A Confidence Region for Zero-Gradient Solutions for Robust Parameter Design Experiments

One of the key issues in robust parameter design is to configure the controllable factors to minimize thevariance due to noise variables. However, it can sometimes happen that the number of control variables isgreater than the number of noise variables. When this occurs, two important situations arise. One is thatthe variance due to noise variables can be brought down to zero The second is that multiple optimal controlvariable settings become available to the experimenter. A simultaneous confidence region for such a locusof points not only provides a region of uncertainty about such a solution, but also provides a statistical testof whether or not such points lie within the region of experimentation or a feasible region of operation.However, this situation requires a confidence region for the multiple-solution factor levels that providesproper simultaneous coverage. This requirement has not been previously recognized in the literature. Inthe case where the number of control variables is greater than the number of noise variables, we show howto construct critical values needed to maintain the simultaneous coverage rate. Two examples are providedas a demonstration of the practical need to adjust the critical values for simultaneous coverage