Research Article Mathematical Modeling and Optimal Blank Generation in Glass Manufacturing Raymond Phillips, 1 Matthew Woolway, 1 Dario Fanucchi, 1 and M. Montaz Ali 1,2 1 School of Computational and Applied Mathematics, University of the Witwatersrand, 1 Jan Smuts Avenue, Private Bag 03, WITS 2050, Johannesburg, South Africa 2 TCSE, Faculty of Engineering and the Build Environment, University of the Witwatersrand, 1 Jan Smuts Avenue, Private Bag 03, WITS 2050, Johannesburg, South Africa Correspondence should be addressed to M. Montaz Ali; montaz.ali@wits.ac.za Received 6 January 2014; Accepted 20 January 2014; Published 27 April 2014 Academic Editor: Aderemi Oluyinka Adewumi Copyright © 2014 Raymond Phillips et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. his paper discusses the stock size selection problem (Chambers and Dyson, 1976), which is of relevance in the loat glass industry. Given a ixed integer N, generally between 2 and 6 (but potentially larger), we ind the N best sizes for intermediate stock from which to cut a roster of orders. An objective function is formulated with the purpose of minimizing wastage, and the problem is phrased as a combinatorial optimization problem involving the selection of columns of a cost matrix. Some bounds and heuristics are developed, and two exact algorithms (depth-irst search and branch-and-bound) are applied to the problem, as well as one approximate algorithm (NOMAD). It is found that wastage reduces dramatically as N increases, but this trend becomes less pronounced for larger values of N (beyond 6 or 7). For typical values of N, branch-and-bound is able to ind the exact solution within a reasonable amount of time. 1. Introduction he cutting and packing of stock are important problems in the metal, paper, wood, and glass industries (amongst others). Consequently, many researchers have considered these problems as mathematical optimization problems and derived good algorithms towards their solutions. In particu- lar, the Stock-Cutting problem is concerned with the cutting of speciic rectangles (orders) desired by customers from larger shapes (blanks) produced during the manufacturing process. his problem was irst treated as a linear programming prob- lem in [1] for one-dimensional stock-cutting and in [2] for two-dimensional stock cutting and has since been extensively studied in various forms. Indeed, Sweeney and Paternoster [3] reviewed more than 400 books, articles, dissertations, and working papers on stock cutting and packing in 1992, and since then new work has appeared (e.g., [4, 5]). In general, the stock cutting problem is concerned with the cutting out of many smaller rectangles (or other shapes) from a ixed larger rectangle. A related, but less well-known, problem is the selection of stock sizes or blanks (the larger rectangle) from which to cut orders. his problem is of particular importance in the loat glass industry, where “holding good stock sizes appears to have at least as big an impact on trim loss as cutting up the stock plates eiciently” [6]. A typical glass manufacturing plant receives hundreds of diferent sized orders per year for a single material and thickness of glass. A single order size will typically need to be cut hundreds, thousands, or tens of thousands of times to satisfy customer demand. In the production of loat glass, a continuous ribbon of lat glass is produced in the manufactur- ing plant. his ribbon is cut on-line into large sizes (blanks) that are stored and cut as needed into speciic order sizes. his two-stage cutting process is carried out for various practical reasons: it is costly and sometimes impossible to cut the many diferent order sizes directly on the loat-line, and it is also sometimes infeasible to store the many diferent order sizes in advance. Given expected order sizes and numbers, the stock size selection problem is the problem of deciding which large sizes (blanks) to cut on the loat line in order to minimize wastage ater all the orders have been cut from these blanks. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 959453, 12 pages http://dx.doi.org/10.1155/2014/959453