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The production planning problem in the Pinus radiata sawmill industry revolves around determining how to cut a set of logs of different diameters to obtain pieces with a rectangular base, typically of the same length as the original log. The primary objective is often to maximize the volumetric yield or the ratio between the total volume of the produced pieces and the available volume of the logs. Given the scarcity of timber forests, small and medium-sized (SME) sawmills must optimize their operations to maximize the volumetric yield of logs, usually procured from third parties and may vary in quality and dimensions. In this context, the absence of decision support tools directly contributes to inefficient raw material utilization, consequently impacting the business’s profit. This study introduces a multi-objective mixed-integer linear programming model incorporating log availability and product demand as input parameters. The objective functions aim to minimize the total volumetric loss of utilized logs and the surplus quantity associated with products exceeding demand. The model integrates cutting patterns pre-determined for each log diameter as an additional input. The Cutlog© software was used to identify all the optimal cutting patterns. The $\varepsilon $ -constraint method, implemented in the CPLEX© solver, was employed to solve the model. The model was validated using representative instances designed to emulate the challenges faced by SME sawmills. Real industry data, surveys, and commercial records from SME sawmills in southern Chile were utilized. The results confirm the effectiveness of the proposed model in addressing the multi-objective challenges encountered by these businesses. The model successfully identifies multiple solutions on the Pareto frontier, offering valuable insights for decision-making.

Machine learning (ML) methods often require large volumes of labeled data to achieve meaningful performance. The expertise necessary for labeling data in medical applications like pathology presents a significant challenge in developing clinical-grade tools. Crowdsourcing approaches address this challenge by collecting labels from multiple annotators with varying degrees of expertise. In recent years, multiple methods have been adapted to learn from noisy crowdsourced labels. Among them, Gaussian Processes (GPs) have achieved excellent performance due to their ability to model uncertainty. Deep Gaussian Processes (DGPs) address the limitations of GPs using multiple layers to enable the learning of more complex representations. In this work, we develop Deep Gaussian Processes for Crowdsourcing (DGPCR) to model the crowdsourcing problem with DGPs for the first time. DGPCR models the (unknown) underlying true labels, and the behavior of each annotator is modeled with a confusion matrix among classes. We use end-to-end variational inference to estimate both DGPCR parameters and annotator biases. Using annotations from 25 pathologists and medical trainees, we show that DGPCR is competitive or superior to Scalable Gaussian Processes for Crowdsourcing (SVGPCR) and other state-of-the-art deep-learning crowdsourcing methods for breast cancer classification. Also, we observe that DGPCR with noisy labels obtains better results ( $\text{F}1=81.91$ %) than GPs ( $\text{F}1=81.57$ %) and deep learning methods ( $\text{F}1=80.88$ %) with true labels curated by experts. Finally, we show an improved estimation of annotators’ behavior.