One-Dimensional Heuristics Adapted for Two-Dimensional Rectangular Strip Packing G. Belov, G. Scheithauer, E.A. Mukhacheva
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1 One-Dimensional Heuristics Adapted for Two-Dimensional Rectangular Strip Packing G. Belov, G. Scheithauer, E.A. Mukhacheva Dresden University Preprint MATH-NM , March 1, 2006 Abstract We consider two-dimensional rectangular strip packing without rotation of items and without the guillotine cutting constraint. We propose a single-pass heuristic which fills every free space in a onedimensional knapsack fashion, i.e. considering only item widths. It appears especially important to assign suitable heuristic pseudo-values as profits in this knapsack problem. This simple heuristic improves the results for most of the test classes from the literature, compared to the results of Bortfeldt (2004) and Lesh and Mitzenmacher (2004). Moreover, we describe a simple modification of the Bottom-Left heuristic and call it Bottom-Left-Right. Executing it iteratively with different input sequences generated by the randomized framework BubbleSearch of Lesh and Mitzenmacher (2004), we obtain the best results in some classes with smaller number of items (20, 40). For larger instances, the pseudo-value-based algorithm is the best one in most cases. Keywords: strip packing, heuristics, greedy, stochastic search, knapsack 1 Introduction The Two-Dimensional Rectangular Strip Packing Problem (2D- SPP) (Bortfeldt, 2004, Lesh and Mitzenmacher, 2004) considers a vertical strip of fixed width. The goal is to minimize the height needed to pack a given set of rectangular items. The practical value of the problem comes not only in cutting and packing but also in modeling of scheduling problems. Let us denote W R w = (w 1,..., w n ) R n l = (l 1,..., l n ) R n The width of the strip, The widths of the items i I = {1, 2,..., n}, The lengths (or heights) of the items. The problem is known to be NP-hard. Bortfeldt (2004) describes a genetic algorithm SPGAL ( L stands for layer approach) where solutions themselves are used as genetic information. He shows advantages of this coding type and Technical University of Dresden, capad Ufa State Aviation Technical University, elita@vmk.ugatu.ac.ru
2 2 G. Belov, G. Scheithauer, E.A. Mukhacheva improves the results on many common test classes. His algorithm also works for the options of item rotation and guillotine cutting. Lesh and Mitzenmacher (2004) introduce a generic approach BubbleSearch for iterative greedy-like heuristics. For 2D-SPP, they use the Bottom-Left (BL) heuristic (Baker et al., 1980) which inputs a sequence of items and packs each one to the most bottom-left position according to that sequence. Lesh and Mitzenmacher observe that some certain permutations of items, e.g. according to non-increasing height or area, give good results and they propose to randomly modify these basic sequences. Bortfeldt and Lesh and Mitzenmacher are unaware of each other s work so we compare their results. Lodi et al. (2002) present an overview of two-dimensional packing. For 2D-SPP, they describe the layer-based heuristics next-fit decreasing, first-fit decreasing, and best-fit decreasing. These pack the items in layers across the strip which is in fact a one-dimensional principle producing 2-stage guillotine layouts. Bortfeldt (2004) uses the same, in fact the layers are the alleles of the genes in SPGAL, but he sophisticates a lot of effort to properly construct and post-optimize the layers to obtain good results also for non-guillotine cutting. In this paper, we describe a new principle to construct a packing, to our knowledge coming from russian origins (Mukhacheva and Mukhacheva, 2004), called Substitution (Sub). It fills free portions of the cross-sections of the strip in a greedy one-dimensional fashion considering only item widths. In this work, we propose the idea to fill the free portions by solving a 1D knapsack problem (SubKP). Moreover, we investigate heuristic principles to assign item profits in these knapsack problems. Here we adapt the method sequential value correction (SVC), already well-known e.g. for 1D stock cutting (Mukhacheva et al., 2000, Belov and Scheithauer, 2003). Moreover, we develop another idea from the same russian source (Mukhacheva and Mukhacheva, 2004): modify Bottom-Left to heuristically decide on left or right positioning of each item in its lowest possible height. We call this algorithm Bottom-Left-Right (BLR). It is well known (Baker et al., 1980) that BL is unable to obtain an optimal solution, even exhaustively trying all item permutations, for some instances where optimum is not waste-free. We show the same for BLR and Sub. One principle capable of representing all possible layouts is sequence pairs (Murata et al., 1995). With two sequences, we can exactly specify mutual position of the items. However, sequence pairs inherit much symmetry. Moreover, Murata et al. (1995) consider a slightly different problem, namely packing with the goal to minimize occupied area, and they do not propose any principle to generate feasible solutions for a container with fixed width.
