# Abstract

Packing Curved Objects / 396

*Ignacio Antonio Salas Donoso, Gilles Chabert*

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This paper deals with the problem of packing two-dimensional objects of quite arbitrary shapes including in particular curved shapes (like ellipses) and assemblies of them. This problem arises in industry for the packaging and transport of bulky objects which are not individually packed into boxes, like car spare parts. There has been considerable work on packing curved objects but, most of the time, with specific shapes; one famous example being the circle packing problem. There is much less algorithm for the general case where different shapes can be mixed together. A successful approach has been proposed recently in Martinez et al. (T. Martinez, L. Vitorino, F. Fages, and A. Aggoun. On Solving Mixed Shapes Packing Problems by Continuous Optimization with the CMA Evolution Strategy. In Proceedings of the first BRICS countries congress on Computational Intelligence, 2013) and the algorithm we propose here is an extension of their work. Martinez et al. use a stochastic optimization algorithm with a fitness function that gives a violation cost and equals zero when objects are all packed. Their main idea is to define this function as a sum of n!/(2!(n-2)!) elementary functions that measure the overlapping between each pair of different objects. However, these functions are ad-hoc formulas. Designing ad-hoc formulas for every possible combination of object shapes can be a very tedious task, which dramatically limits the applicability of their approach. The aim of this paper is to generalize the approach by replacing the ad-hoc formulas with a numerical algorithm that automatically measures the overlapping between two objects. Then, we come up with a fully black-box packing algorithm that accept any kind of objects.