INFORMS Journal on Computing Articles in Advance, pp. 1–11 issn 1091-9856  eissn 1526-5528 inf orms ® doi 10.1287/ijoc.1100.0394 © 2010 INFORMS A Parallel Branch-and-Bound Approach to the Rectangular Guillotine Strip Cutting Problem Sławomir B˛ ak, Jacek Bła ˙ zewicz, Grzegorz Pawlak, Maciej Płaza Institute of Computing Science, Pozna ´ n University of Technology, 60-965 Pozna´ n, Poland {sbak@skno.cs.put.poznan.pl, jacek.blazewicz@cs.put.poznan.pl, grzegorz.pawlak@cs.put.poznan.pl, mplaza@skno.cs.put.poznan.pl} Edmund K. Burke, Graham Kendall School of Computer Science, University of Nottingham, Nottingham NG8 1BB, United Kingdom {ekb@cs.nott.ac.uk, gxk@cs.nott.ac.uk} T his paper presents a parallel branch-and-bound method to address the two-dimensional rectangular guil- lotine strip cutting problem. Our paper focuses on a parallel branching schema. We present a series of computational experiments to evaluate the strength of the approach. Optimal solutions have been found for some benchmark instances that had unknown solutions until now. For many other instances, we demonstrate that the proposed approach is time effective. The efficiency of the parallel version of the algorithm is compared and the speedup, when increasing the number of processors, is clearly demonstrated with an upper bound calculated by a specialised heuristic procedure. Key words : production scheduling; cutting stock; material handling; parallel branch-and-bound method; analysis of algorithms History : Accepted by Michel Gendreau, former Area Editor for Heuristic Search and Learning; received November 2007; revised August 2008, May 2009, January 2010; accepted February 2010. Published online in Articles in Advance. 1. Introduction The cutting and packing family of problems affects several different industries and motivates many areas of research. Research going back at least 50 years has led to the development of many models and math- ematical tools. The diversity of this type of prob- lem has made it necessary to introduce a consistent typology that was proposed by Dyckhoff (1990) and further developed by Wäscher et al. (2007). The earli- est work in this area was conducted by Gilmore and Gomory (1961), where they solved one-dimensional problems to optimality using linear programming. An example of a one-dimensional problem is the divi- sion of steel bars or rods into smaller lengths for fabrication or resale. However, only small problem instances could be solved in reasonable time. Gilmore and Gomory (1966) characterised knapsack functions and used them to develop more efficient methods. Generalisations of these methods were also applied to two-dimensional problems. Two-dimensional problems can be modelled as a set of pieces that must be arranged on a predefined stock sheet so that each piece does not overlap with another, and of course, each piece must fit within the bounds of the sheet. The main objective is to maximise space utilisation and therefore minimise wastage. The complexity of the problem is increased by different constraints within various manufacturing industries, including paper, wood, glass, and metal cutting. For example, paper cutting is generally concerned with guillotine cutting (where only vertical or horizontal straight cuts, across the entire sheet, are allowed) of rectangular items from a stock roll of fixed width, whereas applications in metal and shipbuilding are often concerned with the cutting of irregular shapes from a stock sheet (see, for example, Bła ˙ zewicz et al. 1993). Despite their industrial relevance, irregular cut- ting problems have not been widely researched, but the area has gained popularity in recent years (see, for example, Bennell and Oliveira 2006). In some applica- tions the pieces cannot be rotated—for instance, cut- ting pieces from wooden boards that have to take the wood grain into account—but in other applications, such as cutting pieces from steel sheets, rotation is often allowed. The two-dimensional cutting stock problem is a generalisation of the one-dimensional knapsack prob- lem. In two dimensions, a large stock rectangle S of dimensions L × W and n types of smaller rectangular pieces are presented. Each smaller piece has an asso- ciated profit. The problem is to cut from S a set of small rectangles so that the overall profit is maximised. 1 Copyright: INFORMS holds copyright to this Articles in Advance version, which is made available to institutional subscribers. The file may not be posted on any other website, including the author’s site. Please send any questions regarding this policy to permissions@informs.org. Published online ahead of print July 2, 2010