page_title: Weber problem
section_title: Tellier’s trigonometric solution of the triangle attraction-repulsion problem
hierarchical_section_title: Weber problem / Tellier’s trigonometric solution of the triangle attraction-repulsion problem
caption_attribution_description: English: The angles of the attraction-repulsion triangle problem.
context_page_description: 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.
context_section_description: Tellier (1985) extended the Fermat–Weber problem to the case of repulsive forces. Let us examine the triangle case where there are two attractive forces A1w and A2w, and one repulsive force Rw. Here as in the previous case, the possibility exists for the origins of the three vectors not to coincide. So the solution must require their coinciding. Tellier's trigonometric solution of this problem is the following:
  1– Determine angle ∠e :
  cos ∠e = -( A1w² + A2w² − Rw²) / (2 A1w A2w)  ;
  2– Determine angle ∠p :
  cos ∠p = -( A1w² + Rw² − A2w²) / (2 A1w Rw)  ;
  3– Determine angle ∠c :
  ∠c = 180° − ∠p ;
  4– Determine angle ∠d :
  ∠d = ∠e − ∠c ;
  5– Determine the value of angle ∠3 (this equation derives from the requirement that point D must coincide with point E):
  tan ∠3 = x / y ;
  where	x = sin ∠f – (RA₁/RA₂)(sin ∠d sin [∠e − ∠b] / sin ∠c) ; 
  and	y = (RA₁/RA₂)(sin ∠d cos [∠e − ∠b] / sin ∠c) − cos ∠f ;
  6– Determine ∠1 :
  ∠1 = 180° − ∠e − ∠3 ;
  7– Determine ∠5 :
  ∠5 = 180° − ∠b − ∠c − ∠1 ;
  8– Determine ∠2 :
  ∠2 = ∠a − ∠5 .
caption: The angles of the attraction-repulsion triangle problem.
exif: {}