This study explores the complexities of intuitionistic fuzzy (hyper) graphs, considering them as complex (hyper) networks, and presents a unique idea for intuitionistic fuzzy (quasi) superhypergraphs. The extension considers intuitionistic fuzzy superhypergraphs to be complicated superhyper networks to establish particular and general links between labeled items. These intuitionistic fuzzy (quasi) superhypergraphs arrange labeled object groups and analyze them in several relational aspects at the same time, including part-to-part, part-to-whole, and whole-to-whole groupings. The research investigates the characteristics of intuitionistic fuzzy (quasi) superhypergraphs utilizing positive real numbers, such as valued intuitionistic fuzzy (quasi) superhypergraphs and their complements, permutation-based isomorphism notation, and isomorphic (self-complemented) valued intuitionistic fuzzy (quasi) superhypergraphs. It also presents the concept of impact membership value for intuitionistic fuzzy (quasi) superhypergraphs and demonstrates how it may be used to solve real-world problems. Finally, the research demonstrates the use of intuitionistic fuzzy valued quasi superhyper graphs in addressing social network analysis, emphasizing their practical use.

Complex neutrosophic graphs are created by combining complex neutrosophic sets with graph theory ideas. This provides a flexible framework for tackling complex problem-solving circumstances. Various processes, such as union, join, and composition, are thoroughly investigated to improve the administration of complicated neutrosophic graphs. This research also looks into the area of complicated neutrosophic graph homeomorphisms, investigating the transformations and mappings that occur inside these structures. This investigation advances our knowledge of the intrinsic features and linkages found in complicated neutrosophic graphs. Finally, our methodology’s practical usefulness is demonstrated by its use in the design of hospital infrastructure. We illustrate the effectiveness of complicated neutrosophic graphs in addressing difficult design concerns and optimizing infrastructure solutions for practical deployment in this real-world scenario.

Production systems prioritize economic efficiency, often overlooking environmental sustainability. Growing environmental awareness has led to circular production systems that integrate economic and ecological goals. This study presents a sustainable production–inventory model for a two-echelon supply chain with a manufacturer and retailer, incorporating remanufacturing under carbon emissions regulations. The manufacturing process follows a price and time-dependent demand while remanufacturing relies on a circularity index demand. Two model variants are analyzed using Hessian algorithms, combining cap-and-trade and carbon tax policies with different demand rates and profit functions. Results show that maximum profit occurs under cap-and-trade, despite an 82.20% circularity index. Carbon regulations effectively reduce emissions but slightly lower profitability. This study highlights the need to optimize inventory and remanufacturing decisions to balance economic and environmental goals within a circular economy.

The main concept involved in this study explains the theme of complex t-intuitionistic fuzzy graphs (CTIFGs), which act as a powerful tool in analyzing and displaying the relationships among various applications that are difficult to recognize. The manuscript also demonstrates the capability of CTIFGs to create complex associations with multiple domains when considering a physical situation. Following this, the basic set of operations for CTIFGs is projected. The ideas on isomorphism and homomorphism of the CTIFGs are also presented. Moreover, the manuscript describes the importance of the above-mentioned technique in an effective way, giving a solution to the practical application associated with rubber processing industrial wastewater. The contributing factors and corresponding interdependencies are considered when calibrating the complex nature of industrial wastewater associated with the CTIFGs. The results highlight the adaptability and possible efficiencies of CTIFGs, which act as a decision-making tool and also indicate their importance for policy planners in important societal issues.

A graph or molecular graph’s overall resilience and connectedness are gauged by the Sombor Index (SOI). It is an improved version of the conventional SOI that is designed to capture ambiguity in the relationships between nodes or atoms. This indicator aids in assessing the network’s resilience and stability in erratic circumstances. We have investigated the charac-teristics of the SOI of neutrosophic graphs (N Gs) in this study. The association between the neutrosophic first Zagreb Index (N FZI) of N Gs and the neutrosophic Sombor Index (N SI) was revealed in the study. The application of SOI based on Site Selection for Thermal Power Plants in N Gs is finally covered.

The exponential fuzzy set ($\mathcal{EFS}$) is a new modification that allows for a more flexible representation of uncertainty by using an exponential function to define membership degree. In this work, we define basic operations on $\mathcal{EFS}$, such as complement, union, intersection, simple difference, and limited difference functions. The equivalency formula, symmetrical difference formula, disjoint sets, disjoint sum, and disjunctive sum are further important qualities that we examine. We analyse essential laws in the exponential fuzzy framework, such as the idempotent law of union. In addition, we present a few theorems that govern the relational and algebraic structures of $\mathcal{EFS}$. $\mathcal{EFS}$ has been compared against traditional approaches and the resulting studies showcase its advantages in modeling uncertainty, artificial intelligence, and decision making. This paper studies the use of exponential fuzzy sets in AI driven investment decision processes using the weighted mean method of multifactor investment analysis.