Optimized Packings in Space Engineering Applications: Part I

  • Yuriy Stoyan
  • Alexandr Pankratov
  • Tatiana Romanova
  • Giorgio FasanoEmail author
  • János D. Pintér
  • Yurij E. Stoian
  • Andrey Chugay
Part of the Springer Optimization and Its Applications book series (SOIA, volume 144)


Packing optimization problems have a wide spectrum of real-word applications, including transportation, logistics, chemical/civil/mechanical/power/aerospace engineering, shipbuilding, robotics, additive manufacturing, materials science, mineralogy, molecular geometry, nanotechnology, electronic design automation, very large system integration, pattern recognition, biology, and medicine. In space engineering, ever more challenging packing optimization problems have to be solved, requiring dedicated cutting-edge approaches.

Two chapters in this volume investigate very demanding packing issues that require advanced solutions. The present chapter provides a bird’s eye view of challenging packing problems in the space engineering framework, offering some insight on possible approaches. The specific issue of packing a given collection of arbitrary polyhedra, with continuous rotations and minimum item-to-item admissible distance, into a convex container of minimum size, is subsequently analyzed in depth, discussing an ad hoc mathematical model and a dedicated solution algorithm. Computational results show the efficiency of the approach proposed. The following (second) chapter examines a class of packing optimization problems in space with consideration to balancing conditions.

MSC 2010

05B40 52C17 11H16 90XX 90CX 90C30 49M37 90C06 90C11 90C26 90C59 90C90 


  1. 1.
    Fasano, G., Pintér, J.D.: Optimized Packings and their Applications. Springer Optimization and its Applications. Springer, New York (2015)Google Scholar
  2. 2.
    Cagan, J., Shimada, K., Yin, S.: A survey of computational approaches to three-dimensional layout problems. Comput. Aided Des. 34, 597–611 (2002)CrossRefGoogle Scholar
  3. 3.
    Dyckhoff, H., Scheithauer, G., Terno, J.: Cutting and Packing. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliographies in Combinatorial Optimization, pp. 393–412. Wiley, Chichester (1997)Google Scholar
  4. 4.
    Ibaraki, T., Imahori, S., Yagiura, M.: Hybrid Metaheuristics for Packing Problems. In: Blum, C., Aguilera, M.J., Roli, A., Sampels, M. (eds.) Hybrid Metaheuristics: an Emerging Approach to Optimization. Studies in Computational Intelligence (SCI), vol. 114, pp. 185–219. Springer, Berlin (2008)CrossRefGoogle Scholar
  5. 5.
    Faroe, O., Pisinger, D., Zachariasen, M.: Guided local search for the three-dimensional bin packing problem. INFORMS J. Comput. 15(3), 267–283 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Fekete, S., Schepers, J.: A combinatorial characterization of higher-dimensional orthogonal packing. Math. Oper. Res. 29, 353–368 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Fekete, S., Schepers, J., van der Veen, J.C.: An exact algorithm for higher-dimensional orthogonal packing. Oper. Res. 55(3), 569–587 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Martello, S., Pisinger, D., Vigo, D.: The three-dimensional bin packing problem. Oper. Res. 48(2), 256–267 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Martello, S., Pisinger, D., Vigo, D., Den Boef, E., Korst, J.: Algorithms for general and robot-packable variants of the three-dimensional bin packing problem. ACM Trans. Math. Softw. 33(1), 7 (2007)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pisinger, D.: Heuristics for the container loading problem. Eur. J. Oper. Res. 141(2), 382–392 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Addis, B., Locatelli, M., Schoen, F.: Efficiently packing unequal disks in a circle: a computational approach which exploits the continuous and combinatorial structure of the problem. Oper. Res. Lett. 36(1), 37–42 (2008a)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Birgin, E., Martinez, J., Nishihara, F.H., Ronconi, D.P.: Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization. Comput. Oper. Res. 33(12), 3535–3548 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Egeblad, J., Nielsen, B.K., Odgaard, A.: Fast neighborhood search for two-and three-dimensional nesting problems. Eur. J. Oper. Res. 183(3), 1249–1266 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Gomes, A.M., Olivera, J.F.: A 2-exchange heuristics for nesting problems. Eur. J. Oper. Res. 141, 359–570 (2002)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Scheithauer, G., Stoyan, Y.G., Romanova, T.Y.: Mathematical modeling of interactions of primary geometric 3D objects. Cybern. Syst. Anal. 41, 332–342 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Teng, H., Sun, S., Liu, D., Li, Y.: Layout optimization for the objects located within a rotating vessel a three-dimensional packing problem with behavioural constraints. Comput. Oper. Res. 28(6), 521–535 (2001)CrossRefzbMATHGoogle Scholar
  17. 17.
