Abstract— In this paper we consider the two dimensional strip

and bin packing problem with guillotine cuts. The problem consists in packing a set of rectangular items on one strip of width W and infinite height or in bins of width W and height H. The items packed without overlapping, must be extracted by a series of cuts that go from one edge to the opposite edge (guillotine constraint). To solve this problem, we proposed tow heuristics, BSHF (Best Shelf Heuristic Filling) and NSHF (Non-Shelf Heuristic Filling). We compare the two heuristics with other heuristics from the literature. Computational results show that the two algorithms are complementary and they outperform the other algorithms in most of the instances.

Index Terms— Strip packing, guillotine cuts, heuristic.

I. INTRODUCTION HE two-dimensional strip packing problem (2SP) and bin packing problem (2BP) are a well-known combinatorial optimization problems. They have several industrial

applications like the cutting of rolls of paper or textile material. Consider a set of n rectangular items. Each item has a width wi and a height hi (i ∈ {1, 2,..., n}). The 2SP consists in packing all the items in a strip of width W and infinite height. The dimensions of the items and the strip are supposed integers. The objective is to minimize the total height used to pack the items without overlapping. The 2BP consists in packing all the items in identical bins of width W and height H. The objective is to minimize the number of used bins to pack all items. The orientation of items is assumed to be fixed, i.e., they cannot be rotated. These problems are NP-hard in the strong sense since the special case where all items have the same height is equivalent to the one dimensional bin packing problem. We consider the n items as objects of size (wi, i = 1 . . . n) and the bins capacity is W. The problem is to pack all objects in the minimum number of bins. The one dimensional bin packing problem is proved to be strongly NP-hard (see M. R. Garey and D. S.

ICD-LOSI, (CNRS FRE 2848) Université de Technologie de Troyes France {abdelghani.bekrar, imed.kacem}@utt.fr

Johnson (1979) and S. Martello and al., (2003)). An additional constraint considered in this paper is the guillotine cut: All items must be extracted by cuts that go from one edge to the opposite edge. Figure 1 shows a guillotine pattern where all items can be extracted by guillotine cuts. When items cannot extracted by guillotine cuts, the pattern is called non-guillotine as shown in Figure 2.

Most of papers considering the 2SP and 2BP problems are approximation algorithms. Fernandez de la Vega and V. Zissimopoulos (1998), N. Lesh et al.,(2003), C. Kenyon and E. Remila (1996) and D. Zhang et al., (2006) presented approximation algorithms for the strip packing problem, whereas A. Gomes and J. F. Oliveira (2006), A. Bortfeldt (2006) used meta-heuristics. E. Hopper and B. Turton (2001) provided an overview on metaheuristic algorithms applied to 2D strip packing problem. Chung et al., Chazelle (1981), Lodi et al. (1999) proposed heuristics for the 2BP problem. Lodi et al. (2002) presented a survey on two-dimensional packing problems.

Some papers used exact algorithms to solve 2SP and 2BP

A Comparison study of heuristics for solving the 2D Guillotine Strip and Bin Packing Problems

Abdelghani Bekrar and Imed Kacem

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Figure 2. Non-guillotine pattern.

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Figure 1. Guillotine pattern

978-1-4244-1672-1/08/$25.00 ©2008 IEEE.

problems (Hifi (1998), Martello et al. (1998), Martello et al. (2003), Cui et al. (2006), Bekrar et al. (2006). Cintra et al. (2007), Bekrar et al. (2007)). In this paper we study the two dimensional strip and bin packing problem with guillotine cut. We propose two heuristics to solve the problem. In Section 2 we present the heuristic Best Fit Decreasing Height and our first heuristic BSHF. In Section 3, we explain our second heuristic NSHF. The comparative study is presented in section 4, we compare our algorithms to other literature heuristics for bin packing and strip packing problems. Finally, we discuss some perspectives of our work.

