Models for the two-dimensional open-dimension rectangular packing problem

Abstract

This study examines the two-dimensional open-dimension rectangular packing problem, which involves packing rectangular items with known dimensions into a rectangular container with variable dimensions. The objective is to pack all items orthogonally while minimizing the container's area. Practical applications include facility layout design and cutting and packing scenarios to optimize container sizes or process large raw materials. Mathematical formulations are proposed, including an integer linear programming (ILP) model based on grid positioning and a constraint programming (CP) model using logic operators. As a reference model from the literature, an ILP formulation based on free positioning, adapted from prior work, is presented for comparison purposes. Computational experiments assess the performance of these models in benchmark instances. The results show that the grid-position formulation performs best with fewer item types, while the free-position formulation performs best with fewer total items, and the CP model excels in proven optimality and solution quality.