Geometric Operator Research Articles

Intuitionistic fuzzy social network hybrid MCDM model for an assessment of digital reforms of manufacturing industry in China

• A new intuitionistic fuzzy aggregation method is proposed, called the IF hybrid average and geometric operator. • A multi-dimensional trust propagation model is built to reflect decision makers’ social relationships. • A social network MCDM model is proposed under an IFS environment. • The MCDM model is used to evaluate the effects of digital reform in the communication equipment manufacturing enterprises in China. • The influence of the parameters in the proposed method on the result is considered to ensure the robustness. Digital reform requires enterprises to use digital technology to create a deep integration between their production, management, and operational processes, and generate a data chain for the entire process, thereby meeting the personalized requirements and expectations of customers. The achievements of digital reform in manufacturing enterprises need to be evaluated scientifically, which can help the enterprises adjust their development strategies for a digital reform in a timely manner. We therefore propose a multi-criteria model based on a social network for assessing a digital reform under an intuitionistic fuzzy environment, wherein an intuitionistic fuzzy hybrid average and geometric operator is proposed to aggregate evaluation information more effectively than with existing methods. In addition, because the trust relationships between experts can affect their decisions, a social network is introduced to determine the weights assigned to these experts. Finally, a case study of four manufacturing enterprises is presented to verify the effectiveness of the proposed method.

A novel binary classification approach based on geometric semantic genetic programming

Geometric semantic genetic programming (GSGP) is a recent variant of genetic programming. GSGP allows the landscape of any supervised regression problem to be transformed into a unimodal error surface, thus it has been applied only to this kind of problem. In a previous paper, we presented a novel variant of GSGP for binary classification problems that, taking inspiration from perceptron neural networks, uses a logistic-based activation function to constrain the output value of a GSGP tree in the interval [0,1]. This simple approach allowed us to use the standard RMSE function to evaluate the train classification error on binary classification problems and, consequently, to preserve the intrinsic properties of the geometric semantic operators. The results encouraged us to investigate this approach further. To this aim, in this paper, we present the results from 18 test problems, which we compared with those achieved by eleven well-known and widely classification schemes. We also studied how the parameter settings affect the classification performance and the use of the F-score function to deal with imbalanced data. The results confirmed the effectiveness of the proposed approach.

Fermatean fuzzy soft aggregation operators and their application in symptomatic treatment of COVID-19 (case study of patients identification).

The main focus of this paper is the application of aggregation operators (AOs) in the environment of Fermatean fuzzy soft sets (FFSS). The unique feature of the work is its application in the symptomatic treatment of the COVID-19 disease. For this purpose, the idea of FFSS is introduced which is based on the Senapati and Yagar’s Fermatean fuzzy set. Next we have defined Fermatean fuzzy soft aggregation operators (FFSAOs) like, Fermatean fuzzy soft weighted averaging (FFSWA) operator, Fermatean fuzzy soft ordered weighted averaging (FFSOWA) operator, Fermatean fuzzy soft weighted geometric (FFSWG) operator and Fermatean fuzzy soft ordered weighted geometric (FFSOWG). The prominent properties of these operators are given in details. We have also developed some approaches to solve multi-criteria decision making (MCDM) problems in Fermatean fuzzy soft (FFS) information. An introduction to the novel pandemic, safety measures, and then its possible symptomatic treatment is also provided. The developed operators are utilized in the symptomatic treatment of COVID-19 disease in order to show the practical applications and importance of these AOs as well as Fermatean fuzzy soft information. The stability of the proposed work is also proved by the comparative analysis.Graphical abstract

Novel Aczel–Alsina operations-based hesitant fuzzy aggregation operators and their applications in cyclone disaster assessment

