A hybrid algorithm for the two-dimensional layout problem: the cases of regular and irregular shapes

Transcription

1 Intl. Trans. in Op. Res. 10 (2003) INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH A hybrid algorithm for the two-dimensional layout problem: the cases of regular and irregular shapes M. Hifi a and R. M Hallah b a CERMSEM, Maison des Sciences Economiques, Universite Paris 1 Panthe on-sorbonne, boulevard de l Hoˆpital, Paris Cedex 13, France b Department of Quantitative Methods and Information Technology, Institut Supe rieur de Gestion de Sousse, B.P. 763, Sousse 4000, Tunisia hifi@laria.u-picardie.fr [Hifi]; Rym.Mhallah@irsit.rnrt.tn [M Hallah] Received 14 September 2001; received in revised form 27 May 2002; accepted 26 July 2002 Abstract The two-dimensional layout (TDL) optimization problem consists of finding the minimal length layout of a set of both regular and irregular two-dimensional shapes on a stock sheet of finite width but infinite length. The layout should not contain any overlap. In this paper, we solve the TDL problem by proposing two approximate algorithms. The first one is a new heuristic based on a constructive approach specifically designed for irregular shapes, but that works well with regular ones. The second is a hybrid method in which the constructive approach displays the layouts corresponding to the chromosomes yielded by the genetic algorithm. The resulting algorithm yields satisfactory results in relatively short computational times as shown by the extensive computational testing on problems from the literature. Keywords: Genetic algorithms; heuristics; packing and cutting; sequential placement. 1. Introduction Solving complex combinatorial optimization problems efficiently and optimally is highly important for many applications in the field of science and engineering. Using today s exact methods, it is often impossible to solve many of these instances because of the large amount of computational time required. One seemingly attractive solution is to design robust approximate algorithms that solve some of these problem instances efficiently; i.e. produce good solutions within reasonable computing time. Cutting and packing problems belong to a well-known family of natural combinatorial optimization problems (Dyckhoff, 1990; Dyckhoff and Finke, 1992). These problems are encountered in many real-world applications such as computer science, industrial engineering, r 2003 International Federation of Operational Research Societies. Published by Blackwell Publishing Ltd.

2 196 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) logistics, manufacturing, production processes, etc. Long and intensive research on the problems of this family leads to models and mathematical tools, interesting by themselves, but surpassing the framework of cutting and packing problems. The two-dimensional layout (TDL) problem consists of finding the minimal length layout of a set of (ir)regular two-dimensional pieces on a stock sheet of finite width but infinite length. The layout must not contain any overlap. For example, Downsland, Downsland and Bennell (1998) provide some valid near optimal layouts of irregular cutting patterns. Even though NP-hard (Milenkovic, Daniels and Li, 1991), the layout problem has been widely studied in the literature. Surveys of different approaches to this problem and its variations are available in Cheng, Feiring and Cheng (1994), Dowsland and Dowsland (1995) and Dyckhoff and Finke (1992). Dori and Ben-Bassat (1983) discussed an efficient nesting approach of congruent convex figures based on approximating a convex polygon by a polygon with fewer sides. Li and Milenkovic (1995) implemented compaction and separation algorithms for non-convex polygons and discussed their applications. In Blazewicz, Hawryluk and Walkowiak (1994), the authors solved the TDL irregular-cutting problem using an approach based upon tabu search. Poshyanonda and Dagli (1992) proposed a hybrid neural network genetic algorithm (GA) approach. Dighe and Jakiela (1996) used GA while employing task decomposition and contact detection. Hopper and Turton (2000) investigated meta-heuristic and heuristic algorithms for the TDL rectangular cutting problem. Fujita, Akagi and Hirokawa (1993) detailed a hybrid approach for optimal nesting using GA and local minimization. Bounsaythip and Maouche (1996) and Jakobs (1996) proposed a modified genetic approach to a nesting problem. Liu and Teng (1999) improved Jakobs (1996) bottom-left procedure and implemented a GA for the orthogonal packing of rectangles. Ramesh Babu and Ramesh Babu (1999) provided a different bottom left procedure and employed a hybrid genetic algorithm to nest rectangular shapes on multiple sheets. To our knowledge, no exact solution procedure for the irregular TDL problem exists in the literature. For rectangular and circular cases, Stoyan and Yaskov (1998) developed a mathematical model for optimally solving both versions of the problem. In this paper, we present a hybrid approach to solve the (ir)regular TDL problem. The hybrid approach uses GA to determine the (near)optimal ordering of the pieces while a constructive method searches for the best layout of the ordered set of pieces. The proposed GA implementation eliminates the problem of overlap computation. It uses meaningful operators, and offers a close mapping to the physical problem. The constructive method attempts to mimic human placement strategy. It uses very simple geometric computations based on projections of the lower and left sides of the piece to be placed on the upper and right sides of the placed pieces. Therefore, it avoids the shortcomings of convex enclosure based methods (for more details, see Milenkovic, Daniels and Li (1991)). This paper is organized as follows. In Section 2, we provide a heuristic based on a constructive approach (CA). CA consists of sequentially positioning a set of ordered pieces. Each piece is tested for a set of potential positions defined with respect to already positioned pieces. The best position, following a series of horizontal and vertical translations, is one that minimizes the overall length while maximizing the packing of the already-placed pieces. In Section 3, we propose a hybrid GA and CA. CA evaluates the chromosomes generated by GA, where a chromosome defines an ordering of the pieces. In Section 4, we evaluate the two algorithms and analyze their performance on a set of problems taken from the literature.

