Location problems with costs being sums of powers of euclidean distances

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The Cooper problem of minimizing the sum of weighted powers of the Euclidean distances is further studied. The practical significance of this problem is that instead of the Weber problem where the weighted sum of the Euclidean distances is minimized, here economies and diseconomies of scale are allowed for, which results in a problem of minimizing the sum of weighted powers of the Euclidean distances. The original Cooper solution has been a modification of the Weiszfeld method for the Weber problem; it can be shown to be a steepest descent method with a step size determined by an expression depending on the weights and distances, which is not optimal in any sense. In the present work, the matter of the best step size to be taken is investigated. It is shown that multiplying the step size given by Cooper by 2 n where n is the power of the Euclidean distances, improves the iterative process substantially. For the cases with n≥ l considered by Cooper, namely 1≤ n≤3, the number of steps needed to reach a certain termination criterion is reduced. For higher values of n, the original Cooper step size is usually too large and there is no convergence. With the present amended method, the problem can be solved, using an appropriate normalization, even for powers of 100 and more. A semi-intuitive proof for this step size is given as well as numerical examples of solutions with n = 1,10 and 100. The method can easily be extended to location problems in three (and more) dimensions.