As a fuzzy set (FS) expansion, the hesitant fuzzy set (HFS) is successfully employed to demonstrate circumstances where it is admissible to ascertain a few potential membership degrees (MDs) of a component in a set because of the uncertainty between various values. Considering that there is still no research on Aczel–Alsina triangular norms and conorms in a hesitant fuzzy (HF) environment, in this article we introduce for the first time, Aczel–Alsina operations on HFSs. Then, based on these operations, we originate a few new aggregation operators for aggregating HF information namely HF Aczel–Alsina weighted averaging (HFAAWA) operator, HF Aczel–Alsina ordered weighted averaging (HFAAOWA) operator, HF Aczel–Alsina hybrid averaging (HFAAHA) operator, HF Aczel–Alsina weighted geometric (HFAAWG) operator, HF Aczel–Alsina ordered weighted geometric (HFAAOWG) operator, HF Aczel–Alsina hybrid geometric (HFAAHG) operator and HF Aczel–Alsina weighted Bonferroni mean (HFAAWBM) operator. Some essential characteristics of those suggested operators are shown, and the interrelatedness between them is displayed exhaustively. Then, we take advantage of those operators to produce a methodology to interpret the HF multiple attribute decision making (MADM) difficulties. We present a functional model for cyclone disaster assessment to certify the produced approaches and to establish their effectiveness and practicality. Further, we conduct comparison analysis for the legitimacy of our produced methodologies.

Pythagorean fuzzy prioritized aggregation operators with priority degrees for multi-criteria decision-making

PurposeThe authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined. Further, based on these operators, a decision-making approach is presented under the Pythagorean fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.Design/methodology/approachIn real-world situations, Pythagorean fuzzy numbers are exceptionally useful for representing ambiguous data. The authors look at multi-criteria decision-making issues in which the parameters have a prioritization relationship. The idea of a priority degree is introduced. The aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees.FindingsThe authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined. Further, based on these operators, a decision-making approach is presented under the Pythagorean fuzzy set environment. An illustrative example related to the selection of the best alternative is considered to demonstrate the efficiency of the proposed approach.Originality/valueThe aggregation operators are formed by awarding non-negative real numbers known as priority degrees among strict priority levels. Consequently, the authors develop some prioritized operators named Pythagorean fuzzy prioritized averaging operator with priority degrees and Pythagorean fuzzy prioritized geometric operator with priority degrees. The properties of the existing method are routinely compared to those of other current approaches, emphasizing the superiority of the presented work over currently used methods. Furthermore, the impact of priority degrees on the aggregate outcome is thoroughly examined.

A study on weighted aggregation operators for q‐ rung orthopair m‐ polar fuzzy set with utility to multistage decision analysis

Modeling uncertainties with multipolar information is an important tool in computational intelligence to address complexities in real-world circumstances. An m-polar fuzzy set (mPFS) is the strong model to express multipolarity with m $m$ membership grades (MGs) in the unit closed interval [ 0 , 1 ] $[0,1]$ . A q-rung orthopair fuzzy set (qROFS) is the strong model to express vague and uncertain information with MGs and nonmembership grades (NMGs). The notion of q-rung orthopair m-polar fuzzy set is a new hybrid extension of both mPFS and qROFS. An ROmPFS is a generalized concept that has the ability to deal with multipolarity with m $m$ ordered pairs of MGs and NMGs. Motivated by these robust concepts, in this article, various aggregation operators (AOs) for the aggregation of q-rung orthopair m-polar fuzzy numbers are proposed, including q-rung orthopair m-polar fuzzy weighted averaging operator, symmetric q-rung orthopair m-polar fuzzy weighted averaging operator, q-rung orthopair m-polar fuzzy weighted geometric operator, symmetric q-rung orthopair m-polar fuzzy weighted geometric operator, and q $q$ -rung orthopair m-polar fuzzy Maclaurin symmetric mean operator. On the basis of proposed AOs, a robust multicriteria decision-making approach is proposed. An application of proposed AOs is presented to address economic crises during COVID-19. Furthermore, the comparison analysis is designed to discuss the validity and rationality of proposed AOs.

