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This thesis focuses on the construction of optimum designs for the estimation of the variance components in multistage variability models. Variance components are the model parameters that represent the different sources of variability that affect the response of a system. A general and highly detailed way to define the linear mixed effects model is proposed. The extension considers the explicit definition of all the elements needed to construct a model. One key aspect of this formulation is that the random part is stated as a functional that individually determines the form of the design matrices for each random regressor, which gives significant flexibility. Further, the model is strictly divided into the treatment structure and the variability structure. This allows separate definitions of each structure but using the single rationale of combining, with little restrictions, simple design arrangements called factor layouts. To provide flexibility for considering different models, methodology to find and select optimum designs for variance components is presented using MLE and REML estimators and an alternative method known as the dispersion-mean model. Different forms of information matrices for variance components were obtained. This was mainly done for the cases when the information matrix is a function of the ratios of variances. Closed form expressions for balanced designs for random models with 3-stage variability structure, in crossed and nested layouts were found. The nested case was obtained when the information matrix is a function of the variance components. A general expression for the information matrix for the ratios using REML is presented. An approach to using unbalanced models, which requires the use of general formulae, is discussed. Additionally, D-optimality and A-optimality criteria of design optimality are restated for the case of variance components, and a specific version of pseudo-Bayesian criteria is introduced. Algorithms to construct optimum designs for the variance components based on the aforementioned methodologies were defined. These algorithms have been implemented in the R language. The results are communicated using a simple, but highly informative, graphical approach not seen before in this context. The proposed plots convey enough details for the experimenter to make an informed decision about the design to use in practice. An industrial internship allowed some the results herein to be put into practice, although no new research outcomes originated. Nonetheless, this is evidence of the potential for applications. Equally valuable is the experience of providing statistical advice and reporting conclusions to a non statistical audience.

Em um ambiente competitivo, sistemas estruturais devem ser projetados levando-se em conta, além da funcionalidade, o custo total de construção e operação da estrutura, bem como a sua capacidade de geração de lucro. Os custos estão diretamente ligados ao risco que a construção e operação da estrutura oferecem. O risco deve ser entendido como o produto de um custo esperado de falha pela probabilidade de que essa falha aconteça. A segurança da estrutura está diretamente relacionada com os coeficientes de segurança adotados em projeto. Verifica-se, portanto, que a minimização do custo total de um sistema estrutural passa, necessariamente, por uma otimização do nível de segurança para o qual o sistema é projetado. Diante desses fatos, neste trabalho realiza-se a busca do coeficiente de segurança parcial ótimo que minimiza o custo esperado total de sistemas estruturais: a chamada otimização de risco baseada em confiabilidade. Para tanto, foi desenvolvido um programa computacional em Fortran que encontra o coeficiente de segurança ótimo. Tal programa está acoplado a um programa de análise de confiabilidade estrutural desenvolvido na EESC/USP e a um programa comercial de elementos finitos. O trabalho inclui alguns estudos de caso, entre eles o de uma torre de telefonia sujeita a cargas de vento e de tornado. Esses estudos mostram que quando parcelas não estruturais dominam o custo, um projeto super-dimensionado não representa grande perda econômica, enquanto um projeto sub-dimensionado pode causar enormes prejuízos. O trabalho aponta que a confiabilidade ótima é altamente dependente das conseqüências e custos de falha. Mostra também que casualidades que possuem grande incerteza e grandes conseqüências de falha tendem a dominar o projeto. O estudo indica que, em situações de projeto diferentes (e.g., carregamentos diferentes), é razoável trabalhar-se com níveis de confiabilidade distintos. Além disso, mostra que a falta de percepção do risco pelas partes envolvidas em um projeto pode levar a contratos inadequados. Finalmente, o estudo revela que para uma estrutura submetida a carregamentos que variam com o tempo, como tornados, o custo e a confiabilidade ótimos são altamente dependentes da vida útil adotada para o projeto. / In a competitive environment, structural systems must be designed considering not only their function, but also their total costs of construction and operation and their ability to generate profit. The costs are directly linked to the risk resulting from construction and operation of the structure. Risk is defined as the product of failure cost by failure probability. The safety of the structure is directly related to safety coefficients adopted in design. Hence, the minimization of the total cost of a structural system necessarily involves an optimization of the safety level for which the system is designed. Based on these facts, the present study investigates the partial safety factor which minimizes the total expected cost of specific structural systems. This is referred to as reliability-based risk optimization. A Fortran computer code is developed to find the optimum safety factor. This code works together with a structural reliability analysis program developed at EESC/USP and with a commercial finite element program. Some case studies are presented, including the design of a steel frame communications tower, subjected to extreme storm and tornado wind loads. These studies show that when non-structural terms dominate the cost function, it is not too costly to over-design, but an under-designed project can cause huge money losses. The study shows that optimum reliability is strongly dependant on limit state exceedance consequences. It also shows that events with high uncertainty and high failure consequences generally dominate the design, and that for different design situations (e.g., multiple hazards) it is reasonable to work with different reliability levels. Moreover, it shows that when risk is not properly understood, it is also not properly dealt with by the parties involved, leading to inadequate contracts. Finally, the study shows that for structures under time-varying loads, such as tornados, the optimum reliability is strongly dependent on the selected design life.

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In this paper we discuss an algorithm for the construction of D-optimal experimental designs for the parameters in a regression model when the errors have a correlation structure. We show that design points can collapse under the presence of some covariance structures and a so called nugget can be employed in a natural way. We also show that the information of equidistant design on covariance parameter is increasing with the number of design points under exponential variogram, however these designs are not D-optimal. Also in higher dimensions the exponential structure without nugget leads to collapsing of the D-optimal design when also parameters of covariance structure are of interest. However, if only trend parameters are of interest, the designs covering uniformly the whole design space are very efficient. For illustration some numerical examples are also included. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

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