II. EXISTING HEURISTICS FOR THE 2D GUILLTINE PACKING PROBLEM

A. Best Fit Decreasing Height Heuristic (BFDH) [17]

The BFDH heuristic is one of the level algorithms. The set of items is sorted by non-increasing height. The first item in the list (the tallest one) is then packed, left justified, in the bottom-left position of the strip. A first level is then created with a height equal to the height of the packed item. The next item is packed, left justified, on that level, among those that can accommodate it, for which the residual horizontal space is a minimum. If no level can accommodate it, a new one is created.

B. Best Shelf Heuristic Filling (BSHF) [3] The SHF heuristic is a generalization of the well-known

two dimensional level algorithm. The main idea of this algorithm is to well exploiting the non-used area in each shelf. It makes it possible to pack items one over the other in the same shelf, which is not permitted in the level algorithms. This packing is released within the respect of the guillotine constraint. For this purpose, SHF uses the definition of available rectangle: the rectangle which has the left-down corner as an available point. This latter can be the down-right or the top-left corner of an item already packed.

Items are sorted in non-increasing order of heights. The first item (the tallest one) is packed into the first available rectangle (the lowest). The leftmost item initializes a shelf with a height equal to the height of this item. After every item-packing, the set of available rectangles is updated in order to maintain the guillotine constraint. The update procedure consists in reducing the dimensions of available rectangles which overlap with the packed items. The item-packing creates two new available rectangles. This process is repeated until the last item is packed.

For most of the instances tested, this algorithm generally gives better results compared with other heuristics used for the problem of packing and cutting in two dimensions with guillotine cuts. The waste rate is estimated at 11%. However, it was noticed that for some particular instances, the algorithm is less efficient (especially when one has flattened items and lengthened items). In order to improve the performance of the

algorithm, we proposed the following improvements in [3]: • Modification of sorting items rule: For the above mentioned

instances (of which the items are flattened and/or lengthened), the sorting of the items in decreasing order of area (SHFDA) or in decreasing order of width (SHFDW) give better results than the basic algorithm (SHF).

• Packing items in the best position SHF-BF (Best Fit): SHF places the current item in the first available rectangle which can contain it (the lowest). In [3], we used the Best fit rule to place the items: Among available rectangles, SHF-BF seeks the available rectangle which can contain the current item and minimizes the residual area. In Figure 3, we present an example where the packing of items by Best Fit rule gives an optimal solution, which is not the case when we use SHF.

• The update of the list of the available rectangles: The aim of the updating is to maintain the guillotine constraint after each iteration. Thus, we proposed another procedure to update this list.

III. A NEW HEURISTIC FOR SOLVING THE 2D STRIP AND BIN PACKING PROBLEM

In this section we present a new heuristic that is able to solve the 2SP and 2BP with guillotine cuts. Unlike the previous heuristics (BSHF and BFDH), the items are not packed in levels.

The items are sorted in different order (non-increasing height, non-increasing width, non-increasing area …). The items are packed then in available rectangle like in BSHF. However the set of available rectangles is not updated in the same manner. When the current item can fill in an available rectangle, we check if the obtained pattern is guillotine using the procedure proposed by Ben Messaoud et al., (2003). If this pattern is guillotine, then the packing is validated and the next item is treated, otherwise, we try with another available rectangle. During the packing, no shelf is created, hence the name of the algorithm: Non-Shelf Heuristic Filling.

The heuristic is adapted for the two dimensional bin packing problem, when the current item cannot be packed in the open bins, a new bin is initialized.

A. Update the set of available rectangles The updating procedure in the BSHF heuristic aims to

a) SHF needs two bins to pack items b) SHFBF needs only one bin

Figure 3. Comparison between SHF and SHFBF

maintain the guillotine propriety of the patterns. In the NSHF heuristic, the updating procedure aims to “correct” the dimension of the available rectangles that are in overlapping with the packed item.