3 2. The constructive approach The TDL problem can be divided into two subproblems: finding the optimal ordering of the pieces, and identifying the best packing of the ordered pieces. For instance, the two different orderings of Fig. 1(a) use the same layout mechanism but as expected yield two different lengths. Similarly, the same ordering yields two different lengths if two different layout mechanisms are used as shown in Fig. 1(b). In this paper, a near optimal order of the pieces is obtained using a GA implementation discussed in Section 3. The best packing of an ordered set of pieces is obtained using a constructive approach The main principle of the CA M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) The constructive approach starts by placing the first piece on the left and bottom-most position of the rectangular stock sheet. The following piece will be placed in one of five possible positions, ðx; 0Þ, ð0; yþ, ðx; yþ, ðx; yþ, and ðx; yþ, where x, x, y, y are the maximum and minimum x and y coordinates of the already placed piece. Each candidate position is adjacent to at least two already positioned pieces: one on its left, and one on its bottom. Only those positions that do not cause an overlap of the piece to be positioned with one that is already-placed are retained as valid candidate positions. The candidate lists differ according to the piece to be placed. Since few of the candidate positions are duplicates, each position will appear only once in the candidate list. For each candidate position, the piece to be placed is packed horizontally and vertically with respect to pieces on its left and bottom. The translation steps are based on the projection of Fig. 1. Two aspects of the TDL problem: ordering the pieces and building a layout.

4 198 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) the left and bottom sides of the piece to be placed on the upper and right sides of already-placed ones. Different orders of translations are considered in the search for the best packing. If the piece violates the width constraint, the candidate position is deleted from the list of feasible positions. The constructive method uses very simple geometric tools, avoiding the computation of minimal convex enclosures. This enhances the speed of layout generation, especially when the constructive method is combined with GA. Among all feasible candidate positions, the one that places the piece to the left and bottommost position is preferred. There are cases where such a position is not chosen. For instance, if placing the piece in this position creates an artificial hole by blocking the possibility of positioning any of the remaining pieces to be placed, then the position is not chosen and the next best position is adopted. Hifi and M Hallah (2002) provided a detailed description of these exceptional cases, and compared the performance of the procedure with and without these additional rules. We emphasize the proposed constructive method is neither the bottom-left (BL) (Jakobs, 1996) nor the improved BL (Liu and Teng, 1999) procedure. The proposed method considers a set of possible positions for each piece to be placed. Only one of theses positions is that used in Jakobs (1996) or Liu and Teng (1999). Ramesh Babu and Ramesh Babu (1999) proposed a different BL procedure that improves the results of Jakobs (1996) and Liu and Teng (1999). This approach (called hereafter RB) chooses the BL position among a set of positions or nodes. The nodes are obtained by projecting the left and bottom sides of each positioned piece onto the right and top sides of already-positioned ones. Even though CA defines a maximum of five positions per placed piece, it obtains the same layout as RB for rectangular shapes. Indeed, RB and CA investigate the same potential positions for rectangular shapes, but using different mechanisms to generate them. However, this similarity does not hold for irregular shapes. Consider the example of Fig. 2, adapted from Ramesh Babu and Ramesh Babu (1999), with the addition of a new non-rectangular piece. The new piece causes an overlap with piece 5 if positioned at node 13, and with piece 1 if positioned at node 14. However, placing this new piece at node 14 and translating it to the bottom will yield a feasible layout. P 7 New piece P 1 P 2 P P 8 13 P 5 Fig. 2. A step of the constructive approach on a simple example.

