A dichotomous search-based heuristic for the three-dimensional sphere packing problem

In this paper, the three-dimensional sphere packing problem is solved by using a dichotomous search-based heuristic. An instance of the problem is defined by a set of n unequal spheres and an object of fixed width and height and, unlimited length. Each sphere is characterized by its radius and the aim of the problem is to optimize the length of the object containing all spheres without overlapping. The proposed method is based upon beam search, in which three complementary phases are combined: (i) a greedy selection phase which determines a series of eligible search subspace, (ii) a truncated tree search, using a width-beam search, that explores some promising paths, and (iii) a dichotomous search that diversifies the search. The performance of the proposed method is evaluated on benchmark instances taken from the literature where its obtained results are compared to those reached by some recent methods of the literature. The proposed method is competitive and it yields promising results. Subjects: Combinatorics; Operational Mathematics; Mathematical Modeling