Advertisement

Applications of Nonlinear Programming to Packing Problems

  • Ernesto G. Birgin
Conference paper
Part of the Mathematics for Industry book series (MFI, volume 11)

Abstract

The problem of packing items within bounded regions in the Euclidean space has multiple applications in a variety of areas, such as, Physics, Chemistry, and Engineering. Problems of this type exhibit various levels of complexity. Nonlinear programming formulations and methods had been successfully applied to a wide range of packing problems. In this review paper, a brief description of the state-of-the-art and an illustrated overview of packing nonlinear programming techniques and applications will be presented.

Keywords

Cutting Packing Nonlinear programming 

References

  1. 1.
    Lang, R.J.: A computational algorithm for origami design. In: 12th ACM Symposium on Computational Geometry, pp. 98–105 (1996)Google Scholar
  2. 2.
    Wong, H.C., Kwan, A.K.H.: Packing density: a key concept for mix design of high performance concrete. In: Proceedings of the Materials Science and Technology in Engineering Conference, HKIE Materials Division, Hong Kong, pp. 1–15 (2005)Google Scholar
  3. 3.
    Osman, K.T.: Foerst Soils—Properties and Management. Springer, Cham (2013)Google Scholar
  4. 4.
    Wu, Q.J., Bourland, J.D.: Morphology-guided radiosurgery treatment planning and optimization for multiple isocenters. Med. Phys. 26, 2151–2160 (1999)CrossRefGoogle Scholar
  5. 5.
    Martínez, L., Andrade, R., Birgin, E.G., Martínez, J.M.: Packmol: a package for building initial configurations for molecular dynamics simulations. J. Comput. Chem. 30, 2157–2164 (2009)CrossRefGoogle Scholar
  6. 6.
    Martínez, J.M., Martínez, L.: Packing optimization for automated generation of complex system’s initial configurations for molecular dynamics and docking. J. Comput. Chem. 24, 819–825 (2003)CrossRefGoogle Scholar
  7. 7.
    Drezner, Z., Erkut, E.: Solving the continuous p-dispersion problem using non-linear programming. J. Oper. Res. Soc. 46, 516–520 (1995)CrossRefMATHGoogle Scholar
  8. 8.
    Fraser, H.J., George, J.A.: Integrated container loading software for pulp and paper industry. Eur. J. Oper. Res. 77, 466–474 (1994)CrossRefMATHGoogle Scholar
  9. 9.
    Chen, D., Jiao, Y., Torquato, S.: Equilibrium phase behavior and maximally random jammed state of truncated tetrahedra. J. Phys. Chem. B 118, 7981–7992 (2014)CrossRefGoogle Scholar
  10. 10.
    Birgin, E.G., Martínez, J.M., Mascarenhas, W.F., Ronconi, D.P.: Method of sentinels for packing items within arbitrary convex regions. J. Oper. Res. Soc. 57, 735–746 (2006)CrossRefMATHGoogle Scholar
  11. 11.
    Mascarenhas, W.F., Birgin, E.G.: Using sentinels to detect intersections of convex and nonconvex polygons. Comput. Appl. Math. 29, 247–267 (2010)MathSciNetMATHGoogle Scholar
  12. 12.
    Stoyan, Y.G., Novozhilova, M.V., Kartashov, A.V.: Mathematical model and method of searching for a local extremum for the non-convex oriented polygons allocation problem. Eur. J. Oper. Res. 92, 193–210 (1996)CrossRefMATHGoogle Scholar
  13. 13.
    Stoyan, Y.G.: On the generalization of the dense allocation function. Rep. Ukr. SSR Acad. Sci. Ser. A 8, 70–74 (1980) (in Russian)Google Scholar
  14. 14.
    Birgin, E.G., Martínez, J.M., Ronconi, D.P.: Optimizing the packing of cylinders into a rectangular container: a nonlinear approach. Eur. J. Oper. Res. 160, 19–33 (2005)CrossRefMATHGoogle Scholar
  15. 15.
    Birgin, E.G., Bustamante, L.H., Callisaya, H.F., Martínez, J.M.: Packing circles within ellipses. Int. Trans. Oper. Res. 20, 365–389 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Birgin, E.G., Lobato, R.D.: Orthogonal packing of identical rectangles within isotropic convex regions. Comput. Ind. Eng. 59, 595–602 (2010)CrossRefGoogle Scholar
  17. 17.
    Birgin, E.G., Martinez, J.M., Nishihara, F.H., Ronconi, D.P.: Orthogonal packing of rectangular items within arbitary convex region by nonlinear optimization. Comput. Oper. Res. 33, 3535–3548 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Martello, S., Toth, P.: Knapsack Problems: Algorithms and Computer Interpretations. Wiley, Chichester (1990)MATHGoogle Scholar
  19. 19.
    Graham, R.L.: Sets of points with given minimum separation (solution to problem El921). Am. Math. Mon. 75, 192–193 (1968)CrossRefGoogle Scholar
  20. 20.
    Kravitz, S.: Packing cylinders into cylindrical containers. Math. Mag. 40, 65–71 (1967)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Birgin, E.G., Gentil, J.M.: New and improved results for packing identical unitary radius circles within triangles, rectangles and strips. Comput. Oper. Res. 37, 1318–1327 (2010)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Birgin, E.G., Sobral, F.N.C.: Minimizing the object dimensions in circle and sphere packing problems. Comput. Oper. Res. 35, 2357–2375 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Birgin, E.G., Martínez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2014)CrossRefMATHGoogle Scholar
  24. 24.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: On augmented Lagrangian methods with general lower-level constraints. SIAM J. Optim. 18, 1286–1309 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Andreani, R., Birgin, E.G., Martínez, J.M., Schuverdt, M.L.: Augmented Lagrangian methods under the constant positive linear dependence constraint qualification. Math. Program. 111, 5–32 (2008)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Birgin, E.G., Martínez, J.M.: Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput. Optim. Appl. 23, 101–125 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil

Personalised recommendations