Let Ri(xi, yi, wi, hi) be an available rectangle that has a width wi and a height hi and position (xi, yi). The bottom-left point coordinates of Ri are xi and yi. Let an item p with a width wp and a height hp that will be packed in the position (xp, yp). We distinguish two possibilities:

1. Case 1: x p≥ xi, the item p is located in the right side of Ri. If p overlapped with Ri (i.e., yp+hp > yi) then the width of Ri is reduced (Figure 4.a): Ri(xi, yi, xp-xi, hi).

2. Case 2: xp ≤ xi, the item p is located in the left side of Ri. If p overlapped with Ri (i.e., xp+wp>xi) then the height of Ri is reduced (Figure 4.b): Ri(xi, yi, wi, yp - yi).

IV. COMPUTATIONAL RESULTS To evaluate the performance of the new heuristic, we have tested some literature instances. BSHF and NSHF are compared to the algorithms proposed by Lodi et al. [1999], Hifi [1998] and Hifi [1999b]. The algorithms were coded in C++ language in Linux environment.

A. Results for the 2SP problem The experimentations for the 2SP problem are carried out on the instances of Hifi [1998] and Hifi [1999b]. Those instances are of different sizes. We compared the following heuristics: BFDH, BSHF, NSHF, ISH (proposed in Hifi [1998]) and the best of Hifi’s heuristics proposed in Hifi [1999] (FIA, SIA, HC/FIA, HC/SIA).

Table 1, shows the results obtained by the different

heuristics on instances of little and medium sizes. For each problem, Table 1 gives:

• Problem name, number of items n and the width of the strip W,

• Lmin: the lower bound calculated by Hifi [1998],

• NSHF, BSHF, ISH: results obtained by the heuristics NSHF, BSHF and those of Hifi (ISH),

• OPT: the optimal solution calculated by the exact algorithm of Hifi [1998].

As we can see in Table 1, NSHF is better than the other

heuristics for 5 problems from 25. BSHF is better for 7 problems and ISH for 14 problems.

a) Case 1 b) Case 2

Figure 4. Update the set of available rectangles

The different heuristics are carried on other instances of

greater size of Hifi [1999]. Table 2 shows the results obtained by the heuristics and compared to other heuristics of the author. The following information is presented in Table 2:

• Problem name, number of items n and the width of strip W,

• BC: values of the continuous lower bound which is the ratio of the sum of items area on W

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• BestH : the best results obtained by the heuristics of Hifi (FIA, SIA, HC/FIA, HC/SIA),

• NSHF, BSHF , BFDH: the results of our heuristiccs NSHF, BSHF and those obtained by BFDH,

• A/BC : the ratio of each heuristic on the continuous lower bound,

As we can see in Table 2, the performance of the BSHF heuristic is confirmed, it is better for 5 problems from 9. NSHF is better in 2 problems. Generally, our heuristics are better compared to Hifi heuristics.

TABLE 2

COMPARISON OF HEURISTICS ON HIFI [1999] INSTANCES

B. Results for the 2BP problem We adapted the NSHF algorithm for the bin packing

problem. To evaluate the heuristic, we tested some literature instances and compared with two algorithms of Lodi et al, [1999]: Floor-Ceiling (FC) and Knapsack-Packing (KP). The instances presented in Table 3 are randomly generated using the methods proposed by Berkey and Wang (1987) and Martello and Vigo (1998) for the two dimensional bin-packing problem. These instances consist of ten classes of

problems. For each class, there are 50 instances: 10 with 20 rectangles, 10 with 40 rectangles, 10 with 60 rectangles, 10 with 80 rectangles and 10 with 100 rectangles. The first six classes have been proposed by Berkey and Wang (1987): j = 1,…, n. n = {20, 40, 60, 80, 100}, Class I: wj and hj uniformly random in [1, 10], H = W =10; Class II: wj and hj uniformly random in [1, 10], H =W=30; Class III: wj and hj uniformly random in [1, 35], H =W=40; Class IV: wj and hj uniformly random in [1, 35], H = W=100;