5 Using CA to generate layouts is faster. It investigates a predefined maximum number of nodes; a number that is independent of the geometry of the piece to be placed. It explores the search space better that RB since it is not limited to positioning pieces at predetermined nodes, but proceeds to a series of horizontal and vertical translations of the piece with respect to adjacent ones. This series of translations is critical when handling non-rectangular shapes, particularly concave irregular shapes, and when positioning pieces in closed holes. This difference is further highlighted by the application of CA on the following example CA on a simple example M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) To illustrate how CA proceeds, we consider the four-piece example of Fig. 3, where the pieces are denoted P i, i 5 1,y, P 1 is placed on the left and bottom-most position of the stock sheet. As illustrated by Fig. 3a, placing P 1 automatically generates five possible positions for future pieces to be placed. These are denoted by p 1j, j 5 1,y, Here p 11 5 p 15,andp 12 5 p 13. This reduces the candidate positions list (L) to three positions. None of these positions is infeasible for P 2.Ifplacedinp 14, P 2 is translated horizontally to the p 22 = p 23 P 24 P 2 p 12 = p 13 p 14 p 12 p 14 = p 25 P 1 P 1 (a) p 11 = p 15 p 11 = p 21 (b) P 3 Width P 2 P 4 P 1 (c) Length of Layout Fig. 3. CA on a small example.

6 200 left or vertically to the bottom. In this case, translating p 14 left yields p 12 while translating it to the bottom yields p 15. This is not always the case. Translating it to the left is preferred since it yields a current layout with minimal length, as shown in Fig. 3b. After placing P 2 at p 12, L is p 11 and p P 2 generates another set of five positions with p 22 5 p 23. Since p 11 5 p 21 and p 14 5 p 25, L has only five positions; two of which are infeasible for P 3 ; p 22 violates the width constraint, while p 12 creates an overlap. The best final position for P 3 is placing it in p 24 and translating it horizontally then vertically. This position cannot be obtained from RB. 4. Finally, the best position for P 4 is obtained by placing it in p 12 and translating it vertically until it comes in contact with P 1. Again, this position cannot be obtained by RB. An optimal layout is given in Fig. 3c (an exhaustive enumeration of all possible orders and possible positions proves optimality of this layout) A constructive heuristic M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) The proposed constructive approach yields a unique layout for every ordering of the pieces. Experimental testing suggests that CA yields near optimal layouts when pieces are packed according to increasing size; that is according to length, width, or area. Therefore, a CA-based heuristic, labeled CH, provides CA with these three orderings and chooses the ordering that yields the minimal length. If the pieces are circular, the constructive heuristic packs the pieces according to decreasing radii. 3. A genetic algorithm-based heuristic approach 3.1. The basic genetic algorithm A genetic algorithm (GA) can be viewed as an intelligent probabilistic search algorithm and can be easily applied to a variety of combinatorial optimization problems (Bachelet, Preux and Talbi, 1996; Beasley and Chu, 1996; Powell and Skolnick, 1993; Reeves, 1993). The idea of GA is based on the evolutionary process of biological organisms in nature. The theoretical foundations of this approach were originally proposed by Holland (1975). During evolution, natural populations evolve according to the principles of natural selection. Individuals which are adapted to the environment have a better chance of reproducing; those which are less fit are eliminated. This natural selection of the fittest implies that the genes from the highly fit individuals will spread to an increasing number of individuals in each successive generation. The combination of good characteristics from highly adapted ancestors may produce even more fit offspring. In this way, species evolve to become better adapted to their environment. GA simulates these processes by generating an initial population of individuals and applying genetic operators on the fittest of those individuals in each reproduction cycle. In optimization terms, each individual of the population is represented by a chromosome; i.e. a possible solution to the problem at hand. The fitness of each individual is evaluated with respect to an objective function. Highly fit individuals, or the best available solutions to the problem, are reproduced by exchanging some information with other solutions according to a predefined crossover procedure. In this way, produced children inherit some characteristics from each of the two parents. The

