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article This appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy Journal of Manufacturing Systems 26 (2007) 44 52 Contents lists available at ScienceDirect Journal of Manufacturing Systems journal homepage: www.elsevier.com/locate/jmansys Technical paper Two-dimensional Minkowski sum optimization of ganged stamping blank layouts for use on pre-cut sheet metal for convex and concave parts Rafael Mulero, Bradley Layton Mechanical Engineering and Mechanics Department, Drexel University, Philadelphia, PA, USA article info abstract With the increasing number of parts that manufacturers need to place on a piece of material such as sheet metal, the need increases for more sophisticated algorithms for part orientation and spacing. With greater part shape complexity, the ability of a skilled worker is challenged to minimize waste. Building on the previous work of Nye, this paper presents a Minkowski sum method for maximizing the number of parts within gangs on a rectangular sheet of material. The example provided uses a simply shaped part to illustrate the presented method, yielding a packing efficiency of 62% that is identical to the efficiency that a skilled worker would produce without the algorithm. The paper also provides results for laying out a more complex part in ganged sections, demonstrating a result that would be difficult for a human to reproduce. This work extends that of Nye by adding practical constraints such as the number of parts that can be blanked at once as well as the amount of horizontal and vertical spacing between ganged blanking sets. Additionally, the paper adds an algorithm for laying out polygons with concave geometries by separating the part into a set of convex polygons. Two examples for optimization, one of a chevronshaped part and one of a complex shape previously used by Nye [Nye TJ. Stamping strip layout for optimal raw material utilization. Journal of Manufacturing Systems 2000;19(4):239 46] and Choi et al. [Choi JC, Kim BM, Cho HY, Kim C. A compact and practical CAD system for blanking or piercing of irregular-shaped sheet metal products and stator and rotor parts. International Journal of Machine Tools & Manufacture 1998;38:931 63], are provided, demonstrating the existence of a local maximum number of parts that may be stamped within a single ganged blank. The algorithm is extendable to a program that may provide stamping manufacturers with a tool that can maximize the total number of parts stamped on stock sheet metal, or for other tiling problems. 2008 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. Article history: Received 26 September 2006 Received in revised form 8 May 2007 Accepted 19 October 2007 1. Introduction The motivation for this paper is to provide die makers with an algorithm for minimizing material waste in ganged blank layouts. The paper demonstrates that the maximum number of parts that may be stamped at once is a function of practical manufacturing constraints such as worksheet length, worksheet width, and the horizontal and vertical boundary distances on the worksheet, as well as the horizontal and vertical spacing requirement between blanking sets. Many prior works in this field, while valid for infinite or semi-infinite sheets, ignore the practical constraints of providing the die engineer with a means of aligning and orienting parts where finite worksheet dimensions (less than infinity) and finite boundary widths (greater than zero) exist. The paper also addresses ganged parts, a consideration not previously considered in the literature. Corresponding author. Tel.: +1 215 895 1752; fax: +1 215 895 1478. E-mail addresses: rafael.mulero@drexel.edu (R. Mulero), blay@drexel.edu, blay@alum.mit.edu (B. Layton). Although many high-volume fabrication companies purchase custom-dimensioned sheet metal for each job, there are still many small companies that rely on pre-cut sheet metal for multiple jobs. These small companies must take into consideration the two-dimensional limitations of the sheet metal when designing a stamping die. In addition to the overall bulk sheet dimensional constraints, other typical sheet metal processes such as stamping, printing, finishing, and punching constrain the number of parts or size of the sheet that can be processed at one time. These smaller-sized sheets of metal, called worksheets, often utilize partless borders around their edges, allowing the safe handling of the parts within the borders during processing. Parts within the bordered worksheet are grouped in sets limited by the number that can be processed at one time on a given machine press. These sets are positioned using flanking holes that ensure accurate registration of the parts during all processes. The sets are spaced at a larger distance between the flanking registration holes so that the worksheet may be sheared into strips used during press operation. This is often required because of machine space limitations and increased operator productivity. These