TABLE 1 COMPARISON OF HEURISTICS ON HIFI [1998] INSTANCES

NAME n W LMIN ISH BSHF NSHF OPT SCP 1 10 5 13 14 13 14 13 SCP 2 11 4 40 41 40 41 40 SCP 3 15 6 14 15 15 18 14 SCP 4 11 6 19 21 22 22 20 SCP 5 8 20 20 20 20 20 20 SCP 6 7 30 38 38 38 39 38 SCP 7 8 15 12 14 15 14 14 SCP 8 12 15 17 18 20 20 17 SCP 9 12 27 68 72 77 69 68 SCP 10 8 50 78 80 80 81 80 SCP 11 10 27 47 53 52 49 48 SCP 12 18 81 34 38 38 39 34 SCP 13 7 70 42 50 57 56 50 SCP 14 10 100 60 69 83 83 69 SCP 15 14 45 34 40 40 41 34 SCP 16 14 6 32 33 35 37 33 SCP 17 9 42 39 41 43 39 39 SCP 18 10 70 89 105 104 105 101 SCP 19 12 5 26 27 27 26 26 SCP 20 10 15 19 22 22 26 21 SCP 21 11 30 135 145 150 151 145 SCP 22 22 90 34 39 43 43 39 SCP 23 12 15 33 35 39 35 35 SCP 24 10 50 103 118 123 141 114 SCP 25 15 25 35 39 43 41 36

Class V: wj and hj uniformly random in [1, 100], H= W=100; Class VI: wj and hj uniformly random in [1, 100], H= W=300; The remainder four classes were inspired from Martello and Vigo (1998). The items are classified into four types: Type 1: wj uniformly random in [2/3W, W], hj uniformly random in [1, H/2]; Type 2: wj uniformly random in [1, W/2], hj uniformly random in [2/3H, H]; Type 3: wj uniformly random in [W/2, W], hj uniformly random in [H/2, H]; Type 4: wj uniformly random in [1, W/2], hj uniformly random in [1, H/2]; The dimensions of the bins are H = W = 100 for all these classes, while the items are as follows: Class VII: type 1 with probability 70%, type 2, 3, 4 with probability 10% each; Class VIII: type 2 with probability 70%, type 1, 3, 4 with probability 10% each; Class IX: type 3 with probability 70%, type 1, 2, 4 with probability 10% each; Class X: type 4 with probability 70%, type 1, 2, 3 with probability 10% each; In Table 3, we present the results obtained by testing the different algorithms on the random generated instances. For each 10 instances, we present the average values of each heuristic. The following information is given:

• Class number, number of items n and the dimensions of the bin W×H,

• A/BC : the ratio of each heuristic value (FC, KP and NSHF) on the continuous lower bound,

As we can see in Table 3, our new heuristic outperforms the two heuristics of Lodi et al,. [1999] in most of problems. Only for some problems of the class 2, 4 and 6 where the NSHF is not better.

V. CONCLUSION AND PERSPECTIVES In this paper we proposed a new heuristic to solve the two

dimensional strip and bin packing problem with guillotine cuts.

In this method, items are packed in the bottom-left position of the bin. No shelf is created. At each iteration we check if the current pattern is guillotine, otherwise, the last packing is cancelled and we try with another position.

The new algorithm with the one proposed in [3] are compared on a literature instances with some heuristics of Hifi and Lodi et al,. The obtained results show that our heuristics outperform the other algorithms in most of the tested instances.

As perspective, we aim to use these heuristics and some of

proposed lower bounds in exact algorithm like branch and bound or dichotomical algorithm to find optimal solution.

TABLE 3 COMPARISON OF HEURISTICS ON BERKEY AND WANG (1987) INSTANCES

TABLE 4

COMPARISON OF HEURISTICS ON MARTELLO AND VIGO (1998) INSTANCES

ACKNOWLEDGMENT Research supported in part by Champagne-Ardenne

Regional Council (district grant) and the European Social Fund.

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