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Solving Nonconvex Planar Location Problems by Finite Dominating Sets

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Abstract

It is well-known that some of the classical location problems with polyhedral gauges can be solved in polynomial time by finding a finite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location.

In this paper it is first established that this result holds for a much larger class of problems than currently considered in the literature. The model for which this result can be proven includes, for instance, location problems with attraction and repulsion, and location-allocation problems.

Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal objective value. For the approximation problem two different approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed ∈ - to polynomial approximation algorithms with accuracy ∈ for solving the general model considered in this paper.

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References

  1. Avriel, M., Diewert, W.E., Schaible, S. and Zang, I. (1988), Generalized Concavity. Plenum Press, New York.

    Google Scholar 

  2. Bentley, J.L. and Ottmann, Th. (1979), Algorithms for reporting and counting geometric intersections. IEEE Trans. Computers C 28: 643-647.

    Google Scholar 

  3. Burkard, E.R., Hamacher, H.W. and Rote, G. (1991), Sandwich approximation of univariate convex functions with an application to separable convex programming. Naval Research Logistics 38: 911-924.

    Google Scholar 

  4. Carrizosa, E. and Rodríguez Chía, A.M. (1997) Weber problems with alternative transportation systems. Eur. J. Operational Research, 97: 87-93.

    Google Scholar 

  5. Chen, P.C., Hansen, P., Jaumard, B. and Tuy, H. (1992), Weber's problem with attraction and repulsion. J Regional Science, 32: 467-486.

    Google Scholar 

  6. Dearing, P.M., Hamacher, H.W. and Klamroth, K. Center problems with barriers and the manhattan metric. Technical Report 667, Department of Mathematical Sciences, Clemson University, 1998. Naval Research Logistics, to appear

  7. Drezner, Z. and Wesolowsky, G.O. (1991), The weber problem on the plane with some negative weights. INFOR, 29: 87-99.

    Google Scholar 

  8. Durier, R. and Michelot, C. (1985), Geometrical properties of the fermat-weber problem. Eur. J. Operational Research 20: 332-343.

    Google Scholar 

  9. Francis, R.L., McGinnes Jr., L.F. and White, J.A. (1992), Facility layout and Location: An Analytical Appoach. Prentice Hall.

  10. Hamacher, H.W. (1995), Mathematische Lösungsverfahren für Standortprobleme. Vieweg Publishers.

  11. Hamacher, H.W. and Klamroth, K. (1997), Planar location problems with barriers under polyhedral gauges. Technical Report 667, Department of Mathematics, University of Kaiserslautern, accepted by Annals of Operations Research.

  12. Hamacher, H.W. and Nickel, S. (1995), Restricted planar location problems and applications. Naval Research Logistics, 42: 967-992.

    Google Scholar 

  13. Hamacher, H.W. and Nickel, S. (1998), Classification of location models. Loc Sci 6: 229-242.

    Google Scholar 

  14. Hamacher, H.W. and Schoebel, A. (1997), A note on center problems with forbidden polyhedra. Operations Research Letters 20: 165-169.

    Google Scholar 

  15. Hansen, P., Jaumard, B. and Tuy, H. (1995), Global optimization in location. In Z. D"rezner (ed.) Facility Location: A Survey of Applications and Methods. Springer-Verlag, New York.

    Google Scholar 

  16. Hansen, P., Peeters, D., Richard, D. and Thisse, J.F. (1985), The minisum and minimax location problems revisited. Operations Research 33: 1251-1265.

    Google Scholar 

  17. Hansen, P., Peeters, D. and Thisse, J.F. (1995), The profit maximizing weber problem. Location Science, 3: 67-85.

    Google Scholar 

  18. Horst, R., Pardalos, P.M. and Van Thoai, N. (1995), Introduction to Global Optimization. Kluwer, Dordrecht.

    Google Scholar 

  19. Idrissi, H.F., Loridan, P. and Michelot, C. (1988), Approximations of solutions for location problems. J. Optimization Theory Applications, 56: 127-143.

    Google Scholar 

  20. Käfer, B. and Nickel, S. (1996), Error bounds for the approximate solution of restricted planar location problems. Technical report, University of Kaiserslautern, Department of Mathematics. Report in Wirtschaftsmathematik No. 17.

  21. Kaufmann, L. and Plastria, F. (1988), The weber problem with supply surplus. Belgian J. Operations Research Statistics Computer Science, 28: 15-31.

    Google Scholar 

  22. Klamroth, K. (1997), Planar Location Problems with Barriers, forthcoming, 2000.

  23. Klein, R. (1997), Algorithmische Geometrie. Addison-Wesley.

  24. Love, R.F., Morris, J.G. and Wesolowsky, G.O. (1988), Facilities location: Models and methods. North Holland.

  25. Michelot, C. (1993), The mathematics of continuous location. Studies in Locational Analysis, 5: 59-83.

    Google Scholar 

  26. Nickel, S. (1995), Discretization of Planar Location Problems. Shaker Publishers.

  27. Peeters, D. and Thisses, J.F. (1995), Economic models of firm location. In Z. Drezner, (ed.) Facility Location: A Survey of Applications and Methods. Springer-Verlag, New York.

    Google Scholar 

  28. Plastria, F. (1992), Gbsss, the generalized big square small method for planar single facility location. Eur J Operational Research 62: 163-174.

    Google Scholar 

  29. Plastria, f. (1995), Continuous location problems. In Z. Drezner, (ed.), Facility Location: A survey of Applications and Methods. Springer-Verlag, New York.

    Google Scholar 

  30. Puerto, J. and Fernandez, F.R. (1995), The symmetrical single facility location problem. Technical report, Prepublicación de la Faculdad de Mathemáticas no. 34, Universidad de Sevilla.

  31. Rodriguez-Chia, A.M., Nickel, S., Puerto, J. and Fernandez, F.R. (2000), A flexible approach to location problems. Math Meth Operations Research, 51: 69-89.

    Google Scholar 

  32. Romero-Morales, M.D., Carrizosa, E. and Conde, E. (1997), Semi-obnoxious location models: A global-optimization approach. Eur J Operational Research 102: 295-301.

    Google Scholar 

  33. Rote, G. (1992), The convergence rate of the sandwich algorithm for approximating convex functions. Computing, 48: 337-361.

    Google Scholar 

  34. Tuy, H. and Al-Khayyal, F. (1992), Global optimization of a nonconvex single facility location problem by sequential unconstrained convex minimization. J. Global Optim 2: 61-71.

    Google Scholar 

  35. Tuy, H., Al-Khayyal, F. and Zhou, F. (1995), A d.c. optimization method for single facility location problems. J Global Optimization, 7: 209-227.

    Google Scholar 

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Authors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla, Spain

    Emilio Carrizosa

  2. Fachbereich Mathematik, Universität Kaiserslautern, Germany

    Horst W. Hamacher

  3. Fachbereich Informatik Fern, Universität-GH Hagen, Germany

    Rolf Klein

  4. Wirtschaftsmathematik Kaiserslautern, Institut für Techno-, Germany

    Stefan Nickel

Authors
  1. Emilio Carrizosa
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  2. Horst W. Hamacher
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  3. Rolf Klein
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  4. Stefan Nickel
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Carrizosa, E., Hamacher, H.W., Klein, R. et al. Solving Nonconvex Planar Location Problems by Finite Dominating Sets. Journal of Global Optimization 18, 195–210 (2000). https://doi.org/10.1023/A:1008395305189

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  • DOI: https://doi.org/10.1023/A:1008395305189