Triangular norms on finite lattices

A reliable and data-based teaching quality evaluation is essential for the continuous improvement of higher-education systems. However, the inherent ambiguity of assessment indicators and the subjectivity of evaluators render traditional, crisp-value models insufficient. To address this challenge, we develop a novel intuitionistic fuzzy multi-attribute decision-making framework that integrates Yager triangular norms (t-norms) with the geometric Heronian mean. Specifically, we first introduce intuitionistic fuzzy operations based on Yager t-norms and Yager t-conorms and subsequently construct two aggregation operators: the intuitionistic fuzzy geometric Heronian mean operators and the intuitionistic fuzzy weighted geometric Heronian mean operators. The idempotency, monotonicity, and boundedness properties of these operators are formally proven. Next, the intuitionistic fuzzy weighted geometric Heronian mean operators are employed to develop an approach for multi-attribute decision-making in classroom teaching quality evaluation under intuitionistic fuzzy information. Moreover, an application case study of teaching quality evaluation in an intuitionistic fuzzy environment is presented to demonstrate the practicality and effectiveness of the proposed approach. Additionally, sensitivity and comparative analyses with other techniques are carried out to further confirm the coherence and superiority of the recommended approach. The research results clearly show that our proposed method is highly effective in accurately evaluating teaching quality and can serve as a valuable tool for educational institutions in enhancing their teaching quality management.

The Aczel-Alsina (AA) aggregation operators (AOs) are highly effective in optimizing complex decision-making (DM) tasks involving extensive datasets, particularly in the context of T-spherical fuzzy (T-SF) environments. This research creates a refined decision framework that integrates the AAT-norm (TN) and T-conorm (T-CN) operations with T-spherical fuzzy soft sets (T-SFSs). Two novel aggregation operators, named the T-SFS AA Weighted Geometric (T-SFSAAWG) and T-SFS AA Weighted Averaging (T-SFSAAWA) operators, are put forward. Their mathematical behavior, properties, and specific instances are thoroughly investigated. The Technique for Alternative Ordering of Variables (TAOV) is incorporated into the model to reinforce the ranking process. With the TAOV method, the time and computational effort required for ranking can be diminished, all while preserving precision and stability. Our proposed methodology is applied to a supplier selection problem, demonstrating its superior optimization effectiveness compared to current T-SFS and AA-based approaches. The model has been shown to yield consistent and efficient results through comparative and sensitivity analyses, establishing it as a practical and trustworthy resource for making data-informed decisions.

Retraction notice to "Decision algorithm for educational institute selection with spherical fuzzy heronian mean operators and Aczel-Alsina triangular norm" [Heliyon 10 (2024) e28383

The choice of lens impacts the quality of vision, comfort, and overall eye health. By correcting vision problems such as astigmatism, nearsightedness, or farsightedness, the right lens can guarantee precise and clear vision. The multi-attribute decision-making (MADM) technique is a valuable method for data aggregation with accuracy. The central theme of this article is to develop novel aggregation operators (AOs) by giving a positive value called the priority degree under a strict priority level. We establish a novel MADM method based on an interval-valued t-spherical fuzzy set (IV-TSFS) framework and the Einstein AOs. First, we established Einstein t-norm (TNM) and t-conorm (TCNM) operations based on the IV-TSFS framework. Next, we proposed new prioritized AOs, such as IV-TSF Einstein prioritized weighted averaging (IV-TSFEPWA), IV-TSF Einstein prioritized ordered weighted averaging (IV-TSFEPOWA), IV-TSF Einstein prioritized hybrid weighted averaging (IV-TSFEPHWA), IV-TSF Einstein prioritized weighted geometric (IV-TSFEPWG), IV-TSF Einstein prioritized ordered weighted geometric (IV-TSFEPOWG), and IV-TSF Einstein prioritized hybrid weighted geometric (IV-TSFEPHWG). Some essential axioms, such as boundedness, idempotency, and monotonicity, were also discussed. Then, we developed the MADM algorithm based on the IV-TSFEPWA and IV-TSFEPWG operators, which was monitored through a real-life case study on the selection of optimal eye lens manufacturing companies based on prioritized evaluation criteria. Additionally, to validate the novelty of the established model, a comparative study with existing approach is also provided, highlighting the applicability of the established work. Ultimately, the practical implications of established AOs and solid conclusions will be discussed.