3 One-Dimensional Heuristics for 2D Rectangular Packing 3 The framework of Fekete and Schepers (2004) for exact algorithms for higher-dimensional packing uses a general principle to specify moredimensional layouts by intersections of item projections on the coordinate axes. This principle covers all feasible layouts and avoids some symmetries. The authors propose a branch-and-bound framework to optimally decide whether a given set of items can be packed into a fixed-size container. They do not describe any special heuristic to construct intersection graphs of feasible solutions, the search completely relies on the branching scheme. Clautiaux et al. (2005) constructed a two-step exact algorithm which tries to determine the x-coordinates first and then, if bounds are not violated, starts an internal enumeration to find the complete solution. The results reported are comparable with (Fekete and Schepers, 2004). The efficiency of these frameworks suggests their comparison with heuristics. Unfortunately, the authors give no results for 2D-SPP. The exhaustive algorithm of Lesh et al. (2004) to find perfect (waste-free) packings needs about 450 seconds for the N3 instances of Hopper (2000) with 29 items; series N4 with 49 items was not tried at all. Other exact approaches, to our knowledge, behave similarly. In the next section we introduce the one-dimensional cutting stock problem (1D-CSP, cf. Belov and Scheithauer, 2003) and define additional conditions for 1D-CSP solutions to represent those of 2D-SPP. Then we describe our SubKP, SVC, and BLR heuristics followed by computational results. 2 Representing 2D-SPP as a Special 1D-CSP Let us consider 1D-CSP instances defined by the following data: (W, w, l), where W is the size of every stock bin, w is the vector of product sizes, and l is the vector of order demands. In the classical 1D-CSP, at least l i items of product type i have to be obtained, i I. The goal is to minimize the number of stock bins used. Usually, a solution is represented by a set A A = {a Z n + : wa W, a l} of cutting patterns. The i-th element a i is the number of items of type i in a. To define a solution using certain patterns a A, we need their usage frequencies x a Z +. In order for 1D-CSP solutions to represent those of 2D-SPP, the former have to satisfy further conditions. First, we demand exactly l i items of each product type i I. Additionally, the solutions have to satisfy Contiguity of items: all items of one type be located in consecutive bins;
4 4 G. Belov, G. Scheithauer, E.A. Mukhacheva Constant location: all items of one type have the same position in each bin. To simplify the formal description of these properties, we use the notions of slice and slice structure (Mukhacheva and Mukhacheva, 2004) which are somewhat related to cross-sections defined in (Healy and Creavin, 1997). Definition 1. A slice corresponds to the two-dimensional shape obtained by replicating a certain cross-section pattern a number of times in the length direction. A slice is represented by the list s = (s.a; s.p; s.x) with s.a {0, 1} n s.p R n + s.x R + The vector representing a cross-section pattern. The vector of item positions in it (in the width direction). The length of the slice, i.e. the number of replications of pattern (a, p). Definition 2. The slice structure corresponding to a 2D-SPP solution is a list (s 1 ;... ; s k ) of slices satisfying j (s j.a i )(s j.x) = l i i (item length) { } maxj<j0 {s s j0.a i min j.a i } i, j max j>j0 {s j.a i } 0 slice 4 (contiguity) slice 3 slice 2 s j.p i = const j : s j.a i > 0, i slice 1 (constant location) We want to distinguish the notion of slice from the notion of layer used e.g. in (Bortfeldt, 2004), where items are contained completely inside a layer. Now we proceed to algorithms based on slice structures. 