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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 624398, 15 pages
http://dx.doi.org/10.1155/2013/624398
Research Article

A Heuristic Algorithm for Constrained Multi-Source Location Problem with Closest Distance under Gauge: The Variational Inequality Approach

Department of Mathematics, College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 1 August 2013; Accepted 27 August 2013

Academic Editor: Abdellah Bnouhachem

Copyright © 2013 Jian-Lin Jiang et al. This 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.

Abstract

This paper considers the locations of multiple facilities in the space , with the aim of minimizing the sum of weighted distances between facilities and regional customers, where the proximity between a facility and a regional customer is evaluated by the closest distance. Due to the fact that facilities are usually allowed to be sited in certain restricted areas, some locational constraints are imposed to the facilities of our problem. In addition, since the symmetry of distances is sometimes violated in practical situations, the gauge is employed in this paper instead of the frequently used norms for measuring both the symmetric and asymmetric distances. In the spirit of the Cooper algorithm (Cooper, 1964), a new location-allocation heuristic algorithm is proposed to solve this problem. In the location phase, the single-source subproblem with regional demands is reformulated into an equivalent linear variational inequality (LVI), and then, a projection-contraction (PC) method is adopted to find the optimal locations of facilities, whereas in the allocation phase, the regional customers are allocated to facilities according to the nearest center reclassification (NCR). The convergence of the proposed algorithm is proved under mild assumptions. Some preliminary numerical results are reported to show the effectiveness of the new algorithm.