constraints all must be taken into consideration during the process of designing tool sets. 0278-6125/$ see front matter 2008 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jmsy.2007.10.001 Author's personal copy R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 45 Notation a ax ay A Ap b bx by B BL BW C C1 C2 D EL EW Fmax KL KW Ln L n nmax NL NW OL OW Pn Ps Pt rx ry R sx sy S SL T Tmax t UTS Vs Vf vt W xf vector used in the definition of the Minkowski sum x-coordinate of vector a y-coordinate of vector a point-set representing a polygon in the definition and example of the Minkowski sum area of the part vector used in the definition of the Minkowski sum x-coordinate of vector b y-coordinate of vector b point-set representing a polygon in the definition and example of the Minkowski sum bulk material length bulk material width vertices of the example chevron (Fig. 3a) upper convex portion of the example chevron (Fig. 3b) lower convex portion of the example chevron (Fig. 3b) overlap distance horizontal boundary distance on the worksheet vertical boundary distance on the worksheet maximum punch force of the press worksheet length worksheet width overall length of an n-part series blanking set sheared edge length for one part number of parts within a blanking set maximum number of parts that can be blanked in one set number of parts within blanking sets that fit on the bulk sheet s horizontal L-dimension number of blanking sets that fit on the bulk sheet s vertical W -dimension horizontal spacing requirement between blanking sets vertical spacing requirement between blanking sets pitch dimension of optimized part orientation, including spacing between parts magnitudes of the stationary part s projection on the sweepline magnitude of the projections of a vertex on the first translating part on the sweepline x-coordinate of the vertices in the first rotating part y-coordinate of the vertices in the first rotating part region of the first translating part x-coordinates of an individual vertices on the stationary part y-coordinates of an individual vertices on the stationary part region of the stationary part sweepline total number of parts that fit on one bulk sheet for a given orientation maximum number of parts that fit on one bulk sheet for a given orientation sheet thickness ultimate tensile strength of the sheet a starting event point on the Minkowski sum a finishing event point on the Minkowski sum width event points width of the blanking set x-coordinate(s) of the finishing event points of an edge of the Minkowski sum perimeter xs xt X yf ys yt Y x-coordinate(s) of the starting event points of an edge of the Minkowski sum perimeter x-coordinate(s) of the maximum perpendicular distance from the sweepline to the hull of the MS intersecting x-coordinate of the sweepline and the Minkowski sum y-coordinate(s) of the finishing event points of an edge of the Minkowski sum perimeter y-coordinate(s) of the starting event points of an edge of the Minkowski sum perimeter y-coordinate(s) of the maximum perpendicular distance from the sweepline to the hull of the MS intersecting y-coordinates of the sweepline and the Minkowski sum material usage sweepline angle dummy variable to represent dimensions associated with horizontal or vertical dimension In this paper, the above constraints are considered, as well as the number and orientation of parts on the tool sets, with the goal of affecting the overall material utilization and thus profitability. Since material cost is the primary cost in sheet metal stamping, any attempt to minimize its waste may lead to significant savings in both material and net production costs. This paper develops and describes an algorithm that searches all possible angular orientations of a set of identical parts, laying them out first in the horizontal direction and then in the vertical direction in order to maximize the material utilization on a standard-sized sheet of metal. The paper also provides an example for laying out a concave polygon by dividing it into a set of convex polygons, finding the Minkowski sum, and then laying out the resulting parts in an optimized manner. The paper s primary extension beyond the recent work of Nye is to minimize material waste for ganged parts under practical manufacturing constraints. The algorithm may be extended to parts with curved edges by approximating the curved portions with a finite number of sides, rendering it as a polygon. The more vertices the approximation has, the greater the accuracy of the algorithm. An example of the general problem being solved is given in Fig. 1. A square part is laid out on a blank of finite dimensions. In the first example, the orientation is 45 , yielding a material usage of 50%. In the 30 example, the material usage is 61.8%. The final example, where an orientation of 0 is used, 100% material usage is achieved. The example given in Fig. 1 may be thought of as an individual gang within a larger worksheet, as described in the example section of the paper. The