Einstein Operations Research Articles

Some geometric Choquet aggregation operators using Einstein operations under intuitionistic fuzzy environment

This paper investigates the methods of dealing with the information aggregation problems in the context of intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set respectively. As anything is highly relevant in the real world, it is important and significant for researchers to descript the relationships between any two subjects mathematically. The Choquet integral is a powerful tool to depict the interrelationships between objects. In this paper, we first propose the Einstein based intuitionistic fuzzy Choquet geometric EIFCG operator and Einstein based interval-valued intuitionistic fuzzy Choquet geometric EIIFCG operator. Then, the relationship between the EIFCG operator and the IFCG operator is investigated. We also study series of the desirable properties of the EIFCG operator and EIIFCG operator. Furthermore, an intuitionistic fuzzy multi-criteria decision making method based on the EIFCG operator is proposed. We also focus on the correlations with other intuitionistic fuzzy multi-criteria decision making method.

Multiple Attribute Decision Making Based on Hesitant Fuzzy Einstein Geometric Aggregation Operators

We first define an accuracy function of hesitant fuzzy elements (HFEs) and develop a new method to compare two HFEs. Then, based on Einstein operators, we give some new operational laws on HFEs and some desirable properties of these operations. We also develop several new hesitant fuzzy aggregation operators, including the hesitant fuzzy Einstein weighted geometric (HFEWGε) operator and the hesitant fuzzy Einstein ordered weighted geometric (HFEWGε) operator, which are the extensions of the weighted geometric operator and the ordered weighted geometric (OWG) operator with hesitant fuzzy information, respectively. Furthermore, we establish the connections between the proposed and the existing hesitant fuzzy aggregation operators and discuss various properties of the proposed operators. Finally, we apply the HFEWGεoperator to solve the hesitant fuzzy decision making problems.

Models for multiple attribute decision making based on the Einstein correlated aggregation operators with interval-valued intuitionistic fuzzy information

In this paper, we investigate the multiple attribute decision making MADM problems for selecting the construction project in which attribute values take the form of interval-valued intuitionistic fuzzy information. Firstly, some operational laws of interval-valued intuitionistic fuzzy numbers and score function and accuracy function of interval-valued intuitionistic fuzzy numbers are introduced. Then, we utilize Einstein operations to develop the interval-valued intuitionistic fuzzy Einstein correlated averaging IVIFECA operator. The prominent characteristic of this operator is that they can not only consider the importance of the elements or their ordered positions, but also reflect the interactions phenomena among the decision making attributes or their ordered positions. We have applied the IVIFECA operator to multiple attribute decision making problems for selecting the construction project with interval-valued intuitionistic fuzzy information. Finally an illustrative example for involving a construction project selection has been given to show the developed method.

Model for multiple attribute decision making based on the Einstein correlated information fusion with hesitant fuzzy information

We investigate the multiple attribute decision making (MADM) problems in which attribute values take the form of hesitant fuzzy information. Firstly, some operational laws of hesitant fuzzy elements and score function of hesitant fuzzy elements are introduced. In this paper, we utilize Einstein operations to develop some hesitant fuzzy correlated aggregation operators: hesitant fuzzy Einstein correlated averaging (HFECA) operator and hesitant fuzzy Einstein correlated geometric (HFECG) operator. The prominent characteristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the HFECA and HFECG operators to multiple attribute decision making with hesitant fuzzy information. Finally an illustrative example has been given to show the developed method.

