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Improved formulation of the latent variable model inversion–based optimization problem for quality by design applications

Latent variable regression model (LVRM) inversion is a relevant tool for finding, if they exist, different combinations of manufacturing conditions that yield the desired process outputs. Finding the best manufacturing conditions can be done by optimizing an appropriately formulated objective function using nonlinear programming. To this end, different formulations of the optimization problem based on LVRM inversion have been proposed in the literature that allow the use of happenstance data (eg, historical data) for this purpose, present lower computational costs than optimizing in the space of the original variables, and guarantee that the solution will conform to the correlation structure of available data from the past. However, these approaches, as presented, suffer from some limitations, such as having to actively modify the constraints imposed on the solution to achieve different sets of conditions to those available in the LVRM calibration dataset, or the lack of a standardized approach for optimizing a linear combination of variables. Furthermore, when minimizing or maximizing one or more outputs, a severe handicap is also present related to the definition of arbitrarily low or high “desired” values. This paper aims at tackling all of these issues. The resulting proposed formulation of the optimization problem is illustrated with three case studies.