Some new logarithmic aggregation operators and their application to group decision making problem based on t-norm and t-conorm

In this paper, a logarithmic operational law for intuitionistic fuzzy numbers is defined, in which the based \(\ell\) is a real number such that \(\ell \in \left( {0,1} \right)\) with condition \(\ell \ne 1\). Some properties of logarithmic operational laws have been studied and based on these, several Einstein averaging and Einstein geometric operators, namely logarithmic intuitionistic fuzzy Einstein weighted averaging (LIFEWA) operator, logarithmic intuitionistic fuzzy Einstein ordered weighted averaging (LIFEOWA) operator, logarithmic intuitionistic fuzzy Einstein hybrid averaging (LIFEHA) operator, logarithmic intuitionistic fuzzy Einstein weighted geometric (LIFEWG) operator, logarithmic intuitionistic fuzzy Einstein ordered weighted geometric (LIFEOWG) operator, and logarithmic intuitionistic fuzzy Einstein hybrid geometric (LIFEHG) operator have been introduced, which can overcome the weaknesses of algebraic operators. Furthermore, based on the proposed operators a multi-attribute group decision-making problem is established under logarithmic operational laws. Finally, an illustrative example is used to illustrate the applicability and validity of the proposed approach and compare the results with the existing methods to show the effectiveness of it.

Performance Measure Using a Multi-Attribute Decision-Making Approach Based on Complex T-Spherical Fuzzy Power Aggregation Operators

Performance measurement has a vital role in almost every field of life especially when uncertainty is involved in processing information. The purpose of this research is to use the concept of power aggregation operators (PAOs) of complex T-spherical fuzzy sets (CTSFSs) to analyze the performance measures of software packages. In comparison to other studies, the main advantage of the PAOs of CTSFSs is its ability to feature four possible aspects of the information under uncertainty. In CTSFSs, each component has further two aspects denoted by their amplitude and phase terms and hence provides us a better ground to deal with practical problems. Another advantage of using PAOs for performance evaluation is the involvement of the relationship of the input arguments that play an essential role in aggregation. Other traditional aggregation operators (AOs) do not have such capabilities. We aim to develop complex T-spherical fuzzy (TSF) power weighted averaging and complex TSF power weighted geometric (CTSFPWG) operators and evaluated their validity using some tests. Ultimately, using the recommended operators, multi-attribute decision-making (MADM) algorithm is established for the performance evaluation of the software packages. With the help of a numerical example, we demonstrated the proposed MADM algorithm using complex uncertain information. The results show a positive impact by analyzing them after a comparative analysis where the results obtained using complex TSF PAOs seem to be more reliable than the results obtained using previously developed AOs. To see the effectiveness of the algorithm, the results are numerically compared using some existing approaches, and conclusions are drawn.

Multicriteria q-Rung orthopair fuzzy decision analysis: a novel approach based on Archimedean aggregation operators with the confidence levels

The confidence levels can reduce the influence of the unreasonable evaluation value that was given by the decision-maker on the decision-making results. The Archimedean t-norm and t-conorm (ATS) also have many advantages for the processing of uncertain data. Under this environment, the confidence q-rung orthopair fuzzy aggregation operator based on ATS is one of the most successful extensions of confidence q-rung orthopair fuzzy numbers in which we decrease the deviation caused by the subjective perspective of the decision-maker in the multicriteria group decision-making problems. In this paper, we propose weighted, ordered weighted averaging aggregation operators and weighted, ordered weighted geometric aggregation operators based on ATS, respectively. Moreover, the properties and four specific forms associated with aggregation operators are also investigated. In this study, a novel MCGDM approach is introduced by using the proposed operator. A reasonable example is proposed and compared the results that are obtained by our operators and that in the existing literature, so as to verify the rationality and flexible of our method. From the study, we concluded that the proposed method can reduce the impact of extreme data, and make decision-making results more reasonable by considering the attitudes of decision-makers.