    Caprara, A., Monaci, M.: On the 2-dimensional knapsack problem. Oper. Res. Lett. 32(1), 5–14 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Egeblad, J., Pisinger, D.: Heuristic approaches for the two- and three-dimensional knapsack packing problems. DIKU Technical-Report 2006–13, SSN: 0107-8283, Department of Computer Science, University of Copenhagen, Denmark (2006)Google Scholar
  19. 19.
    Egeblad, J., Pisinger, D.: Heuristic approaches for the two- and three-dimensional knapsack packing problem. Comput. Oper. Res. 36, 1026–1049 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Fekete, S.P., Schepers, J.: A New Exact Algorithm for General Orthogonal D-dimensional Knapsack Problems. In: Algorithms ESA’97, Springer Lecture Notes in Computer Science, vol. 1284, pp. 144–156 (1997)Google Scholar
  21. 21.
    Bortfeldt, A., Wäscher, G.: Container loading problems - A state-of-the-art review. FEMM Working Papers 120007, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management (2012)Google Scholar
  22. 22.
    Kang, M.K., Jang, C.S., Yoon, K.S.: Heuristics with a new block strategy for the single and multiple container loading problems. J. Oper. Res. Soc. 61, 95–107 (2010)CrossRefzbMATHGoogle Scholar
  23. 23.
    Parreño, F., Alvarez-Valdes, R., Oliveira, J.F., Tamarit, J.M.: A maximal-space algorithm for the container loading problem. INFORMS J. Comput. 20(3), 412–422 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Lodi, A., Martello, S., Monaci, M., Vigo, D.: Two-dimensional Bin Packing Problems. In: Paschos, V.T. (ed.) Paradigms of Combinatorial Optimization, pp. 107–129. Wiley/ISTE, Hoboken (2010)Google Scholar
  25. 25.
    Pisinger, D., Sigurd, M.: Using decomposition techniques and constraint programming for solving the two-dimensional bin packing problem. INFORMS J. Comput. 19(1), 36–51 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Iori, M., Martello, S., Monaci, M.: Metaheuristic Algorithms for the Strip Packing Problem. In: Pardalos, P.M., Korotkikh, V. (eds.) Optimization and Industry: New Frontiers, pp. 159–179. Kluwer, Hardbound (2003)CrossRefGoogle Scholar
  27. 27.
    Kenmochi, M., Imamichi, T., Nonobe, K., Yagiura, M., Nagamochi, H.: Exact algorithms for the two-dimensional strip packing problem with and without rotations. Eur. J. Oper. Res. 198, 73–83 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Zhang, D., Kang, Y., Deng, A.: A new heuristic recursive algorithm for the strip rectangular packing problem. Comput. Oper. Res. 33, 2209–2217 (2006)CrossRefzbMATHGoogle Scholar
  29. 29.
    Li, H.L., Chang, C.T., Tsai, J.F.: Approximately global optimization for assortment problems using piecewise linearization techniques. Eur. J. Oper. Res. 140, 584–589 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Pan, P., Liu, C.L.: Area minimization for floorplans. IEEE Trans. Comput. Aided Design Integr. Circuits Syst. 14(1), 123–132 (2006)MathSciNetGoogle Scholar
  31. 31.
    Pintér, J.D., Kampas, F.J.: MathOptimizer Professional: Key Features and Illustrative Applications. In: Liberti, L., Maculan, N. (eds.) Global Optimization: From Theory to Implementation, pp. 263–279. Springer, Berlin (2006)CrossRefGoogle Scholar
  32. 32.