3 The Knapsack Substitution Heuristic (SubKP) Algorithm Sub (Mukhacheva and Mukhacheva, 2004) constructs a 2D-SPP solution slice after slice. It fills empty spaces in each slice considering only item widths, selecting the items by the 1D greedy heuristic. A similar algorithm from (Mukhacheva and Valeeva, 2000) fills every hole by solving a subset-sum problem with item widths; then, items are ordered by non-increasing heights. Inspired by Sub and the 1D-CSP heuristic SVC (Mukhacheva et al., 2000, Belov and Scheithauer, 2003), we propose the following modification SubKP of Sub. Namely, to fill the empty spaces solving a one-dimensional knapsack problem. Especially for waste-free instances, the following idea seems very sound: as item profits for the knapsack problem,
5 One-Dimensional Heuristics for 2D Rectangular Packing 5 select item widths and slightly modify them. This gives something very close to a subset-sum problem. In the next subsection we describe how we select these modifying pseudo-values. Figure 1 gives a formal description. As input, the algorithm obtains some values d i of the items. To solve a knapsack problem in the procedure FillHole (Figure 2), we transform these values into item profits. In fact, as profits we set item widths and add to them those values d i normalized and scaled by a random number generated according to truncated Pareto distribution. What remains yet is to order the packed items in the free space which is also done in FillHole. Inspired by Mukhacheva and Mukhacheva (2000), we accept the following heuristic rule: either we sort the items by non-increasing lengths starting from the higher wall of the hole or we build a tower. An example showing the advantage of sorting the items is shown in Figure 3. Then, the sequence may be permuted using the BubblePermute neighborhood construction procedure from (Lesh and Mitzenmacher, 2004) described in Figure 4. We slightly simplified the BubblePermute procedure by doing only one selection pass for each element (the for-loop in j). The algorithm has a pseudo-polynomial complexity. In Section 5 we show that the set of packings obtainable by SubKP is a true subset of BLR packings. 4 Sequential Value Correction (SVC) A thorough investigation of SVC for 1D cutting has been done in (Mukhacheva et al., 2000, Belov and Scheithauer, 2003). The idea is to fill every bin by solving a knapsack problem while assigning some heuristic pseudo-values to the items. These represent a kind of intuitive per-item stock consumption in previous solutions. Especially if an item occurs in the end of the solution, it might produce a lot of waste. This suggests to pack it earlier in subsequent solutions. The same might apply to 2D-SPP. When a long item is packed in the end, it may rise high having a lot of free space around. In (Belov and Scheithauer, 2003), we proposed to make pseudo-values overproportional to item lengths because smaller items create good packings in the beginning leaving no good combinations for the end of a solution. Here we applied the same idea. However solving a two-dimensional problem, our pseudo-values are correlated to item areas. Moreover, overproportionality appeared most effective when related to the item portions in separate slices. An iteration of the algorithm works as follows (Figure 5): given the slice structure of a solution, the pseudo-values are updated and given as input to a heuristic, e.g. SubKP. The overproportionality parameter slicep is generated as a normally distributed random number.