Minkowski sum (MS) is named after the German mathematician Hermann Minkowski (1864 1909), who developed a representation and set of formulae for performing transformations in multiple dimensions of non-homogeneous units. This representation and many of its associated functions proved useful for the expression of Einstein s theory of general relativity where three spatial dimensions and one temporal dimension were combined to describe a space time continuum. Minkowski s formulae, including those for addition, are extendable to an arbitrarily large number of dimensions [4]. Conversely, the formulae also work for lower dimensions. In their simplest form, Minkowski s addition formulae degenerate to scalar addition in a single dimension. More recent examples of a Minkowski representation include three-dimensional modeling of prosthetic teeth [19], onedimensional tiling of discrete lines [3], two-dimensional packing of polygons of dissimilar shapes [6], and laying out of shapes on a fabric or material with specific orientation constraints and tolerances [12,8]. The most relevant of these to the current work are Author's personal copy 46 R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 2. Methods The method employs a geometric algorithm designed to streamline computational effort for part orientation optimization in a blanking set of finite dimensions. Nye s Minkowski sum strip (MSS) layout algorithm [13] is extended to calculate the overall length of a finite stamping set. The presented algorithm finds the optimal number of parts and part orientation of a ganged stamping set within the pre-cut sheet metal and process dimensional limitations. The maximum number of parts, nmax , that can be blanked in one set is a function of the maximum punch force of the press used, the sheared edge length, the metal thickness, and material properties. An empirically based formula [10] for estimating nmax is given by nmax = 1.43 Fmax UTS t L (1) Fig. 1. Example of Material Usage. For a simple polygon such as a square, the optimum alignment is 0 . Adapted from [14]. those working in two dimensions, as this paper s problem is a twodimensional layout problem. While the work of Milenkovic, Deza and Grishukhin are mathematically valid, they lack approachability from an algorithmic approach such as this work and the works of Nye. The work of [6], while relevant, considers sets of polygons of dissimilar shapes, whereas the industrial processes considered here are those of stamping large sets of identical shapes. Advances in computer technology coupled with mass production and competitive manufacturing markets have created an environment where more sophisticated placement and nesting algorithms became necessary and possible (see [16,18]). This work has taken the place of traditional methods where part layout and nesting processes are performed with the help of templates cut to the exact shape of the proposed blank. The templates are placed in a variety of positions and orientations with respect to the stock sheet so as to maximize material utilization [17]. Traditionally this has been a trial-and-error process, learned by tool design experts through experience and observation [15]. Early work in the field by Adamowicz and Albano [1] generated algorithms for minimizing waste when placing dissimilar parts in a rectangular domain. Lozano and Perez [11] developed a general theory for non-overlapping polyhedra in n-space with applications in both obstacle avoidance and robot motion planning. More recently, a general solution for laying out a number of identical shapes, including non-convex shapes, for both a bounded and an unbounded domain with varying strip width was developed by Joshi and Sudit [9], who included a proof in the form of providing the set of all possible configurations and showing that a local minimum is present. This method was then applied to the practical application of jointly finding optimal orientation and strip width, where the strip width was not specified [13]. Nye then extended his own work to include irregular but identical convex blanks [14]. The present paper s solution adds the practical parameters such as the number of parts that may be stamped at one time, ganging, and the spacing among worksheets within a bulk sheet. Knowledge of these practical constraints is critical to the real-world application of blank layout. where Fmax is the maximum punch force of the press used, UTS is the ultimate tensile strength of the sheet metal, t is the sheet metal thickness, and L is the sheared edge length for one part. The first step in Nye s algorithm for finding the overlap between two adjacent parts of identical size and shape is to calculate the Minkowski sum (MS) of the two shapes. The summation is represented by the symbol (see [7]). Summing two polygons, represented as point-set A and point-set B, results in a new set of non-unique points in Euclidean space, A B = {a + b | a A, b B} , (2a) where a + b denotes the vector sum of the vectors a and b, that is, if