Hesitant triangular fuzzy information aggregation based on Einstein operations and their application to multiple attribute decision making

In this paper, we investigate the multiple attribute decision making (MADM) problems in which attribute values take the form of hesitant triangular fuzzy information. Firstly, definition and some operational laws of hesitant triangular fuzzy elements and score function of hesitant triangular fuzzy elements are introduced. Then, we have developed some hesitant triangular fuzzy aggregation operators based on the Einstein operation: the hesitant triangular fuzzy Einstein weighted averaging (HTFEWA) operator, hesitant triangular fuzzy Einstein weighted geometric (HTFEWG) operator, hesitant triangular fuzzy Einstein ordered weighted averaging (HTFEOWA) operator, hesitant triangular fuzzy Einstein ordered weighted geometric (HTFEOWG) operator, hesitant triangular fuzzy Einstein hybrid average (HTFEHA) operator and hesitant triangular fuzzy Einstein hybrid geometric (HTFEHG) operator. We have applied the hesitant triangular fuzzy Einstein weighted averaging (HTFEWA) operator, hesitant triangular fuzzy Einstein weighted geometric (HTFEWG) operators to multiple attribute decision making with hesitant triangular fuzzy information. Finally an illustrative example has been given to show the developed method.

The induced intuitionistic fuzzy Einstein aggregation and its application in group decision-making

Einstein product and Einstein sum are good alternatives to the algebraic product and algebraic sum, respectively. Nevertheless, it seems that in the literature there is little investigation on aggregation techniques using the Einstein operations on IFS for aggregating a collection of intuitionistic fuzzy value (IFV). The induced ordered weighted averaging operator is more suitable for aggregating individual preference relations into a collective fuzzy preference relation. In order to deal with more complex group decision-making problem with intuitionistic fuzzy information, the paper develops an I-IFOWA ϵ operator which generalizes some of the intuitionistic fuzzy Einstein aggregation operators, such as the IFWA ϵ operator, IFOWA ϵ operator, IFA ϵ , as well as IF maximum operator and IF minimum operator. Furthermore, the authors apply the I-IFOWA ϵ operator and the IFWA ϵ operator to multiple attribute group decision-making with intuitionistic fuzzy information, and give an illustrative example to show the effectiveness of the developed methods.

Interval-valued intuitionistic fuzzy hybrid weighted averaging operator based on Einstein operation and its application to decision making

The notion of interval-valued intuitionistic fuzzy set IVIFS is a generalization of that of Atanassov's intuitionistic fuzzy set AIFS. The fundamental characteristic of IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers. In this paper, we define some Einstein operations on IVIFS and develop three arithmetic averaging operators, such as the interval-valued intuitionistic fuzzy Einstein weighted averaging IVIFWAe operator, interval-valued intuitionistic fuzzy Einstein ordered weighted averaging IVIFOWAe operator, and interval-valued intuitionistic fuzzy Einstein hybrid weighted averaging IVIFHWAe operator, for aggregating interval-valued intuitionistic fuzzy information. The IVIFHWAe operator generalizes both the IVIFWAe and IVIFOWAe operators. Moreover, we establish various properties of these operators and derive the relationship between the proposed operators and the exiting aggregation operators. Finally, we apply the IVIFHWAe operator to multiple attribute decision making with interval-valued intuitionistic fuzzy information.

Intuitionistic Fuzzy Information Aggregation Using Einstein Operations

Aggregation of fuzzy information is a new branch of Atanassov's intuitionistic fuzzy set (AIFS) theory, which has attracted significant interest from researchers in recent years. In this paper, we treat the intuitionistic fuzzy aggregation operators with the help of Einstein operations. We first introduce some new operations of AIFSs, such as Einstein sum, Einstein product, and Einstein scalar multiplication. Then, we develop some intuitionistic fuzzy aggregation operators, such as the intuitionistic fuzzy Einstein weighted averaging operator and the intuitionistic fuzzy Einstein ordered weighted averaging operator, which extend the weighted averaging operator and the ordered weighted averaging operator to aggregate Atanassov's intuitionistic fuzzy values, respectively. We further establish various properties of these operators and analyze the relations between these operators and the existing intuitionistic fuzzy aggregation operators. Moreover, we give some numerical examples to illustrate the developed aggregation operators. Finally, we apply the intuitionistic fuzzy Einstein weighted averaging operator to multiple attribute decision making with intuitionistic fuzzy information.

