The Fermat-Weber problem of optimally locating a service facility in the Euclidean continuous two-dimensional space is usually solved by the iterative process first suggested by Weiszfeld or by later versions thereof. The methods are usually rather efficient, but exceptional problems are described in the literature in which the iterative solution is exceedingly long. These problems are such that the solution either coincides with one of the demand points or nearly coincides with it. We describe a noniterative direct alternative, based on the insight that the gradient components of the individual demand points can be considered as pooling forces with respect to the solution point. It is demonstrated that symmetrical problems can thus be optimally solved with no iterations, in analogy to finding the equilibrium point in statics. These include a well-known ill-conditioned problem and its variants, which can now be easily solved to optimality using geometrical considerations.

The modern problem of finding the optimal location of a service facility with respect to a given set of demand points has emerged from an old geometrical problem. According to Kuhn [

In order to study the properties of the solution, one can write the first derivatives of (

Setting (

In some cases, the iterative procedure lands on a demand point which is not the optimum, and since the function is not differentiable at such point, the process cannot proceed. It should be pointed out that Ostresh [

Note that several authors (e.g., [

It has been mentioned that one of the possibilities is that the solution coincides with one of the demand points. Since the cost function is not differentiable at a demand point, this may cause difficulty if the iterative process steps into a demand point, be it the optimum or not. The possibility of a coincidence of the solution with a demand point has been discussed by Katz [

In the next section, we briefly describe the mechanical analog to the Weber problem and then proceed to the class of problems which can be solved noniteratively, using the concept of the attractive force and a specific symmetry of the problem.

In the Mathematical Appendix to the book by Weber [

As explained, the force elements due to each of the demand points are radial in direction, but independent of the distance between the demand point and the location of the service facility. An interesting implication of this fact is that once the location of the solution is known, it is in fact also the solution of other problems in which each demand point can be anywhere along the ray connecting the service facility point and the demand point, and without changing its weight (see [

As pointed out above, one kind of Weber problem that can be solved noniteratively is the scenario where the solution coincides with a demand point. In the present context, this takes place when the weight

The first example is a problem given by Kuhn [_{1}, which is not the optimum, and gets “stuck” there. As pointed out by Kuhn, the optimal solution is at the origin, _{2} since all the weights are pulling it to the left. Considering points between A_{1} and A_{2}, we can readily see that the forces of these demand points counterbalance each other, but A_{3} and A_{4} have force components that pull to the left. It is thus obvious that the solution must be between −20 and 20, at a point denoted _{3} is of 13 units along the line connecting it to _{4}. The solution point should be, as stated above, an equilibrium point where the resultant force is nil. With the given weights the components of the forces along the

The second example has a similar symmetry, but it seems to give a deeper insight into the possibilities of getting the solution noniteratively for symmetric problems. As shown in Figure _{1} is

The same is true for A_{2}, so together, they exert a force of

Similar considerations apply for other problems, where, for example, A_{1} and A_{2} are on the _{3} and A_{4} can be moved along the _{3} is still on the negative side and A_{4} on the positive side), and the solution does not change. The only difference may relate to the end point associated with (_{4}. If A_{4} is located farther to the right, using (_{4} located at

Similar considerations can be made for problems with more demand points along the

We turn now to a problem presented by Drezner [

From the symmetry, it is evident that the solution should be along the diagonal. If we consider points on the diagonal close to the point at

Let us consider now what happens in cases where the weight at

By the use of the concept of virtual forces, the components of the gradient of the objective function in the Fermat-Weber problem, some problems of this kind can be solved directly, with no iterations. A number of examples in which the demand points are distributed symmetrically with respect to some axis are shown. This includes problems for which the known iterative procedures perform poorly. In particular, the problem given in Figure