Geometric Operator Research Articles

Schweizer–Sklar Power Aggregation Operators Based on Complex Interval-Valued Intuitionistic Fuzzy Information for Multi-attribute Decision-Making

This manuscript proposes the concept of Schweizer–Sklar operational laws under the consideration of the complex interval-valued intuitionistic fuzzy (CIVIF) set theory, where the Schweizer–Sklar norms are the essential and valuable modification of many norms, such as algebraic, Hamacher, and Lukasiewicz norms. Moreover, keeping the dominancy of the presented laws, we derive the concept of CIVIF Schweizer–Sklar power averaging (CIVIFSSPA), CIVIF Schweizer–Sklar power ordered averaging (CIVIFSSPOA), CIVIF Schweizer–Sklar power geometric (CIVIFSSPG), and CIVIF Schweizer–Sklar power ordered geometric (CIVIFSSPOG) operators, which are the combination of the three different structures for evaluating three different problems. Further, some reliable and feasible properties and results for derived work are also invented. Additionally, we also illustrate an application, called multi-attribute decision-making (MADM) scenario for evaluating some real-world problems with the help of discovered operators for showing the reliability and stability of the evaluated operators. Finally, we compare our mentioned operators with various prevailing operators for enhancing the worth and stability of the evaluated approaches.

Prioritized aggregation operators based on Schweizer-Sklar t-norm for linear Diophantine fuzzy sets and their application in green sustainable chain

Schweizer and Sklar invented the idea of the Schweizer-Sklar t-norm and Schweizer-Sklar t-conorm in 1960. Additionally, prioritized aggregation operators were derived by Yager in 2008. Further, linear Diophantine fuzzy (LDF) information is a most recent generalization of q-rung orthopair, Pythagorean, and intuitionistic fuzzy sets. In this article, we invent the Schweizer-Sklar operational laws for LDF set based on the Schweizer-Sklar t-norm and t-conorm. Furthermore, we expose the LDF Schweizer-Sklar prioritized averaging (LDFSSPoA) operator, LDF Schweizer-Sklar weighted prioritized averaging (LDFSSWPoA) operator, LDF Schweizer-Sklar prioritized geometric (LDFSSPoG) operator, LDF Schweizer-Sklar weighted prioritized geometric (LDFSSWPoG) operator and evaluate their fundamental properties and results. Moreover, after evaluation of these operators, we evaluate the problem of the green sustainable chain to enhance the worth of the invented theory. Finally, we show the supremacy and accuracy of the proposed theory with the help of a comparative analysis between the proposed techniques with some existing techniques.

An Integrated Group Decision-Making Framework for the Evaluation of Artificial Intelligence Cloud Platforms Based on Fractional Fuzzy Sets

Due to the rapid development of machine learning and artificial intelligence (AI), the analysis of AI cloud platforms is now a key area of research. Assessing the wide range of frameworks available and choosing the ideal AI cloud providers that may accommodate the demands and resources of a company is mandatory. There are several options, all having their own benefits and limitations. The evaluation of artificial intelligence cloud platforms is a multiple criteria group decision-making (MCGDM) process. This article establishes a collection of Einstein geometric aggregation operators (AoPs) and a novel Fractional Fuzzy VIKOR and Fractional Fuzzy Extended TOPSIS based on the entropy weight of criteria in fractional fuzzy sets (FFSs) for this scenario. The FFSs provide an evaluation circumstance containing more information, which makes the final decision-making results more accurate. Finally, this framework is then implemented in a computational case study for the evaluation of artificial intelligence cloud platforms and comparison of this model with other existing approaches, such as the extended GRA approach, to check the consistency and accuracy of the proposed technique. The most optimal artificial intelligence cloud platform is I1

A robust algorithmic framework for the evaluation of international cricket batters in ODI format based on q-rung linguistic neutrosophic quantification

