Technische Universität Dresden Herausgeber: Der Rektor

Transcription

1 Als Manuskript gedruckt Technische Universität Dresden Herausgeber: Der Rektor The Two-Dimensional Strip Packing Problem: A Numerical Experiment with Waste-Free Instances Using Algorithms with Block Structure G. Belov, A.V. Chiglintsev, A.S. Filippova, E.A. Mukhacheva, G. Scheithauer, R.R. Shirgazin MATH-NM January 18, 2005

2 Contents 1 Introduction 1 2 Modeling of packings by block structures 3 3 Block structure based algorithms Decoder algorithms Substitution decoder (Sub) Improved block design decoder (IBD) Dual Local Search (DLS) Local search algorithms Genetic Block Algorithm (GBA) Sequential Value Correction (SVC) Genetic Multiple Method (GMM) Numerical experiments Experiment 1: Investigation of decoders, esp. checking of correctness Experiment 2: Comparison of stochastic local search algorithms Conclusions 15

3 1 Chiglintsev A.V., Filippova A.S., Mukhacheva E.A., Shirgazin R.R. Ufa State Aviation Technical University, Dept. of Numerical Math. and Cybernetics Belov G., Scheithauer G. Technische Universität Dresden, Institute of Numerical Mathematics The Two-Dimensional Strip Packing Problem: A Numerical Experiment with Waste-Free Instances Using Algorithms with Block Structure Abstract The collection of instances from E. Hopper has two waste-free classes, namely C and N. The absence of waste in the optimum means that the optimal solution value is known. The number of items in the instances varies from 16 to 197. We applied several single-pass algorithms as well as local search methods and metaheuristics, both new and known from the literature. For the single-pass algorithms, we retained the optimal order of items in the list in order to test their correctness. With this ordering, optimal solutions could be obtained by some of the single-pass algorithms including some of the proposed new ones. Among the other algorithms, without knowing the optimal order, only the sequential value correction method has found optima for all the instances. Keywords: strip packing, heuristics 1 Introduction The Two-Dimensional Strip Packing Problem (2D-SPP) [12] is to minimize the strip length used to pack a given set of rectangular items and is defined by the following data: (W ; m; w; l) with W, m Z +, w = (w 1, w 2,..., w m ) T, l = (l 1, l 2,..., l m ) T Z m +, where W is the strip width, m is the number of rectangles, and w i, l i are the sizes of the i-th rectangle, i I = {1,..., m}. Let us introduce a coordinate system where the x-axis coincides with the horizontal (infinite-length) strip side and the y-axis coincides with the vertical (fixed-width) side, the direction is upwards. The small rectangles are to be positioned without overlapping and with their sides parallel to the strip sides. No rotation is allowed. The reference point of each rectangle be its bottom-left corner with

4 2 coordinates (x i, y i ). In the 2D-SPP we are searching for a feasible packing with minimal length L, i. e. the following problem has to be solved: subject to L = max{x i + l i : i = 1,..., m} min (1) [x i x j + l j ] [x j x i + l i ] [y i y j + w j ] [y j y i + w i ] for i, j I, i j, (2) [x i 0] [y i 0] [y i + w i W ] for i I. (3) Conditions (2) guarantee non-overlapping of the items, whereas conditions (3) guarantee containment of the items in the strip. A recent collection of related work can be found in Bortfeldt (2004), [5]. A list of bottom-left corner coordinates (x i, y i ) i I of the rectangles can be seen as a direct coding scheme of the packing. We are interested in coding schemes which can be translated into the direct scheme by some algorithms. A survey on heuristics (decoders) for rectangle packing is given in Lodi, Martello and Vigo (2002), [12]. The following deterministic, level strategy-based heuristics are considered there: Next Fit Decreasing (NFD), First Fit Decreasing (FFD), Best Fit Decreasing (BFD). The complexity of the algorithms for a fixed priority list is O(m log m). The main strategy among those which are not levelbased is Bottom-Left (BL) [2]. It is defined on a vertical strip of unlimited height. Given a priority list of items, it packs every next item into the bottom-most position, including holes, and aligns it to the left. In this paper, we apply this approach to a strip of fixed width and open length, but maintain the well-known BL notion. In Murata (1995) [19], a coding scheme is used, called sequence pairs (SP) to build packings with the minimal area of the containing rectangle. Decoders based on block coding schemes were developed in Mukhacheva and Mukhacheva (2004) [17]. These include: Substitution (Sub) and Dual Local Search (DLS). Furthermore, Mukhacheva and Mukhacheva (2000) [15] developed the Reconstructions Decoder (Rec), and on its basis a hybrid decoder SubRec [17] which combines Sub and Rec. These decoders will be briefly explained below. The application of these decoders in combination with an appropriate metaheuristic procedure is investigated in this paper by means of benchmark instances known in the related literature. Here we are concentrated on waste-free instances. In our comparing experiments we use the following criteria of algorithm effectiveness: the length L of the occupied strip part and the material utilization ratio CC which is the ratio of the total area of the packed items to the whole used area, i. e. CC = i I l i w i /(LW ). Another important indicator is the deviation of the objective value from the lower bound. The calculation of the latter is an important task, especially for exact algorithms. In our

