Translational Placement using Simulated Annealing and Collision Free Region with Parallel Processing André Kubagawa Sato, Rogério Yugo Takimoto, Thiago de Castro Martins, Marcos de Sales Guerra Tsuzuki Computational Geometry Laboratory Departamento de Engenharia Mecatrônica e de Sistemas Mecânicos Escola Politécnica da Universidade de São Paulo e-mail: andre.kubagawa@gmail.com takimotoyugo@gmail.com tcmartins@gmail.com mtsuzuki@usp.br Abstract The irregular shape packing problem is a combinatorial optimization problem that consists of arranging items on a container in such way that no item overlaps. In this paper we adopt a solution that places the items sequentially, touching the already placed items or the container. To place a new item without overlaps, the collision free region for the new item is robustly computed using non manifold Boolean operations. A simulated annealing algorithm controls the items sequence of placement, the item s placement and orientation. In this work, the placement occurs at collision free region s vertices. Several results with benchmark datasets obtained from the literature are reported. Some of them are the best already reported in the literature. To improve the computational cost performance of the algorithm, a parallelization method to determine the collision free region is proposed. We demonstrated two possible algorithms to compute the collision free region, and only one of them can be parallelized. The results showed that the parallelized version is better than the sequential approach only for datasets with very large number of items. The computational cost of the non manifold Boolean operation algorithm is strongly dependent on the number of vertices of the original polygons. I. INTRODUCTION The irregular shape packing problem is a part of a wider set of combinatorial optimization problems, the cutting and packing problems. The field of cutting and packing is important in several industries such as clothing, wood and sheet metal among others. Wäscher et al. [1] have proposed a typology that classifies the problem types by dimension, objective, assortment of item type, as well as the number and nature of the containers. In this paper, we consider the two-dimensional problem with the objective of packing all items inside a container. The items and the container can be any simple polygon, and they are not necessarily convex. There are several proposals in the literature to solve this type of problem. However, one can identify two main approaches for the irregular shape packing problem. The first consists in representing the solution as an ordered list of items and the configuration is constructed by applying a placement rule. This way, a feasible layout is guaranteed, i.e. layouts where items do not overlap and fit inside the container. The second approach represents the solution as a physical layout on the container and moves the items within the layout [2], [3]. The collision free region is a construction very useful for defining collision free positions. Several works used the collision free region to ensure feasible layouts [4], [5], [6], [7], [8], [9], [10]. The second strategy generates an initial configuration on the container either through a simple constructive heuristic or randomly. This strategy may permit overlap between items and imposes a penalty in the objective function, also called as external penalization [11]. Martins and Tsuzuki [9] showed that the use of external penalization can result in an optimal solution whose layout has overlaps. The irregular shape packing problem is NP-Hard, even in its restricted variations (e.g. limiting the polygon shape to rectangle only). Most published solutions make use of probabilistic heuristic methodologies to solve the combinatorial problem. Other researchers combine these heuristics with deterministic ones to constrain the feasible region to a subset of the entire solution space [2], [12]. The metaheuristic simulated annealing was almost simultaneously introduced by [13] for statistic mechanics and by [14] in the area of combinatorial optimization, i.e. when the cost function is defined in a discrete domain. [15] modified the simulated annealing algorithm to apply to the optimization of functions defined in a continuous domain, using distinct steps according to temperature intervals. This work uses a simulated annealing algorithm with adaptive neighborhood proposed by [5], [8]. The adaptive neighborhood heuristic defines the densities of the distribution of the next candidate and it can be simultaneously used in discrete and continuous domains. In this work, the collision free region is determined by a robust implementation of two dimensional non manifold Boolean operations. The items are placed sequentially, one at a time on the vertices of the collision free region. Simulated annealing controls the order of the items and the placement position. Some of the controls can be turned off and the simulated annealing is combined with deterministic heuristics: larger first and bottom left. To improve the performance of the algorithm, the determination of the collision free region is parallelized using a multi-core processor. Several tests were executed to show the impact in the performance of the algorithm.
