Optimal Packing Problems: From Knapsack Problem to Open Dimension Problem

The paper considers a packing problem of arbitrary shaped objects into an optimized container (OPP) formulated as a knapsack problem. Mathematical model of OPP in the form of a knapsack problem (KP) is provided. A new approach of reducing the knapsack problem (KP) to a sequence of the open dimension problems (ODP) is proposed. The key idea of the approach is based on the homothetic transformations of the container. The approach is most efficient for optimization packing problems into containers of complex geometry.