A Decision-Making Framework Using q-Rung Orthopair Probabilistic Hesitant Fuzzy Rough Aggregation Information for the Drug Selection to Treat COVID-19

In our current era, a new rapidly spreading pandemic disease called coronavirus disease (COVID-19), caused by a virus identified as a novel coronavirus (SARS-CoV-2), is becoming a crucial threat for the whole world. Currently, the number of patients infected by the virus is expanding exponentially, but there is no commercially available COVID-19 medication for this pandemic. However, numerous antiviral drugs are utilized for the treatment of the COVID-19 disease. Identification of the appropriate antivirus medicine to treat the infection of COVID-19 is still a complicated and uncertain decision. This study’s key objective is to develop a novel approach called q-rung orthopair probabilistic hesitant fuzzy rough set (q-ROPHFRS), which incorporates the q-rung orthopair fuzzy set, probabilistic hesitant fuzzy set, and rough set structures. New q-ROPHFR aggregation operators have been established: the q-ROPHFR Einstein weighted averaging (q-ROPHFREWA) operator and the q-ROPHFR Einstein weighted geometric (q-ROPHFREWG) operator. In this study, we explored some basic features of the developed operators. Afterward, to demonstrate the viability and feasibility of the established decision-making approach in real-world applications, a case study related to selecting drugs for COVID-19 pandemic is addressed. Furthermore, a comprehensive comparison with the q-rung orthopair probabilistic hesitant fuzzy rough TOPSIS technique is also presented to illustrate the benefits of the new framework. The obtained results confirm the reliability and effectiveness of the proposed approach for finding uncertainty in real-world decision-making.

On the evolution and monotonicity of First eigenvalues under the Ricci flow

In this paper, we derive evolution equations for the first eigenvalue of geometric operators $-\Delta+cR^a$ under the Ricci flow and the normalized Ricci flow, respectively. As applications, we obtain some monotonicity results.

Novel Decision-Making Techniques in Tripolar Fuzzy Environment with Application: A Case Study of ERP Systems.

The intuitionistic fuzzy set (IFS) and bipolar fuzzy set (BFS) are all effective models to describe ambiguous and incomplete cognitive knowledge with membership, non-membership, negative membership, and hesitancy sections. But in daily life problems, there are some situations where we cannot apply the ordinary models of IFS and BFS, separately. Hence, there is a need to combine both the models of IFS and BFS into a single one. A tripolar fuzzy set (TFS) is a generalization of IFS and BFS. In circumstances where BFS and IFS models cannot be used individually, a tripolar fuzzy model is more dependable and efficient. Further, the IFS and BFS models are reduced to corollaries due to the proposed model of TFS. For this purpose in this article, we first consider some novel operations on tripolar fuzzy information. These operations are formulated on the basis of well-known Dombi T-norm and T-conorm, and the desirable properties are discussed. By applying the Dombi operations, arithmetic and geometric aggregation operators of TFS are proposed, and we introduce the concepts of a TF-Dombi weighted average (TFDWA) operator, a TF-Dombi ordered weighted average (TFDOWA) operator, and a TF-Dombi hybrid weighted (TFDHW) operator and explore their fundamental features including idempotency, boundedness, monotonicity, and others. In the second part, we propose TF-Dombi weighted geometric (TFDWG) operator, TF-Dombi ordered weighted geometric (TFDOWG) operator, and TF-Dombi hybrid geometric (TFDHG) operator. The features and specific cases of the mentioned operators are examined. Enterprise resource planning (ERP) is a management and integration approach that organizations employ to manage and develop many aspects of their operations. The study's primary contribution is to employ TFS to create certain decision-making strategies for the selection of optimal ERP systems. The proposed operators are then used to build several techniques for solving multiattribute decision-making (MADM) issues with TF information. Finally, an example of ERP system selection is investigated to demonstrate that the techniques suggested are trustworthy and realistic.

Einstein-Ordered Weighted Geometric Operator for Pythagorean Fuzzy Soft Set with Its Application to Solve MAGDM Problem

The Pythagorean fuzzy soft set (PFSS) is the most influential and operative tool for maneuvering compared to the Pythagorean fuzzy set (PFS), which can accommodate the parameterization of alternatives. It is also a generalized form of intuitionistic fuzzy soft sets (IFSS), which delivers healthier and more exact valuations in the decision-making (DM) procedure. The primary purpose is to extend and propose ideas related to Einstein’s ordered weighted geometric aggregation operator from fuzzy structure to PFSS structure. The core objective of this work is to present a PFSS aggregation operator, such as the Pythagorean fuzzy soft Einstein-ordered weighted geometric (PFSEOWG) operator. In addition, the basic properties of the proposed operator are introduced, such as idempotency, boundedness, and homogeneity. Moreover, a DM method based on a developed operator has been presented to solve the multiattribute group decision-making (MAGDM) problem. A real-life application of the anticipated method has been offered for a capitalist to choose the most delicate business to finance his money. Finally, a brief comparative analysis with some current methods demonstrates the proposed approach’s effectiveness and reliability.

