Skip to main content
SpringerLink
Log in

Cutting and Packing Beyond and Within Mathematical Programming

  • Chapter
  • First Online:
Business Optimization Using Mathematical Programming

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 307))

Abstract

This chapter, based on material provided and written by Prof. Dr. Yuriy Stoyan & Prof. Dr. Tatiana Romanova ( The National Academy of Sciences of Ukraine, Institute of Mechanical Engineering Problems, Department of Mathematical Modeling and Optimal Design, Kharkiv, Ukraine & Kharkiv National University of Radioelectronics, Department of Applied Mathematics.), is devoted to the phi-function technique used for mathematical modeling of cutting and packing (C&P) problems. Phi-functions are constructed here for some 2D and 3D geometric objects. Phi-functions can be described by quite simple formulas. A general solution strategy using phi-functions is outlined. Conceptually, the Phi-function approach exploits NLP and MINLP and can be understood as cutting and packing beyond and within Mathematical Programming. It also exploits polylithic modeling and solution techniques.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The National Academy of Sciences of Ukraine, Institute of Mechanical Engineering Problems, Department of Mathematical Modeling and Optimal Design, Kharkiv, Ukraine & Kharkiv National University of Radioelectronics, Department of Applied Mathematics.

  2. 2.

    In general topology, closed sets that are closures of their interior are said to be canonically closed; this is what phi-objects are.

References

  1. Agarwal, P.K., Flato, E., Halperin, D.: Polygon decomposition for efficient construction of Minkowski sums. Comp. Geom. Theory Appl. 21, 29–61 (2002)

    Google Scholar 

  2. Albano, A., Sapuppo, G.: Optimal allocation of two-dimensional irregular shapes using heuristic search methods. IEEE Transl. Syst. Man Cybernet. 10, 242–248 (1980)

    Article  Google Scholar 

  3. Art, R.C.: An Approach to the Two-Dimensional Irregular Cutting Stock Problem. Tech. Report 36.008, IBM Cambridge Scientific Centre (1966)

    Google Scholar 

  4. Bennell, J.A., Dowsland, K.A.: Hybridising Tabu search with optimisation techniques for irregular stock cutting. Manage. Sci. 47, 1160–1172 (2001)

    Article  Google Scholar 

  5. Bennell, J.A., Oliveira, J.F.: The geometry of nesting problems: a tutorial. Eur. J. Oper. Res. 184, 397–415 (2008)

    Article  Google Scholar 

  6. Bennell, J.A., Song, X.: A comprehensive and robust procedure for obtaining the nofit polygon using Minkowski sums. Comput. Oper. Res. 35, 267–281 (2008)

    Article  Google Scholar 

  7. Bennell, J., Scheithauer, G., Stoyan, Y., Romanova, T.: Tools of mathematical modelling of arbitrary object packing problems. J. Ann. Oper. Res. 179, 343–368 (2010)

    Article  Google Scholar 

  8. Bennell, J., Scheithauer, G., Stoyan, Y., Romanova, T., Pankratov, A.: Optimal clustering of a pair of irregular objects. J. Global Optim. 61, 497–524 (2015)

    Article  Google Scholar 

  9. Burke, E., Hellier, R., Kendall, G., Whitwell, G.: A new bottom-left-fill heuristic algorithm for the two-dimensional irregular packing problem. Oper. Res. 54, 587–601 (2006)

    Article  Google Scholar 

  10. Chernov, N., Stoyan, Y., Romanova, T.: Mathematical model and efficient algorithms for object packing problem. Comput. Geom. 43, 535–553 (2010)

    Article  Google Scholar 

  11. Chernov, N., Stoyan, Y., Romanova, T., Pankratov, A.: Phi-functions for 2D objects formed by line segments and circular arcs. Adv. Oper. Res. (2012). Article ID 346358

    Google Scholar 

  12. Dighe, R., Jakiela, M.J.: Solving pattern nesting problems with genetic algorithms employing task decomposition and contact. Evolut. Comput. 3, 239–266 (1996)

    Article  Google Scholar 

  13. Duriagina, Z., Lemishka, I., Litvinchev, I., Marmolejo, J., Pankratov, A., Romanova, T., Yaskov, G.: Optimized filling a given cuboid with spherical powders for additive manufacturing. J. Oper. Res. Soc. China 2020, 1–16 (2020)

    Google Scholar 

  14. Fasano, G.: Solving Non-Standard Packing Problems by Global Optimization and Heuristics. Springer, Cham (2014)

    Book  Google Scholar 

  15. Gan, M., Gopinathan, N., Jia, X., Williams, R.A.: Predicting packing characteristics of particles of arbitrary shapes. KONA 22, 2–93 (2004)

    Article  Google Scholar 

  16. Ghosh, P.K.: An algebra of polygons through the notion of negative shapes. CVGIP: Image Understand. 54, 119–144 (1991)

    Article  Google Scholar 

  17. Gomes, A.M., Oliveira, J.F.: A 2-exchange heuristic for nesting problems. Eur. J. Oper. Res. 141, 359–370 (2002)

    Article  Google Scholar 

  18. Gomes, A.M., Oliveira, J.F.: Solving irregular strip packing problems by hybridising simulated annealing and linear programming. Eur. J. Oper. Res. 171, 811–829 (2006)

