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Bin Packing Problems (Promising Models and Examples)

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Abstract

The paper is devoted to the promising bin packing problems and some applications. A system view on the formulation of problems based on a set of elements (objects), a set of containers, and binary relations over the specified sets (precedence, dominance, and correspondence of elements to containers) is suggested. Special versions of packing problems with estimations of elements (objects) based on multisets are described. Examples in communication networks, such as the choice of information messages and two-dimensional packaging of messages in WiMAX systems are also briefly considered.

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References

  1. S. Alouf, E. Altman, J. Galtier, J.-F. Lalande, and C. Touati, “Quasi-optimal resource allocation in multispot MFTDMA satellite networks,” in Combinatorial Optimization in Communication Networks, Ed. by M. X. Cheng, Y. Li, and D.-Z. Du (Springer-Verlag, New York, 2006), pp. 325–365.

    Chapter  Google Scholar 

  2. N. Bansal, A. Caprara, and M. Sviridenko, “A new approximation method for set covering problems with applications to multidimensional bin packing,” SIAM J. Comput. 39, 1256–1278 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. A. Bennell and J. F. Oliveira, “A tutorial in irregular shape packing problems. J. ORS 60, 93–105 (2009).

    MATH  Google Scholar 

  4. J. A. Bennell and X. Song, “A beam search implementation for the irregular shape packing problem,” J. Heuristics 16, 167–188 (2010).

    Article  MATH  Google Scholar 

  5. E. G. Birgin, J. M. Martinez, and D. P. Ronconi, “Optimizing the packing of cylinders into a rectangular container: A nonlinear approach,” Eur. J. Oper. Res. 160, 19–33 (2005).

    Article  MATH  Google Scholar 

  6. A. R. Brown, Optimal Packing and Depletion (American Elsevier, New York, 1971).

    Google Scholar 

  7. J. I. Bruno and P. J. Downey, “Probabilistic bounds for dual bin packing,” Acta Inform. 22, 333–345 (1985).

    MathSciNet  MATH  Google Scholar 

  8. D. G. Cattrisse and L. N. Van Wassenhove, “A survey of algorithms for the generalzied assignment problem,” Eur. J. Oper. Res. 60, 260–272 (1992).

    Article  Google Scholar 

  9. Y. Chung, “Inverse bin-packing number problems: NP-hardness and approximation algorithms,” Manag. Sci. and Fin. Eng. 18 (2), 19–22 (2012).

    Google Scholar 

  10. C. Cicconetti, L. Lenzini, A. Lodi, S. Martello, E. Mingozzi, and M. Monaci, “Efficient two-dimensional data location in IEEE.802.16 OFDMA,” in Proc. IEEE INFOCOM 2010 (IEEE, Piscataway, 2010), pp. 2160–2168.

    Google Scholar 

  11. C. Cicconetti, L. Lenzini, A. Lodi, S. Martello, E. Mingozzi, and M. Monaci, “A fast and efficient algorithm to exploit multi-user diversity in IEEE 802.16 BandAMC,” Comp. Netw. 55 (16), 3680–3693 (2011).

    Article  Google Scholar 

  12. C. Cicconetti, L. Lenzini, A. Lodi, S. Martello, E. Mingozzi, and M. Monaci, “Efficient two-dimensional data allocation in IEEE 802.16 OFDMA,” IEEE/ACM Trans. Netw. 22 1645–1658 (2014).

    Article  Google Scholar 

  13. E. G. Coffman, Jr., and J. Y.-T. Leung, and D. W. Ting, “Bin packing: Maximizing the number of pieces packed. Acta Informatica,” 9, 263–271 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  14. E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, “An application of bin-packing to multiprocessor scheduling,” SIAM J. Comput 7 (1), 1–17 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  15. E. G. Coffman, Jr. and J. Y.-T. Leung, “Combinatorial analysis of an efficient algorithm for processor and storage allocation,” SIAM J. Comput. 8, 202–217 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  16. E. G. Coffman, Jr., M. R. Garey, and D. S. Johnson, “Approximation algorithms for bin packing: A survey,” in Approximation Algorithms, Ed. by D. Hochbaum (PWS Publishing Company. 1996), pp. 46–93.

    Google Scholar 

  17. E. G. Coffman, Jr., C. Courcoubetis, M. R. Garey, D. S. Johnson, P. W. Shor, R. R. Weber, and M. Yannkakis, “Bin packing with discrete item sizes, part I: Perfect packing theorems and the average case behavior of optimal packings,” SIAM J. Discr. Math. 13, 384–402 (2000).

    Article  MATH  Google Scholar 

  18. E. G. Coffman, Jr., C. Courcoubetis, M. R. Garey, D. S. Johnson, P. W. Shor, R. R. Weber, and M. Yannakakis, “Perfect packing theorem and the averagecase behavior of optimal and online bin packing,” SIAM Review 44, 95–108 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  19. E. G. Coffman, Jr. and J. Csirik, “Classification scheme for bin packing theory,” Acta Cybern. 18, 47–60 (2007).

    MATH  Google Scholar 

  20. E. G. Coffman, Jr., G. Galambros, S. Martello, and D. Vigo, “Bin packing approximation algorithms: combinatorial analysis,” in Handbook of Combinatorial Optimization, Ed. by P. M. Pardalos, D.-Z. Du, and R. L. Graham, 2nd ed., (Springer-Verlag, 2013), pp. 455–531.

    Chapter  Google Scholar 

  21. R. W. Conway, W. L. Maxwell, and L. W. Miller, Theory of Scheduling (Addison-Wesley, Reading, Mass., 1967).

    MATH  Google Scholar 

  22. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed., (MIT Press and McGraw-Hill, Boston, 2001).

    MATH  Google Scholar 

  23. M. Delorme, M. Iori, and S. Martello, Bin Packing and Cutting Stock Problems: Mathematical Models and Exact Algorithms, Res. Report OR-15-1 (Univ. of Bologna, 2015).

    MATH  Google Scholar 

  24. H. Dyckhoff, “A typology of cutting and packing problems,” Eur. J. Oper. Res. 44, 145–159 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Dyckhoff and U. Finke, Cutting and Packing in Production and Distribution: a Typology and Bibliography (Springer-Verlag, Berlin, 1992).

    Book  Google Scholar 

  26. L. Epstein and L. M. Favrholdt, “On-line maximizing the number of item packed in variable-sized bins,” Acta Cybern. 16, 57–66 (2013).

    MathSciNet  MATH  Google Scholar 

  27. L. Epstein and A. Levin, “On bin packing with conflicts,” SIAM J. Optim. 19, 1270–1298 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Epstein, L. M. Favrholdt, and J. S. Kohrt, “Comparing online algorithms for bin packing problems,” J. Scheduling 15, 13–21 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  29. A. E. Muritiba, M. Iori, E. Malaguti, and P. Toth, “Algorithms for the bin packing problem with conflicts,” INFORMS J. on Comput. 22, 401–415 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  30. A. S. Fukunaga and R. E. Korf, “Bin completion algorithms for multicontainer packing, knapsack, and covering problems,” J. Art. Intell. Res. 28, 393–429 (2007).

    MathSciNet  MATH  Google Scholar 

  31. E. M. Furems, Models of Packing in Multicriterion Decision-Making Problems at Limited Resources, Preprint, (VNIISI, Moscow, 1986).

    Google Scholar 

  32. E. M. Furems, “Inverse problem of packing with qualitative criteria–statements and review of methods,” Iskusstv. Intel. i Prinyatie Resh., No. 3, 31–41 (2016).

    Google Scholar 

  33. G. Galambos, H. Kellerer, and G. J. Woeginger, “A lower bound for on-line vector packing algorithms,” Acta Cybern. 11, 23–34 (1994).

