Facility location in the presence of forbidden regions, I: Formulation and the case of Euclidean distance with one forbidden circle

A constrained form of the Weber problem is formulated in which no path is permitted to enter a prespecified forbidden region R of the plane. Using the calculus of variations the shortest path between two points x, y ∉ R which does not intersect R is determined. If d( x, y) is unconstrained distance, we denote the shortes distance along a feasible path by d( x y) . The constrained Weber problem is, then: given points x j∉ R and positive weights w j , j = 1,2,…, n, find a point x∉ R such that f( x)= Σ n j=1 d( x, x j) is a minimum. An algorithm is formulated for the solution of this problem when d( x, y) is Euclidean distance and R is a single circular region. Numerical results are presented.