Sampling properties of optimal operating conditions of single and multiple response surface systems

When optimizing a process using Response Surface Methodology (RSM), it is a common practice to treat the empirical models obtained through a series of designed experiments as if the models were deterministic. However, even assuming that there is no model bias, the parameter estimates of the models exhibit random variability due to sampling errors that occur throughout the course of the experimentations. Therefore, the resulting analysis based on these empirical models has to take into account the inherent sampling variability of the parameter and response estimates. This sampling variability can be assessed through confidence statements.; One of the most important confidence statements in RSM is the confidence region for the location of the optimal operating conditions of a process, since the most common objective in RSM is to find the values of the controllable factors that give rise to an optimal process performance. These confidence regions can help the analyst making more realistic assessments on the quality of the optimal point estimate and the resulting statistical inferences on the optimality conditions. Albeit the valuable information rendered by such confidence regions, they are seldom used by RSM practitioners. One of the main reasons for this is the lack of efficient methodologies for computing the confidence regions, particularly when the operating conditions are constrained to lie inside a given region of the controllable factor space and when multiple responses are of interest.; The objective of this dissertation is to investigate and develop efficient algorithms for computing confidence regions for the location of the optimal point in unconstrained, constrained, and multiresponse optimizations. Existing methodologies for computing the confidence regions and their limitations are reviewed. Solutions that avoid these limitations are then proposed.; For the unconstrained case, a MAPLE computer algebra program is presented that automates the construction and display of Box-Hunter confidence regions. It is shown by example that these regions can be composed of disjoint subsets, a possibility not reported previously in the literature. For the constrained case, a methodology that avoids the limitations of the Lagrange multiplier approach, an existing approach for computing the confidence regions, is proposed. The proposed approach is based on finding the set of points over a grid for which a particular hypothesis cannot be rejected. The accuracy of the method, however, is made independent of the resolution of the grid of points thanks to the several computational devices implemented in the final algorithm. Coverage rate simulations are conducted to assess the quality of the proposed and existing methodologies. A MATLAB program implements the proposed algorithm for the computation and display of constrained confidence regions. The computer implementation can find the confidence regions for any linear statistical model. Finally, the proposed methodology is extended to the multiple response case by investigating its application to the so-called desirability function approach, a popular method useful in multiple response optimization problems. With this extension, confidence regions on the most desirable operating conditions can be obtained.