Fig. 1. Set of translations of a shape represented as a region containing all possible translations from its reference point. Fig. 2. The inner fit polygon for given item and container. II. NO-FIT POLYGONS AND COLLISION-FREE REGION No-fit polygons represent set of translations of an item and are mathematically represented by a set of vectors. For a better understanding of the properties of the no-fit polygons, the set of translations of an item is represented by polygons in the plane. Every item has a reference point that can be internal or external to it. The no-fit polygon represents a set of forbidden translations that when applied to the item, will move the reference point to be internal to the no-fit polygon, as shown in Fig. 1. For an item P, let i(p ) be its interior, P be its boundary and c(p ) be its complement. Definition 2.1: The no-fit polygon induced by item P i to item P j, denoted by Υ(P i, P j ), is the set of translation vectors applied to P j that leads it to a collision with P i. Thus, Υ(P i, P j ) = i(p i ) i(p j ) (1) = { v a i(p j ), a + v i(p i )} The no-fit polygon can be obtained by the Minkowski sum algorithm [16], that can be calculated very efficiently for convex polygons. The result of a Minkowski sum of two convex polygons is a convex polygon built from the original polygon edges ordered counterclockwise. Non convex polygons can be decomposed into convex polygons on a preprocessing step, as the transformations applied (rotations and translations) do not affect such a decomposition. Definition 2.2: The Minkowski sum of two polygons P i and P j, noted P i P j, is defined as the set of points {O + v + w O + v P i, O + w P j }. Definition 2.3: The opposed polygon for a given polygon P j, denoted by P j, is defined as the set of points P j = {O w O + w P j }. From the above definitions, one can see that i(p i ) i(p j ) = i(p i ) ( i(p j )) (2) meaning that the no-fit polygon is produced by the Minkowski sum of the fixed item with the opposed item to be placed. A. The Inner Fit Polygon The inner fit polygon is another important concept frequently used, which is derived from the no-fit polygon and represents a set of translations for the placement of items Fig. 3. The collision free region is filled with a hatch pattern. The item to be placed is not filled and the already placed items are filled with a solid pattern. inside a container C. The inner fit polygon can be computed by sliding an item along the internal contour of the container [17] (see Fig. 2). Definition 2.4: The inner fit polygon induced by container C to item P j, denoted by Λ(C, P j ), is the set of translation vectors applied to P j that leads it to be inside the container. Thus, Λ(C, P i ) = c(c(c) ( i(p i ))) (3) B. The Collision Free Region = { v a i(p i ), a + v C} Considering a container C and a set of items P = {P 1,, P n } already placed, as shown in Fig. 3, a new item P m, m > n, will be placed in the interior of the container without collision with the already placed items. The feasible set of translations for item P m is given by the collision free region (see Fig. 3). Definition 2.5: The collision free region is the set of all translations, that, when applied to a specific item, place the specific item in the interior of a container without colliding with the already placed items. When the container is empty, the collision free region represents all translations that place the item in the interior of the container. This specific collision free region is the already defined inner fit polygon [17]. For a given item, the calculation of the inner fit polygon is the first step in the determination of the collision free region. The collision free region for a specific item is determined by removing the no-fit polygons generated by the already placed
(a) (b) (c) Fig. 4. (a) No-fit polygon generated from a movable item and two already placed items. The item can be placed in the common boundary edge, as it is not a forbidden placement considering both no-fit polygons. (b) Result of a manifold union operation applied to the no-fit polygons. (c) Correct result obtained by the non manifold union opereation. The item can be placed on the internal edge. (a) items, from the inner fit polygon. Π(C, P, P m ) = Λ(C, P m ) P i P i(p i ) i(p m ) (4) By analyzing expression (4), it is possible to define at least two possible algorithms to compute de collision free region. The first algorithm, removes every no-fit polygon from the inner fit polygon and uses uniquely the difference operator (called as sub-approach). The second algorithm calculates the unions of all no-fit polygons and then subtracts it from the inner fit polygon (called as mixed-approach). The