This paper reviews the most relevant literature on efficient models and methods for packing circular objects/items into Euclidean plane regions where the objects/items and regions are either two- or three-dimensional. These packing problems are NP hard optimization problems with a wide variety of applications. They have been tackled using various approaches-based algorithms ranging from computer-aided optimality proofs, to branch-and-bound procedures, to constructive approaches, to multi-start nonconvex minimization, to billiard simulation, to multiphase heuristics, and metaheuristics.

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate S with length L and width W, to maximize the sum of the profits of the pieces to be cut. Each piece type i, i=1,…,n, is characterized by a length li, a width wi, a profit (or weight) ci, and an upper demand value bi. The upper demand value is the maximum number of pieces of type i that can be cut on S. In this paper, we study the two-staged C_TDC problem, noted C_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two cuts. We solve the C_2TDC problem using an exact algorithm that is mainly based on a bottom-up strategy. We introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns. We evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm.

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The two-dimensional layout (TDL) optimization problem consists of finding the minimal length layout of a set of both regular and irregular two-dimensional shapes on a stock sheet of finite width but infinite length. The layout should not contain any overlap. In this paper, we solve the TDL problem by proposing two approximate algorithms. The first one is a new heuristic based on a constructive approach specifically designed for irregular shapes, but that works well with regular ones. The second is a hybrid method in which the constructive approach displays the layouts corresponding to the chromosomes yielded by the genetic algorithm. The resulting algorithm yields satisfactory results in relatively short computational times as shown by the extensive computational testing on problems from the literature.

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