4 research outputs found

    Split-plot designs: What, why, and how

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    The past decade has seen rapid advances in the development of new methods for the design and analysis of split-plot experiments. Unfortunately, the value of these designs for industrial experimentation has not been fully appreciated. In this paper, we review recent developments and provide guidelines for the use of split-plot designs in industrial applications

    Understanding multistage experiments

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    Abstract: Current advanced manufacturing processes are composed of multiple complex stages which prohibit experimenters from conveniently employing traditional statistical experimental designs due to restrictions on randomisation. In this paper, we demonstrate, and summarise how split plot design and its variants have been used for multistage experimentation, and present several multistage experiment scenarios with comments for practitioners and researchers

    Listing Unique Fractional Factorial Designs

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    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

    脫rdenes de experimentaci贸n en dise帽os factoriales

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    Cuando se plantea un dise帽o factorial la pr谩ctica habitual es recomendar que los experimentos se realicen en orden aleatorio. Esta aleatorizaci贸n tiene como objetivo el proteger de la posible influencia de factores desconocidos, ya que se espera que esa influencia quede difuminada entre todos los efectos de forma que ninguno se vea especialmente afectado y no se cometan errores al valorar su significaci贸n estad铆stica. Pero este proceder tiene dos inconvenientes importantes: 1. El n煤mero de cambios de nivel en los factores que exige la aleatorizaci贸n puede ser grande (bastante mayor que otras posibles ordenaciones) y dif铆cil de llevar a la pr谩ctica, lo que complica y encarece la experimentaci贸n. 2. Si se realizan unas hip贸tesis que parecen muy razonables respecto al tipo de influencia que ejercen los factores desconocidos, existen 贸rdenes claramente mejores que otros para minimizar la influencia de esos factores ajenos a la experimentaci贸n.Numerosos autores han estado trabajando sobre este tema y ya han sido resueltos algunos aspectos, como el de determinar los 贸rdenes de experimentaci贸n que mejor neutralizan la influencia de factores desconocidos, aunque sin tener en cuenta el n煤mero de cambios en los niveles de los factores que esa ordenaci贸n implica. Adicionalmente, se ha resuelto el problema de encontrar 贸rdenes que presentan el m铆nimo n煤mero de cambios de nivel, pero sin considerar de forma simult谩nea la posible influencia de los factores desconocidos.Cuando se considera conjuntamente la influencia de los factores desconocidos y el n煤mero de cambios en los factores, se ha llegado hasta los dise帽os con 16 experimentos, pero no m谩s all谩 porque los procedimientos de b煤squeda empleados son inviables cuando el n煤mero de ordenaciones posibles se hace tan grande (con 32 experimentos el n煤mero de ordenaciones posibles es 32! = 2,6 路 1035)La tesis que se presenta tiene por objetivo encontrar un procedimiento que permita obtener 贸rdenes de experimentaci贸n con el m铆nimo n煤mero de cambios en los factores y que adem谩s minimicen la influencia de los factores desconocidos en la estimaci贸n de los efectos, para cualquier dise帽o factorial a 2 niveles. Adem谩s, se pretende elaborar un procedimiento de forma que dichas ordenaciones, con las propiedades deseadas, puedan ser obtenidas de forma f谩cil por el experimentador. El contenido se presenta estructurado en 7 cap铆tulos y 8 ap茅ndices. El capitulo 1 presenta las motivaciones que se consideraron para seleccionar este tema de investigaci贸n y define los elementos fundamentales que se presentan a lo largo del trabajo, tales como los dise帽os factoriales a dos niveles -completos o fraccionales- los problemas que puede causar la aleatorizaci贸n en estos dise帽os, y c贸mo cuantificar la influencia de los factores ajenos a la experimentaci贸n en la estimaci贸n de los efectos. Asimismo, se plantean las hip贸tesis y el contexto en que se buscar谩n los 贸rdenes de ejecuci贸n que presentan las propiedades