7 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) produced children are then subject to mutation. Mutation alters some genes in the chromosome in an attempt to make it better adapted to its environment. The offspring can either replace the whole population (generational method) or replace less fit individuals (incremental method). This evaluation-selection-reproduction cycle is repeated until a satisfactory solution of the implemented problem is reached. A general outline of a standard GA is described in Box 1. Box 1 A standard genetic algorithm. The performance of GA depends mainly on the adopted solution configuration and the associated fitness function, on the parent selection procedure, on the genetic operators (crossover and mutation), and on the replacement method of populations Solution configuration and fitness function Each individual of the population is a feasible solution to the TDL problem. A solution or a layout is represented by a chromosome. A chromosome is an ordered structure of n integer numbers, where n is the number of pieces. Each integer number, or gene, identifies a piece. The order of a gene in the chromosome is the order in which it will be considered during the construction phase of the layout. Each chromosome corresponds to a unique layout. Since the objective of the TDL problem is to minimize the layout s overall length, the fitness of a chromosome is simply the length of the layout it generates. This configuration is very simple. Not only is it purely symbolic, but it also separates the ordering problem from the layout problem. The fitness function is straightforward since only feasible layouts are generated Parent selection Parent selection consists of assigning reproductive opportunities to each individual in the population based on its relative fitness. In our implementation, each of the best-fit parents is chosen successively to be the first parent. The second parent is chosen randomly among the bestfit parents. This selection procedure can be efficiently implemented. In addition, it guarantees to each fit parent a chance to reproduce at least once while allowing a certain diversity of choice. It, therefore, promotes the choice of the best, while minimizing the risks of duplication and stagnation in local minima.

8 Crossover operators M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) The crossover operator (Golberg and Deb, 1991; Reeves, 1993) takes bits from each parent-string and combines them to create a new child-string. The main idea is to create new strings from substrings of fit-parent strings. In this way, new and promising areas of the search space will be explored. The one-point, two-point, and uniform crossover are frequently used. Beasley and Chu (1996) have proposed another crossover operator called the fusion-operator, which takes into account both the structure and the relative fitness of each of the two parent solutions. The approach used by the authors produces a unique child. However, designed mainly for binary coded GA problems, their approach would yield infeasible solutions in our case. Here, we apply a variation of the OX Davis two point crossover. This crossover guarantees the feasibility of any generated chromosome. 1. Two parents, P 1 and P 2, yield two children, C 1 and C 2, with each C i inheriting a subsequence [ j,y, k] of its genes from its respective P i, i 5 1, The genes of P 2 are used to successively fill, according to their order of appearance in P 2, the empty genes of C If a gene is already in C 1, it is rejected, else it is positioned in the first empty gene of C This iterative process (steps 1 3) stops when all the n genes of C 1 are filled. Similarly, the empty genes of C 2 are filled with the genes of P 1 in their order of appearance. The integer numbers j, k, delimiting the subsequence being inherited are chosen randomly. When j 5 1 or k 5 n, the two-point crossover reduces to a single-point crossover. Choosing a two-point crossover maintains a subsequence of each parent instead of recombining complementary parts from both parts; i.e. it preserves a meaningful part of each chromosome. It is important in this case not to always opt for a single-point crossover to avoid the systematic retransmission of the entire left part. Pieces packed earlier (i.e. those on the left side of the chromosome) affect the packing of those packed later. Thus, through generations, the single-point crossover limits the diversity of the population, and promotes duplication of chromosomes, enhancing the chances of convergence toward local minima. The adopted crossover is somewhat consistent with the philosophy of the fusion operator (Beasley and Chu, 1996) where the child is supposed to inherit the best of both parents. The child inherits a good building block from one parent and the relative position of pieces from the other Mutation operator The mutation operator is generally applied to each newly-created child after crossover. It inverts m, randomly chosen bits in a string, where m is determined experimentally. Generally, it prevents loss of valuable genetic information by re-introducing information lost due to premature convergence, thereby expanding the search space. It offsets any bias created by the crossover operator and guarantees the diversity of the population. Here, four kinds of mutation are used. The first type swaps two random genes. The second swaps two subsequences of genes. The third inverts the order of a subsequence of genes. Finally, the fourth type inverts the whole chromosome.

