Einstein Operations Research Articles

Group decision-making framework under linguistic q-rung orthopair fuzzy Einstein models

The q-rung orthopair fuzzy sets dynamically change the range of indication of decision knowledge by adjusting a parameter q from decision makers, where $$q \ge 1$$ , and outperform the conventional intuitionistic fuzzy sets and Pythagorean fuzzy sets. Linguistic q-rung orthopair fuzzy sets (Lq-ROFSs), a qualitative type of q-rung orthopair fuzzy sets, are characterized by a degree of linguistic membership and a degree of linguistic non-membership to reflect the qualitative preferred and non-preferred judgments of decision makers. Einstein operator is a powerful alternative to the algebraic operators and has flexible nature with its operational laws and fuzzy graphs perform well when expressing correlations between attributes via edges between vertices in fuzzy information systems, which makes it possible for addressing correlational multi-attribute decision-making (MADM) problems. Inspired by the idea of Lq-ROFS and taking the advantage of the flexible nature of Einstein operator, in this paper, we aim to introduce a new class of fuzzy graphs, namely, linguistic q-rung orthopair fuzzy graphs (Lq-ROFGs) and further explore efficient approaches to complicated MAGDM situations. Following the above motivation, we propose the new concepts, including product-connectivity energy, generalized product-connectivity energy, Laplacian energy and signless Laplacian energy and discuss several of its desirable properties in the background of Lq-ROFGs based on Einstein operator. Moreover, product-connectivity energy, generalized product-connectivity energy, Laplacian energy and signless Laplacian energy of linguistic q-rung orthopair fuzzy digraphs (Lq-ROFDGs) are presented. In addition, we present a graph-based MAGDM approach with linguistic q-rung orthopair fuzzy information based on Einstein operator. Finally, an illustrative example related to the selection of mobile payment platform is given to show the validity of the proposed decision-making method. For the sake of the novelty of the proposed approach, comparison analysis is conducted and superiorities in contrast with other methodologies are illustrated.

Cubic linguistic uncertain Einstein averaging operators and decision-making problems

We introduce the Einstein operations based on cubic linguistic uncertain fuzzy numbers. We discuss three arithmetic averaging operators, viz. the cubic linguistic uncertain fuzzy Einstein weighted averaging operator, the cubic linguistic uncertain fuzzy Einstein ordered weighted averaging operator, and the cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging operator, for gathering cubic linguistic uncertain fuzzy data. Moreover, we investigate the association between the current aggregation operators and the suggested operators. We also establish several properties of these operators. To illustrate the proposed method and demonstrate its efficiency, an example is constructed and the results compared with those obtained using existing methods.

Intuitionistic Fuzzy Calculus Based on Einstein Operations

The intuitionistic fuzzy set (A-IFS) was introduced by Atanassov to generalize the concept of Zadeh’s fuzzy set. The basic elements of an A-IFS are the intuitionistic fuzzy numbers (IFNs). In this paper, we deal with the intuitionistic fuzzy set with the help of Einstein operations. First we introduce some operations of A-IFS, such as the Einstein sum, Einstein product, Einstein scalar multiplication and Einstein exponentiation, and redefine two new and important operation as Einstein subtraction and Einstein division. Then we investigate the limit, continuity and derivatives of intuitionistic fuzzy functions (IFFs) based on the Einstein operations. Furthermore and what is more, some infinite and definite integrals for the IFFs are discussed importantly, and an aggregation method based on the definite integral was introduced to solve some actual decision making problems, which focus on the original data and obtain better overall evaluations in information aggregation.

Minimum Parametrization of the Cauchy Stress Operator

When ${\cal{D}}:\xi \rightarrow \eta$ is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that ${\cal{D}}\xi=\eta$ implies ${\cal{D}}_1\eta=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ when the ground manifold has dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a more difficult " inverse problem " is to look for an operator ${\cal{D}}: \xi \rightarrow \eta$ having the generating CC ${\cal{D}}_1\eta=0$. If this is possible, that is when the differential module defined by ${\cal{D}}_1$ is torsion-free, one shall say that the operator ${\cal{D}}_1$ is parametrized by ${\cal{D}}$ and there is no relation in general between ${\cal{D}}$ and ${\cal{D}}_2$. The parametrization is said to be " minimum " if the differential module defined by ${\cal{D}}$ has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchy stress operator in arbitrary dimension $n$ has attracted many famous scientists (G.B. Airy in 1863 for $n=2$, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for $n=3$, A. Einstein in 1915 for $n=4$) . This paper proves that all these works are using the Einstein operator and not the Ricci operator. As a byproduct, they are all based on a confusion between the so-called $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator which is the formal adjoint of the Killing operator ${\cal{D}}$ parametrizing the Riemann operator ${\cal{D}}_1$ for an arbitrary $n$. Like the Michelson and Morley experiment, it is an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment.

