Towards the solution of the (generalised) multi-Weber problem

Despite the long history of the multi-Weber problem and the generalised multi-Weber problem, there is no general optimal method for its solution. This paper has two aims. The first aim is a study of the shape of the objective function of the single and multi-Weber problem. The form of the objective function and the relation of a heuristic to it are critical to the decision to use a heuristic or to seek an optimal solution. The objective function of the multi-Weber problem is shown to be extremely steep in the neighbourhood of the optimal solution, indicating the importance of an optimal rather than heuristic solution. The second aim is a description of a two-step algorithm which, although it cannot guarantee optimality, will frequently terminate optimally and thus appears to be superior to other available heuristics.