9 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) The first two kinds of mutation are particularly successful for local improvements once the algorithm starts converging. The third induces a major shuffling of the genes. Its main objective is to avoid any local stagnation that may be caused by the crossover operator, which transmits subsequences of genes among generations. Finally, unless the layout is a perfect puzzle, either the initial chromosome or its reverse mutation will necessarily be better than the other. The mutant can be regarded as a packing starting from the upper right most position while the chromosome is a packing starting from the lower left most position. In our implementation, each chromosome generated by crossover is subject to these four kinds of mutation operators. This enhances the quality of the generated solutions and promotes good characteristics of chromosomes. So, each chromosome generates a total of five different children Population replacement Our GA implementation uses the incremental replacement method in lieu of the wider spread generational replacement method, where the new children population replaces the whole parent population (for more details, see Beasley, Bull and Martin (1993)). Once constructed, a child either replaces the least fit member of the population or dies if its fitness function is worse than that of the least-fit member of the population. Since the children have better fitnesses than the solutions they are replacing, the average fitness of the population must improve through the generations. In this way, the best solutions are always in the population, and newly created ones are immediately available for selection and reproduction. We choose the best m fit individuals among a population of 11m individuals: m parents, 2m children from crossover, and 8m chromosomes from mutation of the 2m children. Limited computational experience showed that our GA implementation using these parameters with a population of 16 individuals produced satisfactory solutions for the TDL problem. When replacing a solution, care must be taken to prevent a duplicate solution from entering the population. A duplicate solution is identical to any one of the solution structures in the population. Allowing duplicate solutions to exist in the population is undesirable because it could yield a population consisting of all identical solutions; thus, would severely limit GA s ability to generate new solutions. The initial population consists of non-identical chromosomes. The first k individuals are obtained by ordering the pieces according to size, where k 5 3 for rectangular and irregular shapes, and k 5 1 for circles. The following m k individuals are random permutations of the pieces. These m chromosomes are subject to the crossover and mutation operators discussed above. The proposed hybrid heuristic is summarized in Box 2. This implementation uses the same chromosome configuration as in Jakobs (1996) and Liu and Teng (1999). However, it subjects every chromosome at every generation to crossover at least once and to four types of mutation. This strategy favors exploring local neighborhoods versus too much diversification of the search. The choice of this strategy has been motivated by the good quality of the chromosomes generated according to size. In addition, experimental testing has shown that mutating the order of a piece or a subsequence of pieces in a chromosome can be of great effect. As such, intensifying mutation

10 204 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) seemed necessary. However, this intensification can lead to premature convergence if crossover is not used. The incremental replacement used here and not in Jakobs (1996) and Liu and Teng (1999) guarantees that good chromosomes are readily available for reproduction, which shortens the run time. Box 2 Steps of the hybrid CAGA Algorithm.

11 4. Computational study In this section, we present the performance of the proposed constructive heuristic, CH, and constructive approach using genetic algorithm, noted (CAGA). The proposed algorithms, coded in Fortran, wererunonapentiumiii(733mhzand128mbofram),withruntimelimitedto30minutes Discussions M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) The code takes advantage of the libraries available in Fortran to represent shapes. For instance, a circle is identified via its radius and the coordinates of its center. So translating a circle consists simply in translating its center. An (ir)regular piece on the other hand is identified by the number and coordinates of its extreme points enumerated counter-clockwise. Arcs are defined by a minimum of five points and a maximum of nine points. Using less than five points hinders the estimation of the curvature of the arc, while using more than nine increases the run time. The larger the number of extreme points the larger the computation time of overlap detection and translation steps. Translating a(n) (ir)regular piece consists in translating its reference point where the reference point is defined by the left bottom-most point of the piece. The reference point may be a virtual point that does not belong to the piece. It is true that a rectangle can be represented simply by a reference point, a length, and a width. This representation decreases the run time; however, this representation scheme was not retained in an attempt to maintain the code as standard as possible. The procedure used to check for overlap depends on the shape of the pieces being considered. For circles, checking for overlap consists in checking if either of the circles is included in the second one and if they intersect on exactly one or two points. Checking for overlap of irregular pieces, on the other hand, is more complex. Let P j be the piece to be placed and P i,i51,y n p,an already-placed piece with n p the number of currently placed pieces. Checking for overlap of P i and P j is undertaken only if there is a chance that the two pieces overlap; i.e., if x j or x j 2½x i ; x i Š and y j or y j 2½y i ; y i Š, where x, x, y and y are the minimal and maximal x and y coordinates. Checking for overlap proceeds as follows: 1. Set Overlap 5.FALSE. 2. Define the four sides of each piece: Upper U, left L, bottom B, and right R. 3. Set i While(Overlap 5.FALSE..AND. i4n p ) If any of the following conditions is true for P i any point of B j is below U i and above B i while being between L i and R i ; any point of U i is below U j and above B j while being between L j and R j ; any line segment of the B j intersects any line segment of U i ; any point of the L j is on the left of R i while being between B i and U i ; any point of the L i is on the left of R j while being between B j and U j ; any line segment of L j intersects any line segment of R i ; any point of B i is below U j and above B j while being between L j and R j ; any point of U j is below U i and above B i while being between L i and R i ;

