (SVNS) is a well-known model for handling uncertain and indeterminate information. Information measures such as distance measures, similarity measures and entropy measures are very useful tools to be used in many applications such as multi-criteria decision making (MCDM), medical diagnosis, pattern recognition and clustering problems. A lot of such information measures have been proposed for the SVNS model. However, many of these measures have inherent problems that prevent them from producing reasonable or consistent results to the decision makers. In this paper, we propose several new distance and similarity measures for the SVNS model. The proposed measures have been verified and proven to comply with the axiomatic definition of the distance and similarity measure for the SVNS model. A detailed and comprehensive comparative analysis between the proposed similarity measures and other well-known existing similarity measures has been done. Based on the comparison results, it is clearly proven that the proposed similarity measures are able to overcome the shortcomings that are inherent in existing similarity measures. Finally, an extensive set of numerical examples, related to pattern recognition and medical diagnosis, is given to demonstrate the practical applicability of the proposed similarity measures. In all numerical examples, it is proven that the proposed similarity measures are able to produce accurate and reasonable results. To further verify the superiority of the suggested similarity measures, the Spearman’s rank correlation coefficient test is performed on the ranking results that were obtained from the numerical examples, and it was again proven that the proposed similarity measures produced the most consistent ranking results compared to other existing similarity measures.

Topological Data Analysis is a new theoretical trend using topological techniques to mine data. This approach allows the determination of topological data structures. It focuses on investigating the global shape of data rather than on local information of high-dimensional data. The Mapper algorithm is considered a sound representative approach in this area. It is used to cluster and identify concise and meaningful global topological data structures that are out of reach from many other clustering methods. In this paper, we propose a new method called the Shape Fuzzy C-Means (SFCM) algorithm, which is constructed based on the Fuzzy C-Means algorithm with particular features of the Mapper algorithm. The SFCM algorithm can not only exhibit the same clustering ability as the Fuzzy C-Means, but also unlock some relationships through visualizing the global shape of data supplied by the Mapper. We present a formal proof and include experiments to confirm our claims. The efficiency of the enhanced algorithm is demonstrated by a comparative analysis involving the Mapper algorithm for synthetic and real-world data. The comparison is conducted with respect to output visualization in the topological sense and clustering stability

Plithogenic set is an extension of the crisp set, fuzzy set, intuitionistic fuzzy set, and neutrosophic sets, whose elements are characterized by one or more attributes, and each attribute can assume many values. Each attribute has a corresponding degree of appurtenance of the element to the set with respect to the given criteria. In order to obtain a better accuracy and for a more exact exclusion (partial order), a contradiction or dissimilarity degree is defined between each attribute value and the dominant attribute value. In this paper, entropy measures for plithogenic sets have been introduced. The requirements for any function to be an entropy measure of plithogenic sets are outlined in the axiomatic definition of the plithogenic entropy using the axiomatic requirements of neutrosophic entropy. Several new formulae for the entropy measure of plithogenic sets are also introduced. The newly introduced entropy measures are then applied to a multi-attribute decision making problem related to the selection of locations.