This article presents a novel approach for decision-making problems in which the criteria and alternatives are evaluated under a q-rung linguistic neutrosophic set (QRLNS) environment. QRLN sets are an extension of the traditional linguistic variables, which allow more flexibility and accuracy in modeling complex decision-making situations. We introduce several QRLN weighted aggregation operators, including QRLN weighted averaging operator, QRLN weighted geometric operator, and QRLN weighted hybrid operator, which can be used to aggregate the QRLN information provided by decision-makers. The properties and characteristics of these operators are analyzed, and their performance is compared with other existing aggregation operators. Finally, this system is able to handle the uncertainty and imprecision in the data and provide a more reliable assessment of the performance of cricket players. Our study demonstrates the potential of QRLN-based approaches for ranking assessment in other fields and provides insights for future research in this area.

Weyl conformal geometry vs Weyl anomaly

Weyl conformal geometry is a gauge theory of scale invariance that naturally brings together the Standard Model (SM) and Einstein gravity. The SM embedding in this geometry is possible without new degrees of freedom beyond SM and Weyl geometry, while Einstein gravity is generated by the broken phase of this symmetry. This follows a Stueckelberg breaking mechanism in which the Weyl gauge boson becomes massive and decouples, as discussed in the past [1–3]. However, Weyl anomaly could break explicitly this gauge symmetry, hence we study it in Weyl geometry. We first note that in Weyl geometry metricity can be restored with respect to a new differential operator ( hat{nabla} ) that also enforces simultaneously a Weyl-covariant formulation. This leads to a metric-like Weyl gauge invariant formalism that enables one to do quantum calculations directly in Weyl geometry, rather than use a Riemannian (metric) geometry picture. The result is the Weyl-covariance in d dimensions of all geometric operators ( hat{R} , etc) and of their derivatives ( hat{nabla} μ hat{R} , etc), including the Euler-Gauss-Bonnet term. A natural (geometric) Weyl-invariant dimensional regularisation of quantum corrections exists and Weyl gauge symmetry is then maintained and manifest at the quantum level. This is related to a non-trivial current of this symmetry, the divergence of which cancels the trace of the energy-momentum tensor. The “usual” Weyl anomaly and Riemannian geometry are recovered in the (spontaneously) broken phase. The relation to holographic Weyl anomaly is discussed.

Probabilistic Interval-Valued Fermatean Hesitant Fuzzy Set and Its Application to Multi-Attribute Decision Making

It is difficult to describe the hesitation and uncertainty of experts by single-valued information, and the differences in the importance of attributes are often ignored during the decision-making process. This paper introduces the probability and interval values into Fermatean hesitant fuzzy set (FHFS) and creatively proposes the probabilistic interval-valued Fermatean hesitant fuzzy set (PIVFHFS) to deal with information loss. This new fuzzy set allows decision makers to use interval-valued information with probability to express their quantitative evaluation, which broadens the range of information expression, effectively reflects the important degree of different membership degrees, and can describe uncertain information more completely and accurately. Under the probabilistic interval-valued Fermatean hesitant fuzzy environment, several new aggregation operators based on Hamacher operation are proposed, including the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher weighted averaging (PIVFHFHWA) operator and geometric (PIVFHFHWG) operator, and their basic properties and particular forms are studied. Then, considering the general correlation between different attributes, this paper defines the probabilistic interval-valued Fermatean hesitant fuzzy Hamacher Choquet integral averaging (PIVFHFHCIA) operator and geometric (PIVFHFHCIG) operator and discusses related properties. Finally, a multi-attribute decision-making (MADM) method is presented and applied to the decision-making problem of reducing carbon emissions of manufacturers in the supply chain. The stability and feasibility of this method are demonstrated by sensitivity analysis and comparative analysis. The proposed new operators can not only consider the correlation between various factors but also express the preference information of decision makers more effectively by using probability, thus avoiding information loss in decision-making progress to some extent.