deseadas.En el capitulo 2, se realiza una revisi贸n bibliogr谩fica exhaustiva de las propuestas existentes respecto a los 贸rdenes de ejecuci贸n de este tipo de dise帽os para conseguir que las conclusiones del an谩lisis no se vean afectadas por la influencia de factores ajenos a la experimentaci贸n y/o que el n煤mero de cambios a realizar en los niveles de los factores sea m铆nimo. Al final del cap铆tulo se comentan las debilidades del estado del arte actual y se plantean los aportes previstos en esta tesis. En el capitulo 3, se presenta un procedimiento original que permite encontrar 贸rdenes de experimentaci贸n para dise帽os factoriales a 2 niveles con el m铆nimo n煤mero de cambios en los factores y un sesgo conocido. A este procedimiento lo denominamos m茅todo de duplicaci贸n, ya que duplicando las filas de un dise帽o 2k y agregando un factor adicional con una determinada secuencia de signos, permite obtener un dise帽o 2k+1 que conserve las caracter铆sticas del dise帽o anterior. Una importante propiedad de este m茅todo es que puede aplicarse a cualquier n煤mero de factores. Este procedimiento garantiza el m铆nimo n煤mero de cambios de nivel, pero no siempre garantiza el m铆nimo sesgo (medida de la influencia de los factores desconocidos en la estimaci贸n de los efectos). En el capitulo 4, se utilizan diferentes m茅todos de b煤squeda para hallar 贸rdenes de experimentaci贸n que presenten un sesgo menor al proporcionado por el m茅todo de la duplicaci贸n.Estos m茅todos son:- B煤squeda aleatoria con restricciones: Se utiliza un procedimiento que va generando aleatoriamente el orden de ejecuci贸n de los experimentos pero de forma que a una condici贸n experimental solo puede seguirle otra condici贸n que presente solo un cambio en los niveles de los factores (para garantizar el m铆nimo n煤mero de cambios). Una vez completada una ordenaci贸n se calcula su sesgo, y se almacenan las ordenaciones con un sesgo por debajo de un cierto umbral.- B煤squeda exhaustiva: Se utiliza un algoritmo planteado por Dickinson (1974) y que fue adaptado por De Le贸n (2005). Es similar al algoritmo anterior, pero no genera las condiciones de experimentaci贸n de forma aleatoria sino que sigue una sistem谩tica para encontrar todas las ordenaciones posibles. De esta forma se ha encontrado la mejor ordenaci贸n para dise帽os con 32 experimentos, hasta ahora desconocida, entendiendo por mejor la que presenta m铆nimo n煤mero de cambios en los niveles de los factores y de entre estas, la que presenta menor sesgo. - B煤squeda exhaustiva con alimentaci贸n forzada. Es imposible explorar exhaustivamente todas las ordenaciones posibles si el n煤mero de experimentos es mayor a 32. Para explorar la zona de ordenaciones m谩s prometedora se ha aplicado el procedimiento de b煤squeda exhaustiva en tono a una ordenaci贸n que ya es buena y que ha sido obtenida por los m茅todos anteriores.Para dise帽os con m谩s de 32 experimentos los mejores 贸rdenes se obtienen de una combinaci贸n de los diferentes m茅todos propuestos. As铆, para dise帽os con 64 experimentos el mejor orden se obtiene con el m茅todo de b煤squeda exhaustiva con alimentaci贸n forzada, alimentando el algoritmo con una ordenaci贸n obtenida a trav茅s de la b煤squeda aleatoria con restricciones. La mejor ordenaci贸n para 128 experimentos se obtiene de la misma forma, pero alimentando el algoritmo con una ordenaci贸n obtenida por duplicaci贸n del orden obtenido para 64 experimentos.Los m茅todos descritos en el cap铆tulo 4 proporcionan lo que denominamos "贸rdenes semilla", ya que a partir de estos 贸rdenes se pueden deducir otros con sus mismas propiedades. En el capitulo 5, se presentan dos procedimientos para obtener 贸rdenes con las caracter铆sticas deseadas a partir de los 贸rdenes semilla. Estos m茅todos los denominamos m茅todo de permutaci贸n y cambios de signo y el m茅todo de las columnas de expansi贸n. Ambos m茅todos han sido programados en macros de Minitab, lo cual permite generar de forma autom谩tica y aleatoria (de entre todos los posibles) los 贸rdenes con las caracter铆sticas propuestas. En el capitulo 6, se presenta un nueva medida de atenuaci贸n de la influencia de los factores ajenos a la experimentaci贸n que permite comparar la atenuaci贸n entre dise帽os factoriales con diferente n煤mero de factores, mostrando que el procedimiento de duplicaci贸n presentado en el capitulo 3, es adecuado