The compartmentalized knapsack problem arose from two‐phased cutting stock problems, especially with steel roll cutting. In its original formulation, it refers to an integer nonlinear optimization model, which, up to now, has been solved through decomposition heuristics. The objective of this article is to show that the constrained compartmentalized knapsack problem has a linear optimization model. Therefore, we have considered the original nonlinear model and propose a linear model for the problem, demonstrating their equivalence. We also propose a new decomposition heuristic and perform numerical tests to verify the limits of the proposed model and the quality of the heuristic solutions.
Este artigo apresenta resultados de um levantamento de pesquisas fundamentadas no referencial teórico do Pensamento Matemático Avançado (PMA) no contexto da licenciatura em Matemática, que resultou em um corpus de dezessete estudos que utilizaram o referencial teórico do PMA para verificar indícios de pensamento matemático em produções de estudantes. A partir da análise desse corpus, procurou-se entender como essas pesquisas relacionam a aprendizagem em Matemática com os indícios de pensamento matemático. São apresentadas concepções diferentes, mas não contraditórias, tais como: que a aprendizagem pode ser avaliada pelo desenvolvimento do PMA; que o PMA é necessário para a aprendizagem de Matemática Avançada e que o referencial teórico do PMA pode auxiliar o professor a diagnosticar dificuldades e contribuir para a aprendizagem de seus alunos.
Background: In Mathematics Education, the theoretical reference of Advanced Mathematical Thinking has contributed to some research. However, others opt for a different but somehow related theoretical reference. Identifying the similarities and differences between those references can elucidate the researcher’s motivations when adopting one or another reference. Objective: to synthesise three studies involving Mathematical Creativity, Advancing Mathematical Activity and Advanced Mathematical Knowledge, respectively, and to illustrate possible contributions of those theoretical references in the analysis of a written production. Design: the research is qualitative and theoretical and speculative so that theoretical relationships were carried out and used later in the analysis of the written production. Setting and Participants: the research involves a participant with a degree in mathematics with whom the researchers had a virtual interaction to present the questions. Data collection and analysis: the procedures performed in the resolution of questions prepared based on the OBMEP question bank are analysed and, with the support of this example of analysis, possible contributions of each reference in the analysis of a written production are discussed. Results: none of the references covers all the thinking mobilised in the activity, but each one favours the focus on specific aspects that were verified in the written production of the research participant, in line with the similarities and divergences emphasised in the theoretical appreciation of each reference. Conclusions: the knowledge of different theoretical frameworks provides the teacher with foundations to exercise their teaching practice and the researcher with options for an adequate choice for their research.
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