Numerical Optimization
Nooshin Hakamipour
Abstract
Purpose: In this paper k- level constant stress accelerated life test under Type-I progressive censoring for Lomax distribution with non-constant shape and scale parameters is investigated. The purpose of this paper is to estimate the model parameters using the EM algorithm and optimize the test design.Methodology: ...
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Purpose: In this paper k- level constant stress accelerated life test under Type-I progressive censoring for Lomax distribution with non-constant shape and scale parameters is investigated. The purpose of this paper is to estimate the model parameters using the EM algorithm and optimize the test design.Methodology: Life testing often consumes a very long time for testing and this is a fundamental problem in test design. This problem is solved by accelerated life tests. There is a recommended method for reducing the time of failure, such that the stress level of the test units will increase and then they will fail earlier than normal operating conditions. Therefore, these approaches reduced the running time. In this paper, the k-level constant stress accelerated life test under progressive Type-I censoring used. The Expectation-Maximization (EM) algorithm is used to determine the maximum likelihood estimates of the unknown parameters. This algorithm is a very powerful tool in handling the incomplete data problem. Two different criteria used to optimize the test plan. And the effect of the sample size, number of stress levels and inspection and the intermediate censoring proportion are assessed on the design efficiency.Findings: based on the simulation study and a real data set, it is demonstrated that the EM estimator is good. Also, under the optimization criterion II, a more efficient test was obtained than the optimization criterion I. In addition, the small sample size, the small number of stress levels, the small number of inspections and the large intermediate censoring proportion lead to a more efficient test.Originality/Value: In this paper, the periodic inspection is used to collect lifetime data. Although continuous is an ideal mode. But sometimes due to technical limitations and/or budgetary constraints, the continuous inspection is not possible in practice and the experimenter has to use the periodic inspection. In this case, the exact times of test units may not be available and only the failure counts are collected at certain time points during the test. Also, in this paper, we assumed that both scale and shape parameters to be log linear model by operating stress
Numerical Optimization
Abbas Parchami; Majid DoostMohammadi; Mashaaallah Mashinchi
Abstract
Newton's method, which is also known as the Newton Raphson algorithm, as one of the most efficient numerical methods in mathematics, is known as a method and approach for the root approximation of nonlinear equations. After reviewing and interpretation of this method, one of the most widely used in the ...
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Newton's method, which is also known as the Newton Raphson algorithm, as one of the most efficient numerical methods in mathematics, is known as a method and approach for the root approximation of nonlinear equations. After reviewing and interpretation of this method, one of the most widely used in the statistics, i.e. estimation of the unknown parameters by maximum likelihood method is presented in this paper. To facilitate the transfer of concepts, the article includes several different numerical examples and computer programs.
Numerical Optimization
Amir Hossein Salehi Shayegan; Ali Zakeri
Abstract
Inverse parabolic problems are the most famous ill-posed problems. Thus using stable inexact numerical methods for approximating these problems, causes a large perturbation. In this paper, we study the problem of determining the source term of the inverse source ...
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Inverse parabolic problems are the most famous ill-posed problems. Thus using stable inexact numerical methods for approximating these problems, causes a large perturbation. In this paper, we study the problem of determining the source term of the inverse source problem[{partial _t}T(x,t) = kappa ,{nabla ^2}T(x,t) + g(t)delta (x - {x^*}),x in {( circ ,,1)^d},t in ( circ ,,{t_f}),]from the measured data given in the form of[T({x_{measure}},{t_i}) = {y_i}, ,i = 1,2, ldots ,I,](additional condition) where d = 1,2 and [delta ] is the Dirac delta function and (T,g) are the unknown functions. Then, using the statistical spline model and applying Levenberg-Marquardt method, we obtain an approximate solution for quasi solution g=g(t). Finally, to show the priority and accuracy of the introduced method some numerical examples are given.