12 206 any line segment of the B i intersects any line segment of U j ; any point of the L i is on the left of R j while being between B j and U j ; any point of the L j is on the left of R i while being between B i and U i ; any line segment of L i intersects any line segment of R j ; then Overlap 5.TRUE. 5. Stop. This overlap detection procedure is a lot simpler for rectangular and convex shapes. However, we used the same procedure for all (ir)regular shapes in an attempt to maintain the code as standard as possible Computational results M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) The Fortran code tests both CA and CAGA on different problems extracted from the literature. We have made these instances publicly available from ftp://panoramix.univ-paris1.fr/pub/ CERMSEM/hifi/OR-Benchmark.html, hoping to aid further development of exact and approximate algorithms for the TDL problem. The problems we considered are summarized in Table 1. We tested a total of 20 problems corresponding to three classes of pieces: circular, rectangular, and irregular (not necessarily convex). For the circular TDL version of the problem, we used the six problems of Stoyan and Yaskov (1998), referred to as SY1,y, SY6 in Table 1. The optimal solution for each of these six problems is known. For the rectangular TDL problem, we tested 11 problems. Nine of these, noted SCPL1,y, SCPL9 in Table 1, are taken from Hifi (1999). Given that the optimal solutions for these nine problems are not available, we compared the solutions we generated to a lower bound. The most natural lower bound is the sum of the surfaces of all pieces divided by the width of the strip. The last two rectangular TDL problems, noted Jak1 and Jak2 in Table 1 are from Jakobs (1996). Finally, for the irregular TDL problem, we considered three problems. The first one, labeled G8 in Table 1, was proposed by Grinde (1996). The last two, cited as Jak4 and Jak3 in Table 1, were suggested by Jakobs (1996) The circular TDL version First, we tested the six problems of Stoyan and Yaskov (1998), for which the optimal solutions are known. Column 2 of Table 2 displays the optimal length L*. Columns 3 5 and 6 9 report the results yielded by CH and CAGA. Column 3 (respectively 6) contains the best length L reached by CH (respectively CAGA). Column 4 shows the gap of the given solution from the optimal one. The gap ¼ 100 L L L. Column 5 (respectively 8) displays the corresponding run time RT in seconds. Finally, column 9 displays TFBS, the time to first reach the final best solution. This time is obviously shorter than the global run time; which implies that the stopping criterion of CAGA can be altered without much affecting the quality of the solution. As Table 2 shows, CH gives good quality results. On average, it is 7.963% from the optimum, with a worst-case of %. CH is very fast, requiring less than a second for small problems

13 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Table 1 Test problem details. Test problems Circular Rectangular Irregular Inst W n Inst W n Inst W n SY SCPL G SY SCPL Jak SY SCPL Jak SY SCPL SY SCPL SY SCPL SCPL SCPL SCPL Jak Jak (SY1,y, SY4). For large problems, e.g. 100 pieces, its average run time is 80 seconds with the largest observed value 84 seconds. This larger computational time is expected because the pieces are more diverse. The more diverse the pieces, the larger the number of feasible candidate positions. Were CA specifically designed for circular pieces, the runtime would have been shorter. The positions would have been directly tangent to already-placed circles eliminating the need for translations. In summary, CH is a useful starting point for more complex procedures. CAGA produces better results, but with a longer computational time as observed in Table 2. Indeed, it produces an average deviation of 2.972%, varying from 2.114% to 5.013%. The average computational time is under 33 seconds for small problems, which is reasonable considering the good quality of the results. This time is, as expected, longer for larger problems. CAGA evaluates a total of 11 m number of iterations chromosomes. Of course, the larger the number of pieces, the longer it takes to evaluate a chromosome. Table 2 Performance of both CH and CAGA on circular TDL problems. CH CAGA Inst L* L Gap (%) RT (s) L Gap (%) RT (s) TFBS (s) SY o SY o SY o SY o SY SY Average