Generalized complex q-rung orthopair fuzzy Einstein averaging aggregation operators and their application in multi-attribute decision making

The recently proposed q-rung orthopair fuzzy set, which is characterized by a membership degree and a non-membership degree, is effective for handling uncertainty and vagueness. This paper proposes the concept of complex q-rung orthopair fuzzy sets (Cq-ROFS) and their operational laws. A multi-attribute decision making (MADM) method with complex q-rung orthopair fuzzy information is investigated. To aggregate complex q-rung orthopair fuzzy numbers, we extend the Einstein operations to Cq-ROFSs and propose a family of complex q-rung orthopair fuzzy Einstein averaging operators, such as the complex q-rung orthopair fuzzy Einstein weighted averaging operator, the complex q-rung orthopair fuzzy Einstein ordered weighted averaging operator, the generalized complex q-rung orthopair fuzzy Einstein weighted averaging operator, and the generalized complex q-rung orthopair fuzzy Einstein ordered weighted averaging operator. Desirable properties and special cases of the introduced operators are discussed. Further, we develop a novel MADM approach based on the proposed operators in a complex q-rung orthopair fuzzy context. Numerical examples are provided to demonstrate the effectiveness and superiority of the proposed method through a detailed comparison with existing methods.

Linear stability of slowly rotating Kerr black holes

We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equations have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications.

Intuitionistic 2-tuple linguistic aggregation information based on Einstein operations and their applications in group decision making

The linguistic information can be expressed as a 2-tuple of a linguistic variable and a real number in an interval $$[-\frac{1}{2}, \frac{1}{2})$$ . The intuitionistic 2-tuple linguistic (I2TL) set accurately deals with the imprecise and unpredictable information in those decision-making problems where experts prefer the degree of membership and non-membership values in the form of 2-tuple. The existing approaches used for the aggregation operations of I2TL sets are extremely complicated. This work aims to develop new aggregation operations for I2TL sets using Einstein operations. We present intuitionistic 2-tuple linguistic Einstein weighted averaging (I2TLEWA), and intuitionistic 2-tuple linguistic Einstein weighted geometric (I2TLEWG) operators. We also discuss their properties and relationship between them. Moreover, we numerically test the feasibility and significance of our proposed operators by solving a multi-criteria group decision making (MCGDM) problem. Finally, we do a comparative analysis with another method to give insights on our designed operators for I2TL sets.

A Robust q-Rung Orthopair Fuzzy Information Aggregation Using Einstein Operations with Application to Sustainable Energy Planning Decision Management

A q-rung orthopair fuzzy set (q-ROFS), an extension of the Pythagorean fuzzy set (PFS) and intuitionistic fuzzy set (IFS), is very helpful in representing vague information that occurs in real-world circumstances. The intention of this article is to introduce several aggregation operators in the framework of q-rung orthopair fuzzy numbers (q-ROFNs). The key feature of q-ROFNs is to deal with the situation when the sum of the qth powers of membership and non-membership grades of each alternative in the universe is less than one. The Einstein operators with their operational laws have excellent flexibility. Due to the flexible nature of these Einstein operational laws, we introduce the q-rung orthopair fuzzy Einstein weighted averaging (q-ROFEWA) operator, q-rung orthopair fuzzy Einstein ordered weighted averaging (q-ROFEOWA) operator, q-rung orthopair fuzzy Einstein weighted geometric (q-ROFEWG) operator, and q-rung orthopair fuzzy Einstein ordered weighted geometric (q-ROFEOWG) operator. We discuss certain properties of these operators, inclusive of their ability that the aggregated value of a set of q-ROFNs is a unique q-ROFN. By utilizing the proposed Einstein operators, this article describes a robust multi-criteria decision making (MCDM) technique for solving real-world problems. Finally, a numerical example related to integrated energy modeling and sustainable energy planning is presented to justify the validity and feasibility of the proposed technique.

T-Spherical Fuzzy Einstein Hybrid Aggregation Operators and Their Applications in Multi-Attribute Decision Making Problems

T-spherical fuzzy set is a recently developed model that copes with imprecise and uncertain events of real-life with the help of four functions having no restrictions. This article’s aim is to define some improved algebraic operations for T-SFSs known as Einstein sum, Einstein product and Einstein scalar multiplication based on Einstein t-norms and t-conorms. Then some geometric and averaging aggregation operators have been established based on defined Einstein operations. The validity of the defined aggregation operators has been investigated thoroughly. The multi-attribute decision-making method is described in the environment of T-SFSs and is supported by a comprehensive numerical example using the proposed Einstein aggregation tools. As consequences of the defined aggregation operators, the same concept of Einstein aggregation operators has been proposed for q-rung orthopair fuzzy sets, spherical fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. To signify the importance of proposed operators, a comparative analysis of proposed and existing studies is developed, and the results are analyzed numerically. The advantages of the proposed study are demonstrated numerically over the existing literature with the help of examples.

