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, 40136 Bologna, Italy Received 9 March 2001 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 In several industrial applications one is required to allocate a set of rectangular items to larger rectangular standardized stock units by minimi- zing the waste. In wood or glass industries, rect- angular 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 ar- ranged in pages. In these applications, the stan- dardized 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 litera- ture 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 prob- lems are then referred to as two-dimensional strip packing problems. As we will see in the follow- ing, the two problems have a strict relation in almost all algorithmic approaches to their solu- tion. 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, re- flects a number of practical contexts, such as the cutting of corrugated or decorated material (wood, glass, cloth industries), or the newspapers paging. European Journal of Operational Research 141 (2002) 241–252 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +39-051-209-3029; fax: +39- 051-209-3073. E-mail addresses: alodi@deis.unibo.it (A. Lodi), smar- tello@deis.unibo.it (S. Martello), mmonaci@deis.unibo.it (M. Monaci). 0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0377-2217(02)00123-6