5 3 experiment, all instances are waste-free which implies that the lower bound equals the total area of items divided by W. In Mukhacheva and Mukhacheva (2000), Mukhacheva, Chiglintsev, Smagin and Mukhacheva (2001), [15, 14], and some others, a local lower bound is used to build estimation functions for neighborhood solutions. For this purpose, the Sequential Value Correction heuristic (SVC) is used which solves approximately a special one-dimensional cutting stock problem with additional constraints. It is common to perform experiments on examples from the OR-Library [3]. In the current paper we investigate the waste-free examples of Hopper and Turton [10]. The optimal solutions are known. Moreover, the source data of the examples contains the items in the order corresponding to their optimal packing with the BL algorithm. The paper is organized as follows. Next, in Section 2, we define the (vertical) block structure of a packing. In Section 3, we describe block structure based algorithms, especially SVC and some block decoders. Results of computational experiments are given in Section 4, followed by some concluding remarks. 2 Modeling of packings by block structures Block structures of rectangular packings, their properties and applications are thoroughly described in Mukhacheva and Mukhacheva (2004) [17]. Here we give only the definitions and some motivation. As above, the material strip has width W, m is the number of items to be packed and the length of the packing is denoted by L. We consider a feasible packing represented by bottom-left corner coordinates (x i, y i ) of the items i I. Now, the vertical lines drawn through the right sides of the packed rectangles divide the packing into a number r (r m) of rectangular blocks having width W and (in general different) lengths λ j, j J = {1,..., r}. It holds that λ 1 = min{l i : x i = 0, i I}, λ 2 = min{x i : x i > λ 1, i I} λ 1, etc. Let us call this vertical division a vertical block structure, cf. Fig. 1. Each vertical block j is characterized by a set I j of indexes of the rectangles crossing it and its length λ j, j J. All blocks together are gathered in a list S: S = { (I j, λ j ) : j J } = {( {i 1j, i 2j,..., i kj,j}, λ j ) : j = 1,..., r }, (4) where i kj I j I is the index of the k-th rectangle crossing block j.

