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The Inverse Weber Problem on the Plane and the Sphere

Author

Listed:
  • Franco Rubio-López

    (Instituto de Investigación en Matemáticas, Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo 13001, Peru)

  • Obidio Rubio

    (Instituto de Investigación en Matemáticas, Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo 13001, Peru)

  • Rolando Urtecho Vidaurre

    (Instituto de Investigación en Matemáticas, Departamento de Matemáticas, Universidad Nacional de Trujillo, Trujillo 13001, Peru)

Abstract

Weber’s inverse problem in the plane is to modify the positive weights associated with n fixed points in the plane at minimum cost, ensuring that a given point a priori becomes the Euclidean weighted geometric median. In this paper, we investigate Weber’s inverse problem in the plane and generalize it to the surface of the sphere. Our study uses a subspace orthogonal to a subspace generated by two vectors X and Y associated with the given points and weights. The main achievement of our work lies in determining a vector perpendicular to the vectors X and Y , in R n ; which is used to determinate a solution of Weber’s inverse problem. In addition, lower bounds are obtained for the minimum of the Weber function, and an upper bound for the difference of the minimal of Weber’s direct and inverse problems. Examples of application at the plane and unit sphere are given.

Suggested Citation

  • Franco Rubio-López & Obidio Rubio & Rolando Urtecho Vidaurre, 2023. "The Inverse Weber Problem on the Plane and the Sphere," Mathematics, MDPI, vol. 11(24), pages 1-23, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:5000-:d:1302423
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    References listed on IDEAS

    as
    1. Pierre Hansen & Dominique Peeters & Denis Richard & Jacques-Francois Thisse, 1985. "The Minisum and Minimax Location Problems Revisited," Operations Research, INFORMS, vol. 33(6), pages 1251-1265, December.
    2. Zvi Drezner & George O. Wesolowsky, 1983. "Minimax and maximin facility location problems on a sphere," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 30(2), pages 305-312, June.
    3. Burkard, Rainer E. & Galavii, Mohammadreza & Gassner, Elisabeth, 2010. "The inverse Fermat-Weber problem," European Journal of Operational Research, Elsevier, vol. 206(1), pages 11-17, October.
    4. Zhang, Jianzhong & Liu, Zhenhong & Ma, Zhongfan, 2000. "Some reverse location problems," European Journal of Operational Research, Elsevier, vol. 124(1), pages 77-88, July.
    5. Zvi Drezner, 1981. "Technical Note—On Location Dominance on Spherical Surfaces," Operations Research, INFORMS, vol. 29(6), pages 1218-1219, December.
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