Single-valued neutrosophic sets (SVNSs) handling the uncertainties characterized by truth, indeterminacy, and falsity membership degrees, area more flexible way to capture uncertainty.

In the epoch of Internet of Things (IoT), we are confronted five challenges (Connectivity, Value, Security, Telepresence and Intelligence) with complex structures. IoT industry decision making is critically important for countries or societies to enhance the effectiveness and validity of leadership, which can greatly accelerate industrialized and large-scale development. In the case of IoT industry decision evaluation, the essential problem arises serious incompleteness, impreciseness, subjectivity and incertitude. Interval neutrosophic set (INS), disposing the indeterminacy portrayed by truth membership T, indeterminacy membership I, and falsity membership F with interval form, is a more viable and effective means to seize indeterminacy.

“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.

Cut sets, decomposition theorem and representation theorem have a great influence on the realization of the transformation of fuzzy sets and classical sets, and the single-valued neutrosophic multisets (SVNMSs) as the generalization of fuzzy sets, which cut sets, decomposition theorem and representation theorem have the similar effects, so they need to be studied in depth. In this paper, the decomposition theorem, representation theorem and the application of a new similarity measures of SVNMSs are studied by using theoretical analysis and calculations. The following are the main results: (1) The notions, operation and operational properties of the cut sets and strong cut sets of SVNMSs are introduced and discussed; (2) The decomposition theorem and representation theorem of SVNMSs are established and rigorously proved.

In the rising information technology trends, cost, time, delivery, space, quality, durability, and price are all paramount in addressing managerial decision-making complexities within the supply chain, transportation, and inventory control. The intensifying competition within imprecise environments further complicates these challenges. Customer demand, influenced by multifaceted factors such as production price and income levels, often remains elusive or unpredictable in the real-world market. Fuzzy sets, while useful, need to catch up in directly capturing such uncertainties due to their numeric membership functions. Multi-Criteria Decision Making Models and Techniques: Neutrosophic Approaches explores the neutrosophic sets as a solution, uniquely poised to accommodate inherent uncertainties. Neutrosophic sets and logic are evolutionary extensions of fuzzy and intuitionistic fuzzy sets and logic. In real-world problems, these diverse uncertain systems demonstrate a capability to manage heightened levels of uncertainty. This book delves into the burgeoning field of neutrosophic theory, elucidating its application in operations research. Neutrosophic sets and logic have emerged as pivotal tools in grappling with uncertainty, impreciseness, vagueness, incompleteness, inconsistency, and indeterminacy.

“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.

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.

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; they have a similar form: (x, neut(x), anti(x), that satisfy some axioms, for each element x in a given set. This book contains the successful invited submissions to a special issue of Symmetry, reporting on 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, and several papers have been published in first rank international journals.

Fuzzy information in venture capital can be well expressed by neutrosophic numbers, and TODIM method is an effective tool for multi-attribute decision-making. The distance measure is an essential step in TODIM method. The keystone of this paper is to define several new distance measures, in particular the improved interval neutrosophic Euclidean distance, and these measures are applied in the TODIM method for multi-attribute decision-making.

The processing of uncertainty information has gradually became one of the hot issues in arti cial intelligence eld, and the infor- mation measures of uncertainty information processing are of importance. Single value neutrosophic sets (SVNSs) provide us a exible mathematical framework to process uncertainty information. In this paper, we mainly consider the measures of SVNSs. The existing information measures mostly are constructed based on the two typical inclusion relations about single value neutrosopgic sets. However, there exist some practical problems that do not apply to the two typical inclusion relations. Therefore, there exists another inclusion relation which is called the type-3 inclusion relation about SVNSs.