    Article  Google Scholar 

  19. Kallrath, J.: Cutting circles and polygons from area-minimizing rectangles. J. Glob. Optim. 43, 299–328 (2009)

    Article  Google Scholar 

  20. Kallrath, J., Pankratov, A., Romanova, T., Litvinchev, I.: Minimal perimeter convex hulls of convex polygons. J. Glob. Optim. (2021, submitted (under review))

    Google Scholar 

  21. Leao, A.A.S., Toledo, F.M.B., Oliveira, J.F., Carravilla, M., Alvarez-Valdes, R.: Irregular packing problems: a review of mathematical models. Eur. J. Oper. Res. 282(3), 803–822 (2019)

    Article  Google Scholar 

  22. Li, Z., Milenkovic, V.: Compaction and separation algorithms for non-convex polygons and their applications. Eur. J. Oper. Res. 84, 539–561 (1995)

    Article  Google Scholar 

  23. Litvinchev, I., Romanova, T., Corrales-Diaz, R., Esquerra-Arguelles, A., Martinez-Noa, A.: Lagrangian approach to modeling placement conditions in optimized packing problems. Mob. Netw. Appl. 25, 2126–2133 (2020) (2020)

    Google Scholar 

  24. Mahadevan, D.A.: Optimization in Computer-Aided Pattern Packing. Ph.D. Thesis, North Carolina State University (1984)

    Google Scholar 

  25. Oliveira, J.F., Ferreira, J.S.: Algorithms for nesting problems. In: Vidal, R. (ed.) Applied Simulated Annealing. Lecture Notes in Economics and Mathematical Systems, vol. 396, pp. 255–274. Springer, Heidelberg (1993)

    Google Scholar 

  26. Pankratov, A., Romanova, T., Litvinchev, I.: Packing ellipses in an optimized convex polygon. J. Glob. Optim. 75(2), 495–522 (2019)

    Article  Google Scholar 

  27. Pankratov, A., Romanova, T., Litvinchev, I.: Packing ellipses in an optimized rectangular container. Wirel. Netw. 26(7), 4869–4879 (2020)

    Article  Google Scholar 

  28. Pankratov, A., Romanova, T., Litvinchev, I.: Packing oblique 3D objects. Mathematics 8(7) (2020)

    Google Scholar 

  29. Romanova, T., Bennell, J., Stoyan, Y., Pankratov, A.: Packing of concave polyhedra with continuous rotations using nonlinear optimisation. Eur. J. Oper. Res. 268, 37–53 (2018)

    Article  Google Scholar 

  30. Romanova, T., Pankratov, A., Litvinchev, I., Pankratova, Y., Urniaieva, I.: Optimized packing clusters of objects in a rectangular container. Math. Probl. Eng. 2019 (2019). https://doi.org/10.1155/2019/4136430

  31. Romanova, T., Stetsyuk, P., Chugay, A., Shekhovtsov, S.: Parallel computing technologies for solving optimization problems of geometric design. Cybernet. Syst. Anal. 55(6), 894–904 (2019)

    Article  Google Scholar 

  32. Romanova, T., Stoyan, Y., Pankratov, A., Litvinchev, I., Avramov, K., Chernobryvko, M., Yanchevskyi, I., Mozgova, I., Bennell, J.: Optimal layout of ellipses and its application for additive manufacturing. Int. J. Prod. Res. 59(2), 560–575 (2019)

    Article  Google Scholar 

  33. Romanova, T., Litvinchev, I., Grebennik, I., Kovalenko, A., Urniaieva, I., Shekhovtsov, S.: Packing convex 3D objects with special geometric and balancing conditions. In: Vasant, P., Zelinka, I., Weber, G.W. (eds.) Modeling an Optimization in Space Engineering Applications. Intelligent Computing and Optimization. Advances in Intelligent Systems and Computing, vol. 1072, pp. 273–281. Springer, Cham (2020)

    Google Scholar 

  34. Romanova, T., Litvinchev, I., Pankratov, A.: Packing ellipses in an optimized cylinder. Eur. J. Oper. Res. 285, 429–443 (2020)

    Article  Google Scholar 

  35. Romanova, T., Pankratov, A., Litvinchev, I., Plankovskyy, S., Tsegelnyk, Y., Shypul, O.: Sparsest packing of two-dimensional objects. Int. J. Prod. Res. (2020). https://doi.org/10.1080/00207543.2020.1755471

  36. Romanova, T., Stoyan, Y., Pankratov, A., Litvinchev, I., Plankovskyy, S., Tsegelnyk, Y., Shypul, O.: Sparsest balanced packing of irregular 3d objects in a cylindrical container. Eur. J. Oper. Res. 285(2), 429–443 (2020)

    Article  Google Scholar 

  37. Scheithauer, G.: Introduction to Cutting and Packing Optimization - Problems, Modeling Approaches, Solution Methods. International Series in Operations Research and Management Science, vol. 263. Springer, Cham (2018)

