On solving mixed shapes packing problems by continuous optimization with the CMA evolution strategy

Bin packing is a classical combinatorial optimization problem which has a wide range of real-world applications in industry, logistics, transport, parallel computing, circuit design and other domains. While usually presented as discrete problems, we consider here continuous packing problems including curve shapes, and model these problems as continuous optimization problems with a multi-objective function combining non-overlapping with minimum bin size constraints. More specifically, we consider the covariance matrix adaptation evolution strategy (CMA-ES) with a non-overlapping and minimum size objective function in either two or three dimensions. Instead of taking the intersection area as measure of overlap, we propose other measures, monotonic with respect to the intersection area, to better guide the search. In order to compare this approach to previous work on bin packing, we first evaluate CMA-ES on Korf's benchmark of consecutive sizes square packing problems, for which optimal solutions are known, and on a benchmark of circle packing problems. We show that on square packing, CMA-ES computes solutions at typically 14% of the optimal cost, with the time limit given by the best dedicated algorithm for computing optimal solutions, and that on circle packing, the computed solutions are at 2% of the best known solutions. We then consider generalizations of this benchmark to mixed squares and circles, boxes, spheres and cylinders packing problems, and study a real-world problem for loading boxes and cylinders in containers. These hard problems illustrate the interesting trade-off between generality and efficiency in this approach.