6 research outputs found

    Solving Irregular Strip Packing Problems With Free Rotations Using Separation Lines

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    Solving nesting problems or irregular strip packing problems is to position polygons in a fixed width and unlimited length strip, obeying polygon integrity containment constraints and non-overlapping constraints, in order to minimize the used length of the strip. To ensure non-overlapping, we used separation lines. A straight line is a separation line if given two polygons, all vertices of one of the polygons are on one side of the line or on the line, and all vertices of the other polygon are on the other side of the line or on the line. Since we are considering free rotations of the polygons and separation lines, the mathematical model of the studied problem is nonlinear. Therefore, we use the nonlinear programming solver IPOPT (an algorithm of interior points type), which is part of COIN-OR. Computational tests were run using established benchmark instances and the results were compared with the ones obtained with other methodologies in the literature that use free rotation

    Sparse convex quadratic programming methods and their applications in projections onto poliedra

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    O problema de minimiza√ß√£o com restri√ß√Ķes lineares e importante, n√£o apenas pelo problema em si, que surge em v√°rias √°reas, mas tamb√©m por ser utilizado como subproblema para resolver problemas mais gerais de programa√ß√£o n√£o-linear. GENLIN e um m√©todo eficiente para minimiza√ß√£o com restri√ß√Ķes lineares para problemas de pequeno e m√©dio porte. Para que seja poss√≠vel a implementa√ß√£o de um m√©todo similar para grande porte, √© necess√°rio ter um m√©todo eficiente, tamb√©m para grande porte, para proje√ß√£o de pontos no conjunto de restri√ß√Ķes lineares. O problema de proje√ß√£o em um conjunto de restri√ß√Ķes lineares pode ser escrito como um problema de programa√ß√£o quadr√°tica convexa. Neste trabalho, estudamos e implementamos m√©todos esparsos para resolu√ß√£o de problemas de programa√ß√£o quadr√°tica convexa apenas com restri√ß√Ķes de caixa, em particular o cl√°ssico m√©todo Mor√©-Toraldo e o \"m√©todo\" NQC. O m√©todo Mor√©-Toraldo usa o m√©todo dos Gradientes Conjugados para explorar a face da regi√£o fact√≠vel definida pela itera√ß√£o atual, e o m√©todo do Gradiente Projetado para mudar de face. O \"m√©todo\" NQC usa o m√©todo do Gradiente Espectral Projetado para definir em que face trabalhar, e o m√©todo de Newton para calcular o minimizador da quadr√°tica reduzida a esta face. Utilizamos os m√©todos esparsos Mor√©-Toraldo e NQC para resolver o problema de proje√ß√£o de GENLIN e comparamos seus desempenhosThe linearly constrained minimization problem is important, not only for the problem itself, that arises in several areas, but because it is used as a subproblem in order to solve more general nonlinear programming problems. GENLIN is an efficient method for solving small and medium scaled linearly constrained minimization problems. To implement a similar method to solve large scale problems, it is necessary to have an efficient method to solve sparse projection problems onto linear constraints. The problem of projecting a point onto a set of linear constraints can be written as a convex quadratic programming problem. In this work, we study and implement sparse methods to solve box constrained convex quadratic programming problems, in particular the classical Mor√©-Toraldo method and the NQC \"method\". The Mor√©-Toraldo method uses the Conjugate Gradient method to explore the face of the feasible region defined by the current iterate, and the Projected Gradient method to move to a different face. The NQC \"method\" uses the Spectral Projected Gradient method to define the face in which it is going to work, and the Newton method to calculate the minimizer of the quadratic function reduced to this face. We used the sparse methods Mor√©-Toraldo and NQC to solve the projection problem of GENLIN and we compared their performance