6 4 In Fig. 1 we see a packing of 6 rectangles and the corresponding vertical block structure is S = {({1, 2}, λ 1 ); ({2, 3}, λ 2 ); ({3, 4}, λ 3 ); ({3, 5}, λ 4 ); ({3, 6}, λ 5 ); ({6}, λ 6 )}. with 4 λ 1 = l 1, λ 2 = l 2 l 1, λ 3 = l 4. λ 4 = l 5, λ 5 = l 3 λ j, λ 6 = l 6 λ 5. j=2 Since in the definition of the block structure the real position of the items is not regarded, a modification is also considered. In the data structure (list) S = {({(i kj, p kj ).k = 1,..., I j }, λ j ), j = 1,..., r} (5) let i kj I j I denote the index of the k-th item in block j and p kj the y-coordinate of its bottom side. This list is called fixed vertical block structure. In the example in Fig. 1 we have S = {({(1, w 2 ), (2, 0)}, λ 1 ); ({(2, 0), (3, w 4 )}, λ 2 ); ({(3, w 4 ), (4, 0)}, λ 3 ); with again ({(3, w 4 ), (5, 0)}, λ 4 ); ({(3, w 4 ), (6, 0)}, λ 5 ); ({(6, 0)}, λ 6 )}. 4 λ 1 = l 1, λ 2 = l 2 l 1, λ 3 = l 4. λ 4 = l 5, λ 5 = l 3 λ j, λ 6 = l 6 λ 5. j=2 Similarly, the horizontal block structure, and the fixed horizontal block structure, can be defined. Here we restrict ourselves to vertical structures. A block structure, i. e. a list S, can be seen as a code of packing. In order to solve the 2D-SPP we are interested in the following problem: For a given instance (W ; m; w; l) find a list S (or S ) which corresponds to a feasible packing of minimal length. For that, let us give some necessary conditions of correspondence of a block structure to a feasible packing. Let I + j be the set of rectangles ending in block j, and I j the set of rectangles beginning in block j. Proposition 1. Let an instance (W ; m; w; l) of the 2D-SPP and a vertical block structure S be given. In order that S correspond to a feasible packing, the following conditions are necessary:

7 5 Figure 1: Block structure of a rectangular packing 1 0. Diversity. The elements i kj in each sublist j J of S are different for different k; 2 0. Continuity. If an element i I j and i / I + j then i I j+1 ; 3 0. Item length. If i Ij and i I p + then l i = p λ k ; 4 0. Non-crossing with the border. k=j i I j w i W, j J. Conditions 1 0 and 2 0 are implied by the integrity of the single rectangles. The coordinates (x i, y i ) of the rectangles can be computed straightforwardly. However, if S is obtained by some other method, the list of rectangles crossing each block may need to be reordered to obtain a feasible packing, if it is possible. Proposition 2. condition If S is a fixed vertical block structure which additionally fulfills the 5 0. For each pair r, s with i rj = i s,j+1 for some j J it holds that p rj = p s,j+1 ; then S represents a feasible packing pattern. Especially Proposition 2 is a basis for constructing decoders (heuristics). 3 Block structure based algorithms For our experiments, three meta-heuristic algorithms which are not based on block structure, are used for comparison:

8 6 the Genetic Algorithm GA of Lodi, Martello and Vigo (cf. [12]) in which the small rectangles are used as genes and their permutation as a chromosome; the Ant Colony Algorithm AC of Dorigo and Gambardella (cf. [7]); and the Tabu Search Algorithm TS of Glover (cf. [8]). These three meta-heuristic applications were compared with the following block structurebased approaches: the Naive Local Search NS where a certain block decoder is applied repeatedly on random generated item sequences, i.e. NS is a kind of random search, the Genetic Block Algorithm GBA where vertical block structures are used as chromosomes (cf. [18]), and the Sequential Value Correction Method SVC, adapted from [13], where SVC yields different item sequences (priority lists), which are described below in more detail. For all these meta-heuristics well-known decoders are applied such as the original Bottom-Left decoder of Baker et al [2], the Polygonal Bottom-Left decoder PBL and its improved version IBL by Liu and Teng [11] who took this approach from non-rectangular packing and used inside a genetic algorithm. This decoder does not fill holes and has further restrictions, for example the placement into the bottom-most position is not guaranteed, the Sequence Pairs decoder SP proposed by Murata et al. [19], which do not relate to block structures, as well as the block structure based Substitution decoder Sub, its improved version ISub, and the decoder Dual Local Search (DLS). Moreover, we propose an improvement of BL called Improved Block Design IBD. These algorithms are described below in more detail. To identify an algorithm we write as follows: NAME of the meta-heuristic (NAME of decoder). For example, GBA(Sub) means, the Genetic Block Algorithm uses the Substitution decoder. Below we discuss the used decoders and metaheuristics and shortly describe original algorithms based on block structure.

