This paper presents a generalized framework termed the multidimensional topological measure space (MDTMS), developed through multidimensional fuzzy sets, multidimensional topology, and an associated distance measure. The suggested framework enhances traditional fuzzy models by facilitating a more nuanced representation and examination of intricate, multiparameter data. Essential core components, including multidimensional fuzzy topology, basis, and subspace, are rigorously described within this framework. We create many mathematical instruments to facilitate study inside the MDTMS framework, encompassing concepts of multidimensional carrier, closure, core, and interior. Additionally, we analyze sequences of multidimensional fuzzy sets and introduce two forms of convergence, together with their fundamental characteristics. To illustrate the practical significance of the proposed framework, we introduce a decision‐making methodology grounded in Euclidean multidimensional distance functions incorporated within the MDTMS structure. Examples are shown to demonstrate the relevance and efficacy of the proposed notions.

This paper addresses critical limitations in the application of intuitionistic fuzzy sets (IFSs) for complex decision‐making problems under uncertainty. While IFSs offer a robust framework for modeling imprecision through membership, nonmembership, and hesitancy degrees, existing methodologies often simplify the multiplication of intuitionistic fuzzy numbers (IFNs) to linear approximations, overlooking potential nonlinear interactions. Furthermore, a comprehensive signed distance measure for triangular IFNs (TIFNs) and a method for intuitionistic fuzzy nonlinear signed distance are largely absent from the literature. To bridge these gaps, this study develops novel equations for the multiplication of two IFNs, treating the result as a nonlinear IFN, thereby enhancing the precision of fuzzy arithmetic. The practical utility of the proposed methodology is demonstrated through an application involving sustainable business model innovations (SBMI) evaluation under ESG principles. The numerical example is provided, showcasing how the developed techniques can provide an accurate and nuanced assessment for complex multicriteria decision‐making problems.

Pythagorean hesitant fuzzy sets (PHFSs) effectively represent uncertain information in decision‐making by accommodating situations where the sum of membership and nonmembership degrees exceeds 1, provided their squared sum remains at most 1. This study proposes a novel distance measure for PHFSs and rigorously demonstrates its rationality and validity. By integrating this distance measure with conflict analysis, we define the conflict degree in Pythagorean hesitant fuzzy environments. Then a new multicriteria decision‐making (MCDM) method is developed to address problems with completely unknown criterion weights, avoiding artificial data manipulation based on decision‐makers’ risk preferences. The proposed distance measure serves as a key tool for quantifying dissimilarity between PHFSs while maintaining decision consistency and reliability. Finally, an illustrative example of habitat selection evaluation for endangered species, accompanied by comparative analysis, validates the effectiveness and feasibility of the proposed method.

This study introduces several operations on n‐dimensional fuzzy sets, including union, intersection, and complement, and examines the validity of De Morgan’s law within this framework. To further elucidate the conceptual underpinnings, the study also presents illustrative examples of t‐norm, t‐conorm, and negation operations on n‐dimensional fuzzy sets. Furthermore, the paper proposes a novel structure termed the n‐dimensional fuzzy soft sets, which extends the concepts of both soft sets and n‐dimensional fuzzy sets. The foundational properties of n‐dimensional fuzzy soft sets are explored in detail, and various operations such as union, intersection, negation, t‐norms, and t‐conorms are defined and analyzed within this context. Additionally, the study discusses relevant laws, including De Morgan’s law as they pertain to the proposed structure. To demonstrate the practical utility of this new model, the study presents its application in decision‐making scenarios, such as the selection of the most suitable vendor for a critical project. A comparative analysis with existing decision‐making strategies based on intuitionistic fuzzy soft sets and interval‐valued fuzzy soft sets highlights the enhanced effectiveness and robustness of the proposed methodology.

Users expect software to be error‐free; however, preventing faults in software while being developed is difficult. Although predicting faults in software is arduous, it radically helps to improve the software quality. Due to the complexity of software, time, and budget limitations, such prediction helps to deliver more robust and error‐free software with lower expenses. This paper introduces an iterative method based on fuzzy systems and machine learning to predict software faults. High interpretability, transparency, balancing data, and finding the best interval for converting numerical features to fuzzy features are basic challenges for predicting software faults. The proposed framework is split into four phases. In the first phase, the crisp inputs are converted to fuzzy sets. In the second phase, a membership function is constructed using triangular fuzzy sets. In the third phase, training data are balanced, and fuzzy rules are generated. In the last phase, the similarity of inputs with the rules’ antecedents is calculated, and the fired rules are aggregated to label the test data. Eclipse, Promise, and Travis repositories are evaluated with the proposed method. The calculated AUC of the proposed method on Promise, Travis, and Eclipse datasets are, respectively, equal to 89%, 62% and 87%, which are comparable to the results obtained by deep learning methods but with higher interpretability and transparency.

Industrial waste involves any hazardous substance, chemical, or byproduct and waste discharged during the operation of an industry. Due to its nature and probable detriment to the environment, industrial waste is a great matter of concern for governments all over the world. Selection of industry‐linked industrial waste management strategies (IWMSs) is the main objective of this research endeavor. Applying greater generality in the dimensions of the abovementioned applications could potentially enhance a lot of their efficiency. Einstein norms/application of Einstein operations have not been effectively applied within the complex spherical fuzzy sets (CSFSs) framework as such; this study now advances the application of Einstein operators in CSFS which would provide an adiaphorous approach to solving issues related to decision‐making problems (DMPs). The actual situations in which decisions are made invariably present numerous conflicting factors, which complicate the process even further. The methodology of multiattribute group decision‐making (MAGDM) is an essential means of handling such situations. This paper deals with the challenge pertaining to MAGDM and utilizes the established Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). In order to aid successful problem‐solving, new aggregation operators are proposed, namely, complex spherical fuzzy Einstein weighted averaging (CSFEWA) operator, complex spherical fuzzy Einstein ordered weighted averaging (CSFEOWA) operator, and complex spherical fuzzy Einstein hybrid weighted averaging (CSFEHWA) operator.

Transportation Problem (TP) is considered a combinatorial optimization problem, and its aim is to minimize the total transportation cost from several sources to different destinations. In this paper, an intensive literature review of the TP is presented in detail with an enhancement of the novel heuristic entitled Dhouib‐Matrix‐TP1 (DM‐TP1) to solve the TP under a hexagonal fuzzy environment. Indeed, all parameters of the TP—such as transportation cost, demand, and supply—are represented using hexagonal fuzzy numbers (HFNs), which offer a richer structure for modeling uncertainty with less information loss compared to triangular or trapezoidal fuzzy numbers. To convert these HFNs to crisp ones, the centroid ranking function is used. After that, the DM‐TP1 method is applied to rapidly find an initial basic feasible solution using the original metric (Average–Min). Experiment results of DM‐TP1 on different literature Hexagonal Fuzzy Transportation Problems (HFTPs) prove its performance. Moreover, DM‐TP1 illustrates graphically the generated solution and allows the decision maker to ergonomically handle the TPs under HFNs.

For sustainable growth, investments in renewable energy must be maximized. Maximizing investments in renewable energy opens the door to a more successful and environmentally friendly future. Analyzing technological viability, cost‐effectiveness, regulatory compliance, and environmental impact are all part of this optimization process. This paper delves into a sophisticated methodology designed to tackle uncertainties in decision‐making by leveraging the innovative concept of Pythagorean hesitant fuzzy sets (PHFSs). We defined aggregation operations and distance measures for PHFS. After that, we introduced Measurement of Alternatives and Ranking According to the Compromise Solution (MARCOS), a novel methodology under PHFS, it is a robust tool acknowledged for navigating complex decision scenarios with multiple criteria. Following that, we showcased a case study on enhancing renewable energy investments through an AI‐based strategy for sustainable development, utilizing the newly developed MARCOS algorithm. The study highlights the significance of its adaptability and efficiency in practical applications. Furthermore, we compared this methodology with the Technique for Establishing Order Performance by Similarity to the Ideal Solution (TOPSIS), offering insights into their respective strengths. This offers a concrete demonstration of its real‐world utility and potential impact in decision‐making scenarios. Finally, in the last, we conclude the whole study.

Type‐3 fuzzy logic has been recently used in many control methods. The type‐3 fuzzy controller enhances the handling of uncertainty and improves robustness by integrating fuzzy sets with fuzzy membership functions. The latest approaches using type‐3 fuzzy logic in the field of control are studied and evaluated. An overview of developments in control methods based on type‐3 fuzzy logic is also provided. It is shown that type‐3 fuzzy system has many advantages compared to type‐1 and type‐2 fuzzy. The advantages and challenges of using type‐3 fuzzy logic are identified and discussed. The studies are classified according to the type of control approach, as well as by the type of control applications. Finally, the main achievements, open challenges, and future directions and impacts are identified, to provide important guidance for interested researchers.