Nonlinear models for mixture experiments including process variables

Abstract

Mixture experiments are applied in a variety of fields, for example, food processing, chemical engineering, and product quality improvement. All these examples have in common that they rely on experiments that mix multiple components. In mixture experiments, the measured response is a function not of the amount of the mixture components but their proportions. This adds a layer of complexity to the modeling of such experiments. In some mixture experiments, the blending properties of the mixture may be affected by the processing conditions, such as temperature and pressure, besides the mixture components’ proportions. Such types of mixture experiments are known as mixture-process variables experiments. This present work is concerned with finding and assessing a class of models that flexibly fits data from mixture experiments and mixture-process variables experiments, and with providing guidelines for how to design mixture experiments and mixture-process variables experiments when these models are fitted. Most models in the literature are either based on polynomials and are therefore not very flexible, or have a large number of parameters that make the response surface interpretation difficult to understand. The modified fractional polynomial models are a recent class of models from the literature that are flexible and parsimonious but quite restrictive. We contribute to mixture experiments by proposing a new class of nonlinear models, the complement mixture fractional polynomial (CMFP) models, by making an additional transformation of the fractional polynomial, which results in less restrictive models while retaining (and indeed exceeding) the advantages of this class. Moreover, we suggest an extended form for the modified fractional polynomial models to fit data from mixture-process variables experiments. A further main concern of this thesis is to optimize the response by finding the optimal combinations of proportions of the mixture components. Therefore, we found the maximum response by determining the corresponding proportions of the mixture components that achieved this. We conducted a simulation study in which we evaluated the performance of the CMFP models and compared them with different models from the literature. An efficient design can greatly improve the analysis of an experiment. We constructed exact and near-optimal mixture designs in constrained experimental regions for CMFP models with respect to the D-optimal criteria. Comparisons between the obtained designs in terms of their efficiency and robustness assessment are illustrated with several examples.

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Identifiers

ORCID for Shroug Abdullah S Alzahrani: orcid.org/0009-0004-3862-7316

ORCID for Stefanie Biedermann: orcid.org/0000-0001-8900-8268

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