New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation

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Abstract

The two-dimensional bin-packing problem (2BP) consists of minimizing the number of identical rectangles used to pack a set of smaller rectangles. In this paper, we propose new lower bounds for 2BP in the discrete case. They are based on the total area of the items after application of dual feasible functions (DFF). We also propose the new concept of data-dependent dual feasible functions (DDFF), which can also be applied to a 2BP instance. We propose two families of Discrete DFF and DDFF and show that they lead to bounds which strictly dominate those obtained previously. We also introduce two new reduction procedures and report computational experiments on our lower bounds. Our bounds improve on the previous best results and close 22 additional instances of a well-known established benchmark derived from literature.

Introduction

The two-dimensional bin-packing problem (2BP) consists of minimizing the number of identical rectangles used to pack a set of smaller rectangles. This problem can occur in industry if pieces of steel, wood, or paper have to be cut from larger rectangles. It can also be used to model the layout of a newspaper. It is NP-hard as it generalizes the classical one-dimensional bin-packing problem (1BP) [1].

A 1BP instance D1 is a pair (A,C). A={a1,,an} is the set of items to pack and CN is the size of the bins. An item ai has a size ciN. A 2BP instance D2 is a pair (A,B). A is the set of items ai to pack. An item ai has a width wi and a height hi (wi,hiN). We consider the version of the problem in which the items cannot be rotated. The position of item ai, denoted by (xi,yi), corresponds to the coordinates of its bottom left-hand corner. The bin B=(W,H) is of width W and height H. OPT(D) denotes the minimum number of bins required for a given instance D.

In this paper, we describe two new reduction procedures which are used to reduce the size of the instance. They can be applied as a preprocessing step before applying a heuristic or an exact method. For this purpose, we introduce identically-feasible functions (IFF) which can be used to modify the size of the items without changing the value of OPT. Thus, we show that the initial instance can be reduced by removing small items and by increasing the size of the large items.

We also deal with lower bounds. The most recent bounds were introduced by Boschetti and Mingozzi [2]: they dominate the previous best bounds [3], [4]. For 1BP, Haouari and Gharbi [5] have proposed a method to improve the value of a lower bound. They plan to generalize their method to 2BP. Several bounds are equal to the total area of the items for an instance obtained after applying a suitable function to the size of the items. This processing can be done using dual-feasible functions (DFF), introduced by Johnson [6] and used by Lueker [7] and Fekete and Schepers [8] for computing lower bounds for 1BP. Fekete and Schepers [3] have shown that these functions can also be applied to both dimensions of a 2BP instance to obtain lower bounds. The same concept is used by Carlier and Néron [9] for cumulative scheduling problems. The functions they use are a subset of the DFF, restricted to the discrete case. We will refer to this concept as discrete dual-feasible functions (Discrete DFF): the only difference is that we consider discrete values and that they are not defined from [0,1] to [0,1] but from [0,X] to [0,X] for X and X integers. Caprara et al. [10] have also dealt with DFF. They propose an exact method to find the pair of DFF which leads to the best continuous bound for a given instance.

We use the concept of DFF and a new type of functions, the data-dependent dual feasible functions (DDFF), which are designed for a specific type of instances. These functions are used in the general framework proposed by Fekete and Schepers [3] for computing bounds for multi-dimensional packing problems. We also describe three families of functions (two families of DFF and a family of DDFF) and show that they lead to bounds which strictly dominate those proposed by Labbé et al. [11] and Martello and Vigo [4] for 1BP and Boschetti and Mingozzi [2] for 2BP. To our knowledge, our bounds are the best polynomial time computable bounds for 2BP.

We report computational results for the reduction procedures and for our lower bounds. We tested our methods on well-known benchmarks derived from literature [4], [12] so we can compare our methods with the best known results [13] and to the method of Fekete and Schepers [3]. Our reduction procedures are very efficient for several classes of instances, as a large number of items are removed for many benchmarks. The results confirm that our method dominates the others for all instances and leads to improved results.

Section 2 is an overview of the methods used in the literature for reduction procedures and lower bounds for 1BP and 2BP, which are used in this article. In Section 3, we introduce the concept of IFF and we propose two new reduction procedures which use IFF. Section 4 is devoted to our new lower bounds. We show in Section 5 that they dominate previous lower bounds [2], [4], [11]. In Section 6, we compare our reduction procedures and our lower bounds to those proposed in literature. The results are strictly better for several instances, and allow us to close 22 additional instances.

Section snippets

Reduction procedures

The following reduction procedure is proposed by Boschetti and Mingozzi [2]. For a given item aj, one can compute the maximum width Wj* less than W which can be reached by packing a set of items including aj one above the other. The width of item aj can be modified as follows: wjwj+(W-Wj*).

The value Wj* is the optimal value of the following subset-sum problem: Wj*=wj+MaxaiA-{aj}wiξi:aiA-{aj}wiξiW-wj,ξi{0,1}.

A classical pseudo-polynomial algorithm can compute Wj* in O(Wn) time. The

Reduction procedures: identically-feasible functions

In this section, we describe cases where some items could be removed in a 2BP. In the general case, if an item is too small, it may have no impact on the value OPT, and if it is of size (W,H) it forms a trivial sub-instance which can be separately treated. There are few such large items in the initial instance but their number can be increased by applying an efficient reduction procedure. This section deals with methods which increase the size of the large items and remove the small items which

New lower bounds

In the literature, DFF have been successfully applied to obtain lower bounds for 2BP. It appears that there are bounds which cannot be computed by the mean of DFF. So, we propose a new concept of functions which are dependent on the data, and lead to improving bounds.

Dominance relations

We now show that the bound LCCM2D dominates LBM2D proposed by Boschetti and Mingozzi [2]. We first show that LCCM1D dominates LMV1D proposed by Martello and Vigo [4] and LLab1D proposed by Labbé et al. [11] for 1BP.

Computational experiments

We tested our reduction procedures and our lower bounds on a classical benchmark described by Berkey and Wang [12] and Martello and Vigo [4]. The benchmark is composed of 10 classes of randomly generated test problems. Each class contains five groups of 10 instances each. These data are available on the following web page: http://www.or.deis.unibo.it/ORinstances. Results for the methods proposed by Boschetti and Mingozzi are known for these benchmarks [13]. All programs are implemented in C on a

Conclusion

We have proposed procedures which reduce the size of the original instance. They can be used together and dramatically simplify the problem for several instances of a well-known benchmark. We have also proposed new lower bounds for 2BP and shown that they dominate those obtained previously. The theoretical dominance of our lower bounds is confirmed by experiments. They are always better than the previous best bounds and they lead to improved results for several instances. Using a fast

Acknowledgements

The authors would like to thank Emmanuel Néron for his research work and his help for the Discrete DFF.

The authors would also like to thank Marco Boschetti who kindly sent them the results for his lower and upper bounds, and Anis Gharbi, who sent them the benchmarks for 1BP.

The authors also thank Khalil Raı¨s for his computational experiments on the lower bounds.

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