Recursive circle packing problems

Abstract

This paper presents a class of packing problems where circles may be placed either inside or outside other circles, the whole set being packed in a rectangle. This corresponds to a practical problem of packing tubes in a container. Before being inserted in the container, tubes may be put inside other tubes in a recursive fashion. A variant of the greedy randomized adaptive search procedure is proposed for tackling this problem, and its performance is assessed in a set of benchmark instances.

Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition

“…For this experiment the specifications from the second experiment are used for out method. We use the Python implementation of GRASP from [27]. Note that SCIP is written in the programming language C, in which an implementation of GRASP would be much faster.…”

“…We consider two different test sets for our experiments. The first one contains 9 real-world instances from the tube industry, which were used in [27], and is in the following called real test set. Because this test set is too limited for a detailed computational study, we created a second test set containing 800 instances, the rand test set.…”

“…for all i = j with c i = c j . RCPP can be equivalently formulated as an MINLP, see Pedroso et al [27]. The formulation consists of four different types of variables:…”

“…The Recursive Circle Packing Problem (RCPP) has been introduced recently by Pedroso et al [27]. The objective of RCPP is to select a minimum number of rectangles of the same size such that a given set of rings can be packed into these rectangles in a non-overlapping way.…”

“…Figure 1 gives two examples of a feasible packing. Pedroso et al [27] present a nonconvex MINLP formulation for RCPP. The key idea of the developed MINLP is to use binary variables in order to indicate whether a ring is packed inside another larger ring or directly into a rectangle.…”