    MathSciNet  MATH  Google Scholar 

  34. M. R. Garey and D. S. Johnson, Computers and Intractability. The Guide to the Theory of NP-Completeness (W. H. Freeman and Company, San Francisco, 1979).

    MATH  Google Scholar 

  35. M. Gendreau, G. Laporte, and F. Semmet, “Heuristics and lower bounds for the bin packing problem with conflicts,” Comp. and Oper. Res. 31, 347–358 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  36. P. C. Gilmore and R. E. Gomory, “A linear programming approach to the cutting stock problem II,” Oper. Res. 11, 863–888 (1963).

    Article  MATH  Google Scholar 

  37. E. Hopper and B. Turton, “Application of genetic algorithms to packing problems -a review,” in Proc. of the 2nd Online World Conf. on Soft Comput. in Engineering Design and Manufacturing, 1997, Ed. by P. K. Chawdry, R. Roy, and R. K. Kant (Springer-Verlag, London, 1997), pp. 279–288.

    Google Scholar 

  38. E. Hopper and B. C. H. Turton, “A review of the application of meta-heuristic algorithms to 2D strip packing problems,” Artif. Intell. Rev. 16, 257–300 (2001).

    Article  MATH  Google Scholar 

  39. R. Hubscher and F. Glover, “Applying tabu search with influential diversification to multiprocessor scheduling,” Comp. and Oper. Res. 21, 877–884 (1994).

    Article  MATH  Google Scholar 

  40. K. Jansen, “An approximation scheme for bin packing with conflicts,” J. Comb. Opt. 3, 363–377 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  41. D. S. Johnson, “Near-optimal bin-packing algorithm,” Doctoral Thesis (Dept. of Mathematics, MIT, Cambridge, Mass., 1973).

    Google Scholar 

  42. D. S. Johnson, Fast algorithms for bin packing. J. Comp. and Syst. Sci. 8, 272–314 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  43. D. S. Johnson, A. Demers, J. D. Ullman, M. R. Garey, and R. L. Graham, “Worst-case performance bounds for simple one-dimensional packing algorithm,” SIAM J. Optim. 3, 299–325 (1974).

    MathSciNet  MATH  Google Scholar 

  44. R. M. Karp, “Reducibility among combinatorial problems.,” in Complexity of Computer Computations, Ed. by R. E. Miller and J. W. Thatcher (Plenum., 1972), pp. 85–103.

    Chapter  Google Scholar 

  45. H. Kellerer, U. Pferschy, and D. Pisinger, Knapsack Problems (Springer-Verlag, Berlin, 2004).

    Book  MATH  Google Scholar 

  46. H. W. Kuhn, “The Hungarian method for the assignment problems,” Nav. Res. Log. 52, 7–21 (2005).

    Article  Google Scholar 

  47. M. Labbe, G. Laporte, and S. Martello, “Upper bounds and algorithms for the maximum cardinality bin packing problem,” Eur. J. Oper. Res. 149, 490–498 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  48. M. Sh. Levin, “Modular design and improvement of the management system in the smart home with the use of interval multiset estimates,” J. Commun. Technol. Electron. 58, 584–593 (2013).

    Article  Google Scholar 

  49. M. Sh. Levin, Decision Support Technology for Modular Systems. Electronic Book (Moscow, 2013). https://doi.org/www.mslevin.iitp.ru/Levin-bk-Nov2013-071.pdf

    Google Scholar 

  50. M. Sh. Levin, Modular System Design and Evaluation (Sprigner-Verlag, New York, 2015).

    Book  Google Scholar 

  51. M. Sh. Levin, “Towards bin packing (preliminary problem survey, models with multiset estimates),” Elec. Preprint, (May 24, 2016). https://doi.org/arxiv.org/abs/1605.07574[cs.AI]

