Stakeholder-Based Deterministic and Probabilistic Quality Control Using Gaussian Mixtures for Circular Factories

Quality Control (QC) in Circular Factorys (CFs) is fundamentally more challenging due to the uncertainty of the incoming prod-
ucts to be reprocessed, compared to linear factories, which have tight tolerances on the input raw materials and parts. One must have an expressive model to manage the multimodality of circular factories. QC and the decision-making process that it entails can be understood as a mapping from system requirements and variation space to discrete decisions, which can be formulated either deterministically or probabilistically. Ideally, the model must also be able to handle the deterministic and probabilistic cases seamlessly. Gaussian Mixture Models (GMMs) and their universal approximation property are considered suitable for the circular factory context. GMM can also address deterministic cases by approximating the Dirac delta as a limiting case. We present a stakeholder-oriented QC framework in which stakeholder requirements, product generations, and alternative solutions for new product generations are represented probabilistically over quality-relevant variation spaces. Two complementary decision perspectives are considered. ... mehrFirst, stakeholder requirements, exemplified by customer requirements, are modeled as acceptance regions, and the probability that a product generation satisfies individual, joint, or alternative requirements is computed by integrating the corresponding GMM density over these regions. Second, stakeholder requirements and alternative solutions are represented as probability density functions, and distance- and divergence-based metrics are used to quantify their dissimilarity. These metrics support threshold-based acceptance and ranking of alternative solutions under uncertainty. Special attention is given to deterministic limits, threshold design, and the interpretation of distances and divergences when comparing GMMs. The proposed framework provides a basis for stakeholder-based deterministic and probabilistic QC and decision support in circular factories.