a = (ax , ay ) and b = (bx , by ), then a + b is defined as a + b := (ax + bx , ay + by ). (2b) To use the MS for the layout of identical parts in series, A ( A) is used, where ( A) is A rotated 180 in-plane about the origin. For example, A and A (A rotated 180 about the origin) may be defined with vertices represented by the sets: A = {(0, 0), (1, 0), (1, 1), (0, 1)}, ( A) = {(0, 0), ( 1, 0), ( 1, 1), (0, 1)}. The 16 non-unique vertices of the resulting Minkowski sum are shown in Fig. 2. A + ( A) = {(0, 0), ( 1, 0), ( 1, 1), (0, 1), (1, 0), (0, 0), (0, 1), (1, 1), (1, 1), (0, 1), (0, 0), (1, 0), (0, 1), ( 1, 1), ( 1, 0), (0, 0)}. The two-dimensional Minkowski sum results in a set of points that may be non-unique, as seen above. A subset of these points known as the convex hull defines the shape of the MS and enables rapid, non-overlapping positioning of parts in series throughout their range of angular orientation possibilities. The convex hull may be thought of physically as the shape that would result if a large rubber band that encircled all of the vertices or points of a figure were to be allowed to contract until it was taut. The perimeter of the MS for a set of points can be found using the convhull command in Matlab, which is based on the Quickhull algorithm for convex hulls [2]. Author's personal copy R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 47 Fig. 2. Minkowski Sum Progression: (a) the square A and its inverse ( A), (b) the convex hull (dotted line) and the 16 non-unique points of the Minkowski sum of the two squares. among the Minkowski sums necessary to find the final MS. C 1 + ( C1) = {(0, 0), ( 3, 0), ( 1, 4), (2, 4), (3, 0), (0, 0), (2, 4), (5, 4), (1, 4), ( 2, 4), (0, 0), (3, 0), ( 2, 4), ( 5, 4), ( 3, 0), (0, 0)}, C 2 + ( C2) = {(0, 0), ( 2, 4), (1, 4), (3, 0), (2, 4), (0, 0), (3, 0), (5, 4), ( 1, 4), ( 3, 0), (0, 0), (2, 4), ( 3, 0), ( 5, 4), ( 2, 4), (0, 0)}, C 1 + ( C2) = {( 1, 4), ( 3, 8), (0, 8), (2, 4), (2, 4), (0, 8), (3, 8), (5, 4), (0, 0), ( 2, 4), (1, 4), (3, 0), ( 3, 0), ( 5, 4), ( 2, 4), (0, 0)}, C 2 + ( C1) = {(1, 4), ( 2, 4), (0, 0), (3, 0), (3, 8), (0, 8), (2, 4), (5, 4), (0, 8), ( 3, 8), ( 1, 4), (2, 4), ( 2, 4), ( 5, 4), ( 3, 0), (0, 0)}. To determine the pitch between the parts, a sweepline is generated. This sweepline originates at the origin (0, 0) and intersects the MS of the parts as it sweeps through 360 . The result is a pitched series of non-overlapping similarly oriented set of parts along a baseline that ultimately becomes a horizontal set of parts. The concept of the sweepline was introduced by Nye [13]. Rotating the sweepline by some angle is equivalent to rotating the orientation of the part in series along the horizontal. The distance of a sweepline from the origin to the MS perimeter is the pitch between blanks in series within a set rotated by the same angle of the sweepline. The set width is the dimension perpendicular to pitch and is the maximum perpendicular distance between the sweepline and the points on the MS perimeter ([13], Fig. 6). These principles allow the set width and pitch to be calculated at any angle and are used to find the optimal orientation for maximum material efficiency on an infinite strip. 4. Calculation of overall set length To find the optimal orientation and the optimal number of parts on a blank strip for a two-dimensional sheet, the length of the blanking set (the projection of the set of parts on the sweepline) must be found. This length will include an overlap distance in addition to the multiplied pitch of the parts (Fig. 5). To find the overlap distance, first the intersecting coordinates of the sweepline and the MS, (X , Y ), are found using X= Y= Fig. 3. (a) Chevron part with integer dimensions used in example of the Minkowski sum of a non-convex part. (b) Chevron part broken into two convex portions (parallelograms) with each of these parallelograms rotated 180 in-plane about the origin. C 1 and C1 share a common point at the origin. 3. Modified method for non-convex parts A non-convex part is defined as any part that has an area less than the area of the convex hull of its vertices. In the modified algorithm, the MS perimeter for concave parts is found as follows: the non-convex part is divided into convex portions, and the MS perimeter of each pair of the convex portions and their negatives is found. The number of Minkowski sums that must be taken is equal to the square of the number of convex portions. Finally, the union of the resulting n2 Minkowski sums from the n convex portions yields the non-convex part s MS. For example, as shown in Fig. 3a, a chevron part is defined as two separate parallelograms and the same parallelograms rotated 180 about the origin (Fig. 3b) with vertices by: represented C 1 = {(0, 0), (3, 0), (1, 4), ( 2, 4)}, C 2 = {(1, 4), (3, 8), (0, 8), ( 2, 4)}, ( C1) = {(0, 0), ( 3, 0), ( 1, 4), (2, 4)}, ( C2) = {( 1, 4), ( 3, 8), (0, 8), (2, 4)}. The vertices of the Minkowski sums of the chevron are given below and shown in Fig. 4. Note