6 6 G. Belov, G. Scheithauer, E.A. Mukhacheva Algorithm SubKP input: W, w = (w 1,..., w n ), l = (l 1,..., l n ) // 2D-SPP instance data d = (d 1,..., d n ) // pseudo-values iter // iteration number output: ss // the slice structure of the constructed packing variables: vscale, s, s 1, p 2, p Tower, p Permute, f Permute, p InvertHole, p bs INITIALIZATION OF VARIABLES: vscale 10 RndUniform( 0.8, 0.5) ; // a scaling factor for pseudo-values if RndUniform(0, 1) < 0.01 or 1 = iter // at the first run and sometimes later: p 2 RndUniform(0, 0.4); // set probability of towers for the run if p 2 > 0 then p Tower RndUniform(0, 1) / p 2 ; // p Tower is... else p Tower 0; //... the prob-ty of tower for each hole if p Tower > 1 then p Tower 0; if RndUniform(0, 1) < 0.01 or 1 = iter // at the first run and sometimes later: p InvertHole RndUniform(0, 0.05); // prob-ty to invert hole ordering if RndUniform(0, 1) < 0.01 or 1 = iter // at the first run and sometimes later: p Permute RndUniform(0.8, 1); // prob-ty to permute all holes in a run f Permute boolean(rnduniform(0, 1) < p Permute ); p bs RndUniform(0.6, 0.9); // the selection prob-ty for hole perm. ss (EmptySlice(W )); // init the slice structure MAIN PART: repeat s ss.lastslice; for each hole h in s FillHole(h, d, vscale, p Tower, p InvertHole, f Permute, p bs ); s.length min{l i : i s}; s 1 s; for each item i s 1 l i l i s.length; if 0 = l i IntegrateNewHole(s 1, i); // just unite neighboring holes ss (ss; s 1 ); until 0 = l i i ss.deletelastslice; END. Figure 1: Algorithm SubKP
7 One-Dimensional Heuristics for 2D Rectangular Packing 7 procedure FillHole(h, d, vscale, p Tower, p InvertHole, f Permute, p bs ) variables: I 0 // the set of unpacked items i with w i Width(h) dw // item profits for the knapsack problem fhigherleft, fshoveleft, ftower; begin /// CHOOSE THE FILLING ITEMS by solving a binary knapsack problem: dw i w i + d i vscale min{w i /d i : i I 0 } i I 0 ; h Solve 1D BinKP(Width(h), (w i, dw i ), i I 0 ); /// ORDER AND PLACE THE SELECTED ITEMS: fhigherleft boolean(if the left wall of h is higher than the right one); fshoveleft fhigherleft; ftower boolean(h.numberofitems > 2 and RndUniform(0, 1) < p Tower ); if RndUniform(0, 1) < p InvertHole // invert fhigherleft with small probability: fhigherleft not fhigherleft; if RndUniform(0, 1) < p InvertHole // the same for fshoveleft: fshoveleft not fshoveleft; h.sortitemsbydecreasingheight; // (from left to right) if not (fhigherleft or ftower) h.reverse Item Order; if ftower // construct a tower using a non-increasing sequence: Iterate through the sequence and move each even-numbered item to the beginning. if fhigherleft and the last item is higher than the first one or vice-versa h.reverse Item Order; if f Permute BubblePermute(h.Items, p bs ); if not fshoveleft h.shoveitemsright; // left is the default. end. Figure 2: Procedure FillHole of SubKP
8 8 G. Belov, G. Scheithauer, E.A. Mukhacheva Figure 3: An example where item sorting in FillHole is advantageous: on the left, item heights decrease starting from the wall, giving a better packing procedure BubblePermute(s, p bs ) begin for i = 1 to size(s)-1 for j = i to size(s) if RndUniform(0, 1) < p bs if i j remove s j from s and insert it before s i ; break; // continues with next i end. Figure 4: Procedure BubblePermute for sequence modification
9 One-Dimensional Heuristics for 2D Rectangular Packing 9 Algorithm SVC(H) input: W, w = (w 1,..., w n ), l = (l 1,..., l n ) // 2D-SPP instance data H(W, w, l, d, iter) // a heuristic such as SubKP parameters: µ 1 = 2.1, σ 1 = 0.1 // parameters of the normal distribution of slicep p 1 = 0.1 // probability to regenerate slicep in an iteration T 2 = // value reset period output: The best-found solution variables: d = (d 1,..., d n ) // pseudo-values ss // the slice structure of a packing returned by the heuristic slicep, qwaste MAIN PART: ss (); // init ss to an empty list next iter value reset = 1; for iter = 1 to MaxIter if iter = next iter value reset next iter value reset += RndUniform(1000, 19000); d i w i l i i; // (re)init pseudo-values at start or later if RndUniform(0, 1) < p 1 or 1 = iter slicep RndNormal(µ 1, σ 1 ); // regenerate slicep for this iteration for each slice s ss qwaste s.waste/s.occupied; // the relation of waste and occupied parts for each item i s // updating pseudo-values: d i d i + qwaste (w i s.length) slicep ; ss ExecuteHeuristic(H(W, w, l, d, iter)); END. Figure 5: Algorithm SVC with some heuristic H