para obtener 贸rdenes de experimentaci贸n con las caracter铆sticas propuestas en dise帽os con m谩s de 128 experimentos. Finalmente, en el capitulo 7 se presentan las principales conclusiones obtenidas y se definen posibles futuras l铆neas de investigaci贸n que podr铆an ampliar los estudios realizados.En el anexo 1 se presentan los 贸rdenes propuestos por De Le贸n (2005) para dise帽os con 8 y 16 experimentos, citados en diversas ocasiones a lo largo de la tesis y que constituyen uno de los puntos de partida. El anexo 2 presenta el lenguaje de programaci贸n FreeBasic, utilizado para implementar los algoritmos de b煤squeda de ordenaciones, y en los anexos 3 y 4 se incluyen 2 de los programas realizados: el de b煤squeda aleatoria para dise帽os con 64 experimentos (anexo 3) y b煤squeda exhaustiva para dise帽o con 32 experimentos (anexo 4). En el anexo 5 se presenta uno de los 贸rdenes obtenidos con las propiedades deseadas para los dise帽os con 128 experimentos, y en los anexos 6 y 7 se incluyen las macros realizadas con el lenguaje de programaci贸n de Minitab y que a partir de las semillas para cada tipo de experimento, propone una ordenaci贸n de entre todas las posibles que tienen las propiedades deseadas. Finalmente, en el anexo 8 se realizan algunas consideraciones sobre el an谩lisis de los 贸rdenes propuestos, con restricciones en la aleatorizaci贸n, y se resumen las propuestas realizadas sobre este tema.A common recommendation when thinking in a factorial design is randomizing the run order. The purpose of this randomization is to protect the response from the possible influence of unknown factors. This influence is expected to be blurred among all the effects, thus none of them is specially affected and no mistakes are made when estimating its statistical significance. But this praxis has two essential problems: 1. The number of factor's level changes due to randomization might be large (much larger than in other sequences). It can be also difficult to conduct, making the experimentation complicated and more expensive. 2. Making some reasonable hypothesis regarding the influence of the unknown factors, there are some sequences clearly better than others for minimizing the influence of this undesirable factors.Many authors have worked on this topic, and some matters have already been solved. For instance, the experimentation sequence that better neutralises the influence of unknown factors is already determined, but without taking into consideration the number of level changes that this sequence implies. It has also been solved the problem of finding sequences that have the minimum number of level changes, but without considering simultaneously the potential influence of unknown factors. When both the influence of unknown factors and the number of level changes is considered, the problem has been solved up to designs with 16 runs. But not further as the searching procedures used are nonviable when the number of possible sequences becomes so huge (with 32 runs the number of different sequences is 32! = 2,6 路 1035) The aim of this thesis is finding a procedure that makes it possible to obtain run sequences with the minimum number of level changes, and that besides minimize the influence of unknown factors in the effect estimation, for any 2 level factorial design.Moreover, the desired run sequence should be obtained easily by the experimenter when using the proposed procedure.The content is structured in 7 chapters and 8 appendixes. Chapter 1 shows the motivation that lead to chose this research topic. It also defines the basic elements of this work (complete and fractional 2 level factorial designs, problems that appear when randomizing this designs, and how to quantify the influence of unknown and undesired factors in the effect estimation). In addition, the hypothesis and context in which the search for run orders with the desired properties will take place are presented.Chapter 2 gives an exhaustive bibliographic review of the current solutions related with run orders in these designs robust to the influenceof factors alien to the experimentationand/or with minimum number of level changes. The end of the chapter lists weaknesses of the current state of the art and advances the expected contributions of this thesis. Chapter 3 presents an original