Multivariate quality control problems

Robust design (parameter design), originally proposed by Taguchi, is a quality engineering method for reducing variation and improving product process quality. The goal of parameter design is to identify settings of product design characteristics that make the product\u27s performance less sensitive to the effects of environmental variables, deterioration, and manufacturing variations. That is, the method finds the settings of design factors that minimize expected loss (Performance Measure) due to variation. By exploiting the relationship between control parameters and noise variables, it reduces the effect of uncontrollable variations on the response. This is done by using statistically designed experiments in off-line situations where settings of the noise variables are controlled and systematically introduced and their relationships with design factors studied. In recent years, Response Surface Methodology has become the most reasonable approach to quality control problems after Taguchi\u27s two-step procedures involving quantities he calls Signal-to-noise (SN) ratios became very controversial. Analyzing problems, which are multivariate in nature, in a univariate manner has been done for a long time because of lack of computing power and lack of available techniques. Industry has started recognizing quality as a multivariate property. In this dissertation, a multivariate extension of the univariate response surface model suggested initially by Box and Jones is developed. A multivariate extension of the univariate quadratic loss function is discussed and chosen in chapter two. The mean vector and variance-covariance matrix of a performance vector characteristic according to such a model are obtained in chapter three. Multivariate Performance Measures are derived for cases in which the aim is to set the product on target and to minimize the variability. Chapter four discusses minimization of such multivariate performance measures. Solutions (optimum control settings) are obtained explicitly for models which on each coordinate contain only linear control factors. In other cases, calculating such optimum settings numerically is discussed. However, there exists an exceptional case in which optimum control settings are obtained even though second order terms in control variables are allowed in the model. Finally, in chapter five, the results obtained in the previous chapters are implemented on a data set provided by a chemical company. In this application, univariate optimum control settings are obtained first to illustrate the need of multivariate analysis. Then optimum settings are investigated for different defining matrices, corresponding to different performance measures