10 10 G. Belov, G. Scheithauer, E.A. Mukhacheva 5 The Bottom-Left-Right Heuristic (BLR) The Bottom-Left heuristic can be viewed as a two-dimensional version of the one-dimensional first-fit heuristic. In both cases, we take the next item from the priority list and place it in the earliest possible position. But in BL we have to consider the second dimension. An improved version of the BL heuristic was proposed in (Mukhacheva and Mukhacheva, 2004). It employed a strategy to enlarge unfilled zones. However, the description was not clear and we propose a simplified rule. Each object is placed in the lowest possible level and in the left-most available hole. But in that hole it is justified left or right as follows, Figure 6. We analyze how much unused area would appear immediately on both sides of the object. We place the object left-most or right-most so that to create larger unused area on the opposite side, with the hope to have a larger probability to place another object(s) in this area. However, this rule is reverted each time with probability p InvertLR and p InvertLR is selected for each run randomly uniformly from [0, 0.03). To construct item orderings for BLR, we used a simplified BubblePermute procedure from (Lesh and Mitzenmacher, 2004), see Figure 4 above. Here we set p bs = 0.6 and base sequences are sorted according to non-increasing height, area, width, or perimeter, with probabilities 0.4, 0.2, 0.2, and 0.2, respectively. Concerning the implementation, for placing the items in a partial packing without the bottom-left property, we used a slice-structure-based procedure similar to (Healy and Creavin, 1997). The whole algorithm BubbleSearch(BLR) or, shortly, BS(BLR) works iteratively, similar to (Lesh and Mitzenmacher, 2004), as follows (Figure 7): in each run, we generate an input sequence, generate p InvertLR and execute BLR. There is no information interchange between iterations as it is in SVC. There are instances for which BL is unable to find an optimum whatever sorting of items it is given. Such instance must have non-waste-free optimal solutions. In Figure 8, instance a) was given in Baker et al. (1980). Instance b) is even BLR-unsolvable. Observe that item sets (7,10), (7,8,9), (4,5,6), and (1,2,3) must be in the same rows, i.e. have non-empty intersections in the horizontal projection. Item sets (1,4,7), (2,5,8,10), and (3,6,9,10) must be in the same columns. If another optimal solution exists then it differs by permutations of columns (2,5,8) and (3,6,9) and/or rows (1,2,3) and (4,5,6) with block (7,8,9,10). None of these permutations fits into the optimal container. To simplify the proof, we might forbid these permutations by adding items similar to 10, as in Figure 8, c). The instances in Figure 8, b) and c) are unsolvable also by SubKP. Moreover, we have the following
11 One-Dimensional Heuristics for 2D Rectangular Packing 11 Algorithm BLR input: W, w = (w 1,..., w n ), l = (l 1,..., l n ) // 2D-SPP instance data π = (π 1 ;... ; π n ) // item sequence let p InvertLR RndUniform(0, 0.03); // probability to invert each justification for i = 1 to n Find the bottom-most and then left-most hole able to contain item π i ; Compute the areas a L and a R of the free space arising immediately to left and to the right of the item, when justifying it right-most or left-most, respectively; if a L > a R, decide to justify the item right-most, otherwise left-most; Invert this decision with probability p InvertLR ; Figure 6: Algorithm BLR Algorithm BubbleSearch input: W, w = (w 1,..., w n ), l = (l 1,..., l n ) H(W, w, l, π) while not (stop) π (select one of the four base sequences); BubblePermute(π, p bs = 0.6); ExecuteHeuristic(H(W, w, l, π)); // 2D-SPP instance data // a sequence-based heuristic such as BL or BLR Figure 7: Algorithm BubbleSearch (BS) with some sequence-based heuristic The dimensions of piece 1 should be increased by a small ɛ a) b) c) Figure 8: BL-unsolvable instances, a) is from (Baker et al., 1980); b) and c) are also BLR-unsolvable