Multi-attribute decision-making approach based on Aczel-Alsina power aggregation operators under bipolar fuzzy information & its application to quantum computing

The main aim of this paper is to present a new multi-attribute decision-making (MADM) approach for solving the problems under the uncertain and complex environment. The key challenges during any MADM problem are how to quantify the objective uncertainty information in the data and how to aggregate such collective information. To answer this, in this paper, we utilize the concept of the bipolar fuzzy information to mark the information in terms of the positive and negative support. To aggregate this different information, we propose some power aggregation operators based on the Aczel-Alsina operators (AAO). The AAOs are the generalized t-norm based operations with an additional parameter to analyze the influence of the expert preferences. Based on these AAO and bipolar fuzzy information, we stated bipolar fuzzy AA power weighted averaging and geometric operators and investigate their features. Later, based on these operators, we establish a MADM algorithm to solve the decision-making problems. The applicability of the stated algorithm is demonstrated through a case study related to quantum computing. The comparative studies and advantages of the study are also analyzed with the various prevailing theories.

Retracted: Some Novel Geometric Aggregation Operators of Spherical Cubic Fuzzy Information with Application

Retracted: Some Novel Geometric Aggregation Operators of Spherical Cubic Fuzzy Information with Application

Intuitionistic uncertain linguistic prioritized aggregation operators and their application to multi-criteria decision making

The intuitionistic uncertain linguistic information aggregation problems considering different priority levels of criteria are investigated. Firstly, we extended the prioritized averaging (PA) operator to intuitionistic uncertain linguistic environment, defined two new prioritized aggregation operators called the intuitionistic uncertain linguistic prioritized weighted average (IULPWA) operator and the intuitionistic uncertain linguistic prioritized weighted geometric (IULPWG) operator, and established various desirable properties of the proposed operators. Secondly, we developed a multi-criteria decision making (MCDM) approach based on the IULPWA operator (or the IULPWG operator) to deal with the MCDM problems in which the criterion values take the form of intuitionistic uncertain linguistic numbers (IULNs) and the criteria are in different priority levels. Finally, an example is given to illustrate the feasibility and effectiveness of the proposed method, and a comparison analysis is conducted to make clear the differences among the IULPWA operator, the IULPWG operator, the intuitionistic uncertain linguistic number weighted averaging (IULNWA) operator and the intuitionistic uncertain linguistic weighted geometric average (IULWGA) operator.

MCDM technique using single-valued neutrosophic trigonometric weighted aggregation operators

Motivated based on the trigonometric t-norm and t-conorm, the aims of this article are to present the trigonometric t-norm and t-conorm operational laws of SvNNs and then to propose the SvNN trigonometric weighted average and geometric aggregation operators for the modelling of a multiple criteria decision making (MCDM) technique in an inconsistent and indeterminate circumstance. To realize the aims, this paper first proposes the trigonometric t-norm and t-conorm operational laws of SvNNs, which contain the hybrid operations of the tangent and arctangent functions and the cotangent and inverse cotangent functions, and presents the SvNN trigonometric weighted average and geometric operators and their properties. Next, a MCDM technique is proposed in view of the presented two aggregation operators in the circumstance of SvNNs. In the end, an actual case of the choice issue of slope treatment schemes is provided to indicate the practicability and effectivity of the proposed MCDM technique.

Innovative discussion of decision-making model based on complex cubic picture fuzzy information and geometric aggregation operators with applications