implementation of the necessary Boolean operations is not an easy task. C. Modified Boolean Operations General manifold Boolean operations between polygons [18], [19] should not be applied to determine the collision free region. If manifold Boolean operation are adopted then feasible positions may be lost for the placement of items. Fig. 4 shows a situation where the results of the manifold and of the non-manifold union operations are different. The correct result for the determination of the collision free region is obtained by non-manifold Boolean operations. The interior of a no-fit polygon (excluding the boundary) represents placements where the item collides. The boundary and exterior of a no-fit polygon represent feasible placements for the item. When an union is applied to no-fit polygons, the result must consider uniquely the union of the interior of both no-fit polygons (see Fig. 4). The inner fit polygon is a normal manifold polygon as its interior and boundary are feasible placements. When a difference operation is applied to no-fit polygon and inner fit polygon, the operation must consider uniquely the interior of the no-fit polygon. As a consequence, a collision free region can result in a set of multiple disconnected polygons with holes, disconnected edges or vertices. In this work, robust two dimensional non-manifold Boolean operations were implemented. The implementation was done using fixed precision. The intersection determination procedure is the main module [20]. After that the intersections are determined, the edges are classified and the appropriated edges are collected to define the final result. (b) Fig. 5. (a) This placement problem has three items on the left and a rectangular container on the right. (b) Considering that the first item to be placed is the item on the middle, its collision free region is a segment shown with a thick line and it must be placed in a internal point of this segment. The second and third items are placed on their collision free region s vertices. It does not matter which item is placed first, the first item cannot be placed on its collision free region s vertices. III. THE PROPOSED APPROACH Simulated annealing is the probabilistic meta heuristic adopted in this work. The cost function to be minimized is the space wasted inside the container. The container considered here has fixed dimensions, and consequently its cost function assumes only discrete values [9]. The placement of an item is controlled by a discrete parameter r, that is a vertex of the collision free region for that item. Previous works have considered the placement as a continuous parameter [4], [5], [7], [8], [9]. r can assume only some discrete values associated with the vertices of the collision free region. In this work, the new item is always placed on the collision free region s vertices. Fig. 5 shows a specific situation that is very rare in practice, where the first item must not be placed in a collision free region s vertex. The simulated annealing also controls the order of the items in the sequence of placement and the item s orientation. A. Parallelization of the Collision Free Region Determination In this work, two methods to determine the collision free region were adopted. Both methods were proposed based on the analysis of equation (4). The first method subtracts each no-fit polygon from the inner fit polygon. This leads to a series of subtractions, as seen in Fig. 6. The second method is divided into two distinct steps. The first step determines the union of all no-fit polygons and the second step subtracts this result from the inner fit polygon. The first method (sub approach) cannot be parallelized. The first step of the second method was parallelized. The unions are processed in parallel, using a multi-core processor. To achieve this, the no-fit polygons are placed in a queue. Each processor pops two no-fit polygons from the queue and computes the result. The result is pushed into the queue. This
Fig. 6. Determining the collision free region using uniquely subtractions. N: no-fit polygon; I: inner fit polygon; C: collision free region. All operations are executed sequentially by one unique processor. TABLE I BENCHMARK DATA SETS. TNP: TOTAL NUMBER OF POLYGONS; ANV: THE AVERAGE NUMBER OF VERTICES; AO: ADMISSIBLE ORIENTATIONS (DEGREES); ML: MINIMUM LENGTH. THE DATA SETS WITH ARE THE BEST RESULTS IN THE LITERATURE case TNP ANV AO ML Density albano 24 7.25 0, 180 9906.45 87.78 dagli 30 6.30 0, 180 58.20 86.97 dighe1 16 3.87 0 1000 100 dighe2 10 4.70 0 1000 100 fu 12 3.58 0, 90, 180, 270 30.97 92.02 jakobs1 25 5.60 0, 90, 180, 270 11.00 89.09 jakobs2 25 5.36 0, 90, 180, 270 24.00 80.42 mao 20 9.22 0, 90, 180, 270 