A Comprehensive study on (α,β)-multi-granulation bipolar fuzzy rough sets under bipolar fuzzy preference relation

The rough set (RS) and multi-granulation RS (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, the bipolar fuzzy sets (BFSs) are effective tools for handling bipolarity and fuzziness of the data. In this study, with the description of the background of risk decision-making problems in reality, we present $(\alpha, \beta)$-optimistic multi-granulation bipolar fuzzified preference rough sets ($(\alpha, \beta)^o$-MG-BFPRSs) and $(\alpha, \beta)$-pessimistic multi-granulation bipolar fuzzified preference rough sets ($(\alpha, \beta)^p$-MG-BFPRSs) using bipolar fuzzy preference relation (BFPR). Subsequently, the relevant properties and results of both $(\alpha, \beta)^o$-MG-BFPRSs and $(\alpha, \beta)^p$-MG-BFPRSs are investigated in detail. At the same time, a relationship among the $(\alpha, \beta)$-BFPRSs, $(\alpha, \beta)^o$-MG-BFPRSs and $(\alpha, \beta)^p$-MG-BFPRSs is given

New Challenges in Neutrosophic Theory and Applications

Neutrosophic theory has representatives on all continents and, therefore, it can be said to be a universal theory. On the other hand, according to the three volumes of “The Encyclopedia of Neutrosophic Researchers” (2016, 2018, 2019), plus numerous others not yet included in Encyclopedia book series, about 1200 researchers from 73 countries have applied both the neutrosophic theory and method. Neutrosophic theory was founded by Professor Florentin Smarandache in 1998; it constitutes further generalization of fuzzy and intuitionistic fuzzy theories. The key distinction between the neutrosophic set/logic and other types of sets/logics lies in the introduction of the degree of indeterminacy/neutrality (I) as an independent component in the neutrosophic set. Thus, neutrosophic theory involves the degree of membership-truth (T), the degree of indeterminacy (I), and the degree of non-membership-falsehood (F). In recent years, the field of neutrosophic set, logic, measure, probability and statistics, precalculus and calculus, etc., and their applications in multiple fields have been extended and applied in various fields, such as communication, management, and information technology. We believe that this book serves as useful guidance for learning about the current progress in neutrosophic theories. In total, 22 studies have been presented and reflect the call of the thematic vision. The contents of each study included in the volume are briefly described as follows. The first contribution, authored by Wadei Al-Omeri and Saeid Jafari, addresses the concept of generalized neutrosophic pre-closed sets and generalized neutrosophic pre-open sets in neutrosophic topological spaces. In the article “Design of Fuzzy Sampling Plan Using the Birnbaum-Saunders Distribution”, the authors Muhammad Zahir Khan, Muhammad Farid Khan, Muhammad Aslam, and Abdur Razzaque Mughal discuss the use of probability distribution function of Birnbaum–Saunders distribution as a proportion of defective items and the acceptance probability in a fuzzy environment. Further, the authors Derya Bakbak, Vakkas Uluc¸ay, and Memet S¸ahin present the “Neutrosophic Soft Expert Multiset and Their Application to Multiple Criteria Decision Making” together with several operations defined for them and their important algebraic properties. In “Neutrosophic Multigroups and Applications”, Vakkas Uluc¸ay and Memet S¸ahin propose an algebraic structure on neutrosophic multisets called neutrosophic multigroups, deriving their basic properties and giving some applications to group theory. Changxing Fan, Jun Ye, Sheng Feng, En Fan, and Keli Hu introduce the “Multi-Criteria Decision-Making Method Using Heronian Mean Operators under a Bipolar Neutrosophic Environment” and test the effectiveness of their new methods. Another decision-making study upon an everyday life issue which empowered us to organize the key objective of the industry developing is given in “Neutrosophic Cubic Einstein Hybrid Geometric Aggregation Operators with Application in Prioritization Using Multiple Attribute Decision-Making Method” written by Khaleed Alhazaymeh, Muhammad Gulistan, Majid Khan, and Seifedine Kadry