9 7 3.1 Decoder algorithms Among simple algorithms we can distinguish those that are used in multiple-pass algorithms as decoders. As input they typically take a priority list (i.e., permutation) of items and produce a feasible solution. An important property of a decoder is its ability to restore an optimal solution using an optimal priority list (i. e. there exists a priority list for which the algorithm yields some optimal solution). Let us call this property correctness. Note that the BL algorithm is not correct in this sense, in [2] a counterexample is given. However, the BL algorithm is correct with respect to waste-free instances. Here we verify this property for all used decoders with respect to the waste-free examples of Hopper and Turton [10], whose optimal solutions are known. Next we (shortly) describe the base versions of three decoders: the Substitution decoder (Sub), the Improved Block Design decoder (IBD), and Dual Local Search decoder (DLS). As we will see, Sub and DLS are correct with respect to the waste-free examples as well as BL Substitution decoder (Sub) A given 2D-SPP instance (W ; m; w; l) is considered as a one-dimensional cutting stock problem (1D-CSP) with m types of items having length w i and order demands l i, i = 1,..., m, and with material size W. When constructing a new block, the algorithm Sub [17] fills empty spaces similar to the 1D first fit heuristic (FF) in the first level. (FF accepts the first item in the list which fits, then the next which fits in the remaining space, etc, with the total effort O(m) for a single empty part of the block.) Based on this relaxation of the 2D problem, algorithm Sub works as follows: Step 1. According to the given priority list compute a first block pattern with index set I 1 using the (1D) FF heuristic. Set j := 1. Step 2. Compute the block length λ j = min{l i : i I j }. Compute the placement positions (coordinates) (x i, y i ) for all rectangles i I j. Set l i := l i λ j for all i I j. If l i = 0 for all i then Stop. Step 3. Set j := j + 1. Construct the next block pattern (index set I j ) which has to contain all i I j 1 \ I + j 1, by filling the empty spaces (determined by the positions of the already placed items) according to the (1D) FF heuristic and the given priority list. Go to 2.

10 8 The computational complexity of Sub is O(m 2 log(m)). Proposition 3. The set of packings produced by Sub is a true subset of BL packings. Proof Note that both Sub and BL produce packings where the position of each item is bottom-left justified (if we think of the vertical container strip originally used to define BL.) Consider a packing produced by Sub, especially the order of items successively packed. Now, applying BL to the corresponding priority list gives the same packing. On the other hand, we can easily construct a BL packing not obtainable by Sub. We consider also two modified versions of Sub. Greedy or Improved Substitution ISub, proposed in [17]: The priority list given to the algorithm is only used to identify the next piece which can be placed on top of all already assigned pieces in the block. But to fill holes, the sorting of the items according to nonincreasing widths is applied. The complexity is O(m 2 ). Substitution with Reconstruction SubRec [17]: The algorithm of reconstruction tries to place the rectangles in such an order that the number of holes in the further blocks (i.e. waste parts) is as small as possible. In Fig. 3 it is enough to place item 4 before item 3 in the second block and item 8 before item 7 in the 4th block. This is a more advanced algorithm but it has a higher time complexity O(m 3 ). Figure 2: Packing by Sub Figure 3: Packing by SubRec Improved block design decoder (IBD) In the improved version of the BL decoder, IBD [17], we use a strategy to enlarge unfilled zones (areas). Before the next object is placed (according to the BL-strategy) we analyze whether and how much unused area will appear below and above the object. Depending on the two rules described below, we place the object bottom-most or top-most to create a larger unused area (with the hope to have a larger probability to place another object in this area). This means, some items may be placed not bottom-left justified.

