This paper considers packing problems with balancing conditions and items consisting of clusters of parallelepipeds (mutually orthogonal, i.e. tetris-like items). This issue is quite frequent in space engineering and a real-world application deals with the Automated Transfer Vehicle project (funded by the European Space Agency), at present under development. A Mixed Integer Programming (MIP) approach is proposed. The three-dimensional single bin packing problem is considered. It consists of orthogonally placing, with possibility of rotation, the maximum number of parallelepipeds into a given parallelepiped. A MIP formulation of the problem is reported together with a MIP-based heuristic approach. Balancing conditions are furthermore examined, as well as the orthogonal placement (with rotation) of tetris-like items into a rectangular domain.

Abstract. This paper is the continuation of a previous work (Fasano 2004), dedicated to a MIP formulation for non-standard three-dimensional packing issues, with additional conditions. The Single Bin Packing problem (Basic Problem) is considered and its MIP formulation shortly surveyed, together with some possible extensions, including balancing, tetris-like items and non-standard domains. A MIP-based heuristic is proposed to solve efficiently the Basic Problem or any possible extension of it, susceptible to a MIP formulation. The heuristic is a recursive procedure based on a non-blind local search philosophy. The concept of abstract configuration, concerning the relative positions between items, is introduced: the relative positions of items, determined by any abstract configuration, give rise to a feasible solution in an unbounded domain. The heuristic generates a sequence of good abstract configurations and solves, step by step, a reduced MIP model by fixing the relative positions of items, corresponding to the current abstract configuration.

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