Advertisement

Locational Problems And Mathematical Programming

  • Harold W. Kuhn
Part of the C.I.M.E. Summer Schools book series (CIME, volume 38)

Abstract

1. Introduction. The most simple locational problem has its mathematical origin in classical geometry, where it is known as Steiner's Problem [l]. It appeared, in a slightly generalized form, in the pioneering work Über den Standort der Industrien of Alfred Weber [2]. This form qf the problem, which we shall call the Steiner-Weber Problem, asks for a point in the plane that will minimize the weighted sum of distances to n given points in the plane. In spite of the simple and explicit form of the problem, relatively little is known about its solution, either analytically or computationally. The purpose of this paper is to discuss the problem from the point of view of mathematical programming. In Section 2, certain general properties and a set of necessary and sufficient conditions for a solution are derived. In Section 3, a problem dual to the Steiner-Weber Problem is formulated. This problem has a linear objective function and quadratic constraints; it possesses all of the desirable properties of the dual in linear programming and its solution yields a solution of the Steiner-Weber Problem trivially. In Section 4, some preliminary conclusions concerning computation are presented. Economic applications are the subject of a joint paper with R.E. Kuenne [3], to be published shortly; detailed computational methods will be treated in a later paper.

2. Statement of the Steiner-Weber Problem. Although the simpler properties of the problem have been amply discussed in the literature, for the sake of completeness and to establish notation, we shall restate the problem and derive some basic results here.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. 1.
    R. Courant and H. Robbins, What Is Mathematics?, New York, 1941, pp. 354–358.Google Scholar
  2. 2.
    Alfred Weber, Über den Standort der Industrien, TÜbingen, 1909; Ch. III. The work has been translated as Alfred Weber's Theory of the Location of Industries, Chicago, 1929, by C. J. Friedrich.Google Scholar
  3. 3.
    H. Steinhaus, Mathematical Snapshots, New York, 1960, p. 119. The use of this physical model was suggested by Georg Pick in his Mathematical Appendix to Weber's book, and the essential idea can be found in more modern treatments of force systems. See, for example, G. Polya, Induction and Analogy in Mathematics, Princeton 1954, pp. 147–148, and W. Miehle, “Link-Length Minimization in Networks, ” Operations Research, 6(1958), pp. 232–243.MathSciNetCrossRefGoogle Scholar
  4. 5.
    Jean A, Ville, “Sur théorie géneral des jeux ou intervient l'ha-bilité des jouers ” Applications aux Jeux de Hasard by Emilie Borel and Jean Ville, Tome IV, Fasicule II, in the Traité du Calcul des Probabilités et de ses Applications by Emile Borel (1938), 105–113.Google Scholar
  5. 6.
    Harold W, Kuhn and A. W. Tucker, “Nonlinear Programming” in Proc. Second Berkeley Symposium on Mathematical Statistics and Probability, Univ. of Calif. Press, Berkeley, 1951, pp. 481–492.Google Scholar
  6. 7.
    Kurt Eisenmann, “ The Optimum Location of a Center”, SIAM Review, 4(1962), 394–5.CrossRefGoogle Scholar
  7. 8.
    G. Zoutendijk, Method of Feasible Directions, Elsevier Publ.Cc, Amsterdam, 1960.Google Scholar
  8. 9.
    J. B. Rosen, “The Gradient Projection Method for Nonlinear Programming, Part II Nonlinear Constrains, ” SIAM Journal, 9 (1961), pp. 514–532.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Harold W. Kuhn

There are no affiliations available

Personalised recommendations