11 9 The following rules are applied: 1. Move to margin. If the y-coordinate (i. e. the distance from the lower margin) of the lower side of the item is smaller than 1/3 of the strip width, the item is moved downwards as far as possible. 2. Comparison of the free space. This rule is applied if the first rule wasn t. We consider all blocks containing the item to be placed. The item is shifted upwards or downwards so that the connected free space arising in all these blocks is maximal. The IBD heuristic is introduced by an example. Consider the 2D-SPP instance W = 10; m = 7; w = (4; 2; 7; 2; 2; 3; 1); l = (7; 9; 5; 11; 7; 2; 6) and the priority list π = (1, 2, 3, 4, 5, 6, 7). Figure 4: Packing rectangles using the IBD decoder In Fig. 4, item 4 is moved to the lower margin of the strip. This can be represented by the following code, where positive numbers identify the item and negative numbers give the (negated) width of waste in the corresponding position. Before the move we have: S = {(1, 2, 1, 4, 1)7; (6, 1, 2, 1, 4, 1)2; (3, 4, 1)2; (3, 3)3} After the move we obtain: S = {(1, 2, 2, 4)7; (6, 1, 2, 2, 4)2; (3, 4, 1)2; (3, 3)} The free zones are enlarged by aggregation of two or more smaller zones, which increases the probability to pack other items into that space. This strategy allowed us to pack item 5 in the first block. Item 6 occupied the free space in block 2. Item 7 was packed last. Thereby, the block which was formed by item 3 only is split into 2 new blocks. Our implementation of BL and IBD is based on block structure and thus very simple Dual Local Search (DLS) This algorithm [16] is based on the solution of two dual problems, vertical block structure (VBS) and horizontal block structure (HBS). At first it constructs a VBS, trying to

12 10 minimize the occupied length of the strip, then the algorithm tries to construct the corresponding HBS, but it may not exist, in that case the algorithm obtains a new priority list and starts from the beginning. 3.2 Local search algorithms As usual, a single application of a heuristic (i. e. of a decoder) is viewed as a local search. The repeated application of a heuristic is controlled by a meta-heuristic. For a detailed description of modern meta-heuristics we refer to [21, 1] Genetic Block Algorithm (GBA) In the traditional genetic algorithms (e.g. in GA of [12]) for two-dimensional packing, the genes represent single rectangles and chromosomes represent their sequences (some priority lists). In difference to GA, in the Genetic Block Algorithm GBA, genes are the blocks and chromosomes are the block structures, i. e. lists of form S. As computational experience shows, it is difficult to obtain chromosomes corresponding to feasible packings. Alleles (alternative forms) of genes are obtained by reordering elements in the blocks, and loci of genes indicate their positions in the block structure. GBA is interpreted as an evolutionary process. For that, the operators crossover and mutation are used. The starting chromosome is obtained by (approximately) solving a special one-dimensional cutting stock problem (which is a relaxation of the 2D problem) regarding the additional constraints 1 0 and 2 0. The SVC heuristic for the 1D CSP, proposed in [13], does that job. Before mutation in GBA is applied, we perform a neighborhood search in the neighborhood constructed by SVC. A detailed description of GBA is given in [18] Sequential Value Correction (SVC) A thorough investigation of SVC for 1D cutting has been done in [4, 13]. For the 2D-SPP we propose the following scheme. A block structure (4) corresponding to a feasible solution is needed as a packing code for SVC. The SVC algorithm constructs heuristic values, the so-called pseudo-values v i for i I, corresponding to intuitive perpiece material consumption. The pseudo-values correlate with the item area plus some portion of waste in the blocks containing this item. Then, a new priority list is constructed according to non-increasing ratios v i /(l i w i ). It is given to a decoder (e. g. Sub) to obtain another solution. This means that we first try to pack items with high average material

13 11 consumption. Then SVC constructs new values and so on. The pseudo-value updating rules in SVC are based on vertical block structures of type (4). The algorithm is as follows: Step 1. Initializing priority list. Construct a starting priority list π according to non-increasing lengths of the rectangles, or just accepting the order given in the instance source data. Step 2. Generating a neighboring solution. Apply decoder D with the current priority list π to build a new packing and its vertical block-structure S = {(I j, λ j ) : j J = {1,..., r}}. Step 3. Generating new pseudo-values 3.1. Initialize the pseudo-values by the item areas: v i = l i w i i I Update the values using the block structure: for j = 1 to r do w i v i v i + λ j (W w k ) i I j. k I j w k k I j 3.3. Construct a new priority list π according to non-increasing order of the current relative pseudo-values v i /(l i w i ), i = 1,..., m. Ties are broken randomly (this is important for practical efficiency.) Step 4. Termination criterion If the time limit or another termination criterion is reached, then stop; otherwise return to Step 2. The intuitive SVC principle can be applied in many forms and variations. For example, the above description treats pseudo-values as two-dimensional quantities. However, the direct adaptation of 1D formulas from [13] regarding only item widths produces the same results on the given class of instances Genetic Multiple Method (GMM) In this algorithm [20] simple heuristics are used as genes for placing an item in the free part of the strip. All heuristics are derivations of Bottom-Left varying by additional preference rules for choosing the item to be placed in the current position. A chromosome is represented as a sequence of algorithms, individually for each item. The advantage of the method is that there is no need for a stand-alone decoder.

