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location-allocation models

  Models used to determine the optimal location of central facilities (hospitals, offices, warehouses, etc.) in order to minimize movement and other costs. A public-sector example is the location of two new hospitals in a city: the objectives would be to minimize the aggregate travel cost to patients and to arrive at an optimal assignment of patients to the hospitals. A private-sector example is the location of warehouse facilities between factories and markets in order to minimize total distribution costs.

Unlike transportation problem models, in which all the locations are fixed and the optimal assignment can be determined by linear programming, in location-allocation models the siting of the central facilities and the assignment of least-cost flows must be determined simultaneously. The choice of location (whether or not to build a hospital at any given location) is a discrete or integer variable, and linear programming only deals with continuous variables. This makes the solution of these models more difficult, and many applications use heuristic or trial-and-error procedures that search for, but cannot be guaranteed to find, the true optimum. The degree of difficulty is a function of the number of unknowns: a two-hospital problem with fixed hospital capacities is much easier than a five-hospital problem with variable hospital sizes.

Two important classes of location-allocation models are distinguished: location on a continuous surface or plane, and location on a network. The continuous surface form has been widely used in theoretical analysis, and is related to Weber\'s industrial location model (cf. industrial location theory), but its limitations are increasingly recognized. Straight-line distance is often a poor proxy for real transport costs, which occur on a route network, and potential locations are usually restricted. Work has increasingly used the network form, where potential locations are defined as nodes and are linked to sources (and destinations in the warehousing case) by a route network. Mixed integer-continuous techniques are now available to find a true optimum for such problems. Recent extensions have very effectively linked location-allocation together with entropy-maximizing models and geographical information systems to plan both commercial (e.g. car showrooms) and public (e.g. hospital) facilities. (LWH)

Suggested Reading Killen, J.E. 1983: Mathematical programming for geographers and planners. London: Croom Helm.



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