Geometric Dirac operator on the fuzzy sphere

We construct a Connes spectral triple or ‘Dirac operator’ on the non-reduced fuzzy sphere mathbb {C}_lambda [S^2] as realised using quantum Riemannian geometry with a central quantum metric g of Euclidean signature and its associated quantum Levi-Civita connection. The Dirac operator is characterised uniquely up to unitary equivalence within our quantum Riemannian geometric setting and an assumption that the spinor bundle is trivial and rank 2 with a central basis. The spectral triple has KO dimension 3 and in the case of the round metric, essentially, recovers a previous proposal motivated by rotational symmetry.

Intuitionistic Fuzzy Prioritized Aggregation Operators Based on Priority Degrees with Application to Multicriteria Decision-Making

In practise, intuitionistic fuzzy numbers (IFNs) are particularly useful for describing ambiguous data. We look at multicriteria decision-making (MCDM) problems with a prioritising relationship between the parameters. The concept of a priority degree is presented. The aggregation operators (AOs) are formed by assigning nonnegative real numbers to stringent priority levels, known as priority degrees. As a result, we construct “intuitionistic fuzzy prioritized averaging operator with priority degrees” and “intuitionistic fuzzy prioritized geometric operator with priority degrees,” which are both prioritized operators. The attributes of the existing method are frequently compared to those of other current approaches, stressing the superiority of the provided work over other methods now in use. In addition, the impact of priority degrees on the overall result is thoroughly investigated. Furthermore, in the intuitionistic fuzzy set (IFS) context, a decision-making strategy is proposed based on these operators. To highlight the efficacy of the proposed approach, an illustrative example relating to the selection of the best choice is considered.

Single-valued neutrosophic Einstein interactive aggregation operators with applications for material selection in engineering design: case study of cryogenic storage tank

Single-valued neutrosophic sets (SVNSs) and their application to material selection in engineering design. Liquid hydrogen is a feasible ingredient for energy storage in a lightweight application due to its high gravimetric power density. Material selection is an essential component in engineering since it meets all of the functional criteria of the object. Materials selection is a time-consuming as well as a critical phase in the design process. Inadequate material(s) selection can have a detrimental impact on a manufacturer’s production, profitability, and credibility. Multi-criteria decision-making (MCDM) is an important tool in the engineering design process that deals with complexities in material selection. However, the existing MCDM techniques often produce conflicting results. To address such problems, an innovative aggregation technique is proposed for material selection in engineering design based on truthness, indeterminacy, and falsity indexes of SVNSs. Taking advantage of SVNSs and smooth approximation with interactive Einstein operations, single-valued neutrosophic Einstein interactive weighted averaging and geometric operators are proposed. Based on proposed AOs, a robust MCDM approach is proposed for material selection in engineering design. A practical application of the proposed MCDM approach for material selection of cryogenic storage containers is developed. Additionally, the authenticity analysis and comparison analysis are designed to discuss the validity and rationality of the optimal decision.

A novel approach on decision support system based on triangular linguistic cubic fuzzy Dombi aggregation operators

The triangular linguistic cubic fuzzy sets (TLCFSs) can express the fuzzy data easily and is also very useful in modeling of uncertain data in decision making (DM) problems. First of all, on the basis of Dombi t-norm and t-conorm (DTT), we propose novel operational rules of triangular linguistic cubic fuzzy numbers (TLCFNs). We propose some new aggregation operators of TLCFNs based on the newly developed operations, i.e., triangular linguistic cubic fuzzy Dombi weighted averaging (TLCFDWA), triangular linguistic cubic fuzzy Dombi weighted geometric (TLCFDWG), triangular linguistic cubic fuzzy Dombi order weighted averaging (TLCFDOWA), triangular linguistic cubic fuzzy Dombi order weighted geometric (TLCFDOWG), triangular linguistic cubic fuzzy Dombi hybrid weighted averaging (TLCFDHWA), and triangular linguistic cubic fuzzy Dombi hybrid weighted geometric (TLCFDHWG) operators. Furthermore, a new method is proposed with the help of the proposed operators to solve the decision making problem. Finally, a numerical example is provided to illustrate the effectiveness of the new method. Comparative analysis is used to demonstrate the proposed method’s superiority.