that the MS of C 1 and C 2 is not ((yf ys )xs + (xs xf )ys ) ( (yf ys ) (xs xf ) tan( )) ((yf ys )xs + (xs xf )ys ) tan( ) ( (yf ys ) (xs xf ) tan( )) (3a) (3b) Author's personal copy 48 R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 Fig. 4. (a) The Minkowski sums of both of the convex portions of the chevron with their respective negatives. Solid lines represent C 1 and C1. Dashed lines represent C 2 and C2. (b) The Minkowski sums of C 1 with C2 (solid) and C 2 with C1 (dashed). (c) The resulting union of convex hulls from (a) and (b) forms the Minkowski sum of the original non-convex part. Fig. 5. Illustration of necessary Overlap dimension needed to calculate the overall length of a discrete number of parts using the Nye Minkowski Algorithm [13]. Black represents C + ( C), gray represents the stationary part, and white represents the translating parts. SL is the sweepline, P is the pitch, W is the width, L is the gang length, and D is the overlap dimension. Fig. 6. Illustration depicting the starting and finishing event points on Minkowski Sum Vs and Vf , the Intersecting Coordinates of the Sweepline and the Minkowski Sum Perimeter (X, Y ), and the remaining vertices of the translating part using the intersecting coordinates as a reference. The stationary part is filled in gray and the translating part is filled in white. where (xs , ys ) and (xf , yf ) are the coordinates of the starting and finishing event points of an edge of the MS perimeter and is the angle of the sweepline. These intersecting coordinates are also the base point of the first translating part (Fig. 6). The remaining coordinates of the first translating part are found using this base point as a reference; that is, in the case of the chevron of Fig. 3a, the six points are given as, R = {(X, Y ), (X 3, Y ), (X 5, Y + 4), (X 3, Y + 8), (X, Y + 8), (X 2, Y + 4)}. Once the coordinates of each vertex on the first translating part are known, the magnitudes of their projections Pt onto the sweepline are found using the normal form for an equation of a line, Pt = rx cos( ) + ry sin( ) (4) where (rx , ry ) are the individual vertices of the first rotating part; that is, in the case of the chevron, this is the part filled in white in Fig. 6. Next, the magnitude of the projection of the farthest vertex of the stationary part (the gray-filled part in Fig. 6) along the sweepline is found. The coordinates of the stationary part are found similarly to the coordinates of the translating part, except that the reference coordinate becomes the origin. For example, the stationary part s coordinates in the case of the chevron are S = {(0, 0), ( 3, 0), ( 5, 4), ( 3, 8), (0, 8), ( 2, 4)}. The magnitudes of the stationary part s projections Ps onto the sweepline are found using the normal form for an equation of a Author's personal copy R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 49 line, Ps = sx cos( ) + sy sin( ) (5) where (sx , sy ) are the individual vertices of the stationary part. The overlap distance, D, is then calculated by finding the maximum value of D = max (max(Ps ) Pt ) . (6) Once the overlap distance is calculated, it is added to n-times the pitch to find overall length, Ln , of an n-part series blanking set: Ln = n (ys yf )xs + (xf xs )ys + D. (ys yf ) cos( ) + (xf xs ) sin( ) (7) The width of the part strip, W , is calculated using width event points vt = (xt , yt ), which are the maximum perpendicular distance from the sweepline to the hull of the MS ([13], Fig. 6). These event points shift counterclockwise when the sweepline becomes parallel to the edge of the convex hull. The width is calculated as W = xt sin( ) + yt cos( ). (8) Fig. 7. Illustration of sheet optimization variables. The number of blanking sets that fit onto the bulk sheet s W dimension (vertical) is denoted by NW . It is dependent on W , the width of the blanking set in the W -direction of the bulk sheet. Its other variables are dictated by the manufacturer s custom layout criteria, NW = BW KW fix KW 2EW + OW W + OW (10) 5. Optimization of the blanking set configuration within a worksheet Due to safety and registration, a press operator will not blank parts using only a portion of the blanking set. Additionally greater spacing is required between blanking sets than between parts. Thus, a worksheet is typically sheared into smaller strips of blanking sets used during press operation. These smaller strips increase operator productivity, but often require additional optimization steps. Using the following variables, optimization of the blanking set can be found within a bulk pre-cut metal sheet. 5.1. Objective Find the angular orientation, , and number of parts, n, per blanking set that results in the maximum number of parts (Fig. 7) on a standard pre-cut bulk BW BL metal sheet given the following constraints: 1. Parts are oriented in series. 2. The bulk sheet must be divided into identically sized KW KL worksheets. 3. A boundary dimensioned by EL and EW around the worksheet must be free of parts. 