Method of MAGDM based on pythagorean trapezoidal uncertain linguistic hesitant fuzzy aggregation operator with Einstein operations

Method of MAGDM based on pythagorean trapezoidal uncertain linguistic hesitant fuzzy aggregation operator with Einstein operations

Linguistic Pythagorean Einstein Operators and Their Application to Decision Making

Linguistic Pythagorean fuzzy (LPF) set is an efficacious technique to comprehensively represent uncertain assessment information by combining the Pythagorean fuzzy numbers and linguistic variables. In this paper, we define several novel essential operations of LPF numbers based upon Einstein operations and discuss several relations between these operations. For solving the LPF numbers fusion problem, several LPF aggregation operators, including LPF Einstein weighted averaging (LPFEWA) operator, LPF Einstein weighted geometric (LPFEWG) operator and LPF Einstein hybrid operator, are propounded; the prominent characteristics of these operators are investigated as well. Furthermore, a multi-attribute group decision making (MAGDM) approach is presented on the basis of the developed operators under an LPF environment. Ultimately, two application cases are utilized to demonstrate the practicality and feasibility of the developed decision approach and the comparison analysis is provided to manifest the merits of it.

Protraction of Einstein operators for decision-making under q-rung orthopair fuzzy model

An q-rung orthopair fuzzy set is a generalized structure that covers the modern extensions of fuzzy set, including intuitionistic fuzzy set and Pythagorean fuzzy set, with an adjustable parameter q that makes it flexible and adaptable to describe the inexact information in decision making. The condition of q-rung orthopair fuzzy set, i.e., sum of qth power of membership degree and nonmembership degree is bounded by one, makes it highly competent and adequate to get over the limitations of existing models. The basic purpose of this study is to establish some aggregation operators under the q-rung orthopair fuzzy environment with Einstein norm operations. Motivated by innovative features of Einstein operators and dominant behavior of q-rung orthopair fuzzy set, some new aggregation operators, namely, q-rung orthopair fuzzy Einstein weighted averaging, q-rung orthopair fuzzy Einstein ordered weighted averaging, generalized q-rung orthopair fuzzy Einstein weighted averaging and generalized q-rung orthopair fuzzy Einstein ordered weighted averaging operators are defined. Furthermore, some properties related to proposed operators are presented. Moreover, multi-attribute decision making problems related to career selection, agriculture land selection and residential place selection are presented under these operators to show the capability and proficiency of this new idea. The comparison analysis with existing theories shows the superiorities of proposed model.

Some Generalised Einstein Hybrid Aggregation Operators and Their Application to Group Decision-making Using Pythagorean Fuzzy Numbers

Pythagorean fuzzy set (PFS) is one of the prosperous extensions of the intuitionistic fuzzy set (IFS) for handling the fuzziness and uncertainties in the data. Under this environment, in this paper, we introduce the notion of two generalised Einstein hybrid operators namely, generalised Pythagorean fuzzy Einstein hybrid averaging (in short GPFEHA) operator and generalised Pythagorean fuzzy Einstein hybrid geometric (in short GPFEHG) operator along with their desirable properties, such as idempotency, boundedness and monotonicity. The main benefit of the proposed operators is that these operators deliver more general, more correct and precise results as compared to their existing methods. Generalised Einstein operators combine Einstein operators with some generalised operators using Pythagorean fuzzy values. Therefore these methods play a vital role in real world problems. Finally, the proposed operators have been applied to decision-making problems to show the validity, practicality and effectiveness of the new attitude.

Ranking methodology of induced Pythagorean trapezoidal fuzzy aggregation operators based on Einstein operations in group decision making

The Pythagorean fuzzy number is a new tool for uncertainty and vagueness. It is a generalization of fuzzy numbers and intuitionistic fuzzy numbers. In this paper, we define some Einstein operations on Pythagorean trapezoidal fuzzy set and develop two averaging aggregation operators, which is an induced Pythagorean trapezoidal fuzzy Einstein ordered weighted averaging operator and an induced Pythagorean trapezoidal fuzzy Einstein hybrid averaging (I-PTFEHA) operator. We presented some new methods to deal with the multi-attribute group decision-making problems under the Pythagorean trapezoidal fuzzy environment. Finally, we used some practical examples to illustrate the validity and feasibility of the proposed methods by comparing with existing method. It shows that the proposed I-PTFEHA operator is much better and reliable than the existing one.