1767.0 83.61 marques 24 7.37 0, 90, 180, 270 77.30 89.49 shapes0 43 8.75 0, 180 60.0 65.41 shapes1 43 8.75 0, 180 55.0 72.55 shapes2 28 6.29 0, 180 26.51 81.48 shirts 99 6.63 0, 180 62.50 86.40 trousers 64 5.06 0, 180 242.5 89.82 TABLE II SERIAL GENERATION OF ALL COLLISION FREE REGION IN BENCHMARK DATA SETS WITH 1000 RUNS. NNFP: NUMBER OF NO-FIT POLIGONS; SUB: PROCESSING TIME (SEC) USING SUBTRACTION ONLY; US: PROCESSING TIME (SEC) USING UNION AND SUBTRACTION. Fig. 7. Determining the collision free region using unions and subtractions. N: no-fit polygon; I: inner fit polygon; C: collision free region. This approach was implemented in two different ways: sequential and parallel. Considering the situation where two processors are used, the first processor executes N 1 N 2 and the second processor executes N 3 N 4. The operations R 1 R 2 and I R are executed sequentially. process is repeated n 1 times (see Fig. 7), where n is the total number of no-fit polygons. Finally, the subtraction of this result from the inner fit polygon occurs in order to obtain the resulting collision free region. IV. RESULTS In this work, we evaluated the algorithm using 14 benchmark problems gathered from the literature. These data sets can be found on the EURO Special Interest Group on Cutting and Packing (ESICUP) website 1. These sets are irregular strip packing problems with the objective of minimizing the length of the container with a fixed width. To allow the possibility of a comparison, the container dimensions (width and length) were considered fixed. The objective was to find the minimum length for the container such that all items fit inside the container and they do not overlap. All tests were executed on i7 2.8GHz processor with 4 GB RAM. 1 http://paginas.fe.up.pt/ esicup/tiki-index.php. case NNFP SUB US albano 20 2.33 3.13 dagli 27 2.20 2.53 dighe1 9 0.63 1.03 dighe2 6 0.58 0.93 fu 9 0.62 0.85 jakobs1 21 1.54 2.37 jakobs2 21 1.91 2.34 mao 16 1.77 2.31 marques 19 2.23 3.24 shapes0 37 4.24 4.57 shapes1 40 4.38 4.66 shapes2 23 1.39 2.06 shirts 94 12.53 6.54 trousers 57 12.00 6.93 Table I shows the minimum length obtained for the studied problems and the density of the layouts. Figs. 8(a)-(h) and Figs. 9.(a)-(f) shows the optimized layouts. Other tests were executed to evaluate the computational speed of the proposed approach. We tested both approaches proposed to determine the collision free region with one unique processor. Each test was repeated 1000 times and the results are shown on Table II. One can observe that the sub approached computed the collision free region faster, excepting the trousers and shirts benchmark. The trousers (shirts) benchmark was executed by the mixed approach 1.71 (1.94) times faster than the sub approach. The OpenMP was used to develop the parallel algorithms. The results are shown in Table III. Up to four processors were used. It can be observed that the Dighe1, Dighe2, Fu, Jakobs1, Mao, Marques and Shapes2 benchmarks were executed slower by the parallel algorithm when compared to the best sequential algorithm. Fig. 10 shows graphs comparing the time processing of some benchmark datasets for sequential and parallelized versions of the collision free region computation algorithms. It
(a) (a) (b) (b) (c) (c) (d) (d) (e) (f) (e) (g) (h) Fig. 8. The best solutions obtained by the proposed algorithm. (a) albano. (b) shapes0. (c) dagli. (d) fu. (e) dighe1. (f) dighe2. (g) mao. (h) marques. (f) Fig. 9. The best solutions obtained by the proposed algorithm. (a) shapes1. (b) shapes2. (c) shirts. (d) trousers. (e) jakobs1. (f) jakobs2.
TABLE III PARALLEL GENERATION OF ALL COLLISION FREE REGION POLYGONS IN BENCHMARK DATA SETS 1000 TIMES. TIME IN SECONDS. case number of processors 2 3 4 albano 2.66 2.47 2.48 dagli 2.01 1.83 1.65 dighe1 0.96 0.98 0.94 dighe2 0.89 0.93 0.93 fu 0.88 0.79 0.78 jakobs1 1.95 1.87 1.71 jakobs2 1.98 1.89 1.75 mao 2.08 2.08 1.98 marques 2.64 2.55 2.54 shapes0 3.43 3.11 3.05 shapes1 3.44 2.96 2.68 shapes2 1.73 1.63 1.52 shirts 4.47 3.95 3.41 trousers 5.02 4.88 4.74 Fig. 10. Graphs comparing some time processing of benchmark datasets for the sequential (blue lines) and parallelized algorithms (yellow lines). The datasets that showed improvement in the parallelized version of algorithms are shirts and trousers solely. is convenient to observe that the parallelized algorithm is based on unions and differences and the sequential algorithm was based uniquely on differences. Considering both approaches with single processing, the approach based uniquely on differences was faster. The conclusion is that even the parallelization was not enough to get smaller time processing in all situations. This is because the complexity