procedure for finding run orders for 2 level factorial designswith the minimum number of changes in the level factors and a known bias. We called this procedure duplication method, as duplicating the rows of a 2k design and adding a factor with a specific sign sequence, a 2k+1 design with the same properties as the first design is achieved. An important property of this method is that it can be applied to any number of factors. This procedure guarantees the minimum number of level changes, but not always guaranties the minimum bias (measure of the influence that unknown factors have in the effect estimation). Chapter 4 shows different methods for finding run orders with less bias than the one produced by the duplication method. These methods are: - Random search with restrictions: The procedure randomly generates the run order, but in a way that a run is followed by another one that has only one change in the factor levels (the minimum number of changes is then guaranteed). Once the sequence is completed its bias is calculated, and the sequences with a bias under a threshold are stored.- Exhaustive search: An algorithm proposed by Dickinson (1974) and adapted by De Le贸n (2005) is used. It is similar to the previous algorithm, but it does not generate the runs in a random manner. Instead, it behaves systematically in order to find all the possible run orders. With this algorithm the best run order for designs with 32 experiments has been found (and it was unknown until now). The best run order means the one that has minimum number of changes in the levels and, among these, the one with less bias.- Exhaustive search with forced feeding. The exhaustive exploration of all possible run orders with more than 32 runs is impossible. The procedure of exhaustive search around a good run order already found with one of the previous methods allowed the exploration of the most promising run order area. For designs with more than 32 runs the best run orders are obtained from a combination of the proposed methods. For designs with 64 runs the best order comes from the exhaustive search with forced feeding method, feeding the algorithm with a run order obtained from the random search with restrictions method. We used the same procedure for obtaining the best run order for 128 runs, but feeding the algorithm with a run order obtained from duplication of the one for 64 runs.Methods described in chapter 4 provide the so called "seed orders": from this orders new ones with the same properties can be deduced. Chapter5 shows two procedures for obtaining orders with the expected properties from the seed orders. These methods are called permutation and sign change method, and expansion columns method. Both methods have been programmed as Minitab macros, making it possible to automatically and randomly generate (among all possible ones) the orders with the desired properties. A new measure for attenuating the influence of factors alien to experimentation is presented in chapter 6. This allows the comparison among the attenuation of factorial designs with different number of factors, thus showing that the duplication procedure shown in chapter 3 is appropriate for obtaining run orders with the properties desired in designs with more than 128 runs. Finally, chapter 7 gives the main conclusions and defines possible future research areas that could extend our studies.Appendix 1 shows the orders proposed by De Le贸n (2005) for designs with 8 and 16 experiments, cited several times in the thesis and one of our starting points. Appendix 2 explains the FreeBasic programming language, used for implementing the search algorithms. Appendixes 3 and 4 include 2 programs: random search for designs with 32 runs (appendix 3) and exhaustive search for designs with 32 experiments (appendix 4). Appendix 5 shows one of the obtained orders with the desired properties for designs with 128 runs. Appendixes 6 and 7 have the Minitab macros that using the seed orders for each kind of experiment proposes an order among all the possible ones with the desired properties. Finally, appendix 8 has some comments about the proposed run orders, with restrictions in the randomization, and summarizes the proposals about this topic
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