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Given: Blanks A and B (where B=–A when a blank is paired with itself at 180º)
1. Select the relative position of B with respect to A. The Minkowski sum defines the set of feasible relative positions (Fig.2).
2. ‘Join’ A and B at this relative position. Call the combined blank C.
3. Nest the combined blank C on a strip using the Minkowski sum with the algorithm given in [14] or [15].
4. Repeat steps 1-3 to span a full range of potential relative positions of A and B. At each potential position, evaluate if a local optima may be present. If so, numerically optimize the relative positions to maximize material utilization.
Layout Optimization of One Part Paired with Itself
The first step in the above procedure is to select a feasible position of blank B relative to A. This position is defined by translation vector t from the origin to a point on , as shown in Fig.3. During the optimization process, this translation vector traverses the perimeter of .
Relative Part Translation Nodes on , showing Translation Vector t.
Initially, a discrete number of nodes are placed on each edge of . The two parts are temporarily ‘joined’ at a relative position described by each of the translation nodes, then the combined blank is evaluated for optimal orientation and strip width using a single-part layout procedure (e.g., as in [14] or [15]). In this example, consists of 12 edges, each containing 10 nodes, for a total of 120 translation nodes. The position of each node is found via linear interpolation along each edge , where is vertex I on the Minkowski sum with a coordinate of ( , ). Defining a position parameter s such that s = 0 at and s = 1 at , coordinates of each translation node can be found as:
If m nodes are placed on each edge, ,the position parameter values for the node, , are found as:
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.