A Comprehensive Overview of the ELECTRE Method in Multi-Criteria Decision-Making

The ELECTRE(ELimination Et Choix Traduisant la REalite)method has gained widespread recognition as one of the most effective multi-criteria decision-making(MCDM)methods.Its versatility allows it to be applied in a wide range of areas such as engineering,economics,business,environmental management and many others.This paper aims to provide an overview of the ELECTRE method,including its fundamental concepts,applications,advantages,and limitations.At its core,the ELECTRE method is an outranking family of MCDM techniques,which allows for the direct comparison of alternatives based on a set of criteria.The method takes into account the preferences and importance of decision-makers and generates a ranking of the alternatives based on their relative strengths and weaknesses.The ELECTRE method is a powerful tool for decision-making,and its applicability to a wide range of fields demonstrates its versatility and adaptability.By understanding its concepts,applications,merits,and demerits,decision-makers can use the ELECTRE method to make informed and effective decisions in a variety of contexts.

Stability of the MGS-like elimination method for equality constrained least squares problems

This paper proves that the weighting method via modified Gram-Schmidt（MGS） for solving the equality constrained least squares problem in the limit is equivalent to the direct elimination method via MGS（MGS-elimination method）. By virtue of this equivalence, the backward and forward roundoff error analysis of the MGS-elimination method is proved. Numerical experiments are provided to verify the results.

Study on the variation characteristics of the geoelectric field preceding earthquakes

Earthquake events occurred in Nantong, Jiangsu Province on Nov. 3, and Dec. 25, 2001 in which the biggest mag-nitudes were ML=3.8 and ML=4.1, respectively. This paper firstly explains the principle of the eliminating noise method by the multi-dipole observation system of geoelectric field. Then based on the observation data of the multi-dipole observation system obtained by ZD9A telluric current monitors installed in Chongming and Nanjing stations, we study the abnormal variation of the geoelectric field preceding the earthquakes. The study shows that: a) Eliminating noise method of multi-dipole observation is an excellent method by which many kinds of geoelec-tric field noises can be eliminated successfully and the geoelectric precursor information can be recognized; b) The geoelectric precursor signals for the events were recorded on the NS and NE dipoles in Chongming station 42 days, 20 days and 2 days before the earthquakes respectively, in which the station is near the epicenter, and the longest time of persisting period was 9 days. The abnormal variation signals of geoelectric field observed in Nanjing sta-tion are all the noises but not the seismic electric signals, in which the station is not near the epicenter; c) Dipole distribution method of common electrode is not good in the multi-dipole observation system of the geoelectric field.

Kinematics Analysis of Mechanisms Based on Virtual Assembly

Currently, virtual assembly technology has attracted increasing attention due to considerations of solving assembly problems in virtual environment before actual assembly in manufactory. Previous studies on kinematic analysis of mechanism only aim at analyzing motion law of single mechanism, but can not simulate the multi-mechanisms motion process at the same time, let alone simulating the automatic assembly process of products in a whole assembly workshop. In order to simulate the assembly process of products in an assembly workshop and provide effective data for analyzing mechanical performance after finishing assembly simulation in virtual environment, this study investigates the kinematics analysis of mechanisms based on virtual assembly. Firstly, in view of the same function of the kinematic pairs and the assembly constraints on restricting the motion of components （subassembly or part）, the method of identifying kinematic pairs automatically based on assembly constraints is presented. The information of kinematic pairs can be obtained through calculating the constraint degree of the assembly constraints. Secondly, the incidence matrix eliminating element method is proposed in order to search the information and establish the models of mechanisms automatically after finishing assembly simulation in virtual environment. Both methods have important significance for reducing the workload of pretreatment and promoting the level of automation of kinematics analysis. Finally, the method of kinematics analysis of mechanisms is presented. Based on Descartes coordinates, three types of kinematics equations are formed. The parameters, like displacement, velocity, and acceleration, can be obtained by solving these equations. All these data are important to analyze mechanical performance. All the methods are implemented and validated in the prototype system virtual assembly process planning（VAPP）. The mechanism models are established and simulated in the VAPP system, and the result curves are shown accurately. The proposed kinematics analysis of mechanisms based on virtual assembly provides an effective method for simulating product assembly process automatically and analyzing mechanical performance after finishing assembly simulation.

