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 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
The Fermat problem The Weber problem The attraction-repulsion problem First formulated by Fermat (before 1640) Simpson (1750) Tellier (1985) Geometrical solution of the triangle problem Torricelli (1645) Simpson (1750) Tellier (2013) Direct numerical solution of the triangle problem Tellier (1972) Tellier (1972) Tellier (1985) Iterative numerical solution of the problem E. Weiszfeld (1937), Kuhn and Kuenne (1962) E. Weiszfeld (1937), Kuhn and Kuenne (1962) Chen, Hansen, Jaumard and Tuy (1992)
In the triangle case, the Fermat problem consists in locating a point with respect to three points in such a way that the sum of the distances between 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. E. Weiszfeld published a paper in 1937 with an algorithm for the Fermat-Weber problem. As the paper was published in Tohoku Mathematical journal, and Weiszfeld immigrated to USA and changed his name to Vaszoni, his work was not widely known. Kuhn and KuenneKuhn, 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. independently found a similar iterative solution for the general Fermat problem in 1962, and, in 1972, TellierTellier, 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 with respect to three points in such a way that the sum of the transportation costs between 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 with respect to three points and in such a way that the attractive forces exerted by points , and the repulsive force exerted by point 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.
Torricelli’s geometrical solution of the Fermat triangle problem
Evangelista Torricelli’s geometrical solution of the Fermat triangle problem stems from two observations:
Point is at its optimal location when any significant move out of that location induces a net increase of the total distance to reference points , which means that the optimal point is the only point where an infinitesimal movement towards one of the three reference points induces a reduction of the distance to that point that is equal to the sum of the induced changes in the distances to the two other points; in fact, in the Fermat problem, the advantage to reduce the distance from by one kilometer is equal to the advantage to reduce the distance from by one kilometer or the distance from by the same length; in other words, the activity to be located at is equally attracted by ; According to an important theorem of Euclidean geometry, in a convex quadrilateral inscribed in a circle, the opposite angles are supplementary (that is their sum is equal to 180°); that theorem can also take the following form: if we cut a circle with a chord , we get two circle arcs, let us say ; on arc , any angle is the same for any chosen point , and, on arc , all the angles are also equal for any chosen point ; moreover, the angles are supplementary.
It can be proved that the first observation implies that, at the optimum, the angles between the straight lines must be equal to 360° / 3 = 120°. Torricelli deduced from that conclusion that:
If any triangle , whose angle is equal to 120°, generates an convex quadrilateral inscribed in a circle, the angle of the triangle must be equal to (180° − 120°) = 60°; One way to determine the set of locations of for which the angle is equal to 120° is to draw an equilateral triangle (because each angle of an equilateral triangle is equal to 60°), where is located outside the triangle, and draw a circle round that triangle; then all the points of the circumference of that circle that lie within the circle are such that the angle is equal to 120°; The same reasoning can be made with respect to triangles ; This leads to draw two other equilateral triangles , where are located outside the triangle, as well as two other circles round these equilateral triangles, and to determine the location where the three circles intersect; at that location, the angles between the straight lines is necessarily equal to 120°, which proves that it is the optimal location.
Simpson’s geometrical solution of the Weber triangle problem
Simpson's geometrical solution of the so-called "Weber triangle problem" (which was first formulated by Thomas Simpson in 1750) directly derives from Torricelli's solution. Simpson and Weber stressed the fact that, in a total transportation minimization problem, the advantage to get closer to each attraction point or depends on what is carried and on its transportation cost. Consequently, the advantage of getting one kilometer closer to or varies, and the angles no more need to be equal to 120°.
Simpson demonstrated that, in the same way as, in the Fermat triangle problem case, the constructed triangles were equilateral because the three attractive forces were equal, in the Weber triangle problem case, the constructed triangles , where are located outside the triangle, must be proportional to the attractive forces of the location system.
The solution is such that:
In the constructed triangle , the side is proportional to the attractive force pointing towards , the side is proportional to the attractive force pointing towards , and the side is proportional to the attractive force pointing towards ; In the constructed triangle , the side is proportional to the attractive force pointing towards , the side is proportional to the attractive force pointing towards , and the side is proportional to the attractive force pointing towards ; The optimal point is located at the intersection of the two circumferences drawn round the constructed triangles.
A third triangle of forces , where is located outside the triangle, can be drawn based on the side, and a third circumference can be traced round that triangle. That third circumference crosses the two previous ones at the same point .
Tellier’s geometrical solution of the attraction-repulsion triangle problem
A geometrical solution exists for the attraction-repulsion triangle problem. Its discovery is rather recent.Tellier, Luc-Normand, 2013, « Annexe 1 : Solution géométrique du cas triangulaire du problème d’attraction-répulsion », annex of the paper of Pierre Hansen, Christophe Meyer and Luc-Normand Tellier, « Modèles topodynamique et de la Nouvelle économie géographique : compatibilité, convergence et avantages comparés », in Marc-Urbain Proulx (ed.), 2013, Sciences du territoire II : méthodologies, Québec, Presses de l’Université du Québec. That geometrical solution differs from the two previous ones since, in this case, the two constructed force triangles overlap the location triangle (where and are attraction points, and , a repulsion one), while, in the preceding cases, they never did.
This solution is such that:
In the constructed triangle , which partly overlaps the location triangle, the side is proportional to the attractive force pointing towards , the side is proportional to the attractive force pointing towards , and the side is proportional to the repulsive force pushing away from point ; In the constructed triangle , which partly overlaps the location triangle, the side is proportional to the attractive force pointing towards , the side is proportional to the attractive force pointing towards , and the side is proportional to the repulsive force pushing away from point ; The optimal point is located at the intersection of the two circumferences drawn round the and constructed triangles. This solution is useless if one of the forces is greater than the sum of the two other ones or if the angles are not compatible. In some cases, no force is larger than the two other ones, and the angles are not compatible; then, the optimal location lies at the point that exerts the greater attractive force.
Tellier’s trigonometric solution of the Fermat and Weber triangle problems
More than 332 years separate the first formulation of the Fermat triangle problem and the discovery of its non-iterative numerical solution, while a geometrical solution existed for almost all that period of time. Is there an explanation for that? That explanation lies in the possibility of the origins of the three vectors oriented towards the three attraction points not coinciding. If those origins do coincide and lie at the optimum location , the vectors oriented towards , and the sides of the location triangle form the six angles , and the three vectors form the angles. It is easy to write the following six equations linking six unknowns (the angles ) with six known values (angles , whose values are given, and angles , whose values depend only on the relative magnitude of the three attractive forces pointing towards the attraction points):