14 208 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Fig. 4. The Stoyan and Yaskov s example (SY5): the CH solution, of length corresponding to a 4.517% gap. Figure 4 shows the structure of the CH solution for SY5. This solution corresponds to a 4.517% gap. Finding this solution requires 84 seconds. Figure 5 shows the improved solution realized by CAGA after five iterations. Finally, Fig. 6 shows the structure of the CAGA final solution. It was first reached after 760 seconds The rectangular TDL version Next, we tested our approaches on 11 rectangular TDL problems: SCPL1 SCPL9 taken from Hifi (1999), and Jak1 and Jak2 from Jakobs (1996). The computational results appear in Table 3. Column 2, labeled L* or LB contains either the optimal solution, if it is known, or a lower bound (that has not been proven optimal). Whenever a lower bound is used, the problem is marked with a * sign. LB ¼ P n i¼1 l iw i b i w, where bi denotes the demand value of the i-th rectangular piece of length l i and width w i. These lower bounds equal the optimal solution only if the pieces form a perfect rectangular puzzle of width w. Only the optimal solutions for SCPL, Jak1 and Jak2 are known. SCPL7 has two pieces, out of the 139 patterns, with length 121 each; thus, any optimal layout has to be as long as any of the two pieces. Jakobs (1996) artificially created his two problems by cutting a rectangle into a number of pieces. Columns 3 5 (respectively 6 8) report the best solution reached by CH (respectively CAGA), its deviation from L* or LB, and the corresponding RT expressed in seconds. Column 9 reports TFBS, measured in seconds, for CAGA. Table 3 shows that: 1. The constructive heuristic produces good quality results. It is on average 9.029% from the optimum. Occasionally, it obtains seemingly poor results, with a worst case of %. It is suspected that the lower bound is not very tight in these cases. Even though the problem sizes are relatively large, CH remains very fast. The average run time is 4 seconds for small problems (less than 100 pieces) and for large problems with the longest time being 34 seconds (SCPL8 with 156 pieces). This run time can be decreased by (i) using simpler overlap detection procedures that take advantage of the rectangularity of the pieces being considered, and (ii) eliminating the translation procedure since the positions defined here are adjacent to at least two already placed pieces; so no further translation is required.

15 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Fig. 5. The Stoyan and Yaskov s example (SY5): a CAGA intermediate solution of length corresponding to a 2.430% gap. Fig. 6. The Stoyan and Yaskov s example (SY5): the CAGA final solution, of length with a 2.114% gap. 2. The results of CAGA globally improve the results of CH for the test problems. The average deviation is reduced to 4.326%. It varies from 0 to The TFBS is again shorter than the average run time of CAGA. This suggests that the stopping criterion can be changed without risking premature convergence of GA. Despite its relatively longer (than CH) run time, CAGA appears to be a good choice when a high quality solution is needed. Figures 7 and 8 show the solution of the SCPL5 produced by both CH and CAGA. CH creates the packing shown in Fig. 7, whose length is 135. Generating this packing takes approximately 10 seconds. This solution, corresponding to a deviation of 4.745%, is a good starting solution. Figure 8 illustrates the structure of the (final) packing pattern created by CAGA after ten iterations. The solution obviously improved, being only 3.193% from the optimum The irregular TDL version Last, we tested CH and CAGA on three irregular TDL problems. The first example uses nonconvex irregular pieces while the last two use non convex regular pieces. The results obtained by

16 210 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Table 3 Performance of both CH and CAGA on rectangular TDL test problems. CH CAGA Inst L* or LB L Gap (%) RT (s) L Gap (%) RT (s) TFBS (s) SCPL * SCPL * SCPL * SCPL * SCPL * SCPL * SCPL SCPL * SCPL * Jak Jak Average Note: The symbol * means that the reported value denotes the lower bound of the treated instance. Fig. 7. The CH initial solution pattern of instance SCPL5. It has a 135 length and a 4.745% gap. both CH and CAGA are given in Table 4. Column 2, labeled UB (Upper Bound) contains the best-known solution. The negative gap values in columns 4 and 7 indicate that the considered algorithm (CH or CAGA) improves the best-known solution by gap%. CH succeeds in improving the

17 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Fig. 8. The CAGA final packing pattern of instance SCPL5: It has a 133 length and a 3.193% gap. Table 4 Performance of both CH and CAGA on irregular test problems. CH CAGA Inst UB L Gap (%) RT (s) L Gap (%) RT (s) TFBS (s) G Jak Jak Fig. 9. Grinde s example: the CH solution.

18 212 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Fig. 10. Grinde s example: the CAGA final solution. Table 5 Coordinates of the reference points of the final CAGA solution of Jak4 and Jak3. Instance Jak4 Instance Jak3 No piece X Y X Y known solutions for the three problems (G8, Jak4, and Jak3). Moreover, CAGA further improves each of the solutions. A thorough discussion of the results, displayed in Table 4, follows. Instance G8, extracted from Grinde (1996), corresponds to the pattern of a single size woman s dress with a total of eight non-convex pieces to be displayed on 59-inch-wide cloth. The results for this instance are given in the first line of Table 4. CH took approximately two seconds to find the