Symmetries of linearized gravity from adjoint operators

Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and, in particular, the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new operator of differential order six. The corresponding operator identities are based on the Teukolsky equation and the Teukolsky-Starobinsky identities, respectively. The method applies to other field equations as well, which is illustrated with the Maxwell equation. The resulting symmetry operators are connected to Hertz and Debye potentials as well as to the separability of the Teukolsky equation for both Maxwell and linearized gravity.

Picture fuzzy aggregation information based on Einstein operations and their application in decision making

The information aggregation operator plays a key rule in the group decision making problems. The aim of this paper is to investigate the information aggregation operators method under the picture fuzzy environment with the help of Einstein norms operations. The picture fuzzy set is an extended version of the intuitionistic fuzzy set, which not only considers the degree of acceptance or rejection but also takes into the account of neutral degree during the analysis. Under these environments, some basic aggregation operators namely picture fuzzy Einstein weighted and Einstein ordered weighted operators are proposed in this paper. Some properties of these aggregation operators are discussed in detail. Further, a group decision making problem is illustrated and validated through a numerical example. A comparative analysis of the proposed and existing studies is performed to show the validity of the proposed operators.

New work of trapezoidal cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging operator and decision making

In this paper, we define some Einstein operations on trapezoidal cubic linguistic uncertain fuzzy numbers and develop two arithmetic averaging operators, that is, trapezoidal cubic linguistic uncertain fuzzy Einstein weighted averaging operator and trapezoidal cubic linguistic uncertain fuzzy Einstein hybrid weighted averaging (TrCLUFEHWA) operator, for aggregating trapezoidal cubic linguistic uncertain fuzzy information. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. We apply the TrCLUFEHWA operator to multiple-attribute decision making with trapezoidal cubic linguistic uncertain fuzzy information. Finally, a numerical example is provided to demonstrate the submission of the established approach.

Cubic uncertain linguistic powered Einstein aggregation operators and their application to multi-attribute group decision making

The cubic uncertain fuzzy linguistic variable can easily express the fuzzy information, and the power average (PA) operator is a useful tool which provides more versatility in the information aggregation procedure. In this paper, we will combine the PA operator and Einstein operations to cubic uncertain linguistic environment and propose some new PA operators. Firstly, the definition and some basic operations of cubic uncertain linguistic number, power aggregation (PA) operator and Einstein operations are introduced. Then, we propose cubic uncertain linguistic fuzzy powered Einstein averaging operator, cubic uncertain linguistic fuzzy powered Einstein weighted (CULFPEWA) operator, cubic uncertain linguistic fuzzy Einstein geometric operator and cubic uncertain linguistic fuzzy Einstein weighted geometric (CULFPEWG) operator and discuss some properties of these in detail. Furthermore, we develop the decision-making methods for multi-attribute group decision-making problems with cubic uncertain linguistic information and give the detail decision steps. At last, an illustrate example is given to show the process of decision making and the effectiveness of the proposed method.

Some induced interval-valued Pythagorean trapezoidal fuzzy averaging aggregation operators based on Einstein operations and their application in group decision-making

The aim of this paper is to investigate the information aggregation methods under interval-valued Pythagorean trapezoidal fuzzy environment. Some Einstein operational laws on Pythagorean trapezoidal fuzzy numbers are defined based on Einstein sum and Einstein product. We define interval-valued Pythagorean trapezoidal fuzzy aggregation operators, induced interval-valued Pythagorean trapezoidal fuzzy Einstein ordered weighted averaging (I-IVTFEOWA) operator and induced interval-valued Pythagorean trapezoidal fuzzy Einstein hybrid averaging (I-IVPTFEHA) operator. We discuss some basic properties of the proposed operator, including idempotency, commutativity and monotonicity. We construct an algorithm for multiple-attribute group decision-making problem, and apply the proposed aggregation operator to deal with multiple-attribute group decision-making. Finally, we construct a numerical example for multiple-attribute group decision-making.

Linguistic Neutrosophic Numbers Einstein Operator and Its Application in Decision Making

Linguistic neutrosophic numbers (LNNs) include single-value neutrosophic numbers and linguistic variable numbers, which have been proposed by Fang and Ye. In this paper, we define the linguistic neutrosophic number Einstein sum, linguistic neutrosophic number Einstein product, and linguistic neutrosophic number Einstein exponentiation operations based on the Einstein operation. Then, we analyze some of the relationships between these operations. For LNN aggregation problems, we put forward two kinds of LNN aggregation operators, one is the LNN Einstein weighted average operator and the other is the LNN Einstein geometry (LNNEWG) operator. Then we present a method for solving decision-making problems based on LNNEWA and LNNEWG operators in the linguistic neutrosophic environment. Finally, we apply an example to verify the feasibility of these two methods.