Physical education and traditional sports are gaining increasing importance for comprehensive development, making the systemic design of classrooms necessary. Well-designed physical education rooms encourage students to participate in both curricular and extracurricular activities. With a well-structured classroom, opportunities for various teaching methods are available, and everyone is encouraged to participate. The research presents a novel approach to applying the Analytical Hierarchy Process (AHP) within multi-criteria decision-making (MCDM) to design physical education classroom environments. The MCDM approach facilitates the successful evaluation of superior options among multiple alternatives. Through AHP processes, decision-makers can evaluate the importance of criteria by making pairwise comparisons until they identify the most suitable choice. The q-rung ortho pair fuzzy set (q-ROFS) framework extends fuzzy sets with a more sophisticated system that shows both the degree of membership (DoM) and degree of non-membership (DoNM) throughout the left[0, 1right] interval with a large range. The Sugeno-Weber t-norm (TNM) and t-conorm (TCNM) play a significant role in modeling flexible fuzzy systems, offering more control over how fuzzy sets interact. Due to their adjustable parameters, they can easily aggregate complex information. We aimed to develop aggregation operators (AOs) based on the AHP, combined with Sugeno-Weber TNM and TCNM, known as q-rung orthopair fuzzy Sugeno-Weber weighted averaging (q-ROFSWWA) and q-rung orthopair fuzzy Sugeno-Weber weighted geometric (q-ROFSWWG) operators, to address current needs. Additionally, the paper describes an improved combined compromise solution (CCS) methodology to determine the precise ranking of alternatives. Then, we presented the MCDM algorithms of the developed q-ROFSWWA and COCOSO approach under AHP and q-ROFS information. Through practical examples, we rank the four different classes for physical education, including versatile equipment, integration of outdoor spaces, adaptable and versatile environments, and safety and comfort features. We apply our proposed theory to investigate four alternatives, and the obtained result is “integration of outdoor spaces.” To demonstrate the importance of the proposed approach, we conduct a comparative analysis to investigate authenticity and validation. Then, we provide a conclusion. The graphical abstract of the article is provided in Fig. 1.

Ordinal sums of triangular norms on a bounded trellis

Blockchain technology is a decentralized method of securely storing information in blocks. Over the past few decades, sustainability has gained importance and been shown to be a crucial component in building robust supply chains. Manufacturers and suppliers are pressured to become more globally sustainable by customers who want their products verified for sustainability. Furthermore, problems with social sustainability are becoming more complex. Multi-criteria decision-making (MCDM) is the best tool for addressing this issue. In this article, we aimed to propose a combined compromise solution (CoCoSo) method for ranking blockchain platform selection using spherical fuzzy set (SFS) information. SFS is a well-known framework that provides greater freedom to decision-makers and better reliability than intuitionistic fuzzy sets, Pythagorean fuzzy sets, q-rung ortho-pair fuzzy sets, and picture fuzzy set-based frameworks. The CoCoSo approach under the SFS framework is a renowned tool for complicated decision-making problems. In this method, we consider both linear and non-linear correlations among criteria; the technique aims to give decision-makers a more comprehensive assessment of the alternatives. In the MCDM field, the idea of a power aggregation operator (PAO) is a valuable tool for investigating the weightage of alternatives. We also introduce Sugeno Weber tnorm (TN) and tconorm (TCN) operations under the SFS framework. MCDM Sciences utilizes The Sugeno-Weber operations because it provides a solution for managing ambiguous yet unclear information. The approach enables users to mix diverse inputs through a versatile mechanism that proves valuable for selecting from multiple factors. The cartel theme of this article is to construct a new family of aggregation operators (AOs) called spherical fuzzy (SF) Sugeno Weber power weighted averaging (SFSWPWA) and SF Sugeno Weber power weighted geometric (SFSWPWG) operators, including investigation of some fundamental axioms of AOs. The MCDM algorithm for the CoCoSo method will also be constructed, and the SFSWPWA and SFSWPWG operators will be established. We solve a real-life numerical problem for selecting the best blockchain platform using diagnosed approaches. In the numerical example, we rank blockchain platforms like Hyperledger Fabric, Ethereum, IBM Food Trust, and VeChain. Applying the proposed CoCoSo method, SFSWPWA and SFSWPWG operators “Ethereum” are the best options. The proposed methodologies also apply in real-life Sanrio’s where decision-making is involved. To check the applicability of the developed technique, we made a comparison with existing approaches such as interval-valued intuitionistic fuzzy Heronian mean AOs, Pythagorean fuzzy Hamy mean AO, complex q-rung ortho-pair fuzzy Aczel Alsina AOs, and q-rung ortho-pair fuzzy Sugeno-Weber AOs. These discussed AOs are unable to handle SFS-based information. So, we concluded that the suggested framework is more reliable and superior to other discussed environments due to the presence of abstinence grade. Robust conclusions are given in the end.

