Invited Review
Two-dimensional packing problems: A survey

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Abstract

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. We discuss mathematical models, and survey lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches. The relevant special cases where the items have to be packed into rows forming levels are also discussed in detail.

Introduction

In several industrial applications one is required to allocate a set of rectangular items to larger rectangular standardized stock units by minimizing the waste. In wood or glass industries, rectangular components have to be cut from large sheets of material. In warehousing contexts, goods have to be placed on shelves. In newspapers paging, articles and advertisements have to be arranged in pages. In these applications, the standardized stock units are rectangles, and a common objective function is to pack all the requested items into the minimum number of units: the resulting optimization problems are known in the literature as two-dimensional bin packing problems. In other contexts, such as paper or cloth industries, we have instead a single standardized unit (a roll of material), and the objective is to obtain the items by using the minimum roll length: the problems are then referred to as two-dimensional strip packing problems. As we will see in the following, the two problems have a strict relation in almost all algorithmic approaches to their solution.

Most of the contributions in the literature are devoted to the case where the items to be packed have a fixed orientation with respect to the stock unit(s), i.e., one is not allowed to rotate them. This case, which is the object of the present article, reflects a number of practical contexts, such as the cutting of corrugated or decorated material (wood, glass, cloth industries), or the newspapers paging. For variants allowing rotations (usually by 90°) and/or constraints on the items placement (such as the “guillotine cuts”), the reader is referred to Lodi et al. [41], [42], where a three-field classification of the area is also introduced. General surveys on cutting and packing problems can be found in Dyckhoff and Finke [17], Dowsland and Dowsland [16] and Dyckhoff et al. [18]. Results on the probabilistic analysis of packing algorithms can be found in Coffman and Shor [12] and Coffman and Lueker [11].

Let us introduce the problems in a more formal way. We are given a set of n rectangular items jJ={1,…,n}, each defined by a width, wj, and a height, hj:

  • (i)

    in the Two-Dimensional Bin Packing Problem (2BP), we are further given an unlimited number of identical rectangular bins of width W and height H, and the objective is to allocate all the items to the minimum number of bins;

  • (ii)

    in the Two-Dimensional Strip Packing Problem (2SP), we are further given a bin of width W and infinite height (hereafter called strip), and the objective is to allocate all the items to the strip by minimizing the height to which the strip is used.


In both cases, the items have to be packed with their w-edges parallel to the W-edge of the bins (or strip). We will assume, with no loss of generality, that all input data are positive integers, and that wjW and hjH (j=1,…,n).

Both problems are strongly NP-hard, as is easily seen by transformation from the strongly NP-hard (one-dimensional) Bin Packing Problem (1BP), in which n items, each having an associated size hj, have to be partitioned into the minimum number of subsets so that the sum of the sizes in each subset does not exceed a given capacity H.

A third relevant case of rectangle packing is the following. Each item j has an associated profit pj>0, and the problem is to select a subset of items, to be packed in a single finite bin, which maximizes the total selected profit. This problem is usually denoted as (Two-Dimensional) Cutting Stock, although it had been introduced by Gilmore and Gomory [29] as (Two-Dimensional) Cutting Knapsack.

In this survey we concentrate on two-dimensional problems in which all items have to be packed, i.e., on 2SP and 2BP. The reader is referred to Dyckhoff et al. [18, Section 5] for an annotated bibliography on two-dimensional cutting stock problems. For both 2SP and 2BP, we also consider the special case where the items have to be packed into rows forming levels.

In Section 2 we discuss mathematical models for the various problems introduced above. In Section 3 we survey classical approximation algorithms as well as more recent heuristic and metaheuristic methods. In Section 4 we introduce lower bounding techniques, while in Section 5 we describe exact enumerative approaches.

Section snippets

Modeling two-dimensional problems

The first attempt to model two-dimensional packing problems was made by Gilmore and Gomory [29], through an extension of their approach to 1BP (see [27], [28]). They proposed a column generation approach (see [53] for a recent survey) based on the enumeration of all subsets of items (patterns) that can be packed into a single bin. Let Aj be a binary column vector of n elements aij (i=1,…,n) taking the value 1 if item i belongs to the jth pattern, and the value 0 otherwise. The set of all

Approximation algorithms

In this section we concentrate on off-line algorithms, i.e., algorithms which have full knowledge of the input. For on-line algorithms, which pack the items in the order they are encountered in the scan of the input (without knowledge the next items), the reader is referred to the survey by Csirik and Woeginger [13]. In the next two sections we consider classical constructive heuristics for 2SP and 2BP, whereas metaheuristic approaches are presented together in Section 3.3.

Lower bounds for bin and strip packing

Obvious lower bounds for our problems are obtained by allowing each item to be split into unit squares. We get, respectively for 2BP and 2SP, the geometric bound (computable in linear time)Lgb=j=1nwjhjWH,Lgs=j=1nwjhjW.Let L(I) denote the value produced by a lower bound L for an instance I of the problem. Martello and Vigo [49] proved that, for any instance I, Lgb(I)⩾14OPT(I) while Martello et al. [46] showed that max(Lgs(I),maxj∈J{hj})⩾12OPT(I). Both worst-case bounds are tight.

Specialized

Exact algorithms

An enumerative approach for finding the optimal solution of 2SP was proposed by Martello et al. [46]. Initially, the items are sorted by non-increasing height, and a reduction procedure determines the optimal packing of a subset of items in a portion of the strip, thus possibly reducing the instance size.

The branching scheme is an adaptation of the branch-and-bound algorithms proposed by Scheithauer [50] and Martello et al. [47] for 2BP. At each decision node the current partial solution packs

Acknowledgements

We thank the Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR) and the Consiglio Nazionale delle Ricerche (CNR), Italy, for the support to the project.

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