on Convergence in the Single Facility

-The single facility minisum location problem requires finding a point in ~N that minimizes a sum of weighted distances to m given points. The distance measure is typically assumed in the literature to be either Euclidean or rectangular, or the more general Ip norm. Global convergence of a well-known iterative solution method named the Weiszfeld procedure has been proven under the proviso that none of the iterates coincide with a singular point of the iteration functions. The purpose of this paper is to examine the corresponding set of "bad" staxting points which result in failure of the algorithm for a general Ip norm. An important outcome of this analysis is that the set of bad starting points will always have a measure zero in the solution space (7~N), thereby validating the global convergence properties of the Weiszfeld procedure for any Ip norm, p E [1, 2]. Keywordsms ing l e facility minisum location problem, lp norm, Convergence, Singular points. 1. I N T R O D U C T I O N The continuous single facility minisum location problem is one of the classical models in location theory. It comes under various names such as the Fermat-Weber problem. (For an interesting review of the long history of this problem, the reader is referred to [1].) The objective is to find a point in 7~ N which minimizes the sum of weighted distances to m given points. The given points represent customers or demands in a practical setting, the new point denotes a new facility such as a warehouse which needs to be located, and the weighted distances are cost components associated with the interactions such as flow of goods between the new facility and its customers. We consider the case where distances are measured by an Ip norm. The resulting minisum model is formulated as follows: *Author to whom all correspondence should be addressed. Typeset by ,4.Ad~TEX