    Solving irregular packing problems using non-linear programming techniques

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    Os problemas de empacotamento de itens irregulares s√£o problemas de corte e empacotamento, nos quais pe√ßas irregulares de menor tamanho (que chamamos de itens) devem ser empacotados inteiramente em uma pe√ßa grande (que chamamos de placa), obedecendo a restri√ß√Ķes de n√£osobreposi√ß√£o e minimizando as dimens√Ķes da placa. Para garantir a n√£o-sobreposi√ß√£o, fazemos uso de retas separadoras, quer dizer, retas que separam um item de outro. Apresentamos modelos de programa√ß√£o n√£o-linear para problemas de empacotamentos de itens regulares e irregulares que rotacionam livremente. Os itens podem ser c√≠rculos, pol√≠gonos convexos e n√£o-convexos. A principal vantagem dos modelos √© a simplicidade, j√° que estes utilizam somente conceitos b√°sicos de geometria. Usamos o algoritmo de programa√ß√£o n√£o-linear IPOPT (um algoritmo de tipo de pontos interiores), que faz parte da COIN-OR, para a resolu√ß√£o dos problemas. Testes computacionais foram executados usando inst√Ęncias conhecidas da literatura e os resultados foram comparados com resultados apresentados na literatura, obtidos com outras metodologias que tamb√©m usam rota√ß√Ķes livre, mostrando que nossos modelos s√£o competitivos. Propomos tamb√©m o uso de par√°bolas separadoras para a verifica√ß√£o de n√£o-sobreposi√ß√£o na modelagem do problema, o que pode trazer ganhos computacionais e melhor qualidade de solu√ß√Ķes.The irregular packing problems are cutting and packing problems, in which smaller irregular pieces (which we call items) should be packaged entirely in one large piece (which we call a plate), obeying non-overlapping constraints and minimizing the dimensions of the plate. To ensure non-overlapping, we make use of separation lines, that is, lines that separate one item from another. We present nonlinear programming models for problems of packing regular and irregular items that rotate freely. The items can be circles, convex and nonconvex polygons. The main advantage of the models is their simplicity, because they use only basic geometry concepts. We use the nonlinear programming algorithm IPOPT (an algorithm of interior points type), which is part of COIN-OR, to solve the problems. Computational tests were performed using known instances of the literature and the results were compared with results presented in the literature, obtained with other methodologies that also use free rotations, showing that our models are competitive. We also propose the use of separating parabola to avoid items overlaping in the models, which could provide greater computational eficiency as well as solutions with better quality

    Dos enfoques matemáticos epidemiológicos para modelar el comportamiento de los decesos causados por el COVID-19

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    Objective: To compare two deterministic mathematical epidemiological models from the literature, to simulate the death curve in the Atlantic department caused by COVID-19.¬† Methodology: The first model proposed to simulate the number of deaths by COVID-19 is that of Tappe. This approach is based on the exponential behavior of the death number curve, and was initially used by the author with the available data on the number of deaths in China. The other model proposed is the SIRD, an extension of the SIR model, which divides the population between susceptible, infected, dead and recovered. Results: From the obtained results of both models, comparing with the available data from INS, both models reach similar behavior. However, when analyzing the projection for the next 30 days from the 26th may, it is observed that the curve of number of deaths is greater in the SIRD model than the Tappe‚Äôs model, probably due to the adding of more variables on the model. ¬†Conclusions: As SIRD is a more complete model that involves a wide number of variables of population, the projection made with this model is more reliable than that made with the Tappe‚Äôs model. For future studies, the aim is to incorporate the population of those exposed to describe the number of deaths, in a SEIRD model, in the department of Atl√°ntico.Objetivo: Comparar dos modelos epidemiol√≥gicos matem√°ticos determin√≠sticos de la literatura, para simular la curva de decesos en el departamento del Atl√°ntico causados por el COVID-19. Metodolog√≠a: El primer modelo propuesto para simular el n√ļmero de decesos por el COVID-19 es el de Tappe. Este enfoque se basa en el comportamiento exponencial de la curva del n√ļmero de decesos, e inicialmente fue usado por el autor con los datos disponibles del n√ļmero de muertos en China. El otro modelo propuesto es el SIRD, una extensi√≥n del modelo SIR, que divide la poblaci√≥n entre susceptibles, infectados, muertos y recuperados. Resultados: Los resultados obtenidos a partir de los dos modelos, en las fechas estipuladas, mostraron que, comparados con los datos tomados del INS, ambos describen un comportamiento relativamente similar. Sin embargo, al analizar una proyecci√≥n realizada para noventa d√≠as, treinta d√≠as despu√©s de la fecha final de an√°lisis (26 de mayo), se observa que el modelo SIRD describe una curva de crecimiento mayor que la del modelo de Tappe, esto se debe, probablemente, a la inserci√≥n de m√°s variables en el modelo. Conclusiones: Al ser SIRD un modelo m√°s completo, con mayor n√ļmero de variables representativas de la poblaci√≥n, la proyecci√≥n realizada con √©ste es m√°s confiable que la realizada con el modelo de Tappe. Para estudios futuros se pretende incorporar la poblaci√≥n de los expuestos para describir el n√ļmero de decesos, en un modelo SEIRD, en este departament

