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Lagrangian Approach to Modeling Placement Conditions in Optimized Packing Problems

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Abstract

In optimized packing problems a set of objects have to be allocated completely inside a number of containers (containment condition) without overlapping, while optimizing a certain objective. In some applications, a distance between the objects (and/or the objects and the container) has to be at least a certain given threshold. Analytical representations for non-overlapping, containment and distant conditions are proposed for the objects and containers defined by systems of inequalities. The placement constraints are transformed to optimization problems, corresponding optimality conditions are stated using Lagrangian multipliers technique and then are used as constraints to the overall optimized packing problem. The objects can be freely rotated and translated. For the objects presented by convex polytopes, rotations and translations are reduced to defining positions of the vertices subject to shapes preservation. Numerical results are provided to illustrate the proposed approach.

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Authors and Affiliations

  1. Faculty of Mechanical and Electrical Engineering, Nuevo Leon State University, San Nicolas de los Garza, Mexico

    Igor Litvinchev, Rogelio Corrales-Diaz, Aned Esquerra-Arguelles & Alberto Martinez-Noa

  2. Department of Mathematical Modeling and Optimal Design, A. Pidgorny Institute of Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine

    Tatiana Romanova

Authors
  1. Igor Litvinchev
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  2. Tatiana Romanova
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  3. Rogelio Corrales-Diaz
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  4. Aned Esquerra-Arguelles
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  5. Alberto Martinez-Noa
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Correspondence to Igor Litvinchev.

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Litvinchev, I., Romanova, T., Corrales-Diaz, R. et al. Lagrangian Approach to Modeling Placement Conditions in Optimized Packing Problems. Mobile Netw Appl 25, 2126–2133 (2020). https://doi.org/10.1007/s11036-020-01556-w

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