Journal
JOURNAL OF GLOBAL OPTIMIZATION
Volume 65, Issue 2, Pages 283-307Publisher
SPRINGER
DOI: 10.1007/s10898-015-0331-2
Keywords
Quasi-phi-functions; Object continuous rotations; Non-overlapping; Distance constraints; Ellipse packing; Mathematical model; Nonlinear optimization
Funding
- Science and Technology Center in Ukraine
- National Academy of Sciences of Ukraine [5710]
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We further develop our phi-function technique for solving Cutting and Packing problems. Here we introduce quasi-phi-functions for an analytical description of nonoverlapping and containment constraints for 2D- and 3D-objects which can be continuously rotated and translated. These new functions can work well for various types of objects, such as ellipses, for which ordinary phi-functions are too complicated or have not been constructed yet. We also define normalized quasi-phi-functions and pseudonormalized quasi-phi-functions for modeling distance constraints. To show the advantages of our new quasi-phi-functions we apply them to the problem of placing a given collection of ellipses into a rectangular container of minimal area. We use radical free quasi-phi-functions to reduce it to a nonlinear programming problem and develop an efficient solution algorithm. We present computational results that compare favourably with those published elsewhere recently.
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