    Pintér, J.D., Kampas, F.J., Castillo, I.: Globally optimized packings of non-uniform size spheres in ℝd: a computational study. Optim. Lett. 12(3), 585–613 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Allen, S.D., Burke, E.K., Kendall, G.: A hybrid placement strategy for the three-dimensional strip packing problem. Eur. J. Oper. Res. 209(3), 219–227 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Bennel, J.A., Lee, L.S., Potts, C.N.: A genetic algorithm for two-dimensional bin packing with due dates. Int. J. Prod. Econ. 145(2), 547–560 (2013)CrossRefGoogle Scholar
  35. 35.
    Bennell, J.A., Han, W., Zhao, X., Song, X.: Construction heuristics for two-dimensional irregular shape bin packing with guillotine constraints. Eur. J. Oper. Res. 230(3), 495–504 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Bennell, J.A., Oliveira, J.F.: A tutorial in irregular shape packing problems. J. Oper. Res. Soc. 60(S1), S93–S105 (2009)CrossRefzbMATHGoogle Scholar
  37. 37.
    Bortfeldt, A., Gehring, H.: A hybrid genetic algorithm for the container loading problem. Eur. J. Oper. Res. 131(1), 143–161 (2001)CrossRefzbMATHGoogle Scholar
  38. 38.
    Burke, E.K., Hellier, R., Kendall, G., Whitwell, G.: A new bottom-left-fill heuristic algorithm for the 2D irregular packing problem. Oper. Res. 54(3), 587–601 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 39.
    Burke, E.K., Guo, Q., Hellier, R., Kendall, G.: A hyper-heuristic approach to strip packing problems. PPSN. 1, 465–474 (2010)Google Scholar
  40. 40.
    Coffman, E., Garey, J.M., Johnson, D.: Approximation Algorithms for Bin Packing: A Survey. PWS Publishing, Boston (1997)zbMATHGoogle Scholar
  41. 41.
    Dowsland, K.A., Herbert, E.A., Kendall, G., Burke, E.: Using tree search bounds to enhance a genetic algorithm approach to two rectangle packing problems. Eur. J. Oper. Res. 168(2), 390–402 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  42. 42.
    Gehring, H., Bortfeldt, A.: A parallel genetic algorithm for solving the container loading problem. Int. Trans. Oper. Res. 9(4), 497–511 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Gonçalves, J.F., Resende, M.G.: A parallel multi-population biased random-key genetic algorithm for a container loading problem. Comput. Oper. Res. 39(2), 179–190 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  44. 44.
    Hopper, E., Turton, B.C.: A review of the application of meta-heuristic algorithms to 2D strip packing problems. Artif. Intell. Rev. 16(4), 257–300 (2001)CrossRefzbMATHGoogle Scholar
  45. 45.
    Hopper, E., Turton, B.C.: An empirical study of meta-heuristics applied to 2D rectangular bin packing - part II. Studia Inform. Universalis. 2(1), 93–106 (2002)Google Scholar
  46. 46.
    López-Camacho, E., Ochoa, G., Terashima-Marín, H., Burke, E.: An effective heuristic for the two-dimensional irregular bin packing problem. Annals OR. 206(1), 241–264 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    Mack, D., Bortfeldt, A., Gehring, H.: A parallel hybrid local search algorithm for the container loading problem. Int. Trans. Oper. Res. 11(5), 511–533 (2004)CrossRefzbMATHGoogle Scholar
  48. 48.
    Oliveira, J.F., Gomes, A.M., Ferreira, J.S.: TOPOS - a new constructive algorithm for nesting problems. OR Spectr. 22(2), 263–284 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  49. 49.
    Ramakrishnan, K., Bennel, J.A., Omar, M.K.: Solving Two Dimensional Layout Optimization Problems with Irregular Shapes by Using Meta-Heuristic. In: 2008 IEEE International Conference on Industrial Engineering and Engineering Management, pp. 178–182 (2008)CrossRefGoogle Scholar
  50. 50.