This article presents a novel concept of complex cubic picture fuzzy sets (CCPFS) and introduces one more new idea of complex interval-valued picture fuzzy sets (CIVPFS) as foundational framework of CCPFS. The proposed CCPFS combines CIVPFS and complex picture fuzzy sets (CPFS), where CPFS extends the complex intuitionistic fuzzy set by incorporating a neutral membership degree. This unique model offers an expanded range of values using degrees of membership, neutral membership, and non-membership, within the unit disk of a complex plane. Additionally, we introduce two more new ideas of internal complex cubic picture fuzzy sets (ICCPFS) and external complex cubic picture fuzzy sets (ECCPFS) to further enhance the versatility of the approach. To facilitate practical applications, complement, score, and accuracy functions are developed and defined for CCPFS. Moreover, three types of averaging aggregation operators based on complex cubic picture fuzzy sets are introduced, including complex cubic picture fuzzy weighted geometric (CCPFWG) operators, complex cubic picture fuzzy ordered weighted geometric (CCPFOWG) operator, and complex cubic picture fuzzy hybrid weighted geometric (CCPFHWG) operator. The CCPFHWG operator generalizes both CCPFWG and CCPFOWG operators, providing a comprehensive framework for aggregating complex cubic picture fuzzy data. To demonstrate the practicality of the proposed approach, a multi-criteria decision-making (MCDM) problem is presented, showcasing its effectiveness in addressing today's complex decision structures. The utilization of complex cubic picture fuzzy sets and the corresponding aggregation operators in MCDM highlights their applicability and relevance in tackling real-world complexities.

Applications Aczel-Alsina t-norm and t-conorm for the assessment of fire extinguishers using Pythagorean fuzzy information

Prioritization is usually required in problems involving multi-attribute group decision-making (MAGDM). Several strategies and procedures have been introduced in fuzzy systems to apply prioritization. This study examines the MAGDM problem in a Pythagorean fuzzy (PF) setting with varying amounts of demand for specialists and attributes. We regard the novel Aczel Alsina aggregation operators (AOs) as the most addition to fuzzy mathematics that can deal with uncertainties significantly. We suggest a few PF AOs based on Aczel Alsina t-norm and t-conorm, including the PF-prioritized Aczel Alsina averaging (PFPAAA) and PF-prioritized Aczel Alsina geometric (PFPAAG) operators. It is proven that these AOs fulfil the aggregation criteria by investigating the properties of monotonicity, boundedness, and idempotency. The weights for prioritization are derived from the knowledge of experts, and the proposed operators can capture the phenomenon of prioritization among the aggregated arguments. The proposed AOs are then applied to assess fire extinguishers using a MAGDM technique. The importance of PFPAAA and PFPAAG operators is verified by comparing the proposed AOs with other well-known AOs.

Assessment actions for improving railway sector performance using intuitionistic fuzzy-rough multi-criteria decision-making model

The selection and prioritization of appropriate actions to improve the performance of the railway sector pose a common challenge for policymakers and transport planners. This challenge is further complicated by factors such as political power shifts, policy changes, and socio-economic and environmental impacts, introducing various uncertainties. To address this issue, a novel multi-criteria decision-making framework was developed in this study. The framework introduces the weighted geometric Dombi Maclaurin Symmetric Mean operator within an intuitionistic fuzzy environment, enabling the identification of the most appropriate action from nine different alternatives to enhance the performance of railway sector. Moreover, the study makes an additional contribution by combining this approach with an interval rough PIvot Pairwise RElative Criteria Importance Assessment to determine the weight coefficients of the criteria. To assess the robustness of the proposed methodology, sensitivity analysis and a comparative study were conducted. The findings of the study indicate that “enhancing the governance of the transport sector” emerged as the most appropriate action to improve the performance of the railway sector.

Retracted: Multiple-Attribute Decision-Making Method Based on Normalized Geometric Aggregation Operators of Single-Valued Neutrosophic Hesitant Fuzzy Information

Retracted: Multiple-Attribute Decision-Making Method Based on Normalized Geometric Aggregation Operators of Single-Valued Neutrosophic Hesitant Fuzzy Information

Fermatean Fuzzy Aggregation Operators with Priority Degrees and their Applications