Evolution of eigenvalues of a geometric operator under Ricci flow on a Riemannian manifold

The behavior of the eigenvalues of a geometric operator closely related to the Laplacian under Ricci flow is investigated. These depend on a coupling parameter in the operator as well as an evolution parameter which gives a flow on a compact manifold of finite dimension. The main objective is to study the monotonicity properties of the eigenvalues.

Intuitionistic fuzzy-valued neutrosophic multi-sets and numerical applications to classification

A single-valued neutrosophic multi-set is characterized by a sequence of truth membership degrees, a sequence of indeterminacy membership degrees and a sequence of falsity membership degrees. Nature of a single-valued neutrosophic multi-set allows us to consider multiple information in the truth, indeterminacy and falsity memberships which is pretty useful in multi-criteria group decision making. In this paper, we consider sequences of intuitionistic fuzzy values instead of numbers to define the concept of intuitionistic fuzzy-valued neutrosophic multi-set. In this manner, such a set gives more powerful information. We also present some set theoretic operations and a partial order for intuitionistic fuzzy-valued neutrosophic sets and provide some algebraic operations between intuitionistic fuzzy-valued neutrosophic values. Then, we develop two types of weighted aggregation operators with the help of intuitionistic fuzzy t-norms and t-conorms. By considering some well-known additive generators of ordinary t-norms, we give the Algebraic weighted arithmetic and geometric aggregation operators and the Einstein weighted arithmetic and geometric aggregation operators that are the particular cases of the weighted aggregation operators defined via general t-norms and t-conorms. We also define a simplified neutrosophic valued similarity measure and we use a score function for simplified neutrosophic values to rank similarities of intuitionistic fuzzy-valued neutrosophic multi-values. Finally, we give an algorithm to solve classification problems using intuitionistic fuzzy-valued neutrosophic multi-values and proposed aggregation operators and we apply the theoretical part of the paper to a real classification problem.

Novel multiple criteria decision-making analysis under m-polar fuzzy aggregation operators with application.

Aggregation is a very efficient indispensable tool in which several input values are transformed into a single output value that further supports dealing with different decision-making situations. Additionally, note that the theory of m-polar fuzzy (mF) sets is proposed to tackle multipolar information in decision-making problems. To date, several aggregation tools have been widely investigated to tackle multiple criteria decision-making (MCDM) problems in an m-polar fuzzy environment, including m-polar fuzzy Dombi and Hamacher aggregation operators (AOs). However, the aggregation tool to deal with m-polar information under Yager's operations (that is, Yager's t-norm and t-conorm) is missing in the literature. Due to these reasons, this study is devoted to investigating some novel averaging and geometric AOs in an mF information environment through the use of Yager's operations. Our proposed AOs are named as the mF Yager weighted averaging (mFYWA) operator, mF Yager ordered weighted averaging operator, mF Yager hybrid averaging operator, mF Yager weighted geometric (mFYWG) operator, mF Yager ordered weighted geometric operator and mF Yager hybrid geometric operator. The initiated averaging and geometric AOs are explained via illustrative examples and some of their basic properties, including boundedness, monotonicity, idempotency and commutativity are also studied. Further, to deal with different MCDM situations containing mF information, an innovative algorithm for MCDM is established under the under the condition of mFYWA and mFYWG operators. After that, a real-life application (that is, selecting a suitable site for an oil refinery) is explored under the conditions of developed AOs. Moreover, the initiated mF Yager AOs are compared with existing mF Hamacher and Dombi AOs through a numerical example. Finally, the effectiveness and reliability of the presented AOs are checked with the help of some existing validity tests.