12 12 G. Belov, G. Scheithauer, E.A. Mukhacheva Proposition 1. The set of solutions obtainable by SubKP is in general a true subset of BLR packings. Proof. Note that both SubKP and BLR produce packings where the position of each item is bottom-left or bottom-right justified. Consider a packing produced by SubKP, especially the order of items successively packed. Now, applying BLR to the corresponding item ordering can produce the same packing, if the right or left justification is chosen randomly. On the other hand, we can easily construct a BLR packing not obtainable by SubKP. For example, in SubKP there can be no free space on the bottom line of an item in the justification direction. Similarly one can prove that the solution set of Sub (Mukhacheva and Mukhacheva, 2004) where hole fillings are justified to the left is a true subset of BL solutions. Moreover, the method of filling the lowest free space in SubKP, namely always to the full width, can exclude good solutions. Despite these theoretical concerns, SVC(SubKP) performs better than BLR in most test classes because SubKP packs items in groups. 6 Implementation To solve the 1D binary knapsack problems in SubKP, we used the algorithm combo of Martello et al. (1999), available from the website of D. Pisinger ( pisinger/). The program is designed for integer item profits. However, we have real-valued profits. Thus, we scaled all profits dw i so that the simple upper bound max{dw i W /w i : i I 0 } on the solution value was 10 6, where W is the length of the knapsack, and rounded them down. We discovered that the ordering in which items are given to combo is important. For example, if we accept the ordering from the data file where a BL-optimal sequence is given (this is the case for Hopper and Turton instances), then combo fills the bottom of the container with the optimal items which leads to an unclean experiment. Thus, we sorted the items according to the sequences produced for BLR. Random number generation according to approximate normal distribution was done by the so-called 12 uniforms method. The values of algorithm parameters were chosen after some experimentation. For example, for waste-free instances, it is advantageous to select a smaller factor for the pseudo-values when computing the knapsack profits. We applied some compromise values which seemed good on average.
13 One-Dimensional Heuristics for 2D Rectangular Packing 13 7 Numerical Experiments The experiment was run on Linux workstations with AMD Opteron TM 250 CPU (2.4 GHz), 2GB of memory. The software was compiled in GNU C The algorithm SPGAL of Bortfeldt (2004) was executed on a 2GHz Pentium with 16 minutes time per instance (10 runs 95 sec. each). The algorithm BubbleSearch(BL) of Lesh and Mitzenmacher (2004) had 1 hour per instance, thus we chose our time limit according to Bortfeldt. The SPECint2000 ratio is about 1700 for Opteron 250 and about 650 for Pentium 2GHz. Thus, we allowed 6 minutes = 360 seconds of total CPU time for each instance. Hopper and Turton presented several series of waste-free instances of the 2D-SPP, series N and T (Hopper, 2000) and series C (Hopper and Turton, 2000). Each series contains 7 instance groups. In series C, each group contains 3 instances. Series N and T have 5 instances per group and optimal solution sizes are The results are presented in Table 1. The columns of the table have the following meaning. inst is the instance group, n is the average number of items, Opt is the optimal value for each instance. Section SVC(SubKP) shows iter best, iter all, and