Decision-making is a crucial process that involves selecting a course of action from multiple alternatives. It requires evaluating different options, weighing their potential outcomes, and making a choice based on available information and objectives. In physical education, decision-making plays a vital role in instructing teachers and students to interact with each other, facilitating understanding and retention of previously learned material. In this article discusses decision-making in physical education for teaching and learning, aiming to create engaging and inspiring opportunities for students to learn and improve their thinking skills. To achieve this, we utilize a circular Pythagorean fuzzy set (CPyFS), which captures membership and non-membership grades while exhibiting circular behavior. This approach is particularly useful for handling uncertainty in assessment processes, allowing decision-makers ([Formula: see text]) to discuss their opinions within a large domain. The CPyFS technique builds upon fuzzy set (FS) theory, with Pythagorean FS (PyFS) being a special case. Our goal is to define weighted averaging and geometric operators within the CPyFS framework, based on triangular norms. We demonstrate the effectiveness of these aggregation operators (AOs) through examples and establish key mathematical properties, such as idempotency and boundedness. Furthermore, we propose a new model using the weighted aggregated sum product assessment (WASPAS) method, exploring its application in multi-criteria group decision-making (MCGDM). The effectiveness and reliability of this technique are validated through sensitivity analysis and comparative study, showcasing its potential in handling uncertainties in physical education.

The network of millions of linked devices that exchange resources is known as the Internet. The rapid expansion of digital connectivity necessitates an assessment and understanding of its multifaceted impact on younger generations. There are many benefits and drawbacks to using the Internet, including issues with human health, education and learning habits, communication and relationships, and privacy and security. Almost every area of life involves some degree of uncertainty and fuzziness. The best method for practically reducing ambiguity when determining the best option from fuzzy and uncertain data is multi-attribute decision-making (MADM). The research aims to develop a comprehensive decision framework for examining Internet effects on youth generations through fundamental behavioral and social elements. So, in this regard, the Intuitionistic Hesitant Fuzzy set (IHFS) is an effective and flexible tool for the investigation of uncertain information. The Sugeno-Weber operational (SWO) rules are a more generalized and flexible approach than other existing triangular norms (TN) and triangular conorm (TCN). Using SWOs and IHFS information, we construct the intuitionistic hesitant fuzzy Sugeno-Weber weighted averaging (IHFSWWA) and intuitionistic hesitant fuzzy Sugeno-Weber weighted geometric (IHFSWWG) operators. Also, we investigate some fundamental axioms of aggregation operators (AOs), like monotonicity, boundedness, and idempotency. We construct the MADM algorithm based on the developed theory, and some solve real-life numerical examples for assessment of factors affecting the new generations given the internet. To validate the effectiveness and practical applicability, a comparison analysis was conducted between the proposed IHFSWWA and IHFSWWG approaches with existing AOs. The conclusion is discussed in the last section.

