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.

Multivariate statistical methods are used to analyze data from an industrial batch drying process. The objective of the study was to uncover possible reasons for major problems occurring in the quality of the product produced in the process. Partial least-squares (PLS) methods were able to isolate which group of variables in the chemistry, in the timing of the various stages of the batch, and in the shape of the time-varying trajectories of the process variables were related to a poor-quality product. The industrial study illustrates the approach and the power of these multivariate methods for troubleshooting problems occurring in complex batch processes. Several new variations in the multivariate PLS methodology for the analysis of batch data are also implemented. In particular, an application utilizing a novel approach to the time warping of the trajectories for batches, and the subsequent use of the time-warping information, is presented. The use of the time history of the PLS weights of the process variable trajectories to diagnose problems in the dynamic operation of the batches is also clearly illustrated, as is the use of contribution plots for finding features which distinguished between the operating histories of good and bad batches.

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Following a model-centric strategy in the development of a manufacturing process for a new medicine empowers the simultaneous study of a large number of process parameters, which is large enough to exceed the capability of a graphic representation of the interactions across them. This work presents a discussion regarding the identification, description, and communication of multidimensional design spaces of high order. It introduces the reader to mathematical tools developed by the process systems engineering community that become relevant in the challenge to replace graphics as a means to describe and communicate a design space. Concepts like process f lexibility are discussed and illustrated. The paper also introduces geometric projection as a way to capture and describe the shape of the design space in an easier form (than that of the complete mechanistic model) that can be communicated to the regulator. An assessment is presented regarding the key elements communicated by a graphical representation of a design space, and alternate ways of conveying the same information using mathematics are suggested. These ideas are illustrated by applying them to the identification and definition of a design space for a chemical reaction step and the digital risk assessment for a packed bed adsorption step.

In the procedure to monitor a new batch using the method proposed by Nomikos and MacGregor [AIChE J. 1994[AIChE J. , 40 (8), 1361[AIChE J. -1375, an assumption about the unknown future samples in the batch has to be taken. This work demonstrates that using the missing data option and solving the score estimation problem with an appropriate method are equivalent to the use of an accurate adaptive forecast model for the future samples over the shrinking horizon of the remainder of the batch. The dynamic properties of the principal component analysis (PCA) model are illustrated by re-expressing the projection model as a time-varying multivariate prediction model. The benefits of using the missing data estimation option are analyzed by contrasting it with other options on the basis of (i) the accuracy of the forecast done for the unknown samples, (ii) the quality of the score estimates, and (iii) the detection performance during monitoring. Because of the tremendous structural information built into these multivariate PCA models for batch processes, the missing data option is shown to yield the best performance by all measures in predicting the future unknown part of the trajectory, even from the beginning of the batch. However, for the purpose of online detection of process faults (in process monitoring), the differences among the trajectory estimation methods appear to be much less critical because the control charts used in each case are tailored to the filling-in mechanism employed. All of the approaches appear to provide powerful charting methods for monitoring the progress of batch processes.