A linear-time crystal-growth algorithm for discretization of continuum approximation

Abstract

Continuous approximation is regarded as a scalable and insightful method for acquiring near-optimum solutions to various location problems. A continuous approximation solution is nonetheless only a continuous density function and a further discretization procedure is needed to obtain a discrete location solution for engineering practice. Inspired by the process of “crystal growth”, this paper proposes a constructive heuristic algorithm as an alternative to the classic meta-heuristic disk algorithm (Ouyang and Daganzo, 2006) in discretizing a continuous solution from a continuum approximation location model. The main idea of this algorithm is to rasterize the space into a set of small cells (either regular triangles or squares) and repeatedly grow a core cell into a full-service area according to a certain visiting sequence. Thus, it has only linear time complexity proportional to the number of cells in the space. Numerical examples are conducted to test the performance of the proposed algorithm. The results indicate that this algorithm can solve discrete facility locations more efficiently and exhibits robust performance compared with the disk algorithm.