European Journalof OperationalResearch62 (1992) 163-174 163 North-Holland Theory and Methodology GBSSS: The Generalized Big Square Small Square method for planar single-facility location Frank Plastria Centrum voor Bedrijfsinformatica, Faculteit ESP, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium Received March 1989; revisedDecember 1990 Abstract: Big Square Small Square is a geometrical branch-and-bound algorithm, devised by P. Hansen et al. (OR 33, 1251-1265), for the solution of constrained planar minisum single-facility location problems with Lp norms and continuous non-decreasing costfunctions. The method basically works by splitting the studied planar region into squares, and either rejecting or further processing these squares by the evaluation of a lower bound. We present a modified version of this algorithm aimed at correcting a small failure to converge, accelerating the calculations, minimising the information to be stored, and, most importantly, determining a region of near-optimality. Furthermore the method is applicable to any planar single-facility problem with distances measured by mixed norms and as an objective any continuous function of the distances. This includes nearly all the models which have been proposed in the literature. Keywords: Facility location; global optimisation; near-optimality 1. Introduction Continuous location problems have attracted the attention of many scholars during the last decades, and in their simplest form are among the first optimisation problems ever studied. This is probably mainly due to two reasons: they are easy to state and often difficult to solve. The best known model, the Fermat-Weber problem, has given rise to a seemingly unending series of solu- tion methods, and has often been used as test problem for new non-linear optimisation meth- ods, especially as a well-understood but neverthe- less non-differentiable model. It was used by Kuhn (1967) to help clarify duality theory in non-linear optimisation. For actual applications in real-life locational decision making, however, it seems that many authors are unsatisfied by the continuous frame- work and prefer making use of discrete location models, although the use of continuous models has often been advocated as a useful approxima- tion to large-scale (often discrete) problems, see e.g. Erlenkotter (1989) and Love et al. (1988), where the distinction between both families of models is described as site-generating vs site- selecting. Part of the criticism is as follows: - The objective functions yielding continuous models solvable by the usual non-linear optimisa- tion techniques are rather restrictive and mostly unrealistic. As an example, in real life (and in 0377-2217/92/$05.00 © 1992 - ElsevierSciencePublishers B.V. All rightsreserved