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In geometry, the Weber problem, named after Alfred Weber, is one of the most famous problems in location theory. It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to ''n'' destination points, where different destination points are associated with different costs per unit distance. The Weber problem generalizes the geometric median, which assumes transportation costs per unit distance are the same for all destination points, and the problem of computing the Fermat point, the geometric median of three points. For this reason it is sometimes called the Fermat–Weber problem, although the same name has also been used for the unweighted geometric median problem. The Weber problem is in turn generalized by the attraction–repulsion problem, which allows some of the costs to be negative, so that greater distance from some points is better. == Definition and history of the Fermat, Weber, and attraction-repulsion problems == In the triangle case, the Fermat problem consists in locating a point D with respect to three points A, B, and C in such a way that the sum of the distances between D and each of the three other points is minimized. It was formulated by the famous French mathematician Pierre de Fermat before 1640, and it can be seen as the true beginning of both location theory, and space-economy. Torricelli found a geometrical solution to this problem around 1645, but it still had no direct numerical solution more than 325 years later. Kuhn and Kuenne〔Kuhn, Harold W. and Robert E. Kuenne, 1962, "An Efficient Algorithm for the Numerical Solution of the Generalized Weber Problem in Spatial Economics." Journal of Regional Science 4, 21–34.〕 found an iterative solution for the general Fermat problem in 1962, and, in 1972, Tellier〔Tellier, Luc-Normand, 1972, “The Weber Problem: Solution and Interpretation”, Geographical Analysis, vol. 4, no. 3, pp. 215–233.〕 found a direct numerical solution to the Fermat triangle problem, which is trigonometric. Kuhn and Kuenne’s solution applies to the case of polygons having more than three sides, which is not the case with Tellier’s solution for reasons explained further on. The Weber problem consists, in the triangle case, in locating a point D with respect to three points A, B, and C in such a way that the sum of the transportation costs between D and each of the three other points is minimized. The Weber problem is a generalization of the Fermat problem since it involves both equal and unequal attractive forces (see below), while the Fermat problem only deals with equal attractive forces. It was first formulated, and solved geometrically in the triangle case, by Thomas Simpson in 1750.〔Simpson, Thomas, 1750, The Doctrine and Application of Fluxions, London.〕 It was later popularized by Alfred Weber in 1909.〔Weber, Alfred, 1909, Über den Standort der Industrien, Tübingen, J.C.B. Mohr) — English translation: The Theory of the Location of Industries, Chicago, Chicago University Press, 1929, 256 pages.〕 Kuhn and Kuenne’s iterative solution found in 1962, and Tellier’s solution found in 1972 apply to the Weber triangle problem as well as to the Fermat one. Kuhn and Kuenne’s solution applies also to the case of polygons having more than three sides. In its simplest version, the attraction-repulsion problem consists in locating a point D with respect to three points A1, A2 and R in such a way that the attractive forces exerted by points A1 and A2, and the repulsive force exerted by point R cancel each other out as it must do at the optimum. It constitutes a generalization of both the Fermat and Weber problems. It was first formulated and solved, in the triangle case, in 1985 by Luc-Normand Tellier.〔Tellier, Luc-Normand, 1985, Économie spatiale: rationalité économique de l'espace habité, Chicoutimi, Gaëtan Morin éditeur, 280 pages.〕 In 1992, Chen, Hansen, Jaumard and Tuy found a solution to the Tellier problem for the case of polygons having more than three sides. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Weber problem」の詳細全文を読む スポンサード リンク
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