An effective recursive partitioning approach for the packing of identical rectangles in a rectangle

Abstract

In this work, we deal with the problem of packing (orthogonally and without overlapping) identical rectangles in a rectangle. This problem appears in different logistics settings, such as the loading of boxes on pallets, the arrangements of pallets in trucks and the stowing of cargo in ships. We present a recursive partitioning approach combining improved versions of a recursive five-block heuristic and an L-approach for packing rectangles into larger rectangles and L-shaped pieces. The combined approach is able to rapidly find the optimal solutions of all instances of the pallet loading problem sets Cover I and II (more than 50 000 instances). It is also effective for solving the instances of problem set Cover III (almost 100 000 instances) and practical examples of a woodpulp stowage problem, if compared to other methods from the literature. Some theoretical results are also discussed and, based on them, efficient computer implementations are introduced. The computer implementation and the data sets are available for benchmarking purposes.

Mathematical models and decomposition algorithms for cutting and packing problems

“…In all such cases, it turned out that the lower bound found by the most powerful PLP heuristics was the optimal solution value, enforcing the conjecture of Birgin et al [41] that their recursive partitioning approach always finds the optimal solution.…”

“…This problem, whose complexity status is currently unknown, was introduced in the late sixties by Barnett and Kynch [22] and then extensively studied by many researchers, who proposed both exact methods (see, e.g., Dowsland [109], AlvarezValdes et al [5], or Ahn et al [2]) and powerful heuristics (see, e.g., Lins et al [190] or Birgin et al [41]). The number of references being huge, we refer the interested reader to the recent, very complete survey by Silva et al [260].…”

“…Attempts to exactly solve such instances, identified by Birgin et al [41], is currently underway (see the web page http://lagrange.ime.usp.br/~lobato/packing/cover3.php they maintain);…”

Una revisión literaria del Problema de Carga del Pallet

“…In other methods, long time consuming and high demand of computational resources for big instances end up getting no efficient results. The positive results obtained with the use of the above mentioned techniques caused the interest of other researchers to develop hybrid methods which combined Lagrangean and Surrogate Relaxations; Lagrangean Relaxation and Clusters; Tabu Search with G4, Five-block Heuristic with L-approach; or Profit-based Heuristic with Genetic algorithms as in [3], [9], [13], [15] and [23].…”

Orthogonal packing of identical rectangles within isotropic convex regions

“…When the convex region takes the particular form of a rectangle, we are faced with the well known pallet loading problem [6,14,20,30], for which dedicated solution methods exist. Numerical experiments presented in [9] show that nonlinear-based methods, such as the one presented here, are not competitive with clever methods developed for this particular case.…”