t best the averages of the number of iterations and time for the best/all solutions. For the upper bound UB, we show a time row and the percentage gap for the best solution. For BS(BLR), we show only the objective value and gap for the time limit of 360 seconds. In our implementation, BS(BLR) performs nearly triple number of iterations per time unit compared to SVC(SubKP). The minimal gaps are presented in bold. We see that SVC improves the results for all classes and BS(BLR) for some classes. Another standard test set is that of Berkey and Wang (1987), classes C01 C06 in Tables 2, 3, which is usually considered together with the set of Martello and Vigo (1998), classes C07 C10 in Table 3. Here we compare the results also with Iori et al. (2002). The instances are very well described in (Lodi et al., 1999). We see that the only class where SVC(SubKP) does not improve the average value is C10. Its instances were generated with 70% of items with both sizes smaller than half the container width. BS(BLR) has the best values on some classes with smaller number of items. In the column LB we see the lower bound produced by the bar relaxation from (Scheithauer, 1999) which always dominated the bound given in (Bortfeldt, 2004). The third set is from (Martello et al., 2003) whose objective values are in the column UB-prev. SVC(SubKP) improves the results on all the classes except NGCUT01 12, however the values of BS(BLR) match those of (Martello et al., 2003) in this class. Note that these instances have up to 22 items. The column lb best shows the maximum of the lower bounds produced by the contiguous 1D relaxation (Martello et al., 2003) and the bar relaxation
14 14 G. Belov, G. Scheithauer, E.A. Mukhacheva (Scheithauer, 1999). 8 Conclusions We proposed the effective modifications SubKP and BLR of the heuristics Sub and BL, respectively. The iterative framework SVC(SubKP) improves the results for most of the test classes from the literature and BS(BLR) does it for some smaller instances. However, both heuristics are in some way restricted because they cannot produce optima for certain instances. Future research could try to overcome this drawback.
15 One-Dimensional Heuristics for 2D Rectangular Packing 15 References B. S. Baker, E. G. Coffman, and R. L. Rivest. Orthogonal packings in two dimensions. SIAM J. on Computing, 9(4): , G. Belov and G. Scheithauer. Setup and open stacks minimization in onedimensional stock cutting. Preprint MATH-NM , Institute of Numerical Mathematics, Technische Universität Dresden, To appear in INFORMS J. on Computing. J. O. Berkey and P. Y. Wang. Two dimensional finite bin packing algorithms. J. Oper. Res. Soc., 38(5): , A. Bortfeldt. A genetic algorithm for the two-dimensional strip packing problem with rectangular pieces. Working paper, Fernuniversität Hagen, Germany, To appear in EJOR. F. Clautiaux, J. Carlier, and A. Moukrim. A new exact method for the orthogonal packing problem. Technical report, Université de Technologie de Compiègne, France, Under revision for EJOR. S. P. Fekete and J. Schepers. A combinatorial characterization of higherdimensional orthogonal packing. Mathematics of Operations Research, 29 (2): , P. Healy and M. Creavin. An optimal algorithm for rectangle placement. TR UL-CSIS-97-1, Department of Computer Science and Information Systems, University of Limerick, Ireland, E. Hopper. Two-dimensional packing utilising evolutionary algorithms and other meta-heuristic methods. PhD thesis, Cardiff University, E. Hopper and B. Turton. An empirical investigation of metaheuristic and heuristic algorithms for a 2D-packing problem. European Journal of Operational Research, 128(1):34 57, M. Iori, S. Martello, and M. Monaci. Metaheuristic algorithms for the strip packing problem. In P. Pardalos and V. Korotkich, editors, Optimizations and Industry: New Frontiers. Kluwer Academic Publishers, N. Lesh, A. McMahon, J. Marks, and M. Mitzenmacher. An exhaustive approach to 2D strip packing. Information Processing Letters, 90(1):7 14, 2004.
