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http://dx.doi.org/10.14317/jami.2015.157

ALGORITHM FOR WEBER PROBLEM WITH A METRIC BASED ON THE INITIAL FARE  

Kazakovtsev, Lev A. (Department of Information Technologies, Siberian State Aerospace University)
Stanimirovic, Predrag S. (Faculty of Sciences and Mathematics, University of Nis)
Publication Information
Journal of applied mathematics & informatics / v.33, no.1_2, 2015 , pp. 157-172 More about this Journal
Abstract
We introduce a non-Euclidean metric for transportation systems with a defined minimum transportation cost (initial fare) and investigate the continuous single-facility Weber location problem based on this metric. The proposed algorithm uses the results for solving the Weber problem with Euclidean metric by Weiszfeld procedure as the initial point for a special local search procedure. The results of local search are then checked for optimality by calculating directional derivative of modified objective functions in finite number of directions. If the local search result is not optimal then algorithm solves constrained Weber problems with Euclidean metric to obtain the final result. An illustrative example is presented.
Keywords
Location problem; Weber problem; Radar metric;
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1 R.G. Brown, Advanced Mathematics: Precalculus with discrete Mathematics and data analysis, (A.M.Gleason, ed.), Evanston, Illinois: McDougal Littell, 1997.
2 R. Chen, Noniterative Solution of Some Fermat-Weber Location Problems, Advances in Operations Research (2011), Article ID 379505, Published online. 10 pages doi:10.1155/2011/379505, http://downloads.hindawi.com/journals/aor/2011/379505.pdf.
3 M.M. Deza and E. Deza, Encyclopedya of Distances, Springer Verlag, Berlin, Heilderberg, 2009.
4 Z. Drezner, A. Mehrez and G.O. Wesolowsky, The facility location problem with limited distances, Transportation Science, 25 (1991), 183-187.   DOI
5 Z. Drezner and G.O. Wesolowsky, A maximin location problem with maximum distance constraints, AIIE Transact., 12 (1980), 249-252.   DOI
6 Z. Drezner, K. Klamroth, A. Schobel and G.O. Wesolowsky, The Weber problem, in Z. Drezner and H.W. Hamacher (editors), Facility Location: Applications and Theory, Springer-Verlag, 2002, 1-36.
7 Z. Drezner, C. Scott and J.S. Song, The central warehouse location problem revisited, IMA Journal of Managemeng Mathematics, 14 (2003), 321-336.   DOI   ScienceOn
8 Z. Drezner and M. Hamacher, Facility location: applications and theory, Springer-Verlag, Berlin, Heidelberg, 2004.
9 S. Gordon, H. Greenspan, J. Goldberger, Applying the Information Bottleneck Principle to Unsupervised Clustering of Discrete and Continuous Image Representations, Computer Vision. Proceedings. Ninth IEEE International Conference on, Vol.1 (2003), 370-377.
10 R.Z. Farahani and M. Hekmatfar editors, Facility Location Concepts, Models, Algorithms and Case Studies, Springer-Verlag Berlin Heidelberg, 2009.
11 I.F. Fernandes, D. Aloise, D.J. Aloise, P. Hansen and L. Liberti, On the Weber facility location problem with limited distances and side constraints, Optimization Letters, issue of 22 August 2012, 1-18, published online, doi:10.1007/s11590-012-0538-9.
12 P. Hansen, D. Peeters and J.F. Thisse, Constrained location and the Weber-Rawls problem, North-Holland Mathematics Studies, 59 (1981) 147-166.   DOI
13 L.A. Kazakovtsev, Adaptation of the probability changing method for Weber problem with an arbitrary metric, Facta Universitatis, (Nis) Ser. Math. Inform., 27 (2012), 289-254.
14 H. Idrissi, O. Lefebvre and C. Michelot, A primal-dual algorithm for a constrained Fermat-Weber problem involving mixed norms, Revue francaise d'automatique, d'informatique et de recherche operationnelle. Recherche Operationnelle, 22 (1988), 313-330.
15 L.A. Kazakovtsev, Wireless coverage optimization based on data provided by built-in measurement tools, WASJ, 22, Special Volume on Techniques and Technologies (2013), 8-15.
16 A.N. Antamoshkin and L.A. Kazakovtsev, Random search algorithm for the p-median problem, Informatica (Ljubljana), 37 (2013), 267-278.
17 L.A. Kazakovtsev, Algorithm for Constrained Weber Problem with feasible region bounded by arcs, Facta Universitatis, (Nis) Ser. Math. Inform., 28 (2013), 271-284.
18 K. Klamroth, Single-facility location problems with barriers, Springer Verlag, Berlin, Heilderberg, 2002.
19 K. Liao, D. Guo, A Clustering-Based Approach to the Capacitated Facility Location Problem, Transactions in GIS, 12 (2008), 323-339.   DOI   ScienceOn
20 H. Minkowski, Gesammelte Abhandlungen, zweiter Band, Chelsea Publishing, 2001.
21 J. Perreur and J.F. Thisse, Central metric and optimal location, J. Regional Science, 14 (1974), 411-421.   DOI
22 I.P. Stanimirovic, Successive computation of some efficient locations of the Weber problem with barriers, J. Appl. Math. Comput., 42 (2013), 193?11. DOI 10.1007/s12190-012-0637-x   DOI
23 A. Vimal, S.R. Valluri, K. Karlapalem, An Experiment with Distance Measures for Clustering, International Conference on Management of Data COMAD 2008, Mumbai, India, 241-244 (2008)
24 P.S. Staminirovic, M. Ciric, L.A. Kazakovtsev and I.A. Osinuga, Single-facility Weber location problem based on the Lift metric, Facta Universitatis, (Nis) Ser. Math. Inform., 27 (2012), 31-46.
25 E. Taillard, Location problems, web resource available at http://mistic.heig-vd.ch/taillard/problemes.dir/location.html
26 E.D. Thaillard, Heuristic Methods for Large Centroid Clustering Problems, Journal of Heuristics, 9 (2003), 51-73.   DOI
27 K. Voevodski, M.F. Balcan, H. Roglin, S.H. Teng, Y. Xia, Min-sum Clustering of Protein Sequences with Limited Distance Information, Proceedings of the First International Conference on Similarity-based Pattern Recognition (SIMBAD'11), Venice, Italy (2011), 192-206.
28 E. Weiszfeld, Sur le point sur lequel la somme des distances de n points donnes est minimum, Tohoku Mathematical Journal, 43 (1937), 335-386.
29 G. Wesolowsky, The Weber problem: History and perspectives, Location Science, 1 (1993), 5-23.
30 Y. Ying, P. Li, Distance Metric Learning with Eigenvalue Optimization, Journal of Machine Learning Research, 13 (2012), 1-26.