This study explores an innovative application of the analytical hierarchy process (AHP) within the multi-criteria decision-making (MCDM) technique. It focuses on the factors prompting career satisfaction in sports participation. As the importance of physical education and traditional sports in promoting complete development continues to increase, the need for systematic strategies becomes paramount. The MCDM approach based on Pythagorean fuzzy set (PyFS) helps as an effective tool for assessing the best alternatives among several choices. AHP helps with pairwise comparisons and assesses the relative importance of each criterion, enabling decision-makers to identify the most suitable option. The PyFS model provides a more advanced version of the fuzzy set, allowing for the meaning of both the degree of membership (DoM) and the degree of non-membership (DoNM) within the interval of . In this, we proposed some new aggregation operators (AOs) of Sugeno-Weber t-norm (TNM) and t-conorm (TCNM) based on PyFS, which are Pythagorean fuzzy Sugeno-Weber weighted averaging (PyFSWWA) and Pythagorean fuzzy Sugeno-Weber weighted geometric (PyFSWWG) operators. Also, we proposed a model of AHP to calculate the weight vector for criteria. We present the MCDM algorithm for AHP and the derived AOs, offering solutions to practical numerical examples and identifying optimal sports options that improve career happiness. In the case study, take four alternatives: Football, Swimming, Tennis, and Golf, based on physical health, mental health, character development, and discipline. When we use our proposed AOs to select the best alternatives. Tennis is the best alternative. To highlight the significance of the established model, we perform a comparative analysis with existing methods, concluding with a discussion of the key findings.

Strategic school policy formulation is critical in guaranteeing the balanced growth of education systems that facilitate both academic and extracurricular objectives. Physical education makes a considerable contribution towards this vision through the promotion of lifelong healthy living, physical well-being, and general student development. This research proposes a new decision-making model using Circular Intuitionistic Fuzzy Bonferroni Mean (CIFBM) operators, which combine Aczel-Alsina triangular norms and Bonferroni aggregation rules to handle uncertainty in policy assessment. The essential properties of the proposed operators are investigated and proved. A multi-attribute group decision-making (MAGDM) approach is established to evaluate and rank strategic school policies to meet physical education goals. The methodology focuses on the importance of physical activity in promoting student achievement, cooperation, and educational viability. The case study illustrates the real-world application of the model in maximizing resource utilization, reconciling various interests of stakeholders, and enhancing student success. The findings verify the validity of the developed methodology in optimizing decision-making processes in policy development. This publication provides useful information and guidance for policymakers and educators who want to bring strategic planning into harmony with the development of school-based physical education programs. Abbreviations section is for the symbols, parameters, sets, and indices used throughout the paper.

This paper introduces and explores the concept of fuzzy ϕ-contraction to establish the existence of n-tupled coincidence points in partially ordered GV-fuzzy metric spaces. Using the unique properties of the H-type triangular norm, we integrate this norm into GV-fuzzy metric spaces. The mappings under consideration exhibit the mixed monotone property with respect to partial ordering. Furthermore, we employ the mixed g-monotone property of mappings to analyze their behavior within the given framework. To validate our theoretical findings, we present an illustrative example, which demonstrates the higher-dimensional analysis of the ϕ-contraction principle used in our primary results. As an application, we investigate the existence of solutions for a system of second-kind Fredholm nonlinear integral equations. Employing the developed fixed-point results, we establish sufficient conditions that guarantee the solvability of such integral equations within the GV-fuzzy metric space setting.

Intelligent decision analysis for green supplier selection with multiple attributes using circular intuitionistic fuzzy information aggregation and frank triangular norms

The development and use of urban land with sustainable planning is a challenging task. In the optimization of urban land usage plans, this article investigates a decision-making framework based on the Analytical Hierarchy Process (AHP) combined with an intuitionistic fuzzy set (IFS) for a multi-criteria decision-making (MCDM) algorithm. The model considers several factors, including infrastructural demands, social effects, economic viability, and environmental sustainability, to manage the complexities and uncertainties inherent in urban land development. The framework of IFS is a more generalized and superior format of a fuzzy set. It can express both aspects of information in the form of degree of membership (DoM) and degree of non-membership (DoNM) under the range of interval. To address this type of situation, we aimed to develop AHP and Sugeno-weber t-norm (TNM) and t-conorm (TCNM) based aggregation operators (AOs) called intuitionistic fuzzy Sugeno-weber weighted averaging (IFSWWA) operators. The MCDM algorithm for AHP and derived AOs is presented, including the solution of real-life numerical examples for selecting the best plane for urban land development and utilization. To highlight the significance of the proposed approach, we will compare it with existing methodologies. Then, we discussed some solid conclusions.