Kinematic Solution of Spherical Stephenson-Ⅲ Six-bar Mechanism

A closed-form solution can be obtained for kinematic analysis of spatial mechanisms by using analytical method.However,extra solutions would occur when solving the constraint equations of mechanism kinematics unless the constraint equations are established with a proper method and the solving approach is appropriate.In order to obtain a kinematic solution of the spherical Stephenson-III six-bar mechanism,spherical analytical theory is employed to construct the constraint equations.Firstly,the mechanism is divided into a four-bar loop and a two-bar unit.On the basis of the decomposition,vectors of the mechanism nodes are derived according to spherical analytical theory and the principle of coordinate transformation.Secondly,the structural constraint equations are constructed by applying cosine formula of spherical triangles to the top platform of the mechanism.Thirdly,the constraint equations are solved by using Bezout’ s elimination method for forward analysis and Sylvester’ s resultant elimination method for inverse kinematics respectively.By the aid of computer symbolic systems,Mathematica and Maple,symbolic closed-form solution of forward and inverse displacement analysis of spherical Stephenson-III six-bar mechanism are obtained.Finally,numerical examples of forward and inverse analysis are presented to illustrate the proposed approach.The results indicate that the constraint equations established with the proposed method are much simpler than those reported by previous literature,and can be readily eliminated and solved.

ON INCREASING THE PARALLELISM IN NUMERICAL ALGORITHMS

Parallel algorithms have been designed for the past 20 years initially by parallelising existing sequential algorithms for many different parallel architectures. More recently parallel strategies have been identified and utilized 'resulting in many new parallel algorithms. However the analysis of such algorithms reveals that further strategies can be applied to increase the parallelism. One of these, i.e., increasing the computational capacity in each processing node can reduce the congestion/communicgtion for shared memory/distributed memory multiprocessor systems and dramahcally improve the Performance of the algorithm. Two algorithms are identified and studied, i.e., the Cyclic reduction method for solving large tridiagonal linear systems in which the odd/even sequence is increased to a 'stride of 3' or more resulting in an improved algorithm. Similarly the Gaussian Elimination method for solving linear systems in which one element is eliminated at a time can be adapted to parallel form in which two elements are simultaneously eliminated resulting in the Parallel Implicit Elimination (P.I.E.) method. Numerical results are presented to support the analyses.

Space Discretization of Time-Fractional Telegraph Equation with Mamadu-Njoseh Basis Functions

In this paper, we examine the space discretization of time fractional telegraph equation (TFTE) with Mamadu-Njoseh orthogonal basis functions. For ease and convenience, we deal with the fractional derivative by first converting from Caputo’s type to Riemann-Liouville’s type. The proposed method was constrained to precise error analysis to establish the accuracy of the method. Numerical experimentation was implemented with the aid of MAPLE 18 to show convergence of the method as compared with the analytic solution.

The Application of Sylvester Matrix to Analyze in Robotic Position

The elimination method is a general method to solve equations. The resultant method is an important tool in elimination theory. In the analysis of robot pose, equations are solved according to its graph. Resultants can simplify to solve system equation.

Discussion on the application of Ginseng efficacy from"Treatise on Spleen and Stomach"

According to the original works of an ancient Chinese medicine book"Treatise on Spleen and Stomach"(Detailed Analysis of Spleen and Stomach Diseases;1249 C.E.)written by Li Dongyuan(1180-1251 C.E),and related explanations of later physicians,the efficacy and application of Ginseng(Ginseng Radix et Rhizoma)in related prescriptions were investigated from the aspects of dosage,usage and treatment methods.Ginseng appeared in 27 named prescriptions,including 20 decoctions,4 pill and 3 powder.In terms of dosage and usage,the common dosage of Ginseng is five points,mainly decoction.Most of the processing methods are"Fuju"(Bite up the medicine with your mouth so it can be taken.)and decoction with water.In terms of treatment methods,Ginseng is used in the sweet temperature and heat removing method and Nourishes Qi and Eliminates Evils Method(build your body to cure disease)in Li Dongyuan's"Treatise on Spleen and Stomach".Explaining Sweet Temperature Heat Removal Method(Use sweet and warm medicines to treat Qi-deficiency fever or Blood-deficiency fever;there are many reasons for fever,and cold and cooling medicines with heat-clearing effect are generally used for treatment.Use bitter cold medicine,so as not to consume the body's Yang.)takes Buzhong Yiqi Decoction(BYD)as an example,and explaining the Nourishes Qi and Eliminates Evils Method takes Mahuang-renshen-shaoyao decoction as an example.At the same time,"Treatise on Spleen and Stomach"matured in the Jin and Yuan Dynasties(1115-1368 C.E.).Its system of weights and measures,terminology and medical theory are quite different from those of modern Chinese medicine.Therefore,when researching the dosage,usage and treatment methods of Ginseng,we will explore the controversial points,and explain them by using modern Chinese medicine theory,so as to guide the clinical medication more accurately.

Method for eliminating mismatching error in monolithic triaxial ring resonators

Several results on optical-axis perturbation and elimination of the mismatching error C of a monolithic triaxial ring resonator （MTRR） are reported. Based on the augmented 5×5 ray matrix method, by simultaneously considering axial displacement of a mirror and the misalignments in three planar square ring resonators of a MTRR, the rules of optical-axis perturbation are obtained. The mismatching error C of the MTRR is eliminated. The results obtained are important for cavity design, as well as in the improvement and alignment of MTRR.

