A new lower bound for the non-oriented two-dimensional bin-packing problem

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Abstract

We propose a new scheme for computing lower bounds for the non-oriented bin-packing problem when the bin is a square. It leads to bounds that theoretically dominate previous results. Computational experiments show that the bounds are tight. We also discuss the case where the bin is not a square.

Introduction

The two-dimensional discrete bin-packing problem (2BP) consists in minimizing the number of identical rectangles used to pack a set of smaller rectangles. This problem is NP-complete. It occurs in industry if pieces of steel, wood, or paper have to be cut from larger rectangles. It belongs to the family of cutting and packing (C & P) problems, more precisely Two-dimensional single bin size bin-packing problems (2SBSBPP), following the typology of Wäscher et al. [10].

A 2BP instance D is a pair (I,B). It is composed of the set I={1,,n} of items i to pack, and a bin B=(W,H) of width W and height H (W,HN). For the oriented case (2BP|O), an item i is of width wi and height hi (wi,hiN). We consider the non-oriented version of the problem (2BP|R), i.e. where the items can be rotated. So we consider the two possible orientations: wi1=hi2 and wi2=hi1. Before rotation, an item is of size (wi1,hi1), otherwise it is of size (wi2,hi2). An orientation of the set of items is an application r from I to {1,2} that associates an orientation r(i) to each item i in I.

Many methods [9], [7], [2], [5], [4] have been proposed for computing lower bounds for 2BP|O. Most of them can be computed by means of so-called dual-feasible functions. These functions cannot be directly applied if the orientation of the items is not known. Fewer results exist for 2BP|R [6], [3]. Two schemes were proposed for computing such bounds: cutting the items into squares [6], [3], or explicitly using dual-feasible functions [3]. It would appear that the difference between upper and lower bounds is larger than for the oriented case.

In this paper, we propose a new general scheme for computing lower bounds for 2BP|R when the bin is a square. It is based on the construction of a 2BP|O instance composed of 2n items: each item appears once for each orientation. We show that if the original instance needs z bins, the 2n-items instance needs at most 2z bins. When this instance is computed, bounds designed for the oriented case can be applied. We show that when the bin is a square, the bounds of Carlier et al. [5] used in this general framework dominate those of Dell’ Amico et al. [6], as well as those of Boschetti and Mingozzi [3]. When the bin is not a square, the above method can be used, if a suitable preprocessing is performed.

We tested our method against well-known benchmarks proposed by [1], [9]. Computational results show that the dominance is not only theoretical, as the bounds are improved for several instances.

In Section 2, we describe several results from the literature for the oriented case, which are used in our new bounds. We also recall previous results for the non-oriented case. Section 3 deals with the new lower bounds. We present several dominance results in Section 4. Section 5 is devoted to computational experiments.

Section snippets

Literature review

In this section, we recall the lower bounds that are useful in this paper. First we describe the bounds of [5] for the oriented case. These bounds are used in our new lower bounds described in Section 3. Next we describe the bounds of [2] for the oriented case. Finally, we describe the frameworks used to compute bounds for the non-oriented case [6], [3]. In Section 4, these frameworks are compared to our new method.

A new lower bound for 2BP|R

In this section we propose a new scheme for computing lower bounds for the non-oriented case when the bin is a square. It is based on the creation of a new 2BP|O instance. We also show how this instance can be additionally constrained for obtaining better results. At the end of the section, we discuss how our method can be generalized to rectangular bins.

Let DR=(I,B) be an instance of 2BP|R. I={1,,n} is the set of items i and B=(W,W) a bin. Now consider the following 2BP|O problem: D^F=(I^F,B)

Dominance results

In this section, we show that when the bin is a square, the lower bound LCJER proposed above dominates the methods of [6], [3]. For this purpose, we introduce a new bound LCJER,t, which dominates several previous bounds, and is dominated by our new method.

Let F be the set of valid functions f0k and f2k for an instance D. The bound LCJER,t generalizes the bound LBMR,2 of [3]. LCJER,t=maxf,gFiImin{f(wi1)g(hi1),f(wi2)g(hi2)}f(W)g(W).

Proposition 4.1

LCJER,t is a valid lower bound for 2BP|R.

Proof

If one computes all

Computational experiments

We tested our method against well-known benchmarks derived from the literature [1], [9]. There are 10 classes of randomly generated test problems. Each class contains five groups of 10 instances each.

In Table 1 we compare our results to those of [3], which are the best results known so far. For this purpose we report the number of times each bound is equal to the upper bound of [3] (columns Opt). To allow further comparisons, we also report the average value of each lower bound (column Avg).

Concluding remarks

Our bounds are tight when the bin is a square. However, the drawback of our method is that it does not take into account the fact that some items may not be rotated. This may lead to weaker results when the bin is not a square, or when it is used within an enumerative method.

The framework we propose can be adapted to the three-dimensional bin-packing problem. In this case, the instance to construct would have 6n items.

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This work was carried out while François Clautiaux was associate researcher at Heudiasyc lab.

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