16 16 G. Belov, G. Scheithauer, E.A. Mukhacheva N. Lesh and M. Mitzenmacher. BubbleSearch: A simple heuristic for improving priority-based greedy algorithms. Technical report, Mitsubishi Electric Research Labs, A. Lodi, S. Martello, and D. Vigo. Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems. INFORMS Journal on Computing, 11(4): , A. Lodi, S. Martello, and D. Vigo. Recent advances on two-dimensional bin packing problems. Discrete Applied Mathematics, 123: , S. Martello, M. Monaci, and D. Vigo. An exact approach to the strip-packing problem. INFORMS Journal on Computing, 15(3): , S. Martello, D. Pisinger, and P. Toth. Dynamic programming and strong bounds for the 0-1 knapsack problem. Management Science, 45: , S. Martello and D. Vigo. Exact solution of the finite two-dimensional bin packing problem. Management Science, 44: , A. Mukhacheva and A. Valeeva. Dynamic enumeration for two-dimensional packing. Information Technologies, (5):30 37, In Russian. E. Mukhacheva, G. Belov, V. Kartak, and A. Mukhacheva. Linear onedimensional cutting-packing problems: numerical experiments with sequential value correction method (svc) and a modified branch-and-bound method (mbb). Pesquisa Operacional, 20(2): , E. Mukhacheva and A. Mukhacheva. Method of reconstruction for rectangular packing. Information Technologies, (4):30 36, In Russian. E. Mukhacheva and A. Mukhacheva. Local search methods with block structures in rectangular packing. Automation and Telemechanics, (2): , In Russian. H. Murata, K. Fujiyoshi, S. Natake, and Y. Kajitani. Rectangle-packingbased module placement. In Proc. IEEE/ACM International Conf. on Computer-Aided Design, pages , G. Scheithauer. LP-based bounds for the container and multi-container loading problem. Int. Trans. Opl. Res., 6: , 1999.
17 One-Dimensional Heuristics for 2D Rectangular Packing 17 Table 1: Results for the waste-free instances of Hopper (2000) and Hopper and Turton (2000) SVC(SubKP) UB, time row at sec.: BS(BLR) Other alg. inst n Opt iterbest iterall tbest Gap% UB Gap% SPGAL SVC(BL) 3600s C C C C C C C N N N N N N N T T T T T T T
18 18 G. Belov, G. Scheithauer, E.A. Mukhacheva Table 2: Results for the instances of Berkey and Wang (1987), classes C01 C05 SVC(SubKP) UB, time row at seconds: BS(BLR) Other alg. Inst n iterbest iterall tbest UB SPGAL BS(BL) Iori et al. LB C Ave C Ave C Ave C Ave C Ave
19 One-Dimensional Heuristics for 2D Rectangular Packing 19 Table 3: Results for the instances of Berkey and Wang (1987), Martello and Vigo (1998), classes C06 C10 SVC(SubKP) UB, time row at seconds: BS(BLR) Other alg. Inst n iterbest iterall tbest UB SPGAL BS(BL) Iori et al. LB C Ave C Ave C Ave C Ave C Ave
20 20 G. Belov, G. Scheithauer, E.A. Mukhacheva Table 4: Results for the instances of Martello et al. (2003) Instance n lb best UB-prev UB-SVC UB-BS(BLR) BENG BENG BENG BENG BENG BENG BENG BENG BENG BENG Geom. mean: CGCUT CGCUT CGCUT Geom. mean: GCUT GCUT GCUT GCUT Geom. mean: HT HT HT HT HT HT HT HT HT Geom. mean: NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT NGCUT Geom. mean: Arithm. mean: Geom. mean:
21 One-Dimensional Heuristics for 2D Rectangular Packing 21 List of Figures 1 Algorithm SubKP Procedure FillHole of SubKP An example where item sorting in FillHole is advantageous: on the left, item heights decrease starting from the wall, giving a better packing Procedure BubblePermute for sequence modification Algorithm SVC with some heuristic H Algorithm BLR Algorithm BubbleSearch (BS) with some sequence-based heuristic BL-unsolvable instances, a) is from (Baker et al., 1980); b) and c) are also BLR-unsolvable List of Tables 1 Results for the waste-free instances of Hopper (2000) and Hopper and Turton (2000) Results for the instances of Berkey and Wang (1987), classes C01 C Results for the instances of Berkey and Wang (1987), Martello and Vigo (1998), classes C06 C Results for the instances of Martello et al. (2003)
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