Influence of vibration on the grain flow sensor during the harvest and the difference elimination method

The grain yield data collected by the intelligent yield measurement system of the combine harvester is generated into a field plot yield distribution map,which is of great significance for guiding agricultural production.However,in the process of drawing the yield map,the combine harvester is affected by vibration during operation and the generated error data in the process of collecting data which will cause the drawing results to be inaccurate.This study researched two factors that cause errors,then,the influence of vibration interference on the measurement signal was eliminated by filtering,vibration isolation,and designing a double-plates differential grain flow sensor.Three methods were taken to eliminate random errors,gross errors and systematic errors,including using the arithmetic average value to replace the true value,the 3σcriterion,and removing the filling time data and the delaying time data.Finally,the grain yield distribution map was obtained through Matlab and Excel.The results showed that the interference frequency above 50 Hz could be eliminated by filtering,but it was difficult to filter the low-frequency signal which was close to the grain impact frequency.The vibration amplitude was reduced to 14.29%by adding a vibration isolation plate,and the SNR was increased from−4.67 dB to 29.21 dB by combining low-pass filtering and damping vibration isolation.When the grain feeding rate was 2 kg/s,the natural vibration amplitude of the sensor after difference was about 0.02 V and evenly distributed around the zero voltage 0.2 V.The influence of positive and negative offset on the average value of grain impact signal could be ignored,and the signal-to-noise ratio was increased from 29.21 dB to 62.49 dB.The results of field experiments showed that the yield map drawn can clearly display the yield value of the harvest area,which is used to guide agricultural production.

Realizations of BELS as WIV method in both direct and indirect closed-loop system identification

The bias eliminated least squares （BELS） method, which is known as efficient for unknown parameter estimation of transfer function in the correlated noise case, has been developed and applied effectively to the closed-loop system identification. In this paper, under the general settings, the realizations of the BELS method as a weighted instrumental variables （WlV） method in both direct and indirect closed- loop system identification are established through constructing an appropriate weighting matrix in the WlV method. The constructed structures are similar in both cases, which reveals that all the proof procedures of the two realizations are the same. Thus, the unified realizations of the BELS as the WlV method for the closed-loop system identification can be built. A simulation example is given to validate our theoretical analysis.

The Core-EP, Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

We study the constrained systemof linear equations Ax=b,x∈R(A^(k))for A∈C^(n×n)and b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^(D)b.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min_(x∈R(A^(k)))||b−Ax||2,where||·||2 is the 2-norm.For the inconsistent system with a matrix A of index one,it was proved recently that the solution is A^(■)b using the core inverse A^(■)of A.For matrices of an arbitrary index and an arbitrary b,we show that the solution of the constrained system can be expressed as A^(■)b where A^(■)is the core-EP inverse of A.We establish two Cramer’s rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index.Using these expressions,two Cramer’s rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper.We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.

Certifying the Global Optimality of Quartic Minimization over the Sphere

The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.

A MAPLE PACKAGE OF AUTOMATED DERIVATION OF HOMOTOPY ANALYSIS SOLUTION FOR PERIODIC NONLINEAR OSCILLATIONS

Based on the homotopy analysis method, a general analytic technique for strongly nonlinear problems, a Maple package of automated derivation （ADHO） for periodic nonlinear oscillation systems is presented. This Maple package is valid for periodic oscillation systems in rather general, and can automatically deliver the accurate approximations of the frequency co and the mean of motion δof a nonlinear periodic oscillator. Based on the homotopy analysis method which is valid even for highly nonlinear problems, this Maple package can give accurate approximate expressions even for nonlinear oscillation systems with strong nonlinearity. Besides, the package is user-friendly： One just needs to input a governing equation and initial conditions, and then gets satisfied analytic approximations in few seconds. Several different types of examples are given in this paper to illustrate the validity of this Maple package. Such kind of package provides us a helpful and easy-to-use tool in science and engineering to analyze periodic of this Maple package from the is published publicly. nonlinear oscillations. And it is free address http：//numericaltank.sjtu to download the electronic version edu.cn/sjliao.htm once the paper

Trigonometric tension B-spline collocation approximations for time fractional Burgers’equation

This manuscript’s aim is to form and examine the numerical simulation of Caputo-time fractional nonlinear Burgers’equation via collocation approach with trigonometric tension B-splines as base functions.First,L 1 discretization formula is utilized for the time fractional derivative and after linearizing the nonlinear term,the trigonometric tension B-spline interpolants are utilized to get a system of simultaneous linear equations that are solved via Gauss elimination method.Thus,numerical approximation at the desired time level is obtained.It is demonstrated via von-Neumann approach that proposed scheme produces stable solutions.The results of six different test examples having their analytical solutions are compared with the results in the literature to validate the accuracy and efficiency of the scheme.