In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty. The N-soft set theory is arising as an effective mathematical tool for dealing with precision and uncertainties more than the soft set theory. In this regard, we extend the concept of belief interval-valued soft set to possibility belief interval-valued N-soft set (by accumulating possibility and belief interval with N-soft set), and we also explain its practical calculations. To this objective, we defined related theoretical notions, for example, belief interval-valued N-soft set, possibility belief interval-valued N-soft set, their algebraic operations, and examined some of their fundamental properties. Furthermore, we developed two algorithms by using max-AND and min-OR operations of possibility belief interval-valued N-soft set for decision-making problems and also justify its applicability with numerical examples.

Dealing with erroneous, unexpected, susceptible, flawed, vulnerable, and intricate information is simplified with the use of a single-valued neutrosophic set (svns). This is because of the fact that these types of information are more sensitive to error. This is due to the fact that these particular kinds of information are more prone to error. The ideas of fuzzy sets and intuitionistic fuzzy sets have both undergone further development as a direct result of the development of this new theory. In svns, indeterminacy is quantified in a way that is both obvious and unambiguous, and truth membership, indeterminacy membership, and falsity membership are all completely independent of one another. In algebraic analysis, certain binary operations can be thought of as interacting with algebraic modules. These modules are intricate and ubiquitous structures. There are many different applications for modules to be used in. Modules find use in an extremely wide variety of different kinds of businesses and market segments. We investigate the idea of \((\mu ,\nu ,\omega )\)-svns and relate it to \((\mu ,\nu ,\omega )\)-single-valued neutrosophic module and \((\mu ,\nu ,\omega )\)-single-valued neutrosophic submodule, respectively. The goals of this research are to comprehend the algebraic structures of a \((\mu ,\nu ,\omega )\)-single-valued neutrosophic submodule of a classical module and enhance the legitimacy of this technique by discussing numerous essential aspects. Both of these goals will be accomplished through the course of this study. The strategies that we have developed in this manuscript are more generalizable than those that have been utilized in the past. These strategies include fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets.

The use of a single-valued neutrosophic set (svns) makes it much easier to manage situations in which one must deal with incorrect, unexpected, susceptible, faulty, vulnerable, and complicated information. This is a result of the fact that the specific forms of material being discussed here are more likely to include errors. This new theory has directly contributed to the expansion of both the concept of fuzzy sets and intuitionistic fuzzy sets, both of which have experienced additional development as a direct consequence of the creation of this new theory. In svns, indeterminacy is correctly assessed in a way that is both subtle and unambiguous. Furthermore, membership in the truth, indeterminacy, and falsity are all completely independent of one another. In the context of algebraic analysis, certain binary operations may be regarded as interacting with algebraic modules. These modules have pervasive and complicated designs. Modules may be put to use in a wide variety of different applications. Modules have applications in a diverse range of industries and market subsets due to their adaptability and versatility. Under the umbrella of the triplet $(\mu ,\nu ,\omega )$ structure, we investigate the concept of svns and establish a relationship between it and the single-valued neutrosophic module and the single-valued neutrosophic submodule, respectively. The purpose of this study is to gain an understanding of the algebraic structures of single-valued neutrosophic submodules under the triplet structure of a classical module and to improve the validity of this method by analyzing a variety of important facets. In this article, numerous symmetrical features of modules are also investigated, which demonstrates the usefulness and practicality of these qualities. The results of this research will allow for the successful completion of both of these objectives. The tactics that we have devised for use in this article are more applicable to a wide variety of situations than those that have been used in the past. Fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets are some of the tactics that fall under this category.

The ability to monitor the safety of natural gas pipelines is guaranteed by leak detection. Systems are capable of responding quickly and very precisely to events because delays during such events can lead to serious environmental consequences, hurt, damage, or even danger. Federated is a special framework that exists in this work. Leak Detection of Natural Gas pipelines Edge-A-enabled autonomous drone swarms will be real-time, where smart drones will be able to cooperate and reduce latency, keep sensitive data, and improve detection of anomalies in dynamic operational conditions. Such complex decentralized control systems also need advanced systems of decision-making that are capable of dealing with uncertainty, shifting goals, and information gaps or ambiguous information. The research will, in an attempt to achieve this, emphasize the Multi-Criteria Decision-Making (MCDM) methods that have been used over the years as an alternative method of analysis, which is systematic and founded on alternative performance measures. The precedent versions of MCDM, which applied the theory of fuzzy set, allowed the analysts to convey their judgments with vagueness and partial truth. As uncertainty and conflicting decision environments increased, however, neutrosophic sets were included to describe the degree of truth, falsity, and indeterminacy on their own. This was a later representation that was refined to describe hesitation more by using ambiguous membership and non-membership functions of intuitionistic fuzzy sets (IFS). The combined paradigms led to Intuitionistic Neutrosophic Set (INS), a paradigm of powerful mathematics that can reflect the complexity and ambiguity of the decision-making problems of the real world. In this research, the INS framework is isused with Sugeno-Weber (SW) aggregation operators to come up with a hybrid DM framework that is optimally designed to respond to the real-time leak detection and assessment in pipeline networks. The proposed INS-SW solution is contrasted with the time-tested approach to the evaluation of performance, the Weighted Aggregated Sum Product Assessment (WASPAS), as it is easy to operate and can generate credible ranks. The comparative outcomes indicate that the INS-SW model can be better adapted to uncertain, interdependent, and dynamic operation environments and is more robust and precise in that case. In general, the results suggest that the suggested framework adds to the fact and veracity of the drone-based leakage detection to a substantial degree and can provide a scalable and intelligent decision-making tool related to the imperative energy infrastructure. Besides this application in specific, the paper would also applicable in developing uncertainty-sensitive decision science further, besides offering an insight into how to develop sustainable, intelligent, and resilient energy systems in future industrial processes