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Abstract In stamping, operating cost are dominated by raw material costs, which can typically reach 75% of total costs in a stamping facility. In this paper, a new algorithm is described that determines stamping strip layouts for pairs of parts such that the layout optimizes material utilization efficiency. This algorithm predicts the jointly-optimal blank orientation on the strip, relative positions of the paired blanks and the optimum width for the strip. Examples are given for pairing the same parts together with one rotated 180º, and for pairs of different parts nested together. This algorithm is ideally suited for incorporation into die design CAE systems.36299
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Keywords: Stamping, Die Design, Optimization, Material Utilization, Minkowski Sum, Design Tools
Introduction
In stamping, sheet metal parts of various levels of complexity are produced rapidly, often in very high volumes, using hard tooling. The production process operates efficiently, and material costs can typically represent 75% of total operating costs in a stamping facility [1]. Not all of this material is used in the parts, however, due to the need to trim scrap material from around irregularly-shaped parts. The amount of scrap produced is directly related to the efficiency of the stamping strip layout. Clearly, using optimal strip layouts is crucial to a stamping firm’s competitiveness.论文网
Previous Work
Originally, strip layout problems were solved manually, for example, by cutting blanks from cardboard and manipulating them to obtain a good layout. The introduction of computers into the design process led to algorithmic approaches. Perhaps the first was to fit blanks into rectangles, then fit the rectangles along the strip[2]. Variations of this approach have involved fitting blanks into non-overlapping composites of rectangles [3], convex polygons [4,5] and known interlocking shapes[6]. A fundamental limitation exists with this approach, however, in that the enclosing shape adds material to the blank that cannot be removed later during the layout process. This added material may prevent optimal layouts from being found.
A popular approach to performing strip layout is the incremental rotation algorithm [6-10, 16]. In it, the blank, or blanks, are rotated by a fixed amount, such as 2º[7], the pitch and width of the layout determined and the material utilization calculated. After repeating these steps through a total rotation of 180º (due to symmetry), the orientation giving the best utilization is selected. The disadvantage of this method is that, in general, the optimal blank orientation will fall between the rotation increments, and will not be found. Although small, this inefficiency per part can accumulate into significant material losses in volume production.
Meta-heuristic optimization methods have also been applied to the strip layout problem, both simulated annealing [11, 12] and genetic programming [13]. While capable of solving layout problems of great complexity (i.e. many different parts nested together, general 2-D nesting of sheets), they are not guaranteed to reach optimal solutions, and may take significant computational effort to converge to a good solution.
Exact optimization algorithms have been developed for fitting a single part on a strip where the strip width is predetermined [14] and where it is determined during the layout process [15]. These algorithms are based on a geometric construction in which one shape is ‘grown’ by another shape. Similar versions of this construction are found under the names ‘no-fit polygon’, ‘obstacle space’ and ‘Minkowski sum’. Fundamentally, they simplify the process of determining relative positions of shapes such that the shapes touch but do not overlap. Through the use of this construction (in this paper, the particular version used is the Minkowski sum), efficient algorithms can be created that find the globally optimal strip layout.
For the particular problem of strip layout for pair s of parts, results have been reported using the incremental rotation algorithm [7, 16] and simulated annealing [11], but so far no exact algorithm has been available. In what follows, the Minkowski sum and its application to strip layout is briefly introduced, and its extension to nesting pairs of parts is described.