Optimized Object Packings Using Quasi-Phi-Functions

Yuriy Stoyan, Tatiana Romanova, Alexander Pankratov and Andrey Chugay ()
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Yuriy Stoyan: National Academy of Sciences of Ukraine
Tatiana Romanova: National Academy of Sciences of Ukraine
Alexander Pankratov: National Academy of Sciences of Ukraine
Andrey Chugay: National Academy of Sciences of Ukraine

Chapter Chapter 13 in Optimized Packings with Applications, 2015, pp 265-293 from Springer

Abstract: Abstract 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.

Keywords: Packing 2D- and 3D-objects; Continuous rotations; Mathematical model development; Quasi-phi-functions; Nonlinear optimization (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/978-3-319-18899-7_13

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