    Google Scholar 

  38. Stoyan, Y.: Mathematical methods for geometric design. In: Advances in CAD/CAM, Proceedings of PROLAMAT82 (Leningrad, USSR, May 1982), pp. 67–86. North-Holland, Amsterdam (1983)

    Google Scholar 

  39. Stoyan, Y., Romanova, T.: Mathematical models of placement optimisation: two- and three-dimensional problems and applications. In: Fasano, G., Pinter, J.D. (eds.) Modeling and Optimization in Space Engineering. Lecture Notes in Economics and Mathematical Systems, vol. 73, pp. 363–388. Springer, New York (2013)

    Chapter  Google Scholar 

  40. Stoyan, Y., Terno, J., Scheithauer, G., Gil, N., Romanova, T.: Phi-functions for primary 2D-objects. Stud. Inf. Univ. 2, 1–32 (2001)

    Google Scholar 

  41. Stoyan, Y., Gil, M., Terno, J., Romanova, T., Scheithauer, G.: Construction of a Phi-function for two convex polytopes. Appl. Math. 2, 199–218 (2002)

    Google Scholar 

  42. Stoyan, Y., Scheithauer, G., Gil, N., Romanova, T.: Φ-functions for complex 2D-objects. 4OR: Quart. J. Belg. French Ital. Oper. Res. Soc. 2, 69–84 (2004)

    Google Scholar 

  43. Stoyan, Y., Gil, N.I., Scheithauer, G., Pankratov, A., Magdalina, I.: Packing of convex polytopes into a parallelepiped. Optimization 54, 215–235 (2005)

    Article  Google Scholar 

  44. Stoyan, Y., Romanova, T., Scheithauer, G., Krivulya, A.: Covering a polygonal region by rectangles. Comput. Optim. Appl. 48(3), 675–695 (2011)

    Article  Google Scholar 

  45. Stoyan, Y., Romanova, T., Pankratov, A., Chugay, A.: Optimized object packings using quasi-phi-functions. In: Fasano, G., Pinter, J.D. (eds.) Optimized Packings with Applications, pp. 265–293. Springer International Publishing, Cham (2015)

    Chapter  Google Scholar 

  46. Stoyan, Y., Romanova, T., Pankratov, A., Kovalenko, A., Stetsyuk, P.: Modeling and optimization of balance layout problems. In: Fasano, G., Pinter, J.D. (eds.) Space Engineering. Modeling and Optimization with Case Studies. Optimization and its Applications, vol. 114, pp. 369–400. Springer, New York (2016)

    Google Scholar 

  47. Stoyan, Y., Pankratov, A., Romanova, T.: Cutting and packing problems for irregular objects with continuous rotations: mathematical modelling and non-linear optimization. J. Oper. Res. Soc. 67, 786–800 (2016)

    Article  Google Scholar 

  48. Stoyan, Y., Pankratov, A., Romanova, T.: Quasi-phi-functions and optimal packing of ellipses. J. Glob. Optim. 65, 283–307 (2016)

    Article  Google Scholar 

  49. Stoyan, Y., Pankratov, A., Romanova, T.: Placement problems for irregular objects: Mathematical modeling, optimization and applications. In: Butenko, S., Pardalos, P.M., Shylo, V. (eds.) Optimization Methods and Applications, pp. 521–559. Springer International Publishing, Cham (2017)

    Chapter  Google Scholar 

  50. Stoyan, Y., Pankratov, A., Romanova, T., Fasano, G., Pinter, J.D., Stoian, Y.E., Chugay, A.: Optimized packings in space engineering applications: part I. In: Fasano, G., Pinter, J.D. (eds.) Modeling and Optimization in Space Engineering. Springer Optimization and its Applications, vol. 144, pp. 395–437. Springer, Cham (2019)

    Chapter  Google Scholar 

  51. Stoyan, Y., Grebennik, I., Romanova, T., Kovalenko, A.: Optimized packings in space engineering applications: Part II. In: Fasano, G., Pinter, J.D. (eds.) Modeling and Optimization in Space Engineering. Springer Optimization and its Applications, vol. 144, pp. 439–457. Springer, Cham (2019)

    Chapter  Google Scholar 

  52. Stoyan, Y., Yaskov, G., Romanova, T., Litvinchev, I., Yakovlev, S., Velarde Cantu, J.M.: Optimized packing multidimensional hyperspheres: a unified approach. Math. Biosci. Eng. 17, 6601–6630 (2020)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Astronomy, University of Florida, Gainesville, FL, USA

    Josef Kallrath

Authors
  1. Josef Kallrath
    View author publications

    You can also search for this author in PubMed Google Scholar

15.1 Electronic Supplementary Material

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kallrath, J. (2021). Cutting and Packing Beyond and Within Mathematical Programming. In: Business Optimization Using Mathematical Programming. International Series in Operations Research & Management Science, vol 307. Springer, Cham. https://doi.org/10.1007/978-3-030-73237-0_15

Download citation

Publish with us

Policies and ethics

Search

Navigation