European Journal of Operational Research141 (2002) 241–252 www.elsevier.com/locate/dsw Invited Review Two-dimensional packing problems: A survey Andrea Lodi *, Silvano Martello, Michele Monaci Dipartimento di Elettronica, Informatica e Sistemistica, University of Bologna, Viale Risorgimento 2, 40136Bologna, Italy Received 9 March2001 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. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Two-dimensional packing; Bin packing problems; Strip packing problems 1. Introduction into the minimum number of units: the resulting optimization problems are known in the litera- In several industrial applications one is required ture as two-dimensional bin packing problems. In to allocate a set of rectangular items to larger other contexts, such as paper or cloth industries, rectangular standardized stock units by minimi- we have instead a single standardized unit (a roll of zing the waste. In wood or glass industries, rect- material), and the objective is to obtain the items angular components have to be cut from large by using the minimum roll length: the prob- sheets of material. In warehousing contexts, goods lems are then referred to as two-dimensional strip have to be placed on shelves. In newspapers packing problems. As we will see in the follow- paging, articles and advertisements have to be ar- ing, the two problems have a strict relation in ranged in pages. In these applications, the stan- almost all algorithmic approaches to their solu- dardized stock units are rectangles, and a common tion. objective function is to pack all the requested items 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 * Corresponding author. Tel.: +39-051-209-3029; fax: +39- case, which is the object of the present article, re- 051-209-3073. E-mail addresses: [email protected] (A. Lodi), smar- flects a number of practical contexts, suchas the [email protected] (S. Martello), [email protected] (M. cutting of corrugated or decorated material (wood, Monaci). glass, clothindustries), or thenewspapers paging. 0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S0377-2217(02)00123-6 242 A. Lodi et al. / European Journal of Operational Research 141 (2002) 241–252 For variants allowing rotations (usually by 90°) Stock, although it had been introduced by Gilmore and/or constraints on the items placement (such as and Gomory [29] as (Two-Dimensional) Cutting the ‘‘guillotine cuts’’), the reader is referred to Lodi Knapsack. et al. [41,42], where a three-field classification of In this survey we concentrate on two-dimen- the area is also introduced. General surveys on sional problems in which all items have to be cutting and packing problems can be found in packed, i.e., on 2SP and 2BP. The reader is re- Dyckhoff and Finke [17], Dowsland and Dows- ferred to Dyckhoff et al. [18, Section 5] for an land [16] and Dyckhoff et al. [18]. Results on the annotated bibliography on two-dimensional cut- probabilistic analysis of packing algorithms can be ting stock problems. For both2SP and 2BP, we found in Coffman and Shor [12] and Coffman and also consider the special case where the items have Lueker [11]. to be packed into rows forming levels. Let us introduce the problems in a more formal In Section 2 we discuss mathematical models way. We are given a set of n rectangular items for the various problems introduced above. In j 2 J ¼f1; ...; ng, eachdefined by a width, wj, and Section 3 we survey classical approximation algo- a height, hj: rithms as well as more recent heuristic and meta- heuristic methods. In Section 4 we introduce lower (i) in the Two-Dimensional Bin Packing Problem bounding techniques, while in Section 5 we de- (2BP), we are further given an unlimited num- scribe exact enumerative approaches. ber of identical rectangular bins of width W and height H, and the objective is to allo- cate all the items to the minimum number of 2. Models bins; (ii) in the Two-Dimensional Strip Packing Problem 2.1. Modeling two-dimensional problems (2SP), we are further given a bin of width W and infinite height (hereafter called strip), The first attempt to model two-dimensional and the objective is to allocate all the items packing problems was made by Gilmore and Go- to the strip by minimizing the height to which mory [29], through an extension of their approach the strip is used. to 1BP (see [27,28]). They proposed a column generation approach(see [53] for a recent survey) In both cases, the items have to be packed with based on the enumeration of all subsets of items their w-edges parallel to the W-edge of the bins (or (patterns) that can be packed into a single bin. Let strip). We will assume, withno loss of generality, Aj be a binary column vector of n elements aij that all input data are positive integers, and that (i ¼ 1; ...; n) taking the value 1 if item i belongs to 6 6 wj W and hj H (j ¼ 1; ...; n). the jth pattern, and the value 0 otherwise. The set Bothproblems are strongly NP-hard, as is of all feasible patterns is then represented by the easily seen by transformation from the strongly matrix A, composed by all possible A columns Bin Packing Problem j NP-hard (one-dimensional) (j ¼ 1; ...; M), and the corresponding mathemati- (1BP), in which n items, eachhaving an associ- cal model is ated size hj, have to be partitioned into the mini- mum number of subsets so that the sum of the XM sizes in eachsubset does not exceed a given ca- ð2BP-GGÞ min xj ð1Þ pacity H. j¼1 A third relevant case of rectangle packing is the subject to following. Eachitem j has an associated profit XM pj > 0, and the problem is to select a subset of items, to be packed in a single finite bin, which aijxj ¼ 1 ði ¼ 1; ...; nÞ; ð2Þ maximizes the total selected profit. This problem j¼1 is usually denoted as (Two-Dimensional) Cutting xj 2f0; 1gðj ¼ 1; ...; MÞ; ð3Þ A. Lodi et al. / European Journal of Operational Research 141 (2002) 241–252 243 8 where xj takes the value 1 if pattern j belongs to the < 1 if item i is placed withits bottom solution, and the value 0 otherwise. Observe that xipq ¼ : left-hand corner at ðp; qÞ; (1)–(3) is a valid model for 1BP as well, the only 0 otherwise difference being that the A ’s are all columns sat- P j 4 n 6 ð Þ isfying i¼1 aijhi H. Due to the immense number of columns that can appear in A, the only way for handling the for i ¼ 1; ...; n, p ¼ 0; ...; W À wi and q ¼ 0; ...; model is to dynamically generate columns when H À hi. A similar model, in which p and q coor- needed. While for 1BP Gilmore and Gomory dinates are handled through distinct decision [27,28] had given a dynamic programming ap- variables, has been introduced by Hadjiconstanti- proachto generate columns by solving, as a slave nou and Christofides [33]. Both models are used to problem, an associated 0–1 knapsack problem, for provide upper bounds through Lagrangian relax- 2BP they observed the inherent difficulty of the ation and subgradient optimization. two-dimensional associated problem. Hence they A completely different modeling approachhas switched to the more tractable case where the been recently proposed by Fekete and Schepers items have to be packed in rows forming levels (see [20], through a graph-theoretical characterization Section 2.2), for which the slave problem was of the packing of a set of items into a single bin. solved through a two-stage dynamic programming Let Gw ¼ðV ; EwÞ (resp. Gh ¼ðV ; EhÞ) be an inter- algorithm. val graphhavinga vertex vi associated witheach Beasley [4] considered a two dimensional cut- item i in the packing and an edge between two ting problem in which a profit is associated with vertices (vi; vj) if and only if the projections of eachitem, and theobjective is to pack a maximum items i and j on the horizontal (resp. vertical) axis profit subset of items into a single bin (cutting overlap (see Fig. 1). It is proved in [20] that, if the stock problem). He gave an ILP formulation based packing is feasible then on the discrete representation of the geometrical (a)P for eachstable setP S of Gw (resp. Gh), space and the use of coordinates at which items w 6 W (resp. h 6 H); vi2S i vi2S i may be allocated, namely (b) Ew \ Eh ¼;. Fig. 1. The Fekete and Schepers modeling approach. 244 A. Lodi et al. / European Journal of Operational Research 141 (2002) 241–252 This characterization easily extends to packings in Problem 2LBP can be efficiently modeled by higher dimensions.