Discrete Optimization Approximately global optimization for assortment problems using piecewise linearization techniques Han-Lin Li a, * , Ching-Ter Chang b , Jung-Fa Tsai a a Institute of Information Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC b Business Department of Education, National Changhua University of Education, Hsinchu 30050, Taiwan, ROC Received 25 January 2000; accepted 23 May 2001 Abstract Recently, Li and Chang proposed an approximate model for assortment problems. Although their model is quite promising to find approximately global solution, too many 0–1 variables are required in their solution process. This paper proposes another way for solving the same problem. The proposed method uses iteratively a technique of piecewise linearization of the quadratic objective function. Numerical examples demonstrate that the proposed method is computationally more efficient than the Li and Chang method. Ó 2002 Elsevier Science B.V. All rights reserved. Keywords: Cutting; Assortment; Global optimization 1. Introduction Assortment problems occur when a number of small rectangular pieces need to be cut from a large rectangle to get minimum area. Recently, Li and Chang [1] developed a method for finding the global optimal solution of the assortment problem. Li and Chang’s method, however, requires to use numerous 0–1 variables to linearize the polynomial objective function in their model, which would cause heavy computational burden. This paper proposes instead a piecewise linearization method. The major advantage of this method is that it uses much less number of 0–1 variables to linearize the quadratic objective function than used in Li and Chang’s model. The computational efficiency can therefore be improved significantly. The numerical examples demonstrate that the computation time of the proposed method is much less than that in Li and Chang’s model. 2. Problem formulation Given n rectangles with fixed lengths and widths. An assortment optimization problem is to allocate all of these rectangles within an envelop- ing rectangle, which has minimum area. Denote x and y as the width and the length of the enveloping rectangle (x > 0, y > 0), the assortment optimiza- tion problem is stated briefly as follows: European Journal of Operational Research 140 (2002) 584–589 www.elsevier.com/locate/dsw * Corresponding author. Tel.: +886-3-5728709; fax: +886-3- 5723792. E-mail address: hlli@cc.nctu.edu.tw (H.-L. Li). 0377-2217/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII:S0377-2217(01)00194-1