An efficient deterministic heuristic for two-dimensional rectangular packing

This paper proposes a deterministic heuristic, a best fit algorithm (BFA), for solving the NP-hard two-dimensional rectangular packing problem to maximize the filling rate of a rectangular sheet. There are two stages in this new approach: the constructive stage and the tree search stage. The former aims to rapidly generate an initial solution by employing the concepts of action space and fit degree in evaluating different placements. The latter seeks to further improve the solution and searches for promising placements by a partial tree search procedure. We then compare BFA with other approaches in terms of solution quality and computing time. We carry out computational experiments on two sets of well-known benchmark instances, C21 proposed by Hopper and Turton, and N13 proposed by Burke et al. BFA gained an average filling rate of 100% for the C21 instances within short times, indicating that all the layouts obtained are optimal. To the best of our knowledge, this is the first time that optimal layouts on all the 21 instances were obtained by a deterministic algorithm. As for the N13 instances, to date, researchers have found optimal solutions to the first three instances, whereas BFA solved seven, including the first three, within a reasonable period. An additional work is to adapt BFA to solve a relevant problem, the constrained two-dimensional cutting (or packing) problem (CTDC). Though BFA is not for the CTDC in the original design such that some specific characteristics of CTDC are not considered, the adapted algorithm still performed well on 21 public CTDC instances.