Heuristic approaches for the two- and three-dimensional knapsack packing problem

The maximum profit twoor three-dimensional knapsack packing problem asks to pack a maximum profit subset of some given rectangles or boxes into a larger rectangle or box of fixed dimensions. Items must be orthogonally packed, but no other restrictions are imposed to the problem. The problem could also be considered as a knapsack problem generalized to two or three dimensions. In this paper we present a new heuristic based on the sequence pair representation proposed by Murata et al. (1996) using a semi-normalized packing by Pisinger (2006) for the two-dimensional knapsack problem. A local search algorithm maintains a pair of sequences given as permutations of the item numbers. In each step a neighbor solution is generated by making a small permutation in one or both sequences. The new sequence pair is transformed to a packing and the corresponding objective function is evaluated. Solutions are accepted based on a Simulated Annealing. The heuristic is also able to handle problem instances where rotation is allowed. A similar approach with a novel abstract representation of box placements, called sequence tripple, has been developed for the threedimensional knapsack problem. Comprehensive computational experiments comparing the developed heuristics with previous approaches indicate that the results are very promising for both twoand three-dimensional problems.