Fermatean fuzzy numbers (FrFNs) have demonstrated significant utility in real-world scenarios for handling uncertain data. In this study, we focus on multi-criteria decision-making (MCDM) problems with prioritized parameters. To address this, we introduce the notion of "priority levels." By assigning non-negative real numbers, known as "priority degrees," to these priority levels, we establish aggregation operators (AOs). Our work puts forth a diverse set of prioritized operators, notably the Fermatean fuzzy prioritized averaging (FrFPAd) operator with priority degrees and the Fermatean fuzzy prioritized geometric (FrFPGd) operator with priority degrees. Through systematic comparisons, we highlight the superiority of our proposed methodology over other contemporary approaches already in use. We place particular emphasis on thoroughly investigating the influence of priority degrees on the overall decision-making outcomes. This analysis yields valuable insights into the implications and benefits of incorporating prioritization in MCDM. Furthermore, we provide a decision-making strategy based on the aforementioned operators, within the Fermatean fuzzy set environment. This strategy offers a practical framework for effective decision-making when faced with uncertainty.

A Novel Interval-Valued Decision Theoretic Rough Set Model with Intuitionistic Fuzzy Numbers Based on Power Aggregation Operators and Their Application in Medical Diagnosis

Intuitionistic fuzzy information is a potent tool for medical diagnosis applications as it can represent imprecise and uncertain data. However, making decisions based on this information can be challenging due to its inherent ambiguity. To overcome this, power aggregation operators can effectively combine various sources of information, including expert opinions and patient data, to arrive at a more accurate diagnosis. The timely and accurate diagnosis of medical conditions is crucial for determining the appropriate treatment plans and improving patient outcomes. In this paper, we developed a novel approach for the three-way decision model by utilizing decision-theoretic rough sets and power aggregation operators. The decision-theoretic rough set approach is essential in medical diagnosis as it can manage vague and uncertain data. The redesign of the model using interval-valued classes for intuitionistic fuzzy information further improved the accuracy of the diagnoses. The intuitionistic fuzzy power weighted average (IFPWA) and intuitionistic fuzzy power weighted geometric (IFPWG) aggregation operators are used to aggregate the attribute values of the information system. The established operators are used to combine information within the intuitionistic fuzzy information system. The outcomes of various alternatives are then transformed into interval-valued classes through discretization. Bayesian decision rules, incorporating expected loss factors, are subsequently generated based on this foundation. This approach helps in effectively combining various sources of information to arrive at more accurate diagnoses. The proposed approach is validated through a medical case study where the participants are classified into three different regions based on their symptoms. In conclusion, the decision-theoretic rough set approach, along with power aggregation operators, can effectively manage vague and uncertain information in medical diagnosis applications. The proposed approach can lead to timely and accurate diagnoses, thereby improving patient outcomes.

Algorithms for decision-making process using complex Pythagorean fuzzy set and its application to hospital siting for COVID-19 patients

Since the start of COVID-19, a fair amount of work has been undertaken by scholars around the world to model its progression. It became clear from the start of pandemic that its progression is affected by various factors within different communities. Subsequently, the necessary means and the range of measures used to effectively control the virus would vary from place to place. And we have been witness to different approaches adopted around the world to maintain the virus under check both in the short term and the long term. So, in this unexpected situation, it is a great challenge for the world health organization (WHO) to save the lives of COVID-19 patients. For this, several mathematical models have been made for better understanding the coronavirus contagion. Mostly, these models are based on classical integer-order derivative using real numbers which cannot capture the fading memory. Thus, in this unexpected situation, fuzzy sets (FSs) are considered due to their inherent capability to deal with uncertainty. Fuzzy sets (FSs) theory has the ability to manage uncertain situations. Thus, the goal of this research is to present newly mathematical methods based on complex Pythagorean fuzzy sets (CPyFSs) and their operators, namely complex Pythagorean fuzzy Einstein weighted geometric operator, and induced complex Pythagorean fuzzy Einstein hybrid geometric operator to reduce the spreading rate of COVID-19. At the end of the paper an illustrative example is constructed to show the effectiveness, reliability of the new techniques.