    Fundamentos del c√°lculo vectorial con algunas aplicaciones

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    This text is aimed at students of Engineering programs and related fields of the Universidad Nacional Abierta y a Distancia (UNAD for its Spanish acronym) and other universities. It has basic material for a course of Introduction to Linear Algebra and elements of Vector Calculus. It emerges as a didactic proposal that encourages students to learn autonomously, by providing them with conceptual tools and detailed answers to exercises and problems, and involving previous fundamental concepts seen in the courses of Differential Calculus and Integral Calculus. This proposal has been designed for students not to stop their learning process due to conceptual gaps or obstacles, or concepts they may have forgotten. On the contrary, by mobilizing all the knowledge through the well-detailed example, we guarantee conceptual maturity in the constructions projected in the Linear Algebra and Vector Calculus courses. This plan has been developed with the participation of the networks of tutors in the last few years at the School of Basic Sciences, Technology, and Engineering of the Universidad Nacional Abierta y a Distancia (UNAD).El presente texto est√° dirigido a estudiantes de los programas de Ingenier√≠a y √°reas afines de la Universidad Nacional Abierta y a Distancia (UNAD) y otras universidades. Contiene el material b√°sico para un curso de Introducci√≥n al √Ālgebra Lineal y elementos del C√°lculo Vectorial; surge como propuesta did√°ctica que propicia la autonom√≠a del aprendizaje en el estudiante, el cual le brinda las herramientas conceptuales, junto con las soluciones detalladas de ejercicios y problemas involucrando conceptos previos fundamentales vistos en los cursos de C√°lculo Diferencial y C√°lculo Integral. Como propuesta, queremos que el estudiante no detenga su proceso de aprendizaje por vac√≠os u obst√°culos conceptuales, los cuales posiblemente haya olvidado. Al movilizar todos los conocimientos a trav√©s del ejemplo bien detallado, garantizamos una madurez conceptual en las construcciones que se proyectan en los cursos de √Ālgebra Lineal y C√°lculo Vectorial, propuesta que se ha construido con la participaci√≥n de las redes de tutores en los √ļltimos a√Īos en la Escuela de Ciencias B√°sicas, Tecnolog√≠a e Ingenier√≠a de la Universidad Nacional Abierta y a Distancia (UNAD).Este texto √© destinado aos estudantes de programas de Engenharia e √°reas afins da Universidade Nacional Aberta e √† Dist√Ęncia (UNAD) e de outras universidades. Cont√©m o material b√°sico para um curso de Introdu√ß√£o √† √Ālgebra Linear e elementos de C√°lculo Vetorial; surge como uma proposta did√°tica que favorece a autonomia de aprendizagem no aluno, que fornece as ferramentas conceituais, juntamente com as solu√ß√Ķes detalhadas de exerc√≠cios e problemas envolvendo conceitos fundamentais anteriores vistos nos cursos de C√°lculo Diferencial e C√°lculo Integral. Como proposta, n√£o queremos que os estudantes interrompam seu processo de aprendizagem por causa de lacunas ou obst√°culos conceituais, que eles podem ter esquecido. Ao mobilizar todo o conhecimento atrav√©s de um exemplo bem detalhado, garantimos maturidade conceitual nas constru√ß√Ķes que s√£o projetadas nos cursos de √Ālgebra Linear e C√°lculo Vetorial, uma proposta que foi constru√≠da com a participa√ß√£o das redes de tutores nos √ļltimos anos na Escola de Ci√™ncias B√°sicas, Tecnologia e Engenharia da Universidade Nacional Aberta e √† Dist√Ęncia (UNAD)
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