4. The number of parts in a single blanking set cannot exceed nmax because of press and force requirements defined in (1). 5. Each blanking set must be separated by a distance OW , vertically, and OL , horizontally, to avoid the possibility of a partially stamped part, and in some cases to ensure worker safety. The number of parts within blanking sets that fit onto the bulk sheet s L-dimension (horizontal), NL , is dependent on n, the number of parts within the blanking set, and Ln , the length of an n-part blanking set. Its other variables are dictated by the manufacturer s custom layout criteria, NL = nBL KL fix KL 2EL + OL Ln + OL (9) where BW is the bulk sheet width, KW is the worksheet width, EW is the vertical spacing between blanks and the worksheet edge, and OW is the vertical spacing between blank sets within a worksheet. The objective of this entire algorithm is to maximize the equation for the total number, T , of parts per sheet T = NW NL . (11) The material utilization, , may be found by the following equation, = TAP BL BW (12) where AP is the area of the part. The overall optimization algorithm in total is as follows: 1. Calculate the MS, A ( A), of the polygonal part A. 2. Find the angles between the origin and the vertices on the Minkowski sum for a range of 180 . These angles are the event points for vs and v. . f 3. Calculate the convex hull of the vertices on the MS. 4. Find the angles of the edges of the convex hull. These angles are the event points for vt that determine the required strip width. 5. Calculate Ps , Pt , and D. 6. Calculate the maximum number of parts per strip, nmax , constrained by maximum press force. 7. Calculate L( , n) and w( ) for a range of n parts. 8. Calculate T ( , n) and find n and for Tmax . 9. Substitute the values of the L dimensions with the corresponding W dimensions and repeat steps 1 8. 6. Simple example An example of optimized layout of the chevron part of Fig. 3 is now given, where the objective is to find the maximum number of parts on a standard American bulk BW BL (91.44 cm 182.88 cm, 36 in. 72 in.) metal sheet given the following constraints: 1. Parts are oriented in series. 2. The bulk sheet must be divided into KW KL (30.48 cm 30.48 cm, 12 in. 12 in.) worksheets. 3. A (0.635 cm, 1/4 in.) EL and EW boundary around the worksheet must be free of parts. where BL is the bulk sheet length, KL is the worksheet length, EL is the horizontal spacing between blanks and the worksheet edge, and OL is the horizontal spacing between blank sets within a worksheet. The function fix truncates its argument. This avoids the unsafe condition of a die making contact with the sheet metal blank over only a portion of its perimeter. Author's personal copy 50 R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 Table 1 Event points and angles between the vertices of the Minkowski sum convex hull and the origin Event point Angle MS vertices x 1 2 3 4 5 6 0.00 38.66 69.44 110.56 141.34 180.00 3 5 3 y 0 4 8 8 4 0 3 5 3 Table 2 Event points, angles, and event point coordinates used in the calculation of width Event point Angle vt x 1 2 3 4 0.00 63.43 90.00 116.56 y 8 4 3 5 5 3 4 8 4. For simplicity, the number of parts in a single ganged blanking set cannot exceed six, nmax = 6. This condition simulates press force limitations. 5. Each blanking set must be separated by OW (0.3175 cm, 1/8 in.) vertically and OL (0.9525 cm, 3/8 in.) horizontally. The first step is the calculation of the MS of Fig. 4. The angles between the vertices of the MS and the origin are then calculated as shown in Table 1. The angles of the convex hull of the MS sum are then calculated over the 180 range as given in Table 2. Graphs of T vs. are then created for the valid range of n (Fig. 8). The optimal number of parts and their orientation for this particular part are found to be n = 4 and = 0 respectively. The material utilization, , of this blank setup is found to be 62% for the given dimensions and specifications. In this relatively straightforward example, an experienced die maker would also arrive at this utilization percentage. In more complex examples, however, it is expected that the algorithm will not only find more optima than the die maker, but that it is likely to find the optimum solution that the die maker is likely to miss. 7. Complex example Using the same complex part used in [13,5], the layout of the complex part of Fig. 9 on a bulk sheet of aluminum is optimized, with the following goals and constraints: 7.1. Objective Find the maximum number of parts on a standard American bulk metal sheet with dimensions BW BL = 91.44 cm 182.88 cm (36 in. 72 in.), given the following constraints: 1. Parts are oriented in series. 2. The bulk sheet will not be divided. 3. A 1.27 cm (1/2 in.) EL and EW boundary around the worksheet must be free of parts. 4. For simplicity, the number of parts in a single blanking set cannot exceed four (nmax = 4). This condition simulates press force limitations. 