Systematic review of decision making algorithms in extended neutrosophic sets

The Neutrosophic set (NS) has grasped concentration by its ability for handling indeterminate, uncertain, incomplete, and inconsistent information encountered in daily life. Recently, there have been various extensions of the NS, such as single valued neutrosophic sets (SVNSs), Interval neutrosophic sets (INSs), bipolar neutrosophic sets (BNSs), Refined Neutrosophic Sets (RNSs), and triangular fuzzy number neutrosophic set (TFNNs). This paper contains an extended overview of the concept of NS as well as several instances and extensions of this model that have been introduced in the last decade, and have had a significant impact in literature. Theoretical and mathematical properties of NS and their counterparts are discussed in this paper as well. Neutrosophic-set-driven decision making algorithms are also overviewed in detail

Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets

Neutrosophy (1995) is a new branch of philosophy that studies triads of the form (, , ), where is an entity {i.e. element, concept, idea, theory, logical proposition, etc.}, is the opposite of , while is the neutral (or indeterminate) between them, i.e., neither nor .Based on neutrosophy, the neutrosophic triplets were founded, which have a similar form (x, neut(x), anti(x)), that satisfy several axioms, for each element x in a given set.This collective book presents original research papers by many neutrosophic researchers from around the world, that report on the state-of-the-art and recent advancements of neutrosophic triplets, neutrosophic duplets, neutrosophic multisets and their algebraic structures – that have been defined recently in 2016 but have gained interest from world researchers. Connections between classical algebraic structures and neutrosophic triplet / duplet / multiset structures are also studied. And numerous neutrosophic applications in various fields, such as: multi-criteria decision making, image segmentation, medical diagnosis, fault diagnosis, clustering data, neutrosophic probability, human resource management, strategic planning, forecasting model, multi-granulation, supplier selection problems, typhoon disaster evaluation, skin lesson detection, mining algorithm for big data analysis, etc

Quadripartitioned single valued neutrosophic sets with covering based rough sets and their matrix representation

The notion of a quadripartitioned single-valued neutrosophic set (in short QSVNS) is considered to be the more general mathematical framework of the neutrosophic set to model indeterminacy. In QSVNS, the indeterminacy component is divided into two parts, namely, “unknown” and “contradiction”. In a real-life scenario, while handling indeterminacy, we may have some hesitation about whether the indeterminacy occurs due to the belongingness or the non-belongingness of an object. This leads to the introduction of QSVNS. On the other hand, the theory of rough set (RS) is introduced to depict the incomplete data hidden in nature with an aid of equivalence relation. So, by combining the QSVNS and the RS, a new mathematical structure known as a quadripartitioned single-valued neutrosophic rough set is formed. The main purpose of this article is to present two types of quadripartitioned single-valued neutrosophic covering rough set models. Also, we have introduced QSVN β-covering approximation space and studying some of its properties. Based on QSVN β -covering approximation spaces, two types of QSVN covering rough set models are investigated. Furthermore, a matrix representation of the QSVN covering-based rough set model is developed. Finally, an algorithm-based model under QSVN covering-based rough set is developed and employed in a case study to diagnose a patient/s that is more likely suffering from a disease having certain symptoms

Some New Operations of ( alpha, , ) Interval Cut Set of Interval Valued Neutrosophic Sets

In this paper, we define the disjunctive sum, difference and Cartesian product of two interval valued neutrosophic sets and study their basic properties

Algebraic structures of neutrosophic triplets, neutrosophic duplets, or neutrosophic multisets. Volume II

The topics approached in this collection of papers are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; fuzzy measure; clustering algorithm; and many more

A Historical Account of Types of Fuzzy Sets and Their Relationships

In this paper, we review the definition and basic properties of the different types of fuzzy sets that have appeared up to now in the literature. We also analyze the relationships between them and enumerate some of the applications in which they have been used