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(4)
Calculating the utilization at each of the 120 nodes on Fig.3 gives the results shown in Fig.4. In this figure, the curve is broken as the translation vector passes the end of each edge of to show how utilization can change during the traversal of each edge. While some edge traversals show monotonic changes in utilization, others show two or even three local maxima. Discovering these local optima is the reason why a number of translation nodes are needed.
Optimal Material Utilization for Various Translations Between Polygons A and –A.
As a progression is made around , when local maxima are indicated, a numerical optimization technique is invoked. Since derivatives of the utilization function are not available(without additional computational effort),an interval-halving
Approach was taken [19]. The initial interval consists of the nodes bordering the indicated local maximal point. Three equally-spaced points are placed across this interval (i.e. at 1/4, 1/2 and 3/4 positions), and the utilization at each is calculated. By comparing the utilization values at each point, a decision can be made as to which half of the interval is dropped from consideration and the process is repeated. This continues until the desired accuracy is obtained.
Applying this method to the example leads to the optimal translation vector of (747.894, 250.884), giving the strip layout shown in Fig.5, with a material utilization of 92.02%.
Interestingly, while it appears that the pairs of parts could be pushed closer together for a better layout, doing so decreases utilization.
Optimal Strip Layout for Part A Paired with Itself Layout Optimization of Different Parts Paired Together
Very often parts made from the same material are needed in equal quantities, for example, when left-and right-hand parts are needed for an assembly. Blanking such parts together can speed production, and can often reduce total material use. This strip layout algorithm can be applied to such a case with equal ease. Consider a second sample part, B, shown in Fig.6. The relevant Minkowski sum for determining relative position translations, , is shown in Fig.7. In this case, contains 15 edges, whose utilization values are shown in Fig.8. Again, multiple local maxima occur while traversing particular edges of . The optimal layout occurs with a translation vector of (901.214, 130.314), shown in Fig.9, giving a utilization value of 85.32%. Strip width is 1229.74 and pitch is 1390.00 in this example.
Sample Part B to be Nested.
Minkowski Sum (heavy line) of Sample Parts (light and dashed lines).
Optimal Material Utilization for Various Translations Between Polygons A and B.
Conclusions
In the stamping operation, production costs are dominated by material costs, so even tiny per-part gains in material utilization are worth pursing. This paper has presented a new algorithm for creating optimal strip layouts for pairs of parts nested together. This algorithm takes advantage of the Minkowski sum calculation to both find feasible relative positions between the pairs of parts, and to determine the optimal orientation and strip width for the strip layout.
When evaluating combinations of layouts, it should be kept in mind that all permutations should be considered. For example, the strip layout process for the sample parts in this paper would consider strip layouts for A alone, A paired with itself, B alone, B paired with itself, and A and B paired together. The designer would then consider total raw material costs, tooling construction costs and press operating costs since blanking parts together requires larger tools and presses and changes production rates.
There are opportunities to extend this algorithm, as well. One obvious extension is to include optimization over relative rotations between the pairs of parts, i.e., changing the orientation of part B relative to A on the strip. A second opportunity is to study the utilization function more deeply.
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