    Terashima-Marín, H., Ross, P., Farías-Zárate, C.J., López-Camacho, E., Valenzuela-Rendón, M.: Generalized hyper-heuristics for solving 2D regular and irregular packing problems. Ann. Oper. Res. 179, 369–392 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Wang, Z., Li, K.W., Levy, J.K.: A heuristic for the container loading problem: a tertiary-tree-based dynamic space decomposition approach. Eur. J. Oper. Res. 191(1), 86–99 (2008)CrossRefzbMATHGoogle Scholar
  52. 52.
    Yeung, L.H., Tang, W.K.: A hybrid genetic approach for container loading in logistics industry. IEEE Trans. Ind. Electron. 52(2), 617–627 (2005)CrossRefGoogle Scholar
  53. 53.
    Allen, S.D., Burke, E.K., Mareček, J.: A space-indexed formulation of packing boxes into a larger box. Oper. Res. Lett. 40(1), 20–24 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    Chen, C.S., Lee, S.M., Shen, Q.S.: An analytical model for the container loading problem. Eur. J. Oper. Res. 80, 68–76 (1995)CrossRefzbMATHGoogle Scholar
  55. 55.
    Chernov, N., Stoyan, Y.G., Romanova, T.: Mathematical model and efficient algorithms for object packing problem. Comput. Geom. Theory Appl. 43(5), 535–553 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Fasano, G.: Satellite Optimal Layout, Application of Mathematical and Optimization Techniques, pp. 22–26. IBM Europe Institute, Garmisch (1989)Google Scholar
  57. 57.
    Fischetti, M., Luzzi, I.: Mixed-integer programming models for nesting problems. J. Heuristics. 15(3), 201–226 (2009)CrossRefzbMATHGoogle Scholar
  58. 58.
    Hadjiconstantinou, E., Christofides, N.: An exact algorithm for general, orthogonal, two-dimensional knapsack problems. Eur. J. Oper. Res. 83(1), 39–56 (1995)CrossRefzbMATHGoogle Scholar
  59. 59.
    Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Glob. Optim. 43(2–3), 299–328 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  60. 60.
    Padberg, M.W.: Packing Small Boxes into a Big Box. Office of Naval Research, N00014-327, New York University, New York (1999)zbMATHGoogle Scholar
  61. 61.
    Pisinger, D., Sigurd, M.: The two-dimensional bin packing problem with variable bin sizes and costs. Discret. Optim. 2(2), 154–167 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  62. 62.
    Daughtrey, R.S.: A Simulated Annealing Approach to 3-D Packing with Multiple Constraints. Cosmic Program MFS28700. Boeing Huntsville AI Center, Huntsville (1991)Google Scholar
  63. 63.
    Fasano, G.: Solving Non-Standard Packing Problems by Global Optimization and Heuristics. In: Springer Briefs in Optimization. Springer, New York (2014)Google Scholar
  64. 64.
    Fasano, G., Lavopa, C., Negri, D., Vola, M.C.: CAST: A Successful Project in Support of the International Space Station Logistics. In: Fasano, G., Pintér, J.D. (eds.) Optimized Packings and their Applications. Springer Optimization and its Applications. Springer, New York (2015)Google Scholar
  65. 65.
    Fasano, G., Pintér, J.D.: Modeling and Optimization in Space Engineering. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  66. 66.
    Fasano, G., Pintér, J.D.: Space Engineering. Modeling and Optimization with Case Studies. Springer Optimization and its Applications. Springer, New York (2016)zbMATHGoogle Scholar
  67. 67.
    Fasano, G., Vola, M.C.: Space Module On-Board Stowage Optimization Exploiting Containers’ Empty Volumes. In: Fasano, G., Pintér, J.D. (eds.) Modeling and Optimization in Space Engineering, pp. 249–269. Springer, New York (2013)CrossRefGoogle Scholar
  68. 68.
    Takadama, A.K., Shimomura, K.: Cargo Layout Optimization in Spacecraft: Exploring Heuristics for Branch-and- Bound Method. In: The 8th International Symposium on Artificial Intelligence, Robotics and Automation in Space (2005)Google Scholar
  69. 69.
    Addis, B., Locatelli, M., Schoen, F.: Disk packing in a square: a new global optimization approach. INFORMS J. Comput. 20(4), 516–524 (2008b)CrossRefMathSciNetzbMATHGoogle Scholar
  70. 70.