computation of the non manifold Boolean operation is strongly based on the number of vertices of the original polygons. The early use of the inner fit polygon, which has a rectangular shape, by the approach based exclusively on differences showed to have a smaller number of vertices. V. CONCLUSION This work deals with the problem of minimizing the waste of space that occurs on translational placements of a set of irregular bi dimensional items inside a bi dimensional container with fixed dimensions. The collision free region concept was proposed and implemented using robust two dimensional non manifold Boolean operations. The results obtained in this work were very competitive when compared with previous works. A parallelization method to determine the collision free region was proposed. The speed of this algorithm was compared against the serial algorithm. Only problems with very large number of pieces showed gain in speed. Other problems showed that the overhead of the parallelization caused it to run considerably slower. REFERENCES [1] G. Wäscher, H. Haussner, and H. Schumann, An improved typology of cutting and packing problems, European Journal of Operational Research, vol. 183, pp. 1109 1130, 2007. [2] A. M. Gomes and J. F. Oliveira, Solving irregular strip packing problems by hybridising simulated annealing and linear programming, European Journal of Operational Research, vol. 171, pp. 811 829, 2006. [3] J. Egeblad, B. K. Nielsen, and A. Odgaard, Fast neighborhood search for two- and three dimensional nesting problems, European Journal of Operational Research, vol. 183, pp. 1249 1266, 2007. [4] T. C. Martins and M. S. G. Tsuzuki, Irregular rotational placement of shapes over non-convex containers with fixed dimensions, in Proceedings of the 8 th IFAC Workshop on Intelligent Manufacturing Systems, 2007. [5], Simulated annealing with adaptive neighborhood applied to the placement over containers with fixed dimensions, in Proceedings of the IFAC Workshop on Intelligent Manufacturing Systems, 2008. [6], Rotational placement of irregular polygons over containers with fixed dimensions using simulated annealing and no-fit polygons, Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 30, pp. 196 203, 2008. [7], Comparison of deterministic heuristics and simulated annealing for the rotational placement problem over containers with fixed dimensions, in Proceedings of the IFAC Symposium Information Control Problems in Manufacturing, 2009. [8], Placement over containers with fixed dimensions solved with adaptive neighborhood simulated annealing, Bulletin of the Polish Academy of Sciences Technical Sciences, vol. 57, pp. 273 280, 2009. [9], Simulated annealing applied to the irregular rotational placement of shapes over containers with fixed dimensions, Expert Systems with Applications, vol. 37, pp. 1955 1972, 2010. [10] A. K. Sato, T. C. Martins, and M. S. G. Tsuzuki, Rotational placement using simulated annealing and collision free region, in Proceedings of the 10 th IFAC Workshop on Intelligent Manufacturing Systems, 2010, pp. 253 258. [11] R. Heckmann and T. Lengauer, A simulated annealing approach to the nesting problem in the textile manufacturing industry, Annals of Operations Research, vol. 57, pp. 103 133, 1995. [12] J. A. Bennell and K. A. Dowsland, Hybridising tabu search with optimisation techniques for irregular stock cutting, Manage. Sci., vol. 47, pp. 1160 1172, 2001. [13] S. Kirkpatrick, C. D. Gellat, and M. P. Vecchi, Optimization by simulated annealing, Science, vol. 220, pp. 671 680, 1983. [14] V. Cerny, Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm, Journal of Optimization Theory and Applications, vol. 45, pp. 41 51, 1985. [15] A. Corana, M. Marchesi, C. Martini, and S. Ridella, Minimizing multimodal functions of continuous variables with the simulated annealing algorithm, ACM Transactions on Mathematical Software, vol. 13, pp. 262 280, 1987. [16] P. K. Agarwal, E. Flato, and D. Halperin, Polygon decomposition for efficient construction of minkowski sums, Computational Geometry, vol. 21, pp. 39 61, 2002. [17] K. A. Dowsland, S. Vaid, and B. W. Dowsland, An algorithm for polygon placement using a bottom left strategy, European Journal of Operational Research, vol. 141, pp. 371 381, 2002. [18] M. Mäntylä, An Introduction to Solid Modeling. Computer Science Press, 1988. [19] M. S. G. Tsuzuki and M. Shimada, Geometric classification tests using interval arithmetic in b rep solid modeling, Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 25, pp. 396 402, 2003. [20] J. L. Bentley and T. A. Ottmann, Algorithms for reporting and counting geometric intersections, IEEE Transactions on Computers, vol. C-28, pp. 643 647, 1979.