A System for the Compactation of Two-Dimensional Irregular Shapes based on Simulated Annealing

The problem of compacting a given number of twodimensional shapes minimizing the area of t,he enclosing rectangle, i.e., minimizing the waste prodiiced, arises quite often in some industrial processes like the automotive industry, clothing manufacturing, steel construction, electronic engineering and leat.licr cutting. A simulated annealing approach for the compactation of two-dimensional irregular shapes is presented. The energy function is defined by considering three components: 1) A measure of the enclosing rectangle area; 2) A measure of the distances between each piece and the center of the board, weighed by parameters reflecting the desired width/height ratio of the enclosing rectangle; 3) A measure of the quality (goodness) of local solutions. The results show that the annealing algorithm performs rather well dealing with irregular patterns allocation, even though leading to higher computation times than those needed to run some heuristic methods. However, there is some evidence in the results obtained so far that near-optimal solutions may be reached in polinomial time. 1 Int r o d u c t io 11 The problem of compacting a given number of twodimensional regular or irregular shapes minimizing the waste produced, is certainly of relevant interest to some industries, where the cost of the wasted material can reach surprisingly high values. To find the optimal allocation of two-dimensional irregular shapes by complete enumeration is clearly an exponencial time algorithm: given a set of N pieces to allocate over a grid of L x W points, the cardinality 'Centro de A&se e Processamento de Sin+/ Lahoratdrio de RoMtica e Processamento de Informag&o, Complexo I IST, AV. Rovirco Pais, 1096 Lisboa Codex, Portugal. of the set of configurations, including the ones where overlapping occurs, is IC1 = ( L x W x R),', ( 1) where R denotes the number 31 possible orientations for each piece. If the set of feasible configiirations 3 is consitlered, by excluding llie coiifigirratioiis witti ovcrlappcil pieces, the total number of solutions is heavily dependent on the shapes of the pieces to be allocated. A n upper bound for 131 is given by where Aj is the number of grid points covered by the j r h piece. Since A, > 0 for j = 1. . .N 1, JFI < ICI, as expected. Unfortunately, there is no known method for the generation of 3, avoiding the search over the whole set C. During the second half of this century, some approaches have been tried to find good solutions for the allocation problem, partly due to worldwide industry development, and also because computers emerged as excellent tools for the solution of these problems. The methods which have been used belong basically to one of three categories: