Optimized Object Packings Using Quasi-Phi-Functions

  • Yuriy Stoyan
  • Tatiana Romanova
  • Alexander Pankratov
  • Andrey Chugay
Part of the Springer Optimization and Its Applications book series (SOIA, volume 105)


In this chapter we further develop the main tool of our studies,phi-functions. We define new functions, called quasi-phi-functions, that we use for analytic description of relations of geometric objects placed in a container taking into account their continuous rotations, translations, and distance constraints. The new functions are substantially simpler than phi-functions for some types of objects. They also are simple enough for some types of objects for which phi-functions could not be constructed. In particular, we derive quasi-phi-functions for certain 2D&3D-objects. We formulate a basic optimal packing problem and introduce its exact mathematical model in the form of a nonlinear continuous programming problem, using our quasi-phi-functions. We propose a general solution strategy, involving: a construction of feasible starting points, a generation of nonlinear subproblems of a smaller dimension and decreased number of inequalities; a search for local extrema of our problem using subproblems. To show the advantages of our quasi-phi-functions we apply them to two packing problems, which have a wide spectrum of industrial applications: packing of a given collection of ellipses into a rectangular container of minimal area taking into account distance constraints; packing of a given collection of 3D-objects, including cuboids, spheres, spherocylinders and spherocones, into a cuboid container of minimal height. Our efficient optimization algorithms allow us to get local optimal object packings and reduce considerably computational cost. We applied our algorithms to several inspiring instances: our new benchmark instances and known test cases.


Packing 2D- and 3D-objects Continuous rotations Mathematical model development Quasi-phi-functions Nonlinear optimization 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Yuriy Stoyan
    • 1
  • Tatiana Romanova
    • 1
  • Alexander Pankratov
    • 1
  • Andrey Chugay
    • 1
  1. 1.Department of Mathematical Modeling and Optimal Design, Institute for Mechanical Engineering ProblemsNational Academy of Sciences of UkraineKharkivUkraine

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