Reviewing extensions and solution methods of the planar Weber single facility location problem

One of the classic foundational constructs of Location Science was proposed by Alfred Weber in 1909. His construct involved finding the location for a single production facility which minimized the sum of weighted distances of transporting the needed raw materials from localized sources along with the sum of weighted distances in delivering the final product to one or more markets.

The first objective of this paper is to review the major advancements in this simple classic single facility location problem and its variations. One can find in the literature a very large number of algorithms to solve the standard Weber problem. Some are iterative and others are finite even for geometric Euclidean and rectilinear spaces. Moreover, some schemes are efficient (theoretically) and others are practically quite fast.

The second goal of this paper is to show that many extensions of the standard Weber problem can be solved by solving a polynomial number of standard Weber problems. This unifying result implies, in particular, that all these extensions are polynomially solvable since the standard Weber problem can be solved in polynomial time. In addition, with this unifying approach we solve some important planar non-convex Euclidean location problems in polynomial time.