5. Each blanking set must be separated by OW = 0.635 cm (1/4 in.) vertically and OL = 1.905 cm (3/4 in.) horizontally. The first step is the calculation of the MS of the shape of Fig. 9. While involving more steps than the chevron shape, the procedure is the same and results in the shape shown in Fig. 10. Fig. 8. Optimization curves for a series of n parts. This result demonstrates that local maxima exist within each part number, but that the global maximum, Tmax , is found uniquely for n = 4, = 0. Fig. 9. Plot of a complex sample blank used by Choi et al. [5] and Nye [13] broken into convex portions. Units are in centimeters. The angles between the vertices of the MS and the origin are then calculated as in previous examples. These are summarized in Table 3. The angles of the convex hull of the MS sum are then calculated over the 180 range (Table 4). As was done with the simple example, graphs of T vs. are then created for the valid range of n in both directions of the bulk sheet. Author's personal copy R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 51 Fig. 10. Minkowski sum of all convex portions and their respective negatives of the complex blank of Fig. 9. Table 3 Event points and angles between the vertices of the Minkowski sum hull and the origin Event point Angle MS vertices x 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 0.00 7.52 12.38 15.08 29.51 37.14 37.98 60.89 67.47 72.98 78.42 92.88 97.16 97.56 98.33 111.01 117.42 156.09 157.59 164.60 170.12 180.00 18.26 18.26 16.95 13.81 12.03 10.72 10.40 10.40 14.09 10.40 7.86 y 0.00 2.41 3.72 3.72 6.81 8.12 8.12 18.68 33.97 33.97 38.37 38.37 34.89 33.02 32.57 31.56 28.08 6.46 5.69 3.80 3.18 0.00 Fig. 11. Total number of parts that fit on a 36 in. 72 in. sheet under the given constraints for sets of four, three, two, and one parts. 1.93 4.38 4.38 4.77 12.12 14.57 14.57 13.8 13.8 18.26 18.26 Table 4 Event points, angles, and event point coordinates used to calculate width Event point Angle vt x 1 2 3 4 5 6 7 0.00 33.80 54.83 81.59 89.95 97.40 144.39 Y 38.37 31.56 28.08 3.18 1.93 12.12 14.57 13.81 12.03 10.72 10.40 2.41 33.97 38.37 of the constraint of spacing between individual stampings, fewer total pieces may be stamped than when they are ganged. Contrary to what occurs when the parts are oriented with their major axes parallel to that of the sheet, when their starting major axis is perpendicular to the sheet, the gang of four does not go through a local minimum of zero parts (Fig. 12). In this particular case, the maximum, 24, is equal to that of the case shown in Fig. 11, however, the bandwidth of angles where this occurs is slightly diminished. Rather than a geometry that allows for six rows of the gangs of four, the geometry of Fig. 12, where the maximum occurs, consists of three rows of two columns of the gangs of four. A maximum of 24 parts per sheet are found in both orientations using four-part sets at an angle of 0.9 rad (51.6 ). The reason = why the four-part ganged sets yield a greater number of parts than the gangs with fewer parts is because the four-part sets have closer spacing within individual gangs than that existing between gangs. This is a realistic consideration for practical applications where alignment and boundary effects dictate the inter-gang spacing. When performing this algorithm for complex parts, scenarios arise where the pitch is not a true function, that is, the pitch and consequently the overall length of the blanking set at a given sweepline angle is not unique. This is demonstrated in Fig. 13, which shows the length and width of a two-part gang as a function of sweepline angle for the complex shape of [5]. As shown, these non-unique pitch values have no effect on the gang s associated width, and the minimum pitch can be selected for evaluation. 8. Discussion and conclusions Fig. 11 indicates that when a gang of four parts is used, a maximum is found around 51 . This occurs when the gang is tilted to occupy as much horizontal space as possible within a given row, allowing for a total of six rows to occupy the sheet. The precipitous drop after this angle occurs because the parts can no longer fit on the sheet in a gang of four once they have been rotated further. The gang of one never passes through this minimum of zero because a single part can fit the sheet when rotated 90 . However, because A modified version of the Minkowski sum for two-dimensional stamping optimization is presented. The method offers the sheet metal die designer an algorithm for minimizing part waste in two dimensions. A method is presented for finding the MS of non-convex parts whereby the larger part is broken into smaller convex parts and finally laid out on a sheet in a ganged manner. The approach extends previous work by Nye in that it includes Author's personal copy 52 R. Mulero, B. Layton / Journal of Manufacturing Systems 26 (2007) 44 52 and horizontal and vertical boundary distances on the worksheet, as well as the horizontal and vertical spacing requirement between blanking sets. The primary contribution of this work beyond that previously performed by Nye and Choi et al. [5] is the addition of a method for ganging parts into groups for stamping whereby the intra-gang spacing and the inter-gang spacing are accounted for. References [1] Adamowicz M, Albano A. Nesting two-dimensional shapes in rectangular modules. Computer-Aided Design 1976;8:27 33. [2] Barber CB, Dobkin DP, Huhdanpaa HT. The Quickhull algorithm for convex hulls. ACM Transactions on Mathematical Software 1996;22:469 83. [3] Bodini O, Rivals E. Tiling an interval of the discrete line. In: 17th annual symp. on combinatorial pattern matching, vol. 4009. 