Analysing a few trigonometric solutions for the Fermat-Weber facility location triangle problem with and without repulsion and generalising the solutions (Q2204279)

Summary: In location theory, the Weber problem [\textit{A. Weber}, Ueber den Standort der Industrien. Tübingen: J. C. B. Mohr (1909)] is one of the most popular problems. It requires locating a unique point that minimises the sum of transportation costs from that point. In a triangular case, the problem is to locate a point with respect to three points in such a way that the sum of distances between the point and the other three points is the minimum. \textit{L.-N. Tellier} [``The Weber problem: solution and interpretation'', Geographic. Anal. 4, No. 3, 215--233 (1972; \url{doi:10.1111/j.1538-4632.1972.tb00472.x})] found a geometric solution to the problem. In the attraction-repulsion problem, some of the costs may be negative. This was formulated and geometrically solved for a triangular case by \textit{L.-N. Tellier} [Économie spatiale: rationalité économique de l'espace habité. Chicoutimi: Gaëtan Morin (1985)]. Gruulich (1999) discussed about the barycentric coordinates solution applied to the optimal road junction problem. \textit{Nguyen Minh Ha} and \textit{Bui Viet Loc} [Forum Geom. 11, 131--138 (2011; Zbl 1294.51003)] proposed an easy geometrical solution to the repulsive force and two attraction forces triangular problem. This paper generalises the solutions for both the attractions and repulsion problems and analytical formulae are derived.