In response surface methodology, the central composite design is the most popular choice for fitting a second order model. The choice of the distance for the axial runs, alpha, in a central composite design is very crucial to the performance of the design. In the literature, there are plenty of discussions and recommendations for the choice of alpha, among which a rotatable alpha and an orthogonal blocking alpha receive the greatest attention. Box and Hunter (1957) discuss and calculate the values for alpha that achieve rotatability, which is a way to stabilize prediction variance of the design. They also give the values for alpha that make the design orthogonally blocked, where the estimates of the model coefficients remain the same even when the block effects are added to the model. In the last ten years, people have begun to realize the importance of a split-plot structure in industrial experiments. Constructing response surface designs with a split-plot structure is a hot research area now. In this dissertation, Box and Hunters' choice of alpha for rotatablity and orthogonal blocking is extended to central composite designs with a split-plot structure. By assigning different values to the axial run distances of the whole plot factors and the subplot factors, we propose two-strata rotatable splitplot central composite designs and orthogonally blocked split-plot central composite designs. Since the construction of the two-strata rotatable split-plot central composite design involves an unknown variance components ratio d, we further study the robustness of the two-strata rotatability on d through simulation. Our goal is to provide practical recommendations for the value of the design parameter alpha based on the philosophy of traditional response surface methodology. / Ph. D.

Fractional factorial designs are a popular choice in designing experiments for studying the effects of multiple factors simultaneously. The first step in planning an experiment is the selection of an appropriate fractional factorial design. An appro- priate design is one that has the statistical properties of interest of the experimenter and has a small number of runs. This requires that a catalog of candidate designs be available (or be possible to generate) for searching for the "good" design. In the attempt to generate the catalog of candidate designs, the problem of design isomor- phism must be addressed. Two designs are isomorphic to each other if one can be obtained from the other by some relabeling of factor labels, level labels of each factor and reordering of runs. Clearly, two isomorphic designs are statistically equivalent. Design catalogs should therefore contain only designs unique up to isomorphism. There are two computational challenges in generating such catalogs. Firstly, testing two designs for isomorphism is computationally hard due to the large number of possible relabelings, and, secondly, the number of designs increases very rapidly with the number of factors and run-size, making it impractical to compare all designs for isomorphism. In this dissertation we present a new approach for tackling both these challenging problems. We propose graph models for representing designs and use this relationship to develop efficient algorithms. We provide a new efficient iso- morphism check by modeling the fractional factorial design isomorphism problem as graph isomorphism problem. For generating the design catalogs efficiently we extend a result in graph isomorphism literature to improve the existing sequential design catalog generation algorithm. The potential of the proposed methods is reflected in the results. For 2-level regular fractional factorial designs, we could generate complete design catalogs of run sizes up to 4096 runs, while the largest designs generated in literature are 512 run designs. Moreover, compared to the next best algorithms, the computation times for our algorithm are 98% lesser in most cases. Further, the generic nature of the algorithms makes them widely applicable to a large class of designs. We give details of graph models and prove the results for two classes of designs, namely, 2-level regular fractional factorial designs and 2-level regular fractional factorial split-plot designs, and provide discussions for extensions, with graph models, for more general classes of designs.

Orientadores: Roy Edward Bruns, Isabel Cristina S. F. Jardim / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Quimica / Made available in DSpace on 2018-08-09T16:30:39Z (GMT). No. of bitstreams: 1 Breitkreitz_MarciaCristina_M.pdf: 914312 bytes, checksum: 69f8afa4f7adf7b6e40961bef5aed1ee (MD5) Previous issue date: 2007 / Resumo: Este trabalho teve como objetivo o desenvolvimento de modelos combinados considerando simultaneamente o tipo de Fase Estacionária (FE) como variável de processo e diferentes composições de Fase Móvel (FM) como variáveis de mistura para descrever a influência de cada uma destas variáveis, bem como a interação entre elas, na separação de diversos compostos presentes em duas amostras: uma mistura de compostos neutros e uma mistura de agrotóxicos. Os experimentos necessários para a determinação dos coeficientes dos modelos foram realizados de acordo com um planejamento split-plot, no qual os tipos de FE, C 8 e C 18 foram considerados main-plots e as composições de FM, sub-plots. Os resultados foram tratados de duas maneiras: de acordo com a estrutura split-plot do planejamento e supondo completa aleatorização na realização dos experimentos. Para descrever a qualidade da separação, foi utilizada uma função objetiva e o procedimento de otimização simultânea de várias respostas, descrito por Derringer e Suich, empregando, neste caso, critérios elementares como fator de retenção, resolução e fator de separação como respostas. Os modelos foram avaliados empregando-se Análise da Variância quanto à significância dos tratamentos e falta de ajuste. Na descrição da qualidade da separação dos compostos nas duas misturas, o procedimento de Derringer e Suich se mostrou superior às funções objetivas, pois permitiu a construção de modelos tendo como respostas parâmetros cromatográficos, os quais são, de fato, funções da composição da FM e da FE. Estes modelos, quando combinados através da desejabilidade global, permitiram que as condições para a melhor separação de todos os compostos em cada mistura fossem alcançadas, sem perda de informações sobre a separação individual dos pares de picos. Todos os modelos apresentaram capacidade preditiva para as respostas avaliadas ¿ fatores de retenção/resolução/fatores de retenção relativos - nas duas misturas, sem ou com pequena falta de ajuste. Embora todos os planejamentos tenham sido realizados de acordo com um procedimento split-plot, não foram verificadas diferenças nos valores dos erros dos coeficientes dos modelos matemáticos nos cálculos split-plot e supondo completa aleatorização e isto se deveu ao fato do erro main-plot ter sido muito menor que o erro sub-plot / Abstract: The aim of this work was to develop combined statistical models including the stationary phase (SP) as process variables and different compositions of the mobile phase (MP) as mixture variables in order to describe the influence of each type of variable as well as their interactions for the separation of compounds in two samples sets: one containing ten neutral compounds and another containing eleven pesticides. The experiments required to determine the coefficients of the models were carried out according to a split-plot approach, in which the stationary phases, C 8 or C 18 were considered as main-plots and the mobile phase compositions as sub-plots. The results were treated according to the split-plot approach and also supposing a completely random setup. The results provided by an objective function were compared to those obtained by Derringer¿s desirability functions constructed with simple chromatographic criteria such as resolution and relative retention factors as responses. The models were evaluated by means of Analysis of Variance, regarding regression significance and lack of fit. In order to describe the quality of the separation of the compounds in the two mixtures, the desirability procedure was preferred over the objective functions because the responses used in the latter were, in fact, functions of the stationary and mobile phases. The models combined into a global desirability function allowed the best conditions to the separation of all compounds to be found, without loss of information on the individual peak separation. All models presented predictive capabilities for the responses evaluated with none or little lack of fit. Although the experiments were carried out according to a split-plot approach, no significant differences were found in coefficient errors comparing to the complete random approach, which can be explained based on the low main-plot error / Mestrado / Quimica Analitica / Mestre em Química