Elevating Decision Management in Sustainable Energy Planning through Spherical Fuzzy Aggregation Operators

This article introduces a novel paradigm for enhancing the administration of decisions regarding sustainable energy planning. This is achieved by deploying novel spherical fuzzy aggregation operators that have been meticulously tailored to address the inherent complexities of uncertainty and imprecision prevalent in energy planning datasets. These operators vastly increase the precision and efficacy of decision-making processes, thereby transforming the entire sustainable energy landscape. This study focuses predominantly on the complex domain of multi-attribute decision-making (MADM), in which the interplay of parameters is characterized by a discernible hierarchy of importance. This method generates aggregation operators based on the assignment of non-negative real values to clearly defined priority echelons, a framework known as priority degrees. This effort results in the development of two notable prioritized operators: the “spherical fuzzy prioritized averaging operator with priority degrees” and the “spherical fuzzy prioritized geometric operator with priority degrees”. The efficacy of these conceptual frameworks is vividly demonstrated through the application of extensive case studies, in which observable results clearly demonstrate their superiority over conventional methodologies. The empirical findings unequivocally demonstrate the superiority of the proposed operators, resonating with substantial performance and efficiency improvements. This study not only adds a seminal dimension to the field of sustainable energy management but also reveals a revolutionary application of spherical fuzzy aggregation operators at the forefront of effective decision-making paradigms. The seamless fusion of theoretical innovation and practical utility outlines a path forward, with transformative prospects and far-reaching implications for the sustainable energy landscape.

A two-phase multi-criteria decision-making with interval-valued Pythagorean normal fuzzy information and its application for module hospital site selection in combating COVID-19

During the fight against COVID-19, module hospital saved tens of thousands of lives and families. Therefore, how to effectively select the site of module hospital is a critical issue. The site selection of module hospital is a multi-criteria decision-making (MCDM) problem. Recently, interval-valued Pythagorean normal fuzzy (IPNF) number has become an effective and powerful tool for handling ambiguity in decision-making. However, the weights of experts in existing IPNF decision-making are often directly given, lacking objectivity and persuasiveness. Moreover, the current research on aggregation operators in IPNFN only involves simple weighted average and geometry, while ignoring the interaction among fuzzy information. Motivated by these issues, this paper aims to develop a two-phase IPNF MCDM approach and apply it to the site selection of module hospital. Firstly, IPNF Einstein weighted averaging (IPNFEWA) operator, IPNF Einstein order weighted averaging (IPNFEOWA) operator, IPNF Einstein weighted geometric (IPNFEWG) operator, and IPNF Einstein order weighted geometric (IPNFEOWG) operator are proposed. Moreover, numerical and theoretical proofs of the proposed operators are carried out. Secondly, an algorithm for determining experts’ weights is developed using the proposed IPNF social network in the first phase, and the decision-making process is conducted in the second phase. Finally, a numerical example on the site selection of module hospital is provided to demonstrate the feasibility of the proposed method. The numerical results of comparisons and sensitivity analysis are further analyzed to confirm the advantages of the proposed method.

Evaluation of sustainable strategies for urban parcel delivery: Linguistic q-rung orthopair fuzzy Choquet integral approach

For the first time, the critical problem of selecting the most sustainable strategy for urban parcel delivery is addressed and solved in this paper. It aims to help not only postal and logistics companies but also regulatory agencies and city authorities find an environmentally friendly, socially responsible, and economically viable solution. Three alternatives for organizing urban parcel delivery are defined, which are traditional, united consolidation center, and inner-city hubs. According to the three pillars of sustainability, an assessment framework for practitioners is proposed. This paper introduces a linguistic q-rung orthopair fuzzy Choquet integral approach to evaluate sustainable strategies for urban parcel delivery. The linguistic q-rung orthopair fuzzy Choquet integral averaging and linguistic q-rung orthopair fuzzy Choquet integral geometric operators have been constructed for the first time for this purpose. The operator-related qualities are demonstrated. The operators are used to create a multi-criteria group decision-making approach. The case study of the city of Belgrade illustrates the practical applicability of the linguistic q-rung orthopair fuzzy Choquet integral approach. This research findings show that the united consolidation center is the most sustainable strategy for organizing urban parcel delivery.