    Google Scholar 

  52. M. Sh. Levin, “On reconfiguration of solutions in combinatorial optimization,” Inf. Protsessy 16, 414–429 (2016).

    Google Scholar 

  53. A. Lodi, S. Martello, and D. Vigo, “Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems,” INFORMS J. on Comput 11, 345–357 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  54. A. Lodi, S. Martello, and D. Vigo, “Recent advances on two-dimensional bin packing problems,” Disc. Appl. Mat. 123, 379–396 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Lodi, S. Martello, and M. Monaci, “Two-dimensional bin packing problems: A survey,” Eur. J. of Oper. Res. 141, 241–252 (2002).

    Article  MATH  Google Scholar 

  56. A. Lodi, S. Martello, and D. Vigo, “TSpack: a unified tabu search code for multidimensional bin packing problems,” Ann. Oper. Res. 131 (1-4), 203–213 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  57. A. Lodi, S. Martello, M. Monaci, C. Cicconetti, L. Lenzini, E. Mingozzi, C. Eklund, and J. Moilanen, “Efficient two-dimensional packing algorithms for mobile WiMAX,” Man. Sci. 57, 2130–2144 (2011).

    Article  Google Scholar 

  58. S. Martello and P. Toth, Knapsack Problems (Wiley, Chichester, 1990).

    MATH  Google Scholar 

  59. S. Martello, D. Pisinger, and D. Vigo, “The threedimensional bin packing problem,” Oper. Res. 48, 256–267 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  60. S. Martello, “Two-dimensional packing problems in telecommunications,” Pesquisa Operacional 34, 31–38 (2014).

    Article  Google Scholar 

  61. M. Peeters and Z. Degraeve, “Branch-and-price algorithms for dual bin packing and maximum cardinality bin packing problem,” Eur. J. of Oper. Res. 170, 416–439 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  62. D. Pisinger and M. Sigurd, “The two-dimensional bin packing problem with variable sizes and costs,” Discr. Optim. 2, 154–167 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  63. C. Reeves, “Hybrid genetic algorithms for bin-packing and related problems,” Ann. Oper. Res. 63, 371–396 (1996).

    Article  MATH  Google Scholar 

  64. R. Sadykov and F. Vanderbeck, “Bin packing with conflicts: a generic branch-and-price algorithm,” INFORMS J. on Comput. 25, 244–255 (2013).

    Article  MathSciNet  Google Scholar 

  65. S. S. Seiden, R. van Stee, and L. Epstein, “New bounds for variable sized online bin packing,” SIAM J. Comput. 32, 455–469 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  66. P. E. Sweeney and E. R. Paternoster, “Cutting and packing problems: a categorized, application-orientated research bibliography,” J. OR 43, 691–706.

  67. H. Terashima-Martin, P. Ross, C. J. Farias-Zarate, E. Lopez-Camacho, and M. Valenzuela-Rendon, “Generalized hyper-heuristics for solving 2D regular and irregular packing problems,” Ann. Oper. Res. 179, 369–392 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  68. J. D. Ullman, The Performance of a Memory Allocation Algorithm. Techn. Report 100 (Princeton Univ., Princeton, NJ, 1971).

    Google Scholar 

  69. J. M. de Carvalho, “LP models for bin packing and cutting stock problems,” Eur. J. Oper. Res. 141, 253–273 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  70. G. Wascher, H. Haussner, and H. Schumann, “An improved typology of cutting and packing problems,” Eur. J. Oper. Res. 183, 1109–1130 (2007).

    Article  MATH  Google Scholar 

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Correspondence to M. Sh. Levin.

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Original Russian Text © M.Sh. Levin, 2017, published in Informatsionnye Protsessy, 2017, Vol. 17, No. 1, pp. 43–60.

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Levin, M.S. Bin Packing Problems (Promising Models and Examples). J. Commun. Technol. Electron. 63, 655–666 (2018). https://doi.org/10.1134/S1064226918060177

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