Contributions to quality improvement methodologies and computer experiments

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Tan, Matthias H. Y.
Wu, C. F. Jeff
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This dissertation presents novel methodologies for five problem areas in modern quality improvement and computer experiments, i.e., selective assembly, robust design with computer experiments, multivariate quality control, model selection for split plot experiments, and construction of minimax designs. Selective assembly has traditionally been used to achieve tight specifications on the clearance of two mating parts. Chapter 1 proposes generalizations of the selective assembly method to assemblies with any number of components and any assembly response function, called generalized selective assembly (GSA). Two variants of GSA are considered: direct selective assembly (DSA) and fixed bin selective assembly (FBSA). In DSA and FBSA, the problem of matching a batch of N components of each type to give N assemblies that minimize quality cost is formulated as axial multi-index assignment and transportation problems respectively. Realistic examples are given to show that GSA can significantly improve the quality of assemblies. Chapter 2 proposes methods for robust design optimization with time consuming computer simulations. Gaussian process models are widely employed for modeling responses as a function of control and noise factors in computer experiments. In these experiments, robust design optimization is often based on average quadratic loss computed as if the posterior mean were the true response function, which can give misleading results. We propose optimization criteria derived by taking expectation of the average quadratic loss with respect to the posterior predictive process, and methods based on the Lugannani-Rice saddlepoint approximation for constructing accurate credible intervals for the average loss. These quantities allow response surface uncertainty to be taken into account in the optimization process. Chapter 3 proposes a Bayesian method for identifying mean shifts in multivariate normally distributed quality characteristics. Multivariate quality characteristics are often monitored using a few summary statistics. However, to determine the causes of an out-of-control signal, information about which means shifted and the directions of the shifts is often needed. We propose a Bayesian approach that gives this information. For each mean, an indicator variable that indicates whether the mean shifted upwards, shifted downwards, or remained unchanged is introduced. Default prior distributions are proposed. Mean shift identification is based on the modes of the posterior distributions of the indicators, which are determined via Gibbs sampling. Chapter 4 proposes a Bayesian method for model selection in fractionated split plot experiments. We employ a Bayesian hierarchical model that takes into account the split plot error structure. Expressions for computing the posterior model probability and other important posterior quantities that require evaluation of at most two uni-dimensional integrals are derived. A novel algorithm called combined global and local search is proposed to find models with high posterior probabilities and to estimate posterior model probabilities. The proposed method is illustrated with the analysis of three real robust design experiments. Simulation studies demonstrate that the method has good performance. The problem of choosing a design that is representative of a finite candidate set is an important problem in computer experiments. The minimax criterion measures the degree of representativeness because it is the maximum distance of a candidate point to the design. Chapter 5 proposes algorithms for finding minimax designs for finite design regions. We establish the relationship between minimax designs and the classical set covering location problem in operations research, which is a binary linear program. We prove that the set of minimax distances is the set of discontinuities of the function that maps the covering radius to the optimal objective function value, and optimal solutions at the discontinuities are minimax designs. These results are employed to design efficient procedures for finding globally optimal minimax and near-minimax designs.
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