19 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Fig. 11. Instance Jak4: (a) the CH pattern, of length 15, which is one unit shorter than the layout published by Jakobs (1996); (b) the CAGA final solution of length 14, which is two units shorter than the solution published in Jakobs. solution shown in Fig. 9 with a length of An additional 15 seconds produces the solution shown in Fig. 10 (with a length of 142.3), when CAGA is applied. Both CH and CAGA yield better solutions than the method of Grinde (1996) (the best layout obtained by the author is 156). In this case, CAGA produces a similar solution because of the small number of pieces considered. It is suspected that this solution is the optimal one. A human marker will opt for this solution. Total enumeration of possible orders and positions of the eight pieces gives a solution with the same length. Both CH and CAGA produce better results for Jak3 and Jak4 than those published by Jakobs (1996). For both cases, the coordinates of the reference point of each pattern in the solutions yielded by CAGA are detailed in Table 5. Figure 11 illustrates the two layouts produced by CH and CAGA for Jak4. The best length published by Jakobs (1996) equals 16, which is worse than the length of the patterns obtained by both CH and CAGA. The CH solution of length 15 has a 6.667% improvement. The final solution, of length 14, increases this improvement to %. The same phenomenon is again observed for Jak3. Indeed, CH produces a solution equal to 14. This solution improves the solution published in Jakobs (1996) by one unit. This improvement is increased to two units when CAGA is applied. Figure 12 illustrates the structure of the packing pattern given by CAGA.

20 214 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) Fig. 12. Instance Jak3: illustration of the CAGA solution. Its length is 13, which is two units shorter than the solution published by Jakobs (1996). Finally, a comparison of the run times of Jak1 to those of Jak3 and Jak4 shows that the more complex the shapes, the longer it takes to identify the minimal length layout. The difference is not very pronounced in this case (1 versus 2 seconds for CH and 142 versus 245 and 248 seconds for CAGA). It will be amplified if the shapes were more irregular and/or simpler overlap detection procedures were used for Jak1. 5. Conclusion The two-dimensional layout problem is solved using two heuristics: a constructive heuristic and a hybrid genetic algorithm-based heuristic. The heuristics search for a good ordering of the pieces,

21 M. Hifi and R. M Hallah/Intl. Trans. in Op. Res. 10 (2003) and use a new constructive approach to search for a good layout of the ordered set of the pieces. The constructive approach treats the layout problem as assembling parts into positions and translating them, eliminating the need for the lengthy geometric computations involved in detection of overlaps and/or definition of the minimal enclosure area. Empirical testing, conducted on different problems taken from the literature, shows that the constructive heuristic provides good solutions very quickly while the hybrid approach further improves these solutions within reasonable computing time. Acknowledgements The authors thank two anonymous referees for their helpful comments and suggestions which greatly improved the presentation and the content of this paper. Similarly, the authors thank Professor V. Zissimopoulos for kindly providing some of the data sets. Finally, they thank Professor Robert L. Bulfin for proofreading the final manuscript. References Bachelet, V., Preux, P., Talbi, E-G Parallel hybrid meta-heuristics: application to the quadratic assignment problem. In: Proceedings of the first Parallel Optimization Colloquium, University of Versailles, France, pp Beasley, J.E., Bull, D.R., Martin, R.R An overview of genetic algorithms: Part I fundamentals. University Computing 15, Beasley, J.E., Chu, P.C A genetic algorithm for the set covering problem. European Journal of Operational Research 94, Blazewicz, J., Hawryluk, P., Walkowiak, R Using a tabu search approach for solving the two dimensional irregular cutting problem. Annals of Operations Research 41, Bounsaythip, C., Maouche, S A genetic approach to a nesting problem. Proceedings of the 2NWGA, Vaasa, Finland, August, pp Cheng, C.H., Feiring, B.R., Cheng, T.C.E The cutting stock problem A survey. International Journal of Production Economics 36, Dighe, R., Jakiela, M.J Solving pattern nesting problems with GA employing task decomposition and contact detection. Evolutionary Computation 3, Dori, D., Ben-Bassat, M Circumscribing a convex polygon by a polygon of fewer sides with minimal area addition. Computer Vision, Graphics and Image Processing 24, Dowsland, K.A., Dowsland, W. Solution approaches to irregular nesting problems. European Journal of Operational Research 84, Downsland, K., Downsland, W., Bennell, J Jostling for position-local improvement for irregular cutting patterns. Journal of Operational Research Society 49, Dyckhoff, H A typology of cutting and packing problems. European Journal of Operational Research 44, Dyckhoff, H., Finke U Cutting and packing in production and distribution, a typology and bibliography. Physica Verlag. Fujita, K., Akagi, S., Hirokawa, N Hybrid approach for optimal nesting using a genetic algorithm and a local minimization algorithm. Proceedings of the ASME Conferences on Advances in Design and Automation 65, pp

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