Intuitionistic fuzzy geometric aggregation operators based on einstein operations

Intuitionistic fuzzy information aggregation plays an important part in Atanassov's intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and further develop some new geometric aggregation operators, such as the intuitionistic fuzzy Einstein weighted geometric operator and the intuitionistic fuzzy Einstein ordered weighted geometric operator, which extend the weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. We also establish some desirable properties of these operators, such as commutativity, idempotency and monotonicity, and give some numerical examples to illustrate the developed aggregation operators. In addition, we compare the proposed operators with the existing intuitionistic fuzzy geometric operators and get the corresponding relations. Finally, we apply the intuitionistic fuzzy Einstein weighted geometric operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. © 2011 Wiley Periodicals, Inc.

A noncommutative Friedman cosmological model

AbstractThe closed Friedman cosmological model, based on noncommutative geometry, is presented. Two global effects exhibited by the model are discussed. The first effect is the “generation of matter out of geometry”. Gravitational field equation in this model has the form of the eigenvalue equation for the Einstein operator. It turns out that the eigenvalues of this operator reproduce components of the energy‐momentum tensor. The second effect concerns the existence of the initial and final singularities. Because of the strongly probabilistic character of the noncommutative dynamics on the fundamental level, although singularities do exist, they are probabilistically irrelevant.

Generalized Einstein operator generating functions

We present gauge invariant, self-adjoint Einstein operators for mixed symmetry higher spin theories. The result applies to multi-forms, multi-symmetric forms and mixed antisymmetric and symmetric multi-forms, generalizing previous results for combinations of these cases. It also yields explicit action principles for these theories in terms of their minimal covariant field content. For known cases, these actions imply the mixed symmetry equations of motion of Labastida. The result is based on a calculus for handling normal ordered operator expressions built from quantum generators of the underlying constraint algebras.

Noncommutative closed Friedman universe

Heller et al. (J Math Phys 46:122501, 2005; Int J Theor Phys 46:2494, 2007) proposed a model unifying general relativity and quantum mechanics based on a noncommutative algebra \({\mathcal{A}}\) defined on a groupoid having the frame bundle over space–time as its base space. The generalized Einstein equation is assumed in the form of the eigenvalue equation of the Einstein operator on a module of derivations of the algebra \({\mathcal{A}}\) . No matter sources are assumed. The closed Friedman world model, when computed in this formalism, exhibits two interesting properties. First, generalized eigenvalues of the Einstein operator reproduce components of the perfect fluid energy-momentum tensor for the usual Friedman model together with the corresponding equation of state. One could say that, in this case, matter is produced out of pure (noncommutative) geometry. Second, owing to probabilistic properties of the model, in the noncommutative regime (on the Planck level) singularities are irrelevant. They emerge in the process of transition to the usual space–time geometry. These results are briefly discussed.

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{\cal A} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{\cal A}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{\cal A}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{\cal A}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.

Fuzzy controllers based on some fuzzy implication operators and their response functions *

Abstract The fuzzy controllers constructed by 23 fuzzy implication operators based on CRI algorithm and their response functions are discussed. The conclusions show that the fuzzy controllers constructed by 9 fuzzy implication operators are universal approximators to continuous functions and can be used in practical fuzzy control systems. And these 9 fuzzy implication operators except for Einstein operator intersection are all the adjoint pairs of some fuzzy implication operators. Besides, there are 3 other fuzzycontrollers formed by fuzzy implication operators being regarded approximately as fitted functions.

Teleparallel origin of the Fierz picture for spin-2 particle

A new approach to the description of a spin-2 particle in flat and curved spacetime is developed on the basis of the teleparallel gravity theory. We show that such an approach is in fact a true and natural framework for the Fierz representation proposed recently by Novello and Neves. More specifically, we demonstrate how the teleparallel theory fixes uniquely the structure of the Fierz tensor, discover the transparent origin of the gauge symmetry of the spin-2 model, and derive the linearized Einstein operator from the fundamental identity of the teleparallel gravity. In order to cope with the consistency problem on the curved spacetime, similarly to the usual Riemannian approach, one needs to include the nonminimal (torsion dependent) coupling terms.