General Framework for Latent Variable Model Inversion for the Design and Manufacturing of New Products

Abstract

Latent variable regression model (LVRM) inversion is a useful tool to support the development of new products and their manufacturing conditions. The objective of the model inversion exercise is that of finding the best combination of regressors (e.g., raw material properties, process parameters) that are needed to obtain a desired response (e.g., product quality) from the model. Each of the published applications where model inversion has been applied utilizes a tailored approach to achieve the inversion, given the specific objectives and needs. These approaches range from the direct inversion of the LVRM to the formulation of an objective function that is optimized using nonlinear programming. In this paper we present a framework that aims to give a holistic view of the optimization formulations that can arise from the need to invert an LVRM. The different sets of equations that become relevant (either as a term within the objective function or as a constraint) are discussed, and an example of these scenarios is also provided. Additional to the formulation of the different scenarios and their objective functions, this work proposes a new metric (the P 2 statistic) to cross-validate the ability of the model to reconstruct the regressor vector (analogous to the Q 2 statistic aimed at the predictability of the response). This new metric comes from the need to not only predict the response from the regressor, but to also reconstruct the regressors from the scores values. In this context, a discussion is provided on the effect of uncertainty in the reconstruction of the regressor (the actual design) as these values are normally given upstream as targets to the supplier of materials, or as set points to the process.

Advanced Process Decision Making Using Multivariate Latent Variable Methods

“…The idea of latent variable model inversion, first introduced by Jaeckle and MacGregor [41][42][43], has been recently generalized by Tomba et al [46]. The following sections provide an overview of the theoretical aspects beyond this modeling technique.…”

“…A generalized framework for LVM inversion has been proposed recently by Tomba et al [46]. According to the authors, the inversion problem goes through the following steps:…”

“…For a more thorough discussion on the terms in Eqs. (22)(23)(24), the reader is referred to Yacoub and MacGregor [45], García-Muñoz et al [48,49,51] and Tomba et al [46,52].…”

“…These issues were addressed using LVM inversion by Tomba et al [46], García-Muñoz and Mercado [56] and Tomba et al [52].…”

“…Tomba et al [46] used the experimental data reported by Vemavarapu et al [57], who investigated the role of raw materials properties in the processability and performance of a wet granulated product. Namely, each raw material was characterized and processed at fixed process conditions, providing the information to study the effect of the input properties (H 2 O solubility, contact angle, H 2 O holding capacity, D [3,2], D90/D10, surface area, and pore volume) on the final product quality (loss on drying, oversize, Δ(Flodex), Δ(compactability), D [3,2], D90/D10, and growth ratio).…”