2006. p. 117 28. [4] Cheng S-Y, Yau S-T. Maximal space-like hypersurfaces in the Lorentz Minkowski spaces. Annals of Mathematics 1976;104:407 19. [5] Choi JC, Kim BM, Cho HY, Kim C. A compact and practical CAD system for blanking or piercing of irregular-shaped sheet metal products and stator and rotor parts. International Journal of Machine Tools & Manufacture 1998;38: 931 63. [6] Dean HT, Tu YL, Raffensperger JF. An improved method for calculating the nofit polygon. Computers & Operations Research 2006;33:1521 39. [7] de Berg M, Van Kreveld M, Overmars M, Schwarzkopf O. Computational geometry: Algorithms and applications. Berlin: Springer; 1997. [8] Deza M, Grishukhin V. Voronoi s conjecture and space tiling zonotopes. Mathematika 2004;51:1 10. [9] Joshi S, Sudit M. Procedures for solving single-pass strip layout problems. IIE Transactions 1994;26:27 37. [10] Kalpakjian S, Schmid S. Manufacturing processes for engineering materials. Englewood Cliffs (NJ): Prentice-Hall; 2002. [11] Lozano-Perez T. Spatial planning: A configuration space approach. IEEE Transactions on Computers 1983;C-32:108 20. [12] Milenkovic VJ. Rotational polygon containment and minimum enclosure using only robust 2D constructions. Computational Geometry Theory and Applications 1999;13:3 19. [13] Nye TJ. Stamping strip layout for optimal raw material utilization. Journal of Manufacturing Systems 2000;19(4):239 46. [14] Nye TJ. Stamping blank optimal layout and coil slitting for single and multiple blanks. Journal of Engineering Materials and Technology 2001;123:482 8. [15] Prasad YKDV, Somasundaram S. CADDS: An automated die design system for sheet-metal blanking. Computing & Control Engineering Journal 1992;3: 185 91. [16] Shen WM, Wang LH, Hao Q. Agent-based distributed manufacturing process planning and scheduling: A state-of-the-art survey. IEEE Transactions on Systems, Man and Cybernetics, Part C Applications and Reviews 2006;36: 563 77. [17] Singh R, Sekhon GS. A low-cost modeller for two-dimensional metal stamping layouts. Journal of Materials Processing Technology 1998;84:79 89. [18] Tabakov PY, Walker M. A technique for optimally designing engineering structures with manufacturing tolerances accounted for. Engineering Optimization 2007;39:1 15. [19] Yoo K-H, Ha J-S. An effective modeling of single cores prostheses using geometric techniques. Computer-Aided Design 2005;37:35 44. Fig. 12. Total number of parts that fit on a 72 in. 36 in. sheet under the given constraints for sets of four, three, two, and one parts. Fig. 13. Solution for a two-part gang of the complex shape introduced by Choi et al. [5]. The overall gang length (dashed line) yields a non-unique solution as the sweepline passes through 1 radian. This is caused by the part exploring inlets within the shape of the Minkowski sum. The solid line represents the solution for the width of the gang, which does not experience this double-valued solution. Rafael Mulero is a graduate student in the Mechanical Engineering and Mechanics Department at Drexel University (Philadelphia). Mr. Mulero received his BS in mechanical engineering at Drexel University. His current research interests include optimization techniques for manufacturing and nano/micro-pore based biomolecule sensors. Bradley Layton is an assistant professor in the Mechanical Engineering and Mechanics Department at Drexel University (Philadelphia). Dr. Layton received his BS in mechanical engineering from the Massachusetts Institute of Technology, his MS in mechanical engineering from the University of Michigan, and his Ph.D. in biomedical engineering from the University of Michigan. He completed postdoctoral work at the University of Michigan in neurology and radiology. His research interests include nanoscale biomechanics, nanomanufacturing, and packing problems in structural proteins. Dr. Layton is currently a member of ASME, IEEE, EMBS, and BMES. manufacturing criteria such as maximum press force, maximum worksheet size, the maximum number of parts a press can blank at one time, and the spacing between blanking sets. The primary findings indicate that the maximum number of parts n, within a ganged blanking set may have a local maximum that is not intuitive or obvious to the machinist laying out the blanks. This local maximum, as is demonstrated, depends primarily on several processing variables, such as worksheet length, worksheet width,

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