    Castillo, I., Kampas, F.J., Pintér, J.D.: Solving circle packing problems by global optimization: numerical results and industrial applications. Eur. J. Oper. Res. 191(3), 786–802 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  71. 71.
    Floudas, C.A., Akrotirianakis, I.G., Caratzoulas, S., Meyer, C.A., Kallrath, J.: Global optimization in the 21st century: advances and challenges for problems with nonlinear dynamics. Comput. Chem. Eng. 29, 1185–1202 (2005)CrossRefGoogle Scholar
  72. 72.
    Floudas, C.A., Pardalos, P.M.: A Collection of Test Problems for Constrained Global Optimization Algorithms. Springer, New York (1990)CrossRefzbMATHGoogle Scholar
  73. 73.
    Floudas, C.A., Pardalos, P.M., et al.: Handbook of Test Problems in Local and Global Optimization, Nonconvex Optimization and its Applications Series, vol. 33. Kluwer, Dordrecht (1999)zbMATHGoogle Scholar
  74. 74.
    Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization. Kluwer, Dordrecht (2001)zbMATHGoogle Scholar
  75. 75.
    Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization, vol. 1. Kluwer, Dordrecht (1995)zbMATHGoogle Scholar
  76. 76.
    Horst, R., Pardalos, P.M. (eds.): Developments in Global Optimization. Kluwer, Dordrecht (1997)Google Scholar
  77. 77.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  78. 78.
    Kallrath, J.: Mixed-Integer Nonlinear Applications. In: Ciriani, T., Ghiozzi, S., Johnson, E.L. (eds.) Operations Research in Industry, pp. 42–76. Macmillan, London (1999)CrossRefGoogle Scholar
  79. 79.
    Kallrath, J.: Modeling Difficult Optimization Problems. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn, pp. 2284–2297. Springer, New York (2008)Google Scholar
  80. 80.
    Liberti, L., Maculan, N. (eds.): Global Optimization: from Theory to Implementation. Springer, New York (2005)Google Scholar
  81. 81.
    Locatelli, M., Raber, U.: Packing equal circles into a square: a deterministic global optimization approach. Discret. Appl. Math. 122, 139–166 (2002)CrossRefzbMATHGoogle Scholar
  82. 82.
    Pardalos, P.M., Resende, M.G. (eds.): Handbook of Applied Optimization. Oxford University Press, Oxford (2002)zbMATHGoogle Scholar
  83. 83.
    Pardalos, P.M., Romeijn, H.E. (eds.): Handbook of Global Optimization, vol. 2. Kluwer, Dordrecht (2002)zbMATHGoogle Scholar
  84. 84.
    Pintér, J.D.: Global Optimization in Action. Kluwer, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  85. 85.
    Rebennack, S., Kallrath, J., Pardalos, P.M.: Column enumeration based decomposition techniques for a class of non-convex MINLP problems. J. Glob. Optim. 43(2–3), 277–297 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  86. 86.
    Pintér, J.D.: Global optimization in practice: state of the art and perspectives. In: Gao, D., Sherali, H. (eds.) Advances in Applied Mathematics and Global Optimization. Advances in Mechanics and Mathematics, vol. 17. Springer, Boston (2009)Google Scholar
  87. 87.
    Fasano, G.: A Modeling-Based Approach for Non-standard Packing Problems. In: Fasano, G., Pintér, J.D. (eds.) Optimized Packings and their Applications. Springer Optimization and its Applications. Springer, New York (2015)Google Scholar
  88. 88.
    Fasano, G., Saia, D., Piras, A.: Columbus stowage optimization by CAST (Cargo Accommodation Support Tool). Acta Astronaut. 67(3–4), 489–495 (2010)CrossRefGoogle Scholar
  89. 89.
    Stoyan, Y., Grebennik, I., Romanova, T., Kovalenko, A.: Optimized Packings in Space Engineering Applications - Part II. In: Fasano, G., Pintér, J.D. (eds.) Modeling and Optimization in Space Engineering - 2019. Springer, New York (2019)Google Scholar
  90. 90.