The main aim of this paper is to investigate relationship between ideals or filters of a bounded lattice and T-compatible or S-compatible congruences which are obtained by using the definition of compatible aggregation functions. Additionally, we introduce some congruences and characterize the conditions under which they belong to the sets of T-compatible or S-compatible congruences, denoted as Co n T L and Co n S L , respectively. We define an equivalence class denoted by ( a ) I A (A is an aggregation function and I is a nonempty set) and elaborate on the properties of this class when A is t-norm or t-conorm.

In this paper, we explore the application of fuzzy set theory in the context of triangular norms, with a focus on strong Sheffer stroke NMV-algebras. We introduce the concepts of T-fuzzy subalgebras and T-fuzzy filters, analyze their properties, and provide several illustrative examples. Our study demonstrates that T-fuzzy subalgebras and filters generalize classical subalgebras and filters, with level subsets preserving algebraic structures under t-norms. Notably, T-fuzzy sets exhibit strong closure properties, and homomorphisms between SSNMV-algebras extend naturally to fuzzy settings. Furthermore, we examine the relationships between T-fuzzy subalgebras (or filters) and their classical counterparts, as well as their corresponding level subsets and homomorphisms. These results contribute to refined uncertainty modeling in logical systems, with potential applications in fuzzy control and AI.

This work aims to improve intelligence decision-making using interval-valued picture fuzzy sets (IVPFS). In particular, it explores using Sugeno-Weber (SW) norms in IVPFS data recording, providing reliable estimates important for decision-making. This paper introduces a new class of aggregation operators such as the interval-valued picture fuzzy Sugeno-Weber power average (IVPFSWPA), interval-valued picture fuzzy Sugeno-Weber power geometric (IVPFSWPG), interval-valued picture fuzzy Sugeno-Weber power weighted average (IVPFSWPWA), and interval-valued picture fuzzy Sugeno-Weber power weighted geometric (IVPFSWPWG) operators. The real-life characteristics and specific situations of these operators are described as well as how they adapt to real-life situations. The new multi-attribute decision-making method suitable for many practical applications with different requirements or functions is proposed. An example of an intelligent selection process is given to demonstrate its effectiveness. In addition, a general comparative method is proposed to demonstrate the effectiveness and suitability of the collective strategy by comparing its results with existing methods. The study concludes by summarizing its findings and discussing its prospects, highlighting the potential contribution of the proposed studies to the advancement of cutting-edge technology in a dynamic and complex environment.

Q-fuzzy rings extend the concept of fuzzy rings, originally introduced by W. Liu. In Q-fuzzy rings, the membership values of elements are not confined to the interval [0, 1] but can span any value within the real interval [0, ∞]. This paper explores the fundamental ideas of fuzzy rings, Q-fuzzy subrings, Q-fuzzy soft subrings and an intuitionistic Q-fuzzy soft subrings offering a comprehensive examination of their properties and interrelationships. Generally, imprecise or ambiguous data is prevalent across many social and fundamental disciplines. Such uncertainty can stem from various factors, including insufficient information, limitations of measurement devices, data unpredictability and delayed updates to the data. Given the significance of these fields and the ever-increasing volume of uncertain data, there is a continuous need for effective and efficient techniques to model and manage these uncertainties. Various methods have been explored to address these challenges, including interval mathematics intuitionistic fuzzy set theory, rough set theory, vague set theory, probability theory and fuzzy soft set theory. All these methods have their drawbacks, especially when it comes to parameterization, even though they each have some benefits. To address these issues and offer a fresh perspective, this work uses intuitionistic fuzzy soft theory to examine the mathematical framework of rings. Overall, intuitionistic Q-fuzzy soft rings hold promises for representing and reasoning about uncertainties within ring theory. As research progresses, it can lead to advancements in uncertainty handling, new mathematical discoveries and potential applications in various domains.

Encoding Residuation Principle and Representation Theorem Under Pythagorean Fuzzy Using Triangular Norm and Conorm