14 12 4 Numerical experiments Hopper and Turton presented several series of waste-free instances of the 2D-SPP, [10]. We tested the series C and N available in the OR Library [3]. Each series contains 7 instance groups. In series C, each group contains 3 instances. In series N, these are 5 instances per group. Thus, we had 56 instances (21 in C, 35 in N). The optimal values of L are known. Especially, for N we have always L=200. For series C we report the obtained values of L. For series N we show the average values of the material utilization ratio CC. We conducted two experiments: on the decoders and on the metaheuristics. 4.1 Experiment 1: Investigation of decoders, esp. checking of correctness The following single-pass decoders were used: Sub, ISubRec, BL, IBD, DLS, the Improved Polygonal Bottom-Left decoder IBL of Liu and Teng [11] and the Sequence Pairs decoder SP [19]. As input, the decoders obtained the priority list corresponding to the ordering of items in the data files. The results are shown in Tables 1 and 2. We see that the decoders Sub, BL and DLS have the property of correctness for this instance set, whereas for other algorithms no conclusion can be made because they probably need another sequence. Partially, we shall be looking for such a sequence using metaheuristics in the next experiment. As an estimator for the results, let us introduce the following coefficient of optimality CRest: CRest = n k k=1 n + n where δ k = δ k k=1 { 1, if k = 0, 0, if k > 0. (6) k is the absolute difference to the optimal value. The average deviation 1 n n k does not k=1 n account for the number of optimally solved instances. So we added this number, δ k, in k=1 the denominator. Decoder SP finds an optimum only in two cases but the deviations are smaller than with IBD in the groups C6 and C7. The algorithms SP and IBD have similar values of CRest. IBD has rather non-uniform performance, as opposed to SP.

15 13 Table 1: Series C with decoders Inst. m Opt. Sub ISubRec BL IBD SP IBL DLS C C C C C C C C C C C C C C C C C C C C C CRest The base version of decoder DLS currently needs too much time to solve C7-1, C7-2, C7-3 and the class N7. Table 2: Series N with decoders (all results in percent) Instances m Optimal CC, Sub ISub BL DLS N1a-N1e N2a-N2e N3a-N3e N4a-N4e N5a-N5e N6a-N6e N7a-N7e CC ave

16 Experiment 2: Comparison of stochastic local search algorithms The key difference of this second experiment is that the priority list was considered unknown and at the start the items were ordered according to the length, area, etc. The results are given in Tables 3 and 4. The results of Hopper and Turton are given as well. Table 3: Series C with local search algorithms Inst. m Opt. SVC GA GBA ACP TS NS NS GMM E.H. (Sub) (Sub) (IBD) (IBD) (ISub) (ISubRec) C C C C C C C C C C C C C C C C C C C C C CRest As we see, only SVC(Sub) solves all examples optimally. For series C, the remaining methods can be ranged according to the values of CRest as follows: TS(IBD), NS (ISubRec), GBA(IBD), GMM, E.Hopper, GA(Sub), NS (ISub), ACP. Algorithms with block decoders occupy the first three places. For series N, we can range the remaining methods according to the values of CC as follows: GBA(IBD), GMM, ACP, NS(ISub), which highly correlates with series C.