    Romanova, T., Bennell, J., Stoyan, Y., Pankratov, A.: Packing of concave polyhedra with continuous rotations using nonlinear optimization. Eur. J. Oper. Res. 268(1), 37–53 (2018)CrossRefzbMATHGoogle Scholar
  91. 91.
    Stoyan, Y., Pankratov, A., Romanova, T.: Quasi phi-functions and optimal packing of ellipses. J. Glob. Optim. 65(2), 283–307 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  92. 92.
    Stoyan, Y., Romanova, Т.: Mathematical Models of Placement Optimisation: Two- and Three-Dimensional Problems and Applications. In: Fasano, G., Pintér, J. (eds.) Modeling and Optimization in Space Engineering, Springer Optimization and its Applications, vol. 73, pp. 363–388. Springer, New York (2012)CrossRefGoogle Scholar
  93. 93.
    Stoyan, Y., Romanova, T., Pankratov, A., Kovalenko, A., Stetsyuk, P.: Modeling and Optimization of Balance Layout Problems. In: Fasano, G., Pintér, J. (eds.) Space Engineering. Modeling and Optimization with Case Studies. Springer Optimization and its Applications, vol. 114, pp. 369–400. Springer, New York (2016)zbMATHGoogle Scholar
  94. 94.
    Egeblad, J., Nielsen, B.K., Brazil, M.: Translational packing of arbitrary polytopes. Comput. Geom. Theory Appl. 42(4), 269–288 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  95. 95.
    Liu, X., Liu, J., Cao, A., Yao, Z.: HAPE3D - a new constructive algorithm for the 3D irregular packing problem. Front. Inform. Tech. Elect. Eng. 16(5), 380–390 (2015)CrossRefGoogle Scholar
  96. 96.
    Youn-Kyoung, J., Sang, D.N.: Intelligent 3D packing using a grouping algorithm for automotive container engineering. J. Comput. Des. Eng. 1(2), 140–151 (2014)Google Scholar
  97. 97.
    Pankratov, O., Romanova, T., Stoyan, Y., Chuhai, A.: Optimization of packing polyhedra in spherical and cylindrical containers. East. Eur. J. Enterp. Tech. 4(79), 39–47 (2016)Google Scholar
  98. 98.
    Stetsyuk, P., Romanova, T., Scheithauer, G.: On the global minimum in a balanced circular packing problem. Optim. Lett. 10, 1347–1360 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  99. 99.
    Stoyan, Y.G., Gil, N.I., Pankratov, A.V.: Packing Non-convex Polyhedra into a Parallelepiped. Technische Universitat Dresden, Dresden (2004)Google Scholar
  100. 100.
    Stoyan, Y., Gil, N., Scheithauer, G., Pankratov, A., Magdalina, I.: Packing of convex polyhedra into a parallelepiped. Optimization. 54(2), 215–235 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  101. 101.
    Stoyan, Y., Pankratov, A., Romanova, T., Chugay, A.: Optimized Object Packings Using Quasi-Phi-Functions. In: Fasano, G., Pintér, J. (eds.) Optimized Packings and their Applications, Springer Optimization and its Applications, vol. 105, pp. 265–291. Springer, New York (2015)zbMATHGoogle Scholar
  102. 102.
    Fischer, K., Gärtner, B., Kutz, M.: Fast Smallest-Enclosing-Ball Computation in High Dimensions. In: Algorithms - ESA 2003, vol. 2832, pp. 630–641 (2003)CrossRefGoogle Scholar
  103. 103.
    Wachter, A., Biegler, L.T.: On the implementation of an interior-point filter line-search algorithm for large- scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Yuriy Stoyan
    • 1
  • Alexandr Pankratov
    • 1
  • Tatiana Romanova
    • 1
  • Giorgio Fasano
    • 2
    Email author
  • János D. Pintér
    • 3
  • Yurij E. Stoian
    • 1
  • Andrey Chugay
    • 1
  1. 1.Department of Mathematical Modeling and Optimal DesignInstitute for Mechanical Engineering Problems of the National Academy of Sciences of UkraineKharkovUkraine
  2. 2.Thales Alenia SpaceTurinItaly
  3. 3.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA

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