17 15 Table 4: Series N with local search algorithms (results in percent) Instances m Opt. SVC(Sub) NS(ISub) GBA(IBD) GMM ACP N1a-N1e N2a-N2e N3a-N3e N4a-N4e N5a-N5e N6a-N6e N7a-N7e average Conclusions 1. All the waste-free instances of E.Hopper were optimally solved by the simple decoders Sub, BL, DLS (if the optimal priority list is known) and by the local search algorithm SVC(Sub). The working time of the simple decoders is fractions of a second. SVC(Sub) needs less than a second for all examples except C7-2 (CPU Pentium III 900 MHz). Its solution was obtained in 35 seconds. 2. Application of Sub, BL, DLS, and SVC(Sub) to randomly generated non-waste-free examples does not give such good results which is a topic of future research. References [1] E. Aarts and J. K. Lenstra (eds.). Local search in combinatorial optimization. John Wiley & Sons, Chichester, [2] B.S. Baker, E.G. Coffman, and R.L. Rivest. Orthogonal packings in two dimensions. SIAM J. on Computing, 9(4): , [3] J.E. Beasley. OR-Library: distributing test problems by electronic mail. Journal of the Operational Research Society, 41(11): , [4] G. Belov and G. Scheithauer. Setup and Open Stacks Minimization in One- Dimensional Stock Cutting. Dresden University, Preprint MATH-NM To appear in INFORMS J. on Computing. [5] A. Bortfeldt, A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces, Working paper (2004).

18 16 [6] A. Bortfeldt and H. Gehring. Applying tabu search to container loading problems, in: Operations Research Proceedings, Springer, Berlin, [7] M. Dorigo and L.M. Gambardella. Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem, IEEE Transactions on Evolutionary Computation Vol.1. No [8] F. Glover. Tabu search and adaptive memory programming advances, applications and challenges, in: Interfaces in Computer Science and Operations Research P [9] A. Hinxman. The Trim-Loss and assortment problems: a survey, European Journal of Operational Research Vol. 11. P [10] E. Hopper and B.C.H. Turton. A Review of the Application of Meta-Heuristic Algorithms to 2D Strip Packing Problems. Artif. Intell. Rev Vol. 16(4). P [11] D. Liu and H. Teng. An improved BL-algorithm for genetic algorithm of the orthogonal packing of rectangles, European Journal of Operation Research Vol.112. P [12] A. Lodi, S. Martello and D. Vigo. Recent advances on two-dimensional bin packing problems. Discrete Applied Mathematics Vol P [13] E.A. Mukhacheva, G.N. Belov, V.M. Kartak and A.S. Mukhacheva. Linear onedimensional cutting-packing problems: numerical experiments with sequential value correction method (SVC) and a modified branch-and-bound method (MBB), Pesquisa Operacional Vol. 20. No 2. P [14] A.S. Mukhacheva, A.V. Chiglintsev, M.A. Smagin, E.A. Mukhacheva. Genetic Algorithms for Two-Dimensional Packing Based on Mixed Local Search, Information Technologies 2001, No. 9. Appendix (in Russian) [15] E.A. Mukhacheva and A.S. Mukhacheva. Method of Reconstruction for Rectangular Packing, Information Technologies 2000 No. 4. C (in Russian) [16] A.S. Mukhacheva and E.A. Mukhacheva. Local Search Algorithms in Rectangular Packing Using Dual One-Dimensional Cutting Stock Problems, Information Technologies No. 6, C (in Russian) [17] E.A. Mukhacheva and A.S. Mukhacheva. Local Search Methods with Block Structures in Rectangular Packing, Automation and Telemechanics No C (in Russian)

19 17 [18] E.A. Mukhacheva, A.S. Mukhacheva and A.V. Chiglintsev. Genetic Algorithm with Block Structure for Two-Dimensional Packing, Information Technologies No. 11. C (in Russian) [19] H. Murata, K. Fujiyoshi, S. Natake and Y. Kajitani. Rectangle-Packing-Based Module Placement. in: Proc. IEEE/ACM International Conf. on Computer-Aided Design P [20] I.P. Norenkov. Heuristics and their combinations in genetic methods of discrete optimization. Informacionnye Technologii No [21] V.J. Reyward-Smith, L.H. Osman, C.R. Reeves and C.D. Smith, Modern heuristic search methods, John Wiley & Sons, Chichester, [22] P. Schwerin and G. Wäscher. The Bin-Packing Problem: a Problem Generator and Some Numerical Experiments with FFD Packing and MTP, Int. Transactions in Operational Research Vol.4. P